Handbook of Production Economics [1st ed. 2022] 9811034567, 9789811034565

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Subhash C. Ray Robert G. Chambers Subal C. Kumbhakar Editors

Handbook of Production Economics

Handbook of Production Economics

Subhash C. Ray • Robert G. Chambers • Subal C. Kumbhakar Editors

Handbook of Production Economics With 108 Figures and 20 Tables

123

Editors Subhash C. Ray Department of Economics University of Connecticut Storrs, CT, USA

Robert G. Chambers Department of Agricultural and Resource Economics University of Maryland College Park, MD, USA

Subal C. Kumbhakar Department of Economics Binghamton University Binghamton, NY, USA

ISBN 978-981-10-3454-1 ISBN 978-981-10-3455-8 (eBook) ISBN 978-981-10-3456-5 (print and electronic bundle) https://doi.org/10.1007/978-981-10-3455-8 © Springer Nature Singapore Pte Ltd. 2022 All rights are reserved by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publisher, the authors, and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, expressed or implied, with respect to the material contained herein or for any errors or omissions that may have been made. The publisher remains neutral with regard to jurisdictional claims in published maps and institutional affiliations This Springer imprint is published by the registered company Springer Nature Singapore Pte Ltd. The registered company address is: 152 Beach Road, #21-01/04 Gateway East, Singapore 189721, Singapore

Preface

In recent years, the neoclassical theory of production seems to have lost its appeal among academics and graduate students in microeconomic theory courses. Students in standard economics doctoral programs only receive the minimal exposure to production and cost functions necessary for an exposition of the theory of markets en route to the ultimate goal of game theory, experimental economic issues, and strategic behavior. For example, only 40 of the 971 pages of the Microeconomic Theory book by Mas-Colell, Whinston, and Green (1995) are devoted to production, cost minimization, and profit maximization. While a student admittedly has learnt the basic theory of producer behavior in their “Intermediate Micro Theory” courses, more advanced concepts like Allen-Uzawa partial elasticities of substitution are not covered either at undergraduate or at graduate level. An average student never sees a transcendental logarithmic (Translog) or a Generalized Leontief cost function in class. Yet, the latter half of the twentieth century was an era of spectacular development in production theory within economics. The 1951 Cowles Foundation anthology Activity Analysis of Production and Allocation edited by Koopmans remains one of the richest collection of essays in economic theory. Appearing at about the same time, the duality theory of Hotelling, Roy, Hicks, Samuelson, and Shephard opened up novel ways of analyzing the production technology through cost, revenue, and profit functions. These topics are rarely covered in microeconomics courses, although these topics are covered in the twovolume Production Economics: A Dual Approach to Theory and Applications edited by Fuss and McFadden (1978). In the meantime, Nerlove used the dual cost function to empirically estimate the parameters of a Cobb Douglas production function using data for electric utilities in the USA (1965). Emergence of generalized cost functions (like the Translog, the Generalized Leontief, and the Generalized CES) liberated the empirical analyst from the confines of Cobb Douglas, Leontief, or the CES specifications and enriched both economic theory and econometric analysis in equal measures. These seem to be history now. By the last decade of the past century, interest in production theory had clearly waned. Resurgence of identification of production function in the recent literature mostly focuses on the primal CobbDouglas production function – completely bypassing the duality literature. Papers included in this three-volume handbook focus on both theoretical concepts and empirical issues from neoclassical production economics. Each of the chapters is intended to provide a state-of-the-art survey on a specific topic in v

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Preface

production economics. The objective is to serve as a single unified source of reference for the serious scholar seeking in-depth knowledge of the underlying theory behind the sophisticated empirical analysis appearing in applied papers. The chapters in volumes 1 and 2 of the handbook are devoted exclusively to theory and different analytical methodologies for empirical estimation. By contrast, every chapter in volume 3 offers an overview of empirical applications in the accepted literature that employ the theoretical framework described in volume 1 to analyze the technical and behavioral relations between relevant variables in various industries ranging from banking or air transportation to education or professional sports. Putting together the 45 chapters of the handbook contributed by more than twice as many authors, each somehow contributing their valuable time to write the chapters within their busy schedules already full of numerous commitments, has, naturally, been a long-drawn effort lasting over years. On top of it, the upheaval brought about by the Covid-19 pandemic put the viability of the entire project in jeopardy. Fortunately, however, through the collective effort and cooperation of the contributing authors and the editorial staff at Springer Nature, we managed to overcome all hurdles and completed the project. We are grateful to the editorial staff at Springer Nature for their help and particularly thank Sagarika Ghosh, Nupoor Singh, Audrey Wong-Hillman, Mokshika Gaur, and Salmanul Faris Nedum Palli for their valiant effort to keep the publication on track as much as possible. At the present moment, rapid and sweeping developments in information technology are changing the fundamental character of production in many industries, prompting serious researchers to wonder if there will be any workers left in the workplace in the near future. We hope that the handbook will help to revive interest in production economics and inspire a new generation of scholars to revisit and extend the theory. Only that will make editing this handbook worthwhile. May 2022

Subhash C. Ray Robert G. Chambers Subal C. Kumbhakar

Contents

Volume 1 Part I Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1

1

Neoclassical Production Economics: An Introduction . . . . . . . . . . . . Robert G. Chambers and Subhash C. Ray

3

2

Reminiscences of “Returns to Scale in Electricity Supply” . . . . . . . . Marc Nerlove

49

3

Duality in Production . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . W. Erwin Diewert

57

4

Multiproduct Technologies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Rolf Färe, Daniel Primont, and W. L. Weber

169

5

Functional Structure and Aggregation . . . . . . . . . . . . . . . . . . . . . . . . . Daniel Primont

215

6

Elasticities of Substitution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . R. Robert Russell

259

7

Distance Functions in Production Economics . . . . . . . . . . . . . . . . . . . Robert G. Chambers and Rolf Färe

295

8

Stochastic Frontier Analysis: Foundations and Advances I . . . . . . . Subal C. Kumbhakar, Christopher F. Parmeter, and Valentin Zelenyuk

331

9

Stochastic Frontier Analysis: Foundations and Advances II . . . . . . Subal C. Kumbhakar, Christopher F. Parmeter, and Valentin Zelenyuk

371

10

Data Envelopment Analysis: A Nonparametric Method of Production Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Subhash C. Ray

409

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Contents

11

Activity Analysis in Production Economics . . . . . . . . . . . . . . . . . . . . . Thijs ten Raa

471

12

Bad Outputs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Sushama Murty and R. Robert Russell

483

Volume 2 13

Market Structures in Production Economics . . . . . . . . . . . . . . . . . . . Devin Garcia, Levent Kutlu, and Robin C. Sickles

537

14

Production Under Uncertainty . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Robert G. Chambers

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15

Dynamic Analysis of Production . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Spiro E. Stefanou

611

16

Cost, Revenue, and Profit Function Estimates . . . . . . . . . . . . . . . . . . Levent Kutlu, Shasha Liu, and Robin C. Sickles

641

17

Scale Elasticity and Returns to Scale . . . . . . . . . . . . . . . . . . . . . . . . . . Victor V. Podinovski and Finn R. Førsund

681

18

Nonconvexity in Production and Cost Functions: An Exploratory and Selective Review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Walter Briec, Kristiaan Kerstens, and Ignace Van de Woestyne

19

Index Numbers and Productivity Measurement . . . . . . . . . . . . . . . . . D. S. Prasada Rao

20

Conceptualization and Measurement of Productivity Growth and Technical Change: A Nonparametric Approach . . . . . . . . . . . . . Subhash C. Ray

721 755

821

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Modeling Technical Change: Theory and Practice . . . . . . . . . . . . . . Subal C. Kumbhakar

871

22

Economics of Externalities: An Overview . . . . . . . . . . . . . . . . . . . . . . Jean-Paul Chavas

925

23

Shadow Pricing in Production Economics . . . . . . . . . . . . . . . . . . . . . . Rolf Färe, Shawna Grosskopf and Dimitris Margaritis

951

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Capacity and Capacity Utilization in Production Economics . . . . . . 1001 Dale Squires and Kathleen Segerson

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Aggregation of Efficiency and Productivity: From Firm to Sector and Higher Levels . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1039 Valentin Zelenyuk

Contents

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Volume 3 Part II

Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1081

26

Choice of Inputs and Outputs for Production Analysis . . . . . . . . . . . 1083 Subhash C. Ray

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Airline Economics: A Survey of Applied Issues in the Performance of the US and International Airline Industry . . . . . . . 1117 Levent Kutlu, Daniel Prudencio, and Robin C. Sickles

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Globalization, Innovation, and Productivity . . . . . . . . . . . . . . . . . . . . 1145 Shunan Zhao and Man Jin

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Empirical Analysis of Production Economics: Applications to Banking . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1165 Stephen M. Miller

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Applications of Production Economics in Education . . . . . . . . . . . . . 1193 Jill Johnes

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Dairy Farming from a Production Economics Perspective: An Overview of the Literature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1241 Boris E. Bravo-Ureta, Alan Wall, and Florian Neubauer

32

Performance Evaluation of Mutual Funds Using Frontier Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1281 Subrata Sarkar

33

Performance of Microfinance Institutions: A Review . . . . . . . . . . . . 1309 Christopher F. Parmeter and Valentina Hartarska

34

The Economics of Production in Marine Fisheries . . . . . . . . . . . . . . . 1339 Dale Squires and John Walden

35

Production Economics in Spatial Analysis . . . . . . . . . . . . . . . . . . . . . . 1379 Luis Orea and Inmaculada C. Álvarez

36

Technical Efficiency and Its Determinants in the Manufacturing Sector: What We Know and What We Should Know . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1411 Sumon Kumar Bhaumik

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Application of Production Economics in the Electricity Distribution Sector . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1433 Ørjan Mydland and Gudbrand Lien

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Production and the Environment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1463 Moriah Bostian and Tommy Lundgren

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39

Applications of Production Theory in Transportation . . . . . . . . . . . . 1491 Phill Wheat, Kristofer Odolinski, and Andrew Smith

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Productivity in Global Aquaculture . . . . . . . . . . . . . . . . . . . . . . . . . . . 1525 Frank Asche, Ruth Beatriz Mezzalira Pincinato, and Ragnar Tveteras

41

Benchmarking in the European Water Sector . . . . . . . . . . . . . . . . . . 1563 Alan Horncastle, Joseph Duffy, Chien Xen Ng, and Peter Krupa

42

The Economics of Sports . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1629 Joshua Congdon-Hohman and Victor Matheson

43

The Effects of Management on Production: A Survey of Empirical Studies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1651 Alecos Papadopoulos

44

Production Economics in the Telecommunications Industry . . . . . . 1699 Arun Bhattacharyya

45

Cost Assessment of (Un)bundling: Separation of Vertically Integrated Public Utilities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1751 Pablo Arocena and Subal C. Kumbhakar

Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1787

About the Editors

Subhash C. Ray is Professor of Economics at the University of Connecticut, USA. His principal area of research is nonparametric measurement of productivity and efficiency using Data Envelopment Analysis (DEA). His reference textbook Data Envelopment Analysis: Theory and Techniques for Economics and Operations Research (Cambridge University Press) was published in 2004. He is an associate editor of the Journal of Productivity Analysis. He has served as guest editor of special issues of the Journal of Productivity Analysis and Indian Economic Review. He was a member of the editorial board of Indian Economic Review. He has lectured and conducted workshops on DEA in different countries including China, India, Korea, England, Brazil, Peru, Germany, Malaysia, and Turkey, among others. He received the W.W. Cooper Lifetime Contribution Award from International DEA Society in 2016.

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About the Editors

Robert G. Chambers was born in Washington, DC and raised in nearby Rockville, Maryland. He received his undergraduate training at Georgetown University, his MS degree from the University of Maryland, and his PhD from the University of California (Berkeley). He joined the faculty at the University of Maryland in 1979 and has been there ever since apart from leave to serve as senior economist at the US President’s Council of Economic Advisers. He is a fellow of the Agricultural and Applied Economics Association. His areas of interest include production economics, microeconomic theory, decision-making under uncertainty, and agricultural economics. He is married and has three sons, Christopher, Geoffrey, and Timothy. He currently resides in Maryland, New York, and New Mexico with his wife Michelle, his youngest son Tim, and their Portuguese Water Dogs, Nelson and Skipper. Professor Subal C. Kumbhakar (http://bingweb. binghamton.edu/~kkar/) is a University Distinguished Research Professor of Economics at the State University of New York at Binghamton. His main area of research is applied microeconomics with a focus on estimation of efficiency in production using crosssectional and panel data. Professor Kumbhakar is a fellow of the Journal of Econometrics (1998) and a distinguished author of Journal of Applied Econometrics (2017). He holds an honorary doctorate degree (Doctor Honoris Causa) from Gothenburg University, Sweden (1997). Professor Kumbhakar is currently a co-editor of Empirical Economics, associate editor of Empirical Economics since 2001, and former associate editor of the American Journal of Agricultural Economics (1997–1999). He is serving in the board of editors of the Journal of Productivity Analysis since 1998; Technological Forecasting and Social Change: An International Journal since 1991; International Journal of Business and Economics since 2002; Macroeconomics and Finance in Emerging Market Economies since 2007; Applied Econometrics, http://appliedeconometrics. cemi.rssi.ru/AppEc_en.html, since 2016; and Ecos de Economía: A Latin American Journal of Applied Economics, http://publicaciones.eafit.edu.co/index.php/ ecos-economia/index, since 2016. He is a board member of the Journal of Regulatory Economics since 2015.

About the Editors

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Professor Kumbhakar is the co-author (with Knox Lovell) of Stochastic Frontier Analysis (2000), A Practitioner’s Guide to Stochastic Frontier Analysis Using Stata (with Hung-Jen Wang and A. Horncastle) (2015) both published by the Cambridge University Press. Google Scholar Citations: https://scholar.google. com/citations?user=-rB5HVsAAAAJ&hl=en Wikipedia: https://en.wikipedia.org/wiki/Subal_ Kumbhakar Personal webpage: https://sites.google.com/ binghamton.edu/subalckumbhakar/homes

Contributors

Inmaculada C. Álvarez Oviedo Efficiency Group, Department of Economics, Universidad Autónoma de Madrid, Madrid, Spain Pablo Arocena Universidad Pública de Navarra (UPNA), Institute for Advanced Research in Business and Economics (INARBE), Pamplona, Navarra, Spain Frank Asche School of Forest, Fisheries and Geomatics Sciences, Institute for Sustainable Food Systems and Fisheries and Aquatic Sciences, University of Florida, Gainesville, FL, USA Arun Bhattacharyya Director of Strategic Forecasting at Pfizer Inc. NYC., New York, NY, USA Sumon Kumar Bhaumik Sheffield University Management School, University of Sheffield, Sheffield, UK IZA – Institute of Labor Economics, Bonn, Germany Global Labor Organization, Geneva, Switzerland Moriah Bostian Department of Economics, Lewis & Clark College, Portland, OR, USA Department of Economics, Centre for Environmental and Resource Economics (CERE), Umeå University, Umeå, Sweden Boris E. Bravo-Ureta Agricultural and Resource Economics, University of Connecticut, Storrs, CT, USA Walter Briec University of Perpignan, LAMPS, Perpignan, France Robert G. Chambers Department of Agricultural and Resource Economics, University of Maryland, College Park, MD, USA Jean-Paul Chavas University of Wisconsin, Madison, WI, USA Joshua Congdon-Hohman College of the Holy Cross, Worcester, MA, USA

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Contributors

W. Erwin Diewert Vancouver School of Economics, University of British Columbia, Vancouver, BC, Canada School of Economics, UNSW, Sydney, NSW, Australia Joseph Duffy Oxera Consulting, Oxford, UK Rolf Färe Department of Economics and Department of Agricultural and Resource Economics, Oregon State University, Corvallis, OR, USA Department of Economics and Department of Applied Economics, School of Public Policy, Oregon State University, Corvallis, OR, USA Department of Agricultural Economics, University of Maryland, College Park, MD, USA Finn R. Førsund Department of Economics, University of Oslo, Oslo, Norway Devin Garcia Ernst and Young, LLP, Houston, TX, USA Shawna Grosskopf Department of Economics, School of Public Policy, Oregon State University, Corvallis, OR, USA Valentina Hartarska Auburn University, Auburn, AL, USA Alan Horncastle Oxera Consulting, Oxford, UK Man Jin Department of Economics, Oakland University, Rochester, MI, USA Jill Johnes Huddersfield Business School, University of Huddersfield, Huddersfield, UK Kristiaan Kerstens IESEG School of Management, CNRS, Université de Lille, UMR 9221-LEM, Lille, France Peter Krupa Oxera Consulting, Oxford, UK Subal C. Kumbhakar Department of Economics, State University of New York at Binghamton, Binghamton, NY, USA Inland Norway University of Applied Sciences, Lillehammer, Norway Levent Kutlu Department of Economics and Finance, University of Texas Rio Grande Valley, Edinburg, TX, USA Gudbrand Lien Inland School of Business and Social Sciences, Inland Norway University of Applied Sciences, Lillehammer, Norway Shasha Liu Enterprise Model Risk, Freddie Mac, McLean, VA, USA Tommy Lundgren Department of Economics, Centre for Environmental and Resource Economics (CERE), Umeå University, Umeå, Sweden Dimitris Margaritis Department of Accounting and Finance, University of Auckland Business School, Auckland, New Zealand

Contributors

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Victor Matheson College of the Holy Cross, Worcester, MA, USA Ruth Beatriz Mezzalira Pincinato UiS Business School, University of Stavanger, Stavanger, Norway Stephen M. Miller Department of Economics, Lee Business School, University of Nevada, Las Vegas, Las Vegas, NV, USA Sushama Murty Centre for International Trade and Development, School of International Studies, Jawaharlal Nehru University, New Delhi, India Ørjan Mydland Inland School of Business and Social Sciences, Inland Norway University of Applied Sciences, Lillehammer, Norway Marc Nerlove Department of Agricultural and Resource Economics, College of Agriculture and Natural Resources, University of Maryland, College Park, MD, USA Florian Neubauer Agricultural and Resource Economics, University of Connecticut, Storrs, CT, USA Kristofer Odolinski Institute for Transport Studies, University of Leeds, Leeds, UK Society, Environment, and Transport, The Swedish National Road and Transport Research Institute (VTI), Stockholm, Sweden Luis Orea Oviedo Efficiency Group, Department of Economics, University of Oviedo, Oviedo, Spain Alecos Papadopoulos Athens University of Economics and Business, Athens, Greece Christopher F. Parmeter Department of Economics, University of Miami, Miami, FL, USA Victor V. Podinovski School of Business and Economics, Loughborough University, Loughborough, UK Daniel Primont Department of Economics, Southern Illinois UniversityCarbondale, Carbondale, IL, USA Daniel Prudencio Department of Economics, Rice University, Houston, TX, USA D. S. Prasada Rao School of Economics, The University of Queensland, Brisbane St. Lucia, QLD, Australia Subhash C. Ray Department of Economics, University of Connecticut, Storrs, CT, USA R. Robert Russell Department of Economics, University of California, Riverside, Riverside, CA, USA Subrata Sarkar Indira Gandhi Institute of Development Research, Mumbai, India

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Contributors

Kathleen Segerson Department of Economics, University of Connecticut, Storrs, CT, USA Robin C. Sickles Department of Economics, Rice University, Houston, TX, USA Andrew Smith Society, Environment, and Transport, The Swedish National Road and Transport Research Institute (VTI), Stockholm, Sweden Dale Squires NMFS, Southwest Fisheries Science Center, La Jolla, CA, USA Department of Economics, University of California San Diego, La Jolla, CA, USA Spiro E. Stefanou Food and Resource Economics Department, University of Florida, Gainesville, FL, USA Wageningen University, Wageningen, Netherlands Thijs ten Raa Utrecht School of Economics, Utrecht University, Utrecht, The Netherlands Ragnar Tveteras UiS Business School, University of Stavanger, Stavanger, Norway Ignace Van de Woestyne Research Unit MEES, KU Leuven, Brussel, Belgium John Walden NMFS, Northeast Fisheries Science Center, Woods Hole, MA, USA Alan Wall Department of Economics, University of Oviedo, Oviedo, Spain W. L. Weber Department of Accounting, Economics and Finance, Southeast Missouri State University, Cape Girardeau, MO, USA Phill Wheat Institute for Transport Studies, University of Leeds, Leeds, UK Chien Xen Ng Oxera Consulting, Oxford, UK Valentin Zelenyuk School of Economics and Centre for Efficiency and Productivity Analysis (CEPA), The University of Queensland, Brisbane, QLD, Australia Shunan Zhao Department of Economics, Oakland University, Rochester, MI, USA

Part I Theory

1

Neoclassical Production Economics: An Introduction Robert G. Chambers and Subhash C. Ray

Contents Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . An Overview of Neoclassical Production Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The Primal Perspective . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The Dual Perspective . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Restricted Profit Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The Search for a Practical Production Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The Cobb-Douglas Production Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The Constant Elasticity of Substitution (CES) Production Function . . . . . . . . . . . . . . . . . . . Homothetic and Non-homothetic CES Production Functions . . . . . . . . . . . . . . . . . . . . . . . . Additive Implicit Multiple Input Production Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Constant Ratio of Elasticities of Substitution (CRES) Production Functions . . . . . . . . . . . . Indirect Production Function: An Aside . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Additive Implicit Indirect Production Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Flexible Functional Forms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Appendix 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Appendix 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Appendix 3 Elasticity of Substitution Derived from the Dual Cost Function . . . . . . . . . . . . . . Cross-References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

4 4 6 9 21 24 24 26 28 30 32 36 37 41 44 44 45 47 47

We thank Chuang Li for his technical assistance in preparing the final version of this manuscript. R. G. Chambers Department of Agricultural and Resource Economics, University of Maryland, College Park, MD, USA e-mail: [email protected] S. C. Ray () Department of Economics, University of Connecticut, Storrs, CT, USA e-mail: [email protected] © Springer Nature Singapore Pte Ltd. 2022 S. C. Ray et al. (eds.), Handbook of Production Economics, https://doi.org/10.1007/978-981-10-3455-8_18

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R. G. Chambers and S. C. Ray

Abstract

To emphasize the nexus between the theory and the empirics of production, this chapter is split into two parts. The first presents a brief overview of the state of neoclassical production theory as it exists in the third decade of the twenty-first century. The second part presents an overview of the history of the development of functional forms for the production function. Keywords

Production · Primal · Dual · Profit-maximization

Introduction This handbook is divided into two volumes. The first volume focuses on theoretical issues of production economics. The second volume focuses on empirical applications of the theories to applied production analysis. This split, hopefully, clarifies the presentation. In practice, however, no such clear separation exists. Throughout its history, production theory has responded to empirical exigency. An early exemplar is von Thünen’s [25] induction of the principle of diminishing marginal returns from records for his farming estate. So, too, are the Cobb and Douglas [6] development of their production function to fit observed trends in US macroeconomic data and Gorman’s [10] theorem on the aggregation of fixed factors of production. To emphasize this nexus between production theory and production empirics, we split this volume’s introductory chapter into two parts. The first presents a brief overview of the state of neoclassical production theory as it exists in the third decade of the twenty-first century. The second part presents an overview of the history of the development of functional forms for the production function

An Overview of Neoclassical Production Theory Neoclassical theory gradually emerged from classical economics during the last quarter of the nineteenth century. And, as illustrated by Marshall’s Principles, the early twentieth-century discussions of producer behavior were heavily sprinkled with neoclassical notions such as marginal productivity (returns), marginal cost, and diminishing marginal returns. Often, these ideas were not wholly original to neoclassical thinkers. For example, classical writers had recognized the principle of diminishing returns. But they often attributed it to different causes (e.g., deteriorating quality of inputs) than the neoclassical school. What distinguished the early neoclassical writers was their emphasis on the marginal principle defined in terms of identical units of inputs and outputs. As the economic analysis of productions systems developed, it became increasingly evident that identical was to be interpreted in the narrowest possible terms to mean identical in all aspects (e.g., quality, time, place, state of nature, etc.).

1 Neoclassical Production Economics: An Introduction

5

An example from early empirical studies of economic growth illustrates. Growth accounting seeks to attribute output growth to its basic causes. In the neoclassical framework, that means attributing output growth to increased resource use and technical advances in know-how. Abramovitz [1] observed that, according to then current measurement techniques, the primary source of US national product growth in the preceding eight decades was not measured resource growth. Instead, it was an increase in total factor productivity defined as the residual between measured output growth and input growth. This challenged received neoclassical production theory because it implied that the neoclassical model of rational producers reacting to a technology could only explain observed growth patterns in a deus ex machina fashion as unexplained shifts in production frontiers. Jorgenson and Griliches [14] showed, however, that once mismeasurements of inputs, outputs, and prices were eliminated to ensure a closer concordance between measured variates and their theoretical counterparts, all but a small residual of output growth could be explained by input adjustments along a given production frontier.1 The essential point is that neoclassical production theory works with precisely defined mathematical variates, while the extramathematical reasoning [8] that we attach to it often does not. That theory is a way of organizing our thinking about how producers behave in real life. Or it’s a set of formal stories about producers based on a model of real-world decision settings. If the mathematics are correct, so too must be those formal stories. But because models are not replicas, those formal stories can and will differ from reality. The ultimate test of a model and its derived theory is whether they allow us to say something useful about reality. Ideally, models would allow us to predict exactly observed or measured behavior. But that seems beyond our reach. So while confronting theoretical models with data is crucial, one needs to remember that observed data are only truly informative about a model’s accuracy to the extent that measured variates correspond to the model’s theoretical variates. Of course, the converse is also true; a model’s usefulness is circumscribed by the correspondence between the measured variates of interest in real-world settings and the model’s theoretic variates. The ultimate challenge is to strike the proper balance. Our theoretical overview is broken into two parts that roughly accord with different analytic approaches popularly known as the primal and the dual approaches. This terminology stems from four sources: the mathematical notion of a dual space, the economist’s choice of market value as the producer’s objective function, the economist’s expression of behavioral relations for quantities as functions of prices with both treated as objects falling in the same dimensional real space, and the different perspectives adopted in analyzing these relations.

1 At

roughly the same that Abramovitz [1] wrote, Schultz [21] articulated the idea that an ideal input-output formula would have measured output growth completely explained by measured input growth. If that could be achieved, it would suggest that a complete economic explanation of output growth had been accomplished. Schultz [21], in a footnote, attributed the idea to Zvi Griliches, who was a graduate student at the time.

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For an arbitrary space, X, its dual space, X∗ , is defined as the space of linear operators for X. An important property of finite dimensional real spaces, denoted by RM , is that they are self-dual in the sense that RM∗ = RM and RM∗∗ = RM . Thus, as purely mathematical objects, an M−dimensional vector of quantities, denoted z ∈ RM , and an M−dimensional vector of prices, q ∈ RM , are both recognizable as linear operators. The z ∈ RM is given by the linear  market value of bundleM∗ function of quantities m qm zm for prices q ∈ R . But because the dual space of M also be interpreted as the linear RM∗ is RM , the same  market value of z ∈ R can function of prices m zm qm for quantities z ∈ RM∗∗ . Originally, economists focused their attention on value maximization viewed as a problem of choosing quantities to optimize market values of commodity bundles.  That placed the focus on m qm zm viewed as a linear function of quantities. It was only with the development of the theory of optimization in the decades following World War  II that researchers apprehended and exploited the analytic advantages of viewing m zm qm as a linear function of prices. The quantity-based approach was identified with the primal terminology and the price-based approach with the dual approach. It is to be emphasized that the analytic difference between the two approaches is one of perspective and technique. The same substantive results are available from either approach. However, the dual approach quickly proved particularly popular because it offered an econometrically advantageous way to model production systems. The discussion that follows first considers the primal approach. Then the model is generalized and analyzed from a dual perspective. A brief demonstration of how the general model can accommodate both long-run and short-run producer behavior follows.

The Primal Perspective Prior to roughly 1970, the approach that economists used to analyze productive systems closely followed Samuelson’s [18] classic treatment of the profit-maximizing firm. Competitive profit-maximizing producers were modeled as facing a technological constraint treated typically as a smooth transformation or production function, and programming techniques were used to characterize profit-maximizing solutions and to make inferences about producer responses to price perturbations. N M M Let x ∈ RN + , y ∈ R+ , w ∈ R++ , and p ∈ R++ denote, respectively, a vector of inputs, a vector of outputs, the input price vector, and the output price vector. We denote by t (x, y) a function of inputs and outputs that is nondecreasing in y and nonincreasing in x such that x can produce y if and only if t (x, y) ≤ 0. Posed formally, the producer’s problem is to choose (x, y) to   max p y − w  x : t (x, y) ≤ 0 . (1) (Here, and in what follows, x  y for x, y ∈ RN denotes the usual inner product,  n xn yn .) The Lagrangian associated with this problem is written

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L (x, y, w, p, λ) = p y − w  x − λt (x, y) where λ is a nonnegative Lagrangian multiplier, and the first-order conditions for an optimal interior solution are ∇y L = p − λ∇y t (x, y) = 0, ∇x L = −w − λ∇x t (x, y) = 0, ∇λ L = −t (x, y) = 0. Here ∇h f (h) denotes the gradient of f with respect to the argument h. These conditions are familiarly interpretable as: output price ratios, (pk /pj ), are ∂t equated to marginal rates of transformation between yk and yj , dy / ∂t ; input price k dyj ∂t ratios, (wn /wi ), are equated to marginal rates of technical substitution, ∂x / ∂t ; the n ∂xi ∂t marginal product of xn in producing ym , − ∂x / ∂t is equated to (wn /pm ); and n ∂ym x and y are technically efficient (i.e., fall on the relevant frontiers). Figure 1a–c illustrate visually. The associated second-order conditions for a local maximum (under the assumption that λ > 0) require that

Fig. 1 (a): Input equilibrium (b): Output equilibrium (c): Efficient production

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⎡

⎤ ∇yy  t (x, y) ∇yx  t (x, y) ∇y t (x, y) H = − ⎣ ∇xy  t (x, y) ∇xx  t (x, y) ∇x t (x, y) ⎦ ∇y t (x, y) ∇x t (x, y) 0 be negative semi-definite. Then under the assumptions that (a) the first-order conditions are satisfied, (b) the second-order conditions are satisfied, and (c) the conditions of the implicit function theorem are met, optimal producer behavior is differentially characterized by ⎤ ⎡ ⎡ ⎤ dy (p, w) dp ⎣ dx (p, w) ⎦ = −H −1 ⎣ −dw ⎦ , (2) 0 λˆ (p, w) where y (p, w) and x (p, w) denote, respectively, optimal (profit-maximizing) supply and derived demand (for inputs) vectors, λ (p, w) denotes the optimal value of the Lagrangian multiplier, and λˆ (p, w) = d ln λ (p, w) . Using assumptions (a)–(c) and (2) establishes that: Behavioral Prediction

(1) ∂ym∂p(p,w) ≥ 0, m = 1, 2, . . . , M; m

Behavioral Prediction

(p,w) ≤ 0, n = 1, 2, . . . , N; (2) ∂xn∂w n

Behavioral Prediction

(p,w) = (3) ∂ym∂p k

Behavioral Prediction

(p,w) (4) ∂xn∂w = j

Behavioral Prediction j = 1, 2, . . . , N.

(p,w) (5) ∂ym∂w = j

∂yk (p,w) ∂pm , k, m = 1, 2, . . . , M; ∂xj (p,w) ∂wn , j, n = 1, 2, . . . , N ; and ∂x (p,w) − j∂pm , m = 1, 2, . . . , M,

Because the objective function for the profit-maximizing firm is linear in (p, w) and t (x, y) is independent of (p, w) , any solution to the problem, y (p, w) and x (p, w) , for (p, w) must also be a solution for (μp, μw) where μ > 0, whence y (μp, μw) = y (p, w) and x (μp, μw) = x (p, w) . Combined with Behavioral Predictions (1) − (5), zero-degree homogeneity summarizes the core results for neoclassical production theory. Producers do not suffer from money illusion, profitmaximizing supplies slope upward in their own prices, profit-maximizing demands slope downward in their prices, and in a smooth world differential supply and demand adjustments possess an essential symmetry. A number of related results, for example, the Le Chatelier principle relating long- and short-run supply responsiveness, follow from suitable modifications of these basic techniques. Some comments are relevant: First, the setting of the problem and the assumed motivation behind producer behavior are key to its analysis. Assuming that producers are price takers and small relative to the market eliminates the possibility for strategic interactions that would complicate analyses. Second, assuming producers are profit seekers lets them be modeled “as if” they solve a maximization problem with a clearly articulated objective function and constraints. Optimal or rational behavior is then identified with conditions required for profit maximization. And, more suggestively, producers are said to be “in equilibrium” if their behavior is consistent with that optimum. As Samuelson [18] explained, the resemblance to physical systems being “in equilibrium” when entropy is maximized is not

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accidental. Neither is the fact that the tools of analysis, variational techniques applied to differentiably smooth systems, closely parallel those used in classical thermodynamics. Consequently, equilibrium behavior is described primarily in terms of conditions which the physical technology must satisfy in an optimum. Marginal rates of transformation and marginal rates of substitution, which characterize tradeoffs between inputs and outputs internal to the technology, must be equated to real market prices. Moreover, maxima (at least local) are distinguished from inflection points or minima by conditions that the technology must satisfy (see the second-order conditions). Again the resemblance to classical mechanics and thermodynamics is not accidental. And once these conditions are assured, producers can be shown to behave in a manner that accords with the most familiar parables of microeconomic theory as captured by Behavioral Predictions (1) and (2) and zerodegree homogeneity. Third, optimal producer behavior is characterized in infinitesimal terms. To be sure, directional results are obtained, but they only strictly apply in tiny neighborhoods of the identified equilibria. Individuals are modeled “as if” they will perceive and respond (smoothly) to even the tiniest perturbations in market prices. And to make inferences about how individuals respond to discrete changes, these differential results must be augmented by a combination of integral analysis and the correspondence principle.

The Dual Perspective In this section, we continue to treat producers who are profit maximizers and face given prices and a technological constraint. But we alter the framing of the problem and its mathematical analysis. We now model the producer’s technological constraints as a closed and nonempty subset of M-dimensional real space, T ⊂ RM , that we shall refer to interchangeably as the technology set or the technology. We relax the distinction between inputs and outputs and work instead with net outputs (netputs for short) denoted as z ∈ RM . Using netputs accommodates the possibility that in differing circumstances the same commodity can function variously as an input or an output. The technology set, T ⊂ RM , is defined as

T = z ∈ RM : z is technically feasible . In principle, the technology subsumes all feasible productive activities.2 The convention is that zk < 0 denotes a netput functioning as an input and zk > 0

2 This

raises a semantic point. One often reads or hears references to individuals or firms facing different technologies. For example, a hand-push, reel lawn mower and a self-propelled lawn mower might be referred to as two different technologies for cutting grass. This is not our interpretation of T , which we take to encompass all technically feasible activities. In what follows, we shall discriminate between different productive activities (e.g., growing wheat as opposed to producing steel) not as different technologies but as different production processes that fall in T .

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Fig. 2 Netput technology

one functioning as an output. That role can change. Figure 2 illustrates one possible T . Netput prices are denoted as q ∈ RM ++ . We now show that the behavioral predictions for producers derived indirectly from t (x, y) via H and the Lagrangian approach can be deduced directly from the following postulates: Postulate Postulate Postulate

(a) Producers face competitive pricing for netputs. (b) Producers are profit maximizers. (c) Finite solutions exist for   π (q) ≡ max q  z : z ∈ T z

(3)

for all q ∈ RM ++ . In what follows, we refer to π (q) as the profit function. More formally, it is the support function for T [17, p. 28]. Let   Z (q) = arg max q  z : z ∈ T denote the correspondence3 giving the profit-maximizing solutions to (3) and z (q) ∈ RM denote a particular element of Z (q). Because q  z is linear in q and T is independent of q, any solution for (3) for q is also a solution for μq with μ > 0. Hence, for μ > 0, Z (μq) = Z (q) (optimal netput supplies are homogeneous of degree zero in q) and π (μq) = μπ (q) (π (q) is positively homogeneous in q). The homogeneity properties of Z (q) and π (q)

correspondence represents a point to set mapping. Thus, Z (q) ⊂ RM denotes the set of profitmaximizing solutions associated with the point q ∈ RM . We use the correspondence notation to remind the reader that profit maximization problems may have multiple, global solutions.

3A

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manifest the familiar adage that “only real prices matter” in making economic choices. Because z (q o ) , z (q ∗ ) ∈ T , the definition of π (q) ensures that π (q o ) = o q z (q o ) ≥ q o z (q ∗ ) and π (q ∗ ) = q ∗ z (q ∗ ) ≥ q ∗ z (q o ) . These inequalities, which characterize maxima, are the fundamental source of the behavioral results that follow. Geometrically, require for all q ∈ RM ++ that z (q) ∈ T and that  they M T fall in the half-space z ∈ R : q  z ≤ q  z (q) generated by the hyperplane with normal q that passes through z (q) . Visually, that translates into a hyperplane with normal q being tangent to T (from above) at z (q) . Adding the inequalities and rewriting gives4   o    o z q − z q∗ q − q ∗ ≥ 0.

(4)

Formally, expression (4) says that Z (q) is (positively) cyclically monotone in q [17, p. 228]. Cyclical monotonicity represents one multidimensional generalization of univariate monotonicity.5 As an example, when M = 1, cyclical monotonicity requires z (q) to be nondecreasing in q. More generally, setting qko = qk∗ for all k = m in (4) gives   o   o ∗ − qm ≥ 0. zm q − zm q ∗ qm The economic interpretation of this basic characteristic of maxima is that optimal netput supplies must be nondecreasing in their own prices. This condition, which applies for discrete as well as infinitesimal price changes, is to be compared to Behavioral Predictions (1) and (2). Now observe that for all q o , q ∗   π q o ≥ q o z q ∗ 

π q

o

⇓

    ≥ π q∗ + z q∗ qo − q∗ ,

(5)

  where the second inequality follows by adding zero in the form of π (q ∗ ) −q ∗ z(q ∗ ) to the right-hand side of the first inequality. In words, expression (5) requires that any element of Z (q) must belong to the subdifferential correspondence for π (q) , which we denote by ∂π (q) ⊂ RM , at q ∗ [13, p. 220].

4 Samuelson

[18, pp. 80–1] established an equivalent result in the one-output, multiple-input case.  not the only one. Another stronger notion of monotonicity is that q  ≥ q ⇒ z q  ≥ z(q). This version implies cyclical monotonicity but is not implied by cyclical monotonicity. One intuitive way to discriminate between the two notions of monotonicity is that cyclical monotonicity means that price and quantity movements are positively correlated. The stronger notion requires that any price increase be matched by all quantities at least weakly increasing.

5 But

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The subdifferential notion generalizes the more familiar gradient to accommodate nonsmooth (nondifferentiable) functions. Where a gradient is interpreted geometrically as the unique normal of a hyperplane that is tangent to the graph of a function, the subdifferential is interpreted as the set of normals to the hyperplanes that support the graph of the function from below. Panels (a) and (b) of Fig. 3 illustrate. Thus, h ∈ RM belongs to the subdifferential of f (v) for f : RM → R at v o if the hyperplane h v satisfies h v o = f (v o ) and f (v) ≥ h v for all v so that the graph of f (v) lies above that of h v while sharing a point in common. A function that possesses a nonempty ∂f (v) is differentiable if and only if ∂f (v) is a nonempty singleton set. Intuitively, therefore, expression (5) says that tangent hyperplanes that support the graph of π (q) from below at q ∗ must belong to Z (q ∗ ). Figure 4 illustrates. This is a general version of Hotelling’s lemma, which says that profit-maximizing netput supplies correspond to partial derivatives of the profit function, π (q), zm (q) =

∂π (q) , ∂qm

m = 1, 2, . . . , M

when π (q) is smooth (differentiable). Conversely, π (q) is differentiable in qm only if there is a unique optimal solution zm for (3). A familiar geometric characterization of smooth convex functions is that they are always underapproximated by first-order Taylor series around the point of approximation. Cyclical monotonicity of subdifferentials is the generalization of that characterization that covers nonsmooth convex functions. Hence, cyclical monotonicity of ∂π (q) ensures that π (q) is convex as a function of q [17, Theorems 24.8 and 24.9]. Or as Fig. 4 illustrates, all of π (q) s tangent hyperplanes support it from below for all q. (One can show directly that π (q) is a convex function of q. Versions of that argument are presented in several of the chapters that follow, and so we do not repeat it here.)

Fig. 3 (a): Gradients as unique tangents for smooth function (b): Elements of subdifferential for nonsmooth function

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Fig. 4 Subdifferential as profit-maximizing netput

Having demonstrated zero-degree homogeneity of z (q) and the discrete generalization of Behavioral Predictions (1) and (2), all that remains is to derive the netput analogues of Behavioral Predictions (3)–(5). This is done easily by assuming that, in addition to being convex, π (q) is twice-continuously differentiable and applying Hotelling’s lemma and the Schwarz-Young symmetry theorem. Thus, Postulates (a)–(c) suffice to establish generalized versions of the results established using techniques borrowed from classical thermodynamics. While the robustness of those results is to be remarked, it is essential to recognize that they derive not from conditions placed upon T but from Postulates (a) and (b) that require that producers maximize profit in a competitive market setting. As long as T admits a maximum, the behavioral predictions are robust to T  s actual structure. Thus, they are best recognized as extramathematical reasoning applied to fundamentally mathematical results about solutions to optimization problems. So, what we interpret as an economic result involving upward-sloping supply curves is a mathematical characteristic of optimal solutions to a class of optimization problems, cyclical monotonicity, that has intuitively plausible economic implications. The powerful corollary is that models of producer behavior need not be constructed from the “ground up,” so to speak, by starting with t (x, y) and then laboring through first- and second-order conditions to obtain behavioral economic relations. Instead, if we accept Postulates (a)–(c), a direct specification of behavioral models in terms of π (q) and ∂π (q) is available that provides observationally equivalent behavioral predictions to those derived via t (x, y). A natural consequence is that production economists have become less focused on the technical aspects of the underlying technology. Beyond manifesting the theorem of comparative advantage relative to more physically oriented disciplines, such as engineering and the biological sciences, that change in focus also emphasizes the

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true role T , and its representations play in economic analysis. Clearly, producers must understand the details of T if they are to prove successful. On the other hand, production economists are not themselves producers. Their job is to develop models that accurately depict producer behavior. And that requires an understanding of the characteristics of T that play an essential role in conditioning that behavior but not of all of its technical details. Our arguments have shown that, as a mathematical object, π (q) is positively homogeneous and convex (sublinear) as a function of q ∈ RM ++ . Under weak continuity restrictions, those properties ensure that π (q) is the support function for the closed, convex subset T¯ of RM [17, Theorems 13.1 and 13.2], [13, Theorem C.3.1.1] given by

T¯ = z ∈ RM : q  z ≤ π (q) for all q ∈ RM ++ . Recall that a set B ⊂ RM is convex if b0 , b1 ∈ B implies λb0 + (1 − λ) b1 ∈ B for all 0 < λ < 1. The construction of T¯ is illustrated in Fig. 5. There the hyperplanes labeled q¯  z = π (q) ¯ and qˆ  z = π qˆ , respectively, represent the upper boundaries for the closed,      convex half-spaces z ∈ RM : q¯  z ≤ π (q) ¯ and z ∈ RM : qˆ  z ≤ π qˆ generated M ¯ by q¯ ∈ RM ++ and qˆ ∈ R++ . T must be contained in

 

 q, ¯ qˆ = z ∈ RM : q¯  z ≤ π (q) ¯ ∩ z ∈ RM : qˆ  z ≤ π qˆ   and in all other half-spaces z ∈ RM : q  z ≤ π (q) generated by the remaining q ∈  RM ¯ qˆ , as Fig. 5 illustrates, and these other half-spaces are closed ++ . Because  q, convex, T¯ must be as well. Moreover, as is visually apparent from Fig. 5, z˜ ≤ z ∈ T¯ Fig. 5 Constructing T¯

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implies z˜ ∈ T¯ (free disposability of netputs) because z˜ ≤ z requires q  z˜ ≤ q  z for all q ∈ RM ++ . These observations show that T¯ cannot correspond to technology sets that are nonconvex and that do not exhibit free disposability of netputs. (Recall the only restrictions which have been placed upon T are that it be nonempty and closed.) Convex technologies exhibit what economists refer to as diminishing marginal returns and decreasing returns to scale. (Strictly speaking, diminishing and decreasing should be replaced with nonincreasing.) Free disposability of netputs, on the other hand, is the mathematical generalization of t (y, x) nondecreasing in y and nonincreasing in x (nonnegative marginal productivities). Therefore, as a general rule, T¯ = T . Nevertheless, given Postulates (a)–(c), T ⊂ T¯ necessarily. This claim follows from the observation that z ∈ T only if q  z ≤ π (q) for all q ∈ RM ++ . (If this were not true, there would exist a z ∈ T such that q  z > π (q) for some q, which violates the definition of π (q) .) On the other hand, as z˜ in Fig. 6 illustrates, there can exist points falling in T¯ that do not belong to T . Thus, T¯ represents an over or outer approximation to T . All technically feasible points necessarily fall in T¯ , but T¯ can contain technically infeasible   netputs.    For example, in Fig. 6, it is visually obvious that Z qˆ = z1 qˆ , z2 qˆ   and z˜ ≤ μz1 qˆ + (1 − μ) z2 qˆ for all μ ∈ [0, 1] . In words, for qˆ ∈ RM ++ , multiple solutions exist for the profit maximization problem, and there exist points dominated by all the convex combination of those multiple solutions that fall in T¯ but not in T . The existence of such points  manifests  the nonconvex nature of the boundary of T between the points z1 qˆ and z2 qˆ . These nonconvexities can be interpreted, respectively, as the technology exhibiting increasing returns in z1 and z2 in the neighborhood of the origin. It is well known,  however,  that Postulates (a) and (b) guarantee that points falling between z1 qˆ and z2 qˆ on the nonconvex portion of the boundary of T will never be utilized by profit-maximizing producers (e.g., [15, 18]). One sees this visually in Fig. 6 by noting that any hyperplane with

Fig. 6 Non-convex T

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a strictly positive normal cannot support T from above in a region where it is nonconvex. In more traditional terms, profit maximizers never operate in region 1 of the production function. Thus, while T¯ may not characterize T if the latter is nonconvex, the information lost in going from T to T¯ is economically superfluous under Postulates a) and b) because T and T¯ yield the same π (q) . Put another way, π (q) derived from T and T¯ are observationally equivalent to one another in the sense that the profit-maximizing choices made by an individual facing T cannot be distinguished from those of an individual facing T¯ . In more formal terms, T¯ is the free disposal convex hull of T . That is, T¯ is the smallest convex set consistent with free disposability for which T ⊂ T¯ . It consists of all convex combinations of T and all elements of RM dominated (in the sense of ≤) by those convex combinations of T [17, Theorem 2.3]. The practical import is that choosing to model producer behavior in terms of a positively homogeneous and convex π (q) whose subdifferentials depict profitmaximizing netput supplies is mathematically equivalent to assuming that producers solve (3) for convex T satisfying free disposability of netputs. The importance of this observation is hard to overestimate. Few economists would argue that either convexity of the technology or free disposability of netputs is realistic. As just one example, intermediate micro students are routinely introduced to lazy-S-shaped production functions that violate both convexity and free disposability. Nevertheless, in the form of convexity and monotonicity assumptions on t (x, y) , these properties are routinely imposed in more formal analyses of the profit maximization problem because they provide a mathematical foundation for using calculus tools and Kuhn-Tucker theory. What our arguments show is that Postulates (a)–(c) ensure that observed profit and netput supply functions can be treated “as if” they come from a T satisfying these conditions, even if the true T does not exhibit these “nice” regularity conditions, without losing any economically relevant information. When translated from netput space to more familiar input-output space, a convex T exhibiting free disposability would resemble the area on or below the graph of t (x, y) illustrated in Fig. 1c. Figure 1c represents a technology where complete inaction is possible, so that if economic circumstances dictate, the individual is free to operate at (0, 0) . We have not yet endowed T with a parallel property, and as Fig. 7 illustrates, it is possible to specify closed, convex T for which π (q ∗ ) < 0 for ∗ some q ∗ ∈ RM ++ . If one were to confront such a T and q in the real world, common sense would dictate that, in a free market economy, the best long-run solution is not to produce. (Of course, one can easily imagine circumstances in which short-run circumstances would dictate rationally accepting short-term negative profits.) Our Postulate (b), properly interpreted, allows for such decisions. But it is essential to recognize that in formal terms maximizing profit is not the same thing as mechanically solving (3). On the other hand, if T is allowed to permit inaction, formally 0M ∈ T where 0M denotes the traditional origin in M-dimensional real space, then π (q) ≥ 0 for all q ∈ RM ++ .

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Fig. 7 Negative π (q)

Our discussion motivates the following definition. Definition 1. A technology set, T ⊂ RM , is neoclassical if: (a) (b) (c) (d)

T is closed and nonempty. z ∈ T ⇒ z ∈ T for all z ≤ z (free disposability of netputs). 0 < λ < 1 (convexity). z0 , z1 ∈ T ⇒ λz0 + (1 − λ) z1 0M ∈ T (inaction is possible).

The following theorem presents the basic duality results that establish the equivalence of approaching the firm’s problem either via π (q) or via T and then developing π (q). Theorem 1 (Fundamental Duality). If T is neoclassical, π (q) ≥ 0 for all q ∈ RM ++ , π (q) is positively homogeneous and convex as a function of q, and

T = z ∈ RM : q  z ≤ π (q) for all q ∈ RM ++ . In many practical settings, working with set-based concepts can prove challenging. And cardinal functional representations of T then prove attractive. Define

z γ (z) = inf λ > 0 : ∈ T λ as the gauge function for T . Figure 8 illustrates γ (z) visually as the maximal radial expansion of z that is consistent with z/λ remaining technically feasible. If z ∈ T , it’s geometrically obvious from Fig. 8 that γ (z) ≤ 1. Conversely, if γ (z) ≤ 1, then by the definition of γ (z) , z ≤ z/γ (z) ∈ T under free disposability of netput. Hence, if T is neoclassical, γ (z) is a complete function representation of T in the sense that

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Fig. 8 Gauge function for neoclassical T

z ∈ T ⇔ γ (z) ≤ 1.

(Indication property)

Knowing γ (z) is mathematically equivalent to knowing neoclassical T . Moreover, a basic property of gauge functions [13, Theorem C.1.2.5] ensures that if T is neoclassical, then γ (z) is nondecreasing, positively homogeneous, and convex (sublinear) as a function of z . There are a variety of ways to interpret γ (z) . For example, as Fig. 8 illustrates, if a point, z∗ , falls in the interior of T , then γ (z∗ ) < 1. In a production context, that means that z∗ does not lie on the frontier of T and, as such, is not technically efficient. Thus, γ (z) can be viewed as a measure of technical efficiency. (In particular, it is closely related to the distance-function concept discussed in detail in Chambers and Färe, this volume). Another natural interpretation of γ (z) is as a formally derived version of the transformation function used in our Lagrangian formulation of the profit maximization problem. Slightly abusing notation, this can be seen by defining t (z) ≡ γ (z) − 1. Adopting this interpretation and applying the indication property, problem (3) can be reformulated in equivalent mathematical programming terms as   π (q) = max q  z : γ (z) ≤ 1 z

If T is neoclassical, it follows from standard Kuhn-Tucker theory in the smooth case that the first- and second-order conditions developed for the Lagrangian formulation are necessary and sufficient conditions for a global solution to the profit maximization problem. Given Theorem 1, it is natural to suspect that the existence of a positively homogeneous and convex π (q) implies the existence of a γ (z) consistent with

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a neoclassical T . A simple perspective on that relationship is offered by observing the definition of γ (z) requires that z/γ (z) ∈ T , whence q z ≤ π (q) γ (z)

M for all q ∈ RM ++ , z ∈ R .

Therefore, q z ≤ γ (z) π (q)

for all q ∈ RM ++

so long as γ (z) and π (q) > 0. (When T exhibits constant returns to scale, T is a cone, so that both γ (z) and π (q) equal zero.) Thus, if Z (q) = ∅, it must be true z that there exists z (q) such that q  z (q) = π (q) γ (z) so that the upper bound of πq(q) in the inequality must be achieved. Hence,  γ (z) = max

q∈RM ++

= max

q∈RM ++



q z π (q)



 q  z : π (q) ≤ 1 ,

(6)

where the normalization after the second equality follows from the homogeneity of degree zero in q of the objective function. Thus, Theorem 2 (Profit-Gauge Duality). If T is neoclassical,   π (q) = max q  z : γ (z) ≤ 1 z∈RM

and γ (z) = max

q∈RM ++



 q  z : π (q) ≤ 1 .

Figure 9 illustrates the solutions to the two programming problems posed in Theorem 2. For ease of illustration, we have assumed that z2 always acts as an input in T and that the boundary of T is smooth. Otherwise, the visual demonstration is general. From this perspective and the fact that π (q) is the support function for T , it is apparent that an alternative interpretation of γ (z) is as a support function for the dual technology set

T ∗ = q ∈ RM ++ : π (q) ≤ 1 . And, although we do not present a formal demonstration, it also follows that π (q) has a natural interpretation as the gauge function for T ∗ that is associated with a neoclassical T .

20

R. G. Chambers and S. C. Ray

Fig. 9 Support function for T ∗

Modifying earlier arguments, Theorem 2 allows us to establish the following generalized version of Shephard’s lemma for neoclassical technologies (Hotelling’s lemma is a special case): z ∈ ∂π (q) ⇔ q ∈ ∂γ (z)

(Shephard’s lemma)

(7)

As was shown earlier, z ∈ ∂π (q) implies z ∈ Z (q), so that z ∈ Z (q) requires that q support the graph of γ from below at that z. (In other words, market price ratios are equated to marginal rates of substitution (transformation).) Going the other way, Shephard’s lemma shows that if market price ratios are equated to marginal rates of substitution (transformation), then z ∈ Z (q) . This is a generalization of the familiar envelope theorem that can be cast more formally as saying that Z (q) and   Q (z) = arg max q  z : π (q) ≤ 1 , the set of price-dependent netput supplies, are (lower) inverses of one another. Shephard’s lemma links multiplicity of solutions to the profit-maximizing problem to the smoothness properties of γ (z) and π (q) . To illustrate, suppose that ∂π (q ∗ ) is a nonsingleton set so that at q ∗ , π (q ∗ ) possesses a continuum of supporting hyperplanes. Geometrically, this is manifested by π (q ∗ ) being kinked as illustrated in Fig. 10. By Shephard’s lemma, that implies that q ∗ must be a supporting hyperplane to γ (z) for all z ∈ ∂π (q ∗ ) . That can be true only if γ (z) is linear over that continuum of z. Conversely, if γ (z) is kinked at z∗ , z∗ must provide a supporting hyperplane to π (q) over a continuum of q. In short, flats in primal space map into kinks in dual space, and kinks in primal space map into flats in dual

1 Neoclassical Production Economics: An Introduction

21

Fig. 10 Kinks and flats

space, and vice versa. It follows that γ (z) is strictly convex in z if and only if π (q) is strictly convex in q. Apart from the slight difference in their domains, RM versus RM ++ , γ (z) and π (q) , as mathematical objects, are both positively homogeneous and convex (sublinear) functions. That they should be derivable from similar calculating formulae should not be surprising. The recognition, however, that they are essentially natural inverses of one another for neoclassical technologies is fundamental and reflects the mathematical principles behind Theorem 1. In fact, the first formal demonstration of a duality between a dual economic object (a cost function) and a primal representation of the technology (a distance function) due to Shephard [22] assumed virtually the same structure as Theorem 2. Ultimately, however, the analytic message is the same. If there exists a netput supply system, z (q) , which satisfies the usual neoclassical postulates of homogeneity of degree zero (absence of money illusion), upward-sloping output supplies, and downward-sloping demand, there must exist a neoclassical technology that is consistent with it. We close this section with Fig. 11 that illustrates the triadic relationship between T , π (q) , and γ (z).

Restricted Profit Functions To this point, all M netputs have been treated as freely variable in solving the profit maximization problem. Thus, π (q) represents a long-run profit function. Economists, however, are frequently interested in examining short-run behavior that is characterized by some of netputs being fixed. Two important cases in analyses that maintain a split between inputs and outputs are offered by the cost function   c (w, y) = min w  x : (x, y) ∈ T x

and the revenue function

22

R. G. Chambers and S. C. Ray

Fig. 11 A triad of production relations

  R (p, x) = max p y : (x, y) ∈ T . y

Both the cost and the revenue function are manifestations of the more general notion of a restricted (short-run) profit function. Restricted profit functions give the maximal profit available conditional on holding a subvector of z fixed. Partition z as z = z0 , z1 , where z0 ∈ RN with N < M and z1 ∈ RM−N . We refer to z0 as variable netputs and z1 as (potentially) fixed netputs. Given this partition  

 π (q) = max q 0 z0 + q 1 z1 : z0 , z1 ∈ T z0 ,z1

   

 = max max q 0 z0 : z0 , z1 ∈ T + q 1 z1 , z1

z0

where the second equality follows by Bellman’s Principle, which in the current setting simply means that long-run profit maximization always implies variable profit maximization. Defining the variable profit function as

  

π 0 q 0 , z1 ≡ max q 0 z0 : z0 , z1 ∈ T , z0

it is easy to identify c (w, y) with the case where z0 = −x and z1 = y and R (p, x) with z0 = y and z1 = x.  It is immediate from preceding developments that π 0 q 0 , z1 is positively homo geneous and convex in q 0 and that variable-profit-maximizing netputs z0 q 0 , z1   satisfy z0 q 0 , z1 ∈ ∂π 0 q 0 , z1 (where the subdifferential is understood to be in terms of q 0 ). Moreover, a suitably  modified version of Theorem 1 implies the existence of a closed convex T 0 z1 ⊂ RN

1 Neoclassical Production Economics: An Introduction

23





 T 0 z1 = z0 ∈ RN : q 0 z0 ≤ π 0 q 0 , z1 for all q 0 ∈ RN ++ . Making these arguments only requires modifying our previous notation, and so we do not pursue it here. Straightforward corollaries, therefore, are as follows: c (w, y) is positively homogeneous and concave in w, the subdifferentials of −c (w, y) in w represent minus cost-minimizing demands, and those demands are downward sloping in their own input prices; R (p, x) is positively homogeneous and convex in p, the subdifferentials of R (p, x) represent revenue-maximizing supplies, and those supplies are upward sloping in their own prices. From the functional equation 

 π (q) = max π 0 q 0 , z1 + q 1 z1 , z1

we can infer for all q, ˆ q˜ that

   π qˆ ≥ qˆ 0 z0 qˆ 0 , z1 (q) ˜ + qˆ 1 z1 (q) ˜ whence

    π qˆ − π (q) ˜ ≥ π 0 qˆ 0 , z1 (q) ˜ − π 0 q˜ 0 , z1 (q) ˜ + qˆ 1 − q˜ 1 z1 (q) ˜ .

(8)

Figure 12 illustrates. Expression (8) confirms that z1 (q) belongs to the subdifferential of π (q) in the subvector q 1 and that the short-run profit function and π (q) are tangent to one another when z1 is evaluated at z1 (q) . Setting qˆ 1 equal to q˜ 1 , so that only variable netput prices change, gives

   π qˆ − π (q) ˜ ≥ π 0 qˆ 0 , z1 (q) ˜ − π 0 q˜ 0 , z1 (q) ˜ . This expression manifests the Le Chatelier principle that characterizes optima. As commonly interpreted in economics, that principle requires unconstrained (longrun) optima to respond more to parametric changes than constrained (short-run) optima. A direct corollary from it and Hotelling’s lemma is that long-run optimal netput supplies are more responsive to own price changes than their short-run counterparts.  Intuitively, the Le Chatelier principle simply reflects the fact π 0 qˆ 0 , z1 (q) ˜ −  ˜ is the producer’s best response to the price change holding the π 0 q˜ 0 , z1 (q) ˜ . Because this alternative is always available to the fixed netput constant at z1 (q) producer, her optimal long-run response must at least weakly dominate that strategy.

24

R. G. Chambers and S. C. Ray

Fig. 12 Le Chatelier principle

The Search for a Practical Production Function The Cobb-Douglas Production Function The Cobb-Douglas production function remains a classic example of empirical evidence inspiring a theoretical formulation of a production function that has served as the gold standard in neoclassical production economic theory for decades and has retained much of its popular appeal despite the advent of more flexible functional forms even as it nears its centenary. It started from the remarkable observation by Douglas that when plotted on semi-log paper with output (Y), labor (L), and capital (K) measured on the logarithmic scale along the vertical axis against time (t) measured along the horizontal axis, over the years 1899 through 1922, the time series plot of the three series exhibited the tendency that the distance between log(Y) and log(L) was approximately one quarter of the distance between log(L) and log(K). At Douglas’s request, Cobb (a mathematician at Amherst College) came up with the specification 3

1

Y = AL 4 K 4 ; A = 1.01. All of its theoretical properties including constant factor shares, constant returns to scale, diminishing marginal productivities, and unitary elasticity of substitution between the inputs were validated rather than imposed as prior restrictions on the technology. Samuelson [19] offers the following direct derivation of the Cobb-Douglas form as a simple back-of-the-envelope calculation: ln Yt −ln Lt ln Kt −ln Lt

= 14 ; 1899 ≤ t ≤ 1922 ⇒ ln Yt = 34 ln Lt + 14 ln Kt

(9)

1 Neoclassical Production Economics: An Introduction

25

Using the initial conditions ln Yt = ln Lt = ln Kt = 0 (t = 1899), one gets 3

1

Yt = ALt 4 Kt 4 ; A = 1.0. Deviation of the value of the constant (A) from 1 in the formula fitted by Cobb and Douglas can be attributed to measurement errors. As noted in Samuelson [19], one could easily have started from the profitmaximizing behavior of a competitive firm where at the optimal input-output choice, factor prices are equated to their respective values of the marginal product. Denoting the prices of the output, labor, and capital by p, w, and r, respectively, the shares of the two inputs are sL =

∂Y L wL ∂ ln Y = ∂L = f or labor, pY Y ∂ ln L

and sK =

rK = pY

∂Y ∂K K

Y

=

∂ ln Y f or capital. ∂ ln K

Further, zero normal profit in a competitive market implies sL + sK = 1. Now, if the factor shares remain constant, we can set sL = α and sK = β; α + β = 1. Hence, a solution of the partial differential equations (i) ∂ ln Y = α ∂ ln L and (ii) ∂ ln Y = β ∂ ln K yields ln Y = γ + α ln L + β ln K ⇒ Y = ALα K β ; A = eγ , α + β = 1.

(10)

Elasticity of Substitution One of the most important characteristics of the production technology is the degree of substitutability between inputs allowing the producer to change input proportions in response to changes in relative prices of inputs.6 In a two-input case, it relates to the curvature of the isoquant. At one extreme is the Leontief production function Y = min {aL; bK}

(11)

 The cost-minimizing input bundle is (L∗ , K ∗ ) = Ya , Yb which depends only on the output level and does not change with changes in the relative price of inputs. ∗ a The optimal capital-labor ratio is K L∗ = b , and isoquant is L-shaped with zero substitutability between inputs. The other extreme is the linear production function Y = aL + bK

6 See

“Elasticity of Substitution” in the chapter by Russell, this volume.

(12)

26

R. G. Chambers and S. C. Ray

where the marginal rate of substitution along the isoquant is − dK dL = constant at every point on the isoquant. The elasticity of substitution between the two inputs is σ =

d ln( K d ln( K FK .FL L) L) =− =− F L d ln(MRT S) F.FKL d ln( )

a b,

which is

(13)

FK

where Y = F (K, L), FL =

∂F ∂L ,

FK =

∂F ∂K ,

FKL =

∂2F ∂K∂L

(14)

For the Cobb-Douglas production, function σ equals 1 along the entire isoquant. An implication of this unitary elasticity of substitution is that for every 1% increase in w relative to r, the capital labor ratio will always increase by 1%!

The Constant Elasticity of Substitution (CES) Production Function Arrow, Chenery, Minhas, and Solow (ACMS) [2] questioned the validity of the Cobb-Douglas production function across all industries within the manufacturing sector. The basis of their disagreement was the empirical evidence that value added per worker (y = YL ) in any specific industry varied with the real wage rate (w) widely across countries. They used country-level cross-sectional data on different two-digit manufacturing industries to estimate the regression ln yi = a + b ln wi + ui

(15)

The estimated value of the parameter b representing the elasticity of value added per worker with respect to the wage rate differed widely across industries and in most cases was statistically significantly different from both 0 and 1. This prompted them to question the universal validity of both the Leontief (fixed input proportions) production function and the Cobb-Douglas production function exhibiting unitary elasticity of substitution everywhere on the isoquant. Equivalence between the parameter b in (15) and the elasticity of substitution σKL in (13) is not apparent. ACMS [2] first show that the elasticity of the output per worker does, indeed, measure the elasticity of substitution between the two inputs and use their results to develop the new CES production function. Under the assumption of constant returns to scale, the production function Y = F (K, L) is homogeneous of degree 1 so that y = YL = F (k, 1) ≡ f (k) where k=K L is the capital-labor ratio. Under competitive profit maximization w = FL = f (k) − kf  (k) = y − k dy dk .

(16)

1 Neoclassical Production Economics: An Introduction

27

This may be inverted to obtain y = ϕ(w) = ϕ(y − k dy dk )

(17)

From (16)  d(y−kf  (k)) dy = dw − d(kfdw(k)) dw dk dk dk = dfdk(k) dw − f  (k) dw − kf  (k) dw dk dk  = −kf (k) dy dw

(18)

=1 But

dk 1 1 = dy =  dy f (k) dk

Hence, dk dy −kf  (k) dy dw 

(k) dy = − kff  (k) dw = 1

⇒

dy dw

=

(19)

 − kff (k) (k)

Therefore, dy w d ln y f  (k) f (k) − kf  (k) = = −  dw y d ln w kf (k) f (k)

(20)

Finally, as shown in Appendix 1, the expression on the right-hand side of (20) is the elasticity of substitution σKL . Combining (15) and (20), one gets d ln y = b = σ. d ln w

(21)

Substituting (16) and (21) into (15), one gets ln y = ln a + σ ln(y − k dy dk )

(22)

Thus, b y = a(y − k dy dk ) 1

1

⇒ y b = a b (y − k dy dk ) ⇒

dy dk

=

1 1 a b y−y b 1 ka b

=

1 1 − y−a b y b k

=

1 1 − ( −1) y(1−a b y b ) k

(23)

28

R. G. Chambers and S. C. Ray 1

Defining a b = α and

1 b

− 1 = ρ, ρ−1

dy dy αy dy = y(1−αy ρ ) = y + (1−αy ρ ) ⇒ d ln k = d ln y − ρd ln(1 − αy ρ ) dk k

(24)

Upon integration, (24) leads to ln k = ln y + ln(1 − α ln y)−ρ + ln B Setting the constant of integration ln B = kρ =

1 ρ

(25)

ln β yields

βy ρ . 1 − αy ρ

(26)

Thus, ρ

βy k ρ = 1−αy ρ ⇒ y ρ (β + αk ρ ) = k ρ

⇒ y = k(β + αk ρ ) = (βk −ρ + α)

1 −ρ

(27)

1 −ρ

In terms of the aggregate variable, (27) can be expressed as  1 −ρ + β − ρ = α( K ) L − 1  −ρ ⇒ Y = αK + βL−ρ ρ ⇒ Y −ρ = αK −ρ + βL−ρ Y L

Finally, defining γ = (α + β) and δ =

α α+β ,

(28)

(28) can be expressed as

− 1  Y = γ δK −ρ + (1 − δ)L−ρ ρ

(29)

ACMS describe ρ as the substitution parameter, δ as the distribution parameter, and γ as the efficiency parameter.

Homothetic and Non-homothetic CES Production Functions n is said to be homothetic if it can be expressed as A function y = f (x); x ∈ R+

y = g(h(x))

(30)

1 Neoclassical Production Economics: An Introduction

29

where g is strictly increasing and h(x) is homogeneous of degree r. An important property of a homothetic production function is that the marginal rate of substitution between a pair of inputs depends only on the input proportion and is independent of the level of output. This implies that for homothetic technologies, isoquants are radially parallel and output expansion paths are straight lines.7

Homotheticity of the CES Function From (28), −ρY −(ρ+1)

∂Y = −αρK −(ρ+1) ∂K

−ρY −(ρ+1)

∂Y = −βρL−(ρ+1) ∂L

and

Hence, ∂Y ∂L ∂Y ∂K

α = β



K L

(ρ+1) (31)

Thus, the marginal rate of substitution remains unchanged so long as the capitallabor ratio remains constant even as the output level changes.

Non-homothetic CES Function Sato [20] considered two implicit production functions: F (K, L, Y ) = K −ρ + C(Y )L−ρ − H (Y ) = 0 for σ = 1

(32)

F (K, L, Y ) = ln K + C(Y ) ln L − H (Y ) = 0 for σ = 1

(33)

and

Both C(Y) and H(Y) are monotone functions of Y. The function (32) is the nonhomothetic CES (NH-CES), and (33) is the non-homothetic Cobb Douglas (NHCD) production function. For the NH-CES function in (32),

7 See

∂Y ∂K

=

⇒

∂Y ∂L ∂Y ∂K

Appendix 2 for proof.

−(1+ρ) C(Y ) ρK −(1+ρ) ; ∂Y = ρL ; C −ρ −H  (ρ) ∂L C −ρ −H  (ρ)  K (1+ρ) = C(Y ) L

(34)

30

R. G. Chambers and S. C. Ray

Thus, for this production function, the marginal rate of substitution depends on both the level of output (Y) and the capital-labor ratio K L . Hence, the expansion path of the firm is not a straight line. But the elasticity of substitution is still a constant.

Non-homothetic Cobb-Douglas Production Function For (33) ∂Y ∂K

⇒

= ∂Y ∂L ∂Y ∂K

1 ∂Y K(C  (Y ) ln L−H  (Y )) ; ∂L  = C(Y ) K L

=

C(Y ) L(C  (Y ) ln L−H  (Y )) ;

(35)

For the NH-CD even though the elasticity of substitution remains 1 (as in the case of the standard Cobb Douglas function), the marginal rate of substitution between capital and labor declines as one moves to a higher isoquant even as the capital-labor ratio is unchanged. An example of the NH-CD (implicit) production function is ln K +

√

Y ln L − Y 2 = 0

(36)

It can be seen that for (36) ∂Y ∂K

=

⇒

∂Y ∂K ∂Y ∂L

1  ; ∂Y ∂L ln√L K −2Y 2 Y  √ = K Y L 

=

√ Y  ; ln√L L −2Y 2 Y

(37)

Additive Implicit Multiple Input Production Functions Consider an additively separable implicit production function F (Y, x1 , x2 , . . . , xn ) =

n 

F i (Y, xi ) = 1

(38)

i=1

Minimizing cost, conditions

n

i=1 wi xi ,

for this production function yields the first-order i

wi = λ ∂F ∂xi (i = 1, 2, . . . , n),

1−

n 

F i (Y, xi ) = 0,

i=1

where λ is a nonnegative Lagrangian multiplier. From (40)

(39)

(40)

1 Neoclassical Production Economics: An Introduction n 

31

∂xi Fii ∂w = 0 (j = 1, 2, . . . , n) j

(41)

i=1

Here, Fii =

∂F i ∂xi .

Similarly, from (39),

∂xi ∂λ λFiii ∂w + Fii ∂w =0 ⇒ j j j ∂x

j ∂λ ∂wj

λFjj ∂wjj + Fj Here Fiii =

∂Fii ∂xi

=

=1 ⇒

∂xi ∂wj

= − λ1

∂xj ∂wj

=

Fii ∂λ Fiii ∂wj

1 j λFjj

−

; (i = j ) (42)

j

1 Fj ∂λ λ F j ∂wj jj

∂2F i . ∂xi2

Substitution of (42) into (41) yields ∂λ − λ1 ∂w j

n

∂λ ∂wj

=

Fjj j

Fj

⎜ ⎝

1 Fjj λ Fj j

+

⎛

j

⇒

j

(Fkk )2 k=1 F k kk

1

=0 ⎞

(43)

⎟ ⎠ (F k )2

n k k=1 F k kk

Hence, ⎛ ∂xi ∂wj

= − λ1

Fii Fiii

∂λ ∂wj

j

Fi1 Fj

∂xi = − λ1 ⎝ i j ∂wj Fii Fjj

⎞⎛ ⎠⎜ ⎝

⎞ 1 (Fkk )2 n k=1 F k kk

⎟ ⎠

(44)

Define ak ≡ −

Fkk k xk Fkk

(k = 1, 2, . . . , n)

(45)

Using (39), the share of input k in the total cost can be written as sk =

nwk xk i=1 wi xi

=

F k xk n k i i=1 Fi xi

(46)

Thus, ak sk = − ⇒

n

k Fkk

(F k )2 nk i i=1 xi Fi n

k=1 ak sk = −

1 k 2 k (Fk ) Fkk i i=1 xi Fi

k=1

n

(47)

32

R. G. Chambers and S. C. Ray

Hence,8 

aa n i j k=1 ak sk

=− =− =

=−

Fii i x Fi xi Fii i ii ⎛ ⎞ 1 k 2 n k (F ) ⎜ k=1 Fkk k ⎟ ⎝ n ⎠ i i=1 xi Fi



 n

i i=1 xi Fi xi xj

n

i=1 wi xi xi xj

n

i=1 wi xi xi xj



Fii



∂xi ∂wj

=

Fii Fii Fiii Fiii



i i 1 Fi Fi λ Fi Fi ii ii

⎞

⎛

⎝



⎛

n k=1

⎝

1 ⎠ (42) 1 (Fkk )2 k Fkk ⎞

n k=1

C ∂2C ∂C ∂C ∂wi ∂wj . ∂wi ∂wj

1 ⎠ 1 (Fkk )2 k Fkk

= σij .

Constant Ratio of Elasticities of Substitution (CRES) Production Functions Consider the implicit production function  F (Y, x1 , x2 , . . . , xn ) = ni=1 F i (Y, xi ) = 1; F i (Y, xi ) = Di Y −ei di xidi ; (i = 1, 2, . . . , n)

(48)

The Lagrangian for the constrained cost minimization problem is L = w x + λ(1 −

n 

Di Y −ei di xidi ),

i=1

and the first-order conditions are wi = λDi di xi(di −1) Y −ei di (i = 1, 2 . . . , n)

(49)

Hence 1−dj

xj

xi1−di

8 See

=

Dj dj (ej dj −ei di ) wi Y Di di wj

Appendix 3 Elasticity of Substitution Derived from the Dual Cost Function.

(50)

1 Neoclassical Production Economics: An Introduction

33

Thus  xj =

Dj dj Di di



1 1−dj

Y

−

ej dj −ei di 1−dj

(xi )

1−di 1−dj



wi wj



1 1−di

(51)

Define A=

  ei di − ej dj Dj d 1 ;h= ln ; 1 − dj Di di 1 − dj

Further, from (48) ai = −

Fii xi Fiii

j

Fj 1 1 = ; aj = − = . j 1 − di 1 − dj xj Fjj

Then (51) reduces to 

wi ln xj = A + h ln Y + ai ln wj



 +

aj ai

 ln xi

(52)

The equation in (52) can be estimated econometrically although the presence of the endogenous variable ln xi on the right-hand side rules out using OLS. It is clear from (48) that ai aj aj σij 1 − dk = = = . σik ai ak ak 1 − dk

(53)

Because the ratio of the elasticities of the substitution between inputs i and j and between i and k depends on the parameters dj and dk , the ratio is a constant. The implicit production function in (48) can be described as the constant ratio of elasticity of substitution (CRES) production function [12].9 For an intuitive interpretation of (53) consider a production technology with four inputs: unskilled labor (x1 ), skilled labor(x2 ), energy(x3 ), and capital(x4 ).Suppose that σ13 = 2σ14 . That is, the elasticity of substitution between unskilled labor and capital is twice as large as the elasticity of substitution between unskilled labor and energy. Then, by virtue of (53) it must be true that σ23 = 2σ24 . That is, exactly the same proportionality must hold for elasticities of substitution between skilled labor and capital and between skilled labor and energy. For this to be true for any triplet of inputs (i, j, k), the function must be additively separable in every input. We may now consider some special cases with appropriate restrictions on the parameters ei and di .

9 See

also Mukerji [16], Gorman [9], and Hanoch [11].

34

R. G. Chambers and S. C. Ray

Case 1: di = d, ei = e ; (i = 1, 2, . . . , n) In this case, (48) becomes n

i=1 Di Y

−ed x d i

 n

=1

− 1 ed −d D x i i=1 i 1  n −d e = D x i=1 i i

⇒Y = ⇒ Y −d

(54)

Further, if e=1, we get the ACMS/CES production function with constant returns to scale. In this formulation, d is the substitution parameter and e is the returns to scale parameter while the Di s are the distribution parameters. Case 2: di = d but ei not constant In this case, (48) is n 

Di Y −ei d xid = 1.

(55)

i=1

Cost minimization subject to (55) as a constraint leads to = Di Y −ei d xi  1

d−1 ⇒ xi = Di Y −ei d wCi (d−1)

wi C

(56)

Thus,  C=

n 

1 1−d

Di

e d

Y

i − 1−d

− d wi 1−d

− 1−d d

(57)

i=1

This cost function is not multiplicatively separable in output and input prices and, hence, the underlying technology is not homothetic. At the same time, because di = d (i = 1, 2, . . . , n) it is a CES function. Hence, (55) above is a non-homothetic CES production function. Case 3: Consider the special case  F i (Y, xi ) = Di ln xY −ei n i i=1 F (Y, xi ) = 1

(58)

1 Neoclassical Production Economics: An Introduction

35

Thus n

i=1 Di

ln xi − ln Y

⇒ ln Y = − n

1

i=1 ei Di

n

i=1 ei D i = 1  n n Di ln xi i=1 e D k=1 k k

(59)

Di , (i = 1, 2, . . . , n) k=1 ek Dk i

(60)

+

Define α = − n

1

i=1 ei Di

; βi = n

Then (59) reduces to the usual Cobb-Douglas production function ln Y = α +

n 

βi ln xi .

(61)

F i (Y, xi ) = Di Y −ei ln xi n i i=1 F (Y, xi ) = 1.

(62)

i=1

Case 4: Next consider

Now Di Y −ei i xi ; Fii i F ai = − i i = xi Fi

−ei

Fii =

= − Di Y2

⇒

1 , (i = 1, 2, . . . , n)

xi

; xi Fiii =

Di Y −ei xi

(63)

This time ai aj 1 σij = n = n = 1. a s k=1 k k k=1 sk

(64)

Thus it is the Cobb Douglas technology. However wi C

⇒

Fii 1  Di Y −ei n −ek k = xi k=1 Dk Y k=1 xk Fk C  Di Y −ei xi = wi n D Y −ek k=1 k

=

n

(65)

Similarly, xj =

Di Y −ei C n wj k=1 Dk Y −ek

(66)

36

R. G. Chambers and S. C. Ray

Thus, xj = xi



Dj Di



Y −(ej −ei )



wi wj

 (67)

Clearly, the input proportions depend on both the marginal rate of substitution (i.e., the input price ratio) and the output level. Hence, the technology is non-homothetic, even though the constant elasticity of substitution equals unity. This is an example of a non-homothetic Cobb-Douglas production function.

Indirect Production Function: An Aside An alternative specification of the production technology is in terms of the indirect production function that defines the maximum output producible by a firm with a given amount of budgeted expenditure (C) facing a specific vector of input prices (w). This is particularly relevant for nonprofit organizations and public sector enterprises which operate under hard budget constraints. Define the normalized input price vector q = w C . The indirect production function can be expressed as 

Y ∗ = G(q) = max F (x) s.t. q x ≤ 1.

(68)

Consider the Lagrangian L = F (x) + λ(1 − q  x) ∂F > 0 for at least one input, output is not maximized until the budgeted So long as ∂x i expenditure is completely spent. Hence, the (normalized) budget restriction will be strictly binding, and the first-order conditions for a maximum will be ∂F ∂xi q x

= λqi ; (i = 1, 2, . . . , n) =1

(69)

This implies qi =

wi C

= n

∂F ∂xi

∂F k=1 xk ∂xk

(i = 1, 2, . . . , n)

(70)

The relation in (70) is known as Wold’s theorem. One can solve the system of n equations above for the optimal demand functions xi∗ = xi (q) (i = 1, 2, . . . , n)

(71)

1 Neoclassical Production Economics: An Introduction

37

Therefrom, the indirect production function is obtained as Y ∗ = F (x(q)) = g(q).

(72)

Roy’s Identity One of the duality relations in production theory is between the indirect and the direct production functions10 F (x) = minq g(q) s.t. q  x = 1 ↓↑ g(q) = maxx F (x) s.t. q  x = 1

(73)

The Lagrangian for the constrained minimization problem in (73) is L = g(q) + θ (1 − q  x)

(74)

The first-order conditions for a minimum are ∂g ∂qi q x

= θ xi (i = 1, 2, . . . , n) =1

(75)

 ∂g =θ qi xi = θ ∂qi

(76)

Thus n 

n

qi

i=1

i=1

and xi = n

qi

∂g k=1 qk ∂qk

(i = 1, 2, . . . , n)

(77)

The relation in (77) is an example of Roy’s identity.

Additive Implicit Indirect Production Functions Hanoch [12] introduced a class of additive implicit production functions of the form

10 For

an intuitive explanation of a similar duality between the direct and the indirect utility functions see Varian [24] page 129–130.

38

R. G. Chambers and S. C. Ray

G(q, Y ) =

n 

Gi (Y, qi ) = 1

(78)

j =1

Define Gii =

∂Gi ∂qi

∂ 2 Gi . ∂q 2 i

and Giii =

Then, by Roy’s identity,

Gii

xi = n

, (i = 1, 2, . . . , n)

(79)

 ∂ ln Gii ∂ ln( nk=1 qk Gkk ) ∂ ln xi = − ∂wj ∂wj ∂wj

(80)

k k=1 qk Gk

Next, consider

Now, wi ∂ ln Gii ∂ ln Gii ∂qi Giii ∂ C ∂wj = ∂qi ∂wj = Gi ∂wj i Giii  wj xj Giii x = − i C C = − i qj Cj G G i

(81)

i

Also, ∂ ln(

∂(

= =

n

∂(

n

k k=1 qk Gk )

k ∂wj k=1 qk Gk ) = n k ∂wj k=1 qk Gk

n

(82)

n ∂Gkk k ∂qk k=1 qk ∂wj + k=1 Gk ∂wj n n ∂Gkk ∂qk k ∂qk k=1 qk ∂qk ∂wj + k=1 Gk ∂wj n  k k ∂qk k=1 qk Gkk + Gk ∂wj k k=1 qk Gk ) ∂wj

=

n

(83)

Further, ∂w ∂C C ∂wk −wk ∂w ∂qk j j = ∂wj C2 δ w x = Cj k − Ck 2 j

(84)

In (84), δj k = 1 for j = k and 0 otherwise. Thus, ∂(

n

=−

=

C

 δj k k + Gk q G k k=1 kk k C −

n

k k=1 qk Gk ) ∂wj  xj nk=1 qk2 Gkkk

−

xj

n

k k=1 qk Gk

C

j

+

qj Gjj C

+

wk xj C2 j Gj C

 (85)

1 Neoclassical Production Economics: An Introduction

39

But j

xj = n

Gj

k k=1 qk Gk

⇒

xj

n

k k=1 qk Gk

C

j

=

Gj

(86)

C

Hence, ∂(

n

k k=1 qk Gk )

∂wj

j

=

Gj

−

C

xj

n

2 k k=1 qk Gkk

(87)

C

Substitution of (81) and (87) into (80) and multiplying both sides by

C xj

give

2

C C ∂w∂i ∂w C ∂ ln xi j = = σij xj ∂wj xi xj

=−

Giii Gii

qj −

n 2 k k=1 qk Gkk q +  j n k Gii k=1 qk Gk

Giii

(88)

qk Gkkk

(89)

Define ak = −

Gkk

Then, ak Gkk qk Gkkk = − n  j j n j =1 qj Gj j =1 qj Gj

(90)

Hence, by (79), q Gkkk

q 2 Gkkk

ak xk = − n k

j j =1 qj Gj

⇒

⇒ ak xk qk = ak wCk = ak sk = − n k

n

k=1 ak sk

=−

n

2 k j =1 qk Gkk n j j =1 qj Gj

j j =1 qj Gj

(91)

Hence, (88) can be expressed as σij = ai + aj −

n 

ak sk ,

(i = j )

(92)

k=1

An implication of (92) is that for the additive indirect production function considered above, σij − σik = aj − ak which is independent of the input i.

40

R. G. Chambers and S. C. Ray

Hanoch [12] described this as the constant difference of elasticity of substitution (CDE) production function. Consider the specification  G(q, Y ) = ni=1 Gi (qi , Y ) = 1; Gi (qi , Y ) = Bi q bi Y ei bi (i = 1, 2, . . . , n)

(93)

Then, by Roy’s identity, (b −1)

Gii

Bi bi qi i Y ei bi = (i = 1, 2, . . . , n) n k bk e k bk k=1 qk Gk k=1 qk Bk qk Y

xi = n

(94)

Hence, 

xi ln xj





Bi bi = ln Bj bj

 + (ei bi − ej bj ) ln Y + (bi − 1) ln qi − (bj − 1) ln qj (95)

Similarly, 

xk ln xj





Bk bk = ln Bj bj

 + (ek bk − ej bj ) ln Y + (bk − 1) ln qi − (bj − 1) ln qj (96)

Combining (95) and (96), we get 

xi ln xj





xk = Kij + ε ln xj

 + ai ln qi − ak ε ln qk − aj (1 − ε) ln qj

(97)

where  Kij = ln

Bi bi Bj bj

 ; ε=

ei bi − ej bj ; ai = bi − 1 (i = 1, 2, . . . , n) ek bk − ej bj

(98)

This is a system on (n − 1) simultaneous equations which are nonlinear in parameters. Special Cases: Case 1: bi = b (i = 1, 2, . . . , n). In this case,  G(q, Y ) = ni=1 Bi bqib Y ei b = 1  ⇒ C = ni=1 Bi bwib Y ei b

(99)

The cost function is not multiplicatively separable in output and input prices, which implies non-homotheticity of the technology. However,

1 Neoclassical Production Economics: An Introduction

41

σij − σik = bk − bj = 0 ⇒ σij = σik = σ.

(100)

Hence, it is a NH-CES production function. Case 2: bi = b; ei = e; (i = 1, 2, . . . , n) This time, G(q, Y ) = ⇒C=

n

b eb = i=1 Bi bqi Y   1 Y e b ni=1 Bi wib b

1

(101)

This is the dual cost function for the homothetic CES production function. Returns to scale depends, however, on the parameter e. Case 3: ei = e = 1; bi = b = 1; (i = 1, 2, . . . , n) Now,  G(q, Y ) = ni=1 Bi qY = 1 n ⇒C=Y i=1 Bi wi

(102)

This is the cost function for the fixed coefficients Leontief production function. Case 4: Gi (qi , Y ) = Bi ln (qi Y ei ) ; i = 1, 2, . . . , n Thus, Gi (ql , Y ) = Bi ln qi + Bi ei ln Y ⇒ Gii = Bqii , Giii = − B2i ⇒ ai =

q Gi − i i ii Gi

qi

(103)

= 1; i = 1, 2, . . . , n

Hence, σij = ai + aj −

n 

ak sk = 1

(104)

k=1

This corresponds to the Cobb-Douglas technology.

Flexible Functional Forms Although the implicit additive direct and indirect production functions allow more flexibility in respect of input substitution compared to the Cobb-Douglas and the ACMS functions, they still impose considerable structure on the technology a priori. The parallel development of the flexible functional forms, mainly the transcendental logarithmic (translog) production and cost functions developed in [3–5] and the generalized Leontief cost function by Diewert [7], has revolutionized empirical analysis of production by allowing the use of the cost or profit function as the primary analytical format.

42

R. G. Chambers and S. C. Ray

The main properties of the production function like the level of the output, the marginal productivities of inputs, and the pairwise substitution elasticities can be derived from the function itself, its vector of partial derivative, and the Hessian matrix of second and cross partial derivatives. The flexible functional forms permit the analyst to extract these properties from a second-order Taylor’s series approximation of an unspecified underlying production, cost, or profit function. Consider, for example, the function y = f (x) and its second-order approximation at some point x 0 . Thus,



2  ∂ f 1 0 ) 0 f (x)≈g(x) = f (x 0 )+(x − x 0 ) ∂f + (x − x ∂x x=x 0 2 ∂x∂x  x=x 0 (x − x )x=x 0 



 

 

 ∂2f ∂f 1 0 0 + 0 x = f (x 0 ) − x 0 ∂f ∂x x=x 0 + 2 x ∂x x=x 0 x ∂x∂x  x=x 0 x

2  ∂ f + 12 x  ∂x∂x x  0 x=x

≡ a0 + a  x + 12 x  Ax

(105)

Here



2   ∂ f 1 0 0 a0 = f (x 0 ) − x 0 ∂f + x ∂x x=x 0 2 ∂x∂x  x=x 0 x ;

  a = x 0 ∂f ;

2 ∂x x=x 0 ∂ f A = ∂x∂x  0

(106)

x=x

It can be verified that at the point of approximation x 0  f (x) = g(x);

∂f ∂x



 =

∂g ∂x



 ;

∂ 2f ∂x∂x 



 =

∂ 2g ∂x∂x 

 (107)

Translog Cost Function The m-output n-input translog cost function is  m  ln C = α0 + m αi ln Yi + 12 m Y ln Y + i=1 r=1 s=1 αrs ln mr s n n 1 n n β ln w + β ln w ln w + i i j i=1 i i=1 j =1 ij r=1 j =1 γrj ln Yr ln wj 2 (108) Using Shephard’s lemma, one can get si =

  ∂ ln C wi xi = = βi + βij ln wj + γrm ln Yr ; (i = 1, 2, . . . , n) C ∂ ln wi n

m

j =1

r=1

(109) Now, βij ∂si ∂si ∂ ln wj = = ∂wj ∂ ln wj ∂wj wj

(110)

1 Neoclassical Production Economics: An Introduction

43

while ∂

 wi xi C

∂wj

⎛ = wi ⎝

∂xi ∂C C ∂w − xi ∂w j j

C2

⎞

 ⎠ = wi CCij − xi xj 2 C

(111)

Hence,

⇒



wi xi wj xj CCij C C xi xj − 1 CCij β σij = xi xj = 1 + si ijsj .

βij =

(112)

It should be noted that the pairwise substitution elasticities vary across data points because the cost shares of the inputs depend on both the input prices and the output quantities.

Generalized Leontief Cost Function Another popular flexible function form is the generalized Leontief cost function introduced by Diewert [7]: ⎛ ⎞ n n   √ C(w, Y ) = ⎝ bij wi wj ⎠ Y

(113)

i=1 j =1

Using Shephard’s lemma, one gets the conditional input demand functions: ⎛ xi = ⎝bii +



n 

bij

j =1,j =i

It can be seen that if bij = 0 for alli = 0, technology. From (114), Cij =

xi Y

⎞ wj ⎠Y wi

(114)

= bii , and we get the standard Leontief

bij ∂xi = √ Y ; (i = j ) ∂wj 2 wi wj

(115)

Hence,



√ n n i=1 j =1 bij wi wj



bij  σij =       w wi bii + nj=1,j =i bij wji bjj + ni=1,i =j bij w j Clearly, σij = 0 if bij = 0.

(116)

44

R. G. Chambers and S. C. Ray

The homothetic and CRS version of the cost function may be generalized further by allowing the input ratios to depend on both the output level and the input prices. One such formulation that is both non-homothetic and allows nonconstant returns to scale is ⎛ ⎞ ! n n n    √ 1 C(w, Y ) = ⎝ bij wi wj ⎠ Y + 2 di wi Y 2 (117) i=1 j =1

i=1

This time, the conditional input demands are ⎛

n 

xi = ⎝bii +

j =1,j =i

 bij

⎞ wj ⎠ Y + di Y ; (i = 1, 2, . . . , n) wi

(118)

Now at the cost-minimizing bundle, the input proportions depend both on input price ratios and the level of output. While the flexible functional forms impose few prior restrictions on the nature of the technology instead of allowing the empirical analyst to statistically test such restrictions, in applied research, one too often encounters violation of regularity conditions like positive marginal cost, nonnegative factor demands/shares, negative own price elasticities, concavity of the cost function, and so on. Unfortunately, at what level of such violations does the estimated model become unacceptable remains a judgment call for the analyst.

Appendix 1 y=

Y L

⇒ Y = Ly = Lf (k) = f

K

L ∂y ∂y ∂k ∂Y FK = ∂K = L ∂K = L ∂k ∂K = Lf  (k) L1 = f  (k) ∂y ∂y ∂k K  FL = ∂Y ∂L = L ∂L + y = y + L ∂k ∂L = f (k) − Lf (k) L2   = f (k) − K L f (k) = f (k) − kf (k) ∂k FKL = f  (k) ∂L = L1 kf  (k) f  (k) f (k)−f  (k) K FL = − kf( (k)f  (k) ) ⇒ σKL = − FF·F KL

Appendix 2 Consider the production function z = h(x1 , x2 , . . . , xn ) y = g(z); g  (z) > 0 Then,

1 Neoclassical Production Economics: An Introduction ∂y ∂xi

⇒

45

∂h ∂y ∂h = g  (z) ∂x ; = g  (z) ∂x i ∂xj j

∂y ∂xi ∂y ∂xj

=

∂h ∂xi ∂h ∂xj

But due to homogeneity of degree r, h(tx) = t r h(x). Now suppose that g(h(x)) = y0 and g(h(tx)) = y1 .Then, along the isoquant for output y1 , ∂y ∂xi

⇒

∂h ∂y ∂h = g r ∂x ; = g r ∂x i ∂xj j ! ∂y ∂h ∂xi ∂y ∂xj

= y=y1

∂xi ∂h ∂xj

=

∂y ∂xi ∂y ∂xj

! y=y0

The input vectors x and tx have the same ratio of any two inputs i and j. Hence, the marginal rate of substitution between the inputs remains unchanged so long as the input proportions remain the same irrespective of the output level.

Appendix 3 Elasticity of Substitution Derived from the Dual Cost Function This appendix is based on McFadden [15]. We first consider the two-input case: y = f (x1 , x2 ). Dual cost function: C(w1 , w2 , y) = w1 x1 (w1 , w2 , y) + w2 x2 (w1 , w2 , y) Recall the definition of the elasticity of substitution

σ

By Shephard’s lemma, xi = minimization,

∂C ∂wi

12

=

d ln( xx21 ) d ln( ff21 )

= Ci (i = 1, 2) Also, by the FOC for cost

f1 w1 = f2 w2

46

R. G. Chambers and S. C. Ray

Hence,

σ

12

=

d ln( xx12 ) d ln( ff21 )

=

C2 ) d ln( C 1 w1 d ln( w ) 2

d ln C2 − d ln C1 = = d ln w1 − d ln w2

dC2 C2 dw1 w1

− −

dC1 C1 dw2 w2

≡

A B

Now, dC2 = C21 dw1 + C22 dw2 dC1 = C11 dw1 + C12 dw2 Hence, C11 dw1 + C12 dw2 C21 dw1 + C22 dw2 − C2 C1     C12 C11 C22 C12 dw1 + dw2 = − − C2 C1 C2 C1

A=

Now, because xi (w1 , w2 , y) = Ci (w1 , w2 , y) (i = 1, 2) is homogeneous of degree 0 in w, w2 w1 w1 = −C12 w2

C11 w1 + C12 w2 = 0 ⇒ C11 = −C12 C21 w1 + C22 w2 = 0 ⇒ C22 Hence,

  C12 w1 C12 dw1 + − dw2 − C2 w 2 C1     C12 C12 w2 C12 w1 C12 dw1 − dw2 = + + C2 C1 w 1 C2 w 2 C1 

A=

C12 w2 C12 + C2 C1 w 1



C12 (C1 w1 + C2 w2 ) dw1 C12 (C1 w1 + C2 w2 ) dw2 − C1 C2 w1 C1 C2 w2   C12 (C1 w1 + C2 w2 ) dw1 dw2 = − C1 C2 w1 w2 =

But C = C(w1 , w2 , y) is homogeneous of degree 1 in (w1 , w2 )

1 Neoclassical Production Economics: An Introduction

47

Hence, C1 w1 + C2 w2 = C Therefore, A=

C.C12 C1 C2



dw1 dw2 − w1 w2



Hence, σ 12 =

C.C12 A = . B C1 C2

For the multiple-input case, σ ij =

CCij Ci Cj

Uzawa [23] has shown this to be equivalent to the Allen elasticity of substitution defined above.

Cross-References  Elasticities of Substitution

References 1. Abramovitz M (1956) Resource and output trends in the United States since 1870. Am Econ Rev 46:5–23 2. Arrow KJ, Chenery HB, Minhas BS, Solow RM (1961) Capital-labor substitution and economic efficiency. Rev Econ Stat 43(1):225–250 3. Berndt E, Christensen L (1973) The translog function and the substitution of equipment, structures and labor in U.S. manufacturing 1929–1968. J Econ 1(1):81–114 4. Christensen LR, Jorgenson D, Lau L (1971) Conjugate duality and the transcendental logarithmic production function. Econometrica 39(4):225–256 5. Christensen LR, Jorgenson D, Lau L (1973) Transcendental logarithmic production frontiers. Rev Econ Stat 55:28–45 6. Cobb CW, Douglas PH (1928) A theory of production. Am Econ Rev 18(1):139–165. Papers and Proceedings (Mar) 7. Diewert E (1971) An application of the Shepard duality theorem: a generalized Leontief production function. J Polit Econ 79:489–507 8. Fishburn PC (1972) Mathematics of decision theory. Mouton, The Hague 9. Gorman WM (1965) Production functions in which the elasticities of substitution stand in fixed proportions to each other. Rev Econ Stud 32:217–224 10. Gorman WM (1968) Measuring the quantities of fixed factors. In: Wolfe JN (ed) Value, capital and growth. Edinburgh University Press, Edinburgh 11. Hanoch G (1971) CRESH production functions. Econometrica 39(5):695–712 12. Hanoch G (1975) Production and demand models with direct and indirect implicit additivity. Econometrica 43(3):395–419

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13. Hiriart-Urruty J-B, LeMaréchal C (2001) Fundamentals of convex analysis. Springer, Heidelberg/Berlin 14. Jorgenson DW, Griliches Z (1967) The explanation of productivity change. Rev Econ Stud 34(99):249–283 15. McFadden D (1978) Cost, revenue, and profit functions. In: Fuss M, McFadden D (eds) Production economics: a dual approach to theory and applications, vol I. North-Holland, Amsterdam, pp 3–109 16. Mukerji V (1963) A generalized S.M.A.C. function with constant ratios of elasticity of substitution. Rev Econ Stud 30:273–284 17. Rockafellar RT (1970) Convex analysis. Princeton University Press, Princeton 18. Samuelson PA (1947) Foundations of economic analysis. Harvard University Press, Cambridge, MA 19. Samuelson PA (1979) Paul Douglas’s measurement of production functions and marginal productivities. J Polit Econ 87(5):923–939, Part 1 20. Sato R (1977) Homothetic and non-homothetic CES production functions. Am Econ Rev 67(4):559–569 21. Schultz TW (1958) Output-input relationships revisited. J Farm Econ 40:924–932 22. Shephard RW (1953) Cost and production functions. Princeton University Press, Princeton 23. Uzawa H (1962) Production functions with constant elasticity of substitution. Rev Econ Stud 29:201–299 24. Varian H (1992) Microeconomic analysis, 3rd edn. Norton, New York 25. von Thünen J (1826) Der isolirte Staat in Beziehung auf Landwirtschaft und NationalÖkonomie. Wirtschaft & Finan

2

Reminiscences of “Returns to Scale in Electricity Supply” Marc Nerlove

Contents Introduction: Genesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Development . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Conclusion: Loose Ends and the Aftermath . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

50 51 54 55

Abstract

The origins and further development of a very early empirical application of Shephard’s duality theorem is discussed. The investigation of the “Returns to Scale in Electricity Supply” was based on a cost function derived from a CobbDouglas production function initially. The function was modified to allow for variable “returns to scale.” Duality between cost and production then showed the modified Cobb-Douglas production function behind the cost function which was initially estimated. Keywords

Cost/production functions · Duality · Returns to scale · Modified Cobb-Douglas · Electricity generation

M. Nerlove () Department of Agricultural and Resource Economics, College of Agriculture and Natural Resources, University of Maryland, College Park, MD, USA e-mail: [email protected] © Springer Nature Singapore Pte Ltd. 2022 S. C. Ray et al. (eds.), Handbook of Production Economics, https://doi.org/10.1007/978-981-10-3455-8_2

49

50

M. Nerlove

My paper “Returns to Scale in Electricity Supply” was first issued in 1961 as a Technical Report at Stanford University.1 Later that year, my friend and former office-mate at the University of Chicago, Yehuda Grunfeld, was tragically drowned while swimming off the beach at Tel-Aviv, Israel. Carl Christ and Don Patinkin put together a distinguished list of contributors for a volume of essays in his memory. When asked to contribute, it was an honor and I could not but give it my best work at the time. The volume appeared in 1963.2 Soon thereafter, the paper was reprinted in a compendium of papers in econometrics edited by Arnold Zellner.3 After the paper was made more widely accessible, it was discussed in econometric texts, for example, Berndt,4 Hayashi.5 My 1955 data set was augmented for 1970 by Christensen and Greene, who used a translog cost function to estimate returns to scale in electricity supply in the 2 years.6

Introduction: Genesis How did I come to write this paper? What were my source of ideas, in particular the idea of estimating the parameters of a production function from a cost function, without, at the time, knowing of Shephard’s now famous result on the duality of cost and production functions 7 ? I served in the US Army from April 1, 1957, to February 23, 1959. After basic training, I was assigned to the Program Coordination Office of the labs at Fort Detrick, Maryland. There I met a young private, another economist, Max D. Steuer, who had been studying at the London School of Economics.8 We had some time on our hands and decided to do a research project together; we thought for various reasons that it would be useful to study the roles of capital, labor, and fuel in US production of electric power. A ready source of data was at hand in the annual

1 Technical

Report No, 96, May 25, 1961, of the Institute for Mathematical Studies in the Social Sciences, Stanford University. 2 Carl F. Christ [1] 3 Zellner [2] 4 Berndt [3] 5 Hayashi [4] 6 Christensen and Greene [5]. “Cross-section data for 1955 and 1970 are analyzed using the translog cost function. We find that in 1955 there were significant scale economies available to nearly all firms. By 1970, however, the bulk of U.S. electricity generation was by firms operating in the essentially flat area of the average cost curve. We conclude that a small number of extremely large firms are not required for efficient production and that policies designed to promote competition in electric power generation cannot be faulted in terms of sacrificing economies of scale” (p. 655). 7 Shephard [6] 8 He was the grandson of the famous lawyer for the defense in the Triangle Shirtwaist Factory Fire case – in case you wanted to know. Steuer, Max David [7].

2 Reminiscences of “Returns to Scale in Electricity Supply”

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reports of the Federal Power Commission.9 Steuer left the Army shortly thereafter to return to London and I was left to continue on my own. We never resumed our collaboration. During my last year in the Army, I was asked by George Heberton Evans, then the chairman of the Department of Political Economy of the Johns Hopkins University, to teach the course in “Mathematical Economics.” This was, at the time, essentially a course in calculus with economic examples. I sought and received permission from my commanding officer to teach the course on Saturday mornings through the fall semester. I used R. G. D. Allen’s text supplemented by many exercises of my own devising.10 Although teaching this course did not leave me time to return to research on the production of electric power in the United States, it proved to be a crucial element in my thinking when I later returned to the project after leaving the Army to take a position at the University of Minnesota. Toward the end of my course at Hopkins, I dealt with the maximization (minimization) of functions of several variables subject to one or more constraints (Allen, op. cit., Chap. 19). We covered utility maximization subject to an income (expenditure) constraint and total cost minimization for given output and input prices subject to a production function constraint. In one exercise, Allen (Ex.15, p. 519) uses a Cobb-Douglas utility function to show that demands for the various goods have unitary income and own-price elasticities while having zero crossprice price elasticities. I turned this exercise upside down: Instead of maximizing a nonlinear function (Cobb-Douglas utility function) subject to a linear total expenditure constraint, I asked my students to minimize a total linear cost function subject to a nonlinear Cobb-Douglas production function constraint, to solve the first-order conditions for the derived demands for factors of production, and then substitute these derived demand functions into the total cost function. The total cost was thus obtained as a function of output and factor prices. The coefficients in the original Cobb-Douglas production function, including the degree of returns to scale, were uniquely recoverable from the coefficients in this total cost function. I and my students thus stumbled upon a special case of the Shephard duality theorem, but I was not to know this until later.

Development I left the Army at the end of February 1959 to accept a teaching and research position at the University of Minnesota. I taught the same course I had taught at Hopkins and

9 US

Federal Power Commission, Steam Electric Plants, Construction Costs and: Construction Costs and Annual Production Expenses, Washington, D.C.: annually; and Statistics of Electric Utilities in the United States, Classes A and B Privately Owned Companies, Washington, D.C.: annually. 10 Allen [8]. I am indebted to Sara Seten Berghausen, Associate Curator of Collections. David M. Rubenstein Rare Book & Manuscript Library, who retrieved my class notes and exercises for my course at Hopkins.

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John Chipman and I taught a joint Workshop on Econometrics. Among our students in the Workshop was Daniel McFadden, who also served as an unofficial research assistant. For the Workshop, I prepared a set of lectures on the estimation and identification of Cobb-Douglas production functions, which with the addition of two chapters based on my related work was later to serve as a short book.11 We discussed among other things, Klein’s early work on railroad production functions,12 and John Chipman’s reestimation using a different approach.13 Klein assumes a two-output asymmetric Cobb-Douglas production function. He assumes three inputs with exogenously varying prices, all for a cross-section sample of firms. By asymmetric I mean he normalizes the exponent of one of the output variables to be −1 and so treats it as the “dependent” variable in the production function. He argues that the cross-section data represent shortrun adjustments, and because the industry is regulated, outputs for each firm are exogenous. From the first-order conditions for cost minimization subject to the Cobb-Douglas production function constraint, given outputs and input prices, Klein then derives estimates for the ratios of the production function exponents of two pairs of inputs by taking geometric means, implicitly assuming errors in cost minimization or random differences in the production function exponents. To arrive at estimates of the individual exponents in the production function, Klein constructs a synthetic dependent variable in a logarithmic regression with the two output variables as independent, on the assumption that these are exogenously determined for each railroad. If all this seems somewhat convoluted, it is because it is.14 Chipman’s idea was simple and straightforward: If one is going to treat the two output variables and input prices as exogenous, why not estimate the costminimizing demand for inputs derived from the first-order conditions? Cost minimization under a Cobb-Douglas constraint leads thus to a system of three derived demand equations linear in the logs of factor prices and the two outputs.15 From the coefficients in each of the three equations one can derive a different estimate of returns to scale, and thence estimates of the individual elasticities of input levels. The problem is thus to impose appropriate constraints in the estimation for the three equations. This is where Chipman stopped in 1957.16 As far as I know only one person picked up the thread of Chipman’s approach, using a restricted system of derived demand equations much later.17

11 Nerlove

[9] [10]. This paper is briefly summarized in Klein [11]. There is a more extensive discussion in Nerlove, op. cit., Chap. 4, pp. 61–85. 13 Chipman [12] 14 See Nerlove [9], loc. cit., pp. 78–79, for a more precise exposition of what Klein does. I also argue there that Klein’s estimates of returns to scale are neither unbiased nor consistent. 15 See Nerlove [9], loc. cit., pp. 80–81. 16 Chipman, op. cit. 17 Hasenkamp [13, 14]. 12 Klein

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From the exercise I had given to my students the previous year, namely, to derive the cost minimizing cost function for a single output firm, I knew what to do. The method also worked for the case of multiple outputs. Of course, in the case of electricity supply, I had only one output. That year, 1959, with our research assistant Dan McFadden, I gathered up the data on steam-electric generating plants, made initial estimates of factor prices, and estimated the logarithmic cross-section cost functions. “Construction” of the basic cost, input, and above all price data was one of the most difficult and time consuming tasks. Details of how I did this for 145 firms in 1955 are contained in Appendix B (pp. 190–192). Capital input and capital costs were the most problematic. In the end I never got very good results for the elasticity of output with respect to capital input which depends upon estimates of capital prices in the cost function. Once I had estimates for total costs and the three input prices, and of course total annual output, I could estimate a total cost function. If I assumed that firms had one plant each and minimized cost subject to a Cobb-Douglas production function with fixed input prices and given output, I had estimates of the input elasticities and returns to scale. I wrote these results up and presented them at our workshop, and at seminars at Purdue and Stanford. During the Purdue seminar, Hirofumi Uzawa was present and sitting in the back of the room. He asked a seemingly innocent question, “You can always do this?” To which I replied, “I don’t know if I can always do this but for a Cobb-Douglas production function I can.” He replied, “Not a question,” and called our attention to the Shephard Duality theorem, which was apparently not well-known by economists at the time. But that was not the end of the story. In the summer of 1960 I moved to Stanford University. While working on a submitable draft of the paper, I discovered a number of loose ends. First, I was not happy with the finding that firms in this industry were inefficiently small. But then I realized that increasing returns might prevail in the production of electric power but that nonetheless there might be constant or diminishing returns to scale in the supply of electric power because of transmission losses. I worked all this out in an appendix to the paper. There was another problem with the results: When I plotted the residuals from the logarithmic cost regression, they should have had the usual lens shape about a straight line; they did not. Instead the residuals were more often positive for lower than average outputs and higher than average outputs and correspondingly negative for those in between, that is, they were just the opposite of the lens shape I expected to find. I cudgeled my brain for an explanation until I came upon the obvious and simple one: Returns to scale were not constantly increasing nor constantly diminishing as implied by a logarithmic linear cost function, but were first strongly increasing, then less so, and finally nearly constant or slowly decreasing. I broke the 145 firms into five groups of 29 on the basis of output, smallest in the first group, largest in the last, and so on. Those five regressions enabled me to test the hypothesis of variable returns to scale. The test statistic was highly significant. Looking at the results for the separate cost regressions, it occurred to me that if I added the square of the log of output to the overall regression of total cost on log output and log factor prices I could get a good fit and the normal lens shaped plot of residuals on log output. I did so

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and by Shephard’s duality theorem I had a new production function with variable returns to scale. The logarithmic regression including the square of log output fit extremely well and had overall residuals plotted against log output of the traditional shape. Returns to scale were always increasing but variable and decreasing with log output. I was not able to find a closed parametric form for the production function corresponding to my new cost function, though by Shephard’s duality theorem I knew that one existed and I could find the elasticities of output with respect to each of the three inputs as they varied with scale. That was the end of my work on the paper but not of the questions I had.

Conclusion: Loose Ends and the Aftermath After I had finished the paper in May of 1961, I still had some unanswered questions: First, the Cobb-Douglas form with which I had started and the variable extent of returns to scale production function to which I had finally arrived had elasticities of substitution equal to one between any pair of factors. This did not square with my intuition about how electric power was produced in 1955. In particular, I thought that labor played a rather minimal role in steam electric generation and therefore could not easily substitute for fuel. An alternative, the multifactor CES production function, had an associated cost function that was relatively easy to estimate and the parameters of which might be easily interpretable. Unfortunately, the multifactor CES had elasticities all equal, although different from one. This did not seem helpful. Second, although the CES and its limiting cases, the Cobb-Douglas and the Leontief, had associated cost functions easy to estimate and easy to interpret in terms of the underlying production functions, I was curious as to the most general cases in which such duals existed. McFadden, who was then a graduate student at the University of Minnesota, came out to Stanford to work with me on his dissertation in the summer of 1961.18 Uzawa, who had noted at the Purdue seminar that one could always determine the production function from the cost function, had returned from UC Berkeley as associate professor in 1961. McFadden was also keen to work with him. I raised my concerns with them about generalizations of CES production and cost functions and about duality more generally. Uzawa realized the importance of the Shephard duality theorem and provided a more accessible proof of the result in his 1961 paper.19 In response to my question about a generalization of the usual constant elasticity of substitution production

18 McFadden [15]. Much of the core of this dissertation was later published as McFadden [16], in which McFadden generalizes the elasticity of substitution for the multifactor case. 19 Uzawa [17]

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function developed by Arrow et al.,20 Uzawa showed that the most general form for a CES production function was one for which the indices of the input levels could be partitioned, one partition such that the elasticities of substitution were one, and the other for which the elasticities were a constant for any pair other than one, that is, a Cobb-Douglas of two-factor CES functions.21 The difficulties with extending constant elasticities of substitution production functions and duality theory spawned a whole series of subsequent papers modifying and extending production function forms among which are Christensen, Jorgenson, and Lau, and Diewert.22 I hope I may take at least a small part of the credit for this renaissance of production theory and practice, but I fear that the credit belongs largely to Daniel McFadden.23

References 1. Christ CF et al (1963) Measurement in economics: studies in mathematical economics & econometrics in memory of Yehuda Grunfeld. Stanford University Press, Stanford, pp 167–198 2. Zellner A (ed) (1968) Readings in economic statistics and econometrics. Little Brown and Co, Boston, pp 409–439 3. Berndt ER (1990) Chapter 3, Costs, learning curves and scale economies. In: The practice of econometrics: classic and contemporary. Addison, Wesley, Reading, pp 60–101 4. Hayashi F (2000) Chapter 1, Sec. 7 Application: returns to scale in electricity supply. In: Econometrics. Princeton University Press, pp 60–70 5. Christensen LR, Greene WH (1976) Economies of scale in U.S. Electric Power Generation. J Polit Econ 84(4, Part 1):655–678 6. Shephard RW (1953) Cost and production functions. Princeton University Press, Princeton 7. Steuer MD (1951) Who was Who in America, vol IV. Marquis-Who’s Who, Inc., Chicago 8. Allen RGD (1938) Mathematical analysis for economists. Macmillan, London. reprinted 1953 9. Nerlove M (1965) Estimation and identification of Cobb-Douglas production functions. RandMcNally & Co, Chicago 10. Klein LR (1947) The use of cross-section data in econometrics with application to a study of production of railroad services in the United States. Mimeograph. National Bureau of Economic Research 11. Klein LR (1953) A textbook of econometrics. Row Peterson, Evanston, pp 226–236 12. Chipman JS (1957) Returns to scale in the railroad industry: a reinterpretation of Klein’s data, (abstract). Econometrica 25:607 13. Hasenkamp G (1976) Specification and estimation of multiple output production functions. Lecture notes in economics and mathematical systems, No. 120. Springer, Berlin 14. Hasenkamp G (1976) A study of multiple-output production functions: Klein’s railroad study revisited. J Econ 4(3):253–262 15. McFadden D (1962) Factor substitution in the economic analysis of production. PhD thesis, University of Minnesota

20 Arrow

et al. [18] [19] 22 Christensen et al. [20]; Diewert [21] 23 Fuss and McFadden [22] 21 Uzawa

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16. McFadden D (1963) Constant elasticity of substitution production functions. Rev Econ Stud 30(2):73–83 17. Uzawa H (1964) Duality principles in the theory of cost and production. Int Econ Rev 5(2):216–220 18. Arrow KJ, Chenery HB, Mijnhas BS, Solow RM (1961) Capital-labor substitution and economic efficiency. Rev Econ Stat 43:225–250 19. Uzawa H (1962) Production functions with constant elasticities of substitution. Rev Econ Stud 29(4):291–299 20. Christensen LR,Jorgenson DW, Lau LJ (1971) Conjugate duality and the transcendental logarithmic functions, (abstract). Econometrica 39(4):255–256 21. Diewert WE (1971) An application of the Shephard Duality Theorem: a Generalized Leontief production function. J Polit Econ 79(3):481–507 22. Fuss M, McFadden D (eds) (1978) Production economics: a dual approach to theory and applications. North Holland Publishing, Amsterdam

3

Duality in Production W. Erwin Diewert

Contents Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Cost Functions: The One Output Case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The Duality Between Cost and Production Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The Derivative Property of the Cost Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The Comparative Statics Properties of Input Demand Functions . . . . . . . . . . . . . . . . . . . . . . . The Duality Between Constant Returns to Scale Production Functions and Their Unit Cost Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The Constant Elasticity of Substitution Production Function . . . . . . . . . . . . . . . . . . . . . . . . . . Flexible Functional Forms for Cost Functions: The Generalized Leontief Functional Form . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The Translog Functional Form . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The Normalized Quadratic Unit Cost Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The Konüs Byushgens Fisher Unit Cost Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Semiflexible Functional Forms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The Use of Splines for Modeling Technical Progress . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Allowing for Flexibility at Two Sample Points . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . National Product or Variable Profit Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The Comparative Statics Properties of Net Supply and Fixed Input Demand Functions . . . . The Translog Variable Profit Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The Normalized Quadratic Variable Profit Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The KBF Variable Profit Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Joint Cost Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Flexible Functional Forms for Joint Cost Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Applications of Joint Cost Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Problems that Require Additional Research . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

58 61 67 73 75 82 85 95 100 107 114 120 121 123 127 137 143 146 150 152 159 162 163 165

W. E. Diewert () Vancouver School of Economics, University of British Columbia, Vancouver, BC, Canada School of Economics, UNSW, Sydney, NSW, Australia e-mail: [email protected] © Springer Nature Singapore Pte Ltd. 2022 S. C. Ray et al. (eds.), Handbook of Production Economics, https://doi.org/10.1007/978-981-10-3455-8_21

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Abstract

The chapter reviews the application of duality theory in production theory. Duality theory turns out to be a useful tool for two reasons: (i) it leads to relatively easy characterizations of the properties of systems of producer derived demand functions for inputs and producer supply functions for outputs and (ii) it facilitates the generation of flexible functional forms for producer demand and supply functions that can be estimated using econometrics. The chapter focuses on describing the properties of five functional forms that have been used in the production literature: (i) the constant elasticity of substitution (CES), (ii) the generalized Leontief, (iii) the translog, (iv) the normalized quadratic, and (v) the Konüs Byushgens Fisher functional forms. The applications of GDP functions and joint cost functions to various areas of applied economics are explained. Keywords

Production theory · Duality theory · Cost functions · Production functions · Joint cost functions · National product functions · GDP functions · Variable profit functions · Properties of producer demand and supply functions · Shephard’s Lemma · Hotelling’s Lemma · Samuelson’s Lemma · Flexible functional forms · Estimation of technical progress · The valuation of public sector outputs · Modeling monopolistic behavior · Sunk costs

JEL Classification Numbers

C02, C32, C43, D24, D42, D92, E01, E22, F11, H44, L51, M40, O47

Introduction Duality theory is a very useful tool for estimating production functions or more generally, for estimating production possibilities sets. It is also useful in allowing one to derive the theoretical properties that differentiable derived producer demand for input and supply of output functions must satisfy if the producer is maximizing profits or minimizing costs. This chapter will illustrate these advantages of duality theory in the producer context. Sections “Cost Functions: The One Output Case,” “The Duality Between Cost and Production Functions,” “The Derivative Property of the Cost Function,” “The Comparative Statics Properties of Input Demand Functions” and “The Duality Between Constant Returns to Scale Production Functions and Their Unit Cost Functions” below will focus on the case of one output, N input technologies. The multiple output and multiple input case will be considered in sections “National Product or Variable Profit Functions,” “The Comparative Statics Properties of Net Supply and Fixed Input Demand Functions,”

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“The Translog Variable Profit Function,” “The Normalized Quadratic Variable Profit Function,” “The KBF Variable Profit Function,” “Joint Cost Functions,” “Flexible Functional Forms for Joint Cost Functions,” and “Applications of Joint Cost Functions.”1 The one output, many input cost function is defined in section “Cost Functions: The One Output Case” and in section “The Duality Between Cost and Production Functions,” the conditions on the production function that allow the cost function to completely describe the underlying technology are listed: this establishes the Shephard [81] Duality Theorem between cost and production functions. Section “The Derivative Property of the Cost Function” explains Shephard’s Lemma; that is, it shows why differentiating a cost function with respect to input prices generates the vector of cost minimizing input demand functions. If the cost function is twice continuously differentiable with respect to input prices, then Section “The Comparative Statics Properties of Input Demand Functions” derives the properties that the system of cost minimizing input demand functions must satisfy. Section “The Duality Between Constant Returns to Scale Production Functions and Their Unit Cost Functions” looks at the duality between cost and production functions if production is subject to constant returns to scale, that is, if the production function is homogeneous of degree one in inputs. Sections “The Constant Elasticity of Substitution Production Function,” “Flexible Functional Forms for Cost Functions: The Generalized Leontief Functional Form,” “The Translog Functional Form,” “The Normalized Quadratic Unit Cost Function,” and “The Konüs Byushgens Fisher Unit Cost Function” look at specific functional forms for the cost function. The five functional forms that are studied are (i) the constant elasticity of substitution (CES), (ii) the generalized Leontief, (iii) the translog, (iv) the normalized quadratic, and (v) the Konüs Byushgens Fisher (KBF) functional forms. The last four functional forms are flexible functional forms; that is, they can provide a second order approximation to an arbitrary twice continuously differentiable unit cost function at any arbitrary price point.2 A major problem with

1 In

sections “Cost Functions: The One Output Case,” “The Duality Between Cost and Production Functions,” “The Derivative Property of the Cost Function,” “The Comparative Statics Properties of Input Demand Functions,” “The Duality Between Constant Returns to Scale Production Functions and Their Unit Cost Functions,” “The Constant Elasticity of Substitution Production Function,” “Flexible Functional Forms for Cost Functions: The Generalized Leontief Functional Form,” “The Translog Functional Form,” “The Normalized Quadratic Unit Cost Function,” “The Konüs Byushgens Fisher Unit Cost Function,” “Semiflexible Functional Forms,” “The Use of Splines for Modeling Technical Progress,” “Allowing for Flexibility at Two Sample Points,” “National Product or Variable Profit Functions,” “The Comparative Statics Properties of Net Supply and Fixed Input Demand Functions,” “The Translog Variable Profit Function,” “The Normalized Quadratic Variable Profit Function,” “The KBF Variable Profit Function,” “Joint Cost Functions,” and “Flexible Functional Forms for Joint Cost Functions,” it will be assumed that the producer takes prices as given constants in each period. Section “Applications of Joint Cost Functions” extends the analysis to the case of monopolistic behavior. 2 Diewert and Wales ([45], 89–92) discuss some additional flexible functional forms that are not discussed here. These alternative functional forms have various problems.

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flexible functional forms is the curvature problem; that is, an estimated flexible functional form for a unit cost function may violate the concavity in prices property that cost functions must satisfy. It turns out that the normalized quadratic and KBF functional forms are such that the correct curvature conditions can be imposed without destroying the flexibility of the functional form.3 Section “Semiflexible Functional Forms” introduces the concept of a semiflexible functional form. A major problem with the use of a flexible functional form is that it requires the estimation of roughly N2 /2 parameters if there are N inputs. The semiflexible concept reduces this large number of parameters in a sensible way. Section “The Use of Splines for Modeling Technical Progress” shows how piecewise linear functions of time can be used to model technical progress in a more general manner than just using linear time trends in the demand functions. Section “Allowing for Flexibility at Two Sample Points” shows how a flexible functional form can be generalized to achieve the second order approximation property at two sample points if we are estimating production functions in the time series context.4 Section “National Product or Variable Profit Functions” introduces Samuelson’s [79] National Product Function or the variable profit function. This function conditions on a vector of fixed inputs and maximizes the value of outputs less variable inputs. The comparative statics properties of this function are developed in section “The Comparative Statics Properties of Net Supply and Fixed Input Demand Functions.” Sections “The Translog Variable Profit Function,” “The Normalized Quadratic Variable Profit Function,” and “The KBF Variable Profit Function” look at

3 On

a personal note, I did my thesis on flexible functional forms and, with the help of Daniel McFadden (my thesis advisor), I came up with the Generalized Leontief cost function as my first attempt at finding a “perfect” functional form that was flexible, parsimonious (i.e., had the minimal number of parameters to be estimated that would enable it to be flexible) and generated derived demand (or supply) functions that were either linear or close to linear in the unknown parameters in order to facilitate econometric estimation. I was a graduate student at Berkeley at the time (1964–1968) and I met frequently with Dale Jorgenson. He and his student at the time, Lawrence Lau, realized that instead of taking a quadratic form in the square roots of input prices, one could take a quadratic form in the logarithms of prices as a functional form for the logarithm of the cost function and the translog functional form was born. However, empirical applications of these functional forms soon showed that these functional forms had a drawback: it was not possible to impose the correct concavity or convexity properties on these flexible functional forms without destroying the flexibility of the functional form. In the 1980s, Diewert and Wales came up with the normalized quadratic functional form which was flexible, parsimonious and had the property that the correct curvature conditions could be imposed without impairing the flexibility property. However, in order to preserve the parsimony property, one had to pick a more or less arbitrary alpha vector and imbed it into the functional form as we will see later in this chapter. But different choices of alpha could generate perhaps substantially different estimates for demand and supply elasticities. The last flexible functional form that we will discuss in this chapter, the KBF functional form, overcomes this difficulty and hence completes our quest for the “perfect” flexible functional form. 4 It should be noted that our analysis is geared to the time series context. Much of our analysis can be translated to the cross sectional context.

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three flexible functional forms for this function: (i) the translog, (ii) the normalized quadratic, and (iii) the KBF variable profit functions. The systems of estimating equations that these functional forms generate are also exhibited. Sections “Joint Cost Functions,” “Flexible Functional Forms for Joint Cost Functions,” and “Applications of Joint Cost Functions” develop the properties of joint cost functions; that is, these functions generalize the one output cost function to a cost function for multiple output producers. Section “Flexible Functional Forms for Joint Cost Functions” looks at three flexible functional forms for this function: (i) the translog, (ii) the normalized quadratic, and (iii) the KBF joint cost functions. The latter two functions have the property that the correct curvature conditions can be imposed on them without destroying their flexibility properties. Section “Applications of Joint Cost Functions” looks at applications of joint cost functions to: (i) problems associated with the measurement of the outputs of public sector producers in the System of National Accounts, (ii) the measurement of the efficiency of regulated utilities, and (iii) the estimation of technology sets when producers have some monopoly power. Section “Problems that Require Additional Research” concludes with a listing of three problems that are not addressed in this chapter and require further research. It may be useful to use this chapter as part of a course in microeconomic theory or in production theory. To facilitate this use, the author has added many straightforward problems that the instructor can assign to students. These problems are also an efficient way of extending the results presented in the main text.

Cost Functions: The One Output Case The production function and the corresponding cost function play a central role in many economic applications. In the following section, we will show that under certain conditions, the cost function is a sufficient statistic for the corresponding production function; that is, if we know the cost function of a producer, then this cost function can be used to generate the underlying production function. Let the producer’s production function f(x) denote the maximum amount of output that can be produced in a given time period, given that the producer has access to the nonnegative vector of inputs, x ≡ [x1 , . . . , xN ] ≥ 0N .5 If the production function satisfies the minimal regularity condition of continuity from above,6 then given any positive output level y that the technology can produce and any strictly x ≥ 0N means each component of the vector x is nonnegative,  x > 0N means x ≥ 0N and x = 0N and x  0N means each component of x is positive. pT x ≡ n = 1 N pn xn . Vectors are understood to be column vectors when it matters. 6 We require that f be continuous from above for the minimum to the cost minimization problem to exist; i.e., for every output level y that can be produced by the technology (so that y ∈ Range f), we require that the set of x’s that can produce at least output level y (this is the upper level set L(y) ≡ {x : f(x) ≥ y}) is a closed set in RN . 5 Notation:

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positive vector of input prices p ≡ [p1 , . . . , pN ]  0N , we can calculate the producer’s cost function C(y,p) as the solution value to the following constrained minimization problem:   C (y, p) ≡ minx pT x : f (x) ≥ y; x ≥ 0N .

(1)

It turns out that the cost function C will satisfy the following seven properties, provided that the production function is continuous from above7 : Theorem 1 Diewert ([27], 107–114)8 : Suppose f is continuous from above. Then C defined by Eq. (1) has the following properties: Property 1 C(y,p) is a nonnegative function. Property 2 C(y,p) is positively linearly homogeneous in p for each fixed y; that is, C (y, λp) = λC (y, p)

(2)

for all λ > 0, p >> 0N and y∈Range f (i.e., y is an output level that is producible by the production function f). Property 3 C(y,p) is nondecreasing in p for each fixed y ∈ Range f; that is,     y ∈ Range f, 0N  p1 < p2 implies C y, p1 ≤ C y, p2 .

(3)

Property 4 C(y,p) is a concave function of p for each fixed y ∈ Range f; that is,   y ∈ Range f, p1  0N ; p2  0N ; 0 < λ < 1 implies C y, λp1 + (1 − λ) p2     ≥ λC y, p1 + (1 − λ) C y, p2 . (4) Property 5 C(y,p) is a continuous function of p for each fixed y ∈ Range f. Property 6 C(y,p) is nondecreasing in y for fixed p; that is,     p  0N , y1 ∈ Range f, y2 ∈ Range f, y1 < y2 implies C y1 , p ≤ C y2 , p . (5)

7 Note 8 For

that this minimal regularity condition cannot be contradicted using a finite data set. the history of closely related results, see Diewert ([22], 116–120).

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Property 7 For every p  0N , C(y,p) is continuous from below in y; that is,

y∗ ∈ Range f, yn ∈ Range f for n = 1, 2, . . . , yn ≤ yn+1 , limn→∞ yn     = y∗ implies limn→∞ C yn , p = C y∗ , p .

(6)

Proof of Property 1 Let y ∈ Range f and p  0N . Then

C (y, p) ≡ minx pT x : f (x) ≥ y; x ≥ 0N = pT x∗

where x∗ ≥ 0N and f (x∗ ) ≥ y

≥0

since p  0N and x∗ ≥ 0N .

Proof of Property 2 Let y ∈ Range f, p  0N and λ > 0. Then   C (y, λp) ≡ minx λpT x : f (x) ≥ y; x ≥ 0N   = λminx pT x : f (x) ≥ y; x ≥ 0N since λ > 0 = λC (y, p)

using the definition of C (y, p) .

Proof of Property 3 Let y ∈ Range f, 0N  p1 < p2 . Then     C y, p2 ≡ minx p2T x : f (x) ≥ y; x ≥ 0N   = p2T x∗ where f x∗ ≥ y and x∗ ≥ 0N ≥ p1T x∗ since x∗ ≥ 0N and p2 > p1   ≥ minx p1T x : f (x) ≥ y; x ≥ 0N since x∗ is feasible for this problem   ≡ C y, p1 .

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Proof of Property 4 Let y ∈ Range f, p1  0N ; p2  0N ; 0 < λ < 1. Then 

C y, λp + (1 − λ) p 1

2



 

T 1 2 ≡ minx λp + (1 − λ) p x : f (x) ≥ y; x ≥ 0N

T   = λp1 + (1 − λ) p2 x∗ where x∗ ≥ 0N and f x∗ ≥ y = λp1T x∗ + (1 − λ) p2T x∗   ≥ λminx p1T x : f (x) ≥ y; x ≥ 0N + (1 − λ) p2T x∗

since x∗ is feasible for the cost minimization problem that uses the price vector p1 and using also λ > 0   = λC y, p1 + (1 − λ) p2T x∗ using the definition   of C y, p1     ≥ λC y, p1 + (1 − λ) minx p2T x : f (x) ≥ y; x ≥ 0N since x∗ is feasible for the cost minimization problem that usesthe price vector p2 and using also 1 − λ > 0     = λC y, p1 + (1 − λ) C y, p2 using the definition   of C y, p2 . Figure 1 below illustrates why this concavity property holds. In Fig. 1, the isocost line {x : p1T x = C(y, p1 )} is tangent to the production possibilities set L(y) ≡ {x : f(x) ≥ y, x ≥ 0N } at the point x1 and the isocost line {x : p2T x = C(y, p2 )} is tangent to the production possibilities set L(y) at the point x2 . Note that the point x** belongs to both of these isocost lines. Thus, x** will belong to any weighted average of the two isocost lines. The λ and 1 − λ weighted average isocost line is the set {x : [λp1 + (1 − λ)p2 ]T x = λC(y, p1 ) + (1 − λ)C(y, p2 )} and this set is the dotted line through x** in Fig. 1. This dotted line lies below9 the parallel dotted line that is just tangent to L(y), which is the isocost line {x : [λp1 + (1 − λ)p2 ]T x = [λp1 + (1 − λ)p2 ]T x∗ = C(y, λp1 + (1 − λ)p2 )} and it is this fact that gives us the concavity inequality (4).

9 It

can happen that the two dotted lines coincide.

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Fig. 1 The concavity in prices property of the cost function

Proof of Property 5 Since C(y,p) is a concave function of p defined over the open set of p’s,  ≡ {p : p  0N }, it follows that C(y,p) is also continuous in p over this domain of definition set for each fixed y ∈ Range f.10 Proof of Property 6 Let p  0N , y1 ∈ Range f, y2 ∈ Range f, y1 < y2 . Then     C y2 , p ≡ minx pT x : f (x) ≥ y2 ; x ≥ 0N   ≥ minx pT x : f (x) ≥ y1 ; x ≥ 0N since if y1 < y2 , the set  set

x : f (x) ≥ y1



x : f (x) ≥ y2

 is a subset of the

 and the minimum of a linear function

over a bigger set cannot increase   ≡ C y1 , p . Proof of Property 7 The proof is rather technical and may be found in Diewert ([27], 113–114).

10 See

Fenchel ([52], 75) or Rockafellar ([77], 82).

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Problems 1. In industrial organization,11 it once was fairly common to assume that a firm’s cost function had the following linear functional form: C(y, p) ≡ α + βT p + γy where α and γ are scalar parameters and β is a vector of parameters to be estimated econometrically. What are sufficient conditions on these N + 2 parameters for this cost function to satisfy properties 1–7 above? Is the resulting cost function very realistic? 2. Suppose a producer’s production function, f(x), defined for x ∈ S where S = {x : x ≥ 0N } satisfies the following conditions: (i) f is continuous over S (ii) f(x) > 0 if x  0N and (iii) f is positively linearly homogeneous over S; that is, for every x ≥ 0N and λ > 0, f(λx) = λf(x) Define the producer’s unit cost function c(p) for p  0N as follows: (iv) c(p) ≡ C(1, p) ≡ minx {pT x : f(x) ≥ 1; x ≥ 0N }; that is, c(p) is the minimum cost of producing one unit of output if the producer faces the positive input price vector p. For y > 0 and p  0N , show that (v) C(y, p) = c(p)y Note: A production function f that satisfies property (iii) is said to exhibit constant returns to scale. The interpretation of (v) is that if a production function exhibits constant returns to scale, then total cost is equal to unit cost times the output level.12 3. Shephard ([81], 4) defined a production function F to be homothetic if it could be written as (i) F(x) = g[f(x)]; x ≥ 0N where f satisfies conditions (i)–(iii) in Problem 2 above and g(z), defined for all z ≥ 0, satisfies the following regularity conditions: (ii) g(z) is positive if z > 0 (iii) g is a continuous function of one variable and (iv) g is monotonically increasing; that is, if 0 ≤ z1 < z2 , then g(z1 ) < g(z2 ). Let C(y,p) be the cost function that corresponds to F(x). Show that under the above assumptions, for y > 0 and p  0N , we have (v) C(y, p) = g−1 (y)c(p) where c(p) is the unit cost function that corresponds to the linearly homogeneous f and g−1 is the inverse function for g; that is, g−1 [g(z)] = z for all z ≥ 0. Note that g−1 (y) is a monotonically increasing continuous function of one variable.

11 For

example, see Walters [85]. will study the unit cost function in more detail in section “The Duality Between Constant Returns to Scale Production Functions and Their Unit Cost Functions” below.

12 We

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The Duality Between Cost and Production Functions The material in the previous section shows how the cost function can be determined from knowledge of the production function. We now ask whether knowledge of the cost function is sufficient to determine the underlying production function. The answer to this question is yes, but with some qualifications. To see how we might use a given cost function (satisfying the seven regularity conditions listed in the previous section) to determine the production function that generated it, pick an arbitrary feasible output level y > 0 and an arbitrary vector of positive prices, p1  0N and use the given cost function C to define the following isocost surface: {x : p1T x = C(y, p1 )}. This isocost surface must be tangent to the set of feasible input combinations x that can produce at least output level y, which is the upper level set, L(y) ≡ {x : f(x) ≥ y; x ≥ 0N }. It can be seen that this isocost surface and the set lying above it must contain the upper level set L(y); that is, the following halfspace M(y,p1 ), contains L(y):      M y, p1 ≡ x : p1T x ≥ C y, p1 .

(7)

Pick another positive vector of prices, p2  0N and it can be seen, repeating the above argument, that the halfspace M(y, p2 ) ≡ {x : p2T x ≥ C(y, p2 )} must also contain the upper level set L(y). Thus, L(y) must belong to the intersection of the two halfspaces M(y,p1 ) and M(y,p2 ). Continuing to argue along these lines, it can be seen that L(y) must be contained in the following set, which is the intersection of all of the supporting halfspaces to L(y): M (y) ≡

 p0N

M (y, p) .

(8)

Note that M(y) is defined using just the given cost function, C(y,p). Note also that since each of the sets in the intersection, M(y,p), is a convex set, then M(y) is also a convex set. Since L(y) is a subset of each M(y,p), it must be the case that L(y) is also a subset of M(y); that is, we have L (y) ⊂ M (y) .

(9)

Is it the case that L(y) is equal to M(y)? In general, the answer is no; M(y) forms an outer approximation to the true production possibilities set L(y). To see why this is, see Fig. 1 above. The boundary of the set M(y) partly coincides with the boundary of L(y) but it encloses a bigger set: the backward bending parts of the isoquant {x : f(x) = y} are replaced by the dashed lines that are parallel to the x1 axis and the x2 axis and the inward bending part of the true isoquant is replaced by the dashed line that is tangent to the two regions where the boundary of M(y) coincides with the boundary of L(y). However, if the producer is a price taker in input markets, then it can be seen that we will never observe the producer’s

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nonconvex portions or backwards bending parts of the isoquant.13 Thus, under the assumption of competitive behavior in input markets, there is no loss of generality in assuming that the producer’s production function is nondecreasing (this will eliminate the backward bending isoquants) or in assuming that the upper level sets of the production function are convex sets (this will eliminate the nonconvex portions of the upper level sets). A function has convex upper level sets if and only if it is quasiconcave.14 Putting the above material together, we see that conditions on the production function f(x) that are necessary for the sets M(y) and L(y) to coincide are15 : f (x) is defined for x ≥ 0N and is continuous from above over this domain of definition set;

(10)

f is nondecreasing and

(11)

f is quasiconcave.

(12)

Theorem 2 Shephard Duality Theorem16 : If f satisfies Eqs. (10), (11), and (12), then the cost function C defined by Eq. (1) satisfies the properties listed in Theorem 1 above and the upper level sets M(y) defined by Eq. (8) using only the cost function coincide with the upper level sets L(y) defined using the production function; that is, under these regularity conditions, the production function and the cost function determine each other. We consider how an explicit formula for the production function in terms of the cost function can be obtained. Suppose we have a given cost function, C(y,p), and we are given a strictly positive input vector, x  0N , and we ask what is the maximum output that this x can produce. It can be seen that

13 Hotelling

([62], 74) made this point many years ago.

14 f is a quasiconcave function defined over a convex subset S of RN

if f has the following property: x1 ∈ S, x2 ∈ S, 0 < λ < 1 implies f(λx1 + (1 − λ)x2 ) ≥ min {f(x1 ), f(x2 )}; see Fenchel ([52], 117). 15 Since each of the sets M(y,p) in the intersection set M(y) defined by Eq. (8) are closed, it can be shown that M(y) is also a closed set. Hence if M(y) is to coincide with L(y), we need the upper level sets of f to be closed sets and this will hold if and only if f is continuous from above. 16 Shephard [81, 82] was the pioneer in establishing various duality theorems between cost and production functions. See also Samuelson [79], Uzawa [84], McFadden [72, 73], Diewert ([20, 22], 116–118) and Blackorby et al. [6] for various duality theorems under alternative regularity conditions. Our exposition follows that of Diewert ([27], 107–117). These duality theorems are global in nature; i.e., the production and cost functions satisfy their appropriate regularity conditions over their entire domains of definition. However, it is also possible to develop duality theorems that are local rather than global; see Blackorby and Diewert [5].

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f (x) = maxy {y : x ∈ M (y)}   = maxy y : C (y, p) ≤ pT x for every p  0N using definitions (7) and (8).   = maxy y : C (y, p) ≤ 1 for every p  0N such that pT x = 1 (13) where the last equality follows using the fact that C(y,p) is linearly homogeneous in p as is the function pT x and hence we can normalize the prices so that pT x = 1. We now consider the continuity properties of C(y,p) with respect to p. We have defined C(y,p) for all strictly positive price vectors p and since this domain of definition set is open, we know that C(y,p) is also continuous in p over this set, using the concavity in prices property of C. We would like to extend the domain of definition of C(y,p) from the strictly positive orthant of prices,  ≡ {p : p  0N }, to the nonnegative orthant, Clo  ≡ {p : p ≥ 0N }, which is the closure of . It turns out that it is possible to do this if we make use of some theorems in convex analysis. Theorem 3 Continuity from above of a concave function using the Fenchel closure operation: Fenchel ([52], 78): Let f(x) be a concave function of N variables defined over the open convex subset S of RN . Then there exists a unique extension of f to Clo S, the closure of S, which is concave and continuous from above. Proof Using one of Fenchel’s ([52], 57) characterizations of concavity, the hypograph of f, H ≡ {(y, x) : y ≤ f(x); x ∈ S}, is a convex set in RN + 1 . Hence, the closure of H, Clo H, is also a convex set. Hence, the following function f* defined over Clo S is also a concave function: f∗ (x) = maxy {y : (y, x) ∈ Clo H} ; = f (x)

x ∈ Clo S.

for x ∈ S.

(14)

Since Clo H is a closed set, it turns out that f* is continuous from above. To see that the extension function f* need not be continuous, consider the following example, where the domain of definition set is S ≡ {(x1 , x2 ); x2 ∈ R1 , x1 ≥ x2 2 } in R2 : f (x1 , x2 ) ≡ −x2 2 /x1 if x2 = 0, x1 ≥ x2 2 ; ≡ 0 if x1 = 0 and x2 = 0.

(15)

It is possible to show that f is concave and hence continuous over the interior of S; see problem 5 below. However, it can be shown that f is not continuous at (0,0). Let (x1 ,x2 ) approach (0,0) along the line x1 = x2 > 0. Then

lim x1→0 f (x1 , x2 ) = lim x1→0 −x1 2 /x1 = lim x1→0 [−x1 ] = 0.

(16)

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Now let (x1 ,x2 ) approach (0,0) along the parabolic path x2 > 0 and x1 = x2 2 . Then lim x2→0; x1 = x22 f (x1 , x2 ) = lim x2→0 − x2 2 /x2 2 = −1.

(17)

Thus, f is not continuous at (0,0). It can be verified that restricting f to Int S and then extending f to the closure of S (which is S) leads to the same f* as is defined by Eq. (15). Thus, the Fenchel closure operation does not always result in a continuous concave function. Theorem 4 below states sufficient conditions for the Fenchel closure of a concave function defined over an open domain of definition set to be continuous over the closure of the original domain of definition. Fortunately, the hypotheses of this Theorem are weak enough to cover most economic applications. Before stating the Theorem, we need an additional definition. Definition A set S in RN is a polyhedral set iff S is equal to the intersection of a finite number of halfspaces. Theorem 4 Continuity of a concave function using the Fenchel closure operation; Gale et al. [57], Rockafellar ([77], 85): Let f be a concave function of N variables defined over an open convex polyhedral set S. Suppose f is bounded from below over every bounded subset of S. Then the Fenchel closure extension of f to the closure of S results in a continuous concave function defined over Clo S. The proof of this result is too involved to reproduce here but we can now apply this result. Applying Theorem 4, extend the domain of definition of C(y,p) from strictly positive price vectors p to nonnegative price vectors using the Fenchel closure operation and hence C(y,p) will be continuous and concave in p over the set {p : p ≥ 0N } for each y in the interval of feasible outputs.17 Now return to the problem where we have a given cost function, C(y,p), we are given a strictly positive input vector, x  0N , and we ask what is the maximum output that this x can produce. Repeating the analysis in Eq. (13), we have

f(0N ) = 0 and f(x) tends to plus infinity as the components of x tend to plus infinity, then the feasible y set will be y ≥ 0 and C(y,p) will be defined for all y ≥ 0 and p ≥ 0N .

17 If

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71

f (x) = maxy {y : x ∈ M (y)}   = maxy y : C (y, p) ≤ pT x for every p  0N using definitions (7) and (8).   = maxy y : C (y, p) ≤ 1 for every p  0N such that pT x = 1 where we have used the linear homogeneity in prices property of C   = maxy y : C (y, p) ≤ 1 for every p ≥ 0N such that pT x = 1 where we have extended the domain of definition of C (y, p) to nonnegative prices from positive prices and used the continuity of the extension function over the set of nonnegative prices = maxy {y : G (y, x) ≤ 1} (18) where the function G(y,x) is defined as follows:   G (y, x) ≡ maxp C (y, p) : p ≥ 0N and pT x = 1 .

(19)

Note that the maximum in Eq. (19) will exist since C(y,p) is continuous in p and the feasible region for the maximization problem, {p : p ≥ 0N and pT x = 1}, is a closed and bounded set.18 Property 7 on the cost function C(y,p) will imply that the maximum in the last line of Eq. (18) will exist. Property 6 on the cost function will imply that for fixed x, G(y,x) is nondecreasing in y. Typically, G(y,x) will be continuous in y for a fixed x and so the maximum y that solves Eq. (18) will be the y* that satisfies the following equation19 :   G y∗ , x = 1.

(20)

Thus, Eqs. (19) and (20) implicitly define the production function y∗ = f(x) in terms of the cost function C. Problems 4. Show that the f(x1 ,x2 ) defined by Eq. (15) above is a concave function over the interior of the domain of definition set S. You do not have to show that S is a convex set.

is where we use the assumption that x  0N in order to obtain the boundedness of this set. method for constructing the production function from the cost function may be found in Diewert ([22], 119).

18 Here 19 This

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5. In the case where the technology is subject to constant returns to scale, the cost function has the following form: C(y, p) = yc(p) where c(p) is a unit cost function. For x  0N , define the function g(x) as follows: (i)   g (x) ≡ maxp c (p) : pT x = 1; p ≥ 0N . Show that in this constant returns to scale case, the function G(y,x) defined by Eq. (19) reduces to (ii) G (y, x) = yg (x) . Show that in this constant returns to scale case, the production function that is dual to the cost function has the following explicit formula for x  0N : (iii) f (x) = 1/g (x) . 6. Let x ≥ 0 be input (a scalar number) and let y = f(x) ≥ 0 be the maximum output that could be produced by input x, where f is the production function. Suppose that f is defined as the following step function: (i) f (x) ≡ 0 for 0 ≤ x < 1; ≡ 1 for 1 ≤ x < 2; ≡ 2 for 2 ≤ x < 3; and so on. Thus, the technology cannot produce fractional units of output and it takes one full unit of input to produce each unit of output. It can be verified that this production function is continuous from above. (a) Calculate the cost function C(y,1) that corresponds to this production function; that is, set the input price equal to one and try to determine the corresponding total cost function C(y,1). It will turn out that this cost function is continuous from below in y. (b) Graph both the production function y = f(x) and the cost function c(y) ≡ C(y, 1). 7. Suppose that a producer’s cost function is defined as follows for y ≥ 0, p1 > 0 and p2 > 0: (i)

C (y, p1 , p2 ) ≡ b11 p1 + 2b12 (p1 p2 )1/2 + b22 p2 y where the bij parameters are all positive.

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(a) Show that this cost function is concave in the input prices p1 ,p2 . Note: this is the two input case of the generalized Leontief cost function defined by Diewert [20]. (b) Calculate an explicit functional form for the corresponding production function f(x1 ,x2 ) where we assume that x1 > 0 and x2 > 0. Hint: This part of the problem is not completely straightforward. You will obtain a quadratic equation but which root is the right one?

The Derivative Property of the Cost Function Theorem 2, the Shephard Duality Theorem, is of mainly academic interest: if the production function f satisfies properties (10), (11), and (12), then the corresponding cost function C defined by Eq. (1) satisfies the properties listed in Theorem 1 above and moreover completely determines the production function. However, it is the next property of the cost function that makes duality theory so useful in applied economics. Theorem 5 Shephard’s ([81], 11) Lemma: If the cost function C(y,p) satisfies the properties listed in Theorem 1 above and in addition is once differentiable with respect to the components of input prices at the point (y* ,p* ) where y* is in the range of the production function f and p∗  0N , then   x∗ = ∇p C y∗ , p∗

(21)

where ∇ p C(y∗ , p∗ ) is the vector of first order partial derivatives of cost with respect to input prices, [∂C(y∗ , p∗ )/∂p1 , . . . , ∂C(y∗ , p∗ )/∂pN ]T , and x* is any solution to the cost minimization problem     minx p∗T x : f (x) ≥ y∗ ≡ C y∗ , p∗ .

(22)

Under these differentiability hypotheses, it turns out that the x* solution to Eq. (22) is unique. Proof Let x* be any solution to the cost minimization problem (22). Since x* is feasible for the cost minimization problem when the input price vector is changed to an arbitrary p  0N , it follows that   pT x∗ ≥ C y∗ , p for every p  0N .

(23)

Since x* is a solution to the cost minimization problem (22) when p = p∗ , we must have   p∗T x∗ = C y∗ , p∗ .

(24)

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But Eqs. (23) and (24) imply that the function of N variables, g(p) ≡ pT x∗ − C(y∗ , p) is nonnegative for all p  0N with g(p∗ ) = 0. Hence, g(p) attains a global minimum at p = p∗ and since g(p) is differentiable with respect to the input prices p at this point, the following first order necessary conditions for a minimum must hold at this point:     ∇p g p∗ = x∗ − ∇p C y∗ , p∗ = 0N .

(25)

Now note that Eq. (25) is equivalent to Eq. (21). If x** is any other solution to the cost minimization problem (22), then repeat the above argument to show that   x∗∗ = ∇p C y∗ , p∗ =x

(26)

where the second equality follows using Eq. (25). Hence, x∗∗ = x∗ and the solution to Eq. (22) is unique. The above result has the following implication: postulate a differentiable functional form for the cost function C(y,p) that satisfies the regularity conditions listed in Theorem 1 above. Then differentiating C(y,p) with respect to the components of the input price vector p generates the firm’s system of cost minimizing input demand functions, x(y, p) ≡ ∇ p C(y, p). Shephard [81] was the first person to establish the above result starting with just a cost function satisfying the appropriate regularity conditions.20 However, Hotelling ([61], 594) stated a version of the result in the context of profit functions and Hicks ([60], 331) and Samuelson ([79], 15–16) established the result starting with a differentiable utility or production function. One application of the above result is its use as an aid in generating systems of cost minimizing input demand functions that are linear in the parameters that characterize the technology. For example, suppose that the cost function had the following generalized Leontief functional form 21 : C (y, p) ≡

 i=1

N k=1

N

bik pi 1/2 pk 1/2 y;

bik = bki for 1 ≤ i < j ≤ N (27)

where the N(N + 1)/2 independent bik parameters are all nonnegative. With these nonnegativity restrictions, it can be verified that the C(y,p) defined by Eq. (27) satisfies properties 1–7 listed in Theorem 1.22 Applying Shephard’s Lemma 20 This is why Diewert ([22], 112) called the result Shephard’s Lemma. See also Fenchel ([52], 104). We have used the technique of proof used by McKenzie [74]. 21 See Diewert [20]. 22 Using problem 7 above, it can be seen that if the b are nonnegative and y is positive, then ik the functions bik pi 1/2 pk 1/2 y are concave in the components of p. Hence, since a sum of concave functions is concave, it can be seen that the C(y,p) defined by Eq. (27) is concave in the components of p.

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shows that the system of cost minimizing input demand functions that correspond to this functional form are given by: xi (y, p) = ∂C (y, p) /∂pi =



N k=1

bik (pk /pi )1/2 y;

i = 1, 2, . . . , N.

(28)

Errors can be added to the system of Eq. (28) and the parameters bik can be estimated using linear regression techniques if we have time series or cross sectional data on output, inputs and input prices.23 If all of the bij equal zero for i = j, then the demand functions become: xi (y, p) = ∂C (y, p) /∂pi = bii y;

i = 1, 2, . . . , N.

(29)

Note that input prices do not appear in the system of input demand functions defined by Eq. (29) so that input quantities do not respond to changes in the relative prices of inputs. The corresponding production function is known as the Leontief [71] production function.24 Hence, it can be seen that the production function that corresponds to Eq. (28) is a generalization of this production function. We will consider additional functional forms for a cost function in subsequent sections.

The Comparative Statics Properties of Input Demand Functions Before we develop the main result in this section, it will be useful to establish some results about the derivatives of a twice continuously differentiable linearly homogeneous function of N variables. We say that f(x), defined for x  0N is positively homogeneous of degree α iff f has the following property: f (λx) = λα f (x)

for all x  0N and λ > 0.

(30)

A special case of the above definition occurs when the number α in the above definition equals 1. In this case, we say that f is (positively) linearly homogeneous25 iff

23 Note

that b12 will appear in the first input demand equation and in the second as well using the cross equation symmetry condition, b21 = b12 . There are N(N − 1)/2 such cross equation symmetry conditions and we could test for their validity or impose them in order to save degrees of freedom. The nonnegativity restrictions that ensure global concavity of C(y,p) in p can be imposed if we replace each parameter bik by a squared parameter, (aik )2 . However, the resulting system of estimating equations is no longer linear in the unknown parameters. 24 The Leontief production function can be defined as f(x , . . . , x ) ≡ min {x /b : i = 1, . . . , N}. 1 N i i ii It is also known as the no substitution production function. Note that this production function is not differentiable even though its cost function is differentiable. 25 Usually in economics, we omit the adjective “positively” but it is understood that the λ which appears in definitions (30) and (31) is restricted to be positive.

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W. E. Diewert

f (λx) = λf (x)

for all x  0N and λ > 0.

(31)

Theorem 6 Euler’s Theorems on Differentiable Homogeneous Functions: Let f(x) be a (positively) linearly homogeneous function of N variables, defined for x  0N . Part 1: If the first order partial derivatives of f exist, then the first order partial derivatives of f satisfy the following equation: f (x) =



N

n=1

xn ∂f (x1 , . . . , xN ) /∂xn = xT ∇f (x)

for all x  0N .

(32)

Part 2: If the second order partial derivatives of f exist, then they satisfy the following equations: N 2

 k=1

∂ f (x1 . . . , xN ) /∂xn ∂xk xk = 0

for all x  0N and n = 1, . . . , N. (33)

The N equations in (33) can be written using matrix notation in a much more compact form as follows: ∇ 2 f (x) x = 0N

for all x  0N .

(34)

Proof of Part 1 Let x  0N and λ > 0. Differentiating both sides of Eq. (31) with respect to λ leads to the following equation using the composite function chain rule: f (x) = =

 

N

[∂f (λx1 , . . . , λxN ) /∂ (λxn )] [∂ (λxn ) /∂λ]

N

[∂f (λx1 , . . . , λxN ) /∂ (λxn )] xn .

n=1 n=1

(35)

Now evaluate Eq. (35) at λ = 1 and we obtain Eq. (32). Proof of Part 2 Let x  0N and λ > 0. For n = 1, . . . , N, differentiate both sides of Eq. (31) with respect to xn and we obtain the following N equations: fn (λx1 , . . . , λxN ) ∂ (λxn ) /∂xn = λfn (x1 , . . . , xN ) fn (λx1 , . . . , λxN ) λ = λfn (x1 , . . . , xN ) fn (λx1 , . . . , λxN ) = fn (x1 , . . . , xN )

for n = 1, . . . , N or for n = 1, . . . , N or (36) for n = 1, . . . , N

where the nth first order partial derivative function is defined as fn (x1 , . . . , xN ) ≡ ∂f(x1 , . . . xN )/∂xn for n = 1, . . . , N.26 Now differentiate both sides of the last set 26 Using definition (30) for the case where α = 0, it can be seen that the last set of equations in (36) shows that the first order partial derivative functions of a linearly homogenous function are homogeneous of degree 0.

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of equations in (36) with respect to λ and we obtain the following N equations: 0= =



N

[∂fn (λx1 , . . . , λxN ) /∂xk ] [∂ (λxk ) /∂λ]

N

[∂fn (λx1 , . . . , λxN ) /∂xk ] xk .

k=1



k=1

for n = 1, . . . , N (37)

Now evaluate Eq. (37) at λ = 1 and we obtain the N Eq. (33). The above results can be applied to the cost function, C(y,p). From Theorem 1, C(y,p) is linearly homogeneous in p. Hence, by part 2 of Euler’s Theorem, if the second order partial derivatives of the cost function with respect to the components of the input price vector p exist, then these derivatives satisfy the following restrictions: ∇ 2 pp C (y, p) p = 0N .

(38)

Theorem 7 Diewert ([27], 148–150): Suppose the cost function C(y,p) satisfies the properties listed in Theorem 1 and in addition is twice continuously differentiable with respect to the components of its input price vector at some point, (y,p). Then the system of cost minimizing input demand equations, x(y, p) ≡ [x1 (y, p), . . . , xN (y, p)]T , exists at this point and these input demand functions are once continuously differentiable. Form the N by N matrix of input demand derivatives with respect to input prices, B ≡ [∂xi (y, p)/∂pj ], which has ij element equal to ∂xi (y, p)/∂pj . Then the matrix B has the following properties27,28,29 : B = BT so that ∂xi (y, p) /∂pk = ∂xk (y, p) /∂pi for all i = k;

(39)

B is negative semidefinite and

(40)

Bp = 0N .

(41)

Proof Shephard’s Lemma implies that the firm’s system of cost minimizing input demand equations, x(y, p) ≡ [x1 (y, p), . . . , xN (y, p)]T , exists and is equal to

27 These are the Hicks ([60], 311) and Samuelson ([78], 69) symmetry restrictions. Hotelling ([61], 549) obtained analogues to these symmetry conditions in the profit function context. 28 Hicks ([60], 311) and Samuelson ([78], 69) also obtained versions of this result by starting with the production (or utility) function f(x), assuming that the first order conditions for solving the cost minimization problem held and that the strong second order sufficient conditions for the primal cost minimization problem also held. Thus using duality theory, we obtain the same results under weaker regularity conditions. 29 Hicks ([60], 331) and Samuelson ([78], 69) also obtained this result using their primal technique.

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W. E. Diewert

x (y, p) = ∇p C (y, p) .

(42)

Differentiating both sides of Eq. (42) with respect to the components of p gives us   B ≡ ∂xi (y, p) /∂pk = ∇ 2 pp C (y, p) .

(43)

Now property (39) follows from Young’s Theorem in calculus. Property (40) follows from Eq. (43) and the fact that C(y,p) is concave in p and the fourth characterization of concavity. Finally, property (41) follows from the fact that the cost function is linearly homogeneous in p and hence Eq. (38) holds. Note that property (40) implies the following properties on the input demand functions: ∂xn (y, p) /∂pn ≤ 0

for n = 1, . . . , N.

(44)

Property (44) means that input demand curves cannot be upward sloping. If the cost function is also differentiable with respect to the output variable y, then we can deduce an additional property about the first order derivatives of the input demand functions. The linear homogeneity property of C(y,p) in p implies that the following equation holds for all λ > 0: C (y, λp) = λC (y, p)

for all λ > 0 and p  0N .

(45)

Partially differentiating both sides of Eq. (45) with respect to y leads to the following equation: ∂C (y, λp) /∂y = λ∂C (y, p) /∂y

for all λ > 0 and p  0N .

(46)

But Eq. (46) implies that the function ∂C(y, p)/∂y is linearly homogeneous in p and hence part 1 of Euler’s Theorem applied to this function gives us the following equation: ∂C (y, p) /∂y =

 n=1

N

pn ∂ 2 C (y, p) /∂y∂pn = pT ∇ 2 yp C (y, p) .

(47)

But using Eq. (42), it can be seen that Eq. (47) is equivalent to the following equation30 : ∂C (y, p) /∂y =



N n=1

pn ∂xn (y, p) /∂y.

(48)

30 This method of deriving these restrictions is due to Diewert ([27], 150) but these restrictions were originally derived by Samuelson ([78], 66) using his primal cost minimization method.

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Problems 8. For i = k, the inputs i and k are said to be substitutes if ∂xi (y, p)/∂pk = ∂xk (y, p)/ ∂pi > 0, unrelated if ∂xi (y, p)/∂pk = ∂xk (y, p)/∂pi = 031 and complements if ∂xi (y, p)/∂pk = ∂xk (y, p)/∂pi < 0. (a) If N = 2, show that the two inputs cannot be complements. (b) If N = 2 and ∂x1 (y, p)/∂p1 = 0, then show that all of the remaining input demand price derivatives are equal to 0; that is, show that ∂x1 (y, p)/∂p2 = ∂x2 (y, p)/∂p1 = ∂x2 (y, p)/∂p2 = 0. (c) If N = 3, show that at most one pair of inputs can be complements.32 9. Let N ≥ 3 and suppose that ∂x1 (y, p)/∂p1 = 0. Then show that ∂x1 (y, p)/∂pn = 0 as well for n = 2, 3, . . . , N. Hint: You may need to use the definition of negative semidefiniteness in a strategic way. This problem shows that if the own input elasticity of demand for an input is 0, then that input is unrelated to all other inputs. 10. Recall the definition (27) of the generalized Leontief cost function where the parameters bij were all assumed to be nonnegative. Show that under these nonnegativity restrictions, every input pair is either unrelated or substitutes. Hint: Simply calculate ∂ 2 C(y, p)/∂pi ∂pk for i = k and look at the resulting formula. Comment: This result shows that if we impose the nonnegativity conditions bik ≥ 0 for i = j on this functional form in order to ensure that it is globally concave in prices, then we have a priori ruled out any form of complementarity between the inputs. This means if the number of inputs N is greater than 2, this nonnegativity restricted functional form cannot be a flexible functional form 33 for a cost function; that is, it cannot attain an arbitrary pattern of demand derivatives that are consistent with microeconomic theory, since the nonnegativity restrictions rule out any form of complementarity. 11. Suppose that a producer’s three input production function has the following Cobb Douglas [17] functional form: (a) f (x1 , x2 , x3 ) ≡ x1α1 x2α2 x3α1 where α1 > 0, α2 > 0, α3 > 0 and α1 + α2 + α3 = 1.Let the positive input prices p1 > 0, p2 > 0, p3 > 0 and the positive output level y > 0 be given. (i) Calculate the producer’s cost function, C(y,p1 ,p2 ,p3 ) along with the three input demand functions, x1 (y,p1 ,p2 ,p3 ), x2 (y,p1 ,p2 ,p3 ), and x3 (y,p1 ,p2 ,p3 ). Hint: Use the usual Lagrangian technique for solving

31 Pollak

([76], 67) used the term “unrelated” in a similar context. result is due to Hicks ([60], 311–312): “It follows at once from Rule (5) that, while it is possible for all other goods consumed to be substitutes for x1 , it is not possible for them all to be complementary with it.” 33 Diewert ([22], 115) introduced the term “flexible functional form” to describe a functional form for a cost function (or production function) that could approximate an arbitrary cost function (consistent with microeconomic theory) to the second order around any given point. The Generalized Leontief cost function defined by Eq. (27) above is flexible for the class of cost functions that are dual to linearly homogeneous production functions if we do not impose any restrictions on the parameters bij ; see Diewert [20] or section “Flexible Functional Forms for Cost Functions: The Generalized Leontief Functional Form” below for a proof of this fact. 32 This

80

W. E. Diewert

constrained minimization problems. You do not need to check the second order conditions for the problem. The positive constant k ≡ α1−α1 α2−α2 α3−α3 will appear in the cost function. (ii) Calculate the input one demand elasticity with respect to output [∂x1 (y, p1 , p2 , p3 )/∂y][y/x1 (y, p1 , p2 , p3 )] and the three input one demand elasticities with respect to input prices [∂x1 (y, p1 , p2 , p3 )/∂pn ] [pn /x1 (y, p1 , p2 , p3 )] for n = 1, 2, 3. (iii) Show that −1 < [∂x1 (y, p1 , p2 , p3 )/∂p1 ][p1 /x1 (y, p1 , p2 , p3 )] < 0. (iv) Show that 0 < [∂x1 (y, p1 , p2 , p3 )/∂p2 ][p2 /x1 (y, p1 , p2 , p3 )] < 1. (v) Show that 0 < [∂x1 (y, p1 , p2 , p3 )/∂p3 ][p3 /x1 (y, p1 , p2 , p3 )] < 1. (vi) Can any pair of inputs be complementary if the technology is a three input Cobb Douglas? Comment: The Cobb Douglas functional form is widely used in macroeconomics and in applied general equilibrium models. However, this problem shows that it is not satisfactory if N ≥ 3. Even in the N = 2 case where analogues to (iii) and (iv) above hold, it can be seen that this functional form is not consistent with technologies where the degree of substitution between inputs is very high or very low. 12. Suppose that the second order partial derivatives with respect to input prices of the cost function C(y,p) exist so that the nth cost minimizing input demand function xn (y, p) = ∂C(y, p)/∂pn > 0 exists for n = 1, . . . , N. Define the input n elasticity of demand with respect to input price k as follows: (a)    enk (y, p) ≡ ∂xn (y, p) /∂pk pk /xn (y, p)

for n = 1, . . . , N and k = 1, . . . , N.

 Show that for each n, k = 1 N enk (y, p) = 0. 13. Let the producer’s cost function be C(y,p), which satisfies the regularity conditions in Theorem 1 and, in addition, is once differentiable with respect to the components of the input price vector p. Then the nth input demand function is xn (y, p) ≡ ∂C(y, p)/∂pn for n = 1, . . . , N. Input n is defined to be normal at the point (y,p) if ∂xn (y, p)/∂y = ∂ 2 C(y, p)/∂pn ∂y > 0; that is, if the cost minimizing demand for input n increases as the target output level y increases. On the other hand, input n is defined to be inferior at the point (y,p) if ∂xn (y, p)/∂y = ∂ 2 C(y, p)/∂pn ∂y < 0. Prove that not all N inputs can be inferior at the point (y,p). Hint: Make use of Eq. (48). 14. If the production function f dual to the differentiable cost function C(y,p) exhibits constant returns to scale so that f(λx) = λf(x) for all x ≥ 0N and all λ > 0, then show that for each n, the input n elasticity of demand with respect to the output level y is 1; that is, show that for n = 1, . . . , N, [∂xn (y, p)/∂y][y/xn (y, p)] = 1.

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15. Let C(y,p) be a twice continuously differentiable cost function that satisfies the regularity conditions listed in Theorem 1 in section “Cost Functions: The One Output Case” above. By Shephard’s Lemma, the input demand functions are given by (i) n = 1, . . . , N.

xn (y, p) = ∂C (y, p) /∂pn > 0;

The Allen ([1], 504), Uzawa [83] elasticity of substitution σnk between inputs n and k is defined as follows for 1≤ n, k≤ N: (ii) σnk (y, p) ≡C (y, p) {∂ 2 C (y, p) /∂pn ∂pk }/{∂C (y, p) /∂pn }{∂C (y, p) /∂pk }.  Define = [σnk (y, p)] as the N by N matrix of elasticities of substitution.  (a) Show that has the following properties: (iii)  (iv) (v)



=

T

;

is negative semidefinite and 

s = 0N

where s ≡ [s1 , . . . , sN ]T is the vector of cost shares; that is, sn ≡ pn xn (y, p)/C(y, p) for n = 1, . . . , N. Now define the N by N matrix of cross price elasticities of demand E in a manner analogous to definition (ii) above: (vi) n = 1, . . . , N; k = 1, . . . , N E ≡ enk   = (pk /xn ) ∂xn (y, p) /∂pk

using (i) = (pk /xn ) ∂ 2 C (y, p) /∂pn ∂pk = xˆ −1 ∇ 2 pp C (y, p) p. ˆ  (b) Show that E = sˆ where sˆ is an N by N diagonal matrix with the elements of the share vector s running down the main diagonal.

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The Duality Between Constant Returns to Scale Production Functions and Their Unit Cost Functions In this section, we will add more structure to the production function: we will assume that f(x) is subject to constant returns to scale so that f(λx) = λf(x) for every nonnegative input vector x ≥ 0N and nonnegative scalar λ ≥ 0. In many areas of applied economics, constant returns to scale in production are assumed. Samuelson [80] justified this assumption as an approximation to reality by using a plant replication argument. He assumed that there was a plant size that minimized average cost and showed if this optimal plant size output level was small relative to the size of the market, then by replicating optimal size plants, the industry production function would approximate a constant returns to scale production function.34 Thus, in this section, we will assume constant returns to scale in production and see what additional properties the resulting cost function must satisfy. Before we develop a formal duality theorem, it is necessary to prove a useful mathematical result. Theorem 8 Berge ([3], 208): If f is a positive, linearly homogeneous and quasiconcave function defined over the positive orthant in RN , , then f is also concave over . Proof Let x1  0N , x2  0N and 0 < λ < 1. We need to show that:       f λx1 + (1 − λ) x2 ≥ λf x1 + (1 − λ) f x2 .

(49)

Without loss of generality, we can assume 0 < f(x1 ) ≤ f(x2 ). Let μ > 0 be the scalar that causes f(μx2 ) to equal f(x1 ). Using the constant returns to scale property of f, μ can be defined as follows:     μ ≡ f x1 /f x2 > 0.

(50)

Points on the line segment joining the point x1 to μx2 can be represented by + (1 − α)μx2 where 0 ≤ α ≤ 1. The quasiconcavity property of f implies that the following equality holds for all α such that 0 ≤ α ≤ 1: αx1

    f x1 ≤ f αx1 + (1 − α) μx2 .

(51)

Define β > 0 as the proportionality factor that deflates the point λx1 + (1 − λ)x2 onto the line segment joining the point x1 to μx2 . Thus, we have:

34 Diewert

[26] elaborated on Samuelson’s results.

3 Duality in Production

83

β λx1 + (1 − λ) x2 = αx1 + (1 − α) μx2 .

(52)

Thus, the unknown α and β must be the solution to the following two equations: βλ = α; β (1 − λ) = (1 − α) μ.

(53)

The solution to Eq. (52) is β = μ/(1 − λ + λμ) and α = λμ/(1 − λ + λμ). It is straightforward to show that the solution satisfies β > 0 and 0 ≤ α ≤ 1. Now substitute Eq. (52) into (51) and we obtain the following inequality:     f x1 ≤ f αx1 + (1 − α) μx2

  = f β λx1 + (1 − λ) x2   = βf λx1 + (1 − λ) x2

using (52) (54) using the linear homogeneity of f

  = [μ/ (1 − λ + λμ)] f λx1 + (1 − λ) x2 . Thus, Eq. (54) implies:     f λx1 + (1 − λ) x2 ≥ μ−1 (1 − λ + λμ) f x1     = μ−1 (1 − λ) f x1 + λf x1     = λf x1 + (1 − λ) f x2

using definition (50). (55)

The above result will prove to be useful in what follows. Recall that in section “Cost Functions: The One Output Case” above, we initially assumed that the production function f(x) only satisfied continuity from above. We continue to make this very weak regularity assumption but we now assume that in addition, f satisfies the following linear homogeneity property: f (λx) = λf (x) for all λ ≥ 0 and x ≥ 0N .

(56)

We also assume that there exists an x∗ > 0N such that y∗ ≡ f(x∗ ) > 0; that is, there exists a nonnegative, nonzero input vector x* which can produce a positive output. This assumption along with the constant returns to scale assumption (56) means that the technology can produce any positive output level.

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Let y > 0 and p  0N . We can define the total cost function that corresponds to our homogeneous production function using definition (1) again; that is, define C(y,p) as follows:   C (y, p) ≡ minx pT x : f (x) ≥ y; x ≥ 0N   = minx pT x : y−1 f (x) ≥ 1; x ≥ 0N   = minx pT x : f (x/y) ≥ 1; x ≥ 0N

using (56)

  = minx ypT (x/y) : f (x/y) ≥ 1; x/y ≥ 0N   = y minz pT z : f (z) ≥ 1; z ≥ 0N

(57)

letting z = x/y

= y c (p) where c(p) is the unit cost function that corresponds to f, defined as follows:   c (p) ≡ minz pT z : f (z) ≥ 1; z ≥ 0N .

(58)

We can use the input price properties of the total cost function C(y,p) that were implied by Theorem 1 in section “Cost Functions: The One Output Case” in order to derive the properties of the unit cost function, c(p). Thus, Theorem 1 tells us that c(p) is well defined as a minimum for p  0N and it is nonnegative, positively linearly homogeneous, nondecreasing, and concave in p over the positive orthant. In fact, the continuity from above property of f along with the assumption that f is linearly homogeneous will imply that f(0N ) = 0 and this in turn will imply that c(p) > 0 for p  0N . Since c(p) is concave over the positive orthant, we can also deduce that it is continuous over this domain of definition. The domain of definition of c(p) can be extended to the nonnegative orthant using the Fenchel closure operation as was done in section “Cost Functions: The One Output Case.” The resulting c(p) will be continuous over the nonnegative orthant. Thus, there is no problem in going from the production function to its unit cost function. Can we use the unit cost function to recover the underlying production function? We can get an outer approximation to the true technology using the algebra in section “Cost Functions: The One Output Case.” Let x > 0N be an arbitrary nonzero, nonnegative input vector. The maximum output y that is consistent with using the outer approximation technology and the input vector x must satisfy the inequalities yc(p) ≤ pT x for every p > 0N . Thus, we want the maximum y such that y ≤ pT x/c(p) for every p > 0N . Now the functions pT x and c(p) are both linearly homogeneous so we can normalize one of these functions and minimize or maximize the remaining

3 Duality in Production

85

function to obtain y = f∗ (x), where f* (x) is the production function that corresponds to the outer approximation technology. If we set pT x = 1, then we want to minimize 1/c(p) subject to the constraint pT x = 1 and so in this case, f* (x) is defined as follows:   f∗ (x) ≡ minp 1/c (p) : pT x = 1; p ≥ 0N (59)   T = 1/maxp c (p) : p x = 1; p ≥ 0N . Note that the maximization problem in Eq. (59) is a concave programming problem. On the other hand, we could set c(p) = 1. In this case, f* (x) is (equivalently) defined as follows:   f∗ (x) ≡ minp pT x : c (p) = 1; p ≥ 0N (60)   = minp pT x : c (p) ≥ 1; p ≥ 0N . In order to recover the original production function, f(x), by using the formulae on the right hand sides of Eq. (59) or (60), we need to assume that f is nondecreasing and quasiconcave, as in section “Cost Functions: The One Output Case.” However, using Berge’s Theorem 8 above, it can be seen that when f is linearly homogeneous and quasiconcave (and positive) over the positive orthant, then f is also a concave function over the positive orthant. If in addition, f is continuous over the nonnegative orthant, then f will also be concave over the nonnegative orthant. Thus, f and c satisfy exactly the same regularity conditions, with respect to x and p, respectively, if we assume that f is nondecreasing and quasiconcave. Moreover, the underlying technology can be represented by using either the linearly homogeneous production function or its dual unit cost function. Samuelson ([79], 15) and Shephard [81] were the first to obtain versions of this duality theorem for the homogeneous case.35 In the following sections, we will exhibit various explicit functional forms for a linearly homogeneous f or its dual unit cost function.

The Constant Elasticity of Substitution Production Function The constant elasticity of substitution (CES) production function, f(x), is defined as follows: f (x1 , . . . , xN ) ≡

35 See

 n=1

N

1/s βn xn s

also Diewert ([22], 110–112) for a duality theorem along the present lines.

(61)

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W. E. Diewert

where the parameters βn are positive and s is a parameter which satisfies s = 0 and the inequality s ≤ 1. The two input case of this functional form was introduced into the economics literature by Arrow et al. ([2], 230).36 The problems below show that the CES production function is a well-behaved constant returns to scale production function which satisfies the regularity conditions that were developed in the previous section, provided that s ≤ 1. Problems 16. Let s = 0and rewrite the f(x) defined  by Eq. (61) as f(x) = γfs (x) where N β ∗ x s ]1/s , β ∗ ≡ β / N β for n = 1, . . . , N and fs (x) ≡ [ n = 1 n n n n i=1 i   N 1/s γ ≡ [ i = 1 βi ] . Show that lims → 0 lnfs (x1 , . . . , xN ) = n = 1 N βn ∗ lnxn . Thus the CES production function defined by Eq. (61) tends to a Cobb-Douglas production function as the parameter s tends to 0. Hint: Write lnfs (x1 , . . . , xN )  as g(s)/h(s) where g(s) ≡ ln [ n = 1 N βn ∗ xn s ] and h(s) ≡ s. Let s tend to 0 and apply l’Hospital’s Rule. Note that g(0) = h(0) = 0. 17. Let βT ≡ [β1 , . . . , βN ] where βn > 0 for n = 1, . . . , N. Define βˆ as the N by N diagonal matrix with the elements of the vector β running down the main diagonal. Show that the N by N matrix − βˆ + ββT is a negative semidefinite matrix. Hint: Show that the inequality zT −βˆ + ββT z ≤ 0 for all vectors z is equivalent to the Cauchy-Schwarz inequality (xT y)2 ≤ (xT x)(yT y) with x ≡ βˆ 1/2 1N ; y ≡ βˆ 1/2 z where 1N is a vector of ones of dimension N and βˆ 1/2 is a diagonal matrix with the positive square roots of the elements of β running down the main diagonal. 18. Show that the CES production function f(x) defined by Eq. (61) above is homogeneous of degree one in the components of x. 19. Show that the CES production function f(x) defined by Eq. (61) above is a concave function of x if s = 0 and s ≤ 1 and is a convex function of x if s ≥ 1. Hint: Calculate the matrix of second order partial derivatives of f, ∇ xx 2 f(x), for x  0N and show it is negative semidefinite if s ≤ 1 and positive semidefinite if s ≥ 1. Problem 17 will be useful. We now want to determine the cost minimizing system of input demand functions. We will first calculate the unit cost function that corresponds to the CES production function defined by Eq. (61). We assume that the producer faces the positive input prices p ≡ [p1 , . . . , pN ]  0N . The unit cost minimization problem is the following one:

 authors wrote the CES functional form by Eq. (61) as f(x) = γ[ n = 1 N βn ∗ xn s ]1/s  defined 1/s is a positive efficiency parameter. They N where the βn ∗ now sum up to one and γ ≡ [ n = 1 βn ] noted that the function of x that is defined by [ n = 1 N βn ∗ xn s ]1/s is mean of order s of the inputs, x1 , . . . , xN and they referred to Hardy et al. ([59], 13) for the mathematical properties of this class of means. 36 These

3 Duality in Production

minx

87

 

N n=1

pn xn :



N

n=1



1/s βn xn

s

= 1; x ≥ 0N .

(62)

Ignoring the nonnegativity constraints, x ≥ 0N , and assuming that s < 1 and s = 0, the Lagrangian first order conditions for an interior solution for Eq. (62) are equivalent to the following conditions: pn = λβn xn s−1 ; 

1=

N n=1

n = 1, . . . , N;

(63)

βn xn s

(64)

where the unknowns in Eqs. (63) and (64) are x1 , . . . , xN and the Lagrange multiplier λ. The solution to Eqs. (63) and (64) turns out to be the following one (remember, s = 0 and s = 1)37 : ∗

xn (p) ≡ pn

1/(s−1)

βn

1/(1−s)

 /

N i=1

1/s βi

1/(1−s)

pn

s/(s−1)

;

n = 1, . . . , N. (65)

Once the unit output demand functions have been calculated, the unit cost function, c(p), can be calculated: c (p) ≡ = =



N

n=1



pn xn ∗ (p)

N n=1



N n=1

(s−1)/s βn 1/(1−s) pn s/(s−1)

using (65)

(66)

1/r αn pn

r

where the new parameters r and α1 , . . . , αN are defined as follows38 : r ≡ s/ (s − 1) ; αn ≡ βn 1/(1−s) ; n = 1, . . . , N.

(67)

s = 1, we have a linear production function. Usually, an interior solution to the cost minimization problem defined by Eq. (62) will not occur; i.e., in this case, we have a linear programming problem and the solution will normally be a corner solution.   38 Note that c(p) ≡ [ N α p r ]1/r can be rewritten as γ∗ [ N ∗ r 1/r where n=1  n n n = 1 αn pn ]  αn ∗ ≡ αn / i = 1 N αi and γ∗ ≡ [ i = 1 N αi ]1/r . Thus c(p) is equal to an efficiency parameter γ∗ times a mean of order r. 37 When

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When s takes on the values between 1 and −∞, r = s/(s − 1) goes from −∞ to 1.39 Thus, the range of r and s is the same, but they travel in opposite directions. Hence, the CES unit cost function c(p) defined by Eq. (66) will be a linearly homogeneous, concave, and nondecreasing function and have the same mathematical properties as the CES production function f(x) defined by Eq. (61). Once the CES unit cost function has been defined, the CES total cost function is defined as C(y, p) ≡ yc(p) where c(p) is defined by Eq. (66). Using Shephard’s Lemma, the CES system of cost minimizing demand functions is the following one: xn (y, p) = y αn pn

r−1



= C (y, p) αn pn

N i=1

r−1

(1/r)−1 αi pi

 /

;

r

i=1

n = 1, . . . , N. (68)

N

αi pi

r

Problem 20. Recall problem 15 above which defined the Allen Uzawa elasticity of substitution σnk between inputs n and k. Show that if C(y,p) is the CES total cost function, then σnk (y, p) = 1 − r for all input pairs n,k such that n = k. Thus every elasticity of substitution between any two distinct inputs is equal to the same constant. The above problem shows why the CES functional form is unsatisfactory if the number of inputs N exceeds two, since it is a priori unlikely that all elasticities of substitution between every pair of inputs would equal the same number. Thus, in the following sections, we will look for functional forms for the production or cost function that allow for more flexible patterns of substitution between inputs. We conclude this section by listing some possible methods for estimating the elasticity of substitution if the underlying technology can be adequately described by the CES functional form. We will first look at estimating equations where input prices are exogenous variables and input quantities (and hence output) are endogenous variables. Take logarithms of both sides of the CES input demand functions defined by Eq. (68). Add error terms to each equation, say en t for equation n in period t.40 Subtract the logarithm of the first input demand function from these N equations. Suppose that there are data on inputs, output, and input prices for t periods and the period t data are xt ≡ [x1 t , . . . , xN t ], yt and pt ≡ [p1 t , . . . , pN t ] for t = 1, . . . , T. We obtain the

that when s = 0, r will also  equal 0. Rewrite the c(p) defined by the last line in Eq. (66) as c(p) = γ∗ cr (p) where γ∗ ≡ [ i = 1 N αi ]1/r and cr (p) ≡ [ n = 1 N αn ∗ pn r ]1/r . Using the results of problem 16, it can be seen that the limiting casefor cr (p) as r tends to 0 is the Cobb-Douglas  unit cost function which has the logarithm equal to n = 1 N αn ∗ lnpn where αn ∗ = αn / i = 1 N αi for n = 1, . . . , N. 40 The errors in our models can be due to measurement errors in the prices and quantities, the assumption of incorrect functional forms and errors in optimization. 39 Note

3 Duality in Production

89

following estimating equations41 :     ln xn t /x1 t = lnαn − lnα1 + (r − 1) ln pn t /p1 t (69) + en − e1 ; t

n = 2, . . . , N; t = 1, . . . , T.

t

The above equations are linear in the unknown parameters, the lnαn and r − 1 ≡ − σ. However, not all of the lnαn can be identified. This may not matter if the focus is on the estimation of r (or on the elasticity of substitution, σ). In order to identify all of the parameters, we can add a unit cost function equation to the system defined by Eq. (69) Thus, define observed unit cost in period t as ct ≡ ( n = 1 N pn t xn t )/yt for t = 1, . . . , T. Add the following estimating equations to Eq. (69) where e0 t is the period t error term42 : lnc = (1/r) ln t



N n=1

 αn pn

r

+ e0 t ;

t = 1, . . . , T.

(70)

Of course, the estimating equations in (70) are nonlinear in the unknown parameters so nonlinear regression techniques will have to be used. If the focus is on estimating the elasticity of substitution, Eq. (69) can be differenced again, this time with respect to time. Thus, define the double differenced logarithmic input quantity and price variables, dxn t and dpn t as follows for n = 2, . . . , N; t = 2, . . . , T:

  dxn t ≡ ln xn t /x1 t − ln xn t−1 /x1 t−1 = lnxn t − lnx1 t − lnxn t−1 + lnx1 t−1 ; (71)

  dpn t ≡ ln pn t /p1 t − ln pn t−1 /p1 t−1 = lnpn t − lnp1 t − lnpn t−1 + lnp1 t−1 . (72)

41 Much

of the literature on estimating CES unit cost functions deals with the application of this functional form in the consumer context when aggregating over similar products; e.g., see Broda and Weinstein [8], Bernard et al. [4] and Gábór-Toth and Vermeulen [56]. Almost all of the estimating equations discussed in this section can be applied to the consumer context; i.e., replace the period t output level yt by the period t utility level ut and interpret xt as a vector of cost minimizing consumer demands. Estimating equations which involve yt cannot be used in the consumer context since the utility level ut is not observable. 42 This equation cannot be estimated in the consumer context because unit cost ct is not observable.

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The double differenced counterparts to Eq. (69) are now the following equations43 : dxn t = (r − 1) dpn t + en t − e1 t − en t−1 + e1 t−1 ;

n = 2, . . . , N; t = 2, . . . , T (73)

where r − 1 = −σ . There are (N − 1)(T − 1) estimating equations in the system of equations defined by Eq. (73) and only one economic parameter to estimate, namely, −σ = r − 1. Note that the only exogenous variables in Eqs. (69), (70), and (73) are input prices. Thus, to prevent biased estimates, it is important that these prices be measured with minimal measurement error. There is a problem with the systems of estimating equations defined by Eqs. (69) and (73) and that is that these equations are dependent on the choice of the numeraire input, which in the above algebra is input 1. Looking at the estimating equations, it is evident that it is probably best to choose the numeraire commodity as one where the original error terms, the en t , have means close to 0 and small variances. In practice, it may be difficult to choose the “best” numeraire commodity.44 There is a way to avoid asymmetry in the estimating equations and that is to shift from estimating systems of input demand functions to estimating systems of share equations.  From Eq. (68), it can be seen that sn (y, p) ≡ pn xn (y, p)/C(y, p) = αn pn r / i = 1 N αi pi r  for n = 1, . . . , N. Define the nth input share of cost in period t as sn t ≡ pn t xn t / i = 1 N pi t xi t for n = 1, . . . , N and t = 1, . . . , T. Adding error terms to the above cost share equations leads to the following nonlinear system of estimating equations:

43 The double differencing methodology originated in Feenstra ([49], 163). Equation (73) can be converted into double differenced log input shares equal to a constant times double differenced log input prices plus error terms; see Broda and Weinstein ([7], 564, [8], 714) and Gábór-Toth and Vermeulen [56] and Eq. (75) below for these share equations. The present analysis follows the material in Diewert and Feenstra ([32], 14). Diewert and Feenstra ([32], 76–79) worked out the analogous estimating equations for a CES direct aggregator function where double differenced log shares were equal to double differenced log quantities plus error terms. A potential cost of the double differencing technique is that the variance of the error terms in the system of estimating Eq. (73) can be much larger than the variances in the system of equations defined by Eq. (69) or in a system that just used xn t or lnxn t as the dependent variable for input n in period t. However, the standard error for σ when the very simple estimating system of equations defined by Eq. (76) used by Diewert and Feenstra was very small. 44 Here is a possible strategy for choosing the numeraire input. Take logs of both sides of Eq. (68) and add the error term (with 0 mean) en t to equation n for period t. Run a preliminarysystems nonlinear regression in order to obtain estimates for the variance-covariance matrix of the vector of errors, [e1 t , . . . , eN t ] which is assumed to be distributed independently over time. Use  the estimated matrix, ∗ say, to solve the convex programming problem,  variance-covariance minw {wT ∗ w : wT w = 1}. The ∗ solution to this problem will be a normalized eigenvector that corresponds to the smallest eigenvalue of ∗ . Make a furthernormalization of w* so that the resulting vector, w** , satisfies the constraint w∗ ∗ T 1N = 1. If ∗ is a diagonal matrix, this methodology will pick the numeraire input to be the input which has the smallest error variance.

3 Duality in Production

  r  sn = αn pn t /

91 N

t

i=1

  t r αi pi + en t ;

n = 1, . . . , N; t = 1, . . . , T. (74)

 N t If we sum Eq. (74) over n for a fixed t, we find that n = 1 en = 0 for t = 1, . . . , T. Thus, within each time period, the errors cannot be distributed independently. Thus to prevent exact collinearity, one of the N estimating equations must be dropped. Furthermore, it can be seen that not all of the α n parameters can be identified. Thus, we require a normalization on the αn such as i = 1 N αi = 1 or α1 = 1. Alternatively, Eq. (70) can be added to the (N − 1)T independent estimating equations in (74) as additional estimating equations which will enable all of the αn to be identified. An alternative stochastic specification can  be obtained if we take logarithms of both sides of the equations sn t = [αn (pn t )r / i = 1 N αi (pi t )r ] and add error terms en t* to the resulting equations. Choose input 1 as a numeraire input and consider the following estimating equations:     ln sn t /s1 t = lnαn − lnα1 + rln pn t /p1 t + en t∗ − e1 t∗ ;

n = 2, . . . , N; t = 1, . . . , T.

(75)

If the focus is on estimating the elasticity of substitution, σ = 1 − r, then Eq. (75) can be differenced with respect to time and we obtain the following system of estimating equations: dsn t = rdpn t + en t∗ − e1 t∗ − en t−1∗ + e1 t−1∗ ;

n = 2, . . . , N; t = 2, . . . , T (76)

where the double differenced log price dpn t is defined by Eq. (72) and the double differenced log share dsn t is defined as lnsn t − lns1 t − lnsn t − 1 + lns1 t − 1 . Note that dpn t appears as an exogenous variable on the right hand sides of equation n,t in (73) and (76). We conclude this section by considering the estimation of a system of CES inverse demand functions; that is, we assume that prices are the endogenous variables and output and input quantities are the exogenous variables. Thus, the input prices are regarded as the prices that rationalize the observed choice of inputs, assuming that the CES production function is the “true” production function.45 This may seem to be an odd thing to do but it can turn out that estimating the CES system

45 This was the methodological approach taken by Arrow et al. [2] in their pioneering study on the estimation of CES functional forms. If the CES unit cost function model fits the observed data perfectly, then it will turn out that estimating the direct CES production function using a system of inverse demand functions will also fit the data perfectly.

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of inverse demand functions can lead to a much better fitting model than estimating the CES system of direct input demand functions as was done above.46 Let y > 0 and p  0N and the technology can be described by the CES production  function defined by Eq. (61); that is, f(x1 , . . . , xN ) ≡ [ n = 1 N βn xn s ]1/s where s < 1, s = 0 and βn > 0 for n = 1, . . . , N. Then the producer’s cost minimization problem is equivalent to the following constrained maximization problem: minx



N

n=1

pn xn :



N

n=1

 βn xn s = ys ; x ≥ 0N .

(77)

The first order necessary (and sufficient) conditions for solving Eq. (77) are equivalent to the following conditions: pn = λβn xn s−1 ; ys =

 n=1

N

n = 1, . . . , N;

βn xn s .

(78) (79)

Multiply both sides of equation n in Eq. (78) by xn and sum the resulting equations. We obtain the following equation:  n=1

N

 pn xn = λ

N n=1

βn xn s

(80)

= λys where the secondequation follows using Eq. (79). Use the second equation in (80) to solve for λ = n = 1 N pn xn /ys and substitute this equation back into Eq. (78). The resulting equations evaluated at the period t data are Eq. (81) below. As usual, the period t data are xt ≡ [x1 t , . . . , xN t ], yt and pt ≡ [p1 t , . . . , pN t ] for t = 1, . . . , T. T We obtain the following equations: pn t /

 n=1

N

 pn t xn t

 s−1  t s = βn xn t / y ;

n = 1, . . . , N; t = 1, . . . , T. (81)

Take logarithms of both sides of Eq. (81) and add the error term en t to the resulting equations.47 We obtain the following system of estimating equations:

46 This

was the case in the empirical study of CES estimation undertaken by Diewert and Feenstra [32]. 47 These error terms are different from the error terms defined previously.

3 Duality in Production

  t ln pn /

n=1

93 N

 t

pn xn

= ln βn + (s − 1) ln xn t − slnyt + en t ;

t

(82)

n = 1, . . . , N; t = 1, . . . , T. Choose input 1 as the numeraire input and form the differenced Eq. (83):     ln pn t /p1 t = lnβn − lnβ1 + (s − 1) ln xn t /x1 t + en t − e1 t ;

n = 2, 3, . . . , N; t = 1, . . . , T.

(83)

Not all of the parameters βn can be identified using the (N − 1)T equations in (83). In order to identify all of the βn , we could make y an endogenous variable  that is explained by the exogenous xn , using the production function, y = [ n = 1 N βn xn s ]1/s . Thus, we could add the following estimating Eq. (84) to Eq. (83): lnyt = (1/s) ln



N

n=1

  s + e0 t ; βn xn t

t = 1, . . . , T.

(84)

If the focus is on estimating the elasticity of substitution, then we can time difference Eq. (83) and obtain the following estimating equations: dpn t = (s − 1) dxn t + en t − e1 t − en t−1 + e1 t−1 ;

n = 2, . . . , N; t = 2, . . . , T (85)

where the double log differenced variables dxn t and dpn t are defined by Eqs. (71) and (72). Recall the r which appeared in the CES cost function. The elasticity of substitution that corresponds to r is σ = 1 − r. The s which appears in Eq. (85) corresponds to r = s/(s − 1). Thus s − 1 = − σ−1 . Our previous system of estimating Eq. (73) for r can be written as dxn t = (r − 1)dpn t = − σdpn t , where we have omitted the error terms. Our new system of estimating equations for s, Eq. (85), can be written as dpn t = (s − 1)dxn t = − σ−1 dxn t where we have again omitted the error terms. Thus, if either CES model fits the data perfectly, then the other model will fit the data perfectly and the two estimates for σ will be identical. Note that the two systems of estimating equations both have (N − 1)(T − 1) degrees of freedom and only one (nonvariance) parameter, σ, to estimate. It is useful to obtain a different system of estimating equations. Recall the first order condition Eq. (79) above. If we evaluate these equations using the period t data, we obtain the following equations which will hold if there are no errors in the CES cost minimization model:  t s  y =

N n=1

 s βn xn t

t = 1, . . . , T.

(86)

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W. E. Diewert

Recall our earlier first order condition Eq. (81). Multiply equation n,t by xn t and we obtain the following system of equations after adding error terms, en t : sn t ≡ pn t xn t /



N

i=1

 pi t xi t + en t

n = 1, . . . , N; t = 1, . . . T

 s  s = βn xn t / yt + en t using equations (81)   s N  t s = βn xn t / βi xi + en t ; using equations (86).

(87)

i=1

 If we sum Eq. (87) over n for a fixed t, we find that n = 1 N en t = 0 for t = 1, . . . , T Thus, within each time period, the errors cannot be distributed independently. To prevent exact collinearity, one of the N estimating equations must be dropped. Furthermore, it can be seen that not all of theβn parameters can be identified. Thus we require a normalization on the βn such as i = 1 N βi = 1 or β1 = 1. Alternatively, Eq. (84) can be added to the (N − 1)T independent estimating equations in (87) as additional estimating equations which will enable all of the βn to be identified. Note that the dependent variables in Eq. (87) are exactly the same as the dependent variables in our earlier nonlinear system of share estimating equations, Eq. (74). In Eq. (87), input quantities xn t are the explanatory variables, whereas in Eq. (74), input prices pn t were the explanatory variables. In actual empirical applications of the CES model, the fit in the two systems can differ enormously.48 This explains why we developed the algebra for the estimation of either system. An alternative stochastic specification can  be obtained if we take logarithms of both sides of the equations sn t = [βn (pn t )r / i = 1 N βi (pi t )r ] and add error terms en t* to the resulting equations. Choose input 1 as a numeraire input and consider the following estimating equations:     ln sn t /s1 t = ln βn − ln β1 + s ln xn t /x1 t + en t∗ − e1 t∗ ;

n = 2, . . . , N; t = 1, . . . , T.

(88)

If the focus is on estimating the elasticity of substitution, σ = 1/(1 − s), then Eq. (88) can be differenced with respect to time and we obtain the following system of estimating equations: dsn t = sdxn t + en t∗ − e1 t∗ − en t−1∗ − e1 t−1∗ ;

n = 2, . . . , N; t = 2, . . . , T (89)

where the double differenced log input quantity dxn t is defined by Eq. (73) and the double differenced log share dsn t is defined as lnsn t − lns1 t − lnsn t−1 + lns1 t−1 .

48 See Diewert and Feenstra [32]. The system (87) fit their data much better than the corresponding system (74).

3 Duality in Production

95

Note that dxn t appears as an exogenous variable on the right hand sides of equation n,t in (85) and (89).

Flexible Functional Forms for Cost Functions: The Generalized Leontief Functional Form From the previous section, it can be seen that the CES functional form is not suitable for economic applications where elasticities of substitution are allowed to be different between different pairs of inputs. This leads us to define formally the concept of a flexible functional form. We will define this concept first for a unit cost function c(p) and then for a general cost function C(y,p). Let c* (p) be an arbitrary unit cost function that satisfies the appropriate regularity conditions on unit cost functions and in addition, is twice continuously differentiable around a point p∗  0N . Then we say that a unit cost function c(p) that is also twice continuously differentiable around the point p* is flexible if it has enough free parameters so that the following 1 + N + N2 equations can be satisfied49 :     c p∗ = c∗ p∗ ;

(90)

    ∇c p∗ = ∇c∗ p∗ ;

(91)

    ∇ 2 c p∗ = ∇ 2 c∗ p∗ .

(92)

Thus c(p) is a flexible functional form if it has enough free parameters to provide a second order Taylor series approximation to an arbitrary unit cost function. At first glance, it looks like c(p) will have to have at least 1 + N + N2 independent parameters in order to be able to satisfy all of the Eq. (90), (91), and (92). However, since both c and c* are assumed to be twice continuously differentiable, Young’s Theorem in calculus implies that ∂ 2 c(p∗ )/∂pi ∂pk = ∂ 2 c(p∗ )/∂pk ∂pi for all i = k (and of course, the same equations hold for the second order partial derivatives of c* (p) when evaluated at p = p∗ ). Thus, the N2 equations in (92) can be replaced with the following N(N + 1)/2 equations:     ∂ 2 c p∗ /∂pi ∂pk = ∂ 2 c∗ p∗ /∂pi ∂pk

for 1 ≤ i ≤ k ≤ N.

(93)

Another property that both unit cost functions must have is homogeneity of degree one in the components of p. By part 1 of Euler’s Theorem on homogeneous functions, c and c* satisfy the following equations:

49 Diewert [20] introduced

the concept of a flexible functional form. The actual term “flexible” was introduced in Diewert ([22], 133).

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W. E. Diewert

        c p∗ = p∗T ∇c p∗ and c∗ p∗ = p∗T ∇c∗ p∗ .

(94)

Thus, if c and c* satisfy Eq. (91), then using Eq. (94), we see that c and c* automatically satisfy Eq. (90). By part 2 of Euler’s Theorem on homogeneous functions, c and c* satisfy the following equations:     ∇ 2 c p∗ p∗ = 0N and ∇ 2 c∗ p∗ p∗ = 0N .

(95)

This means that if we have ∂ 2 c(p∗ )/∂pi ∂pk = ∂ 2 c∗ (p∗ )/∂pi ∂pk for all i = k, then Eq. (95) will imply that ∂ 2 c(p∗ )/∂pi ∂pi = ∂ 2 c∗ (p∗ )/∂pi ∂pi as well, for i = 1, . . . , N. Summarizing the above material, if c(p) is linearly homogeneous, then in order for it to be flexible, c(p) needs to have only enough parameters so that the N equations in (91) can be satisfied and so that the following N(N − 1)/2 equations can be satisfied:     ∂ 2 c p∗ /∂pj ∂pk = ∂ 2 c∗ p∗ /∂pi ∂pk ≡ cik ∗ ;

for 1 ≤ i ≤ k ≤ N.

(96)

Thus, in order to be flexible, c(p) must have at least N+N(N−1)/2 = N(N+1)/2 independent parameters. Recall that the generalized Leontief cost function was introduced in section “The Derivative Property of the Cost Function.” The unit cost function that corresponds to this function form is defined as follows50 : c (p) ≡

N

 i=1

N k=1

bik pi 1/2 pk 1/2

(97)

where bik = bki for all i and k. Note that there are exactly N(N + 1)/2 independent bik parameters in the c(p) defined by Eq. (97). For this functional form, the N equations in (91) become:    ∂c p∗ /∂pn =

N

k=1

 1/2   bnk pk ∗ /pn ∗ = ∂c∗ p∗ /∂pn ≡ cn ∗ ;

n = 1, . . . , N. (98)

The N(N − 1)/2 equations in (96) become:     ∗ ∗ 1/2 1 = cik ∗ ; 2 bik / pi pk

1 ≤ i < k ≤ N.

(99)

However, it is easy to solve Eq. (99) for the bik :  1/2 ; 1 ≤ i < k ≤ N. bik = 2cik ∗ pi ∗ pk ∗

50 We

no longer restrict the bij to be nonnegative.

(100)

3 Duality in Production

97

Once the bik for i < k have been determined using Eq. (100), we set bki = bik for i < k and finally the bii are determined using the N equations in (98). The above material shows how we can find a flexible functional form for a unit cost function.51 We now turn our attention to finding a flexible functional form for a general cost function C(y,p). Let C* (y,p) be an arbitrary cost function that satisfies the appropriate regularity conditions on cost functions listed in Theorem 1 above and in addition is twice continuously differentiable around a point (y* ,p* ) where y∗ > 0 and p∗  0N . Then we say that a given cost function C(y,p) that is also twice continuously differentiable around the point (y* ,p* ) is flexible if it has enough free parameters so that the following 1 + (N + 1) + (N + 1)2 equations can be satisfied:     C y∗ , p∗ = C∗ y∗ , p∗ ;

(1 equation)

(101)

    ∇p C y∗ , p∗ = ∇p C∗ y∗ , p∗ ;

(N equation)

(102)

    ∇ 2 pp C y∗ , p∗ = ∇ 2 pp C∗ y∗ , p∗ ;     ∇y C y∗ , p∗ = ∇y C∗ y∗ , p∗ ;

  N2 equation (1 equation)

(103) (104)

    ∇ 2 py C y∗ , p∗ = ∇ 2 py C∗ y∗ , p∗ ;

(N equation)

(105)

    ∇ 2 yp C y∗ , p∗ = ∇ 2 yp C∗ y∗ , p∗ ;

(N equation)

(106)

    ∇ 2 yy C y∗ , p∗ = ∇ 2 yy C∗ y∗ , p∗ ;

(1 equation) .

(107)

Equations (101), (102), (103), (104), (105), (106), and (107) are the counterparts to our earlier unit cost Eqs. (90), (91), and (92). As was the case with unit cost functions, Eq. (102) is implied by the linear homogeneity in prices of the cost functions and Part 1 of Euler’s Theorem on homogeneous functions. Young’s Theorem on the symmetry of cross partial derivatives means that the lower triangle of equations in (103) is implied by the equalities in the upper triangle of both matrices of partial derivatives. Part 2 of Euler’s Theorem on homogeneous functions implies that if all the off diagonal elements in both matrices in Eq. (103) are equal, then so are the diagonal elements. Hence, in order to satisfy all of the equations in (101), (102), and (103), we need only satisfy the N equations in (102) and the

51 This

material can be adapted to the case where we want a flexible functional form for a linearly homogeneous utility or production function f(x): just replace p by x and c(p) by f(x).

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N(N − 1)/2 equations in the upper triangle of the N2 equations in (103). Young’s Theorem implies that if Eq. (105) are satisfied, then so are Eq. (106). However, Euler’s Theorem on homogeneous functions implies that         ∂C u∗ , p∗ /∂y = p∗T ∇ 2 py C∗ y∗ , p∗ = p∗T ∇ 2 py C∗ u∗ , p∗ = ∂C∗ y∗ , p∗ /∂y. (108) Hence, if Eq. (105) are satisfied, then so is the single Eq. (104). Putting this all together, we see that in order for C to be flexible, we need enough free parameters in C so that the following equations can be satisfied: • • • •

Equation (102); N equations The upper triangle in Eq. (103); N(N − 1)/2 equations Equation (105); N equations Equation (107); 1 equation

Hence, in order for C to be a flexible functional form, it will require a minimum of 2N + N(N − 1)/2 + 1 = N(N + 1)/2 + N + 1 parameters. Thus, a fully flexible cost function, C(y,p), will require N + 1 additional parameters compared to a flexible unit cost function, c(p). In the following Sections, we will define several flexible functional forms for unit cost functions c(p). Once we have a flexible functional form for a unit cost function c(p), then the algebra below shows how we can modify c(p) to obtain a flexible total cost function C(y,p).52 Suppose the unit cost function is the generalized Leontief unit cost function c(p) defined by Eq. (97) above. We now show how terms can be added to it in order to make it a fully flexible cost function. Thus, define C(u,p) as follows: C (y, p) ≡ yc (p) + bT p + (1/2) a0 αT py2

(109)

where b ≡ [b1 , . . . , bN ] is an N dimensional vector of new parameters, a0 is a new parameter and α ≡ [α1 , . . . , αN ] > 0N is a vector of predetermined parameters.53 Using Eq. (109) as our candidate for a flexible (total) cost function C, Eqs. (102), (103), (105), and (107) become:

52 The

    y∗ ∇p c p∗ + b + (1/2) a0 αy∗2 = ∇p C∗ y∗ , p∗ ;

(110)

    y∗ ∇ 2 pp c p∗ = ∇ 2 pp C∗ y∗ , p∗ ;

(111)

algebra for converting the translog unit cost function into the translog cost function is different. 53 We have defined the cost function C in this manner so that it has the minimal number of parameters required in order to be a flexible functional form. Thus it is a parsimonious flexible functional form.

3 Duality in Production

99

    ∇p c p∗ + a0 αy∗ = ∇ 2 py C∗ y∗ , p∗ ;

(112)

  a0 αT p∗ = ∇ 2 yy C∗ y∗ , p∗ .

(113)

Use Eq. (111) in order to determine the bik for i = k. Use Eq. (113) in order to determine the single parameter a0 . Use Eq. (112) in order to determine the bii . Finally, use Eq. (110) in order to determine the parameters bn in the b vector. Thus, the cost function C(u,p) defined by Eq. (109), which uses the generalized Leontief unit cost function c(p) defined by Eq. (97) as a building block, is a parsimonious flexible functional form for a general cost function. In fact, it is not necessary to use the generalized Leontief unit cost function in definition (109) in order to convert a flexible functional form for a unit cost function into a flexible functional form for a general cost function. Let c(p) be any flexible functional form for a unit cost function and define C(y,p) by Eq. (109). Use Eq. (113) to determine the parameter a0 . Once a0 has been determined, Eqs. (111) and (112) can be used to determine the parameters in the unit cost function c(p). Finally, Eq. (110) can be used to determine the parameters in the vector b. Differentiating Eq. (109) leads to the following system of estimating equations, where x(y, p) = ∇ p C(y, p) is the producer’s system of cost minimizing input demand functions: x (y, p) = y∇c (p) + b + (1/2) a0 αy2 .

(114)

If the generalized Leontief unit cost function is used as the c(p) in Eq. (114), then the N estimating equations will be linear in the unknown parameters. This will facilitate econometric estimation. The cross equation symmetry restrictions could be tested or imposed. In empirical applications, if we use the generalized Leontief functional form when there are more than two inputs, a problem can occur: one or more of the estimated bik can turn out to be negative numbers (so that inputs i and k are complements). Under these conditions, the estimated cost function can fail to be concave at the observed data points and it will not be globally concave over all positive input prices. Global concavity can be imposed by replacing the off diagonal bik parameters by their squares54 but if this is done, then all pairs of inputs will be either substitutes or be unrelated. Global concavity can be imposed but at the cost of destroying the flexibility of the functional form.55 Thus, the generalized Leontief functional form is not a “perfect” flexible functional form. Finding flexible functional forms where the restrictions implied by microeconomic theory can be

54 The

resulting estimating equations become nonlinear in the parameters when we square the bik . Typically, this does not create any problems: just use a nonlinear estimation method. 55 If there are more than 4 inputs and we allow for complementarity, then experience has shown that complementary input pairs show up almost always.

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W. E. Diewert

imposed on the functional form without destroying its flexibility is a nontrivial task which we will address later in sections “The Normalized Quadratic Unit Cost Function” and “The Konüs Byushgens Fisher Unit Cost Function” below.

The Translog Functional Form The translog unit cost function, c(p), is defined as follows56 : lnc (p) ≡ α0 +



N i=1

αi lnpi + (1/2)

 i=1

N

N k=1

γik lnpi lnpk

(115)

where the parameters αi and γik satisfy the following restrictions: γik = γki ;

1 ≤ i < k ≤ N; 

N

i=1



N

k=1

(N (N − 1) /2symmetry restrictions) (116)

αi = 1;

γik = 0;

(1 restriction)

i = 1, . . . , N

(N restrictions) .

(117)

(118)

Note that the symmetry restrictions (116) and the restrictions (118) imply the following restrictions: 

N i=1

γik = 0;

k = 1, . . . , N.

(119)

There are 1 + N αi parameters and N2 γik parameters. However, the restrictions (116), (117), (118), and (119) mean that there are only N independent αi parameters and N(N − 1)/2 independent γik parameters, which is the minimal number of parameters required for a unit cost function to be flexible. We show that the translog unit cost function c(p) defined by Eqs. (115), (116), (117), and (118) is linearly homogeneous; that is, we need to show that c(λp) = λc(p) for λ > 0 and p  0N . Thus, we need to show that   lnc (λp) = ln λc (p) = lnλ + lnc (p) ;

λ > 0 and p  0N .

(120)

56 This functional form is due to Christensen et al. [14–16]. The material in this Section is due to these authors.

3 Duality in Production

101

Using definition (115), we have  N    lnc λp1 , . . . , λpN = α0 + αi lnλpi + (1/2) = α0 +



N i=1

+ (1/2) = α0 +

i=1

+ (1/2)

N

i=1 N i=1

αi [lnλ] +



N

i=1





+ (1/2) + (1/2) + (1/2)

+ (1/2) + (1/2) + (1/2)

+ (1/2) + (1/2)

N

N

i=1 i=1



N

i=1



N



k=1

 i=1



N i=1



N

k=1

N

k=1



 γik

N

γik





N

i=1

k=1



N i=1

γik [lnλ] [lnλ]



i=1

N

k=1

 γik [lnλ] [lnλ]

  lnpi [lnλ]

   γik lnpi lnpk  i=1

N

[0] [lnλ] [lnλ]

   [0] lnpk [lnλ] + (1/2) N

N

k=1

  lnpk [lnλ]

αi lnpi + (1/2)

N

= lnλ + lnc (p)

   γik lnpi lnpk

N

N

i=1

N

k=1

i=1



  γik lnpi [lnλ]

i=1

N



   γik lnλ + lnpi lnλ + lnpk using(117)

αi lnpi + (1/2)

N



   γik lnλ + lnpi lnλ + lnpk

N

k=1 N

αi lnpi

  γik [lnλ] lnpk

k=1



i=1

N

k=1

N

i=1

N

αi lnpi + (1/2)

N



= lnλ + α0 +

γik lnλpi lnλpk

αi lnpi

k=1 N i=1



= lnλ + α0 +

N

N



= lnλ + α0 +

k=1

   γik lnλ + lnpi lnλ + lnpk



k=1 i=1

i=1

= lnλ + α0 +

N

k=1

= α0 + 1 [lnλ] + + (1/2)

i=1

N

  αi lnλ + lnpi





N

   γik lnpi lnpk

αi lnpi + (1/2)

 i=1

N i=1

  [0] lnpi [lnλ]

using (118)and(119) N k=1

N

   γij lnpi lnpk

using definition(115) (121)

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W. E. Diewert

which establishes the linear homogeneity property (120). Thus the restrictions (116), (117), and (118) imply the linear homogeneity of the translog unit cost function. To establish the flexibility of the translog unit cost function c(p) defined by Eqs. (115), (116), (117), and (118), we need only solve the following system of equations, which is equivalent to the N(N + 1)/2 equations defined by Eqs. (91) and (93):   lnc (p) = lnc∗ p∗ ;     ∂lnc p∗ /∂lnpi = ∂lnc∗ p∗ /∂lnpi ;

1 equation

(122)

i = 1, 2, . . . , N − 1; N − 1 equations (123)

    ∂ 2 lnc p∗ /∂lnpi ∂lnpk = ∂ 2 lnc∗ p∗ /∂lnpi ∂lnpk ; 1 ≤ i < k ≤ N; N (N − 1) /2 equations.

(124)

Upon differentiating the translog unit cost function defined by Eq. (115), we see that Eq. (123) are equivalent to the following equations: αi +



N

k=1

  γik lnpj = ∂lnc∗ p∗ /∂lnpi ;

i = 1, 2, . . . , N − 1.

(125)

Differentiating the translog unit cost function again, we find that Eq. (124) are equivalent to the following equations:   γik = ∂ 2 lnc∗ p∗ /∂lnpi ∂lnpk ;

1 ≤ i < j ≤ N.

(126)

Now use Eq. (126) to determine the γik for 1 ≤ i < k ≤ N. Use the symmetry restrictions (116) to determine the γik for 1 ≤ k < i ≤ N. Use Eq. (118) to determine the γii for i = 1, 2, . . . , N. With the entire N by N matrix of the γij now determined, use Eq. (125) in order to determine the αi for i = 1, 2, . . . , N − 1. Now use Eq. (117) to determine αN . Finally, use Eq. (112) to determine α0 . We turn our attention to the problems involved in obtaining estimates for the unknown parameters αi and γik , which occur in the definition of the translog unit cost function, c(p) defined by Eq. (115). The total cost function C(y,p) is defined in terms of the unit cost function c(p) as follows: C (y, p) ≡ yc (p) .

(127)

Taking logarithms on both sides of Eq. (127) yields, after some rearrangement:   ln C (y, p) /y = lnc (p)  = α0 +

i=1

N

αi lnpi + (1/2)



N i=1

N k=1

γik lnpi lnpk (128)

3 Duality in Production

103

where we have replaced lnc(p) using Eq. (115). The corresponding system of cost minimizing input demand functions x(y,p) is obtained using Shephard’s Lemma: x (y, p) ≡ ∇p C (y, p) = y∇p c (p) .

(129)

Suppose that we have data for a production unit on output in period t, yt , inputs xt ≡ [x1 t , . . . , xN t ] and input prices pt ≡ [p1 t , . . . , pN t ] for t = 1, . . . , T. Thus, the period t observed unit cost is: ct ≡ ptT xt /yt ≡



N

i=1

pi t xi t /yt ;

t = 1, . . . , T.

(130)

Evaluate Eq. (128) at the period t data and add an error term, e0 t . Using Eq. (130), (128) evaluated at the period t data becomes the following estimating equation: lnct = α0 +



N

i=1

+ (1/2)

αi lnpi t



N

i=1

N k=1

(131) γij lnpi lnpk + e0 ; t

t

t

t = 1, . . . , T.

Note that Eq. (131) is linear in the unknown parameters. In order to obtain additional estimating equations, we have to use the input demand functions, xi (y, p) ≡ y∂c(p)/∂pi for i = 1, . . . , N (see Eq. (129) above). The ith input share function, si (y,p), is defined as: si (y, p) ≡ pi xi (y, p) /C (y, p)

i = 1, . . . , N

  = pi y∂c (p) /∂pi /C (y, p)   = pi y∂c (p) /∂pi /yc (p)   = pi ∂c (p) /∂pi /c (p)

using(129) using(127) (132)

= ∂lnc (p) /∂lnpi = αi +

 k=1

N

γik lnpk

where the last equation follows upon differentiating the c(p) defined by Eq. (115). Now evaluate both sides of Eq. (132) at the period t data and add error terms ei t to obtain the following system of estimating equations:

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W. E. Diewert

si t ≡ pi t xi t /Ct = αi +



N

j=1

γij lnpj t + ei t ;

i = 1, . . . , N; t = 1, . . . , T. (133)

Note that Eq. (133) are also linear in the unknown parameters.57 Obviously, the N estimating equations in (133) could be added to the single estimating Eq. (131) in order to obtain N + 1 estimating equations with cross equation equality constraints on the parameters αi and γij . However, since total cost in any period t, Ct , equals the sum of the individual expenditures on the inputs, i = 1 N pi t xi t , the observed input shares si t ≡ pi t xi t /Ct will satisfy the following constraint for each period t:  i=1

N t si

= 1;

t = 1, . . . , T.

(134)

Thus, the stochastic error terms ei t in Eq. (133) cannot all be independent. Hence we must drop one estimating equation from (133). Thus, Eq. (131) and any N − 1 of the N equations in (133) may be used as a system of estimating equations in order to determine the parameters of the translog unit cost function.58 We now turn our attention to the problem of deriving a formula for the price elasticities of demand, ∂xi (y, p)/∂pj , given that the unit cost function has the translog functional form defined by Eqs. (115), (116), (117), and (118). Recall Eq. (132) above. For k = i, differentiate the ith equation in (132) with respect to the log of pk and we obtain the following equations for all k = i: 

 ∂si (y, p) /∂lnpk = pi ∂ xi (y, p) /C (y, p) /∂lnpk = γik .

(135)

Hence upon noting that si (y, p) = pi xi (y, p)/C(y, p) and using Eq. (135), we have for k = i:

also that the cross equation symmetry conditions, γik = γki , could be tested or imposed. situations where N is large relative to the number of observations T, maximum likelihood estimation of Eq. (131) and N − 1 of the Eq. (133) can fail if a general variance covariance matrix is estimated for the error terms in these equations. The problem is that all of the unknown economic parameters are contained in Eq. (131) and as a result, the estimated squared residuals in this equation will tend to be small relative to the estimated squared residuals in Eq. (133), where each equation has only a few unknown economic parameters. Hence Eq. (131) can suffer from multicollinearity problems and the small apparent variance of the residuals in this equation can lead to the maximum likelihood estimation procedure giving too much weight to the unit cost function equation relative to the other equations. Under these conditions, the resulting elasticities may be erratic and they may not satisfy the appropriate curvature conditions. Note that the estimation of the Generalized Leontief unit cost function did not suffer from this problem of having every unknown parameter in a single equation. 57 Note 58 In

3 Duality in Production

105

  γik = pi ∂ xi (y, p) /C (y, p) /∂lnpk   = pi pk ∂ xi (y, p) /C (y, p) /∂pk      2  = pi pk 1/C (y, p) ∂xi (y, p) /∂pk − xi (y, p) 1/C (y, p) ∂C (y, p) /∂pk 

 = pi xi (y, p) /C (y, p) ∂lnxi (y, p) /∂lnpk    − pi xi (y, p) /C (y, p) pk xk (y, p) /C (y, p) using Shephard s Lemma, xk (y, p) = ∂C (y, p) /∂pk

= si (y, p) ∂lnxi (y, p) /∂lnpk − si (y, p) sk (y, p) . (136) Equation (136) can be rearranged to give us the following formula for the cross price elasticities of input demand for all i = k:  −1 ∂lnxi (y, p) /∂lnpk = si (y, p) γik + sk (y, p) .

(137)

Now differentiate the ith equation in (135) with respect to the logarithm of pi and get the following equations:   γii = pi ∂ pi xi (y, p) /C (y, p) /∂pi ; i = 1, . . . , N;      = pi xi (y, p) /C (y, p) + pi /C (y, p) ∂xi (y, p) /∂pi

  − pi xi (y, p) /C(y, p)2 ∂C (y, p) /∂pi      = pi xi (y, p) /C (y, p) + pi /C (y, p) ∂xi (y, p) /∂pi

  − pi xi (y, p) /C(y, p)2 xi (y, p)

(138)

using Shephard s Lemma, xi (y, p) = ∂C (y, p) /∂pi    = pi xi (y, p) /C (y, p) + pi xi (y, p) /C (y, p) ∂lnxi (y, p) /∂lnpi  2 − pi xi (y, p) /C (y, p)   = si (y, p) + si (y, p) ∂lnxi (y, p) /∂lnpi − si (y, p)2 . Equation (138) can be rearranged to give us the following formula for the own price elasticities of input demand:  −1 ∂lnxi (y, p) /∂lnpi = si (y, p) γii + si (y, p) − 1; i = 1, . . . , N.

(139)

Thus, given econometric estimates for the αi and γij , which we denote by αi ∗ and γij , the estimated or fitted shares in period t, si t* are defined using these estimates ∗

106

W. E. Diewert

and Eq. (133) evaluated at the period t data: si t∗ ≡ αi ∗ +



N j=1

γij ∗ lnpj t ; i = 1, . . . , N; t = 1, . . . , T.

(140)

Now use Eq. (137) evaluated at the period t data and econometric estimates to obtain the following formula for the period t cross elasticities of demand, Eik t :    −1 Eik t ≡ ∂lnxi yt , pt /∂lnpk = si t∗ γik ∗ + sk t∗ ; i = k; t = 1, . . . , T.

(141)

Similarly, use Eq. (139) evaluated at the period t data and econometric estimates to obtain the following formula for the period t own elasticities of demand, Eii t :    −1 Eii t ≡ ∂lnxi yt , pt /∂lnpi = si t∗ γii ∗ + si t∗ − 1; i = 1, . . . , N; t = 1, . . . , T. (142) We can also obtain an estimated or fitted period t unit cost, ct* , by using our econometric estimates for the parameters and by exponentiating the right hand side of equation t in (130): 

∗

c ≡ exp α0 + t∗

 i=1

N

∗

αi lnpi + (1/2) t



N i=1

k=1

N

∗



γik lnpi lnpk t

t

;

t = 1, . . . , T. (143) Finally, our fitted period t shares si t* defined by Eq. (50) and our fitted period t costs Ct* defined by Eq. (53) can be used in order to obtain estimated or fitted period t input demands, xi t* , as follows: xi t∗ ≡ yt ct∗ si t∗ /pi t ; i = 1, . . . , N; t = 1, . . . .T.

(144)

Given the matrix of period t estimated input price elasticities of demand, [Eik t ], we can readily calculate the matrix of period t estimated input price derivatives, ∇ p x(yt , pt ) = ∇ 2 pp C(yt , pt ) = yt ∇ 2 pp c(pt ). The estimate for element ik of ∇ 2 pp C(yt , pt ) is: Cik t∗ ≡ Eik t xi t∗ /pk t ; i, k = 1, . . . , N; t = 1, . . . , T

(145)

where the estimated period t elasticities Eik t are defined by Eqs. (141) and (142) and the fitted period t input demands xi t* are defined by Eq. (144). Once the estimated input price derivative matrices [Cik t* ] have been calculated for period t, then we may check whether it is negative semidefinite using determinantal conditions or by checking if all of the eigenvalues of each matrix are zero or negative for t = 1, . . . , T. Unfortunately, very frequently these negative semidefiniteness conditions will fail to

3 Duality in Production

107

be satisfied for both the translog and generalized Leontief functional forms. Thus, the translog and generalized Leontief functional forms both suffer from the same problem: in general, it is not possible to impose concavity on these functional forms without destroying their flexibility property. Hence, in the following two sections, we study functional forms where these curvature conditions can be imposed without destroying the flexibility of the functional form.

The Normalized Quadratic Unit Cost Function The normalized quadratic unit cost function c(p) is defined as follows for p  0N 59 : c (p) ≡ bT p + (1/2) pT Bp/αT p

(146)

where bT ≡ [b1 , . . . , bN ] and αT ≡ [α1 , . . . , αN ] are parameter vectors and B ≡ [bik ] is a matrix of parameters. The vector α and the matrix B satisfy the following restrictions: α > 0N ;

(147)

B = BT ; i.e., the matrix B is symmetric;

(148)

Bp∗ = 0N for some p∗  0N .

(149)

In most empirical applications, the vector of nonnegative but nonzero parameters α is fixed a priori. The two most frequent a priori choices for α are α ≡ 1N , a vector of ones or α ≡ (1/T) t = 1 T xt , the sample mean of the observed input vectors. The two most frequent choices for the reference price vector p* are p∗ ≡ 1N or p∗ ≡ pt for some period t; that is, in this second choice, we simply set p* equal to the observed period t price vector. Assuming that α has been predetermined, there are N unknown parameters in the b vector and N(N − 1)/2 unknown parameters in the B matrix, taking into account the symmetry restrictions (148) and the N linear restrictions in (149). Note that the c(p) defined by Eq (146) is linearly homogeneous in the components of the input price vector p. Another possible way of defining the normalized quadratic unit cost function is as follows:

59 This functional form was introduced by Diewert and Wales ([42], 53) where it was called the Symmetric Generalized McFadden functional form. It is a generalization of a functional form due to McFadden ([73], 279). Additional material on this functional form can be found in Diewert and Wales [43–45]).

108

W. E. Diewert

c (p) ≡ (1/2) pT Ap/αT p

(150)

where the parameter matrix A is symmetric; that is, A = AT ≡ [aik ] and α > 0N as before. Assuming that the vector of parameters α has been predetermined, the c(p) defined by Eq. (150) has N(N + 1)/2 unknown aik parameters. Comparing Eq. (146) with (150), it can be seen that Eq. (150) has dropped the b vector but has also dropped the N linear constraints (149). It can be shown that the model defined by Eq. (146) is a special case of the model defined by Eq. (150). To show this, given Eq. (146), define the matrix A in terms of B, b and α as follows:

A ≡ B + bαT + αbT .

(151)

Substituting Eq. (151) into (150), Eq. (150) becomes: 

 c (p) = (1/2) pT B + bαT + αbT p/αT p

= (1/2) pT Bp/αT p + (1/2) pT bαT + αbT p/αT p   = (1/2) pT Bp/αT p + (1/2) pT bαT p + pT αbT p /αT p   = (1/2) pT Bp/αT p + (1/2) 2pT bαT p /αT p

(152)

= (1/2) pT Bp/αT p + pT b which is the same functional form as (146). However, it is preferable to work with the model (146) rather than with the seemingly more general model (150) for three reasons: • The c(p) defined by Eq. (146) clearly contains the no substitution Leontief functional form as a special case (simply set B = 0N × N ) • The estimating equations that correspond to Eq. (146) will contain constant terms. • It is easier to establish the flexibility property for (146) than for (150). The first and second order partial derivatives of the normalized quadratic unit cost function defined by Eq. (146) are given by:  −1  −2 ∇p c (p) = b + αT p Bp − (1/2) αT p pT Bpα;

(153)

 −1  −2  −2  −3 ∇ 2 pp c (p) = αT p B − αT p BpαT − αT p αpT B + αT p pT BpααT . (154)

3 Duality in Production

109

We now prove that the c(p) defined by Eqs. (146), (147), (148), and (149) (with α predetermined) is a flexible functional form at the point p* . Using the restrictions (149), Bp∗ = 0N , we have p∗ T Bp∗ = p∗ T 0N = 0. Thus, evaluating Eqs. (153) and (154) at p = p∗ yields the following equations:   ∇p c p∗ = b;

(155)

−1    B. ∇ 2 pp c p∗ = αT p∗

(156)

We need to satisfy Eqs. (91) and (92) above to show that the c(p) defined by Eqs. (146), (147), (148), and (149) is flexible at p* . Using Eq. (155), we can satisfy Eq. (91) if we choose b as follows:   b ≡ ∇c∗ p∗ .

(157)

Using Eq. (156), we can satisfy Eq. (92) by choosing B as follows:  −1   B ≡ αT p∗ ∇ 2 c∗ p∗ .

(158)

Since ∇ 2 c∗ (p∗ ) is a symmetric matrix, B will also be a symmetric matrix and so the symmetry restrictions (148) will be satisfied for the B defined by Eq. (158). Moreover, since c* (p) is assumed to be a linearly homogeneous function, Euler’s Theorem implies that   ∇ 2 c∗ p∗ p∗ = 0N .

(159)

Equations (158) and (159) imply that the B defined by Eq. (158) satisfies the linear restrictions (149). This completes the proof of the flexibility property for the normalized quadratic unit cost function. It is convenient to define the vector of normalized input prices, vT ≡ [v1 , . . . , vN ] as follows:  −1 v ≡ pT α p.

(160)

The system of input demand functions x(y,p) that corresponds to the normalized quadratic unit cost function c(p) defined by Eq. (146) can be obtained using Shephard’s Lemma in the usual way: x (y, p) = y∇c (p) .

(161)

Using Eq. (161) and definition (146) evaluated at the period t data, we obtain the following system of estimating equations:

110

W. E. Diewert

xt /yt = b + Bvt − (1/2) vtT Bvt α + et ; t = 1, . . . , T

(162)

where xt is the observed period t input vector, yt is the period t output, vt ≡ pt /αT pt is the vector of period t normalized input prices and et ≡ [e1 t , . . . , eN t ]T is a vector of stochastic error terms. Equation (162) can be used in order to statistically estimate the parameters in the b vector and the B matrix. Note that Eq. (162) are linear in the unknown parameters. Note also that the symmetry restrictions (148) can be imposed when estimating the system of Eq. (162) or their validity can be tested. Once estimates for b and B have been obtained (denote these estimates by b* and * B respectively), then Eq. (162) can be used in order to generate a period t vector of fitted input demands, xt* say:

xt∗ ≡ yt b∗ + B∗ vt − (1/2) vtT B∗ vt α ; t = 1, . . . , T.

(163)

Equations (154) and (161) may be used in order to calculate the matrix of period t estimated input price derivatives, ∇ p x(yt , pt ) = ∇ 2 pp C(yt , pt ). The estimated matrix of second order partial derivatives ∇ 2 pp C(yt , pt ) for t = 1, . . . , T is the following one:  −1 −2 −2    t∗  t αT pt Cij ≡ y B∗ − αT pt B∗ pt αT − αT pt αptT B∗  −3  T t tT ∗ t T + α p p B p αα .

(164)

Equations (163) and (164) may be used in order to obtain estimates for the matrix of period t input demand price elasticities, [Eij t ]:   Eij t ≡ ∂lnxi yt , pt /∂lnpj = pj t Cij t∗ /xi t∗ ; i, j = 1, . . . , N; t = 1, . . . , T

(165)

where xi t* is the ith component of the vector of fitted demands xt* defined by Eq. (163). There is one important additional topic that we have to cover in our discussion of the normalized quadratic functional form: what conditions on b and B are necessary and sufficient to ensure that c(p) defined by Eqs. (146), (147), (148), and (149) is concave in the components of the price vector p? The function c(p) will be concave in p if and only if ∇ 2 c(p) is a negative semidefinite matrix for each p in the domain of definition of c. Evaluating Eq. (154) at p = p∗ and using the restrictions (149) yields: −1    B. ∇ 2 c p∗ = αT p∗

(166)

Since α > 0N and p∗  0N , αT p∗ > 0. Thus, in order for c(p) to be a concave function of p, the following necessary condition must be satisfied:

3 Duality in Production

111

B is a negative semidefinite matrix.

(167)

We now show that the necessary condition (167) is also sufficient to imply that c(p) is concave over the set of p such that p  0N . Unfortunately, the proof is somewhat involved.60 Let p  0N . We assume that B is negative semidefinite and we want to show that ∇ 2 c(p) is negative semidefinite or equivalently, that −∇ 2 c(p) is positive semidefinite. Thus, for any vector z, we want to show that −zT ∇ 2 c(p)z ≥ 0. Using Eq. (154), this inequality is equivalent to: −1  −2  −2  zT Bz + αT p zT BpαT z + αT p zT αpT Bz − αT p −3 pT BpzT ααT z ≥ 0 − α p 

(168)

T

or  −1  −3  2  −2 − αT p zT Bz − αT p pT Bp αT z ≥ −2 αT p zT BpαT z using B = BT . (169) Define A ≡ − B. Since B is symmetric and negative semidefinite by assumption, A is symmetric and positive semidefinite. Thus there exists an orthonormal matrix U such that UT AU = ;

(170)

UT U = IN

(171)

where IN is the N by N identity matrix and  is a diagonal matrix with the nonnegative eigenvalues of A, λi , i = 1, . . . , N, running down the main diagonal. Now premultiply both sides of Eq. (170) by U and postmultiply both sides by UT . Using Eq. (171), UT = U−1 , and the transformed Eq. (170) becomes the following equation: A = UUT = U1/2 1/2 UT = U1/2 UT U 1/2 UT since UT U = IN = SS

60 The

proof is due to Diewert and Wales ([42], 66).

(172)

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where 1/2 is the diagonal matrix that has the nonnegative square roots λi 1/2 of the eigenvalues of A running down the main diagonal and the symmetric square root of A matrix S is defined as S ≡ U1/2 UT .

(173)

If we replace – B in Eq. (169) with A, the inequality that we want to establish becomes  −1  −2  2 2 αT p zT ApαT z ≤ zT Az + αT p pT Ap αT z

(174)

where we have also multiplied both sides of Eq. (169) by the positive number αT p in order to derive Eq. (174) from (169). Recall the Cauchy-Schwarz inequality for two vectors, x and y:  1/2  1/2 yT y . xT y ≤ xT x

(175)

Now we are ready to establish the inequality (174). Using Eq. (172), we have:  −1  −1 αT p zT ApαT z = αT p zT SSpαT z 1/2 1/2 

−2

2  αT p αT z pT ST SP ≤ zT SST z  −1   αT z Sp using (175) with xT ≡ zT S and y ≡ αT p 1/2  1/2 

2−

2 αT p αT z pT SSp = zT SSz using S = ST 1/2 1/2 

−2

2 T T T α p α z p Ap = z Az using (172), A = SS 

T

 

−2

2   αT z pT Ap ≤ (1/2) zT Az + (1/2) αT p (176) where the last inequality follows using the nonnegativity of zT Az, pT Ap, the positivity of αT z and the Theorem of the Arithmetic and Geometric Mean.61 The inequality (176) is equivalent to the desired inequality (174). Thus, the normalized quadratic unit cost function defined by Eqs. (146), (147), (148), and (149) will be concave over the set of positive prices if and only if the

61 This

proof is due to Diewert and Wales [42].

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symmetric matrix B is negative semidefinite. Thus, after econometric estimates of the elements of B have been obtained using the system of estimating Eq. (162), we need only check that the resulting estimated B* matrix is negative semidefinite. However, suppose that the estimated B* matrix is not negative semidefinite. How can one reestimate the model, impose negative semidefiniteness on B, but without destroying the flexibility of the normalized quadratic functional form? The desired imposition of negative semidefiniteness can be accomplished using a technique due to Wiley et al. [86]: simply replace the matrix B by B ≡ −AAT

(177)

where A is an N by N lower triangular matrix; that is, aij = 0 if i < j.62 We also need to take into account the restrictions (149), Bp∗ = 0N . These restrictions on B can be imposed if we impose the following restrictions on A: AT p∗ = 0N .

(178)

To show how this curvature imposition technique works, let p∗ = 1N and consider the case N = 2. In this case, we have: 

a 0 A ≡ 11 a21 a22



 a11 a21 and A = . 0 a22 

T

   0 a11 + a21 = a22 0 and a22 = 0. Thus, in this case, 

The restrictions (178) become: AT 12 = and hence we must have a21 = − a11  B ≡ −AAT = −

a11 0 − a11 0



a11 −a11 0 0



 =−

2 2 −a11 a11 2 2 − a11 a11



 = a11 2

 −1 1 . 1 −1 (179)

Equation (179) shows how the elements of the B matrix can be defined in terms of the single parameter, a11 2 . Note that with this reparameterization of the B matrix, it will be necessary to use nonlinear regression techniques rather than modifications of linear regression techniques. This turns out to be the cost of imposing the correct curvature conditions on the unit cost function.

zT AAT z = (AT z)T (AT z) = yT y ≥ 0 for all vectors z, AAT is positive semidefinite and hence − AAT is negative semidefinite. Diewert and Wales ([42], 53) showed that any positive semidefinite matrix can be written as AAT where A is lower triangular. Hence, it is not restrictive to reparameterize an arbitrary negative semidefinite matrix B as − AAT . 62 Since

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The Konüs Byushgens Fisher Unit Cost Function Define the KBF unit cost function, c(p), as follows63 : 1/2  ; B = BT c (p) ≡ pT Bp

(180)

where B is an N by N symmetric matrix which has one positive eigenvalue (with a strictly positive eigenvector) and the remaining N − 1 eigenvalues are negative or zero. The vector of first order partial derivatives of this unit cost function, ∇c(p), and the matrix of second order partials, ∇ 2 c(p), are equal to the following expressions: 1/2  ; ∇c (p) = Bp/ pT Bp

(181)

   −1/2  −1 B − Bp pT Bp ∇ 2 c (p) = pT Bp pT B .

(182)

At this point, we need to determine the region of price space where the c(p) defined by Eq. (180) is a concave function. In general, the unit cost function defined by Eq. (180) will not be concave for all strictly positive price vectors p.64 In order for a unit cost function to provide a valid global representation of homothetic preferences, it must be a nondecreasing, linearly homogeneous and concave function over the positive orthant. However, in order for c to provide a valid local representation of preferences, we need only require that c(p) be positive, nondecreasing, linearly homogeneous, and concave over a convex subset of prices, say S, where S has a nonempty interior.65 It is obvious that c(p) defined by Eq. (20) is linearly homogeneous. The nondecreasing property will hold over S if the gradient vector ∇c(p) defined by Eq. (181) is strictly positive for p ∈ S and the concavity property will hold if ∇ 2 c(p) defined by Eq. (182) is a negative semidefinite matrix for p ∈ S. We will show how the regularity region S can be determined shortly but first, we will indicate why the c(p) defined by Eq. (20) is a flexible functional form66 since this explanation will help us to define an appropriate region of regularity.

63 This is a special case of a functional form due to Denny [18], which Diewert ([24], 131) called the quadratic mean of order r unit cost function. This functional form with r = 2 was introduced into the economics literature by Konüs and Byushgens ([69], 168) and its connection to the Fisher [53] ideal price index was explained by these authors and Diewert [24]. See Problem 22 below. 64 The following analysis of the regularity conditions for the c(p) defined by Eq. (180) is due to Diewert and Hill [38]. 65 See Blackorby and Diewert [5] for more details on local representations of preferences using duality theory. 66 Diewert ([24], 130) established the flexibility of c(p) defined by Eq. (180) as part of a more general result.

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Let p∗  0N be a strictly positive reference price vector and suppose that we are given an arbitrary unit cost function c* (p) that is twice continuously differentiable in a neighborhood around p* .67 Let x∗ ≡ ∇ c∗ (p)  0N be the strictly positive vector of first order partial derivatives of c* (p* ) and let S∗ ≡ ∇ 2 c∗ (p∗ ) be the negative semidefinite symmetric matrix of second order partial derivatives of c* evaluated at p* . Euler’s Theorem on homogeneous functions implies that S* satisfies the following matrix equation: S∗ p∗ = 0N .

(183)

In order to establish the flexibility of the KBF c defined by Eq. (180), we need only show that there are enough free parameters in the B matrix so that the following equations are satisfied:   ∇c p∗ = x∗ ;

(184)

  ∇ 2 c p∗ = S∗ .

(185)

In order to prove the flexibility of c, it is convenient to reparameterize the B matrix. Thus we now set B equal to: B = bbT + A

(186)

where b  0N is a positive vector and A is a negative semidefinite matrix which has rank equal to at most N − 1 and it satisfies the following restrictions: Ap∗ = 0N .

(187)

Note that bbT is a rank one positive semidefinite matrix with p∗ T bbT p∗ = > 0 and A is a negative semidefinite matrix and satisfies p∗ T Ap∗ = 0. Thus, it can be seen that B is a matrix with one positive eigenvalue and the other eigenvalues are negative or zero. Substitute Eq. (181) into (184) in order to obtain the following equation:

(bT p∗ )2

67 Of

course, in addition, we assume that c* satisfies the appropriate regularity conditions for a unit cost function. Using Euler’s Theorem on homogeneous functions, the fact that c* is linearly homogeneous and differentiable at p* means that the derivatives of c* satisfy the following restrictions: c∗ (p∗ ) = p∗ T ∇ c∗ (p∗ ) and ∇ 2 c∗ (p∗ )p∗ = 0N . The unit cost function c defined by Eq. (180) satisfies analogous restrictions at p = p∗ . These restrictions simplify the proof of the flexibility of c at the point p* .

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1/2  x∗ = Bp∗ / p∗T Bp∗



1/2 = bbT + A p∗ / p∗T bbT + A p∗ using (186) 1/2  using (187) = bbT p∗ / p∗T bbT p∗

(188)

= b. Thus, if we choose b equal to x* , Eq. (184) will be satisfied. Now substitute Eq. (182) into (183) and obtain the following equation:  −1/2   −1  B − Bp∗ p∗T Bp∗ S∗ = p∗T Bp∗ p∗T B  −1/2   −1  ∗T T ∗ T T ∗ ∗T T ∗ ∗T T bb + A − bb p p bb p p bb = p bb p

(189)

using (186) and (187) −1  A using bT p∗ > 0. = bT p∗ Thus, if we choose A equal to (bT p* )S* , Eq. (185) will be satisfied and the flexibility of c defined by Eq. (180) is established.68 Now we are ready to define the region of regularity for c defined by Eq. (180).69 Consider the following set of prices: S ≡ {p : p  0N ; Bp  0N } .

(190)

If p ∈ S, then it can be seen that c(p) = (pT Bp)1/2 > 0 and using Eq. (181), ∇c(p)  0N . However, it is more difficult to establish the concavity of c(p) over the set S. We first consider the case where the matrix B has full rank so that it has one positive eigenvalue and N − 1 negative eigenvalues. Let p ∈ S and using Eq. (182), we see that ∇ 2 c(p) will be negative semidefinite if and only if the matrix M defined as:  −1 M ≡ B − Bp pT Bp pT B

68 We

(191)

need to check that A is negative semidefinite (which it is since it is a positive multiple of the negative semidefinite substitution matrix S* ) and that A satisfies the restrictions in Eq. (187), since we used these restrictions to derive Eq. (188) and the second line in Eq. (189). But A does satisfy Eq. (187) since A satisfies Eq. (183). 69 The region of regularity can be extended to the closure of the set S.

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is negative semidefinite. Note that M is equal to the matrix B plus the rank 1 negative semidefinite matrix – Bp(pT Bp)−1 pT B. B has one positive eigenvalue and the remaining eigenvalues are 0 or negative. Since M is B plus a negative semidefinite matrix, the eigenvalues of M cannot be greater than the eigenvalues of B. Now consider two cases; the first case where B has one positive and N − 1 negative eigenvalues and the second case where B has N − 1 negative or zero eigenvalues in addition to its positive eigenvalue. Consider case 1, let p ∈ S and calculate Mp:    −1 T T Mp = B − Bp p Bp p B p = 0N .

(192)

The above equation shows that p = 0N is an eigenvector of M that corresponds to a 0 eigenvalue. Now the addition of a negative semidefinite matrix to B can only make the N − 1 negative eigenvalues of B more negative (or leave them unchanged) so we conclude that the addition of the negative semidefinite matrix – Bp(pT Bp)−1 pT B to B has converted the positive eigenvalue of B into a zero eigenvalue and hence M is negative semidefinite. Case 2 follows using a perturbation argument. Thus, we have shown that the KBF unit cost function c(p) defined by Eq. (180) is positive, increasing in the components of p and concave in p over the region of prices S defined by Eq. (190). It is useful to show if c(p) ≡ (pT Bp)1/2 is defined by Eq. (180), then we can decompose the matrix B into bbT + A where b  0N and A is a negative semidefinite matrix with Ap∗ = 0N for some p∗  0N . Recall that definition (180) specified that c(p) ≡ (pT Bp)1/2 where B is an N by N symmetric matrix which has one positive eigenvalue (with a strictly positive eigenvector) and the remaining N − 1 eigenvalues are negative or zero. Let λ1 > 0 and λi ≤ 0 for i = 2, 3, . . . , N be the eigenvalues of B and let the column vectors ui be the corresponding eigenvectors, which are orthonormal to each other; that is, uiT ui = 1 for i = 1, . . . , N and uiT uj = 0 for all i = j. Then it is well known that the matrix B has the following representation: B=



N i=1

λi ui uiT .

(193)

Using the regularity conditions in definition (180), it can be seen that the first eigenvector u1 is strictly positive. Make the following definitions: p∗ ≡ u1  0N ; b ≡ (λ1 )1/2 u1 ; A ≡

 i=2

N

λi ui uiT .

(194)

It can be seen that A is a negative semidefinite matrix. Since u1 = p∗ is orthogonal to u2 , . . . , uN , Ap∗ = 0N . Thus, we have B = bbT + A where b is a positive vector and A is negative semidefinite with Ap∗ = 0N . The following problems show the connection of the KBF functional form with Irving Fisher’s [53] ideal index number formula.

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Problems 21. Suppose that a producer’s unit cost function is defined by Eq. (180). Assume cost minimizing behavior on the part of the producer for periods 1 and 2 so that using Shephard’s Lemma, we have: (i)   xt = ∇c pt yt ;

t = 1, 2

where pt , xt , and yt are the period t input price and quantity vectors and yt is the period t output level for t = 1, 2. (a) Show that (ii)     xt /ptT xt = ∇c pt /c pt ;

t = 1, 2.

(b) Show that we also have the following equations: (iii)   xt /ptT xt = Bpt /c pt ;

t = 1, 2.

22. Continuation of 21: The Fisher [53] ideal input price index PF is defined as the following function of the observed input price and quantity vectors for periods 1 and 2: (i)  

1/2 PF p1 , p2 , x1 , x2 ≡ p2T x1 p2T x2 /p1T x1 p1T x2 . Assume that p1 ,p2 ,x1 ,x2 satisfy equations (i) in Problem 21 where the KBF unit cost function c(p) is defined by Eq. (180). Show that (ii)       PF p1 , p2 , x1 , x2 = c p2 /c p1 . Hint: Note that the inner products of p2 with x1 /p1T x1 and p1 with x2 /p2T x2 appear in the formula (i) above for PF (p1 ,p2 ,x1 ,x2 ). Apply part (b) of Problem 21. Comment: The ratio of unit costs, c(p2 )/c(p1 ), can be interpreted as a theoretical input price index, (due originally to Konüs [68] in the consumer context). Equation (ii) above tells us that this theoretical input cost index can be calculated using just observed input price and quantity data for the two periods under consideration using the Fisher index provided that the producer is cost minimizing in the two periods and has the production function that is dual to the unit cost function defined by

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Eq. (180). Thus, no econometric estimation is necessary in order to construct the ratio of unit costs.70 We conclude this section by looking at the problems associated with estimating the unknown parameters in the symmetric B matrix, assuming that we have data on a production unit producing one output and using N inputs for T time periods. Using Eq. (181), Shephard’s Lemma and definition (180) evaluated at the period t data, we obtain the following system of estimating equations: 1/2  xt /yt = Bpt / ptT Bpt + et ; t = 1, . . . , T

(195)

where xt is the observed period t input vector, yt is the period t output, pt is the vector of period t input prices and et ≡ [e1 t , . . . , eN t ]T is a vector of stochastic error terms with 0 means. Equation (195) can be used in order to statistically estimate the N(N + 1)/2 independent bij parameters in the B matrix. However, the system of equations defined by Eq. (195) is nonlinear in the unknown parameters. Define period t unit cost by ct ≡ ptT xt /yt . In theory, ct should equal (ptT Bpt )1/2 plus an error term. Thus, the system of estimating Eq. (195) can be replaced by the following system: ct xt /yt = Bpt + et∗ ; t = 1, . . . .T

(196)

where et∗ ≡ [e1 t∗ , . . . , eN t∗ ]T is a new vector of stochastic error terms with 0 means. Note that the new system of estimating equations defined by Eq. (196) is linear in the unknown bij .71 As was the case when estimating the normalized quadratic unit cost function, it will often turn out that the estimated B matrix will not satisfy the regularity conditions that are associated with definition (180). As we have seen above, B may be estimated as the equivalent expression equal to bbT + A where b is a strictly positive vector and A is a symmetric negative semidefinite matrix with Ap∗ = 0N for some strictly positive reference vector p* . Thus, we need only set A = − CCT where C is a lower triangular matrix with CT p∗ = 0N and the correct curvature conditions 70 This

result is much more important in the consumer context where we interpret f(x) as a utility function defined over consumption vectors x and c(p) is the dual unit expenditure function. Note that utility cannot be observed whereas output can be observed. 71 In the consumer context where output yt is replaced by (unobservable) utility ut and xt is the period t consumption vector, rewrite Eq. (195) as xt = ut Bpt /(ptT Bpt )1/2 where we have dropped the error terms. Total period t expenditure is ptT xt = ut (ptT Bpt )1/2 . Thus we obtain xt /ptT xt = Bpt /ptT Bpt . Premultiply both sides of equation n by pn t and we obtain the following system of estimating equations: pn t xn t /ptT xt ≡ sn t = pn t  i = 1 N bni pi t /ptT Bpt + en t for n = 1, . . . , N and t = 1, . . . , T. We need to impose a normalization on the elements of the B matrix such as b11 = 1 and since we have share equations, we need to drop one of these share equations in the nonlinear estimation procedure. For an example of this methodology in the consumer context, see Diewert and Feenstra [32].

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will be imposed on the resulting functional form for the unit cost function defined as follows:

1/2 

1/2  c (p) ≡ pT bbT − CCT p = pT bbT − i=1 N−1 ci ciT p

(197)

where c1T ≡[c1 1 , c2 1 , . . . ,cN 1 ], c2T ≡ [0, c2 2 , . . . , cN 2 ], c3T ≡ [0, 0, c3 3 , . . . , cN 3 ], . . . , c(N − 1)T ≡ [0, . . . , 0, cN − 1 N − 1 , cN N − 1 ] and cnT p∗ = 0 for n = 1, 2, . . . , N − 1. We have considered four flexible functional forms for a unit cost function: the generalized Leontief, the translog, the normalized quadratic, and the KBF functional forms. The last two functional forms have the advantage that concavity can be imposed on these functional forms without destroying the flexibility of the resulting functions. The normalized quadratic functional form has the disadvantage that it is usually necessary to choose the vector α72 , whereas all of the parameters for the KBF functional form can be estimated endogenously.

Semiflexible Functional Forms In models where the number of commodities N is large, it can be difficult to estimate all of the parameters for a flexible functional form. Thus, when estimating the parameters for the normalized quadratic defined by Eq. (146) above, it was necessary to estimate the elements of the N by N symmetric matrix B and for the KBF functional form, it was necessary to estimate the elements of the N by N symmetric matrix A in Eq. (186). If we impose concavity on these functional forms, then in both of these cases, the B and A matrices are replaced by – CCT where C is lower triangular and Cp∗ = 0N for a reference positive price vector p* . An effective way to estimate the C matrix is to estimate it one column at a time. Thus, consider our estimating Eq. (162) for the normalized quadratic unit cost function. Replace the B matrix in these equations by – CCT where C is lower triangular and Cp∗ = 0N and we obtain the following system of equations: xt /yt = b − CCT vt + (1/2) vtT CCT vt α + et ; t = 1, . . . , T.

(198)

In Stage 1, we set C = 0N × N and use the resulting equations in (198) in order to estimate the vector of parameters b. In Stage 2, set CCT = c1 c1T where c1T ≡ [c1 1 , c2 1 , . . . , cN 1 ] and c1T p∗ = 0. Equation (198) now becomes a nonlinear regression model. For starting parameter values, use the b vector that was estimated in Stage 1 and set the vector c1 = 0N . In Stage 3, set CCT = c1 c1T + c2 c2T where c1T ≡ [c1 1 , c2 1 , . . . , cN 1 ], c2T ≡ [0, c2 2 , . . . , cN 2 ] and ciT p∗ = 0 for i = 1, 2. For 72 Thus different choices for the α vector could lead to different estimates for elasticities of demand.

N − 1 components of the α vector could be estimated along with the remaining parameters but then we would not have a parsimonious flexible functional form.

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starting parameter values, use the b and c1 vectors that were estimated in Stage 2 and set the vector c2 = 0N . This procedure of gradually adding nonzero columns of the lower triangular C matrix can be continued until the full number of N − 1 nonzero columns have been added, provided that the number of time series observations T is large enough compared to N, the number of commodities in the model.73 However, in models where T is small relative to N, the above procedure of adding nonzero columns to A will have to be stopped well before the maximum number of N − 1 nonzero columns has been added, due to the lack of degrees of freedom. Suppose that we stop the above procedure after K < N − 1 nonzero columns have been added. Then Diewert and Wales ([43], 330) called the resulting normalized quadratic functional form a flexible of degree K functional form or a semiflexible functional form. A flexible of degree K functional form for a cost function can approximate an arbitrary twice continuously differentiable functional form to the second order at some point, except the matrix of second order partial derivatives of the functional form with respect to prices is restricted to have maximum rank K instead of the maximum possible rank, N − 1. The cost of using a semiflexible functional form of degree K where K is less than N − 1 is that we will miss out on the part of CCT that corresponds to the smallest eigenvalues of this matrix. In many situations, this cost will be very small; that is, as we go through the various stages of estimating C by adding an extra nonzero column to C at each stage, we can monitor the increase in the final log likelihood (if we use maximum likelihood estimation) and when the increase in Stage k + 1 over Stage k is “small,” we can stop adding extra columns, secure in the knowledge that we are not underestimating the size of CCT by a large amount. This semiflexible technique has not been widely applied but it would seem to offer some big advantages in estimating substitution matrices in situations where there are a large number of commodities in the model.74

The Use of Splines for Modeling Technical Progress Recall the definitions for the generalized Leontief, normalized quadratic, and KBF unit cost functions c(p) given by Eqs. (97), (146), and (180). If these functions are estimated in the time series context for a production unit for say T periods, then a problem will often occur: these functional forms make no allowance for technical progress that may have taken place over the sample time period. This problem can be solved if we add the function dT pt to the unit cost function c(p) where

73 In empirical applications, typically a final stage K < N − 1 will be reached where the addition of another column to the CCT matrix leads to no increase in log likelihood and the last column cK is a column of zeros. 74 Diewert and Lawrence in some unpublished work have successfully estimated semiflexible models for profit functions for 40 commodities. Neary [75] used semiflexible functional forms for 11 commodity groups.

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dT ≡ [d1 , . . . , dN ] is an N dimensional vector of technical progress parameters and t is a scalar time variable which takes on the value t for time period t. Thus, choose a flexible functional form for the unit cost function c(p) and add the function dT pt to it. Using our usual notation for a data set on inputs xt , input prices pt and output levels yt for period t, we obtain the following system of estimating equations using Shephard’s Lemma:   xt /yt = ∇c pt + dt + et ; t = 1, . . . , T

(199)

where et is a suitable error vector. If we choose c(p) to be the normalized quadratic unit cost function, then the resulting estimating Eq. (199) will be linear in the unknown parameters.75 However, in many applications of this model, the results may not be satisfactory. The problem with the model defined by Eq. (199) is that the resulting measures of technical progress are too smooth; that is, typically if one looks at the residuals generated by the model, substantial amounts of autocorrelation will be present in the estimating equations. This is an indication that rates of technical progress are not constant over the sample time period. Under these circumstances, it will be useful to replace the function simple linear function dT pt by the following piece-wise linear spline function, τ(p, t), defined as follows: τ (p, t) ≡ d1T pt if 1 ≤ t ≤ t1∗ ;   ≡ d1T pt1∗ + d2T p t − t1∗ if t1∗ ≤ t ≤ t2∗ ;     ≡ d1T pt1∗ + d2T p t2∗ − t1∗ + d3T p t − t2∗ if t2∗ ≤ t ≤ T

(200)

where d1 , d2 , and d3 are N dimensional technical progress parameters and t1* and t2∗ > t1∗ are two time periods where the piece-wise linear function of time t, τ(p, t), changes from one set of rates of technical progress to another set.76 The estimating equations are now the following ones77 :

75 However, if we impose concavity on the normalized quadratic functional form, then the resulting

estimating equations will be nonlinear in the unknown parameters. For a worked example of this methodology for modelling technical progress, see Diewert and Wales [42]. 76 The break points t1* and t2* can be chosen by running a preliminary regression of the form (199) and examining the regression residuals to see when these turning points occur. In our example, we have three time periods where the rates of technical progress are linear in time. If necessary, additional break points can be added at the cost of having to estimate additional parameter vectors di . 77 If the unit cost function is translog, then the estimating equations will be somewhat different.

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  xt /yt = ∇c pt + d1 t + et ;     = ∇c pt + d1 t1∗ + d2 t − t1∗ + et ;       = ∇c pt + d1 t1∗ + d2 t2∗ − t1∗ + d3 t − t2∗ + et ;

1 ≤ t ≤ t1∗ ; t1∗ < t ≤ t2∗ ; t2∗ < t ≤ T. (201)

If we chose c(p) to be the normalized quadratic unit cost function, then, assuming that it is not necessary to impose concavity, the above estimating equations will be linear in the unknown parameters. For an example of the use of the above spline methodology, see Fox [54].78 The above spline methodology for modeling technical progress can be modified to model nonconstant returns to scale technologies; see Fox and Grafton [55]. The above linear spline model has the disadvantage that rates of technical progress will typically jump in a discontinuous manner as we move from one linear spline segment to the following one. This problem can be remedied (at the cost of a more complicated set of estimating equations) if the linear splines in time t are replaced with quadratic splines in t. For an example of the quadratic spline approach, see Diewert and Wales [44].

Allowing for Flexibility at Two Sample Points There can be a problem with our two flexible functional forms for unit cost functions where the correct curvature conditions can be imposed (the normalized quadratic and the KBF unit cost functions): the elasticities of input demand that these functions generate in the time series context can exhibit substantial trends. We need to derive a formula for the elasticity of demand for input n with respect to a change in the price of input k, say Enk (y,p) where y is output and p is an input price vector. Recall that the normalized quadratic unit cost function was defined by c(p) ≡ bT p + (1/2)pT Bp/αT p where α is predetermined and B is a symmetric matrix which satisfies Bp∗ = 0N .79 The vector of first order partial derivatives and the matrix of second order partial derivatives of this unit cost function are as follows:

78 Fox

∇c (p) = b + Bv − (1/2) vT Bvα;

(202)

 −1 ∇ 2 c (p) = αT p B − BvαT − αvT B + vT BvααT

(203)

used a more scientific method to pick the break points (cross validation). Eqs. (146), (147), (148), and (149) above. We also require that B be negative semidefinite, a property which can be imposed as was explained in section “The Normalized Quadratic Unit Cost Function” above. 79 See

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where v ≡ p/αT p is a vector of normalized input prices. The system of input demand functions that is generated by this functional form is x(y, p) ≡ y ∇ c(p) and the N by N matrix of input demand derivatives with respect to input prices is ∇ p x(y, p) ≡ y∇ 2 c(p). Using Eq. (203), we see that the elasticity Enk (y, p) ≡ [pk /xn ]∂xn (y, p)/∂pk is equal to the following expression:

  Enk (y, p) = pk /αT p y/xn bnk − Bn · vαk − Bk · vαn + vT Bvαn αk ;

(204)

n, k = 1, . . . , N where bnk is the nkth element of the matrix B, Bi· denotes the ith row of the B matrix for i = 1, . . . , N and v ≡ p/αT p is the vector of normalized prices; that is, the components of the input price vector p are divided by αT p. Note that when p = p∗ , the restrictions imply that v∗ T Bv∗ = 0 and Bi· v∗ = 0 for i = 1, . . . , N where v∗ ≡ p∗ /αT p∗ . Thus

    Enk y, p∗ = pk ∗ /αT p∗ y/xn bnk ; n, k = 1, . . . , N.

(205)

The reference price vector p* will usually be a representative input price vector for the sample under consideration. Thus, the price elasticity of input demand when evaluated at these reference prices, Enk (y,p* ), will be equal to the constant term bnk times the price ratio term pk ∗ /αT p∗ times the quantity ratio term y/xn . The remaining three terms on the right hand side of Eq. (204) will be equal to zero when p = p∗ . Thus, the first term will generally be the most significant term that defines Enk (y,p) for a general input price vector. If there are substantial divergent trends in either input prices p or input quantities x, it can be seen that [pk /αT p][y/xn ]bnk will also have substantial trends and hence Enk (yt ,pt ) will, in general, also exhibit substantial trends under these conditions. What can be done to remedy this problem of trending elasticities? If the number of observations τ + 1 is relatively large compared to the number of inputs N, then we can set the unit cost function equal to the following function of time t:     c (p, t) ≡ 1 − τ−1 t b1T p + τ−1 tb2T p + (1/2) pT 1 − τ−1 t B1

+τ−1 tB2 p/αT p; t = 0, 1, 2, . . . , τ

(206)

where B1 p0 = 0N and B2 pτ = 0N .80 Thus the resulting unit cost function evaluated at period 0 is c(p, 0) ≡ b1T p + (1/2)pT B1 p/αT p and evaluated at period τ is

80 We

require that B1 and B2 be symmetric negative semidefinite matrices. If the estimated matrices fail to be negative semidefinite, then we can impose negative semidefiniteness by setting Bi = − Ci CiT for i = 1, 2 where each Ci is an arbitrary lower triangular matrix satisfying C1T p0 = 0N and C2T pT = 0N .

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c(p, τ) ≡ b2T p + (1/2)pT B2 p/αT p; that is, the resulting unit cost function is flexible at two data points. If there are trends in input demand elasticities using this functional form, then these trends are implied by the data rather than by the choice of the functional form.81 Note that the unit cost function defined by Eq. (206) allows for biased technical change over the sample period; that is, it allows for trends in the b ≡ (1 − t)b1 + tb2 vector.82 It is possible to generalize the KBF unit cost function in a similar manner. Recall that this unit cost function was defined by Eq. (180): c(p) ≡ (pT Bp)1/2 where B ≡ bbT + A and A is a negative semidefinite symmetric matrix which satisfies Ap∗ = 0N . The vector of first order partial derivatives was defined by Eq. (181). Using this equation and Shephard’s Lemma, we have x(y, p) = y ∇ c(p) and so x/y = ∇ c(p). Thus, using (181), we obtain the following equations: 1/2  . x (y, p) = y∇c (p) = yBp/ pT Bp

(207)

When p = p∗ , using B ≡ bbT + A and Ap∗ = 0N , it can be seen that   x∗ ≡ x y, p∗ = yb.

(208)

The matrix of input demand derivatives with respect to input prices is ∇ p x(y, p) = y∇ 2 c(p). The matrix of second order partial derivatives of the unit cost function was defined by Eq. (182). Thus, we have: ∇p x (y, p) = y∇ 2 c (p)   1/2  −1  T T T B − Bp p Bp p B = y p Bp −1/2

= yc(p)

   −1 T T B − Bp p Bp p B

using (182)  1/2 using c (p) ≡ pT Bp

  using (207) = yc(p)−1/2 B − y−2 x (y, p) x(y, p)T   = yc(p)−1/2 bbT + A − y−2 x (y, p) x(y, p)T using B=bbT + A. (209) Now evaluate Eq. (209) when p = p∗ . We find that:

81 This

technique of imposing price flexibility at two points is due to Diewert and Lawrence [40].

82 If the residuals in the final model exhibit substantial autocorrelation, then it is possible to replace

the b vector by a piece-wise linear function of time as was done in the previous section. This will allow for a more general pattern of technical change.

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    −1/2    T  A + bbT − y−2 x y, p∗ x y, p∗ ∇p x y, p∗ = yc p∗   −1/2  A + bbT − bbT using (208) = yc p∗

(210)

 −1/2 = yc p∗ A. Using Eq. (209), we see that the elasticity Enk (y, p) ≡ [pk /xn ]∂xn (y, p)/∂pk is equal to the following expression:

   Enk (y, p) = pk /c (p) y/xn (y, p) ank + bn bk − y−2 xn (y, p) xk (y, p) ; n, k = 1, . . . , N (211) where ank is the nkth element of the negative semidefinite matrix A (which satisfies Ap∗ = 0N ), bn is the nth element of the vector b and xn (y,p) is the nth element of the cost minimizing input vector x(y,p) defined by Eq. (207). Using Eq. (210), we see that when p = p∗ , Enk (y, p∗ ) = [pk /c(p∗ )][y/xn (y, p∗ )]ank so that the last two terms on the right hand side of Eq. (211) sum to zero when p = p∗ . Thus, the first term associated with ank will generally be the most significant term that defines Enk (y,p) for a general input price vector. If there are substantial divergent trends in either input prices p or input quantities x, it can be seen that [pk /c(p* )][y/xn (y,p* )]ank will also have substantial trends and hence Enk (yt ,pt ) will, in general, also exhibit substantial trends under these conditions. Again, if the number of observations τ + 1 is relatively large compared to the number of inputs N, then we can set the KBF unit cost function equal to the following function of time t:      c (p, t) ≡ pT 1 − τ−1 t b1T p + τ−1 tb2T p + 1 − τ−1 t A1

1/2 +τ−1 tA2 p ; t = 0, 1, 2, . . . , τ

(212)

where A1 p0 = 0N and A2 pT = 0N .83 Thus, the resulting unit cost function evaluated at period 0 is c(p, 0) ≡ (pT [b1 b1T + A1 ]p)1/2 and evaluated at period τ is c(p, τ) ≡ (pT [b1 b1T + A1 ]p)1/2 ; that is, the resulting unit cost function is flexible at two data points. As was the case for the normalized quadratic, if there are trends in input demand elasticities using this functional form, then these trends are implied by the data rather than by the choice of the functional form. Again, the unit cost

83 We

require that A1 and A2 be symmetric negative semidefinite matrices. Again, if the estimated matrices fail to be negative semidefinite, then we can impose negative semidefiniteness by setting Ai = − Ci CiT for i = 1, 2 where each Ci is an arbitrary lower triangular matrix satisfying C1T p0 = 0N and C2T pT = 0N .

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function defined by Eq. (212) allows for biased technical change over the sample period; that is, it allows for trends in the b ≡ (1 − t)b1 + tb2 vector. We turn our attention to multiple input and multiple output technologies.

National Product or Variable Profit Functions Up to now, we have only considered technologies that produce one output. In reality, production units (firms or industries) usually produce many outputs.84 Hence, in this section, we consider technologies that produce many outputs while using many inputs. Let S denote the technology set of a production unit. We decompose the inputs and outputs of the firm into two sets of commodities: variable and fixed. Let y ≡ [y1 , . . . , yM ] denote a vector of variable net outputs (if ym > 0, then commodity m is an output while if ym < 0, then commodity m is an input) and let x ≡ [x1 , . . . , xN ] denote a nonnegative vector of “fixed” inputs.85 Thus, the technology set S is a set of feasible variable net output and fixed input vectors, (y,x). Let p  0M be a strictly positive vector of variable net output prices that the firm faces during a production period. Then conditional on a given vector of fixed inputs x ≥ 0N , we assume that the firm attempts to solve the following conditional or variable profit maximization problem:   maxy pT y : (y, x) ∈ S ≡ π (p, x) .

(213)

The optimized objective function, π(p, x), has been called many names,86 depending on the context. Alternative names for this function are the national product function Samuelson ([79], 10), the gross profit function Gorman [58], the conditional profit function McFadden [72, 73], the variable profit function Diewert [21], the GDP function Kohli [65, 67], and the value added function Diewert [25]. If there are no intermediate inputs or imports in the outputs, then π(p, x) becomes the revenue function Diewert [23]. Some regularity conditions on the technology set S are required in order to ensure that the maximum in Eq. (213) exists. A simple set of sufficient conditions are87 : (i) S is a closed set in RM + N and (ii) for each x ≥ 0N ,

84 See Bernard et al. [4]. In the sample of US firms considered by Hottman et al. ([63], 1301), the mean number of products (measured by distinct barcodes) was 13 per firm and the maximum number was 388. 85 These “fixed” inputs may only be fixed in the short run. Or we may simply decide to allow outputs and intermediate inputs to be variable and condition on an x vector of primary inputs. 86 The concept of this function is due to Samuelson [79]. 87 Let x ≥ 0 and p  0 . Then by (ii), there exists y such that (y , x) ∈ S. Define the closed N M x x and bounded set B(x, p) ≡ {y : y ≤ b(x)1M ; pT y ≥ pT yx } where b(x) > 0 is an upper bound on all possible net output vectors that can be produced by the technology if the vector of fixed inputs x is

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there exists a y such that (y, x) ∈ S and the set of such y vectors is bounded from above. We will call these conditions the minimal regularity conditions on S. Note that π(p, x) is equal to the optimized objective function in Eq. (213) and is regarded as a function of the net output prices for variable commodities that the firm faces, p, as well as a function of the vector of fixed inputs, x, that the firm has at its disposal. Just as in section “Cost Functions: The One Output Case” above where we showed that the cost function C(y,p) satisfied a number of regularity conditions without assuming much about the production function, we can now show that the profit function π(p, x) satisfies some regularity conditions without assuming much about the technology set S. Theorem 9 McFadden [72, 73], Gorman [58], Diewert [21]: Suppose the technology set S satisfies the minimal regularity conditions (i) and (ii) above. Then the variable profit function π(p, x) defined by Eq. (213) has the following properties with respect to p for each x ≥ 0N : Property 1 π(p, x) is positively linearly homogeneous in p for each fixed x ≥ 0N ; that is, π (λp, x) = λπ (p, x) for all λ > 0, p  0N and x ≥ 0N .

(214)

Property 2 π(p, x) is a convex function of p for each x ≥ 0N ; that is, x ≥ 0M , pi  0M , i = 1, 2; 0 < λ < 1 implies       π λp1 + (1 − λ) p2 , x ≤ λπ p1 , x + (1 − λ) π p2 , x .

(215)

Problem 23. Prove Theorem 9. Hint: Properties 1 and 2 above for π(p, x) are analogues to Properties 2 and 4 for the cost function C(y,p) in Theorem 1 above and can be proven in the same manner. We now ask whether a knowledge of the profit function π(p, x) is sufficient to determine the underlying technology set S. As was the case in section “The Duality Between Cost and Production Functions” above, the answer to this question is yes, but with some qualifications. To see how to use a given profit function π(p, x) can be used to determine the technology set that generated it, pick an arbitrary vector of fixed inputs x ≥ 0N and an arbitrary vector of positive prices, p1  0M . Now use the given profit function π to define the following isoprofit surface: {y : p1T y = π(p1 , x)}. This isoprofit surface must be tangent to the set of net output combinations y that are feasible, given that

available to the producer. It can be seen that the constraint (y, x) ∈ S in Eq. (213) can be replaced by the constraint (y, x) ∈ S ∩ B(x, p). Using (ii), S ∩ B(x, p) is a closed and bounded set so that the maximum in (213) will exist.

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the vector of fixed inputs x is available to the firm, which is the conditional on x production possibilities set, S(x) ≡ {x : (y, x) ∈ S}. It can be seen that this isoprofit surface and the set lying below it must contain the set S(x); that is, the following halfspace M(x,p1 ), contains S(x):      M x, p1 ≡ y : p1T y ≤ π p1 , x .

(216)

Pick another positive vector of prices, p2  0M and it can be seen, repeating the above argument, that the halfspace M(x, p2 ) ≡ {y : p2T y ≤ π(p2 , x)} must also contain the conditional on x production possibilities set S(x). Thus, S(x) must belong to the intersection of the two halfspaces M(x,p1 ) and M(x,p2 ). Continuing to argue along these lines, it can be seen that S(x) must be contained in the following set, which is the intersection over all p  0M of all of the supporting halfspaces to S(x): M (x) ≡ ∩p0M M (x, p) .

(217)

Note that M(x) is defined using just the given profit function, π(p, x). Note also that since each of the sets in the intersection, M(x,p), is a convex set, then M(x) is also a convex set. Since S(x) is a subset of each M(x,p), it must be the case that S(x) is also a subset of M(x); that is, we have S (x) ⊂ M (x) .

(218)

Is it the case that S(x) is equal to M(x)? In general, the answer is no; M(x) forms an outer approximation to the true conditional production possibilities set S(x). Suppose that that there are only two outputs and for a given input vector x, the output production possibilities set is the heart shaped region in Fig. 2. The boundary of the set M(x) partly coincides with the boundary of S(x) but it encloses a bigger set: the backward bending parts of the true production frontier are replaced by the dashed lines that are parallel to the y1 axis and the y2 axis and the inward bending part of the true production frontier is replaced by the dashed line that is tangent to the two regions where the boundary of M(x) coincides with the boundary of S(x). However, if the producer is a price taker in the two output markets, then it can be seen that we will never observe the producer’s nonconvex or backward bending parts of the production frontier. What are conditions on the technology set S (and hence on the conditional technology sets S(x)) that will ensure that the outer approximation sets M(x), constructed using the variable profit function π(p, x), will equal the true technology sets S(x)? It can be seen that the following two conditions on S (in addition to the minimal regularity conditions (i) and (ii)) are the required conditions 88 : N = 1 so that there is only one fixed input, then given a producible net output vector y ∈ RM , we can define the (fixed) input requirements function that corresponds to the technology set S as g(y) ≡ minx {x : (y, x) ∈ S}. In this case, condition (220) becomes the following condition: the

88 If

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W. E. Diewert y2

S(x) M(x)

y1

Fig. 2 The geometry of the two output maximization problem

For every x ≥ 0N , the set S (x) ≡ {x : (y, x) ∈ S} has the following free disposal property : y1 ∈ S (x) , y2 ≤ y1 implies y2 ∈ S (x) ; (219) For every x ≥ 0N , the set S (x) ≡ {y : (y, x) ∈ S} is convex.

(220)

Conditions (219) and (220) are the conditions on the technology set S that are counterparts to the two regularity conditions of nondecreasingness and quasiconcavity89 that were made on the production function, f(x), in section “The Duality Between Cost and Production Functions” above in order to obtain a duality between cost and production functions. If the firm is behaving as a price taker in variable commodity markets, it can be seen that it is not restrictive from an empirical point of view to assume that S satisfies conditions (219) and (220), just as it was not restrictive to assume that the production function was nondecreasing and quasiconcave in the context of the producer’s (competitive) cost minimization problem studied earlier. The next result provides a counterpart to Shephard’s Lemma, Theorem 5 in section “The Derivative Property of the Cost Function” above. Theorem 10 Hotelling’s ([61], 594) Lemma 90 : If the profit function π(p, x) satisfies the properties listed in Theorem 9 above and in addition is once differentiable with respect to the components of the variable commodity prices at the point (p* ,x* )

input requirements function g(y) is quasiconvex in y. For additional material on this one fixed input model, see Diewert [23]. 89 Recall conditions (11) and (12) in section “The Duality Between Cost and Production Functions.” 90 See also Gorman [58] and Diewert ([22], 137).

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where x∗ ≥ 0N and p∗  0M , then   y∗ = ∇p π p∗ , x∗

(221)

where ∇ p π(p∗ , x∗ ) is the vector of first order partial derivatives of variable profit with respect to variable commodity prices and y* is any solution to the profit maximization problem       maxy p∗T y : y, x∗ ∈ S ≡ π p∗ , x∗ .

(222)

Under these differentiability hypotheses, it turns out that the y* solution to Eq. (222) is unique. Proof Let y* be any solution to the profit maximization problem (222). Since y* is feasible for the profit maximization problem when the variable commodity price vector is changed to an arbitrary p  0N , it follows that   pT y∗ ≤ π p, x∗ for every p  0M .

(223)

Since y* is a solution to the profit maximization problem (22) when p = p∗ , we must have   p∗T y∗ = π p∗ , x∗ .

(224)

But Eqs. (223) and (224) imply that the function of M variables, g(p) ≡ pT y∗ − π(p, x∗ ) is nonpositive for all p  0M with g(p) = 0. Hence, g(p) attains a global maximum at p = p∗ and since g(p) is differentiable with respect to the variable commodity prices p at this point, the following first order necessary conditions for a maximum must hold at this point:     ∇p g p∗ ≡ y∗ − ∇p π p∗ , x∗ = 0M .

(225)

Now note that Eq. (225) is equivalent to Eq. (221). If y** is any other solution to the profit maximization problem (222), then repeat the above argument to show that y∗∗ = ∇ p π(p∗ , x∗ ) which in turn is equal to y* . Hotelling’s Lemma may be used in order to derive systems of variable commodity output supply and input demand functions just as we used Shephard’s Lemma to generate systems of cost minimizing input demand functions; for examples of this use of Hotelling’s Lemma, see Diewert ([22], 137–139) and sections “The Translog Variable Profit Function,” “The Normalized Quadratic Variable Profit Function,” and “The KBF Variable Profit Function” below. If we are willing to make additional assumptions about the underlying firm production possibilities set S, then we can deduce that π(p, x) satisfies some additional properties. One such additional property is the following one: S is subject

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to the free disposal of fixed inputs if it has the following property:     x2 > x1 ≥ 0N and y, x1 ∈ S implies y, x2 ∈ S.

(226)

The above property means if the vector of fixed inputs x1 is sufficient to produce the vector of variable inputs and outputs y and if we have at our disposal a bigger vector of fixed inputs x2 , then y is still producible by the technology that is represented by the set S. Theorem 1191 Suppose the technology set S satisfies the weak regularity conditions (i) and (ii) above. (a) If in addition, S has the following property92 : For every x ≥ 0N , (0M , x) ∈ S;

(227)

then for every p  0M and x ≥ 0N , π(p, x) ≥ 0; that is, the variable profit function is nonnegative if Eq. (227) holds. (b) If S is a convex set, then for each p  0M , then π(p, x) is a concave function of x over the set  ≡ {x : x ≥ 0N }. (c) If S is a cone so that the technology is subject to constant returns to scale, then π(p, x) is (positively) homogeneous of degree one in the components of x. (d) If S is subject to the free disposal of fixed inputs, property (226), then     p  0, x2 > x1 ≥ 0N implies π p, x2 ≥ π p, x1 ;

(228)

that is, π(p, x) is nondecreasing in the components of x. Proof of (a) Let p  0M and x ≥ 0N . Then

π (p, x) ≡ maxy pT y : (y, x) ∈ S ≥ pT 0M since by(227), (0M , x) ∈ S and hence is feasible for the problem = 0. (229) Proof of (b) Let p  0M , x1 ≥ 0N , x2 ≥ 0N and 0 < λ < 1. Then

91 The results in this Theorem are essentially due to Samuelson ([79], 20), Gorman [58], McFadden (1968) and Diewert ([21, 22], 136) but they are packaged in a somewhat different form in this chapter. 92 This property says that the technology can always produce no variable outputs and utilize no variable inputs given any vector of fixed inputs x.

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      π p, x1 ≡ maxy pT y : y, x1 ∈ S   = pT y1 where y1 , x1 ∈ S;       π p, x2 ≡ maxy pT y : y, x2 ∈ S   = pT y2 where y2 , x2 ∈ S.

(230)

(231)

Since S is assumed to be a convex set, we have    

λ y1 , x1 + (1 − λ) y2 , x2 = λy1 + (1 − λ) y2 , λx1 + (1 − λ) x2 ∈ S. (232) Using the definition of π, we have:       π p, λx1 + (1 − λ) x2 ≡ maxy pT y : y, λx1 + (1 − λ) x2 ∈ S

≥ pT λy1 + (1 − λ) y2 since by(232), λy1 + (1 − λ) y2 is feasible for the problem = λpT y1 + (1 − λ) pT y2     = λπ p, x1 + (1 − λ) π p, x2 using(230)and(231). (233) Proof of (c) Let p  0M , x∗ ≥ 0N and λ > 0. Then       π p, x∗ ≡ maxy pT y : y, x∗ ∈ S   = pT y∗ where y∗ , x∗ ∈ S.

(234)

Since S is a cone and since (y∗ , x∗ ) ∈ S, then we have (λy∗ , λx∗ ) ∈ S as well. Hence, using a feasibility argument:       π p, λx∗ ≡ maxy pT y : y, λx∗ ∈ S   ≥ pT λy∗ since λy∗ + λx∗ ∈ S and hence is feasible for the problem = λpT y∗ . (235) Now suppose that the strict inequality in Eq. (235) held so that

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π (p, x)

(236)

Since S is a cone, λ > 0, and (y∗∗ , λx∗ ) ∈ S, then we have (λ−1 y∗∗ , x∗ ) ∈ S as well. Thus, λ−1 y∗∗ is feasible for the maximization problem (234) that defined π(p, x∗ ) and so     pT y∗ = maxy pT y : y, x∗ ∈ S using(234) ≥ pT λ−1 y∗∗ since λ−1 y∗∗ is feasible for the problem

(237)

= λ−1 pT y∗∗ or since λ > 0, Eq. (237) is equivalent to λpT y∗ ≥ pT y∗∗ > λpT y∗ using(236).

(238)

But Eq. (238) implies that λpT y∗ > λpT y∗ , which is impossible and hence our supposition is false and the desired result follows. Proof of (d) Let p  0, x2 > x1 ≥ 0N . Using the definition of π(p, x1 ), we have       π p, x1 ≡ maxy pT y : y, x1 ∈ S   = pT y1 where y1 , x1 ∈ S.

(239)

Using the free disposal property (228) for S, since (y1 , x1 ) ∈ S and x2 > x1 , we have   y1 , x2 ∈ S (240) Using the definition of π(p, x2 ), we have       π p, x2 ≡ maxy pT y : y, x2 ∈ S   = pT y1 since by (240), y1 , x2 is feasible   = π p, x1 using(239).

(241)

If the technology set S satisfies the minimal regularity conditions (i) and (ii) plus all of the additional conditions that are listed in Theorem 11 above (we shall call such a technology set a regular technology set), then the associated variable profit function π(p, x) will have all of the regularity conditions with respect to

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its fixed input vector x that a nonnegative, nondecreasing, concave, and linearly homogeneous production function f(x) possesses with respect to its input vector x. Hotelling’s Lemma enabled us to interpret the vector of first order partial derivatives of the variable profit function with respect to the components of the variable commodity price vector p, ∇ p π(p, x), as the producer’s vector of variable profit maximizing output supply (and the negative of variable input demand) functions, y(p,x), provided that the derivatives existed. If the first order partial derivatives of the variable profit function π(p, x) with respect to the components of the fixed input vector x exist, then this vector of derivatives, ∇ x π(p, x), can also be given an economic interpretation as a vector of shadow prices or imputed contributions to profit of adding marginal units of fixed inputs. The following result also shows that these derivatives can be interpreted as competitive input prices for the “fixed” factors if they are allowed to become variable. Theorem 12; Samuelson’s Lemma93 Suppose the technology set S satisfies the minimal regularity assumptions (i) and (ii) above and in addition is a convex set. Suppose in addition that p∗  0M , x∗ ≥ 0N and that the vector of derivatives, ∇ x π(p∗ , x∗ ) ≡ w∗ , exists. Then x* is a solution to the following long run profit maximization problem that allows the “fixed” inputs x to be variable:     maxx π p∗ , x − w∗T x : x ≥ 0N .

(242)

Proof Part (b) of Theorem 11 above implies that π(p∗ , x) is a concave function of x over the set  ≡ {x : x ≥ 0N }The function – w*T x is linear in x and hence is also a concave function of x over . Hence, f(x) defined for x ≥ 0N as   f (x) ≡ π p∗ , x − w∗T x

(243)

is also a concave function in x over the set . Since x∗ ≥ 0N , x∗ ∈ . Hence, using the fact that a differentiable concave function has a Taylor series approximation that provides an upper bound to the function around any point x* where the function is differentiable, we have the following inequality:   T    x − x∗ for all x ≥ 0N f (x) ≤ f x∗ + ∇x f x∗         = π p∗ , x∗ −w∗T x∗ +0N T x−x∗ since ∇x f x∗ =∇x π p∗ , x∗ − w∗ =0N   = π p∗ , x∗ − w∗T x∗ . (244)

National Product function, N(p,v), is the counterpart to our π(p, x) where his v is a vector of primary inputs. Samuelson ([79], 10) derived the equations w = ∇ v N(p, v). Our proof follows that of Diewert ([22], 140).

93 Samuelson’s

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But Eqs. (243) and (244) show that x* solves the profit maximization problem (242). Corollary If in addition to the above assumptions, π(p, x) is differentiable with respect to the components of p at the point (p∗ , x∗ ), so that y∗ ≡ ∇ p π(p∗ , x) exists, then (y∗ , x∗ ) solves the following long run profit maximization problem: 

   p∗ , w∗ ≡ maxy,x p∗T y − w∗T x : (y, x) ∈ S .

(245)

Proof Using Hotelling’s Lemma, we know that y* solves the following variable profit maximization problem:       π p∗ , x∗ ≡ maxy p∗T y : y, x∗ ∈ S = p∗T y∗ .

(246)

Now look at the long run profit maximization problem defined by Eq. (245):     p∗ , w∗ ≡ maxy,x p∗T y − w∗T x : (y, x) ∈ S 

 = maxx maxy p∗T y : (y, x) ∈ S − w∗T x where we have rewritten themaximization problem as a two stage maximization problem 

   = maxx π p∗ , x − w∗T x using the definition of π p∗ , x   = π p∗ , x∗ − w∗T x using Theorem 12. (247) Hence, with x = x∗ being an x solution to Eq. (247), we must have     p∗ , w∗ ≡ maxy,x p∗T y − w∗T x : (y, x) ∈ S  

  = maxy p∗T y : y, x∗ ∈ S − w∗T x∗ letting x = x∗

(248)

= p∗T y∗ − w∗T x∗ using(246). Hotelling’s Lemma and Samuelson’s Lemma can be used as a convenient method for obtaining econometric estimating equations for determining the parameters that characterize a producer’s technology set S. Assuming that S satisfies the minimal regularity conditions on S, we need only postulate a differentiable functional form for the producer’s variable profit function, π(p, x), that is linearly homogeneous and convex in p. Suppose that we have collected data on the fixed input vectors used by the production unit in period t, xt , and the net supply vectors for variable commodities produced in period t, yt , for t = 1, . . . , T time periods as well as

3 Duality in Production

137

the corresponding variable commodity price vectors pt . Then the following MT equations can be used in order to estimate the unknown parameters in π(p, x):   yt = ∇p π pt , xt + ut ; t = 1, . . . , T

(249)

where ut is a vector of errors. If in addition, S is a convex set and it can be assumed that the production unit is optimizing with respect to its vector of “fixed” inputs in each period, where it faces the “fixed” input price vector wt in period t, then the following N equations can be added to Eq. (249) as additional estimating equations:   wt = ∇x π pt , xt + vt ; t = 1, . . . , T

(250)

where vt is a vector of errors.94 We will look at some specific functional forms for π(p, x) and their econometric estimating equations in the final sections of this chapter.

The Comparative Statics Properties of Net Supply and Fixed Input Demand Functions From Theorem 11 above, we know that the firm’s variable profit function π(p, x) is convex and linearly homogeneous in the components of the vector of variable commodity prices p for each fixed input vector x. Thus, if π(p, x) is twice continuously differentiable with respect to the components of p at some point (p,x), then using Hotelling’s Lemma, we can prove the following counterpart to Theorem 7 for the cost function. Theorem 13 Hotelling ([61], 597), Hicks ([60], 321), Samuelson ([79], 10), Diewert ([22], 142–146): Suppose the variable profit function π(p, x) is linearly homogeneous and convex in p and in addition is twice continuously differentiable with respect to the components of p at some point, (p,x). Then the system of variable profit maximizing net supply functions, y(p, x) ≡ [y1 (p, x), . . . , yM (p, x)]T , exists at this point and these net supply functions are once continuously differentiable. Form the M by M matrix of net supply derivatives with respect to variable commodity prices, B ≡ [∂ym (p, x)/∂pk ], which has mk element equal to ∂ym (p, x)/∂pk . Then the matrix B has the following properties95 :

94 If

in addition, the technology set S is subject to constant returns to scale and the data reflect this fact by “adding up” (so that ptT yt = wtT xt for t = 1, . . . , T), then the error vectors ut and vt in Eqs. (249) and (250) cannot be statistically independent. Hence, under these circumstances, one of the M + N equations in (249) and (250) must be dropped from the system of estimating equations. 95 These are the Hotelling ([61], 549) and Hicks ([60], 321) symmetry restrictions on supply functions.

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B = BT so that ∂ym (p, x) /∂pk = ∂yk (p, x) /∂pm for all m = k;

(251)

B is positive semidefinite and

(252)

Bp = 0M .

(253)

Proof Hotelling’s Lemma implies that the firm’s system of variable profit maximizing net supply functions, y(p, x) ≡ [y1 (p, x), . . . , yM (p, x)]T , exists and is equal to y (p, x) = ∇p π (p, x) .

(254)

Differentiating both sides of Eq. (254) with respect to the components of p gives us   B ≡ ∂ym (p, x) /∂pk = ∇ 2 pp π (p, x) .

(255)

Property (251) follows from Young’s Theorem in calculus. Property (252) follows from Eq. (255) and the fact that π(p, x) is convex and twice differentiable in p and hence the matrix of second order partial derivatives ∇ 2 pp π(p, x) must be positive semidefinite. Finally, property (253) follows from the fact that the profit function is linearly homogeneous in p and hence, using Part 2 of Euler’s Theorem on homogeneous functions, (253) holds. Note that property (252) implies the following properties on the net supply functions: ∂ym (p, x) /∂pm ≥ 0 for m = 1, . . . , M.

(256)

Property (256) means that output supply curves cannot be downward sloping. However, if variable commodity m is an input, then ym (p,x) is negative. If we define the positive input demand function as dm (p, x) ≡ −ym (p, x) ≥ 0,

(257)

then the restriction (256) translates into ∂dm (p, x)/∂pm ≤ 0, which means that variable input demand curves cannot be upward sloping. Obviously, if the technology set is a convex cone, then the firm’s competitive fixed input price functions (or inverse demand functions), w(p, x) ≡ ∇ x π(p, x), will satisfy properties analogous to the properties of cost minimizing input demand functions in Theorem 7. Theorem 14 Samuelson ([79], 10), Diewert ([22], 144–146): Suppose that the production unit’s technology set S is regular. Define the variable profit function

3 Duality in Production

139

π(p, x) by Eq. (213). Suppose that π(p, x) is twice continuously differentiable with respect to the components of x at some point (p,x) where p  0M and x ≥ 0N . Then the system of input price functions, w(p, x) ≡ [w1 (p, x), . . . , wM (p, x)]T , exists at this point96 and these input price functions are once continuously differentiable. Form the N by N matrix of input price derivatives with respect to the “fixed” inputs, C ≡ [∂wi (p, x)/∂xk ], which has ik element equal to ∂wi (p, x)/∂xk . Then the matrix C has the following properties: C = CT so that ∂wi (p, x) /∂xk = ∂wk (p, x) /∂xi for all i = k;

(258)

C is negative semidefinite and

(259)

Cx = 0N .

(260)

Proof Using Samuelson’s Lemma, the firm’s system of fixed input price functions, w(p, x) ≡ [w1 (p, x), . . . , wN (p, x)]T , exists and is equal to w (p, x) ≡ ∇x π (p, x) .

(261)

Differentiating both sides of Eq. (261) with respect to the components of x gives us   C ≡ ∂wi (p, x) /∂xk = ∇ 2 xx π (p, x) .

(262)

Now property (258) follows from Young’s Theorem in calculus. Property (259) follows from Eq. (262) and the fact that π(p, x) is concave in x.97 Finally, property (260) follows from the fact that the profit function is linearly homogeneous in x98 and hence, using Part 2 of Euler’s Theorem on homogeneous functions, (260) holds. Note that property (259) implies the following properties on the fixed input price functions: ∂wn (p, x) /∂xn ≤ 0; n = 1, . . . , N.

(263)

Property (263) means that the inverse fixed input demand curves cannot be upward sloping. 96 The

assumption that S is regular implies that S has the free disposal property in fixed inputs property (226), which implies by part (d) of Theorem 11 that π(p, x) is nondecreasing in x and this in turn implies that w(p, x) ≡ ∇ x π(p, x) is nonnegative. 97 The assumption that S is regular implies that S is a convex set and this in turn implies that π(p, x) is concave in x. Concavity in x plus our differentiability assumption implies that ∇ 2 xx π(p, x) is negative semidefinite. 98 The assumption that S is regular implies that S is a cone and this in turn implies that π(p, x) is linearly homogeneous in x.

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If the firm’s production possibilities set S is regular and if the corresponding variable profit function π(p, x) is twice continuously differentiable with respect to all of its variables, then there will be additional restrictions on the derivatives of the variable net output supply functions y(p, x) = ∇ p π(p, x) and on the derivatives of the fixed input price functions w(p, x) = ∇ x π(p, x). Define the M by N matrix of derivatives of the net output supply functions y(p,x) with respect to the components of the vector of fixed inputs x as follows:   D ≡ ∂ym (p, x) /∂xn = ∇ 2 px π (p, x) ; m = 1, . . . , M; n = 1, . . . , N,

(264)

where the equalities in Eq. (264) follow by differentiating both sides of the Hotelling’s Lemma relations, y(p, x) = ∇ p π(p, x), with respect to the components of x. Similarly, define the N by M matrix of derivatives of the fixed input price functions w(p,x) with respect to the components of the vector of variable commodity prices p as follows:   E ≡ ∂wn (p, x) /∂pm = ∇ 2 xp π (p, x) ; n = 1, . . . , N; m = 1, . . . , M,

(265)

where the equalities in Eq. (265) follows by differentiating both sides of the Samuelson’s Lemma relations, w(p, x) = ∇ x π(p, x), with respect to the components of p. Theorem 15 Samuelson ([79], 10), Diewert ([22], 144–146): Suppose that the production unit’s technology set S is regular. Define the variable profit function π(p, x) by Eq. (213). Suppose that π(p, x) is twice continuously differentiable with respect to the components of x at some point (p,x) where p  0M and x ≥ 0N and define the matrices of derivatives D and E by Eqs. (264) and (265), respectively. Then these matrices have the following properties: D = ET so that ∂ym (p, x) /∂xn = ∂wn (p, x) /∂xm for m = 1, . . . , M and n = 1, . . . , N;

(266)

w (p, x) ≡ Ep ≥ 0N ;

(267)

y (p, x) = Dx.

(268)

Proof The symmetry restrictions (266) follow from definitions (264) and (265) and Young’s Theorem in calculus. Since π(p, x) is linearly homogeneous in the components of p, we have π (λp, x) = λπ (p, x) for all λ > 0.

(269)

Partially differentiate both sides of Eq. (269) with respect to xn and we obtain:

3 Duality in Production

141

∂π (λp, x) /∂xn = λ∂π (p, x) /∂xn for all λ > 0 and n = 1, . . . , N.

(270)

But Eq. (270) implies that the functions wn (p, x) ≡ ∂π(p, x)/∂xn are homogeneous of degree one in p. Hence, we can apply Part 1 of Euler’s Theorem on homogeneous functions to these functions wn (p,x) and conclude that 

wn (p, x) =

M

 ∂wn (p, x) /∂pm pm ; n = 1, . . . , N.

m=1

(271)

But Eq. (271) are equivalent to the equations in (267). The inequality in (267) follows from w(p, x) = ∇ x π(p, x) ≥ 0N , which in turn follows from the fact that regularity of S implies that π(p, x) is nondecreasing in the components of x. Since S is regular, part (c) of Theorem 11 implies that π(p,x) is linearly homogeneous in x, so that π (p, λx) = λπ (p, x) for all λ > 0.

(272)

Partially differentiate both sides of Eq. (272) with respect to pm and we obtain: ∂π (p, λx) /∂pm = λ∂π (p, x) /∂pm for all λ > 0 and m = 1, . . . , M.

(273)

But Eq. (273) implies that the functions ym (p, x) ≡ ∂π(p, x)/∂pm are homogeneous of degree one in x. Hence, we can apply Part 1 of Euler’s Theorem on homogeneous functions to these functions ym (p,x) and conclude that ym (p, x) =

 n=1

N

 ∂ym (p, x) /∂xn xn ; m = 1, . . . , M.

(274)

But Eq. (274) are equivalent to Eq. (268). Following up on the pioneering work of Samuelson [79], Diewert and Woodland ([46], 383–390) developed additional comparative statics properties for a consolidated production sector consisting of a finite number of constant returns to scale production units. For additional applications of the National Product Function to the theory of international trade, see Kohli [65, 67], Dixit and Norman [48], Woodland [87] and Feenstra [50]. Problems 24. Under the hypotheses of Theorem 15, show that y(p,x) and w(p,x) satisfy the following equation: (i) pT y (p, x) = xT w (p, x) . 25. Let S be a technology set that satisfies the minimal regularity assumptions and let π(p, x) be the corresponding differentiable variable profit function defined by Eq. (213). Variable commodities m and k (where m = k) are said to be

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substitutes if (i) below holds, unrelated if (ii) below holds and complements if (iii) below holds: (i) ∂ym (p, x) /∂pk < 0; (ii) ∂ym (p, x) /∂pk = 0; (iii) ∂ym (p, x) /∂pk > 0. (a) If the number of variable commodities M = 2, then show that the two variable commodities cannot be complements. (b) If M = 2 and the two variable commodities are unrelated, then show that: (iv) ∂y1 (p, x) /∂p1 = ∂y2 (p, x) /∂p2 = 0. (c) If M = 3, then show that at most one pair of variable commodities can be complements.99 26. Let S be a regular technology set and let π(p, x) be the corresponding twice continuously differentiable variable profit function defined by Eq. (213). Variable commodities m and fixed input n are said to be normal if (i) below holds, unrelated if (ii) below holds and inferior if (iii) below holds (we assume p  0M and x  0N ): (i) ∂ym (p, x) /∂xn = ∂wn (p, x) /∂pm > 0; (ii) ∂ym (p, x) /∂xn = ∂wn (p, x) /∂pm = 0; (iii) ∂ym (p, x) /∂xn = ∂wn (p, x) /∂pm < 0.

99 This type of argument (that substitutability tends to be more predominant than complementarity)

is again due to Hicks ([60], 322–323) but we have not followed his terminology exactly.

3 Duality in Production

143

(a) If wn (p, x) > 0, then there exists at least one variable commodity m such that commodity m and fixed input n are normal. (b) If wn (p, x) ≥ 0, then there exists at least one variable commodity m such that commodity m and fixed input n are either normal or unrelated. (c) If ym (p, x) > 0, then there exists at least one fixed input n such that commodity m and fixed input n are normal. (d) If ym (p, x) < 0, then there exists at least one fixed input n such that commodity m and fixed input n are inferior. In the following three sections, we will look at the properties of some specific functional forms for a variable profit function. We will assume that these profit functions are dual to a regular technology.

The Translog Variable Profit Function Assume that the log of the variable profit function for a regular technology, lnπ(p, x), has the following translog functional form100 : ln π (p, x) ≡ a0 + + +





M

m=1 N

n=1



m=1

am lnpm + (1/2)

bn lnxn + (1/2)

M

N n=1





M m=1

N

n=1

i=1

k=1 N

M

amk lnpm lnpk

bni lnxn lnxi

cmn lnpm lnxn ; (275)

The coefficients must satisfy the following restrictions in order for π(p, x) to be linearly homogeneous in the components of p as well as the components of x101 

M

m=1



N n=1

am = 1;

(276)

bn = 1;

(277)

100 This functional form was suggested by Diewert ([22], 139) as a generalization of the translog functional form introduced by Christensen et al. [14]. Diewert ([22], 139) indicated that this functional form was flexible for regular technologies. For applications of this functional form to international trade theory, see Kohli [65, 67]. For applications to index number theory and the measurement of productivity, see Caves et al. [11, 12], Diewert and Morrison [41], Kohli [66], Feenstra et al. [51] and Inklaar and Diewert [64]. 101 There are additional restrictions on the parameters which are necessary to ensure that π(p, x) is convex in p and concave in x.

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amk = akm for all k, m;

(278)

bni = bin for all n, i;

(279)



M

k=1



amk = 0 for m = 1, . . . , M;

N

i=1



N

n=1



bni = 0 for n = 1, . . . , N;

(281)

cmn = 0 for m = 1, . . . , M;

(282)

M

m=1

(280)

cmn = 0 for n = 1, . . . , N.

(283)

If some of the variable outputs are actually inputs, then the domain of definition of p and x needs to be restricted to p and x such that π(p, x) > 0, since we cannot take the logarithm of a nonpositive number. The proof that the translog profit function defined by Eqs. (275), (276), (277), (278), (279), (280), (281), (282), and (283) is linearly homogeneous in p follows our earlier proof that the translog unit cost function c(p) defined in section “The Translog Functional Form” was linearly homogeneous in p. The proof that π(p, λx) = λπ(p, x) for all λ > 0 follows in an analogous manner. Note that using Hotelling’s Lemma, we have ∂lnπ(p, x)/lnpm = [pm /π(p, x)]∂π(p, x)/∂pm = [pm /π(p, x)]ym (p, x) ≡ sm (p, x) where ym (p,x) is the profit maximizing conditional net supply function for net output m and sm (p,x) is the share of net output m in total variable profits. Thus, differentiating the logarithm of π(p, x) defined by Eq. (275) with respect to the logarithm of pm leads to the following system of net variable output share equations: sm (p, x) = am +

 k=1

M

amk lnpk +



N n=1

cmn lnxn ; m = 1, . . . , M.

(284)

Thus, if we have data on the net outputs for period t, yt , the corresponding net output prices pt  0M and fixed inputs used in period t, xt  0N by a production unit for t = 1, . . . , T, then we can form observed variable profits for period t, πt ≡ ptT yt > 0 and the period t net variable output shares sm t ≡ pm t ym t /πt for m = 1, . . . , M and t = 1, . . . , T. A set of econometric estimating equations is the following very simple system of equations:

3 Duality in Production

ln πt = a0 + + +



M m=1



M n=1

am lnpm t + (1/2)

bn lnxn t + (1/2)

N



sm t = am +

145

m=1



n=1 M

k=1

N



M

m=1



N

n=1

j=1

cmn lnpm t lnxn t + e0 t ;

amk lnpk t +



N

n=1

M

k=1 N

amk lnpm t lnpk t

bnj lnxn t lnxj t

(285)

t = 1, . . . , T;

cmn lnxn t + em t ; m = 1, . . . , M;

(286)

t = 1, . . . , T where the em t are error terms with 0 means for m = 0, 1, . . . , M and t = 1, . . . , T. Note that these equations are linear in the unknown parameters. The cross equation symmetry restrictions, amk = akm for 1 ≤ m < k ≤ M could be imposed on the above equations or these conditions could be tested.102 Suppose now that we have reason to believe that the producer is optimizing with respect to the vector of “fixed” inputs x. Using Samuelsons’s Lemma, we have ∂lnπ(p, x)/lnxn = [xn /π(p, x)]∂π(p, x)/∂xn = [xn /π(p, x)]wn (p, x) ≡ Sm (p, x) where wn (p,x) is the profit maximizing inverse demand function for “fixed” input n and Sn (p,x) is the share of “fixed” input n in total “fixed” input cost.103 Thus, differentiating the logarithm of π(p, x) defined by Eq. (275) with respect to the logarithm of xn leads to the following system of input cost share equations: Sn (p, x) = bn +

 j=1

N

bnj lnxj +

 m=1

M

cmn lnpm ; n = 1, . . . , N.

(287)

Thus if we have data on “fixed” input prices for the T periods in addition to the already mentioned data, then we can form the observed cost shares for “fixed” input n in period t, Sn t ≡ wn t xn t /πt for t = 1, . . . , T. Thus, we can add the following set of estimating equations to the estimating equations defined by Eqs. (285) and (286): Sn t ≡ bn +

 j=1

N

 bnj lnxj t +

m=1

M

cmn lnpm t + un t ; n = 1, . . . , N; t=1, . . . , T (288)

where the un t are error terms with 0 means.104

102 Since

the shares sm t sum to 1 over m for each t, the Eq. (286) cannot have independent error terms and hence one of the M equations in (286) should be dropped when estimating the unknown parameters. 103 From Problem 24, we know that pT y(p, x) = xT w(p, x) = π(p, x) since we have assumed that the underlying production possibilities set is regular. 104 Since the shares S t sum to one over n for each t, one of the N estimating equations in (287) n should be dropped. Typically, the cross equation parameter restrictions defined by Eqs. (278), (279), (280), (281), (282), and (283) would be imposed but in principle, they could be tested.

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The simplicity of the estimating equations given by Eqs. (285), (286), and (288) means that it is relatively easy to estimate the translog variable profit function. However, there are two disadvantages associated with the translog functional form: • Not all of the parameters of the translog π(p, x) can be estimated unless Eq. (285) are included in the estimation procedure. But every parameter is included in each of these equations and this can lead to singularity problems if N + M is large and T is small105 . • It is not possible to impose the convexity in p and concavity in x property for the translog functional form without destroying the flexibility of the functional form. Thus, in the following two sections, we look at functional forms for a regular technology variable profit function where we can impose the correct concavity and convexity properties. Another problem with the translog π(p, x) defined by Eq. (275) is that this functional form does not allow for technical progress. This problem can be readily remedied: terms to the right hand side of definition (275):  simply add the following  α0 t + m = 1 M αm tlnpm + n = 1 N βn tlnxn where t is a scalar time variable and the M new parameters α and β satisfy the additional restrictions m n m = 1 αm = 0 and  N β = 0.106 These restrictions will ensure that the resulting translog π(p, x) n=1 n is linearly homogeneous in p and x separately.107

The Normalized Quadratic Variable Profit Function At this point, it will be useful to list the equations that a twice continuously differentiable functional form for a variable profit function π(p, x) that is dual to a regular technology must satisfy in order to be a flexible functional form at the point p∗  0M and x∗  0N . Let π∗ (p, x) be an arbitrary variable profit function that is dual to a regular technology set and suppose that π∗ (p, x) is twice continuously differentiable at (p* ,x* ). For π(p, x) to be a flexible functional form, it must have enough free parameters so that it can provide a second order approximation to π∗ (p, x) at the point (p* ,x* ). Thus, the candidate function π must have enough parameters so that it can satisfy the following 1 + M + N + (M + N)2 equations:

105 This problem is perhaps not too serious; if Eqs. (286) and (288) are estimated, then all of the parameters that appear in definition (275) can be identified except the parameter a0 . This parameter could be estimated in a second stage where Eq. (285) are used to solve for a0 in terms of lnπt and the fitted values from the first stage for the right hand side of Eq. (285) omitting the term a0 . 106 This extension of the translog function GDP function to allow for technical progress is due to Kohli [65] in a model with four outputs and two inputs. Feenstra ([50], 423) noted these restrictions in the general M outputs and N inputs model. 107 More general specification of technical progress can be made using linear or quadratic splines in the time variable.

3 Duality in Production

147

    π p∗ , x∗ = π∗ p∗ , x∗ ; 1 equation;

(289)

    ∇p π p∗ , x∗ = ∇p π∗ p∗ , x∗ ; M equations;

(290)

    ∇x π p∗ , x∗ = ∇x π∗ p∗ , x∗ ; N equations;

(291)

    ∇ 2 pp π p∗ , x∗ = ∇ 2 pp π∗ p∗ , x∗ ; M2 equations;

(292)

    ∇ 2 xx π p∗ , x∗ = ∇ 2 xx π∗ p∗ , x∗ ; N2 equations;

(293)

    ∇ 2 px π p∗ , x∗ = ∇ 2 px π∗ p∗ , x∗ ; MN equations;

(294)

    ∇ 2 xp π p∗ , x∗ = ∇ 2 xp π∗ p∗ , x∗ ; NM equations.

(295)

However, because π(p, x) and π∗ (p, x) are both linearly homogeneous in p and x separately and both are assumed to be twice continuously differentiable at (p* ,x* ), not all of the equations in (289), (290), (291), (292), (293), (294), and (295) are independent. Equation (289) is implied by the first part of Euler’s Theorem on homogeneous functions and Eq. (290) or (291). Thus, Eq. (289) can be dropped from the list of equations that π(p, x) must satisfy since it will be satisfied if either Eq. (290) or (291) is satisfied. Since p∗ T ∇ p π(p∗ , x∗ ) = x∗ T ∇ x π(p∗ , x∗ ) and p∗ T ∇ p π∗ (p∗ , x∗ ) = x∗ T ∇ x π(p∗ , x∗ ), any one of the M + N equations in (290) and (291) can also be dropped. Young’s Theorem from calculus and the second part of Euler’s Theorem on homogeneous functions imply that if the M(M − 1)/2 equations in the upper triangle of Eq. (292) hold, then all M2 equations in (292) will hold. Similarly, if the N(N − 1)/2 equations in the upper triangle of Eq. (293) hold, then all N2 equations in (293) will hold. Young’s Theorem implies that if the MN equations in (294) hold, then the NM equations in (295) will also hold. Recall Eqs. (254), (261), (267), and (268) in section “The Comparative Statics Properties of Input Demand Functions.” These equations imply that ∇ 2 px π(p∗ , x∗ )x∗ = ∇ p π(p∗ , x∗ ) and p∗ T ∇ 2 px π(p∗ , x∗ ) = ∇ x π(p∗ , x∗ )T . The same equations will apply to the corresponding partial derivatives of π∗ (p∗ , x∗ ). Thus we need only satisfy Eq. (294) for the (M − 1) by (N − 1) submatrix of the N M matrix ∇ 2 px π(p∗ , x∗ ) that drops the last row and column of this matrix. Thus, for π(p, x) to be flexible at (p* ,x* ), we need to satisfy M + N − 1 of the equations in (290) and (291), the M(M − 1)/2 equations in the upper triangle of Eq. (292), the N(N − 1)/2 in the upper triangle of Eq. (293) and the (M − 1)(N − 1) equations in (294) that drop the equations for one row and one column of the matrix equation involving M rows and N columns. Thus, a flexible functional form for a regular variable profit function must have at least M + N − 1 + M(M − 1)/2 + N(N − 1)/2 + (M − 1)(N − 1) independent parameters.

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Recall that the normalized quadratic unit cost function was defined by Eqs. (146), (147), (148), and (149) in section “The Normalized Quadratic Unit Cost Function” above. We will adapt this functional form to our present context. Define the function r(p) for p > 0M as follows: r (p) ≡ bT p + (1/2) pT Bp/βT p

(296)

where β > 0M is a predetermined vector, b is a parameter vector and B is symmetric positive semidefinite parameter matrix that satisfies: Bp∗ = 0M .

(297)

Use the normalized quadratic functional form to define the following function of f(x) for x > 0N : f (x) ≡ aT x + (1/2) xT Ax/αT x

(298)

where α > 0N is a predetermined vector, a is a parameter vector and A is symmetric negative semidefinite parameter matrix that satisfies: Ax∗ = 0N .

(299)

Normalize α and β so that they satisfy the following restrictions: αT x∗ = 1; βT p∗ = 1.

(300)

Use the f(x) and r(p) defined above in the following definition for the normalized quadratic variable profit function 108 : π (p, x) ≡ r (p) f (x) + pT Cx

(301)

where C is an M by N parameter matrix. Using the restrictions defined by Eqs. (297), (299), and (300), the level and first and second order partial derivatives of the π(p, x) defined by Eq. (300) evaluated at (p* ,x* ) are set equal to the corresponding level and derivatives of an exogenously given π∗ (p∗ , x∗ )

108 An

  π∗ p∗ , x∗ = aT x∗ bT p∗ + p∗T Cx∗ ;

(302)

  ∇p π∗ p∗ , x∗ = baT x∗ + Cx∗ ;

(303)

alternative functional form for a variable profit function that used the r(p) and f(x) defined by Eqs. (296) and (298) as building blocks appeared in Diewert and Fox [36]. Note that net outputs y and fixed inputs x are separable if C = OM × N , an M by N matrix of 0’s. See Blackorby et al. [6] on separability concepts.

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  ∇x π∗ p∗ , x∗ = abT p∗ + CT p∗ ;

(304)

  ∇ 2 pp π∗ p∗ , x∗ = BaT x∗ ;

(305)

  ∇ 2 xx π∗ p∗ , x∗ = AbT p∗ ;

(306)

  ∇ 2 px π∗ p∗ , x∗ = baT + C.

(307)

We show that there is an a, b, A, B and C solution to the above equations. Tentatively assume that: aT x∗ = 1; Cx∗ = 0M and p∗T C = 0N T .

(308)

Substitute Eq. (308) into (303) and solve for b = ∇ p π∗ (p∗ , x∗ ). This implies that p∗ T b = π∗ (p∗ , x∗ ). Substitute Eq. (308) into (304) and solve for a = ∇ x π∗ (p∗ , x∗ )/bT p∗ = ∇ x π∗ (p∗ , x∗ )/π∗ (p∗ , x∗ ). Since x∗ T ∇ x π∗ (p∗ , x∗ ) = π∗ (p∗ , x∗ ), it can be seen that aT x∗ = 1. Substitute this equation into (305) and solve for B = ∇ 2 pp π∗ (p∗ , x∗ ), a symmetric positive semidefinite matrix that satisfies Bp∗ = 0M using the linear homogeneity of π∗ (p, x) in p. Using p∗ T b = π∗ (p∗ , x∗ ), Eq. (306) implies that A = [π∗ (p∗ , x∗ )]−1 ∇ 2 xx π∗ (p∗ , x∗ ). Thus, A is a negative semidefinite matrix that satisfies Ax∗ = 0N . Finally, define C ≡ ∇ 2 px π∗ (p∗ , x∗ )−baT =∇ 2 px π∗ (p∗ , x∗ )−[π∗ (p∗ , x∗ )]−1 ∇ p π∗ (p∗ , x∗ )∇ x π∗ (p∗ , x∗ )T . Using ∇ 2 px π∗ (p∗ , x∗ )x∗ = ∇ p π∗ (p∗ , x∗ ), p∗ T ∇ p π∗ (p∗ , x∗ ) = ∇ x π∗ (p∗ , x∗ )T and x∗ T ∇ x π∗ (p∗ , x∗ ) = π∗ (p∗ , x∗ ) = p∗ T ∇ p π∗ (p∗ , x∗ ), it can be seen that Cx∗ = 0M and p∗ T C = 0N T . Thus, the normalized quadratic profit function defined by Eq. (301) is a flexible functional form. Given data on net outputs yt , “fixed” inputs xt and their prices pt and wt for t = 1, . . . , T, econometric estimating equations for a production unit whose technology is (approximately) dual to the profit function π(p, x) defined by Eq. (301) can be obtained by using Hotelling’s Lemma and Samuelson’s Lemma to generate the following nonlinear estimating equations for t = 1, . . . , T:   −1 −2

       yt = b + βT pt Bpt − 1 2 βT pt ptT Bpt β aT x + 1 2 xtT Axt /αT xt + Cxt + ut ; (309)  −1 −2

       Axt − 1 2 αT xt xtT Axt α bT p+ 1 2 ptT Bpt /βT pt wt = a+ αT xt 

+ CT pt + vt (310)

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where the error vectors ut and vt have zero means. The disadvantage of these estimating equations is that they are more complicated than the rather straightforward comparable translog estimating equations that were obtained in the previous section. However, this functional form has the advantage that the appropriate curvature conditions can be imposed; that is, the matrices A and B that appear in the above equations can be replaced by A = − A∗ A∗ T and B = B∗ B∗ T where A* and B* are lower triangular matrices with A∗ T x∗ = 0N and B∗ T p∗ = 0M .109 These substitutions will not destroy the flexibility of the resulting functional form. Semiflexible versions of the A and B matrices can also be estimated in order to conserve on the number of parameters in the model. Finally, technical progress can easily be accommodated in the above model: simply add the time trend vector a* t to the a vector and add the time trend vector b* t to the vector b in the estimating Eqs. (308) and (309) for period t.110

The KBF Variable Profit Function In section The Konüs Byushgens Fisher Unit Cost Function” of this chapter, we studied the KBF unit cost function. This functional form can be used as a basic building block to obtain a flexible functional form for a variable profit function that is dual to a regular production possibilities set.111 Thus define the function r(p) for p > 0M as follows:  1/2  r (p) ≡ pT bbT + B p

(311)

where b is a parameter vector and B is symmetric positive semidefinite parameter matrix that satisfies: Bp∗ = 0M .

(312)

We also use the KBF functional form to define the following function of f(x) for x > 0N :  1/2  f (x) ≡ xT aaT + A x

(313)

109 After making these substitutions for A and B, the resulting π(p, x) will satisfy the convexity and

concavity conditions at the point (p,x) provided that p > 0M , x > 0N , r(p) > 0 and f(x) > 0. identification, add the constraint a∗ T 1N = 0. Of course, to achieve additional flexibility, linear or quadratic splines in time could be added to the a and b vectors; see Fox [54] or Fox and Grafton [55] for empirical examples using the normalized quadratic functional form and piece-wise linear splines to model technical progress. 111 The advantage of using this functional form over using the normalized quadratic as a basic building block is that when using the KBF functional form, we do not have to specify the exogenous vectors α and β which appeared in the normalized quadratic functional form. 110 For

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where a is a parameter vector and A is symmetric negative semidefinite parameter matrix that satisfies: Ax∗ = 0N .

(314)

We use the f(x) and r(p) defined above in the following definition for the KBF variable profit function112 : π (p, x) ≡ r (p) f (x) + pT Cx

(315)

where C is an M by N parameter matrix. Using the restrictions defined by Eqs. (312) and (314), the level and first and second order partial derivatives of the π(p, x) defined by Eq. (315) evaluated at (p* ,x* ) are set equal to the corresponding level and derivatives of an exogenously given π∗ (p∗ , x∗ )   π∗ p∗ , x∗ = aT x∗ bT p∗ + p∗T Cx∗ ;

(316)

  ∇p π∗ p∗ , x∗ = baT x∗ + Cx∗ ;

(317)

  ∇x π∗ p∗ , x∗ = abT p∗ + CT p∗ ;

(318)

  ∇ 2 pp π∗ p∗ , x∗ = BaT x∗ /bT p∗ ;

(319)

  ∇ 2 xx π∗ p∗ , x∗ = AbT p∗ /aT x∗ ;

(320)

  ∇ 2 px π∗ p∗ , x∗ = baT + C.

(321)

It can be seen that these equations are identical to Eqs. (302), (303), (304), (305), (306), and (307) in the previous section except that Eqs. (319) and (320) are slightly different from the corresponding Eqs. (305) and (306). It turns out that this difference does not affect the proof that there is an a, b, A, B, and C solution to the above equations. Thus, it is straightforward to establish that the KBF variable profit function is a flexible functional form. Given data on net outputs yt , “fixed” inputs xt and their prices pt and wt for t = 1, . . . , T, econometric estimating equations for a production unit whose technology is (approximately) dual to the profit function π(p, x) defined by Eq. (315) can be obtained by using Hotelling’s Lemma and Samuelson’s Lemma to generate the following nonlinear estimating equations for t = 1, . . . , T:

112 Net

outputs y will be separable from inputs x if C = OM × N .

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  −1/2   1/2 xtT aaT + A xt yt = bbT pt + Bpt ptT bbT + B pt + Cxt + ut ; (322)

  −1/2   1/2 ptT bbT + B pt + CT pt + vt wt = aaT xt + Axt xtT aaT + A xt (323) where the error vectors ut and vt have zero means. Again, the disadvantage of these estimating equations is that they are a lot more complicated than the rather straightforward comparable translog estimating equations that were obtained for the translog functional form. However, as was the case with the normalized quadratic profit function, this functional form has the advantage that the appropriate curvature conditions can be imposed without destroying the flexibility of the functional form; that is, the matrices A and B that appear in the above equations can be replaced by A = − A∗ A∗ T and B = B∗ B∗ T where A* and B* are lower triangular matrices with A∗ T x∗ = 0N and B∗ T p∗ = 0M .113 As usual, semiflexible versions of the A and B matrices can also be estimated in order to conserve on the number of parameters in the model. And again as usual, flexible forms of technical progress can easily be accommodated in the above model by adding the time trend vector a* t to the a vector and add the time trend vector b* t to the vector b in the estimating Eqs. (322) and (323) for period t.114 The KBF functional form developed in this section is very similar to the normalized quadratic functional form that was developed in the previous section. However, the KBF functional form has the advantage that it is not necessary to specify an α and β vector a priori as was the case for the normalized quadratic profit function. The KBF functional form seems to be the most promising parsimonious functional form that has been developed up to the present.

Joint Cost Functions Instead of maximizing profits with respect to variable inputs and outputs, in this section we minimize cost subject to producing a specified vector of outputs. Thus consider a production unit that produces the output vector y ≥ 0M using an input vector x ≥ 0N . The set of feasible output and input vectors (y,x) is a set S which satisfies the following minimal regularity condition115 :

113 After making these substitutions for A and B, the resulting π(p, x) will satisfy the convexity and

concavity conditions provided at the point (p,x) provided that p > 0M , x > 0N , r(p) > 0 and f(x) > 0. in order to identify all of the parameters, add the constraint a∗ T 1N = 0. To achieve additional flexibility, linear or quadratic splines in time could be added to the a and b vectors. 115 Note that in this section, y is a vector of outputs rather than a vector of net outputs as in previous sections. 114 Again,

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S is closed subset of M + N space such that for every output vector y ≥ 0M , there exists an input vector x ≥ 0N such that (y, x) ∈ S. (324) Let w  0N be a strictly positive vector of input prices and let y ≥ 0M be an output vector. Define the producer’s joint cost function C(y,w) as follows:   C (y, w) ≡ minx wT x : (y, x) ∈ S .

(325)

The regularity conditions (324) on S and the assumption that w  0N imply that the minimum in Eq. (325) will exist. It is frequently useful to assume that S satisfies free disposability of inputs, property (326) below, and/or free disposability of outputs, property (327) below.     y ≥ 0M , 0N ≤ x1 < x2 and y, x1 ∈ S implies y, x2 ∈ S. (326)     0M ≤ y1 < y2 and y2 , x ∈ S implies y1 , x ∈ S.

(327)

Problems 27. Theorem 15: Suppose S satisfies conditions (324) and define C(y,w) by Eq. (325) for y ≥ 0M and w  0N . Show that C(y,p) has the following properties: (i) C(y,w) is a nonnegative function; that is, C(y, w) ≥ 0 for y ≥ 0M and w  0N . (ii) C(y,w) is positively linearly homogeneous in p for each fixed y; that is, C(y, λ, w) = λC(y, w) for all λ > 0, w  0N and y ≥ 0M . (iii) C(y,w) is nondecreasing in w for each fixed y; that is, C(y, w1 ) ≤ C(y, w2 ) for y ≥ 0M and w2 > w1  0N . (iv) C(y,w) is a concave function of w for each fixed y; that is, C(y, λw1 + (1 − λ)w2 ) ≥ λC(y, w1 ) + (1 − λ)C(y, w2 ) for y ≥ 0M , w1  0N ; w2  0N and 0 < λ < 1. (v) C(y,w) is a continuous function of w for each fixed y ≥ 0M . Hint: Adapt the proof of Theorem 1 in section “Cost Functions: The One Output Case” above. 28. Continuation of 27: Suppose S satisfies the free disposability of outputs property (327) in addition to the minimal regularity conditions (324). Show that C(y,w) is nondecreasing in y for fixed w; that is, show that w  0N , 0M ≤ y1 < y2 and (y2 , x) ∈ S implies C(y1 , p) ≤ C(y2 , p). Hint: Use a feasibility argument.

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Thus, the joint cost function C(y,w) has much the same properties with respect to input prices as the single output cost function that was studied in section “Cost Functions: The One Output Case” above. In particular, C(y,w) must be linearly homogeneous and concave in w for fixed y. Under what conditions can a knowledge of the joint cost function, C(y,p), be sufficient to determine the underlying technology set S? We now address this question. Suppose S satisfies the minimal regularity conditions (324). For each y ≥ 0M , define the set of inputs that can produce at least y, L(y), as follows: L (y) ≡ {x : (y, x) ∈ S} .

(328)

If we are given the family of upper level sets, L(y) for every y ≥ 0M , then S can be recovered using S = {(y, x) : y ≥ 0M and x ∈ L(y)}. Thus, the above question can be reduced to the equivalent question: under what assumptions on L(y) can the joint cost function be used to determine L(y) for each y ≥ 0M ? We can use the method explained in section “The Duality Between Cost and Production Functions” above to answer this question. Let y ≥ 0M and w  0N . Use the given joint cost function C(y,w) to define the following half space of inputs:   M (y, w) ≡ x : wT x ≥ C (y, w) .

(329)

The above half space must contain the level set L(y). Thus, L(y) must be contained in the following set, which is the intersection of all of the supporting halfspaces to L(y): M (y) ≡ ∩w0N M (y, p) .

(330)

Since each of the sets in the intersection, M(y,p), is a convex set, then M(y) is also a convex set. Since L(y) is a subset of each M(y,p), it must be the case that L(y) is also a subset of M(y); that is, we have L(y) ⊂ M(y). As was the case in section “The Duality Between Cost and Production Functions,” in order to ensure that M(y) = L(y), we need to add the following two conditions on the family of level sets L(y): For each y ≥ 0M , L (y) satisfies free disposability of inputs; i.e., x1 ∈ L (y) , x2 ≥ x1 implies x2 ∈ L (y) . (331)

For each y ≥ 0M , L (y) is a convex set.

(332)

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Condition (331) on the family of as input level sets L(y) is equivalent to condition (326) on the production possibilities set S. As in section “The Duality Between Cost and Production Functions,” assumptions (331) and (332) rule out backward bending and nonconvex input production possibilities sets L(y). As was the case in section “The Duality Between Cost and Production Functions,” if the producer is a price taker in input markets, then it is not necessary to assume properties (331) and (332) when estimating a joint cost function: a cost minimizing producer will never choose an input vector that belongs to a nonconvex or backward bending upper level set L(y). Thus, an estimated joint cost function can be used to form the upper level sets M(y) and these sets can provide an adequate approximation to the true L(y) for most purposes. If the joint cost function C(y,w) satisfies the conditions listed in Theorem 15 and is differentiable with respect to input prices w, then we can show that Shephard’s Lemma still holds; that is, the producer’s system of cost minimizing input demand functions is equal to x(y, w) ≡ ∇ w C(y, w) for y ≥ 0M and w  0N .116 If the production possibilities set S has additional properties, then we can deduce that the joint cost function C(y,w) has additional properties. Two familiar additional properties for S are the following ones:     S is a convex set; i.e., y1 , x1 ∈ S, y1 , x1 ∈ S and 0 < λ   < 1 implies λy1 + (1 − λ) y2 , λx1 + (1 − λ) x2 ∈ S. S is a cone; i.e., if (y, x) ∈ S and λ > 0, then (λy, λx) ∈ S.

(333)

(334)

The cone assumption (334) means that production is subject to constant returns to scale. The convexity assumption rules out technologies that are subject to increasing returns to scale. Some of the implications of these assumptions are listed in the following problems. Problems 29. Assume S satisfies Eq. (234) and the convexity assumption (333). (i) Show that L(y) ≡ {x : (y, x) ∈ S} is a convex set for each y ≥ 0M . (ii) Show that C(y,w) defined by Eq. (325) is a convex function of y for fixed w  0N . Hint: Look at the proof of part (b) of Theorem 11. 30. Assume S satisfies Eq. (234) and the output free disposability assumption (327). Show that C(y,w) defined by Eq. (325) is a nondecreasing function of y for fixed w  0N . Hint: Use a feasibility argument.

116 The

proof of Theorem 5 in section “The Derivative Property of the Cost Function” can be adapted to prove this result.

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31. Assume S satisfies Eq. (234) and the cone assumption (334). Show that C(y,w) defined by Eq. (325) is a linearly homogeneous function of y for fixed w  0N . Hint: Modify the proof of part (c) of Theorem 11. The above problems show that if S satisfies the output free disposal assumption (327) and the convexity and constant returns to scale assumptions (333) and (334), then the corresponding joint cost function C(y,w) will be a nondecreasing, linearly homogeneous, and convex function of y for fixed w. Assume that C(y,w) is differentiable with respect to y and w. Shephard’s Lemma enables us to interpret the vector of first order partial derivatives of the joint cost function with respect to the input price vector w, ∇ w C(y, w), as the producer’s vector of input demand functions, x(y,w). The vector of first order partial derivatives of the joint cost function with respect to y, ∇ y C(y, w), is obviously the vector of marginal costs for each output. However, if S satisfies the convexity assumption (333), then p = ∇ y C(y, w) can be interpreted as the producer’s system of inverse supply functions; that is, if the producer faced the output price vector p and the input price vector w, then an output vector y which satisfied the system of equations p = ∇ y C(y, w) and the x = ∇ w C(y, w) would be a solution to the following producer’s profit maximization problem:   maxy,x pT y − wT x : (y, x) ∈ S .

(335)

Theorem 16 Suppose the technology set S satisfies the minimal regularity assumptions (324) plus (326) (free disposability of inputs), (327) (free disposability of outputs) and (333) (convexity). Let y∗ ≥ 0N and w∗  0N . Suppose that C(y,w) is differentiable at (y* ,w* ). Define x∗ ≡ ∇ w C(y∗ , w∗ ) and p∗ ≡ ∇ y C(y∗ , w∗ ). Then (y* ,x* ) is a solution to the following profit maximization problem:   maxy,x p∗T y − w∗T x : (y, x) ∈ S .

(336)

Proof The free disposability assumptions imply that p∗ ≥ 0M and x∗ ≥ 0M . The convexity assumption on S implies that C(y,w* ) is a convex function of y. Thus, the function f(y) ≡ C(y, w∗ ) − p∗ T y is also a convex function of y for all y ≥ 0M . Note that ∇f(y∗ ) = ∇ y C(y∗ , w∗ ) − p∗ = 0M using the definition of p* . Since f(y) is a convex function and differentiable at y = y∗ , its first order Taylor series approximation around this point will lie below (or be coincident with) f(y). Thus, we have for all y ≥ 0M :  T     y − y∗ f (y) ≥ f y∗ + ∇f y∗   = f y∗

(337)

where the inequality follows since ∇f(y∗ ) = 0M . Thus, f(y) attains a global minimum at y* . Using the definition of f, we see that y* is a solution to the following

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minimization problem:       miny C y, w∗ − p∗T y; y ≥ 0M = C y∗ , w∗ − p∗T y∗   = w∗T ∇w C y∗ , w∗ − p∗T y∗

(338)

= w∗T x∗ − p∗T y∗ where the second equality follows from the linear homogeneity of C(y,w) in w and the third equality follows from the definition of x∗ ≡ ∇ w C(y∗ , w∗ ). It can be verified that solving the profit maximization problem defined by Eq. (335) is equivalent to solving the following (net) cost minimization problem:   

 miny,x w∗T x − p∗T y : (y, x) ∈ S = miny minx w∗T x : (y, x) ∈ S − p∗T y     = miny C y, w∗ −p∗T y; y≥0M using (325) = w∗T x∗ − p∗T y∗

using (338). (339)

The above result is a joint cost function counterpart to Samuelson’s Lemma, Theorem 12 above. It says that if producers take prices as given on both input and output markets and the technology set is convex, then the producer’s system of inverse supply functions, p(y,x), is equal to ∇ y C(y,w), the producer’s system of marginal cost functions. If the production possibilities set S satisfies all of the regularity conditions on S that are listed in this section (free disposability of inputs and outputs, convexity, and constant returns to scale), we say that S is a regular production possibilities set. Problems 32. Suppose S satisfies the minimal regularity conditions (324). Define the corresponding joint cost function C(y,w) by Eq. (325). Suppose C(y,w) is twice continuously differentiable with respect to w at some point y > 0M and w  0N . Then the system of cost minimizing input demand functions is x(y, w) = ∇ w C(y, w) and the N by N matrix of demand derivatives with respect to input prices, B ≡ [∂xn (y, w)/∂wi ] = ∇ 2 ww C(y, w) exists. Show that the matrix B has the following properties: (i) B = BT . (ii) B is negative semidefinite. (iii) Bw = 0M . Hint: Adapt the proof of Theorem 13. 33. Suppose S is a regular production possibilities set and the corresponding C(y,w) is twice continuously differentiable at the point y  0M and w  0N . Then the system of inverse supply functions, p(y, w) = ∇ y C(y, w) and the M by M matrix of partial derivatives with respect to output quantities,

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A ≡ [∂pm (y, w)/∂yk ] = ∇ 2 yy C(y, w) exist. Show that the matrix A has the following properties: (i) A = AT so that ∂pm (y, w)/∂yk = ∂pk (y, w)/∂ym for all m = k. (ii) A is positive semidefinite and. (iii) Ay = 0M . Hint: Adapt the proof of Theorem 14. 34. Suppose S is a regular production possibilities set and the corresponding C(y,w) is twice continuously differentiable at the point y  0M and w  0N . Then the system of inverse supply functions is p(y, w) = ∇ y C(y, w) and the M by N matrix of partial derivatives supply prices with respect to input prices, D ≡ [∂pm (y, w)/∂wn ] = ∇ 2 yw C(y, w) exists. The system of cost minimizing input demand functions is x(y, w) = ∇ w C(y, w) and the N by M matrix of partial derivatives of input quantities with respect to output quantities, E ≡ [∂xn (y, w)/∂ym ] = ∇ 2 wy C(y, w) exists. Show that the matrices D and E have the following properties: (i) D = ET ; (ii) p (y, w) = Dw ≥ 0M ; (iii) x (y, w) = Ey ≥ 0N . Hint: Adapt the proof of Theorem 15. Shephard’s Lemma and Theorem 16 can be used as a convenient method for obtaining econometric estimating equations for determining the parameters that characterize a producer’s technology set S. Assuming that S satisfies the minimal regularity conditions on S, we need only postulate a differentiable functional form for the producer’s joint cost function, C(y,w), that is linearly homogeneous and concave in w. Suppose that we have collected data on the input vectors used by the unit in period t, xt , and the outputs produced in period t, yt , for t = 1, . . . , T time periods as well as the corresponding input price vectors wt . Then the following NT equations can be used in order to estimate the unknown parameters in C(y,w):   xt = ∇w C yt , wt + ut ; t = 1, . . . , T

(340)

where ut is a vector of errors. If in addition, S is a convex set and the firm is maximizing profits facing the fixed output and input price vectors, pt and wt , respectively, in period t, then the following MT equations can be added to Eq. (340) as additional estimating equations:

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  pt = ∇y C yt , wt + vt ; t = 1, . . . , T

(341)

where vt is a vector of errors.117

Flexible Functional Forms for Joint Cost Functions Specific functional forms for C(y,w) can be found by adapting the functional forms explained in sections “The Translog Variable Profit Function,” “The Normalized Quadratic Variable Profit Function,” and “The KBF Variable Profit Function” above. Adapting the material in section “The Translog Variable Profit Function,” we could assume the log of the joint cost function for a regular technology, lnC(y,w), has the following translog functional form118 : ln C (y, w) ≡ a0 + + +



M m=1



N n=1

am lnym + (1/2)

bn lnwn + (1/2)

M

 m=1

n=1

N



M

m=1



N

n=1

j=1

k=1 N

M

amk lnym lnyk

bni wn lnwi

cmn lnym lnwn . (342)

The unknown coefficients in Eq. (342) must satisfy the restrictions (276), (277), (278), (279), (280), (281), (282), and (283) listed in section “The Translog Variable Profit Function” if S is a regular production possibilities set. Note that using Shephard’s Lemma, we have ∂lnC(y, w)/lnwn = [wn /C(y, w)] ∂C(y, w)/∂wn = [wn /C(y, w)]xn (y, w) ≡ Sn (y, w) where xn (y,w) is the cost minimizing demand function for input and Sn (y,w) is the share of input n in total cost. Assuming that the producer minimizes cost and S is dual to the translog joint cost function defined by Eq. (342), then differentiating the logarithm of C(y,w) defined by Eq. (342) with respect to the logarithm of wn leads to the following system of input share equations: Sn (y, w) = bn +

117 If

 j=1

N

bnj lnwj +

 m=1

M

cmn lnym ; n = 1, . . . , N.

(343)

in addition, the technology set S is subject to constant returns to scale and the data reflect this fact by satisfying ptT yt = wtT xt for t = 1, . . . , T, then the error vectors ut and vt in Eqs. (340) and (341) cannot be statistically independent. Hence one of the M + N equations in (340) and (341) must be dropped from the system of estimating equations. 118 This functional form is due to Burgess [9] who applied it to international trade theory. For applications of this functional form to index number theory, see Diewert and Morrison [41] and Diewert and Fox [34].

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Equations (342) and (343) can be used as estimating equations if the production unit is minimizing costs. Note that these equations are linear in the unknown parameters.119 Suppose that in addition to the assumption that the production unit is minimizing costs, we assume that the technology set is regular and the producer is maximizing profits. Using Theorem 16, ∂lnC(y, w)/lnym =[ym /C(y, w)]∂C(y, w)/∂ym = [ym /C(y, w)]pm (y, w) ≡ sm (p, x) where pm (y,w) is the profit maximizing inverse demand function for output m and sm (p,x) is the share of output m in total profit maximizing revenue. Assume that S is regular. Assuming that the producer maximizes profit and S is dual to the translog joint cost function C(y,w) defined by Eq. (342), then differentiating the logarithm of C(y,w) with respect to the logarithm of ym leads to the following system of revenue share equations: sm (y, w) = am +



M

k=1

amk lnyk +

 n=1

N

cmn lnwn ; m = 1, . . . , M.

(344)

Equations (342), (343), and (344) can be used as estimating equations if the production unit is maximizing profits and has a regular translog technology. The above functional form for the logarithm of joint cost does not allow for technical progress. To remedy this problem,  simply add the following  terms to the right hand side of definition (342): α0 t + m = 1 M tαm lnym + n = 1 N tβn lnwn where t is a scalar  time variable and the new parameters αm and βn satisfy  M α = 0 and N 120 m=1 m n = 1 βn = 0. A problem with the translog joint cost function is that it is not possible to impose concavity in w (and convexity in y if the dual S satisfies convexity) over the region spanned by the sample input prices wt (and the region spanned by the sample output vectors yt if S is a convex set) without impairing the flexibility of the functional form. In order to impose these curvature conditions without destroying the flexibility property, we turn to the functional forms defined in sections “The Normalized Quadratic Variable Profit Function” and “The KBF Variable Profit Function.” Define the normalized quadratic joint cost function C(y,w) for y > 0M and w > 0N as follows: C (y, w) ≡ g (y) c (w) + yT Ew

(345)

where g(y) ≡ bT y + (1/2)yT By/βT y, β > 0M is a predetermined vector that satisfies βT y∗ = 1, b > 0M is a parameter vector, B is symmetric positive semidefinite parameter matrix that satisfies By∗ = 0M , c(w) ≡ aT w + (1/2)wT Aw/αT w, α > 0N is a predetermined vector that satisfies αT w∗ = 1, a is a parameter vector that satisfies 119 If we do not impose constant returns to scale and convexity on S, then the parameter restrictions

(277) and (281), (282), and (283) do not have to be imposed. These restrictions should be imposed if we assume constant returns to scale and convexity. 120 Linear or quadratic spline functions in time can also be added to the estimating equations to better approximate variable rates of technical progress over time.

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aT w∗ = 1, A is symmetric negative semidefinite parameter matrix that satisfies Aw∗ = 0N and E is an M by N parameter matrix. Define the KBF joint cost function C(y,w) using Eq. (345) where E is again an M by N parameter matrix. However, redefine g(y) and c(w) as follows: g(y) ≡ (yT [bbT + B]y)1/2 , where b > 0M is a parameter vector, B is symmetric positive semidefinite parameter matrix that satisfies By∗ = 0M , c(w) ≡ (wT [aaT + A]w)1/2 , a is a parameter vector that satisfies aT w∗ = 1 and A is symmetric negative semidefinite parameter matrix that satisfies Aw∗ = 0N . For both of these joint cost functions, the vector of cost minimizing input demand functions x(y,p) can be obtained by calculating the vector of first order partial derivatives, ∇ w C(y, w). The concavity in input prices property for the joint cost function can be imposed by setting A = − A∗ A∗ T with A* lower triangular and A∗ T w∗ = 0N . In the case where the underlying production possibilities set S is convex, the vector of profit maximizing output prices p(y,w) that is consistent with the production of the vector y of outputs can be obtained by calculating the vector of first order partial derivatives, ∇ y C(y, w). The convexity property in output quantities for C(y,w) can be imposed by setting B = B∗ B∗ T with B* lower triangular and B∗ T y∗ = 0M .121 The normalized quadratic and KBF joint cost functions as defined above do not allow  for technicalprogress. This problem can be remedied by adding the term ( m = 1 M γm ym t) ( n = 1 N δn wn t) to the right hand side of definitions (345) where the γm and δn are technical progress parameters and t is a time trend.122 These additional technical progress terms may not capture the trends in technical progress in series context  the time  ifN the sample period is long. In this case, the terms M γ y t and m=1 m m n = 1 δn wn t can be replaced by piece-wise linear spline functions as was done in section “The Use of Splines for Modeling Technical Progress” above; see Eq. (200).

121 After

making these substitutions for A and B, the resulting C(y,w) will satisfy the convexity and concavity conditions provided at the point (y,w) provided that y > 0M , w > 0N , g(y) > 0 and c(w) > 0. The proof of the flexibility of the normalized quadratic and KBF joint cost functions in the case of a regular technology is entirely analogous to the corresponding proofs of normalized quadratic and KBF variable profit functions that were discussed in sections “The Normalized Quadratic Variable Profit Function” and “The KBF Variable Profit Function.” 122 In order to identify all of these technical progress parameters, we need to impose a normalization  on them such as m = 1 M γm = 1.

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Applications of Joint Cost Functions In this section, we discuss three areas of research where joint cost functions play important roles. Many government outputs are produced in a nonmarket context. The output quantities can usually be measured but typically, there are no market prices for the outputs that are produced by many government production units. However, government producers still have an incentive to minimize costs. If the public sector production unit is minimizing costs and the technology set can be approximated by a constant returns to scale production possibilities set S and econometric estimation of a differentiable dual joint cost function C(y,w) is possible (using just the input demand functions as estimating equations), then approximate output prices can be obtained as the vector of marginal costs, p ≡ ∇ y C(y, w). If production is subject to constant returns to scale, then the resulting output price vector p will have the property that pT y = C(y, w) = wT x; that is, the resulting value of outputs will equal the value of inputs.123 This result is useful in the national income accounting context where government statisticians have to find methods for valuing public sector outputs. Using marginal cost prices is also useful when economists want to measure the productivity performance of public sector production units.124 A second application for the estimation of joint cost functions is in the context of the regulation of utilities that deliver electricity, water, and communications services via networks. Regulators are interested in using marginal costs to aid them in setting utility prices. Utilities may be forced to sell their outputs at regulated prices that do not reflect marginal costs but regulated utility firms will still have an incentive to minimize costs. In this case, joint cost functions can be estimated and the resulting estimates can be used to measure technical progress as well as the total factor productivity of the regulated firms.125 A third area where joint cost functions play an important role is in modeling monopolistic behavior. Typically producers take input prices as fixed and beyond their control. However, they may have some pricing power over their outputs. Recall Eq. (245) which defined a producer’s competitive profit maximization problem. A monopolistic counterpart to this problem is the following problem: maxy.x



= maxy

123 In

M m=1

 fm (ym ) ym − w x : (y, x) ∈ S

 m=1

T

M



(346)

fm (ym ) ym − C (y, w)

practice, the vector of marginal costs may have to be approximated by average costs of production, which in turn will usually require many accounting imputations. 124 See Diewert [29–31] on this topic. 125 For examples of the use of joint cost functions in a regulatory context, see Denny et al. [19], Lawrence and Diewert [70] and Diewert et al. [47].

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where w  0M is a positive input price vector, y ≡ (y1 , . . . , yM ) ≥ 0M is an output vector, S is the producer’s production possibilities set, C(y,w) is the producer’s joint cost function defined by Eq. (325) and pm = fm (ym ) is the (downward sloping) inverse demand function for output m that the producer faces for m = 1, . . . . M. If the inverse demand functions fm (ym ) and the joint cost function C(y,w) are once differentiable when evaluated at the period t data, then under appropriate regularity conditions on the fm (ym ) and S, the following equations will be satisfied by a profit maximizing monopolist using the observed period t data:     pˆ t 1M − μt = ∇y C yt , wt ; t = 1, . . . , T;

(347)

  xt = ∇w C yT , wT ; t = 1, . . . , T

(348)

where 1M is an M dimensional vector of ones, μt ≡ [μ1 t , . . . , μM t ]T ≥ 0M is a period t markup vector where μm t ≡ − [ym t /pm t ][∂fm (ym t )/∂ym ] is the markup of price over marginal cost for output m in period t, yt , and xt are the observed quantity vectors for outputs and inputs in period t, pt and wt are the corresponding observed output and input price vectors for period t = 1, . . . , T and pˆ t is an M by M diagonal matrix with the elements of the vector pt on the main diagonal. If the markups are constant over time, given a suitable functional form for the joint cost function C(y,w), Eqs. (347) and (348) can be used as econometric estimating equations.126 Thus again, joint cost functions play a crucial role in this area of economics.127

Problems that Require Additional Research We conclude this chapter with some comments on three problem areas that have not been addressed in the above sections. The first problem area is the difficulty of distinguishing increasing returns to scale from technical progress if there is general growth of all inputs and outputs for the production unit that is under consideration. Multicollinearity problems usually arise in this situation: the two effects typically cannot be reliably determined using just time series data. The second problem area is the fact that many inputs cannot be varied in the short run and thus producers are not necessarily producing outputs and utilizing inputs on

126 If the markups are not constant, then linear (or piece-wise linear) trends in the markups could be

introduced into the model. See Diewert and Fox [33] for an econometric application of this model and Diewert and Fox [34] for an application of this model to index number theory. 127 If the monopolist provides some goods and services on a competitive basis (i.e., at marginal cost), then the markup for this commodity can be set equal to zero. Alternatively, this commodity could be removed from the y vector and be placed with the x inputs, except the quantity would be indexed with a negative sign in the input demand equations. The resulting input cost would become input cost less the revenue from the sales of goods and services provided at marginal cost.

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the frontiers of their production possibilities sets. For example, suppose a recession occurs in the economy so that demand for the outputs of production units declines. Producers can reduce the demand for their variable inputs but they are more or less stuck with their structure inputs and with other durable capital investments that are “bolted down.”128 Thus, producers end up being in the interior of their production possibilities sets.129 When a producer makes an irreversible investment, the total cost of the investment should not be charged to the period when the investment was made but this cost should be allocated over the useful life of the investment. But how exactly should this cost be allocated? This is the fundamental problem of accounting.130 Note that in addition to structure and network capital inputs, a successful R&D project is another example of a fixed cost input whose input cost must be allocated over time in some manner. If there is only a single sunk cost input (or we aggregate all sunk cost inputs into a single input), then it is possible to set up an intertemporal profit maximization problem that justifies the purchase of the fixed input. The price of this fixed asset at a particular point in time is the discounted net revenue generated by the project over its remaining useful life and if this information on discounted net revenues can be forecasted, then the initial cost of the asset can be amortized in a manner that is proportional to the forecasted net revenues by period.131 The final problem area that has not been addressed in this survey of the applications of duality theory in production theory is the new goods problem and the problem of quality change. Modern economies are subject to tremendous product churn, and in addition, revolutionary new products are constantly being developed.132 Up to this point, we have assumed that the production unit is producing M outputs and N inputs and this set of outputs and inputs remains constant over time (if we are in the time series context) or it remains constant over different production units in the same industry (in the cross sectional context). If the underlying technology set St for a production unit does not change very much when new outputs appear and some old outputs disappear in period t, then the various econometric models proposed above could in theory deal with this problem if we allow for technical change. But if there are many such changes over many periods,

128 Some

labour hoarding may also occur; i.e., the costs of firing and then rehiring workers after the recession is over may be higher than just keeping the workers employed. 129 This inefficiency problem will be addressed in other chapters in this Handbook using nonparametric production analysis or  Chap. 10, “Data Envelopment Analysis: A Nonparametric Method of Production Analysis”; see Charnes and Cooper [13]. Most of the research in this area is applied to cross sectional or panel data. For an application of the nonparametric approach to production theory and the measurement of efficiency in the time series context, see Diewert and Fox [37]. 130 See Cairns [10]. 131 For examples of this methodology, see Diewert [28], Diewert et al. [47], Diewert and Huang [39], Cairns [10] and Diewert and Fox [35]. 132 See Broda and Weinstein [7, 8], Bernard et al. ([4], 82) and Hottman et al. ([63], 1300) for information on the number of products sold in the US (at least 1.6 million). The last three papers have information on the frequency of product entry and exit in the US (about 2% per month).

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obviously, we will not be able to estimate flexible functional forms due to the proliferation of parameters. Even if output changes are infrequent, the production of a new output and the discontinuance of an existing output could lead to a radical change in the use of inputs as the newer technology replaces the existing one and again, we will have a proliferation of parameters, a lack of degrees of freedom and our suggested econometric approaches will fail. Thus, there is a need for further research to address these problems. Acknowledgments The author thanks Robert Cairns, Kevin Fox, John Hartwick, Robert Inklaar, Peter Neary, Subhash Ray, Stephen Redding, Philip Vermeulen and Valentin Zelenyuk for helpful comments and the SSHRC of Canada for financial support.

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4

Multiproduct Technologies Rolf Färe, Daniel Primont, and W. L. Weber

Contents Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The Production Technology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Set Representations of Technologies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Cost and Revenue Indirect Technologies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Functional Representations of the Technology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Radial Distance Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A Joint Production Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Directional Distance Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A Distance Function Tree . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Cost and Revenue Indirect Distance Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Optimization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Profit Maximization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Cost Minimization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Revenue Maximization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Efficiency Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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R. Färe () Department of Economics and Department of Agricultural and Resource Economics, Oregon State University, Corvallis, OR, USA Department of Economics and Department of Applied Economics, School of Public Policy, Oregon State University, Corvallis, OR, USA Department of Agricultural Economics, University of Maryland, College Park, MD, USA e-mail: [email protected] D. Primont Department of Economics, Southern Illinois University-Carbondale, Carbondale, IL, USA e-mail: [email protected] W. L. Weber Department of Accounting, Economics and Finance, Southeast Missouri State University, Cape Girardeau, MO, USA e-mail: [email protected] © Springer Nature Singapore Pte Ltd. 2022 S. C. Ray et al. (eds.), Handbook of Production Economics, https://doi.org/10.1007/978-981-10-3455-8_5

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Duality Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Cost Function Dualities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Revenue Function Dualities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Profit Function Dualities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Calculus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Shadow Pricing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Scale Elasticities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Elasticities of Substitution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Appendix: Production Axioms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Cross-References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Introduction This chapter lays out the theory of multiproduct technologies where inputs are transformed into outputs by producers. The technology underlying production can be represented by various sets as shown in section “The Production Technology”, for instance, the input requirement set which includes all input combinations that produce given outputs or the output set which includes all feasible output combinations that can be produced from given inputs. If input prices or output prices are known, cost or revenue indirect sets can be useful representations of the technology. Section “Functional Representations of the Technology” shows how functions can be used as representations of the various technology sets with the functions inheriting their properties from the sets. Producers pursue various objectives such as maximum profits, minimum costs, and maximum revenues. The optimization of these objectives is covered in section “Optimization” as well as a comparison between observed outcomes of the production process and optimal outcomes in what is known as efficiency measurement. When the technology set is convex, one can move between functions defined on the quantity space and functions defined on the price space, and various duality results are shown in section “Duality Theory”. When differentiable, the functional representations of the technology can be used to recover shadow prices of inputs or outputs and scale elasticities or substitution elasticities as shown in section “Calculus”.

The Production Technology A production technology describes how inputs are transformed into outputs. Sets or functions can be used to represent the technology. In this section, we begin with set representations of the technology and discuss various axioms on the technology sets. These sets serve as the foundation for describing multiproduct technologies, i.e., technologies where multiple inputs are transformed into multiple outputs. Sets describing how input quantities are transformed into output quantities are first described. Then, we show how various technologies can be described by prices and values, for instance, output sets that depend on input prices and costs or input sets that depend on output prices and a target level of revenue. The various alternatives

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allow the empirical economist who is confronted with limited data a meaningful way to represent the technology.

Set Representations of Technologies In this section, we represent the technologies as sets, including the output, input, and M technology sets. Let x ∈ RN + denote inputs and y ∈ R+ outputs. The technology set is given by T = {(x, y) : x can produce y}.

(1)

The input requirement set is denoted by L(y) = {x : x can produce y} = {x : (x, y) ∈ T }

(2)

and the output set is P (x) = {y : x can produce y} = {y : (x, y) ∈ T }.

(3)

From these definitions, it follows that (x, y) ∈ T ⇔ x ∈ L(y) ⇔ y ∈ P (x),

(4)

i.e., each set is a complete representation of a technology and is often associated with a different optimization problem. Profit maximization is best modeled with the technology set, revenue maximization with the output set, and cost minimization with the input set. Activity Analysis1 (AA) or Data Envelopment Analysis2 (DEA) models are examples of a specific set representation. Let (x k , y k ) k = 1 . . . , K represent the K observations of input and output vectors. The AA/DEA model constructed from these data is T = {(x, y) :

K 

zk xkn ≤ xn , n = 1, . . . , N,

k=1 K 

zk ykm ≥ ym , m = 1, . . . , M,

k=1

zk ≥ 0, k = 1, . . . , K}

1 John

von Neumann [26] introduced activity analysis. et al. [6] coined the term Data Envelopment Analysis.

2 Charnes

(5)

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where zk ≥ 0, k = 1, . . . , K are the intensity variables. A corresponding output set is P (x) = {y :

K 

zk xkn ≤ xn , n = 1, . . . , N,

k=1 K 

zk ykm ≥ ym , m = 1, . . . , M,

k=1

zk ≥ 0, k = 1, . . . , K}

(6)

and the corresponding input set is L(y) = {x :

K 

zk xkn ≤ xn , n = 1, . . . , N,

k=1 K 

zk ykm ≥ ym , m = 1, . . . , M,

k=1

zk ≥ 0, k = 1, . . . , K}.

(7)

Following [20], we impose the following conditions on the data (x k , y k ), k = 1, . . . , K: I.

K 

xkn > 0, n = 1, . . . , N

k=1

II.

N 

xkn > 0, k = 1, . . . , K

n=1

III.

K 

ykm > 0, m = 1, . . . , M

k=1

IV.

M 

ykm > 0, k = 1, . . . , K.

(8)

m=1

Condition I states that each input, n = 1, . . . , N , is used by some k DMU (decision-making unit). Condition II requires that each DMU use at least some of one input. On the output side, condition III says that each output, m = 1, . . . , M, is produced by at least one DMU, and finally, condition IV states that each DMU produces a positive amount of at least one output. If the conditions in (8) hold, one can prove that T is a closed set, i.e., if (x l , y l ) ∈ T for all l and (x l , y l ) converges to (x o , y o ), then (x o , y o ) ∈ T .

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We note that if T is closed, so are P (x) and L(y). These results follow by considering the sequences (x o , y l ) ∈ T and (x l , y o ) ∈ T .

(9)

Since (x o , y l ) ∈ T ⇔ y l ∈ P (x o ) and since (x o , y o ) ∈ T , it implies that y o ∈ P (x o ) proving that P (x) is a closed set. The same logic applies to show that L(y) is a closed set. The above conditions also imply that P (x) is a bounded set, a condition often referred to as scarcity. In addition one can prove a no free lunch condition: that is, if y ∈ P (0) then y = 0.

(10)

The M inequalities (≥) for outputs in (6) allow outputs to be freely disposable, i.e., (in terms of the output set) y ∈ P (x), y  ≤ y ⇒ y  ∈ P (x).

(11)

With respect to inputs, the N inequalities (≤) in (7) make inputs freely disposable, i.e., x ∈ L(y), x  ≥ x ⇒ x  ∈ L(y).

(12)

The intensity variables zk , k = 1, . . . , K are restricted to be nonnegative, which implies that the technology exhibits constant returns to scale: λT =T , λ > 0, or P (λx) =λP (x), λ > 0, or L(λy) =λL(y), λ > 0.

(13)

Since the intensity variables are nonnegative, if we consider two vectors of these variables, z0 and z1 , then their convex combination, z = λz0 +(1−λ)z1 , 0 ≤ λ ≤ 1, is also nonnegative. This implies that T is convex, i.e., if (x 0 , y 0 ) ∈ T and (x 1 , y 1 ) ∈ T then (λx 0 + (1 − λ)x 1 , λy 0 + (1 − λ)y 1 ) ∈ T , 0 ≤ λ ≤ 1.

(14)

Convexity of T implies that P (x) is convex. To see this result, just take x 0 = x 1 . Similarly, if we take y 0 = y 1 , then it follows that L(y) is convex. However, the converse does not apply. Figure 1 illustrates the technology set T , the output sets P (x 1 ) and P (x 2 ), and the input sets L(y 1 ) and L(y 2 ). One can easily see that for each fixed x, the set P (x) is convex and for each fixed y, the set L(y) is convex. However, T is not convex. When we extend our model to include bad or undesirable outputs, additional J represent undesirable or bad outputs, and axioms must be introduced. Let b ∈ R+ let the output set take the form

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y

Fig. 1 Nonconvexity of T

T

}} y2

P(x2)

y1

P(x1)

x x1

x2

}}

0

L(y2)

L(y1)

P (x) = {(y, b) : x can produce (y, b)}.

(15)

When good and bad outputs are jointly produced, the assumption of free disposability does not apply; one cannot just throw away the bads. To model disposability in this case, we say that outputs are weakly disposable if radial contractions are feasible, i.e., (y, b) ∈ P (x), 0 ≤ θ ≤ 1 ⇒ (θy, θ b) ∈ P (x).

(16)

To model the condition that y and b are jointly produced, we say that these outputs are null-joint, i.e., if (y, b) ∈ P (x) and b = 0 ⇒ y = 0.

(17)

In words, no fire without smoke. We may extend the AA/DEA model to include “bads.” This model now becomes P (x) = {y :

K 

zk xkn ≤ xn , n = 1, . . . , N,

k=1 K 

zk ykm ≥ ym , m = 1, . . . , M,

k=1 K 

zk bkj = bj , j = 1, . . . , J,

k=1

zk ≥ 0, k = 1, . . . , K}

(18)

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In this model, a third set of linear restrictions are added, and we note that these restrictions are equalities. One can prove that these equality constraints for the bad outputs and inequality constraints for the “good” outputs (ym ) make (y, b) weakly disposable. To model the null-jointness of y and b in the AA/DEA framework, we require that two conditions hold, namely,

V.

K 

bkj > 0, j = 1, . . . , J and

k=1

VI.

J 

bkj > 0, k = 1, . . . , K.

(19)

j =1

The first condition says that each bad output is produced by some k, and the second condition states that each k produces some bad output. To verify that these conditions make good and bad outputs null-joint set bj = 0, j = 1, . . . , J . Then each intensity variable must satisfy zk = 0, k = 1, . . . , K. Thus, ym = 0, m = 1, . . . , M. For a list of axioms, see the Appendix: Production Axioms.

Cost and Revenue Indirect Technologies We now turn to technologies where firms operate under a cost or revenue constraint. These models are often referred to as indirect technologies. Let w ∈ RN + be an input price vector and c ∈ R+ represent total or allowed cost. The budget or cost constraint is the linear function c ≥ wx =

N 

wn xn .

(20)

n=1

The cost indirect output set, I P (w/c), is the union of all output sets P (x) where the input vector does not cost more than c. Formally, I P (w/c) = {y : y ∈ P (x), wx ≤ c} = {y : y ∈ P (x), (w/c)x ≤ 1}.

(21)

Like the indirect output set, the indirect input requirement set can also be obtained. Let p ∈ RM + represent an output price vector and r ∈ R+ be the target or minimum revenue to be attained. The target revenue constraint is r ≤ py =

M  m=1

pm ym .

(22)

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The revenue indirect input requirement set, L(p/r), is the union of all input sets L(y) such that y generates at least revenue r, i.e., L(p/r) = {x : x ∈ L(y), py ≥ r} = {x : x ∈ L(y), (p/r)y ≥ 1}.

(23)

The indirect output set I P (w/c) is useful in modeling production where the producer faces a budget constraint such as in the public sector. Here the producer might be interested in knowing the different combination of outputs that might be produced given their budget and inputs. With respect to outputs, the indirect input requirement set L(p/r) might be useful to a producer with multiple outlets or divisions where each outlet has a revenue target. In this case, the producer gains knowledge of the different input combinations that can meet the revenue target. As we show in section “Cost and Revenue Indirect Distance Functions”, these indirect sets can be combined with behavioral assumptions to construct indirect distance functions.

Functional Representations of the Technology While sets are useful models of the production technology, functional representations of the technology are more generally used in empirical work. In this section, we examine functions that can be used to represent multiproduct technologies. In turn, these functions can be used to measure efficiency and returns to scale. In addition, functions can be used in empirical work to recover shadow or support prices of nonmarket outputs and inputs and to estimate various elasticities.

Radial Distance Functions Next we study radial distance functions which were introduced into production economics by [28, 29]. We start with the input distance function. The input requirement set L(y) consists of all input vectors that can produce y, L(y) = {x : x can produce y}, y ∈ RN +,

(24)

the isoquants of which are defined as I soq L((y) = {x : x ∈ L(y), λ < 1, λx ∈ / L(y)}.

(25)

The input distance function is defined by Di (y, x) = sup{λ > 0 : (x/λ) ∈ L(y)}, y ∈ RM +. λ

(26)

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One property of the input distance function is that it is homogeneous of degree +1 in inputs (the scaled vector), i.e., Di (y, μx) = μDi (y, x). This property can be easily proved as follows:   μx ∈ L(y) Di (y, μx) = sup λ : λ λ  λ μx  = μ sup : ∈ L(y) μ λ λ  λ x : ∈ L(y) = μ sup (λ/μ) λ/μ μ = μDi (y, x), μ > 0.

(27)

Now, assuming weak disposability of inputs (i.e., if x ∈ L(y) and λ ≥ 1, then λx ∈ L(y)), then and only then is the input distance function a functional representation of the input sets, i.e., Di (y, x) ≥ 1 if and only if x ∈ L(y) or L(y) = {x : Di (y, x) ≥ 1}.

(28)

Formally: Proposition: Inputs are weakly disposable if and only if L(y)={x : Di (y, x)≥1}. Proof:3 (⇐) : x ∈ L(y) ⇒ Di (y, x) ≥ 1 ⇒ Di (y, λx) ≥ λ ≥ 1 (using homogeneity of Di in x) ⇒ λx ∈ L(y). (⇒) : x ∈ L(y) ⇒ λ = 1 is feasible in the definition of Di . Thus, Di (y, x) ≥ 1. On the other hand Di (y, x) ≥ 1 ⇒ (x/λ) ∈ L(y) for some λ ≥ 1. Then, using A.7, λ(x/λ) ∈ L(y), i.e., x ∈ L(y). Q.E.D. Also note that Di (y, x) = 1 if and only if x ∈ I soq L(y). From the definition of Di (y, x) and the above proposition, it follows that the production axioms have an equivalent representation in terms of Di (y, x). The radial output distance function is defined in terms of the output set P (x) = {y : x can produce y},

(29)

Do (x, y) = inf{λ : y/λ ∈ P (x)}, x ∈ RN +.

(30)

as λ

3 This

proof is from [15, p.22].

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From its definition, it follows that Do (x, λy) = λDo (x, y), λ > 0

(31)

i.e., it is homogeneous of degree +1 in the (scaled) output vector.4 Using homogeneity and weak output disposability (A.5), one can prove that the output distance function has the representation property, P (x) = {y : Do (x, y) ≤ 1}, x ∈ RN +.

(32)

Let the output isoquant be defined as I soq P (x) = {y : y ∈ P (x), λ > 1, ⇒ λy ∈ / P (x)}, x ∈ RN +.

(33)

Then, one can prove that Do (x, y) = 1 if and only if y ∈ I soq P (x).

(34)

Unlike the production function where multiple inputs produce a single output, Shephard distance functions allow a functional representation of technologies where multiple outputs are produced by multiple inputs. The Shephard distance functions have been widely used as measures of technical efficiency, and we outline these efficiency measures in section “Efficiency Analysis”. Furthermore, in section “Duality Theory”, we show that these distance functions have a dual representation in price space as the cost function and revenue function.

A Joint Production Function M Following [29, p. 212]5 , we define the joint production function J : RN + × R+ → R+ such that:

(a) for y ≥ 0, L(y) = ∅, I soq L(y) = {x : J (x, y) = 0} and (b) for x ≥ 0, P (x) = ∅, I soq P (x) = {y : J (x, y) = 0}.

(35)

Two questions are of interest with respect to J (x, y). First, does it exist? and second, how can it be represented? Regarding the last question, distance functions may be used as its representation, namely,

4 The

proof is similar to that of homogeneity of inputs in the input distance function, and we omit it here. 5 See also [10], pp. 38–40.

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J (x, y) = Di (y, x) − Do (x, y).

(36)

Regarding the existence of J (x, y), we refer to the proof in [10, p. 39]. The conditions required are I soq P (x) ∩ I soq P (λx) = ∅, λ = 1, and I soq L(y) ∩ I soq L(θy) = ∅, θ = 1.

(37)

Directional Distance Functions Luenberger [22, 23] building on [1] introduced directional distance functions using the terminologies, shortage and benefit functions. These functions are additive in structure and generalize the radial Shephard distance functions. Here we follow [4] and term them directional distance functions. Given the technology set T = {(x, y) : x can produce y}

(38)

the directional technology distance function (shortage function) is defined as − → D T (x, y; gx , gy ) = sup{β : (x − βgx , y + βgy ) ∈ T }

(39)

β M where g = (gx , gy ) ∈ RN + × R+ , g = 0 is the directional vector that indicates the direction along which (x, y) is projected onto the boundary of T . This function simultaneously contracts x along gx and expands y along gy . This contraction in inputs and expansion in outputs is in line with profit maximization where it is desirable to use fewer inputs to produce more outputs. If inputs and outputs are strongly disposable, i.e.,

if (x, y) ∈ T and x  ≥ x, y  ≤ y, then (x  , y  ) ∈ T

(40)

then the technology distance function has the representation property6 − → D T (x, y; gx , gy ) ≥ 0 if and only if (x, y) ∈ T .

(41)

This property allows us to express the production axioms in terms of the directional technology distance function. From its definition, this distance function satisfies the translation property:

6A

weaker condition called g = (gx , gy ) disposability will suffice for this property to hold.

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− → − → D T (x − αgx , y + αgy ; gx , gy ) = D T (x, y; gx , gy ) − α, α ∈ R.

(42)

The directional output distance function holds inputs constant (gx = 0) and projects outputs onto the boundary of P (x): − → D o (x, y; gy ) = sup{β : (y + βgy ) ∈ P (x)}.

(43)

β

Similarly, the directional input distance function holds outputs constant (gy = 0) and projects inputs onto the boundary of L(y): − → D i (y, x; gx ) = sup{β : (x − βgx ) ∈ L(y)}.

(44)

β

Each of these functions has properties similar to those of the directional technology distance function, i.e., representation and translation, now of course with respect to the output set P (x) and input set L(y).

A Distance Function Tree To relate the five distance functions introduced so far, we form a tree (see Fig. 2) whose root is the directional technology distance function. It is obvious that by setting gx = 0 the directional output distance function is obtained and setting gy = 0 the directional input distance function is obtained. To derive the radial distance functions, an argument is in place. Recall that these functions meet the representation property. Thus, e.g., the radial output distance function may be used − → in the definition of D o (x, y; gy ):

Fig. 2 Distance function tree

Do (x,y) gy=y Do(x,y;gy)

Di (y,x) g x=x

Di (x,y;gx) gy=0

gx=0 DT(x,y;gx,gy)

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− → D o (x, y; gy ) = sup{β : (y + βgy ) ∈ P (x)} β

= sup{β : Do (x, y + βgy ) ≤ 1}.

(45)

β

Now, set gy = y and recall that Do (x, y) is homogeneous of degree one in y. Then − → D o (x, y; y) = sup{β : Do (x, (1 + β)y) ≤ 1} β

= −1 + sup{(1 + β) : Do (x, y)(1 + β) ≤ 1} β

= −1 + sup{(1 + β) : (1 + β) ≤ β

= −1 +

1 Do (x, y)

1 Do (x, y)

(46)

which can be rearranged as Do (x, y) =

1 . − → 1 + D o (x, y; y)

(47)

A similar argument can be used to establish the relation between Di (y, x) and − → D i (y, x; gx ). Setting gx = x we can write − → D i (x, y; x) = sup{β : Di (y, x − βx) ≥ 1} β

= sup{β : Di (y, (1 − β)x) ≥ 1} β

  Di (y, x) ≥1 = sup β : 1−β β   1 1 ≥ = 1 − inf 1 − β : 1−β 1−β Di (y, x) =1−

1 Di (y, x)

(48)

which can be rearranged as Di (y, x) =

1 . − → 1 − D i (x, y; x)

(49)

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The results in (46) and (48) show that the Shephard output and input distance functions are special cases of the directional output and directional input distance functions when the directional vectors are chosen as gy = y and gx = x. Since other directional vectors may be chosen, the directional distance function provides a more generalized representation of the technology.

Cost and Revenue Indirect Distance Functions All distance functions in sections “Radial Distance Functions”, “A Joint Production Function”, “Directional Distance Functions”, and “A Distance Function Tree” were defined on the “direct” technologies T , P (x), or L(y). Here we turn our attention to distance functions defined on the indirect technologies, I P (w/c) and L(p/r). Again we study radial/Shephard and directional distance functions defined on those indirect technologies. Recall that the cost indirect output set is I P (w/c) = {y : y ∈ P (x), wx ≤ c} = {y : y ∈ P (x), (w/c)x ≤ 1}

(50)

where w ∈ RN + is an input price vector, c is total cost, and P (x) is the “direct” output set. The corresponding (radial) cost indirect output distance function is defined as I Do (w/c, y) = inf{θ : (y/θ ) ∈ I P (w/c)}. θ

(51)

This function is homogeneous of degree +1 in outputs (y): I Do (w/c, λy) = λI Do (w/c, y), λ > 0.

(52)

Under strong disposability of outputs7 , the indirect output distance function meets the representation condition I Do (w/c, y) ≤ 1 ⇔ y ∈ I P (w/c) or I P (w/c) = {y : I Do (w/c, y) ≤ 1}.

(53)

We consider two special cases for I Do (w/c, y): first when y is a scalar and second when the technology exhibits constant returns to scale (CRS). When output is a scalar, the distance function can be written as I Do (w/c, y) = I Do (w/c, 1)/y.

7 Weaker

conditions can be established.

(54)

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To see this result, recall the homogeneity property given in (52), and let λ = 1/y. Now, if y is efficient, we have I Do (w/c, y) = 1 so that (54) can be written as 1 = I Do (w/c, 1)/y or y = I Do (w/c, 1)

(55)

which is the indirect production function or in consumer theory, the indirect utility function. Next we show that under CRS the cost function and I Do (w/c, y) are equivalent. The cost function is defined as C(y, w) = min{wx : x ∈ L(y). x

(56)

If input prices are strictly positive, i.e., wn > 0, n = 1, . . . , N, then one can prove that8 I P (w/c) = {y : C(y, w) ≤ c}.

(57)

If the technology exhibits CRS, the cost function is homogeneous of degree +1 in outputs C(λy, w) = λC(y, w), λ > 0.

(58)

Using (55) and (58), we show that the cost indirect output distance function equals the cost function:9 I Do (w/c, y) = inf{θ : (y/θ ) ∈ I P (w/c)} θ

= inf{θ : C(y/θ, w) ≤ c} θ

= inf{θ : C(y, w/c) ≤ θ } θ

= C(y, w/c)

(59)

establishing our conjecture. One may ask, what is the relation between the “direct” and indirect output distance functions Do (x, y) and I Do (w/c, y)? With their first arguments, the direct distance function is defined in primal/quantity space and the indirect function

8 Färe 9 Färe

and Primont [15], p. 95 and Primont [15], p. 83–84

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defined in dual/price space; hence, they are dual to each other in their first argument, N 10 x ∈ RN + and w/c ∈ R+ . Although we devote a section to duality theory, this nontraditional relation is included here. If L(y) is convex, we can prove that I Do (w/c, y) = inf{Do (x, y) : wx ≤ c} x

and Do (x, y) = sup{I Do (w/c, y) : wx ≤ c}.

(60)

w/c

Hence, the two distance functions are dual with respect to inputs (x) and normalized input prices (w/c). The indirect directional output distance function is − → I D o (w/c, y; gy ) = sup{β : (y + βgy ) ∈ I P (w/c)}

(61)

β

and it meets the standard translation and representation properties. Again, we may ask what the relation between the direct and indirect directional output distance functions is. The following duality theorem illustrates the relation:11 − → − → I D o (w/c, y; gy ) = sup{ D o (x, y; gy ) : wx ≤ c} x

− → − → D o (x, y; gy ) = inf {I D o (w/c, y; gy ) : wx ≤ c}. w/c

(62)

Therefore, under convexity of P (x), the two distance functions model the same technology. Turning to input representations of the technology, recall that the revenue indirect input requirement set is given by I L(p/r) = {x : x ∈ L(y), r ≤ py} = {x : x ∈ L(y), 1 ≤ (p/r)y}

(63)

where p ∈ RM + is a vector of output prices, r is total revenue, and L(y) is the “direct” input requirement set. The radial indirect distance function defined on this set is I Di (p/r, x) = sup{λ : x/λ ∈ I L(p/r)}. λ

that RN and its dual (RN )∗ are equal, RN = (RN )∗ ; see, e.g., [19], p. 80–81. [16, p. 244] for a proof.

10 Note 11 See

(64)

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This distance function is homogeneous of degree +1 in inputs I Di (p/r, λx) = λI Di (p/r, x), λ > 0

(65)

and satisfies the representation condition under strong disposability of inputs, i.e., I Di (p/r, x) ≥ y ∈ I L(p/r) or I L(p/r) = {x : I Di (p/r, x) ≥ 1}.

(66)

The maximum revenue function is defined as R(x, p) = max{ry : y ∈ P (x)}. y

(67)

Using the revenue function and CRS, we can show that I Di (p/r, x) = R(x, p/r).

(68)

First we note that if output prices are positive, pm > 0, m = 1, . . . , M, the indirect input set may be written as I L(p/r) = {x : R(x, p/r) ≥ 1}.

(69)

Second, under CRS, the revenue function is homogeneous of degree +1 in inputs, R(λx, p/r) = λR(x, p/r), λ > 0.

(70)

Using the facts in (69) and (70) and knowing that the revenue function is homogeneous of degree +1 in output prices, our conjecture follows: I Di (p/r, x) = sup{λ : x/λ ∈ I L(p/r)} λ

= sup{λ : R(x/λ, p) ≥ r} λ

= sup{λ : R(x/λ, p/r) ≥ 1} λ

= sup{λ : R(x, p/r) ≥ λ} λ

= R(x, p/r) proving our conjecture.

(71)

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Contrasting the direct and indirect distance functions, Di (y, x) and I Di (p/r, x), we see that their first vector differs. The direct distance function depends on output y ∈ RM + , while the indirect distance function depends on revenue normalized output prices, p/r ∈ RM + . These two concepts are dual to each other, and in particular we have y(p/r) = 1.

(72)

Thus, the two distance functions are dual to each other, and when the output set P (x) is convex, we can prove that I Di (p/r, x) = sup{Di (y, x) : py ≥ r} y

Di (y, x) = inf {I Di (p/r, x) : py ≥ r}. p/r

(73)

Let us define the revenue constrained directional input distance function as − → I D i (p/r, x; gx ) = sup{β : x − βgx ∈ I L(p/r)},

(74)

β

and note that it meets the translation and representation properties (among others). − → − → Comparing I D i (p/r, x; gx ) and D i (y, x; gx ), we again observe that they have dual first vectors, p/r and y, respectively. Thus, the two functions are dual with respect to these vectors. In particular, if P (x) is convex, we can prove the duality theorem12 − → − → I D i (p/r, x; gx ) = sup{ D i (x, y; gx ) : (p/r)y ≥ 1} y

− → − → D i (x, y; gx ) = inf {I D i (p/r, x; gx ) : (p/r) ≥ 1}. p/r

(75)

Optimization The technology can also be represented as the result of optimizing behavior on the part of the producer. In this section, we consider the profit function, cost function, and revenue function as dual representations of the technology, each of which inherits its properties from the technology T , L(y), or P (x). We make these relations clear when discussing duality.

12 Färe

and Primont [16]

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Profit Maximization We first turn our attention to the profit function which is the result of maximizing behavior and is a dual representation of the technology, T . The existence of a maximum, i.e., there is an input/output vector (x, y) ∈ T such that py ∗ − wx ∗ ≥ py − wx for all (x ∗ , y ∗ ) ∈ T

(76)

needs some consideration that we bring up later. Given that (x ∗ , y ∗ ) maximizes profits, it then follows that x ∗ minimizes costs of producing outputs y ∗ given input prices w and y ∗ maximizes revenue given inputs x ∗ and output prices p. To verify the assertion that profit maximization implies cost minimization, assume that x ∗ does not minimize costs. Then, there must exist a feasible input vector, x, ˆ i.e., (x, ˆ y ∗ ) ∈ T such that w xˆ < wx ∗ .

(77)

However, the condition above would then mean that py ∗ − wx ∗ < py ∗ − w x, ˆ

(78)

which contradicts that (x ∗ , y ∗ ) maximize profit. Hence, profit maximization implies cost minimization given input prices and y ∗ . To confirm that profit maximization also implies revenue maximization, assume not. Then there must exist a feasible output vector, y, ˆ such that (x ∗ , y) ˆ ∈ T and py ∗ < py. ˆ

(79)

By the condition above, then it must be that py ∗ − wx ∗ < pyˆ − wx ∗

(80)

contradicting the condition that (x ∗ , y ∗ ) maximizes profit. Thus, profit maximization implies revenue maximization. In addition to defining the profit function in terms of a set, we next show how it can be expressed using distance functions. These formulations are important for duality theory. Recall the following equivalences: (x, y) ∈ T ⇔ y ∈ P (x) ⇔ x ∈ L(y),

(81)

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where P (x) is an output set and L(y) is an input set. We know that the directional technology distance function − → D T (x, y; gx , gy ) = sup{β : (x − βgx , y + βgy ) ∈ T }

(82)

β

meets representation, i.e., − → T = {(x, y) : D T (x, y; gx , gy ) ≥ 0}.

(83)

Hence, we may use this function to define profit maximization − → π(w, p) = max{py − wx : D T (x, y; gx , gy ) ≥ 0}. y,x

(84)

Moreover, we are aware of the fact that y ∈ P (x) ⇔ Do (x, y) ≤ 1.

(85)

P (x) = {y : Do (x, y) ≤ 1}.

(86)

(x, y) ∈ T ⇔ y ∈ P (x),

(87)

Thus, we have

and since

we have the following profit maximization formulation: π(w, p) = max{py − wx : Do (x, y) ≤ 1}. y,x

(88)

With respect to the input distance function, Di (y, x), we have x ∈ L(y) ⇔ Di (y, x) ≥ 1,

(89)

where L(y) = {x : Di (y, x) ≥ 1}. Now, since (x, y) ∈ T ⇔ x ∈ L(y), we can maximize profit with the input distance function as its constraint π(w, p) = max{py − wx : Di (y, x) ≥ 1}. y,x

(90)

In our duality section, we will make use of all these distance function formulations of profit maximization.

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Cost Minimization Let w ∈ RN + be a vector of input prices and L(y) = {x : x can produce y}

(91)

be an input requirement set. The cost minimization problem is min wx subject to x ∈ L(y). x

(92)

When a solution to the cost minimization problem exists, the value function C(y, x) = min{wx : x ∈ L(y)}, x

(93)

is referred to as the cost function. We note that C(y, w) ≤ wx for all x ∈ L(y)

(94)

since C(y, w) is a minimum. Since the input distance function has the representation property x ∈ L(y) ⇔ Di (y, x) ≥ 1, the cost minimization problem may be written as C(y, w) = min{wx : Di (y, x) ≥ 1} x

(95)

which has the Lagrangian formulation C(y, w) = wx − μ(Di (y, x) − 1),

(96)

where μ is the Lagrangian multiplier. We have also indicated that the directional input distance function − → D i (y, x; gx ) = sup{β : (x − βgx ) ∈ L(y)}

(97)

β

meets the representation condition − → x ∈ L(y) ⇔ D i (y, x; gx ) ≥ 0.

(98)

Therefore, we may write the cost minimization problem as − → C(y, w) = min{wx : D i (y, x; gx ) ≥ 0} x

with the Lagrangian formulation

(99)

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− → C(y, w) = wx − μ( D i (y, x; gx ) − 0),

(100)

where μ is the Lagrangian multiplier. Of course we may also formulate the minimization problem with other distance functions that meet the representation property. We leave that exercise to the reader. In the previous section, we proved that profit maximization π(x, p) = py ∗ − wx ∗

(101)

implied that wx ∗ was the minimum cost of producing y ∗ , i.e., C(y ∗ , w) = wx ∗ = min{wx : x ∈ L(y ∗ )}. x

(102)

Thus, we may use the cost function, C(y, w), as part of the profit maximization problem π(w, p) = max{py − C(y, w)}. y

(103)

This formulation holds provided the cost function represents the technology, i.e., if C(y, w) is dual to it. As we show in section “Cost Function Dualities”, representation is true under certain conditions, like the input requirement set L(y) being closed, nonempty, and convex.

Revenue Maximization Denote the output price vector as p ∈ RM + and the output set by P (x) = {y : x can producey}.

(104)

The optimization problem consists of firms choosing a feasible output vector so as to maximize revenues given inputs and output prices. The resulting value function R(x, p) = max{py : y ∈ P (x)} y

(105)

is referred to as the revenue function. Given that P (x) is closed and bounded, i.e., compact, the revenue function exists. We would like to express our duality theorems in terms of functions; hence, we need to express revenue maximization with function constraints, rather than constraints on sets, in this case the output set. Consider first the radial/Shephard output distance function Do (x, y) = inf{λ : y/λ ∈ P (x)}. λ

(106)

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This function meets the representation condition y ∈ P (x) ⇔ Do (x, y) ≤ 1,

(107)

P (x) = {y : Do (x, y) ≤ 1}.

(108)

i.e., we have

Thus, we may substitute the output distance function into the revenue maximization problem so that R(x, p) = max{py : Do (x, y) ≤ 1}. y

(109)

This revenue function may be written as a Lagrangian problem as R(x, p) = py − μ(Do (x, y) − 1),

(110)

where μ is the Lagrangian multiplier. We also know that the directional output distance function meets the representation condition, i.e., − → P (x) = {y : D o (x, y; gy ) ≥ 0},

(111)

so that an alternative constraint in the revenue maximization problem can represent the technology, i.e., − → R(x, p) = max{py : D o (x, y; gy ) ≥ 0}. y

(112)

We use (112) in section “Revenue Function Dualities” on duality theory. We proved in section “Profit Maximization” that profit maximization implies revenue maximization, i.e., π(w, p) = py ∗ − wx ∗ . Thus we have R(x ∗ , p) = py ∗ = max {py : y ∈ P (x ∗ )} y

(113)

where input is restricted to equal the optimal input under profit maximization, x ∗ . Now, if R(x, p) represents the technology, i.e., it is dual to the output distance function, then the profit maximization problem may be formulated as π(w, p) = max {R(x, p) − wx}. x

(114)

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Efficiency Analysis In sections “Functional Representations of the Technology” and “Optimization”, we have discussed distance functions and value functions. Merging these concepts allow us to study efficiency in production, which consists of a technical component, distance functions, and an overall component – value functions. We first establish [18] cost efficiency model and proceed with the corresponding directional distance function analog. Recall that under cost minimization the following inequality holds: C(y, w) ≤ wx for all x ∈ L(y).

(115)

From the discussions of distance functions, we know that Di (y, x) = sup{λ : x/λ ∈ L(y)} = λ∗

(116)

λ

where λ∗ is the maximal (supremal) contraction of the input vector x that can still feasibly produce the output vector y. When this contraction in inputs is achieved, we have x x ∈ L(y), = λ∗ Di (y, x)

(117)

x i.e., Di (y,x) belongs to the input set L(y) and is thus feasible. Combining this condition with the cost minimization inequality from (115) yields

C(y, w) ≤

wx . Di (y, x)

(118)

Interpreting wx as observed cost, then the right-hand side of (118) can be interpreted as the cost of production if the observed inputs are contracted to the input isoquant, I soq L(y).13 Rearranging (118) yields C(y, w) 1 ≤ wx Di (y, x)

(119)

where the left-hand side is the ratio of minimum costs to actual costs which measures overall cost efficiency (OCE) and the right-hand side equals input technical efficiency (T Ei ). To close the inequality, we introduce an allocative efficiency component, AEi , as a residual.14 That is,

13 The 14 See

input isoquant is defined as I soq L(y) = {x : x ∈ L(y), λ < 1 ⇒ λx ∈ / L(y)}. [3] for an approach in which AEi is not a residual.

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1 C(y, w) = × AEi wx Di (y, x) or OCE = T Ei × AEi .

(120)

That is, overall cost efficiency equals the product of input technical efficiency and allocative efficiency. Turning to the directional input distance function − → D i (y, x; gx ) = sup{β : (x − βgx ) ∈ L(y)} = β ∗

(121)

β

where β ∗ is the supremum. We note that − → x − β ∗ gx = x − D i (y, x; gx )gx ∈ L(y).

(122)

Combining (122) with cost minimization yields the inequality − → C(y, w) ≤ w(x − D i (y, x; gx )gx ) − → ≤ wx − D i (y, x; gx )wgx

(123)

which we may rearrange as wx − C(y, w) − → ≥ D i (y, x; gx ), wgx

(124)

where the left-hand side measures the difference between actual costs and minimum −−→ costs normalized by wgx and serves as a measure of overall cost inefficiency (OCI ) − → and the right-hand side measures input technical inefficiency (T I i ). Following [9], we refer to these measures of inefficiency as indicators, while the radial measures of efficiency are termed indexes.15 − → To close the gap, we introduce allocative inefficiency (AI i ) additively; i.e., wx − C(y, w) − → − → = D i (y, x; gx ) + AI i wgx or −−→ − → − → OCI = T I i + AI i

15 See

also [13] for difference indicators using a profit function.

(125)

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−−→ is the decomposition of the cost inefficiency indicator (OCI ) into technical inef− → − → ficiency (T I i ) and allocative inefficiency (AI i ). A producer exhibits no technical − → − → inefficiency if T I i = 0 and no allocative inefficiency if AI i = 0. When both −−→ components equal zero, the producer exhibits no cost inefficiency, i.e., OCI = 0. We note that in the radial case, allocative efficiency is introduced multiplicatively and in the directional case allocative inefficiency is introduced additively. Higher values for the efficiency indexes – OCE, T Ei , and AEi – indicate greater efficiency in the use and allocation of inputs given outputs and input prices. In −−→ − → − → contrast, higher values of the inefficiency indicators – OCI , T I i , and AI i – indicate greater inefficiency in the use and allocation of inputs given outputs, input prices, and the directional vector (gx ). Next we study revenue efficiency with both radial and directional distance functions. Since this analysis closely parallels the cost notions, we keep it to a minimum. Recall the following revenue inequality R(x, p) ≥ py for all y ∈ P (x)

(126)

and the two distance functions Do (x, y) = inf{λ : y/λ ∈ P (x)}, λ

(127)

with y/Do (x, y) ∈ P (x) and − → D o (x, y; gy ) = inf{β : y + βgy ∈ P (x)}, β

(128)

− → with y + D o (x, y; gy )gy ∈ P (x). To construct a measure of overall revenue efficiency (ORE), the revenue inequality is rearranged and then closed by multiplying the output distance function by an index of output allocative efficiency (AEo ), i.e., R(x, p) ≥

py Do (x, y)

py = Do (x, y) × AEo R(x, p) ORE = T Eo × AEo

(129)

where T Eo is output technical efficiency as measured by the output distance function. The indexes ORE, T Eo , and AEo take a maximum value of one when the producer is efficient in the particular index. Similar to overall cost inefficiency, an indicator of overall revenue inefficiency −−→ (ORI ) can be constructed. Overall revenue inefficiency equals the normalized difference between maximum revenues and actual revenues. To derive this indicator,

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start with the revenue inequality using the directional output distance function which − → serves as the indicator of output technical inefficiency (T I o ). Then, rearrange and − → finally, add an indicator of output allocative inefficiency (AI o ): − → R(x, p) ≥ p(y + D o (x, y; gy )gy ) R(x, p) − py − → − → = D o (x, y; gy ) + AI o pgy −−→ − → − → ORI = T I o + AI o .

(130)

− → −−→ − → Efficient producers have ORI = 0, T I o = 0, and AI o = 0 with higher values indicating greater inefficiency in the particular component. Above in this section we showed how the radial/Shephard distance functions could be used as constraints in the profit maximization problem. First, recall that x ∈ L(y) ⇔ (x, y) ∈ T . Second, consider the following expressions: Di (y, x) = sup{λ : x/λ ∈ L(y)} λ

= sup{λ : (x/λ, y) ∈ T }.

(131)

λ

Now, since Di (y, x) meets the representation conditions16 of L(y) = {x : Di (y, x) ≥ 1} and T = {(x, y) : Di (y, x) ≥ 1}, we have shown that the input distance function may be used as the technology constraint for the profit maximization problem. We follow [5] and make use of these representation properties in developing Nerlovian indicators of profit inefficiency. When a profit maximum exists, maximum profits are no less than actual profits π(w, p) ≥ py − wx for all (x, y) ∈ T .

(132)

x , y) ∈ T . Thus From the above analysis, we know that ( Di (y,x)

π(w, p) ≥ py −

wx . Di (y, x)

(133)

By adding and subtracting wx from the right-hand side and rearranging, one obtains π(w, p) ≥ py − wx

16 Say

under strong disposability of inputs.

1 + wx − wx Di (y, x)

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1 π(w, p) − (py − wx) ≥1− wx Di (y, x) π(w, p) − (py − wx) − → ≥ D i (y, x; gx ) wx − → where D i (y, x; x) = 1 −

(134)

when gx = x as shown in (48). Adding − → a directional allocative inefficiency component (AI i ) to the right-hand side of (134) closes the inequality and yields an additive decomposition of overall profit inefficiency into technical inefficiency and allocative inefficiency: 1 Di (y,x)

π(w, p) − (py − wx) − → − → = D i (y, x; x) + AI i wx

(135)

where the left-hand side of (135) measures overall profit inefficiency and is equal to the amount that maximum profits exceed actual profits normalized by actual costs: π(w,p)−(py−wx) . wx As an alternative to (135), one can also develop a multiplicative index. Again, starting from the profit inequality (132), we also know that if (x, y ∗ ) ∈ T , then π(w, p) ≥ py ∗ − wx. Now, since (x, y ∗ ) ∈ T , then ( Di (yx∗ ,x) , y ∗ ) ∈ T which means wx Di (y ∗ , x) wx . π(w, p) − py ∗ ≥ − Di (y ∗ , x) π(w, p) ≥ py ∗ −

(136)

Since π(w, p) = py ∗ − wx ∗ , we have wx Di (y ∗ , x) wx −wx ∗ ≥ − Di (y ∗ , x)

py ∗ − wx ∗ − py ∗ ≥ −

1 wx ∗ ≤ . wx Di (y ∗ , x)

(137)

The inequality can be closed by multiplying the right-hand side by an index of input allocative efficiency, AEi . Thus, we have overall cost efficiency equals the product of input technical efficiency and allocative efficiency: 1 wx ∗ = × AEi wx Di (y ∗ , x) OCEi = T Ei × AEi .

(138)

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The decomposition of OCEi can also be done for values of y other than y ∗ . One can also develop a Nerlovian indicator of profit inefficiency by exploiting the representation properties of either the Shephard output distance function or the directional output distance function. We start with the output distance function. Recall that y ∈ P (x) ⇔ (x, y) ∈ T . The representation property yields P (x) = {y : Do (x, y) ≤ 1} and T = {(x, y) : Do (x, y) ≤ 1}. When a profit maximum exists, maximum profits are no less than actual profits: π(w, p) ≥ py −wx for all (x, y) ∈ y ) ∈ T , we can derive a second Nerlovian T . Thus, since (x, y) ∈ T ⇒ (x, Do (x,y) profit efficiency indicator as π(w, p) ≥

py − wx + (py − py) Do (x, y)

π(w, p) − (py − wx) 1 ≥ −1 py Do (x, y) π(w, p) − (py − wx) − → ≥ D o (x, y; y) py π(w, p) − (py − wx) − → − → = D o (x, y; y) + AI o py − → where D o (x, y; y) =

(139)

− → − 1 when gy = y. Here D o (x, y; y) is an indicator − → of output technical inefficiency, and AI o is an indicative of output allocative inefficiency. In this case, overall profit inefficiency equals the amount that maximum profits exceed actual profits normalized by actual revenues: π(w,p)−(py−wx) . We py leave the derivation of the multiplicative index of output efficiency along the same lines as (138) to the reader. In general, the indicator of Nerlovian profit inefficiency can be decomposed as − → the sum of an indicator of technical inefficiency measured by D T (x, y; gx , gy ) and − → an indicator of allocative inefficiency AI T . Starting with the profit inequality (132) − → and making use of the representation property (x, y) ∈ T ⇔ D T (x, y; gx , gy ) and − → definition of D T (x, y; gx , gy ), the Nerlovian profit inefficiency indicator is derived: 1 Do (x,y)

π(w, p) ≥ py − wx − → π(w, p) ≥ p(y + D T (x, y; gx , gy )gy ) − → − w(x − D T (x, y; gx , gy )gx ) π(w, p) − (py − wx) − → ≥ D T (x, y; gx , gy ) pgy + wgx π(w, p) − (py − wx) − → − → = D T (x, y; gx , gy ) + AI T pgy + wgx

(140)

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− → where the inequality is closed by adding the allocative inefficiency component AI T . Clearly, when gx = x and gy = 0 the indicator of Nerlovian profit inefficiency collapses to (135). Similarly, when gx = 0 and gy = y, the profit inefficiency indicator collapses to (139).

Duality Theory This section is devoted to duality theory. This theory has its roots in the fact that a convex set can be modeled in two ways: one as the set of all convex combinations of points in the set and two as the intersection of all half-spaces containing it. Shephard [28] introduced this theory into economics as a way to make use of price/cost data to model a technology. Statistical studies of cost functions are generally more accessible than corresponding empirical investigations of production functions, because economic data are most frequently in price and monetary terms. [28, p. 28]

In order to illustrate the idea, we make use of a paper by [30]. Assume we are given a Leontief production function with two inputs, y = min{x1 , x2 }. The cost function associated with this production function is C(y, w) = min{wx1 + w2 x2 : min(x1 , x2 ) ≥ y} x

(141)

and can be derived as C(y, w) = y(w1 + w2 ). Our illustration of the duality between the Leontief production function and cost function is represented in Fig. 3 which has a primal space (x1 , x2 ) ≥ 0 and a dual space ( wc1 , wc2 ). The input requirement set is illustrated in the primal space, and the cost function is illustrated in the dual space. In the primal space, the horizontal and vertical intercepts ( wc1 , wc2 ) of the relative price lines give the maximum input quantities that could be purchased if the entire budget was spent on the specific input. The figure shows how the Leontief technology is reflected through asymptotes w1 x1 w2 x2 c = 1 and c = 1 from the primal quantity space into the price space where it is a straight line. Of course, one may also start in the price space with the cost function and derive the production function in the quantity space. We note that duality does not preserve differentiability, but might create it.

Cost Function Dualities We start with [28] duality theory between the cost function and the radial/Shephard input distance function and end this section with a cost function and directional input distance function duality statement. Recall that the cost function is defined as

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x2

Fig. 3 Cost and Production Function Duality

L(y) (w2x2)/c=1 x1

w2/c C(w/c,y)

(w1x1)/c=1

w1/c C(y, w) = min{wx : x ∈ L(y)}, w > 0 and y ∈ Dom L x

(142)

N N where y ∈ RM + is an output vector, w ∈ R+ an input price vector, x ∈ R+ an input vector, and Dom L = {y : L(y) = ∅}. In general we drop the assumption that y ∈ Dom L. One may show that the cost function meets the following conditions:17

C.1 C(y, w) is nonnegative and non-decreasing in w. C.2 C(y, w) is homogeneous of degree +1 in w. C.3 C(y, w) is concave and continuous in w. If strong disposability and convexity are imposed on the input requirement set L(y), one can prove that L(y) = {x : wx ≥ C(y, w) for all w > 0}.

(143)

Hence, we can establish a duality between L(y) and C(y, w) in the sense that each can be derived from the other: C(y, w) = min{wx : x ∈ L(y)} x

and L(y) = {x : wx ≥ C(y, w) for all w > 0}.

17 See

[15] Drop (1995).

(144)

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Earlier we have shown that the radial input distance function, Di (y, x) = sup{λ > 0 : (x/λ) ∈ L(y)}, represents the input requirement set, i.e., L(y) = {x : Di (y, x) ≥ 1}. Thus the cost function duality may use the representation property to yield C(y, w) = min{wx : Di (y, x) ≥ 1}. x

(145)

The second part above may be written in terms of an optimization condition18 which gives us a second duality theorem. C(y, w) = min{wx : Di (y, x) ≥ 1} x

Di (y, x) = inf{wx : C(y, w) ≥ 1} w

(146)

This duality theorem is between the two functions which makes it handy for empirical work on pricing. The two parts are both constraint optimization problems, and next we transform these into unconstrained problems,19 i.e., wx  , w>0 x Di (y, x)  wx  Di (y, x) = inf , x ∈ RN +. w C(y, w)  C(y, w) = min

(147)

These dual relationships play an important role in efficiency calculation where, for 1 example, C(y,w) wx is the [18] measure of cost efficiency and Di (y,x) is the measure of input technical efficiency. Turning to the directional input distance function, we have shown that it meets the representation condition, i.e., − → L(y) = {x : D i (y, x; gx ) ≥ 0}

(148)

given that inputs are freely disposable, where gx is the directional vector showing how the input x is contracted onto the boundary of the input set L(y). The above fact (148) yields an unconstrained duality condition. − → C(y, w) = min{wx − D i (y, x; gx )wgx } x

 wx − C(y, w)  − → . D i (y, x; gx ) = inf w wgx

18 See 19 See

[15]. [15].

(149)

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Revenue Function Dualities Next we develop duality theorems for the revenue function20 which is defined as R(x, p) = max{py : y ∈ P (x)} y

(150)

and the output distance function which is defined as Do (x, y) = inf{λ : y/λ ∈ P (x)}. λ

(151)

Our first theorem is between the revenue function and the output distance function, and it states that R(x, p) = max{py : Do (x, y) ≤ 1} y

Do (x, y) = sup{py : R(x, p) ≤ 1}.

(152)

p

These two constrained optimization problems can be transformed into unconstrained problems as py  y Do (x, y)  py  . Do (x, y) = sup p R(x, p) R(x, p) = max



(153)

Recall that the directional output distance function − → D o (x, y; gy ) = sup{β : (y + βgy ) ∈ P (x)}

(154)

β

− → also completely characterizes the output set: P (x) = {y : D o (x, y; gy ) ≥ 0}. Thus, the revenue maximization problem may be formulated as − → R(x, p) = max{py : D o (x, y; gy ) ≥ 0}. y

(155)

Given appropriate conditions on P (x), namely, convexity and free disposability of outputs, the unconstrained duality conditions between the revenue function and the directional output distance function are

20 This

section mimics section “Cost Function Dualities”, but is terse.

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− → R(x, p) = max{py + D o (x, y; gy )gy } y

 R(x, p) − py  − → . D o (x, y; gy ) = inf p pgy

(156)

Profit Function Dualities In this section, we illustrate three profit function dualities related to the radial input and output distance functions and the directional technology distance function.21 Recall that these distance functions are complete representations of the technology. In particular we have − → T = {(x, y) : D T (x, y; gx , gy ) ≥ 0} P (x) = {y : Do (x, y) ≤ 1}

(157)

L(y) = {x : Di (y, x) ≥ 1} and that (x, y) ∈ T ⇔ y ∈ P (x) ⇔ x ∈ L(y).

(158)

Then, given a convex technology with inputs and outputs freely disposable, the following duality relation can be established:22 − → π(w, p) = sup{py − wx + D T (x, y; gx , gy )(pgy + wgx )} x,y

 π(w, p) − (py − wx)  − → D T (x, y; gx , gy ) = inf . p,w pgy + wgx

(159)

With respect to the radial input distance function, [15] established the duality that  wx  π(w, p) = sup py − Di (y, x) x,y   wx (160) Di (y, x) = inf p,w py − π(w, p) and for the radial output distance function, we have

21 Of course we could also show dualities between the profit function and directional input distance

function and directional output distance function. [22] for a proof.

22 See

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 py − wx x,y Do (x, y)   py . Do (x, y) = inf p,w wx + π(w, p)  π(w, p) = sup

(161)

Calculus A famous calculus statement in economics is Shephard’s lemma [28], which derives the conditional demand function for inputs through calculus. In particular we have ∇w C(y, w) = x(y, w),

(162)

where x(y, w) is the input demand function and the cost function is C(y, w) = minx {wx : x ∈ L(y)} for inputs x ∈ RN + . It is important to note that by applying calculus in price space, one ends up in quantity space, i.e., calculus moves you from one space into its dual. In economics we frequently think of quantity space, say RQ as primal and price space RP as its dual (Q = P ), where (RQ )∗ = RP , i.e., the dual (RQ )∗ is the price space. One can prove that (RQ )∗ = RQ , which of course is not true in general; see, e.g., [21]. Although RQ and its dual (RQ )∗ are equal, in economics one can say that you can eat a donut, but not its price. Thus one needs to distinguish between the two spaces. In order to formalize the idea of Shephard’s lemma, let, for simplicity, F : R → R be a function from the real numbers into the real numbers. Then we want to show that dF ∈ (R)∗ , dx

(163)

i.e., it is in the dual space for R. A function, F (x), is differentiable at x o if there exists a linear functional L ∈ (R)∗ such that lim

h→0

F (x + h) − F (x) − L(h) = 0. h

(164)

Since L(h) is linear, L(h) = hL(1). Thus, lim

h→0

F (x + h) − F (x) hL(1) − =0 h h

(165)

F (x + h) − F (x) = L(1) h

(166)

or lim

h→0

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where lim

h→0

F (x + h) − F (x) dF = . h dx

(167)

dF = L(1) ∈ (R)∗ , dx

(168)

Thus,

justifying our claim that the derivative belongs to the dual space on which the function is defined. Of course, this generalizes to RQ , Q > 1, and hence Shephard’s lemma has a natural explanation, i.e., the gradient of the cost function with respect to input prices yields input quantities. Next one may ask, what are the conditions on the input requirement set L(y) that yield a differentiable cost function (value function). Mas Colell et al. [24, p. 141] show that the cost function is differentiable when L(y) is strictly convex, i.e., for x 0 , x 1 ∈ L(y), then λx 0 + (1 − λ)x 1 ∈ Interior L(y), 0 < λ < 1.

(169)

From our duality section, it is clear that convexity is just a sufficient condition for differentiability. For a necessary and sufficient condition, see [15, 16, 18]. Next we turn to some applications of calculus.

Shadow Pricing Our first application of calculus is shadow pricing. The hints from an earlier section suggest that we use input/output data to estimate a distance function and then derive the shadow prices using calculus. Here we restrict ourselves to Shephard’s input distance function23 and derive expressions to estimate input prices. Let the input distance function be Di (y, x) = sup{λ > 0 : x/λ ∈ L(y)}, y ∈ RM +

(170)

λ

and recall that it is homogenous of degree +1 in inputs, Di (y, μx) = μDi (y, x), μ > 0, and meets the representation property Di (y, x) ≥ 1 if and only if x ∈ L(y). Let w ∈ RN + be a vector of input prices. Then C(y, x) = min{wx : Di (y, x) ≥ 1} x

23 See

[11] in this volume for a comprehensive discussion of the topic.

(171)

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is a cost function. The Lagrangian problem for this cost function is C(y, w) = wx − μ(Di (y, x) − 1),

(172)

where μ is the Lagrangian multiplier. The first-order conditions (FOCs) associated with it are w − μ∇x Di (y, x) = 0.

(173)

M Suppose that one input price, say w1 , is known in addition to x ∈ RN + and y ∈ R+ . Then the shadow prices for the other n = 2, . . . , N inputs can be determined as

wn = w 1

∂Di (y, x)/∂xn ∂Di (y, x)/∂x1

(174)

so that total cost can be computed as c = w1 x1 + w1

N  ∂Di (y, x)/∂xn n=2

∂Di (y, x)/∂x1

xn .

(175)

 The next model assumes that total cost, c = N n=1 wn xn , is known together with M , but no individual price w is known. In this case, we can x ∈ RN and y ∈ R n + + derive the pricing model w=c

∇x Di (y, x) . Di (y, x)

(176)

To derive this pricing model, we need to interpret the Lagrangian multiplier μ. Thus, consider ˜ C(y, w, α) = wx − μ(Di (y, x) − α)

(177)

as a permutation of the distance function. By homogeneity we have ˜ C(y, w, α) = wx − αμ(Di (y, x/α) − 1) = αw xˆ − αμ(Di (y, x) ˆ − 1)

(178)

= αC(y, w) ˜ where xˆ = x/α. The derivative is then ∂ C(y, w, α)/∂α = μ from (177), and ˜ from the last line in (178), we have ∂ C(y, w, α)/∂α = C(y, x). Thus, the optimal Lagrangian multiplier equals the cost function: μ = C(y, w). Inserting C(y, w) into the first-order conditions yields

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w = C(y, x)∇x Di (y, x).

(179)

Now, multiply both sides of (179) by x, and using Euler’s theorem24 yields c = wx = C(y, x)Di (y, x).

(180)

Thus, C(y, w) = c/Di (y, x), and substituting this expression into (179) yields the desired result: w = C(y, w)∇x Di (y, x) =

c∇x Di (y, x) . Di (y, x)

(181)

Assuming that inputs are technically efficient in the sense of [18], i.e., Di (y, x) = 1, and that prices are cost deflated, wˆ = w/c, then our pricing model becomes wˆ = ∇x Di (y, x).

(182)

In the case of a single output, y ∈ R+ , the pricing formula can be found in [28, p. 19] and [29, p. 171]. Finally, suppose we know r = total revenue, output prices p ∈ RM + , and inputs x ∈ RN . Then we can estimate the revenue function, R(x, p), and hence we can + price inputs by maximizing profits by choosing inputs x π(w, p) = max{R(x, p) − wx} x

(183)

and then obtain the pricing rule via the first-order conditions w = ∇x R(x, p).

(184)

Scale Elasticities In production theory as opposed to consumer theory, size matters. A measure of size is scale elasticity. Here we follow [15] in our discussion. As a second application of calculus, we discuss a primal and dual formulation of scale elasticities. Let F (x) = max{y : (x, y) ∈ T } x

(185)

a homogeneous function of degree +1 in x, Euler’s theorem states that x∇x Di (y, x) = Di (y, x).

24 For

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be a production function. Then scale elasticity is defined as (x) =

∂lnF (λx) ∇x F (x)x |λ=1 = . ∂λ F (x)

(186)

In the multi-output case, scale elasticity generalizes as o (x, y) =

∂lnθ |θ=λ=1 ∂lnλ

(187)

where Do (λx, θy) = 1 and N

∂Do (x,y) ∂xn xn ∂Do (x,y) m=1 ∂ym ym n=1

o (x, y) = − M =−

∇x Do (x, y)x applying Euler’s theorem Do (x, y)

= −∇x Do (x, y)x

(188)

since Do (x, y) = 1, when λ = θ = 1. Let p ∈ RM + be an output price vector. Then, recall that the revenue function is dual to the output distance function represented by the duality py  y Do (x, y)  py  . Do (x, y) = sup p R(x, p) R(x, p) = max



(189)

Hence, one may derive an expression for scale elasticity in terms of the revenue function as R (x, p) =

∇x R(x, p)x R(x, p)

(190)

where ∇x R(x, p)x ∇x Do (x, y)x =− R(x, p) Do (x, y) i.e., R (x, p) = o (x, y).

(191)

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Elasticities of Substitution The curvature of input isoquants and output possibility frontiers reveals information about the degree of substitutability between inputs or between outputs. In a consumer context, [7] and [8] show the relation between the Slutsky matrix of compensated price effects derived from an expenditure function and the Antonelli matrix of compensated quantity effects derived from distance functions. To investigate the curvature requires value functions or distance functions to be twice differentiable which we assume. Knowledge of the elasticity of substitution (transformation) reveals how input cost (revenue) shares change as relative input (output) prices change. Dual elasticities of substitution and transformation are derived from distance functions.25 These dual elasticities measure the percent change in relative prices that support a percent change in relative quantities. For the curvature of the input requirement set, we examine elasticities of substitution using the cost function and input distance function. These elasticities of substitution are based on the work of [25] and [2]. Then, we derive output elasticities of transformation from the revenue function and their dual measures from the output distance function and directional output distance function. We begin with the cost function. Input demand functions are derived from Shephard’s lemma: ∇w C(y, w) = x(y, w). The ratio of two inputs, i and j , then equals the ratio of the first derivative of the cost function ∂C(y, w)/∂wi xi (y, w) . = ∂C(y, w)/∂wj xj (y, w)

(192)

Taking the natural log of both sides of (192) and then the partial derivative with respect to log relative input prices yields the Morishima elasticity of substitution (MES):     ∂C(y,w)/∂wi ∂ln ∂C(y,w)/∂w ∂ln xxji (y,w) (y,w) j     MESij = − =− wi wi ∂ln w ∂ln wj j  ∂ 2 C(y,w) = wi

∂wi ∂wj ∂C(y,w) ∂wj

−

∂ 2 C(y,w) ∂wi2 ∂C(y,w) ∂wi

= j i (y, w) − ii (y, w)



 = wi

∂xj (y,w) ∂wi

xj (y, w)

−

∂xi (y,w) ∂wi



xi (y, w) (193)

where j i (y, w) is the cross price elasticity between the j th input demand and the ith input price and ii is own price elasticity of demand for input i. As shown by [2],

25 See

the [27] chapter in this volume on Elasticities of Substitution by R.R. Russell for a detailed discussion of alternative definitions and forms of substitution elasticities.

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the Morishima elasticity is asymmetric in terms of whether the relative price change is due to the price of input i changing or due to the price of input j changing. Furthermore, the elasticity of the relative cost shares of i and j due to relative price change of input i is 1 − MESij . Thus, if MESij is greater (less) than 1, the cost share of input i falls (rises) relative to j due to an increase in the price of input i. i j ) which Next, we turn to the dual Morishima elasticity of substitution (MES measures the percent change in relative shadow prices of inputs i and j due to a percent change in the relative input quantities. We start with the dual Shephard’s lemma found in (179) ∇x Di (y, x) =

w(y, x) C(y, x)

(194)

which for two inputs i and j is ∂Di (y, x)/∂xi wi (y, x) . = ∂Di (y, x)/∂xj wj (y, x)

(195)

Again, we take the natural logarithms of both sides of (195) and then take the partial derivative with respect to log changes in the ratio of the ith input quantity to the j th input quantity     ∂Di (y,x)/∂xi wi (y,x) ∂ln ∂D ∂ln (y,x)/∂x w (y,x) i j j ij = −     MES =− xi ∂ln xj ∂ln xxji  ∂ 2 Di (y,x) = xi

∂xi ∂xj ∂Di (y,x) ∂xj

−

∂ 2 Di (y,x) ∂xi2 ∂Di (y,x) ∂xi

= ˜j i (y, x) − ˜ii (y, x)



 = xi

∂wj (y,x) ∂xi

wj (y, x)

−

∂wi (y,x) ∂xi



wi (y, x) (196)

The Morishima output elasticities of transformation (substitution) provide information on the curvature of the frontier of the output possibility set. The revenue function is used to derive the output elasticity of transformation, and either the Shephard or directional output distance function can be used to derive dual elasticities of transformation in price space. We begin by deriving the output elasticity of transformation from the revenue function and its dual from the output distance function. Then, we reintroduce jointly produced undesirable outputs back into our framework and derive the dual output elasticity of transformation from the directional output distance function. The revenue function is R(x, p) = maxy {py : y ∈ P (x)}. Using the representation property, we can write this revenue function using the output distance function as

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R(x, p) = max{py : Do (x, y) ≤ 1}.

(197)

y

As shown in (153), the revenue function can be written as an unconstrained maximization problem R(x, p) = max



y

py  Do (x, y)

(198)

from which the output supply functions can be derived ∇p R(x, p) =

y(x, p) Do (x, y)

(199)

where y(x, p) ∈ RM + is the vector of output supply functions. For two outputs, s and t, we then have ys (x, p) ∂R(x, p)/∂ps . = ∂R(x, p)/∂pt yt (x, p)

(200)

The Morishima elasticity of transformation between outputs s and t is     (x,p) s ∂ln ∂R(x,p)/∂p ∂ln yyst (x,p) ∂R(x,p)/∂pt     METst = − =− ∂ln ppst ∂ln ppst  ∂ 2 R(x,p) = ps

∂ps ∂pt ∂R(x,p) ∂pt

−

∂ 2 R(x,p) ∂ps2 ∂R(x,p) ∂ps

= δts (x, p) − δss (x, p)



 = ps

∂yt (x,p) ∂ps

yt (x, p)

−

∂ys (x,p) ∂ps



ys (x, p) (201)

where δts is the cross price elasticity of supply between output t and output price s, while δss is the own price elasticity of supply for output s. As our last application, we account for desirable and jointly produced undesirable outputs in the technology and derive the dual Morishima elasticity of transformation.26 This elasticity measures the percent change in the shadow price ratio due to a percent change in the ratio of the desirable to undesirable output. Here the directional output distance function provides a way of modeling the objective of simultaneously expanding desirable outputs while contracting undesirable outputs given the technology as represented by the output possibility set. The output possibility set is P (x) = {(y, b) : (y, b) can be produced by x}. The directional output distance function is defined on this set as

26 This

discussion follows [12].

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− → D o (x, y, b; gy , gb ) = sup{β : (y + βgy , b − βgb ) ∈ P (x)}.

(202)

β

The undesirable output price vector is q ∈ RJ+ , and the revenue function that accounts for the charges or fines that the producer must incur from producing undesirable output is R(x, p, q) = max{py − qb : (y, b) ∈ P (x)} y,b

(203)

which the representation property allows us to write as − → R(x, p, q) = max{py − qb : D o (x, y, b; gy , gb ) ≥ 0}. y,b

(204)

The duality between the revenue function and the directional output distance function allows the directional output distance function to be written as an unconstrained minimization problem  R(x, p, q) − (py − qb)  − → . D o (x, y, b; gy , gb ) = inf p,q pgy + qgb

(205)

Differentiating with respect to y and b yields − → ∇y D o (x, y, b; gy , gb ) = −

p pgy + qgb

and − → ∇b D o (x, y, b; gy , gb ) =

q . pgy + qgb

(206)

Thus, for desirable output s and undesirable output j , we have − → qj (x, y, b) ∂ D o (x, y, b; gy , gb )/∂bj =− − → ps (x, y, b) ∂ D o (x, y, b; gy , gb )/∂ys

(207)

where q(x, y, b) is the vector of inverse (price) supply functions for the undesirable output and p(x, y, b) is the vector of inverse supply functions for the desirable output. The dual Morishima elasticity of transformation can be derived as27

27 We

state the dual Morishima elasticity of transformation in terms of second derivatives of − → − → D o (x, y, b; gy , gb ) since the natural log of D o (x, y, b; gy , gb ) = 0 is not well-defined.

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js = MET

 qj (x,y,b) ∂ln ps (x,y,b) ∂lnbj − →

 ∂ 2 D o (x,y,b;gy ,gb ) ∂ys ∂bj

= bj

− → ∂ D o (x,y,b;gy ,gb ) ∂ys

 = bj

∂ps (x,y,b) ∂bj

ps (x, y, b)

−

−

− → ∂ 2 D ( x,y,b;gy ,gb ) ∂bj ∂bj − → ∂ D o (x,y,b;gy ,gb ) ∂bj

∂qj (x,y,b) ∂bj





qj (x, y, b)

= δ˜j s (x, y, b) − δ˜ss (x, y, b)

(208)

where δ˜j s (x, y, b) is the dual cross elasticity of transformation and δ˜ss (x, y, b) is the dual own elasticity of transformation.

Appendix: Production Axioms In this appendix, we list axioms on the technology sets. There is no particular order of these axioms. Since we have that (x, y) ∈ T ⇔ y ∈ P (x) ⇔ x ∈ L(y)

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each axiom may be given a representation in each of these sets. A.1 T is a closed set, i.e., if (x l , y l ) ∈ T for all l = 1, . . . , ∞ and (x l , y l ) → (x o , y o ), then (x o , y o ) ∈ T . A.2 P (x) is a closed set, i.e., if y l ∈ P (x), for all l = 1, . . . , ∞ and y l → y o , then y o ∈ P (x). A.3 L(y) is a closed set, i.e., if x l ∈ L(y) for all l = 1, . . . , ∞ and x l → x o , then x o ∈ L(y). Comment: A.1 implies A.2 and A.3. A.4 Strong disposability or free disposability of outputs: if y o ∈ P (x) and y  ≤ y o , then y  ∈ P (x). A.5 Weak disposability of outputs: if y o ∈ P (x) and 0 ≤ θ ≤ 1, then θy o ∈ P (x). A.6 Strong or free disposability of inputs: if x o ∈ L(y) and x  ≥ x o , then x  ∈ L(y). A.7 Weak disposability of inputs: if x o ∈ L(y) and θ ≥ 1, then θ x o ∈ L(y). Comment: A.4 implies A.5 and A.6 implies A.7. A.8 Scarcity: P (x) is bounded for each x ∈ RN +. A.9 Possibility of inaction: (0, 0) ∈ T . A.10 Possibility of inaction: 0 ∈ P (x) for each x ∈ RN +. Comment: A.10 implies A.9.

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A.11 T is a convex set, i.e., if (x 0 , y 0 ) ∈ T and (x 1 , y 1 ) ∈ T , then (θ x 0 + (1 − θ )x 1 , θy 0 + (1 − θ )y 1 ) ∈ T , 0 ≤ θ ≤ 1. A.12 P (x) is a convex set, i.e., if y 0 and y 1 ∈ P (x), then (θy 0 +(1−θ )y 1 ) ∈ P (x), 0 ≤ θ ≤ 1. A.13 L(y) is a convex set, i.e., if x 0 and x 1 ∈ L(y), then (θ x 0 + (1 − θ )x 1 ) ∈ L(y), 0 ≤ θ ≤ 1. Comment: A.11 implies A.12 and A.13. A.14 Constant returns to scale: T = θ T , θ > 0. A.15 Non-increasing returns to scale: (x, y) ∈ T and 0 ≤ θ ≤ 1, imply (θ x, θy) ∈ T. A.16 Non-decreasing returns to scale: (x, y) ∈ T and θ ≥ 1, imply (θ x, θy) ∈ T . A.17 Null-jointness: if (y, b) ∈ P (x) and b = 0, then y = 0.

Cross-References  Shadow Pricing in Production Economics

References 1. Allais M (1943) A la Recherche d’une Discipline Economique Premiére Partie, l’Economie Pure Paris: Atliers Industria, vol 1 2. Blackorby C, Russell RR (1989) Will the real elasticity of substitution please stand up? Am Econ Rev 79(4):882–888 3. Bogetoft P, Färe R, Obel B (2006) Allocative efficiency of technically inefficient production units. Eur J Oper Res 168(2):450–462 4. Chambers RG, Chung Y, Färe R (1996) Benefit and distance functions. J Econ Theory 70(2):407–419 5. Chambers RG, Chung Y, Färe R (1998) Profit, directional distance functions, and Nerlovian efficiency. J Optim Theory Appl 98(2):351–364 6. Charnes A, Cooper WW, Rhodes E (1978) Measuring the efficiency of decision making units. Eur J Oper Res 2:429–444 7. Deaton A (1979) The distance function in consumer behaviour with applications to index numbers and optimal tax theory. Rev Econ Stud 46(3):391–405 8. Deaton A, Muellbauer J (1980) Economics and consumer behavior. Cambridge University Press, Cambridge 9. Diewert WE (1998) Index number theory using differences rather than ratios, Department of Economics, University of British Columbia, Discussion paper No.98–10 10. Färe R (1988) Fundamentals of production theory. Springer, Berlin 11. Färe R, Grosskopf S, Margaritis D (2017) Shadow Pricing in Production Economics 12. Färe R, Grosskopf S, Noh D-W, Weber WL (2005) Characteristics of a polluting technology: theory and practice. J Econ 126:469–492 13. Färe R, Grosskopf S, Ray SC, Miller SM, Mukherjee K (2000) Difference measures of profit inefficiency: an application to U.S. Banks, Conference on Banking and Finance, Miguel Hernadez University, Elche, May 2000 14. Fare R, Primont D (1986) On differentiability of cost functions. J Econ Theory 38:233–237 15. Färe R, Primont D (1995) Multi-output production and duality: theory and applications. Kluwer Academic Publishers, Boston

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16. Färe R, Primont D (2006) Directional duality theory. Economic Theory 29(1):239–247 17. Färe R, Primont D, Samuelson L (1990) On differentiability of cost functions: corrigendum. J Econ Theory 52:237 18. Farrell MJ (1957) The measurement of productive efficiency. J R Stat Soc Ser A General 125:(Part 2):252–267 19. Fleming W (1977) Functions of several variables. Springer, New York 20. Kemeny JG, Morgenstern O, Thompson GL (1956) A generalization of the von Neumann Model of an expanding economy. Econometrica 24:115–135 21. Luenberger DG (1969) Optimization by vector space methods. Wiley, New York 22. Luenberger DG (1992) Benefit functions and duality. J Math Econ 21:461–481 23. Luenberger DG (1995) Microeconomic theory. McGraw-Hill, New York 24. Mas-Colell A, Whinston MD, Green JR (1995) Microeconomic theory. Oxford University Press, New York 25. Morishima M (1967) A few suggestions on the theory of elasticity, (in Japanese). Kezai Hyoron (Economic Review) 16:144–150 26. von Neumann J (1937, 1945/1946), Über ein ökonomisches gleichungssytem und eine verallgemeinerung des Brouwerschen Fixpunksatzes. In: Menger K (ed) Ergebnisses eines Mathematischen Kolloquiums reprinted as A Model of General Economic Equilibrium Rev Econ Stud 13(1):1–9 27. Russell R (2020) Elasticities of substitution. In: Chambers R, Kumbhakar S, Ray SC (eds) Handbook of production economics, vol 1. Springer, Singapore 28. Shephard RW (1953) Cost and production functions. Princeton University Press, Princeton 29. Shephard RW (1970) Theory of cost and production functions. Princeton University Press, Princeton 30. Weymark JA (1980) Duality results in demand theory. Eur Econ Rev 14:377–395

5

Functional Structure and Aggregation Daniel Primont

Contents Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Example: Intermediate Inputs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Example: House Building . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Functional Structure with Two Sectors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Defining Separability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Separability and Functional Structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Functional-Structure Equivalences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Functional Structure with More Than Two Sectors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Separability of Dual Representations of Technology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Homothetic Separability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Additive Functional Structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Recursive Functional Structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Application . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Multioutput Technologies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

216 217 218 219 219 222 230 236 237 241 245 248 252 253 256

Abstract

A production function can involve dozens or even hundreds of inputs that are combined to produce a single output. Economists can simplify these complex processes by positing separability restrictions which, in turn, yield (1) a particular functional structure of the production function and (2) aggregates of subsets of the inputs that can be viewed as intermediate inputs. This has the effect of reducing the number of variables that enter the economic analysis.

D. Primont () Department of Economics, Southern Illinois University-Carbondale, Carbondale, IL, USA e-mail: [email protected] © Springer Nature Singapore Pte Ltd. 2022 S. C. Ray et al. (eds.), Handbook of Production Economics, https://doi.org/10.1007/978-981-10-3455-8_22

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Various combinations of separability assumptions are considered, and the resulting forms of functional structure are characterized. This survey considers not only production functions but also dual representations such as cost functions and indirect output functions thereby giving rise to aggregates of both input quantities and input prices. The extension to the case of multiple outputs is briefly considered. Keywords

Separability · Functional structure · Aggregation · Production function · Cost function · Distance function

Introduction Consider a production technology for which three inputs are used to produce a single output. This is typically represented by a production function written as: y = f (x1 , x2 , x3 ) ,

(1)

where x1 , x2 , and x3 denote the quantities of the three inputs used and y denotes the quantity of output. Suppose it is possible to aggregate inputs 2 and 3 into an intermediate input denoted by φ (x2 , x3 ) . Moreover, suppose we could rewrite the production function in (1) as y = f (x1 , x2 , x3 ) = f (x1 , φ (x2 , x3 )) .

(2)

In other words we can also represent the technology as one for which the output is produced using x1 and an intermediate input, φ (x2 , x3 ) . The result of this is an example of what is meant by the term “functional structure.” One can ask the following question: What assumption can be made about the production technology that would justify the functional structure expressed in equation (2)? To answer this question, we compute the marginal rate of substitution between inputs 2 and 3 using equation (2). We get ∂y/∂x2 (∂f (x1 , z) /∂z) (∂φ (x2 , x3 ) /∂x2 ) = ∂y/∂x3 (∂f (x1 , z) /∂z) (∂φ (x2 , x3 ) /∂x3 ) =

∂φ (x2 , x3 ) /∂x2 , ∂φ (x2 , x3 ) /∂x3

where z = φ (x2 , x3 ) is the quantity of the intermediate input. We see that the marginal rate of substitution between inputs 2 and 3 does not depend on the quantity of input 1, i.e.,

5 Functional Structure and Aggregation

∂ ∂x1



217

∂y/∂x2 ∂y/∂x3

 = 0.

(3)

We may conclude that the functional structure expressed in equation (2) implies the restriction in equation (3). What is remarkable is that the restriction in (3) implies the functional structure in (2), i.e., (2) and (3) are equivalent. This equivalence was demonstrated in pioneering papers by Leontief [19, 20] and Sono [26, 27]. Stated in words, the restriction in (3) says that inputs 2 and 3 are separable from 1. Hence, the notion of functional structure is intimately related to the concept of separability. For the remainder of this section, we will look at some specific examples of functional structure that will, hopefully, motivate our study. In section “Functional Structure with Two Sectors” we consider functional structure for the simple case of two subsets, or sectors, of inputs for which the inputs in sector 2 are separable from the inputs in sector 1. In section “Functional Structure with More Than Two Sectors” we consider a more general case for which two or more sectors are separable from all inputs in the other sectors. The examples of steel production and personal computer production given below pertain to section “Functional Structure with More Than Two Sectors”. In section “Additive Functional Structure” we consider a functional structure that involves additivity. In section “Recursive Functional Structure” we study functional structure that is completely recursive. Complete recursivity is another way that the results in section “Functional Structure with Two Sectors” may be generalized. The house building example given below illustrates a completely recursive production function. Finally, in section “Multioutput Technologies” we consider how the results for single-output production may be extended to multioutput production. All applications considered here will involve functional structure of various representations of a production technology. For applications of functional structure to utility or welfare functions, the reader is referred to the survey by Blackorby et al. [4].

Example: Intermediate Inputs One of the uses of functional structure is the modelling of intermediate inputs in production. This was, in fact, the example used by Leontief [19] who considered the production of steel using the intermediate inputs, coal and iron ore, along with a labor input. In this simplified example, inputs 1,2, and 3 are used to produce coal; inputs 4,5,6, and 7 are used to produce iron ore; and input 8 is the labor input that runs the blast furnaces. This results in the following system: y1 = F 1 (x1 , x2 , x3 ) (coal production)

(4)

y2 = F (x4 , x5 , x6 , x7 ) (iron ore production)

(5)

2

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y = F (y1 , y2 , x8 ) (steel production)

(6)

Substituting (4) and (5) into (6) yields the following functional structure   y = F F 1 (x1 , x2 , x3 ) , F 2 (x4 , x5 , x6 , x7 ) , x8 .

(7)

Here is another example. A simple personal computer can be assembled from the following components. 1. 2. 3. 4. 5. 6. 7.

Case Power supply Motherboard Central processing unit Cooling system Random-access memory Hard drive

Since these components are numbered from 1 to 7, we denote the quantities of each component by the variables y1 , y2 , . . . , y7 . Other inputs are denoted by x 0 and would include the labor and tools used to assemble the personal computers. Let y be the number of computers assembled; clearly y is a function of (x 0 , y1 , y2 , . . . , y7 ), and we can write the production function for personal computers as y = F(x 0 , y1 , y2 , . . . , y7 ).

(8)

Of course, each of the components themselves must be produced before they can be assembled. Let the production function for the r-th component be given by yr = F r (x r ) where x r is the vector of inputs used to produce yr , r = 1, 2, . . . , 7. Assume that none of the inputs that produce component r are used in the production of component s where s = r. For example, the labor that is used to produce the power supplies is specialized in that use and is not used in the production of any other component. Substitution of the component production functions into (8) yields        y = F x0, F 1 x1 , F 2 x2 , . . . , F 7 x7 .

(9)

The functional structure exhibited in equation (9) implies that the inputs used to produce yr are separable from all other inputs.

Example: House Building Building a house involves a production process with many inputs of capital, labor, and materials. How might an economist model such a complex undertaking?

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One answer involves breaking down this large problem into a number of smaller subproblems. Consider the following steps in building a house. 1. 2. 3. 4.

Prepare site Pour foundation. Frame floor, walls, and roof. Install plumbing, electrical system, heating, ventilation, air conditioning, and so on.

Each step can be modelled by a subproduction function that depends on specialized inputs of labor, capital, and materials. Let y1 be the site preparation output and let x 1 be the vector of inputs used in site preparation. Then   we can write the production function for this intermediate output as y1 = F 1 x 1 . Foundation output will be a function of prepared sites, y1 , and inputs that are used to pour   foundations, x2 . We get y2 = F 2 y1 , x 2 . Next, let y3 be the framing output and x3 the vector of inputs used in framing. The production function for this intermediate output is y3 = F 3 y2 , x 3 , and so on. For expositional simplicity, suppose that all production steps beyond step 3 are lumped together into a final step 4, viz., finish building house. Then the final output, y, will equal step 4 output, y4 , and we have         y = y4 = F 4 F 3 F 2 F 1 x 1 , x 2 , x 3 , x 4 .   The inputs in x 1 are separable from the inputs in x 2 , x 3 , x 4 , the inputs in  3 4    1 2 x , x are separable from the inputs in x , x , and the inputs in x 1 , x 2 , x 3 are separable from the inputs in x 4 .

Functional Structure with Two Sectors In this section, functional structure and its associated property, separability, are examined in the simple case of two sectors, numbered 1 and 2. In this setting, functional structure arises from the condition that sector 2 is separable from sector 1 but sector 1 is not necessarily separable from sector 2.

Defining Separability Let RN + be the N-dimensional nonnegative Euclidean orthant, i.e.,  RN + = x : x = (x1 , . . . , xN ) , xn  0, n = 1, . . . , N .

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A production function is a real-valued function defined on the domain RN + with image y = F (x) . It is assumed that y = F (x) is the maximum output that can be produced using the input vector x. The range of F is denoted by R (F ) . Let I = {1, 2, . . . , N } be the set of integers that identify the inputs for which the production function is defined. Sort the set of inputs into two groups, i.e., let 

I¯ = I 1 , I 2 be a binary partition of I. In other words I 1 ∪ I 2 = I,

I 1 ∩ I 2 = ∅,

I 1 = ∅,

I 2 = ∅.

Assume that I has been conveniently ordered so that I 1 = {1, 2, . . . , n1 } and I 2 = {n1 + 1, n1 + 2, . . . , N } . Let n2 = N − n1 . The production function is equivalently defined on the domain  n1 n2 RN vector can be written as x = x 1 , x 2 . + = R+ × R+ . Moreover, the input  Output is now given by y = F x 1 , x 2 . According to the Leontief-Sono definition, I 2 is separable from I 1 if and only if ∂ ∂xk



∂y/∂xi ∂y/∂xj

 = 0 for all i, j ∈ I 2 , k ∈ I 1 .

The economic content of this condition is this: Inputs in sector I 2 are separable from inputs in sector I 1 if and only if marginal rates of substitution between inputs in sector I 2 are independent of input quantities in sector I 1 . For example, suppose there are only three inputs. Let x 1 = x1 and x 2 = (x2 , x3 ) . Further suppose that the production function is given by   1/2 1/4 1/4 1/2 1/2 y = F x 1 , x 2 = x1 x2 x3 + x2 x3 . Then     1/2 −3/4 1/4 −1/2 1/2 + x x x x x (1/2) (1/4) 1 2 3 2 3 ∂y/∂x2     = 1/2 1/4 −3/4 1/2 −1/2 ∂y/∂x3 + (1/2) x2 x3 (1/4) x1 x2 x3   ⎞  ⎛ 1/2 −3/4 1/4 −1/2 1/2 + (1/2) x2 x3 x3 x2 ⎝ (1/4) x1 x2 x3   ⎠  = x2 x3 (1/4) x 1/2 x 1/4 x −3/4 + (1/2) x 1/2 x −1/2 1 2 3 2 3

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221

   ⎞ 1/2 1/4 1/4 1/2 1/2 x x + x x x (1/2) (1/4) 1 2 3 2 3 x3 ⎝   ⎠  = x2 (1/4) x 1/2 x 1/4 x 1/4 + (1/2) x 1/2 x 1/2 1 2 3 2 3 ⎛

=

x3 . x2

Thus the marginal rate of technical substitution between inputs 2 and 3 does not depend on the value of input 1 since ∂ ∂x1



x3 x2

 = 0,

and hence inputs 2 and 3 are separable from input 1. It can be verified that inputs 1 and 2 are not separable from input 3 and that inputs 1 and 3 are not separable from input 2. Another way to understand the separablity condition is to define a conditional isoquant in Rn+2 . It is given by the set of group 2 input vectors that yield constant output conditional on group 1 inputs, i.e.,

   x 2 : F x 1 , x 2 = k = constant .

Separability implies the invariance of marginal rates of substitution in group 2 when group 1 inputs change. This will be guaranteed if conditional isoquants do not depend on group 1 inputs. This means that conditional upper level sets in Rn+2 , given by

   x 2 : F x 1 , x 2  k = constant , do not depend on group 1 inputs. We can now state a definition of separability, due to Stigum [28], that does not depend on differentiability. Define the conditional upper level set by  

    β x 1 , x 2 = xˆ 2 : xˆ 2 ∈ Rn+2 , F x 1 , xˆ 2  F x 1 , x 2 .   Inputs in group 2 are separable from inputs in group 1 if and only if β x 1 , x 2 is   independent of x 1 for all x 1 , x 2 in Rn+1 × Rn+2 . In this case,     β x 1 , x 2 = β x˜ 1 , x 2 , for all x 1 ∈ Rn+1 , x˜ 1 ∈ Rn+1 , x 2 ∈ Rn+2 . There is another way of stating the definition of separability. Inputs in group 2 are separable from inputs in group 1 if and only if

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        F x 1 , xˆ 2  F x 1 , x 2 ⇔ F x˜ 1 , xˆ 2  F x˜ 1 , x 2 , for all x 1 , x˜ 1 ∈ Rn+1 , x 2 , xˆ 2 ∈ Rn+2 .

Separability and Functional Structure The Production Function The following theorem is the basic result that links separability with functional structure. Theorem 1 ( [7, 14]). Assume that the production function is continuous. Given the partition, I 1 , I 2 , inputs in group 2 are separable from inputs in group 1 if and only if there exists continuous functions F 2 and F such that      F x1, x2 = F x1, F 2 x2 ,

(10)

   2 2 for all x 1 , x 2 ∈ RN + , where F is increasing in its last argument, F x . The following sketch of the proof of Theorem 1 is short and may yield further insight. To show that the given functional structure implies the separability condition, note that 

     n2 β x 1 , x 2 = xˆ 2 : xˆ 2 ∈ R+ , F x 1 , xˆ 2  F x 1 , x 2      

n2 , F x 1 , F 2 xˆ 2  F x 1 , F 2 x 2 = xˆ 2 : xˆ 2 ∈ R+    

n2 , F 2 xˆ 2  F 2 x 2 , = xˆ 2 : xˆ 2 ∈ R+ where the last equality is implied in its  by the  condition that F is increasing  last argument. This shows that β x 1 , x 2 is independent of x 1 for all x 1 , x 2 in Rn+1 × Rn+2 . Conversely, suppose the separability condition holds. Let     F 2 x 2 = F O 1 , x 2 for all x 2 ∈ Rn+2 for some arbitrarily chosen vector O 1 in Rn+1 . Next, define        F x 1 , F 2 x 2 = F x 1 , x 2 for all x 1 , x 2 ∈ Rn+1 × Rn+2 . We need to show that F is increasing in its last argument. First, note that

(11)

5 Functional Structure and Aggregation

223

    F 2 x 2  F 2 xˆ 2 if and only if     F O 1 , x 2  F O 1 , xˆ 2 for all O 1 ∈ Rn+1 if and only if       F x 1 , F 2 x 2  F x 1 , F 2 xˆ 2 . Hence, we have shown that           F 2 x 2  F 2 xˆ 2 if and only if F x 1 , F 2 x 2  F x 1 , F 2 xˆ 2 ,   which implies that F is increasing in y2 = F 2 x 2 . We will refer to the function F 2 in (10) as an aggregator function since it aggregates the inputs in group 2 into a scalar measure, e.g., an intermediate input.   The function F will be called a macro function since it maps x 1 and y2 = F 2 x 2 into final output, y. The original production function F will be referred to as the master function. The functional structure,      F x1, x2 = F x1, F 2 x2 , may be given a schematic representation as in Fig. 1. Although we began with a one-period production function, Fig. 1 suggests that production takes place over two time periods. In the first period, group 2 inputs are used to produce an intermediate input, y2 . In the second period, group 1 inputs and the intermediate input, y2 , are used to produce final output y. This can be interpreted as a network production model. See Färe and Grosskopf [9] and the discussion that begins on page 4. So far we have only assumed that the master function F is continuous. Theorem 1 tells us that in this case, both the aggregator function F 2 and the macro function F can be chosen to be continuous. Put another way, continuity of the aggregator function and continuity of the macro function are inherited from the master function. x1

Fig. 1 One Separable Sector

x

2

F

2

y2

F

y

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Are there other properties of the master function that are inherited by the aggregator function and the macro function? The answer is generally yes. Here is a list of some of these inherited properties: (i) quasiconcavity, (ii) strictly increasing, (iii) homotheticity, (iv) homogeneity of degree one, and (v) concavity. Proofs of these claims are somewhat scattered in the literature. For example, proofs for properties (i) and (iv) can be found in Solow [25]. A comprehensive set of proofs for all five properties (plus more) are given in Theorem 3.5, page 78, in Blackorby et al. [3]. A simple example that illustrates the construction of F 2 in equation (11) is 1/3 1/3 1/3

F (x1 , x2 , x3 ) = x1 x2 x3 . It is straightforward to show that inputs 2 and 3 are separable from input 1. Let the reference vector, O 1 , be x1 = 1. Then an aggregator function for inputs 1/3 1/3 2 and 3 is just F 2 (x2 , x3 ) = x2 x3 . A correct choice of F in this case   1/3 is F x1 , F 2 (x2 , x3 ) = x1 F 2 (x2 , x3 ) . However, these choices of F and F 2 1/2 1/2 are not unique. Another correct choice would be F 2 (x2 , x3 ) = x2 x3 and     2/3 1/3 F x1 , F 2 (x2 , x3 ) = x1 F 2 (x2 , x3 ) . Note that this latter choice entails an aggregator function that exhibits constant returns to scale in inputs 2 and 3. An additional example, that was previously considered, is   1/2 1/4 1/4 1/2 1/2 y = F x 1 , x 2 = x1 x2 x3 + x2 x3 , where x 1 = x1 and x 2 = (x2 , x3 ) . Let   1/2 1/2 F 2 x 2 = x2 x3 . Then      1/2   1/2 F x 1 , F 2 x 2 = x1 F 2 x 2 + F 2 x2 .

Duality In this section we review some basic results from duality theory. More thorough treatments can be found in Shephard [24], Blackorby et al. [3], Diewert [8], and Färe and Primont [10]. It is assumed that the production function is (i) continuous and nondecreasing in x, i.e.,   x  x  ⇒ F (x)  F x  for all x, x  in RN +; N (ii) satisfies a no-local-maxima condition, i.e., given  any input  vector x in R+  and     any ε > 0, there exists another input vector x with x − x < ε such that F x >

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225

F (x) ; and (iii) quasiconcave in x, i.e.,       F (x)  F x  ⇒ F αx + (1 − α) x   F x  , for all α such that 0  α  1 and for all x, x  in RN + . The no-local-maxima condition implies that isoquants are not “thick”. It is the production theory analog to the assumption of local nonsatiation in utility theory. There are several dual representations of the technology. They include the cost function, the input distance function, and the indirect production function. Each of these is defined below. In a subsequent section, we will look at multioutput technologies and discuss an alternative representation, namely, the output distance function. The cost function is defined by  C (y, w) = min w · x : F (x)  y , w ∈ RN ++ , x

where RN ++ = {w = (w1 , .., wN ) : wn > 0, n = 1, . . . , N} is the set of input price vectors whose components are strictly positive. The cost function is (i) jointly continuous in (y, w) , (ii) increasing in y, and (iii) nondecreasing, concave, and homogeneous of degree one in w. If a function C (y, w) satisfies (i)–(iii), then one may recover the production function that would generate C (y, w) through the cost minimization problem. The input distance function is defined by  D (y, x) = max λ : F (x/λ)  y . λ>0

It is (i) jointly continuous in (y, x) , (ii) decreasing in y, and (iii) nondecreasing, concave, and homogeneous of degree one in x. On the other hand, if we start with a function D (y, x) that satisfies (i)–(iii), then one can find the corresponding production function by solving F (x) = max {y : D (y, x) ≥ 1} . y

As a consequence, the input distance function is an implicit representation of the production function in the sense that y = F (x) ⇔ D (y, x) = 1. The indirect production function is defined by  G (w/c) = max F (x) : (w/c) · x  1 , c > 0 and w ∈ RN ++ . x

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This is the production counterpart to the indirect utility function of consumer theory. It is (i) continuous, (ii) nonincreasing, and (iii) quasiconvex in w/c. (G is quasiconvex if −G is quasiconcave.) The (direct) production function may be recovered from G; it will be the production function that would generate G through the output maximization problem. Details of these results can be found in Chapter 2 and the Appendix in Blackorby et al. [3] and in Diewert [8]. The relationship between the cost function and the indirect production function is analogous to the relationship between the input distance function and the production function. In particular  C (y, w) = min c : G (w/c)  y c>0

and  G (w/c) = min y : C (y, w/c)  1 . y

It follows that the cost function is an implicit representation of the indirect production function, i.e., y = G (w/c) ⇔ C (y, w/c) = 1. There are two important duality relationships. The first of these involves the input distance function and the cost function. The main result is:  C (y, w) = min w · x : D (y, x)  1

(12)

 D (y, x) = min w · x : C (y, w)  1 .

(13)

x

if and only if w

Equation (12) follows from the fact that F (x)  y ⇔ D (y, x)  1. If the solution to (12) is unique, denote it by x ∗ = ζ (y, w) . It is the vector of cost minimizing or Hicksian input demand functions. (If the solution is not unique, then we can reinterpret ζ (y, w) as an arbitrary selection from the set of solutions.) The Hicksian demand functions are homogeneous of degree zero in w. The minimized value of cost is C (y, w) = w · ζ (y, w) . Similarly, let w ∗ = δ (y, x) be the solution to (13) (or an arbitrary selection from the set of solutions). It is the vector of output-constant price-demand functions or, alternatively, the vector of shadow prices of the input vector x. The shadow price functions are homogeneous of degree zero in x. The shadow cost of x is D (y, x) = δ (y, x) · x. The following result is attributed to Hotelling [18]: If C is differentiable at (y, w) , then

5 Functional Structure and Aggregation

ζi (y, w) =

227

∂C (y, w) , i = 1, . . . , N. ∂wi

(14)

A result that is dual to (14) is attributed to Shephard [24]: If D is differentiable at (y, x) , then δi (y, x) =

∂D (y, x) , i = 1, . . . , N. ∂xi

(15)

Another duality relationship exists for the production function and the indirect production function. This is given by:  G (w/c) = max F (x) : (w/c) · x  1

(16)

 F (x) = min G (w/c) : (w/c) · x  1 .

(17)

x

if and only if

w/c

Denote the solution to (16) by x ∗ = φ (w/c) . It is the vector of output maximizing input vectors when the firm is cost-constrained. (If the solution is not unique, then let φ (w/c) be an arbitrary selection from the set of solutions.) It is analogous to the vector of Marshallian demand functions in the theory of the consumer; that label can be carried over to the theory of the firm. The value of maximized output is given by G (w/c) = F (φ (w/c)) . Similarly, let (w/c)∗ = ξ (x) be the solution (or one of the solutions) to (17). It is the vector of cost-normalized price-demand functions or, alternatively, the vector of shadow prices of the input vector x. The following result is attributed to Roy [22, 23]: If G is differentiable at w/c, then

φi (w/c) =

∂G (w/c) ∂ (wi /c) N   ∂G (w/c)    · wj /c ∂ wj /c j =1

, i = 1, . . . , N.

(18)

A result that is dual to (18) is attributed to Wold [29]: If F is differentiable at x, then ∂F (x) ∂xi ξi (x) = N  ∂F (x) j =1

∂xj

, i = 1, . . . , N. xj

(19)

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The Cost Function  For the partition I 1 , I 2 , we define separability of the cost function by first defining the conditional upper level sets  

    γ 2 y, w 1 , w 2 = wˆ 2 ∈ Rn+2 : C y, w 1 , wˆ 2  C y, w 1 , w 2 ,   for each y ∈ R (F ) , w 1 , w 2 ∈ Rn+1 × Rn+2 . Then we say that the input prices in I 2   are separable from the input prices in I 1 if and only if γ 2 y, w 1 , w 2 is independent of w1 , i.e.,     γ 2 y, w 1 , w 2 = γ 2 y, w˜ 1 , w 2 ,   for each y ∈ R (F ) , for all w 1 , w 2 ∈ Rn+1 × Rn+2 , and for any w˜ 1 ∈ Rn+1 . It is important to note that while I 2 is separable from I 1 , it is not necessarily separable from the output variable y. (This case will be considered later.) If C is twice differentiable with positive first partial derivatives, then we can apply the Leontief-Sono condition for separability. In this case I 2 is separable from I 1 if and only if ∂ ∂wk



∂C (y, w) /∂wi ∂C (y, w) /∂wj



∂ = ∂wk



ζi (y, w) ζj (y, w)

 = 0,

for all i, j ∈ I 2 , k ∈ I 1 and for all (y, w) ∈ R (F ) × Rn+1 × Rn+2 . This condition says that ratios of Hicksian demand functions for inputs in sector I 2 do not depend on input prices in sector I 1 . Next is a statement of the functional structure of the cost function that obtains under the above separability conditions. Theorem 2. Assume that C is continuous and nondecreasing in w. Then I 2 is separable in C from I 1 if and only if there exist functions C 2 and C such that    C (y, w) = C y, w 1 , C 2 y, w 2 ,

(20)

  2 2 for all (y, w) ∈ R (F )×RN + , where C is increasing in its last argument, C y, w . In addition, if C is continuous, nondecreasing, homogeneous of degree one, and concave in w, jointly continuous in (y, w) , and increasing in y, then C is continu ous, nondecreasing, homogeneous of degree one, and concave in w 1 , C 2 y, w 2 , and C 2 is continuous, nondecreasing, homogeneous of degree one, and concave in  w 2 and is jointly continuous in y, w 2 and increasing in y. The proof of this theorem is contained in the proof of Theorem 3.4, page 70, in Blackorby, Primont, and Russell [3]. As stated both the macro function C and

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229

2 the aggregator function C  have  the properties of the master function C. It is thus 2 tempting to think of C y, w 2 as a price index for inputs in sector I 2 . However, this is somewhat problematic since C 2 depends on the output level y. We will see later what we get when y is freed from C 2 .

The Input Distance Function To define separability in the context of the input distance function, define the conditional upper level sets  

    η2 y, x 1 , x 2 = xˆ 2 : xˆ 2 ∈ Rn+2 , D y, x 1 , xˆ 2  D y, x 1 , x 2 ,   for each y ∈ R (F ) and for each x 1 , x 2 ∈ Rn+1 × Rn+2 . The set of inputs in I 2 is   separable from the set of inputs in I 1 if and only if η2 y, x 1 , x 2 is independent of x 1 , i.e.,     η2 y, x 1 , x 2 = η2 y, x˜ 1 , x 2 ,   for each y ∈ R (F ) , for all x 1 , x 2 ∈ Rn+1 × Rn+2 , and for any x˜ 1 ∈ Rn+1 . Once again it is important to note that while I 2 is separable from I 1 , it is not necessarily separable from the output variable y. If D is twice differentiable with positive first partial derivatives, then the Leontief-Sono definition is: I 2 is separable from I 1 if and only if ∂ ∂xk



∂D (y, x) /∂xi ∂D (y, x) /∂xj

 =

∂ ∂xk



δi (y, x) δj (y, x)

 = 0,

for all i, j ∈ I 2 , k ∈ I 1 and for all (y, x) ∈ R (F ) × RN + . In words, the ratios of shadow prices for inputs in sector 2 are independent of the quantities of inputs in sector 1. Note that this ratio is not independent of the level of output y. We now state: Theorem 3. Assume D is continuous and nondecreasing. I 2 is separable from I 1 in D if and only if there exist functions D 2 and D such that    D (y, x) = D y, x 1 , D 2 y, x 2 ,

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  2 2 for all (y, x) ∈ R (F ) × RN + , where D is increasing in its last argument, D y, x . The proof of Theorem 3 parallels that of Theorem 2 since C (y, w) and D (y, x) have the same properties (except the monotonicity property in y.)

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The Indirect Production Function The definition of the separability of a sector of normalized price vectors in the indirect production function is based on the lower level sets given by  α

2

w1 w2 , c c



 =

  1   1 wˆ 2 w w2 w wˆ 2 n2 ∈ R+ : G , G , . c c c c c

We say that the normalized input prices in I 2 are separable from the normalized input prices in I 1 if and only if α 2 w 1 /c, w 2 /c is independent of w 1 /c, i.e.,  α2

w1 w2 , c c



 = α2

w˜ 1 w 2 , c c

 ,

  for all w 1 /c, w 2 /c ∈ RN ˜ 1 /c ∈ Rn+1 . If G is twice differentiable + and for all w with negative first partial derivatives, then the Leontief-Sono separability condition is written as:     ∂ φi (w/c) ∂ ∂G (w/c) /∂ (wi /c)   = = 0, ∂ (wk /c) ∂G (w/c) /∂ wj /c ∂ (wk /c) φj (w/c) 2 1 for all (w/c) ∈ RN + , for all (i, j ) ∈ I , and for all k ∈ I , where the second equality follows from Roy’s theorem (18). In this case the ratios of cost-constrained output maximizers (Marshallian demand functions) in I 2 are independent of normalized prices in I 1 .

Theorem 4. Assume G is continuous and nonincreasing. Sector I 2 is separable in G from I 1 if and only if there exist continuous functions G2 and G such that    G (w/c) = G w 1 /c, G2 w 2 /c ,

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 2  2 for all (w/c) ∈ RN + , where G is decreasing in G w /c . If, in addition, G is quasiconvex, then G is nonincreasing, and quasiconvex and G2 is nondecreasing and quasiconcave.

Functional-Structure Equivalences Cost and Input Distance Functions Separability of sector I 2 from I 1 in F is not equivalent to separability of sector I 2 from I 1 in D or in C or in G. Put another way, separability of I 2 from I 1 imposes a different restriction on the production technology than does separability of I 2 from I 1 in D or in C or in G. This statement is generally true for any pair of these four

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231

representations except for one pair, namely, the input distance function and the cost function. The following is proved in Blackorby et al. ( [3], Theorem 3.6, page 83). Their proof is based on Gorman [15, 17]. It is also proved by McFadden [21]. Before stating the theorem notice that we must assume that the production function, F, is continuous, nondecreasing, and quasiconcave in x and satisfies the no-local-maxima condition. This assumption implies that the cost function is jointly continuous in (y, w) , increasing in y, and nondecreasing, concave, and homogeneous of degree one in w and that the input distance function is jointly continuous in (y, x) , decreasing in y, and nondecreasing, concave, and homogeneous of degree one in x. Thus, the duality relationship between C and D holds. Theorem 5. Assume that the production function is continuous, nondecreasing, and quasiconcave in x. Then sector I 2 is separable from sector I 1 in the input distance function, i.e.,    D (y, x) = D y, x 1 , D 2 y, x 2 if and only if sector I 2 is separable from sector I 1 in the cost function, i.e.,    C (y, w) = C y, w 1 , C 2 y, w 2 . Proof. “only if”:

    C (y, w) = min w 1 · x 1 + w 2 · x 2 : D y, x 1 , D 2 y, x 2  1 x  

     = min w 1 · x 1 + min w 2 · x 2 : D 2 y, x 2  d2 : D y, x 1 , d2  1 x 1 ,d2

x2



   = min w 1 · x 1 + d2 min w 2 · x 2 /d2 : D 2 y, x 2 /d2  1 x 1 ,d2

x 2 /d2

   : D y, x 1 , d2  1

     = min w 1 · x 1 + d2 C 2 y, w 2 : D y, x 1 , d2  1 x 1 ,d2

   = C y, w1 , C 2 y, w 2 . “if”: Because of the symmetry between C and D, one can repeat the “only if” proof swapping the roles of Cand D, C and D, and C 2 and D 2 and minimizing over input prices rather than input quantities. It is interesting to ask what it is about the D − C pair that distinguishes it from the other possible pairings of technology representations. The answer seems

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to be that only the D − C pair has the property that both representations are homogenous of degree one in its variables (except y). This means that D 2 (C 2 ) can be chosen to be homogeneous of degree one in x 2 (w 2 ). This suggests that adding the assumption that the aggregator functions are homothetic (and so can be chosen to be homogeneous of degree one) would extend the result of Theorem 5 to all possible pairs of representations. This turns out to be the case as we will see in the next subsubsection.

Homothetic Separability The production function is homothetically separable if it has the functional structure      F x1, x2 = F x1, F 2 x2 ,   where F is increasing in the intermediate input y2 = F 2 x 2 and F 2 is a homothetic function. We want to examine the implications of a homothetically separable production function for the cost, input distance, and indirect production function. As a preliminary step, we will first consider a special case in which the parent function,  F , is homothetic in its arguments x 1 , x 2 Theorem 6. Assume that the production function is continuous, nondecreasing, and quasiconcave in x. Then the following four statements are equivalent: (i) The production function F is homothetic, i.e.,   F (x) = h F¯ (x) , where h is an increasing function of its single argument and F¯ is homogeneous of degree one in x. (ii) There exist functions C and C¯ such that   C (y, w) = C y, C¯ (w) , where C is increasing in c = C¯ (w) and C¯ (w) is homogeneous of degree one in w. In this case   C y, C¯ (w) = h−1 (y) C¯ (w) . (iii) The indirect production function is negatively homothetic, i.e.,  1 , G (w/c) = h C¯ (w/c) 

where h is increasing (as in (i)) and 1/C¯ (w/c) is homogeneous of degree minus one in w/c.

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233

(iv) Their exist functions D and D¯ such that   D (y, x) = D y, D¯ (x) , where D is increasing in its last argument and D¯ is homogeneous of degree one in x. In this case D (y, x) =

F¯ (x) . h−1 (y)

Proof. (i) ⇒ (ii):  C (y, w) = min w · x : F (x)  y x    = min w · x : h F¯ (x)  y x

 = min w · x : F¯ (x)  h−1 (y) x

 = min w · x : F¯ (x)  y , y =h−1 (y) x  = y min w · (x/y) : F¯ (x/y)  1 x/y

= yC (1, w) = h−1 (y) C¯ (w) (ii) ⇒ (iii):  G (w/c) = max y : C (y, w)  c y

 = max y : h−1 (y) C¯ (w)  c y

 = max y : h−1 (y) C¯ (w/c)  1 y

 = max y : h−1 (y) 

 1 y C¯ (w/c)    1 = max y : y  h y C¯ (w/c)   1 . =h ¯ C (w/c) (iii) ⇒ (iv):

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D. Primont

 D (y, x) = min (w/c) · x : G (w/c)  y w/c

  = min (w/c) · x : h w/c

1 ¯ C (w/c)



 y

 1 −1  h (y) w/c C¯ (w/c)   1  y , y = h−1 (y) , = min (w/c) · x : w/c C¯ (w/c)   1 1 1 = min y (w/c) · x : y yw/c C¯ (yw/c)

 = min (w/c) · x :

=

1 ¯ F (x) y

=

F¯ (x) . h−1 (y)

(iv) ⇒ (i):  F (x) = max y : D (y, x)  1 y

  F¯ (x) 1 = max y : −1 y h (y) 

= max y : F¯ (x)  h−1 (y) y

   = max y : h F¯ (x)  y y

  = h F¯ (x) . See Blackorby et al. [3], pages 91–93. It is apparent from the theorem that homotheticity of the production function is equivalent to separability of all input prices from output in the cost function and equivalent to separability of all inputs from output in the input distance function. The following is a generalization of Theorem 6. Theorem 7. Assume that the production function is continuous, nondecreasing, and quasiconcave in x. Then the following four statements are equivalent. (i) The production function is homothetically separable, i.e., it has the functional structure      F x 1 , x 2 = F x 1 , F¯ 2 x 2 ,

5 Functional Structure and Aggregation

235

  where F is increasing in y2 = F¯ 2 x 2 and F¯ 2 is a homothetic function. (ii) The cost function has the following functional structure    C (y, w) = C y, w 1 , C¯ 2 w 2 ,   where C is increasing in c2 = C¯ 2 w 2 and C¯ 2 is homogeneous of degree one in w 2 . (iii) The input distance function has the following functional structure    D (y, x) = D y, x 1 , D¯ 2 x 2 ,   where D is increasing in d2 = D¯ 2 x 2 and D¯ 2 is homogeneous of degree one in x 2 . (iv) The indirect production function is homothetically separable, i.e., it has the following functional structure    ¯ 2 w 2 /c , G (w/c) = G w 1 /c, G   ¯ 2 w 2 /c and G ¯ 2 is homogeneous of degree where G is increasing in y2 = G 2 minus one in w /c. The above theorem is proved in Blackorby et al. ( [3], pages 94–97).

Two Sector Applications It has already been noted in the Introduction that various functional structures may be used to model intermediate inputs in production. Another application involves the study of real valued added. Suppose, for simplicity, that the production function may be written as y = F (m, K, L) = F (m, F (K, L)) , where m is the input of raw materials, K is the capital input, and L is the labor input. The above functional structure embodies the assumption that capital and labor are separable from materials. Then F (K, L) is interpreted as real value added. This is the starting point of a paper by Arrow [2]. He assumes that F is homogeneous of degree one and concave in its arguments. He proves that both F and F can be chosen to be homogeneous of degree one and concave. He then goes on to illustrate two econometric models that could be used to estimate this model. One can also use functional structure results to study the aggregation of inputs. An early example of this appears in Solow [25]. He begins with a simple production function

236

D. Primont

Q = F (L, C1 , C2 ) , where L is labor and C1 and C2 are two types of capital. He then asks the question: When can we write the production function as Q = F (L, C1 , C2 ) = H (L, K) , where K = φ (C1 , C2 ) is a measure of aggregate capital. The topic of capital aggregation is further pursued in Fisher [11–13], Stigum [28], and Blackorby and Schworm [5, 6].

Functional Structure with More Than Two Sectors Theorem 1 can be generalized to the case of several separable subsets of inputs. Suppose the set of input indices, I = {1, 2, . . . , N } , is partitioned into R+1 subsets and let this partition be given by

 I 0, I 1, . . . , I R .

Corresponding to this partition, the input vector may be rewritten as   x = x0, x1, . . . , xR , where input indices have been renumbered appropriately. The following theorem formally states the desired generalization. Theorem 8 ( [7, 14]). Suppose the production function is continuous,  nondecreasing, and quasiconcave in x. For the partition I 0 , I 1 , . . . , I R group r, r = 1, . . . , R is separable from all of the other inputs in I if and only if there exist functions, F, F 1 , . . . , F R , which are continuous, nondecreasing, and quasiconcave in x and      F (x) = F x 0 , F 1 x 1 , . . . , F R x R , (23) where the macro function, F, is increasing in its last R arguments.   yr = F r x r , r = 1, . . . , R. The proof of Theorem 8 follows from a repeated application of Theorem 1. (An example of this functional structure was given in (9), the production of personal

5 Functional Structure and Aggregation

237

x0

Fig. 2 Several Separable Sectors

x1

F1

x2

F2

y1 y2

F

y

y3 x3

F3

computers.) The subset of inputs, I 0 , is often referred to as a free sector, i.e., a nonseparable sector. Since (23) is a direct representation of the technology (as opposed to the indirect production function), we refer to the condition in (23) as direct separability. For R = 3, the functional structure in (23) has the following schematic representation (Fig. 2). Again, one can think of this as a two-period production process. In period 1 inputs x 1 , x 2 , and x 3 are used to produce intermediate inputs y1 , y2 , and y3 , respectively. In period 2, x 0 , y1 , y2 , and y3 are used to produce final output, y. This is another example of a network production model [9]. 0 A special case of the  above arises when I is empty and, thus, the revelant partition is written as I 1 , . . . , I R . For that partition each of the R subsets is separable from all of the other inputs if and only if there exist continuous functions, F, F 1 , . . . , F R , such that      F (x) = F F 1 x 1 , . . . , F R x R , where the macro function, F, is increasing in its R arguments. In this case it is often said that the production function is “weakly separable”. Again, the aggregator functions and the macro functions will inherit properties of the master function. See pages 108–110 in Blackorby et al. [3].

Separability of Dual Representations of Technology  Separability results will be presented for the partition I 0 , I 1 , . . . , I r , . . . , I R where I 0 is a free sector. We first consider separability of the cost function. Theorem 9. Assume that the cost function is continuous, nondecreasing, homogeneous of degree one, and concave in w, jointly continuousin (y, w) , and increasing in y. Then the cost function is separable in the partition I 0 , I 1 , .., I R if and only if it has the functional structure given by      C (y, w) = C y, w 0 , C 1 y, w 1 , . . . , C R y, w R ,

(24)

238

D. Primont

where C is increasing in its last R arguments,   cr = C r y, w r , r = 1, . . . , R. In addition, the macro function, C, is continuous and increasing in (c1 , . . . , cR ), and each aggregator function, C r , is continuous, nondecreasing, concave, and homogeneous of degree one in wr , r = 1, . . . , R. Moreover, each C r is jointly continuous in (y, w r ) and increasing in y. (See Corollary 4.1.4 on page 112 in Blackorby et al. [3].) If the input distance function is increasing in x and twice differentiable, then, according to the Leontief-Sono definition of separability, sector r is separable from all of the other sectors if and only if ∂ ∂xk



∂D (y, x) /∂xi ∂D (y, x) /∂xj

 = 0,

for all i, j ∈ I r and for all k ∈ / I r . A representation theorem that is analogous to Theorem 8 can be proved for the input distance function. Theorem 10. Assume that the input distance function is continuous, nondecreasing, concave, and homogeneous of degree one in x, jointly continuous in (y, x) , and0 decreasing in y. Then the input distance function is separable in the partition I , I 1 , . . . , I R if and only if it has the following functional structure:      D (y, x) = D y, x 0 , D 1 y, x 1 , . . . , D R y, x R ,

(25)

where D is increasing in its last R arguments,   dr = D r y, x r , r = 1, . . . , R. In addition, the macro function, D, is continuous and increasing in (d1 , . . . , dr ) , and each aggregator function, D r ,is continuous, nondecreasing, concave, and homogeneous of degree one in x r , r = 1, . . . , R. Moreover, each D r is jointly continuous in (y, x r ) and decreasing in y. Is the functional structure depicted in equation (23) for the production function equivalent to the functional structure depicted in (25) for the input distance function? In general the answer to this question is no. One way to see why this is so is to note that the aggregator functions, F r , r = 1, . . . , R in (23) depend only on x r , r = 1, . . . , R while the functions D r , r = 1, . . . , R in (25) depend on x r and y, r = 1, . . . , R. Another way to see this is to calculate the marginal rate of technical substitution between two inputs in sector r by applying the implicit function theorem to the input distance function. We get

5 Functional Structure and Aggregation

∂y/∂xi = ∂y/∂xj

239

∂D (y, x) /∂xi ∂D (y, x) /∂y , i, j ∈ I r ∂D (y, x) /∂xj − ∂D (y, x) /∂y −

=

∂D (y, x) /∂xi , i, j ∈ I r ∂D (y, x) /∂xj

=

∂D r (y, x r ) /∂xi , i, j ∈ I r ∂D r (y, x r ) /∂xj

=

∂D r (F (x) , x r ) /∂xi , i, j ∈ I r . ∂D r (F (x) , x r ) /∂xj

for r = 1, . . . , R. The marginal rate of technical substitution depends, in general, on the entire input vector x and thus fails to satisfy the Leontief-Sono condition in the production function. We conclude that a production function that is separable in the partition I 0 , I 1 , . . . , I R will, in general, correspond to a technology that is different than one that corresponds to an input distance function that is separable in the same partition. If the indirect production function is decreasing in w/c and twice differentiable, then, according to the Leontief-Sono definition of separability, sector r is separable from all of the other sectors if ∂ ∂ (wk /c)



∂G (w/c) /∂ (wi /c)   ∂G (w/c) /∂ wj /c

 = 0,

/ I r . A representation theorem that is analogous to for all i, j ∈ I r and for all k ∈ Theorem 8 can be proved for the indirect production function. Theorem 11. Assume that the indirect production function is continuous, nonincreasing, and quasiconvex  in (w/c) . Then the indirect production function is separable in the partition I 1 , . . . , I R if and only if it has the functional structure given by      G (w/c) = G G1 w 1 /c , . . . , GR w R /c

(26)

where G is increasing in its R arguments. In addition, the macro function G is continuous and quasiconvex, and the aggregator functions G1 , . . . , GR , are continuous, nonincreasing, and quasiconcave. We refer to the condition in (26) as indirect separability. Since the cost function can be written as an implicit representation of the indirect production function, one can apply the implicit function theorem to

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D. Primont

C (y, w/c) = 1 to calculate the derivatives  of G. Thus, one can show that separability of the cost function in the partition I 0 , I 1 , .., I R does not necessarily imply separability of   the indirect production function in the partition I 1 , . . . , I R . In particular, ∂C (y, w/c) /∂ (wi /c) ∂y/∂ (wi /c) ∂C (y, w/c) /∂y  =   , i, j ∈ I r ∂y/∂ wj /c ∂C (y, w/c) /∂ wj /c ∂C (y, w/c) /∂y =

∂C (y, w/c) /∂ (wi /c)   , i, j ∈ I r ∂C (y, w/c) /∂ wj /c

=

∂C r (y, w r /c) /∂ (wi /c)   , i, j ∈ I r ∂C r (y, w r /c) /∂ wj /c

=

∂C r (G (w/c) , w r /c) /∂ (wi /c)   , i, j ∈ I r , ∂C r (G (w/c) , w r /c) /∂ wj /c

for r = 1, . . . , R. Thus ratios of partial derivatives of G with respect to normalized prices in group r will, in general, depend on the entire vector of normalized prices, w/c, and hence fail to satisfy the Leontief-Sono conditions for separability of the indirect production function. The production function, the cost function, the input distance function, and the indirect production function form six possible pairs of technology representations. Only one pair exhibits the property that separability in a given partition is equivalent for the two representations. We can state: Theorem 12. Assume that the production function is continuous, nondecreasing, and quasiconcave in x. Then each sector r, r = 1, . . . , R is separable in the input distance function from its complement in I = {1, . . . , N } , i.e.,      D (y, x) = D y, x 0 , D 1 y, x 1 , . . . , D R y, x R if and only if each sector r, r = 1, . . . , R is separable in the cost function from its complement in I = {1, . . . , N } , i.e.,      C (y, w) = C y, w 0 , C 1 y, w 1 , . . . , C R y, w R . Here is a sketch of the proof of the “if” part of this statement.  D (y, x) = min w · x : C (y, w)  1 w

(27)

5 Functional Structure and Aggregation

 = min w x + 0 0

w

R 

241

      0 1 1 R R 1 w · x : C y, w , C y, w , . . . , C y, w r

r

r=1

(28)  = min

w0 x 0 +

w

 = min

w 0 x 0 + cr

w

=

min

w0 ,c1 ,...,cR



R

r r r wr r=1 minwr w · x : C  (y, ) 0 C y, w , c1 , . . . , cr  1

R



  cr :

r w r /cr · x r : C r (y, w r /cr ) r=1 min   w /cr0 C y, w , c1 , . . . , cr  1

(29)  1 :

     w 0 x 0 + cr D r y, x r : C y, w 0 , c1 , . . . , cr  1

     = D y, x 0 , D 1 y, x 1 , . . . , D R y, x R .

(30) (31) (32)

The complete proof of this statement can be found in Gorman [15,17], McFadden [21], and Blackorby et al. ( [3], Theorem 4.2). All of these proofs exploit the fact that each D r is homogeneous of degree one in x r , r = 1, . . . , R and that each C r is homogeneous of degree one in w r , r = 1, . . . , R.. (Note what happens when going from equation (29) to equation (30).) This suggests, for example, that if the aggregator functions F r , r = 1, . . . , R are homogeneous of degree one in x r , then the aggregator functions Gr , r = 1, . . . , R will be homogeneous of degree minus one in w r /c and, moreover, that direct and indirect separability are equivalent. We shall see that this is so in the next subsection.

Homothetic Separability The production technology satisfies homothetic separability in the partition

 I 0, I 1, . . . , I R if the production function can be written as in (23) and each of the aggregator functions are homothetic. In other words      (33) F (x) = F x 0 , F 1 x 1 , . . . , F R x R , where F is continuous and increasing in its last R arguments and each F r is homothetic. If F r is homothetic, then there exists a continuous, increasing function hr and a function F¯ r that is homogeneous of degree one in x r such that F r (x r ) =  hr F¯ r (x r ) , r = 1, . . . , R. Then        F (x) = F x 0 , h1 F¯ 1 x 1 , . . . , hR F¯ R x R

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D. Primont

     = F x 0 , F¯ 1 x 1 , . . . , F¯ R x R where each hr is, in effect, absorbed into F to yield F. Since each hr is continuous and increasing, F is continuous and increasing in its last R arguments. Thus, whenever homothetic separability is assumed, we may also assume, without loss of generality, that the aggregator functions have been normalized to be homogeneous of degree one in (33). If we think of F r as the subproduction function for some intermediate good, yr = F r (x r ) , then the isoquants for yr have been monotonically renumbered in such a way that F r is homogeneous of degree one. Theorem 13. Assume that the production function is continuous, nondecreasing, and quasiconcave in x. Let the output variable y be indexed by 0. Then the following four statements are equivalent. 1. The production function is homothetically separable in the partition

 I 0, I 1, . . . , I R ,

i.e.,      F (x) = F x 0 , F 1 x 1 , . . . , F R x R , where F is increasing in its last R arguments and each F r , r = 1, . . . , R, is homothetic. 2. The cost function is separable in the partition

 {0} ∪ I 0 , I 1 , . . . , I R ,

i.e.,      C (y, w) = C y, w 0 , 1 w 1 , . . . , R w R where C is increasing in its last R arguments and each r , r = 1, . . . , R, is homogeneous of degree one. 3. The input distance function is separable in the partition

 {0} ∪ I 0 , I 1 , . . . , I R ,

i.e.,      D (y, x) = D y, x 0 , X1 x 1 , . . . , XR x R

5 Functional Structure and Aggregation

243

where D is increasing in its last R arguments and each Xr , r = 1, . . . , R, is homogeneous of degree one. 4. The indirect production function is homothetically separable in the partition

 I 0, I 1, . . . , I R ,

i.e.,      G (w/c) = G w 0 /c, G1 w 1 /c , . . . , GR w R /c , where G is increasing in its last R arguments and each Gr , r = 1, . . . , R, is negatively homothetic in wr /c. A sketch of the proof is provided here. 1 ⇒ 2: Assume, without loss of generality, that each F r is homogeneous of degree one. The rth intermediate product is yr = F r (x r ) , and it is produced under constant returns to scale. As is well known, the corresponding cost function for sector r,    r r   C r yr , w r = min w x : F r x r  yr , r x

has the form     C r yr , w r = C r 1, w r yr   = r w r yr ,

(34) (35)

where r (w r ) is the constant marginal and average cost of producing yr . Now we derive the form of the overall cost function.  C (y, w) = min w · x : F (x)  y x

 = min

w x + 0 0

x 0 ,...,x R

x 0 ,...,x R

w x + 0 0

min

x 0 ,y1 ,...,yR

R  r=1

 =

     w x : F x0, F 1 x1 , . . . , F R xR  y



r r

r=1

 = min

R 

w x + 0 0

   r r  r r 0 y min x : F , y , . . . , y w x  y : F x r 1 R r



x

R  r=1

   

w r yr : F x 0 , y1 , . . . , yR  y r

     = C y, w 0 , 1 w 1 , . . . , R w R .

 (using (35))

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2 ⇒ 3: This is easily accomplished by substituting equation (35) into the sketched proof of Theorem 12 that separability of C implies separability of D. 3 ⇒ 4: This will consist of two steps. Step 1 is to show that 3 ⇒ 2, and Step 2 is to show that 2 ⇒ 4. To prove Step 1, we derive the cost function from the input distance function as follows.  C (y, w) = min wx : D (y, x)  1 x



= min w x + 0 0

x

      0 1 1 R R 1 w x : D y, x , X x , . . . , X x r r

r=1

 = min

w0 x 0 +

x

 = min

min

x 0 ,d1 ,...,dR

  R minx r w r x r : Xr (x r )  dr : r=1   D y, x 0 , d1 , . . . , dR  1

  minx r /dr w r x r /dr : Xr (x r /dr )  1 :   D y, x 0 , d1 , . . . , dR  1   R     r 0 0 r 0 dr w : D y, x , d1 , . . . , dR  1 w x +

w0 x 0 +

x

=

R 

R

r=1 dr

r=1

     = C y, w0 , 1 w 1 , . . . , R w R . Thus 3 ⇒ 2. Now we show that 2 ⇒ 4.  G (w/c) = min y : C (y, w/c)  1 y

      = min y : C y, w 0 /c, 1 w 1 /c , . . . , R w R /c  1 y

     = G w 0 /c, 1 w 1 /c , . . . , R w R /c . We can now conclude that 3 ⇒ 4. 4 ⇒ 1: Note that  F (x) = min G (w/c) : w/c · x  1 w/c

 − max −G (w/c) : w/c · x  1 . w/c

We see that the problem

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 max −G (w/c) : w/c · x  1 w/c

is mathematically equivalent to the problem  max F (x) : w/c · x  1 , x

with the roles of x and w/c reversed. Since we have shown that 1 ⇒ 4, then this mathematical equivalence implies that 4 ⇒ 1.

Application Theorem 13 suggests a straightforward way that the firm can decentralize its decision-making. Each firm sector has a constant-returns-to-scale production function given by yr = F r (x r ) , r = 1, . . . , R. A cost function for sector r is then derived and is given by C r (yr , w r ) = C r (1, w r ) yr = r (w r ) yr , r = 1, . . . , R. This price of the intermediate good, r (w r ) , is then reported to the manager of the firm who now solves  min

x 0 ,y1 ,...,yR

w x + 0 0

R 

   

w r yr : F x 0 , y1 , . . . , yR  y



r

r=1

to derive the firm’s demand for the inputs in I 0 and for the intermediate inputs, (y1 , . . . , yR ) . The resulting cost function is      C (y, w) = C y, w 0 , 1 w 1 , . . . , R w R , which can be differentiated to yield all of the other inputs demands via Shephard’s lemma.

Additive Functional Structure Suppose the set of variable indices, I = {1,  2, . . . , N} , is partitioned into R subsets or sectors. This partition is denoted by I 1 , I 2 , . . . , I R . In this section we will examine the conditions under which the production function has the following functional structure:  F (x) = F

R 

  r , F x r

r=1

where F is an increasing function of its single argument. The separability condition that yields this additive structure is called complete separability. A production is

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 completely separable in the partition I 1 , I 2 , . . . , I R if every union of sectors is separable from all of the inputs in the sectors outside the union. We start with a result derived from the theory of functional equations [1] that were effectively used in Gorman [14]. The result is that if there are two separable subsets of I in F that have a nonempty intersection (i.e., are overlapping), then F must have an additive structure. Let us state this more precisely. Suppose there are four sectors, I 1 , I 2 , I 3 , and I 4 , that comprise a partition of I. Let I c be the union of sectors 2, 3, and 4, i.e., I c = I 2 ∪ I 3 ∪ I 4.   In other words, I c is the complement of I 1 in I. Let x c = x 2 , x 3 , x 4 . In order to avoid certain degenerate cases, we define essentiality and strict essentiality as follows. n1 The sector I 1 is said to be essential if, for every xˆ 1 ∈ R+ , the set

    x 1 : F x 1 , x c = F xˆ 1 , x c

is nonempty for at least one point x c ∈ Rn+2 × Rn+3 × Rn+4 , where n2 , n3 , and n4 are the number of elements of I 2 , I 3 , and I 4 , respectively. Sector I 1 is said to be strictly essential if it is essential for all points x c ∈ Rn+2 × Rn+3 × Rn+4 . Essentiality and strict essentiality of each of the other three sectors are defined in the same way. Example. Consider the production structure:    F (x) = min min {x1 , x2 } , F c x c , where I 1 = {1, 2} . Then I 1 is essential since F is increasing in its first c c c argument,  min {x1 , x2 } , for points x such that min {x1 , x2 } < F (x ) . However,  F x 1 , x c = F xˆ 1 , x c for all x c such that min {x1 , x2 }  F c (x c ) . Thus I 1 is not strictly essential. Let I r = I 1 ∪ I 2 and I s = I 2 ∪ I 3 . Set differences are given by I r − I s = {z : z ∈ I r , z ∈ / I s } = I 1 and I s − I r = I 3 . The sectors I r and I s are overlapping separable sectors. Theorem 14 ( [14]). Assume that the production function is continuous and that all four sectors I 1 , I 2 , I 3 , and I 4 are essential. Consider the following three conditions. (a) I r and I s are nonempty and separable from their complements, I 3 ∪ I 4 and I 1 ∪ I 4 , respectively; I r ∩ I s , I r − I s and I s − I r are nonempty; and I s − I r is strictly essential. (b) Each of the following sets is separable from their complements:

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  I r ∩ I s , I r − I s , I s − I r and I r − I s ∪ I s − I r . (c) There exist continuous functions, F, F 1 , F 2 , F 3 such that         F (x) = F F 1 x 1 + F 2 x 2 + F 3 x 3 , x 4 , where F increasing in its first argument. Then (a) ⇒ (b), (a) ⇒ (c), (c) ⇒ (a) and (c) ⇒ (b). The theorem can be extended to R sectors. Theorem 15 ( [14]). Assume that the production function F is continuous. Then F  is completely separable in the partition I 1 , I 2 , . . . , I R , R  3, if and only if there exist function F, F 1 , . . . , F R such that  F (x) = F

R 

  r F x , r

r=1

where F is an increasing function of its single argument. When the production function is twice differentiable with positive first partial derivatives, complete separability can be defined (locally) as: F is completely  separable in the partition I 1 , I 2 , . . . , I R , R  3, if ∂ xk



∂F (x) /∂xi ∂F (x) /∂xj

 = 0,

for all i ∈ I r , j ∈ I s , k ∈ I t , r, s, t = 1, . . . , R, r = t, s = t. An analysis of the differentiable case can be found in Leontief [19, 20]. Theorems 14 and 15 are valid when there are R  3 sectors. What can be said about the case in which there are only two sectors (R = 2)? Put another way, do there exist separability conditions that yield the following functional structure        F x1, x2 = F F 1 x1 + F 2 x2 ?

(36)

The short answer to this question is no. While  it is true that the functional structure in (36) implies separability in the partition I 1 , I 2 , the converse is not true since  separability in the partition I 1 , I 2 only implies that        F x1, x2 = F F 1 x1 , F 2 x2 .

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Thus, alternative conditions are needed to provide an equivalent characterization of (36). In the special case for which I 1 consists of a single variable index, Sono [26, 27] provided a solution to this problem using a condition called independence. Blackorby et al. [3] extended Sono’s proof to the case for which I 1 may include two or more variable indices. We will summarize their result here. First, a definition. Assume that the production function is twice continuously differentiable. The sector I 1 is independent of I 2 if and only if there exist functions ψ j i , i, j ∈ I 1 , such that ∂ ∂xi

      ∂F (x) /xk ∂ ∂F (x) /x ln = ln = ψ j i x1 , ∂F (x) /xj ∂xi ∂F (x) /xj

for all i, j ∈ I 1 and for all k,  ∈ I 2 . Theorem 16. Assume that the production function, F, is twice continuously differentiable. Then I 1 is independent of I 2 and is separable from I 2 if and only if there exist functions F, F 1 , and F 2 such that      F (x) = F F 1 x 1 + F 2 x 2 . For a proof and more details, see Blackorby et al. ( [3], pp. 159–165).

Recursive Functional Structure When a production process takes place over several time periods, it may be appropriate to model this process with an function. The  intertemporal production  input vector may be written as x = x 1 , x 2 , . . . , x t , . . . , x T where x t is the input vector used at time t and production takes place over T time periods. Final production is written as   y = F x1, x2, . . . , xt , . . . , xT . The intertemporal cost function has the usual definition:   C y, w1 , . . . , w t , . . . , w T  = min x

T 

   1 2 t T w · x : F x ,x ,...,x ,...,x  y , t

t

t=1

  where w1 , . . . , w t , . . . , w T is a vector of time-discounted input prices.

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If we are willing to impose separability restrictions on the production function, the cost minimization problem can be broken down into a series of smaller steps. Such separability assumptions might be justified for a production process like the “building a house” example given earlier. In the general case, we have T intermediate steps in the production process. We get the following system of intermediate production functions:   y1 = F 1 x 1   y2 = F 2 y1 , x 2 .. .   yt = F t yt−1 , x t .. .

(37)

  yT = F (x) = F T yT −1 , x T .

structure is equivalent to the condition that the inputs in  1This functional I , . . . , I t−1 are separable from the inputs in I t , . . . , I T , t = 2, . . . , T . These separability assumptions are collectively referred to as complete recursivity. Also note that, since yT is the output of the final step, we have y = yT . It will be convenient to refer the reader to certain proofs in Blackorby et al. ( [3], Chaper 6). There is a slight, but easily surmountable, problem here. Their results pertain to the system of subutility functions given by   uR = U R x R   uR−1 = U R−1 uR , x R .. . ur = U r (ur+1 , x r ) .. .   u1 = U (x) = U 1 u2 , x 1 . This looks just like the production model above except the numbering of sectors is reversed. This is because utility model, a useful  in the intertemporal assumption is that variables in I r .I r+1 , . . . , I R are separable from variables in  1 2 I , I , . . . , I r−1 , r = 2, . . . , R. (The assumption is useful because it paves the way to consistent intertemporal decision-making. For a discussion of this, see Chapter 10 in Blackorby et al. [3].) Nevertheless, we can still appeal to their results for intertemporal utility functions as long as we mentally renumber the sectors. Complete recursivity of the indirect production function is characterized in an analogous fashion (Blackorby et al. [3], p. 225). However, complete recursivity of the production function is not generally equivalent to complete recursivity of the indirect production function. Complete recursivity of the cost function has the functional structure of the following system of functions

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  c1 = C 1 y, w 1   c2 = C 2 y, c1 , w 2 .. .   ct = C t y, ct−1 , w t .. .

(38)

  cT = C (y, w) = C T y, cT −1 , w T ,

is where each C t is increasing in ct−1 , t = 2, . . . , T . This  functional structure equivalent to the separability conditions: Input prices in I 1 , . . . , I t−1 are sepa rable from input prices in I t , . . . , I T , t = 2, .. . , T , Moreover, each C t inherits  t concavity and homogeneity of degree one in ct−1 , w . (See Corollary 6.1.3, page 226 in Blackorby et al. [3].) Similarly, the functional structure of a completely recursive input distance function is given by the system of functions   d1 = D 1 y, x 1   d2 = D 2 y, d1 , x 2 .. .   dt = D t y, dt−1 , x t .. .

(39)

  dT = D (y, x) = D T y, dT −1 , x T ,

where each D t is increasing in dt−1 , t = 2, . . . , T . This functional structure is equivalent to the separability conditions: Inputs in I 1 , . . . , I t−1 are separable  from inputs in I t , . . . , I T , t = 2, .. . , T , In addition, each D t inherits concavity and homogeneity of degree one in dt−1 , x t . (See Corollary 6.1.4, page 226 in Blackorby et al. [3]). Earlier we saw that separability of the cost function is equivalent to separability of the input distance function. An analogous argument establishes that the word “separability” may be replaced by “complete recursivity”. Formally stated: Theorem 17. The cost function has a completely recursive functional structure as in (38) if and only if the input distance function has a completely recursive functional structure as in (39). In the absence of further assumptions, this equivalence result only holds for the cost function – input distance function pair of representations. However, when an assumption of homotheticity is added to the mix, we get equivalence results analogous to those for symmetric structures. First, some formal definitions are in order.

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A production function is homothetically completely recursive if it is completely recursive (37) and each of the following functions, F1 , F2 , . . . , Ft , . . . , FT −1 , is a homothetic function.     y1 = F 1 x 1 = F1 x 1     y2 = F 2 y1 , x 2 = F2 x 1 , x 2 .. .     yt = F t yt−1 , x t = Ft x 1 , x 2 , . . . , x t .. .     yT −1 = F T −1 yT −2 , x T −1 = FT −1 x 1 , x 2 , . . . , x T −1 .

(40)

  Notice that this definition says nothing about the function yT = F T yT −1 , x T =   F x 1 , x 2 , . . . , x T , i.e., it does not include the assumption of (overall) homotheticity of the production function. According to Theorem 6.3, page 228 in Blackorby et al. [3], if the production function is homothetically completely recursive, then there exists a completely recursive representation in which the functions, F 1 , F 2 , . . . , F t , . . . , F T −1 , are  homogeneous of degree  one in their arguments. Given a cost function, C y, w 1 , . . . , w t , . . . , w T , suppose input prices in  1  I , . . . , I t−1 are separable from input prices in I t , . . . , I T , t = 2, . . . , T , and from output, y, in C. Then the resulting functional form of C will satisfy the following system of equations:   c1 = C 1 w 1   c2 = C 2 c1 , w 2 .. .   ct = C t ct−1 , w t .. .

(41)

  cT = C (y, w) = C T y, cT −1 , w T .

The converse is also true (Theorem 6.4, page 230 in Blackorbyet al. [3]). An analogous result holds for input distance functions. Inputs in I 1 , . . . , I t−1  t are separable from inputs in I , . . . , I T , t = 2, . . . , T , and from output, y, in D if and only if D has a functional structure given by   d1 = D 1 x 1   d2 = D 2 d1 , x 2 .. .   dt = D t dt−1 x t .. .

  dT = D (y, x) = D T y, dT −1 , x T .

(42)

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The systems (40), (41), and (42) are related in the following Theorem 18. The following three functional structures are equivalent: (i) The productionfunction is homothetically completely recursive, i.e., (40). (ii) Input prices in I 1 , . . . , I t−1 are separable from input prices in I t , . . . , I T , t = 2, . . . ,T , and from output, y, in C, i.e.,(41).  (iii) Inputs in I 1 , . . . , I t−1 are separable from inputs in I t , . . . , I T , t = 2, . . . , T , and from output, y, in D, i.e., (42).

Application Theorem 18 suggests that, for example, the firm’s cost minimization problem can be broken down into a series of steps described by the following algorithm. Step 1: Solve  

   C 1 w 1 y1 = min w 1 x 1 : F 1 x 1  y1 , x1

where the multiplicative form on the left hand side is implied by the homogeneity of degree one of F 1 in x 1 . Moreover, C 1 is homogeneous of degree one in w 1 . This is a well-behaved price aggregate for the intermediate output y1 . It is used in the next step in minimizing the cost of intermediate input y2 . Step 2: Solve

     min C 1 w 1 y1 + w 2 x 2 : F 2 y1 , x 2  y2

y1 ,x 2

    = C 2 C 1 w 1 , w 2 y2   = C 2 c1 , w 2 y2 ,

    where c1 = C 1 w 1 . Note that C 2 is homogeneous of degree one in c1 , w 2 and, hence, serves as a well-behaved price aggregate in step 3 when minimizing the cost of intermediate output y3 . However, instead of moving to step 3, we will move to step t for t = 3, . . . , T −1. Step t: Solve

min

yt−1 ,x t

     C t−1 ct−2 , w t−1 yt−1 + w t x t : F t yt−1 , x t  yt

    = C t C t−1 ct−2 , w t−1 , w t yt   = C t ct−1 , w t yt ,

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Fig. 3 Complete Recursivity

F1

x1

y1

F2

x2

y2

F3

y

x3

  where ct−1 = C t−1 ct−2 , w t−1 . At the final step  (T ), the cost of final ouput y is minimized using the price aggregate C T −1 cT −2 , w T −1 that was computed in step T − 1. Step T : Solve

     min C T −1 cT −2 , w T −1 yT −1 + w T x T : F t yT −1 , x T  yT = y yT −1 ,x T

    = C T y, C T −1 cT −2 , w T −1 , w T   = cT = C (y, w) = C T y, cT −1 , w T ,

  where cT −1 = C T −1 cT −2 , w T −1 . Note that the resulting functional structure of C (y, w) is precisely the structure given in (41). The above algorithm is a proof that (i) implies (ii) in Theorem 18. Here is another application. Consider the following completely recursive production function:   y1 = F 1 x 1   y2 = F 2 y1 , x 2   y = F 3 y2 , x 3 . This model of production can be depicted schematically by a directed network production diagram (Fig. 3). In period 1, the input vector x 1 is used to produce y1 . In period 2, y1 and x 2 are used to produce y2 . In the third periiod, y2 and x 3 are used to produce the final output y. This conceptual approach to a multiperiod production process has been utilized by Färe and Grosskopf [9], who name their approach dynamic DEA since they model production using Data Envelopment Analysis.

Multioutput Technologies To examine functional structure for multioutput technologies, we will dispense with the use of single-output production and indirect production functions. However, we can still utilize cost and input distance functions when we replace the single output

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y with a vector of outputs. Assume there are N inputs and M outputs. Input vectors are denoted as  x ∈ RN + = (x1 , . . . , xn , . . . , xN ) : xn  0, n = 1, . . . , N and output vectors are denoted as  y ∈ RM + = (y1 , . . . , ym , . . . , yM ) : ym  0, m = 1, . . . , M . We can define the technology set as

 M S = (x, y) : x ∈ RN , y ∈ R , x can produce y . + + It is sometimes convenient to work with input (requirement) sets defined as L (y) = {x : (x, y) ∈ S} or with output (possibility) sets defined by P (x) = {y : (x, y) ∈ S} . It seems reasonable that these three sets should be related by (x, y) ∈ S ⇔ x ∈ L (y) ⇔ y ∈ P (x) , and, indeed, they are. There are some standard axioms that L (y) and P (x) should satisfy. See Färe and Primont [10] for the details. For our purposes we will want to make sure that both the cost function and the input distance function are well-defined and dual to each other. We assume that inputs are at least weakly disposable, i.e., x ∈ L (y) ⇒ λx ∈ L (y) for all λ  1, for all x ∈ RN + . A stronger assumption is that inputs are strongly disposable, i.e., x ∈ L (y) ⇒ x  ∈ L (y) for all x   x, for all x ∈ RN + . Outputs are weakly disposable if y ∈ P (x) ⇒ θy ∈ P (x) for all 0  θ  1, for all y ∈ RM + . The stronger assumption here is that outputs are strongly disposable, i.e.,

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y ∈ P (x) ⇒ y  ∈ P (x) for all 0  y   y, for all y ∈ RM +. We are now in a position to extend the definition of the input distance function in the single-output case to one in which there are multiple outputs. A formal definition of an input distance function is M Di (y, x) = sup {λ > 0 : (x/λ) ∈ L (y)} for all x ∈ RN + , y ∈ R+ . λ

We have made a slight notational change here. For the single-output case, we used the symbol D for the input distance function; now we use the symbol Di in the multioutput case. This is necessary to distinguish the input distance function from the output distance function which is defined by M Do (x, y) = inf {θ > 0 : y/θ ∈ P (x)} for all x ∈ RN + , y ∈ R+ . θ

Weak disposability of inputs is necessary and sufficient for the following relationship between the input set and the input distance function ( [10], page 22): L (y) = {x : Di (y, x) ≥ 1} . Similarly, weak disposability of outputs is necessary and sufficient for the following relationship between the output set and the output distance function ([10], page 15):  P (x) = y : Do (x, y)  1 . In the multioutput framework, the cost function is defined as C (y, w) = min {w · x : x ∈ L (y)} x  = min w · x : Di (y, x)  1 . x

The cost function is nonnegative and nondecreasing, homogeneous of degree one, concave, and continuous in w. We have already seen that separability of the cost function is equivalent to separability of the input distance function, i.e., each sector r, r = 1, . . . , R is separable in the input distance function, i.e.,      Di (y, x) = Di y, x 0 , Di1 y, x 1 , . . . , DiR y, x R if and only if each sector r, r = 1, . . . , R is separable in the cost function, i.e.,      C (y, w) = C y, w 0 , C 1 y, w 1 , . . . , C R y, w R .

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(Theorem 12) This result was proved treating y as a single output. However, the proof would be unchanged if y is viewed as a vector, an observation made by Gorman [16]. Similarly, Theorem 13 would continue to hold for the cost function and the input distance function if the scalar y is replaced by a vector y. Now define the revenue function by R (x, p) = max {p · y : y ∈ P (x)} y

 = max p · y : Do (x, y)  1 . y

The revenue function is nonnegative and nondecreasing, homogeneous of degree one, convex, and continuous in p. The output distance function can found by solving  Do (x, y) = sup p · y : R (x, p)  1 . p

An output-revenue analog of Theorem 12 is given by: Theorem 19. Each sector r, r = 1, . . . , R, is separable in the output distance function, i.e.,      Do (x, y) = Do x, y o , Do1 x, y 1 , . . . , DoR x, y R , if and only if each sector r, r = 1, . . . , R, is separable in the revenue function, i.e.,      R (x, p) = R x, p0 , R 1 x, p1 , . . . , R R x, pR . The proof of this theorem is analogous to the proof of Theorem 12 generalized to the multioutput case.

References 1. Aczèl J (1966) Lectures on functional equations and their applications. Academic, New York 2. Arrow K (1974) The measurement of real value added. In: David PA, Reder MW (eds) Trade, stability, and macroeconomics. Academic, New York, pp 181–202 3. Blackorby C, Primont D, Russell RR (1978) Duality, separability, and functional structure: theory and economic applications. Elsevier North-Holland, New York 4. Blackorby C, Primont D, Russell RR (1998) Separability: a survey. In: Barbarà S, Hammond PJ, Seidl C (eds) Handbook of utility theory, volume I. Kluwer Academic Publishers, Boston, pp 49–92 5. Blackorby C, Schworm W (1984) The structure of economies with aggregate measures of capital: a complete characterization. Rev Econ Stud 51(4):633–650 6. Blackorby C, Schworm W (1988) The existence of input and output aggregates in aggregate production functions. Econometrica 56(3):613–643

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7. Debreu G (1960) Topological methods in cardinal utility theory. In: Arrow K, Karlin S, Suppes P (eds) Mathematical methods in the social sciences. Stanford University Press, Stanford, pp 16–26 8. Diewert WE (1982) Duality approaches to microeconomic theory. In: Arrow KJ, Intriligator MD (eds) Handbook of mathematical economics, vol 2. North-Holland, Amsterdam, pp 535–599 9. Färe, R.G. and S. Grosskopf (1996), Intertemporal Production Frontiers: with Dynamic DEA, Boston: Kluwer Academic Publishers. 10. Färe RG, Primont D (1995) Multi-output production and duality: theory and applications. Kluwer Academic Publishers, Boston 11. Fisher FM (1965) Embodied technical change and the existence of an aggregate capital stock. Rev Econ Stud 32:263–288 12. Fisher FM (1968) Embodied technology and the existence of labor and output aggregates. Rev Econ Stud 35:391–412 13. Fisher FM (1968) Embodied technology and the aggregation of fixed and moveable capital goods. Rev Econ Stud 35:417–428 14. Gorman WM (1968) The structure of utility functions. Rev Econ Stud 35:367–390 15. Gorman WM (1970) Quasi-separable preferences, costs, and technologies. Unpublished manuscript, published as pp 104–114 in Gorman WM (1995) 16. Gorman WM (1987) Separability. In: Eatwell J, Milgate M, Newman P (eds) The new palgrave: a dictionary of economics. Macmillan Press, London. Reprinted in Gorman W.M. (1995) 17. Gorman WM (1995) The collected works of W.M. Gorman, vol 1, Blackorby C, Shorrocks A (eds). Oxford, Oxford University Press 18. Hotelling H (1932) Edgeworth’s taxation paradox and the nature of supply and demand functions. J Polit Econ 40:577–616 19. Leontief WW (1947) A note on the interrelation of subsets of independent variables of a continuous function with continuous first derivatives. Bull Am Math Soc 53:343–350 20. Leontief WW (1947) Introduction to a theory of the internal structure of functional relationships. Econometrica 15:361–373 21. McFadden D (1978) Cost, revenue, and profit functions. In: Fuss M, McFadden D (eds) Production economics: a dual approach to theory and applications. Amsterdam, North-Holland, pp 3–110 22. Roy R (1942) De l’Utilitè. Hermann, Paris 23. Roy R (1947) La Distribution du Revenue Entre Les Divers Biens. Econometrica 15:205–225 24. Shephard RW (1970) Theory of cost and production functions. Princeton University Press, Princeton 25. Solow RN (1955) The production function and the theory of capital. Rev Econ Stud 23: 101–108 26. Sono M (1945) The effect of price changes on the demand and supply of separable goods. (in Japanese). Kokumin Keisai Zasshi 74:1–51. English translation in Sono (1961) 27. Sono M (1961) The effect of price changes on the demand and supply of separable goods. Int Econ Rev 2:239–271 28. Stigum B (1967) On certain problems of aggregation. Int Econ Rev 8(3):349–367 29. Wold HOA (1943) A synthesis of pure demand analysis, I, II. Skandinavisk Akguarietidskrift 26:85–118, 220–263

6

Elasticities of Substitution R. Robert Russell

Contents Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Two-Input Elasticity of Substitution: Early Formulations and Characterizations . . . . . . . . . . Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Comparative Statics of Income Shares . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Constant Elasticity of Substitution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Digression: Dual Representations of Multiple-Input, Multiple-Output Technologies . . . . . . . Allen and Morishima Elasticities of Substitution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Allen Elasticities of Substitution (AES) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Morishima Elasticities of Substitution (MES) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . AES and MES and the Comparative Statics of Income Shares . . . . . . . . . . . . . . . . . . . . . . Constancy of the Allen and Morishima Elasticities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Non-homothetic Technologies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Dual Elasticities of Substitution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Two-Input Elasticity of Substitution Redux . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Dual Morishima and Allen Elasticities of Substitution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Symmetric Elasticity of Complementarity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Gross Elasticities of Substitution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Elasticities of Substitution and Separability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Separability and Functional Structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Elasticity Identities and Functional Structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Concluding Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Cross-References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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R. R. Russell () Department of Economics, University of California, Riverside, Riverside, CA, USA e-mail: [email protected] © Springer Nature Singapore Pte Ltd. 2022 S. C. Ray et al. (eds.), Handbook of Production Economics, https://doi.org/10.1007/978-981-10-3455-8_10

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Abstract

This chapter lays out the theoretical foundation of the measurement of the degree of substitutability among inputs utilized in a production process. It proceeds from the well-settled (Hicksian) notion of this measure for two inputs (typically labor and capital) to the more challenging conceptualization for technologies with more than two inputs (most notably, Allen-Uzawa and Morishima elasticities). Dual elasticities of substitution (also called elasticities of complementarity) and gross elasticities of substitution (measuring substitutability for non-homothetic technologies taking account of output changes) are also covered. Also analyzed are functional representations of two-input technologies with constant elasticity of substitution (CES) and of n-input technologies with constant and identical elasticities for all pairs of inputs. Finally, the chapter explores the relationship between elasticity values and the comparative statics of factor income shares and the relationships between certain elasticity identities and separability conditions rationalizing consistent aggregation of subsets of inputs. Keywords

Duality theory · Income shares · Separability and Functional structure

Introduction In his classic book on the Theory of Wages, the Oxford University economist John R. Hicks [42] introduced two concepts that persist to this day as important components of both microeconomic and macroeconomic analysis: (1) elasticity of substitution1 and (2) input neutrality (alternatively, input bias) of technological change. Each of these constructs is fundamental to the analysis of changing factor income shares as an economy (or other production unit) expands (or, for that matter, contracts). This chapter focuses on the first of these concepts;  Chap. 20, “Conceptualization and Measurement of Productivity Growth and Technical Change: A Nonparametric Approach”. As noted by Blackorby and Russell [14, p. 882] in their discussion of the elasticity of substitution, “Hicks’ key insight was to note that [in a two-factor economy] the effect of changes in the capital/labor ratio (or the factor price ratio) on the distribution of income (for a given output) can be completely characterized by a scalar measure of curvature of the isoquant.”2 This measure, the (two-input)

1 This

concept was independently formulated by Cambridge University economist Joan Robinson [69] in her comparably classic book on The Economics of Imperfect Competition. Abba Lerner [60] and A. C. Pigou [68] also contributed to the understanding of the concept at its genesis. 2 While the use of the word “curvature” in this quote conveys the appropriate intuition, it is nevertheless technically incorrect, in part because curvature, formally defined, is a unit-dependent mathematical concept. See de la Grandville [29] for a clear exposition of this point.

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elasticity of substitution, is a logarithmic derivative of the input-quantity ratio with respect to the technical rate of substitution between the two inputs, holding output constant. The elasticity of substitution and its relationship to the comparative statics of income shares is exposited in section “Two-Input Elasticity of Substitution: Early Formulations and Characterizations”, where the famous SMAC (Arrow, Chenery, Minhaus, and Solow [4]) theorem on the functional characterization of constancy of the elasticity of substitution (CES) is also discussed. Generalization of the elasticity of substitution to allow for more than two inputs began with suggestions by Hicks and Allen [45]. One suggestion was to employ the constructions defining the original Hicksian notion for any two inputs, holding the other input quantities fixed. This idea was further explored by McFadden [61], but since then it has faded from the picture, largely because its failure to allow for optimal adjustment of other inputs means that it generally fails to provide information about the comparative statics of relative income shares of any two inputs.3 The other generalization formulated by Hicks and Allen [45] – and further analyzed by Hicks (1938), Allen [2], and Uzawa [79] – is now known as the Allen elasticity of substitution (AES) or the Allen-Uzawa elasticity of substitution (AUES). This elasticity is a share-weighted (constant-output) cross elasticity of demand. An alternative generalization, first formulated by Morishima [62] (in Japanese and unfortunately never translated into English) and independently discovered by Blackorby and Russell [11], is a constant-output cross elasticity of demand minus a constant-output own price elasticity of demand.4 Blackorby and Russell [13] named this concept the “Morishima elasticity of substitution” (MES) and argued that it, unlike the AES, preserves the salient properties of the original Hicksian notion when the number of inputs is expanded to more than two. As these (and other) elasticity concepts are most evocatively described using dual representations of the technology, section “Digression: Dual Representations of Multiple-Input, Multiple-Output Technologies” presents some useful duality constructs. Features of the AES and the MES are then explored in section “Allen and Morishima Elasticities of Substitution”, particularly their relationships to the comparative statics of factor income shares and functional representations of the technology when the elasticities are invariant with respect to changes in input quantities.

3 As

pointed out by Blackorby and Russell [14, p. 882], “[O]nly if the two variables were separable from all other variables would [this elasticity] provide information about shares; if we were to require all pairs to have this property, the production function would be additive. When combined with homotheticity (an assumption maintained in all these studies . . . ) this implies that the production function is CES, in which case [the elasticities] are constant for all pairs of inputs.” 4 The Morishima elasticity is a generalization of Robinson’s [69] characterization of the two-input elasticity and for this reason is called the “Robinson elasticity of substitution” by Kuga and Murota [54] and Kuga [53].

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The curvature of a two-input isoquant can be equivalently (and again informally5 ) represented by the inverse of the Hicksian elasticity, the logarithmic derivative of the technical rate of substitution – i.e., the shadow price ratio – with respect to the quantity ratio. These concepts are dual to one another. Of course, in the case of only two variables, these two elasticities are simple inverses of one another (and have inverse implications for the curvature of the isoquant). With more than two inputs, however, the analogous dual concepts – duals to the Allen and Morishima elasticities – are not simple inverses of one another. This dual structure, developed by Blackorby and Russell [11, 13], is examined in section “Dual Morishima and Allen Elasticities of Substitution”. Stern [75] points out that the dual Morishima elasticities do not reflect differential movements along an isoquant (essentially because only one input is varied in the calculation). He proposes an elasticity, named the symmetric elasticity of complementarity, that constrains differential changes in quantities to be contained in an isoquant. This elasticity, unlike the dual Morishima elasticity, is symmetric.6 It is presented in section “Symmetric Elasticity of Complementarity”. The Allen and Morishima elasticities of substitution are calculated for differential movements along a constant-output surface. If the technology is homothetic, this is not a restriction, but in general the comparative-static calculations on, say, income shares hold only for cases where outputs are exogenous. The extension of the Allen elasticity to incorporate output effects was broached by Mundlak [63] and formulated in the dual (using the profit function) by Lau [56]. Extensions of these results to Morishima elasticities can be found in Davis and Shumway [28] and Blackorby et al. [16]. In line with the latter, I refer to these concepts as gross elasticities and describe and analyze them in section “Gross Elasticities of Substitution”. In a widely cited paper, Berndt and Christensen [6] were the first to notice that identities among certain pairs of Allen elasticities are equivalent to corresponding separability restrictions on the production function. Essentially, a set of inputs is separable from a distinct input if the technical rates of substitution among inputs in the set are independent of the quantity of the excluded input. Separability is a powerful concept in part because it has implications for the possibility of consistent aggregation of inputs (i.e., aggregation across different types of labor inputs to form an aggregate input in the functional representation of the technology).7 If a subset of inputs is separable from all inputs excluded from the subset, there exists an aggregator over the inputs in the subset, which then is an aggregate input into the production function. Berndt and Christensen discovered that a subset of inputs is separable from a distinct input if and only if the Allen elasticities of

5 See

Footnote 2 above.

6 It is dual, not to the Morishima elasticity, but to McFadden’s

[61] shadow elasticity of substitution. is also a necessary condition for decentralization of an optimization problem (as in, e.g., two-stage budgeting). See Blackorby et al. [15, Ch. 5] for a thorough exposition of the connection between separability and decentralized decision-making.

7 Separability

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substitution between the excluded and each of the inputs in the separable subset are identical. Russell [70] and Blackorby and Russell [12, 13] later generalized these results and extended them to comparable identity restrictions for Morishima elasticities of substitution. These relationships between certain elasticity identities and functional structure are covered in section “Elasticities of Substitution and Separability”. The conclusion contains with a brief discussion of the extensive theoretical and empirical literature in which the elasticity of substitution plays a salient role.

Two-Input Elasticity of Substitution: Early Formulations and Characterizations Definition Hicks was particularly interested in the substitutability between labor and capital and more particularly in the relative income shares of these two inputs. Let us therefore denote the input quantity vector, in an obvious notation, by x , xk  ∈ R2+ and the (scalar) output quantity by y ∈ R+ . The production function, F : R2+ → R+ , is assumed to be increasing, strictly quasi-concave, and homothetic.8 For convenience, we restrict our analysis to the interior of quantity space, R2++ , and assume that F is continuously twice differentiable on this space.9 Homotheticity of F implies that the technical rate of substitution between labor and capital,10 trs,k = F (x , xk )/Fk (x , xk ) =: T RS,k (x , xk ), is homogeneous of degree zero, so that trs,k = T RS,k (1, xk /x ) =: t (xk /x ) and   ln trs,k = ln t (xk /x ) =: θ ln(xk /x ) . 8 As

we shall see in Digression: Dual Representations of Multiple-Input, Multiple-Output Technologies the homotheticity assumption can be dropped when the elasticity concept is formulated in the dual. 9 We could extend our analysis to all of R2 by employing directional derivatives at the boundary + but instead leave this technical detail to the interested reader. 10 Subscripts on functions indicate differentiation with respect to the specified variable. The relation, =, should be interpreted as an identity throughout this chapter (i.e., as holding for all allowable values of the variables). Also, A := B means the relation defines A, and A =: B means the relation defines B.

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Owing to strict quasi-concavity of F , t is strictly monotonic, hence invertible, and we can write xk = t −1 (trs,k ) x or ln

xk =: φ(ln trs,k ). x

(1)

The (two-input) elasticity of substitution is defined as the log derivative of the input-quantity ratio with respect to the technical rate of substitution, σ = φ  (ln trs,k ), or, equivalently, as the inverse of the log derivative of the technical rate of substitution with respect to the input-quantity ratio, σ =

θ



1 . ln(xk /x )

As the production function is strictly quasi-concave, the elasticity of substitution lies in the open interval, (0, ∞). Relatively large values of σ indicate that the rate at which one input can be substituted for the other is relatively insensitive to changes in the input ratio: in the vernacular, substitution is “relatively easy” (isoquants are “relatively flat”). Conversely, lower values of σ reflect “relatively difficult” substitution (and “strong curvature” of isoquants). As σ → ∞, the isoquants converge to (parallel) linear line segments (perfect substitution), and as σ → 0, the isoquants converge to Leontief (fixed proportions) isoquants.

Comparative Statics of Income Shares As shown by Hicks [42], the value of the elasticity of substitution has unambiguous implications for the effects of changes in relative factor prices or in relative factor quantities on relative factor shares in a competitive (price taking) economy. In an obvious notation for factor prices, the share of capital relative to labor is s = pk xk /p x . In a competitive economy, where trs,k = p /pk , ln s = ln(xk /x ) − ln(p /pk )   ˆ = φ ln(p /pk ) − ln(p /pk ) =: S(ln(p  /pk )),   ˜ = ln(xk /x ) − θ ln(xk /x ) =: S(ln(x k /x )), so that the proportional effect on relative shares of a change in the price ratio is

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    Sˆ  ln(p /pk ) = φ  ln(p /pk ) − 1 = σ − 1 and the proportional effect on relative shares of a change in the quantity ratio is     1 S˜  ln(xk /x ) = 1 − θ  ln(xk /x ) = 1 − . σ Thus, the comparative statics of functional income shares when σ is constant is encapsulated in ⎛ ⎞ < ˆ d S(ln(p  /pk )) ⎝ ⎠ = 1 d ln(p /pk ) >

⎛

⇐⇒

⎞ < σ ⎝=⎠ 1 >

or ⎛ ⎞ < ˜ /x )) d S(ln(x k  ⎝ ⎠ = 1 d ln(xk /x )) >

⎛

⇐⇒

⎞ > σ ⎝ = ⎠ 1.
0, 1 > α > 0.

(CD2 )

Thus, constancy of the elasticity of substitution implies that the production function must be belong to the CES family of technologies, (CES2 ) or (CD2 ).11 Moreover, 11 The

Cobb-Douglas production function was well-known at the time of the SMAC derivation, having been proposed much earlier [25]. The (CES2 ) production function made its first appearance in Solow’s [73] classic economic growth paper, but the functional form had appeared much earlier in the context of utility theory: Bergson (Burk) [20] proved that additivity of the utility function and linear Engel curves (expenditures on individual goods proportional to income for given prices)

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the fixed-proportions technology (Leon5ef [59]) is generated as a limiting case of (CES2 ):  lim

ρ→−∞

ρ α x

ρ + αk xk

1/ρ

 = min α x , αk xk .

(L2 )

Some simple calculations establish the following equivalences: (CES2 ) ⇐⇒ σ = 1/(1 − ρ); (CD2 ) ⇐⇒ σ = 1; and (L2 ) ⇐⇒ σ → 0.12 I need not bring to the attention of the reader the extent to which the CES family of production functions has been, over the years, a perennial workhorse in both theoretical and empirical research employing production functions.

Digression: Dual Representations of Multiple-Input, Multiple-Output Technologies The Allen and Morishima elasticities, originally formulated by Hicks and Allen [45] and Morishima [62] in the (primal) context of a single-output production function, are more generally and evocatively exposited in the dual (using the cost function) as first shown, respectively, by Uzawa [79]13 and by Blackorby and Russell [11, 13], Kuga [53], and Kuga and Murota [54]. The dual approach also allows us to move seamlessly from technologies with a single output to those with multiple outputs. Finally, duality theory is needed for the development of the dual elasticities of substitution in section “Dual Elasticities of Substitution”. This section lays out the requisite duality theory.14 Denote the ordered set of inputs by N = 1, . . . n and the ordered set of outputs by M = 1, . . . m. Input and output quantity vectors are denoted x ∈ Rn+ and y ∈ Rm + , respectively. The technology set is the set of all feasible input, output combinations: 

| x can produce y . T := x, y ∈ Rn+m +

implies that the utility function belongs to the CES family. As the SMAC authors point out, the function (CES2 ) itself was long known in the functional-equation literature (see Hardy et al. [41, p. 13]) as the “mean value of order ρ.” 12 The SMAC theorem is easily generalized to homothetic technologies, in which case the production function is a monotonic transformation of (CES2 ) or (CD2 ); in the limiting case as σ → 0 it is a monotonic transformation of (L2 ). 13 Because the Allen elasticities are now typically exposited in the dual, they are often called the “Allen-Uzawa elasticities,” and I resort to that nomenclature on occasion as well. Uzawa’s approach was later extended by Blackorby and Russell [11, 13]. 14 Thorough expositions of duality theory can be found in, e.g., Blackorby et al. [15], Chambers [22], Cornes [26], Diewert [30, 31], Färe and Primont [36], Fuss and McFadden [37], and Russell [71].

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While the nomenclature suggests that feasibility is a purely technological notion, a more expansive interpretation is possible: feasibility could incorporate notions of institutional and political constraints, especially when we consider entire economies as the basic production unit. An input requirement set for a fixed output vector y is

 L(y) := x ∈ Rn+ | x, y ∈ T . We assume throughout that, for all y ∈ Rm + , L(y) is closed and strictly convex (relative to Rn+ )15 and satisfies strong input disposability L(y) = L(y) + Rn+ ∀ y ∈ Rm +, output monotonicity,16 y¯ > y =⇒ L(y) ¯ ⊂ L(y), and “no free lunch,” / L(y). y = 0(m) =⇒ 0(n) ∈ The (input) distance (gauge) function, a mapping from17

 Q := x, y ∈ Rn+m | y = 0(m) ∧ x = 0(n) ∧ L(y) = ∅ + into the positive real line (where 0(n) is the null vector of Rn+ ), is defined by

 D(x, y) := max λ | x/λ ∈ L(y) . Under the above assumptions, D is well defined on this restricted domain and satisfies homogeneity of degree one, positive monotonicity, concavity, and continuity in x and negative monotonicity in y. (See, e.g., Färe and Primont [36] for proofs of these properties and most of the duality results that follow.18 ) Assume, in

15 These

assumptions are stronger than needed for much of the conceptual development that follows, but in the interest of simplicity I maintain them throughout. 16 Vector notation: y¯ ≥ y if y¯ ≥ y for all j ; y¯ > y if y¯ ≥ y for all j and y¯ = y; and y¯  y if j j j j y¯j > yj for all j . 17 We restrict the domain of the distance function to assure that it is globally well defined. An alternative approach (e.g., Färe and Primont [36]) is to define D on the entire non-negative (n+m)dimensional Euclidean space and replace “max” with “sup” in the definition. See Russell [71, footnote 12] for a comparison of these approaches. 18 Whatever is not there can be found in Diewert [30] or the Fuss/McFadden [37] volume.

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addition, that D is continuously twice differentiable in x. The distance function is a representation of the technology, since (under our assumptions) x, y ∈ T ⇐⇒ D(x, y) ≥ 1. In the single-output case (m = 1), where the technology can be represented  by a production function, F : Rn+ → R+ , D x, F (x) = 1 and the production function is recovered by inverting D(x, y) = 1 in y. If (and only if) the technology is homogeneous of degree one (constant returns to scale), D(x, y) =

F (x) . y

The cost function, C : Rn++ × Y → R+ , where

 Y = y | x, y ∈ Q for some x , is defined by

 C(p, y) = min p · x | x ∈ L(y) x

or, equivalently, by

 C(p, y) = min p · x | D(x, y) ≥ 1 . x

(2)

Under our maintained assumptions, D is recovered from C by

 D(x, y) = inf p · x | C(p, y) ≥ 1 , p

(3)

and C has the same properties in p as D has in x. This establishes the duality between the distance and the cost function. On the other hand, C is positively monotonic in y. We also assume that C is twice continuously differentiable in p. By Shephard’s Lemma (application of the envelope theorem to (2)), the (vectorvalued, constant-output) input demand function, δ : Rn++ × Y → Rn+ , is generated by first-order differentiation of the cost function19 : δ(p, y) = ∇p C(p, y). Of course, δ is homogeneous of degree zero in p. The (normalized) shadow-price vector, ρ : Q → R+ , is obtained by applying the envelope theorem to (3):

19 ∇

p C(p, y)

:=  C(p, y)/∂p1 , . . . , ∂C(p, y)/∂pn .

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ρ(x, y) = ∇x D(x, y).

(4)

As is apparent from the re-writing of (3) (using homogeneity of C in p) as D(x, y) = inf

p

p/c

p

· x | C(p/c, y) ≥ 1 = inf · x | C(p, y) ≥ c , p/c c c

(5)

where c can be interpreted as (minimal) expenditure (to produce output y) and the vector ρ(x, y) in (4) can be interpreted as shadow prices normalized by minimal cost.20 In other words, under the assumption of cost-minimizing behavior,   ρ δ(p, y), y =

p . C(p, y)

Clearly, ρ is homogeneous of degree zero in p.

Allen and Morishima Elasticities of Substitution Allen Elasticities of Substitution (AES) The Allen elasticity of substitution between inputs i and j is given by σijA (p, y) : = =

C(p, y)Cij (p, y) Ci (p, y)Cj (p, y)

(6)

ij (p, y) , sj (p, y)

(7)

∀ i, j  ∈ N × N,

where the subscripts on the cost function C indicate differentiation with respect to the indicated variable(s); ij (p, y) :=

pj Cij (p, y) ∂ ln δi (p, y) . = ∂ ln pj Ci (p, y)

is the (constant-output) elasticity of demand for input i with respect to a change in the price of input j ; and sj (p, y) =

20 See Färe and Grosskopf

shadow prices.

pj Cj (p, y) pj δj (p, y) = C(p, y) C(p, y)

[34] and Russell [71] for analyses of the distance function and associated

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is the cost share of input j . Thus, the Allen elasticity is simply a share-weighted cross (i = j ) or own (i = j ) demand elasticity. It collapses to the Hicksian elasticity when n = 2.

Morishima Elasticities of Substitution (MES) To define the Morishima elasticities, let pi be the (n − 1)-dimensional vector of price ratios with pi in the denominator. Zero-degree homogeneity of δ in p allows us to write   δˆ pi , y := δ(p, y). The Morishima elasticity of substitution of input i for input j is defined directly as

σijM (p, y) :=

     i  ∂ ln δˆi pi , y δˆj p , y ∂ ln(pj /pi )

.

(8)

Note that any variation in a single component of pi , say pj  /pi , holding other components (j = j  ) constant, must be entirely manifested in variation in pj  alone. Hence, for all pairs i, j , ∂ δˆi (pi , y) ∂δi (p, y) 1 = . ∂(pj /pi ) ∂pj pi Using this fact, along with Shephard’s Lemma, the MES can be re-written in the dual as  Cij (p, y) Cjj (p, y) − (9) σijM (p, y) = pj Ci (p, y) Cj (p, y) = ij (p, y) − jj (p, y).

(10)

Thus, the MES is simply the difference between the appropriate (constant output) cross price elasticity of demand and the (constant output) own elasticity of demand for the input associated with the axis along which the price ratio is being varied. The Morishima elasticity, unlike the Allen elasticity, is non-symmetric, since the value depends on the normalization adopted in (8) – that is, on the coordinate direction in which the prices are varied to change the price ratio, pj /pi 21 Of course, if there are only two inputs, there is no difference between changing pi and changing

21 See Blackorby and Russell [11, 13, 14] for a discussion of this asymmetry, which (as pointed out by Stern [76]) was recognized much earlier by Pigou [68] in his analysis of Robinson’s [69] less-formal characterization of this elasticity concept.

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pj , and the MES, like the AES, collapses to the standard Hicksian elasticity. But in the general n-input case, the interactions between these two inputs are dependent on whether it is one or the other that is varied.22

AES and MES and the Comparative Statics of Income Shares If σijA (p, y) > 0 (i.e., if increasing the j th price increases the optimal quantity of input i), we say that inputs i and j are Allen-Uzawa substitutes; if σijA (p, y) < 0, they are Allen-Uzawa complements. Similarly, if σijM (p, y) > 0 (i.e., if increasing the j th price increases the optimal quantity of input i relative to the optimal quantity of input j ), we say that input j is a Morishima substitute for input i; if σijM (p, y) < 0, input j is a Morishima complement to input i. As the Morishima elasticity of substitution is non-symmetric, so is the taxonomy of Morishima substitutes and complements.23 The conceptual foundations of Allen-Uzawa and Morishima taxonomies of substitutes and complements are, of course, quite different. The Allen-Uzawa taxonomy classifies a pair of inputs as substitutes (complements) if an increase in the price of one causes an increase (decrease) in the quantity demanded of the other. This is the standard textbook definition of net substitutes (and complements). The Morishima concept, on the other hand, classifies a pair of inputs as substitutes (complements) if an increase in the price of one causes the quantity of the other to increase (decrease) relative to the quantity of the input for which price has changed. For this reason, the Morishima taxonomy leans more toward substitutability (since the theoretically necessary decrease in the denominator of the quantity ratio in (8) (owing to jj (p, y) < 0) helps the ratio to decline when the price of the input in the denominator increases). Put differently, if two inputs are substitutes according to the Allen-Uzawa criterion, theoretically they must be substitutes according to the Morishima criterion, but if two inputs are complements according to the Allen-Uzawa criterion, they can be either complements or substitutes according to the Morishima criterion. This relationship can be seen algebraically from (7) and (10). If i and j are AllenUzawa substitutes, in which case ij (p, y) > 0, concavity of the cost function (and hence negative semi-definiteness of the corresponding Hessian) implies that ij (p, y) − jj (p, y) > 0, so that j is a Morishima substitute for i. Similar algebra establishes that two inputs can be Morishima substitutes when they are Allen-Uzawa complements.

22 “Own” Morishima elasticities are identically equal to zero and hence uninteresting, as one might

expect to be the case for a sensible elasticity of substitution. notions are referred to as “p-substitutes” and “p-complements” in much of the literature (see Stern [76] and the papers cited there), as distinguished from “q-substitutes” and “qcomplements,” which I call dual substitutes and complements in section “Dual Elasticities of Substitution”.

23 These

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Note that, for i = j ,   ∂ ln si (p, y) = ij (p, y) − sj (p, y) = sj (p, y) σijA (p, y) − 1 , ∂ ln pj so that an increase in pj increases the absolute cost share of input i if and only if σijA (p, y) > 1 : that is, if and only if inputs i and j are sufficiently strong net substitutes. Thus, the Allen-Uzawa elasticities provide immediate qualitative comparative-static information about the effect of price changes on absolute shares. To obtain quantitative comparative-static information, one needs to know the share of the j th input as well as the Allen-Uzawa elasticity of substitution. The Morishima elasticities immediately yield both qualitative and quantitative information about the effect of price changes on relative input shares:     ∂ ln(ˆsi pi , y /ˆsj pi , y ) = ij (p, y) − jj (p, y) − 1 = σijM (p, y) − 1, ∂ ln(pj /pi ) where (with the use of zero-degree homogeneity of si in p) sˆi (pi , y) := si (p, y) for all i. Thus, an increase in pj increases the share of input i relative to input j if and only if σijM (p, y) > 1 : that is, if and only if inputs i and j are sufficiently substitutable in the sense of Morishima. Moreover, the degree of departure of the Morishima elasticity from unity provides immediate quantitative information about the effect on the relative factor shares.

Constancy of the Allen and Morishima Elasticities In addition to his dual reformulation of the elasticity proposed by Hicks and Allen [45], Uzawa [79] also extended the SMAC theorem on constancy of the elasticity of substitution to encompass more than two inputs. He conjectured (p. 293) that the “production function which extends the Arrow-Chenery-Minhas-Solow function to the n-factor case may be the following type:” F (x) =



1/ρ ρ

αi xi

,

αi > 0 ∀ i,

1 ≥ ρ = 0.

(CES)

i∈N

This structure does indeed yield a constant Allen-Uzawa elasticity, σ = 1/(1 − ρ), for all pairs of inputs and moreover converges to F (x) = α

 i∈N

β

xi i ,

α > 0,

βi > 0 ∀ i,

(CD)

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with σ = 1 as ρ → ∞. Moreover, Uzawa shows that this structure is necessary as well as sufficient for constancy of the Allen-Uzawa elasticities, but it turns out not to be necessary for the elasticities to be identical for all pairs of inputs. Uzawa went on to establish necessary (as well as sufficient) conditions for constancy and uniformity of the AES for a PLH production function. To explicate these conditions, consider a partition of the set of inputs into m subsets, N = {N 1 , . . . , N S }, with ns inputs in subset s for each s. Uzawa showed that the AES are constant and identical if and only if, for some partition N , the (PLH) production function is given by F (x) =

S 

βs > 0 ∀ s

F s (x s )βs ,

and

s=1

S 

βs = 1,

(11)

αi > 0 ∀ i ∈ N s .

(12)

s=1

where, for all s, F (x ) = s

s



ρ αi xi s

1/ρs 0 = ρs < 1,

,

i∈N s

That is, the production function can be written as a Cobb-Douglas function of CES aggregator functions. Note that the structure (11)–(12) collapses to the (CES) case when S = 1 and to the (CD) case when |N s | = 1 for all s. Moreover, when n = 2, (11) collapses to (CD2 ) and (12) collapses to (CES2 ).24 Analogous results for constancy and uniformity of the Morishima elasticities were established by Blackorby and Russell [11] and Kuga [53] (generalizing the three-input case proved by Murota [66]). Again maintaining positive linear homogeneity of the production function, the MES are constant if and only if the production function takes the form (CES) or (CD) above. 24 McFadden

[61] showed that his direct elasticities of substitution are constant and identical if and

only if F (x) =

 S

1/ρ αs F s (x s )

,

0 = ρ < 1/n∗ ,

0 ≤ ρ < 1/n∗ ,

αs > 0 ∀ s,

s=1

or F (x) = α0

S 

∗

F s (x s )αs /n ,

αs > 0 ∀ s,

and n∗ = max{ns }

s=1

s

where F s (x s ) =

ns 

xi ,

s = 1, . . . , m :

i=1

that is, if and only if the production function can be written as a CES or Cobb-Douglas function of (specific) Cobb-Douglas aggregator functions.

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Blackorby and Russell [14, p. 888] point to the contrast between these representations for the AES and the MES to support their view that the latter is the natural generalization of the Hicks elasticity to encompass more than two inputs: if equation (CES) “is the CES production function, then the MES – and not the AES – is the elasticity of substitution.” Using standard duality theory, constancy of the elasticity of substitution can also be characterized in terms of the structure of the cost and distance functions. For PLH production functions, the Morishima elasticities are globally constant and identical if and only if the cost function takes the form25 C(p, y) = y

 n

ρˆ

αˆ i pi

1/ρˆ ,

1/(1−ρ)

αˆ i = αi

∀i,

ρˆ = ρ/(ρ − 1),

(13)

i=1

or C(p, y) = y

n 

β

αˆ i = αiαi ∀ i.

αˆ i pi i ,

(14)

i=1

The distance function dual to (13) and (14), formally derived using (3), has the following alternative images: D(x, y) = y

−1

 n

1/ρ ρ αi xi

(15)

i=1

or D(x, y) = y −1 α

n 

xiαi .

(16)

i=1

Obviously, setting D(x, y) = 1 in (15) and (16) and inverting in y yields the explicit production function (CES) and (CD). These dual structures, necessary and sufficient for constancy of the MES, are easily modified to encompass the cases where the production function is homogeneous but does not satisfy constant returns to scale: virtually  the same proof goes through if we simply replace y by y 1/α where α = i αi in (13), (14), (15), and (16). The results are similarly extended to homothetic technologies if we replace y with (y) in (13), (14), (15), and (16), where  is an increasing function. In fact, this structure also suffices when there are multiple outputs, in which case  is a mapping from Rm + into R+ , increasing in each output quantity. Intuitively 25 Note

the “self-duality” of this structure, a concept formulated by Houthakker [46] in the context of dual consumer preferences: the cost-function structure in prices mirrors the CES/Cobb-Douglas structure of the production function in input quantities.

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these simple extensions of the basic result reflect the fact that all of the structure implied by constancy of the elasticity of substitution is imbedded in the “shape” (or “curvature”) of an isoquant, and as long as there is no change in its shape as we move from one isoquant to another, the basic structure is preserved.26 The next subsection examines the possibilities when the shape of the isoquant does change when outputs change – i.e., when the technology is not homothetic.

Non-homothetic Technologies Blackorby and Russell [11, 13] generalized the representation result for MES to encompass the case of non-homothetic technologies. Their proof first characterizes the constancy of the MES in terms of the structure of the cost function. In particular, the MES are constant if and only if the cost function has the following structure:  n

C(p, y) = (y)

ρˆ

1/ρˆ

αi (y) pi

0 = ρˆ ≤ 1,

,

(17)

i=1

or C(p, y) = (y)

n 

β

pi i ,

n 

βi > 0 ∀ i,

i=1

βi = 1,

(18)

i=1

where, denoting the range of F by R(F ),  : R(F ) → R++ , and αi : R(F ) → Rn++ , i = 1, . . . , n, are increasing functions. Thus, the basic CES/Cobb-Douglas structure is preserved when we expand the set of allowable technologies to be non-homothetic. An important difference, however, is the dependence on output y of the “distribution coefficients,” αi (y), i = 1, . . . , n in the CES structure in (17).27 This additional flexibility allows the isoquants to “bend” differently for different output vectors while keeping constant the curvature of the isoquant. The distance function dual to (17) is −1

D(y, p) = (y)

 n i=1

26 I

ρ αi (u)−1 xi

1/ρ ,

ρ=

ρˆ . ρˆ − 1

(19)

am unaware of similar explorations of possible generalizations of the results on constancy of the Allen-Uzawa elasticities, but intuition suggests that similar results would go through there as well. 27 Blackorby and Russell [13] proved that the dependence on y of the corresponding coefficients, βi , i = 1, . . . n, in (18) leads to a violation of positive monotonicity of the cost function in y. Thus, generalization to non-homothetic technologies does not expand the Cobb-Douglas technologies consistent with constancy of the MES.

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Owing to the dependence of the distribution coefficients, αi (y), i = 1, . . . n, on y, the explicit production function cannot be derived in closed form in the case of a single output. In the Cobb-Douglas case, however, the dual distance function is D(x, y) = (y)−1 α

n 

xiαi ,

(20)

i=1

which in the case of a single output can be set equal to one and inverted in y to obtain the explicit production function:   n xiαi . y =  −1 α i=1

As pointed out by Blackorby and Russell [13], if the cost structure is given by (17)–(18), the Allen-Uzawa elasticities are constant and equal to the Morishima elasticities. As far as I know, however, necessary structural conditions for constancy of the Allen-Uzawa elasticities have not been worked out, though one might expect some variation on (17)–(18).

Dual Elasticities of Substitution Two-Input Elasticity of Substitution Redux Let us return briefly to the discussion of the two-variable elasticity of substitution in section “Two-Input Elasticity of Substitution: Early Formulations and Characterizations”, where the elasticity is defined as the log derivative of the quantity ratio with respect to the technical rate of substitution: σ = φ  (ln trs,k ). The inverse of φ  , with image σ d := θ  (xk /x ), is the log derivative of a technical rate of substitution with respect to a quantity ratio. This is also an elasticity, one that is dual to σ . In contrast to the (direct) elasticity of substitution σ , large values of σ d reflect “difficult” substitution, or strong complementarity, whereas low values reflect “easy” substitution, or weak complementarity. Following the lead of Sato and Koizumi [72], several papers refer to this concept as an elasticity of complementarity.28 In what follows, I use the terms “dual elasticity of substitution” and “elasticity of complementarity” interchangeably.29 Of course, in the two-input case, σ and σ d convey the same information about the curvature of the isoquant and the degree of substitutability (or complementarity) between the two inputs. In the remainder of this section, I extend this dual concept

28 See

Bertoletti [10], Kim [50], and Stern [76]. another possible assignation is “shadow elasticity of substitution,” since this dual concept is formulated in terms of shadow prices.

29 Yet

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to technologies with more than two inputs, paralleling the development of Allen and Morishima (direct) elasticities in the previous section. The development is facilitated by the use of duality theory, since the trs,k can be interpreted as the relative shadow price of inputs  and k and in fact is equal to the market price ratio, p /pk , under conditions of competitive market pricing.

Dual Morishima and Allen Elasticities of Substitution The dual Morishima elasticity of substitution (Blackorby and Russell [11, 13]) is given by

σijDM (x, y) : =

     i  ∂ ln ρˆi x i , y ρˆj x , y 

= xj

∂ ln(xj /xi ) Dij (x, y) Djj (x, y) − Di (x, y) Dj (x, y)

D = ijD (x, y) − jj (x, y),

(21) (22) (23)

where x i is the (n − 1)-dimensional vector of input quantity ratios with xi in the denominator and ijD (x, y) =

∂ ln ρi (x, y) ∂ ln xj

is the (constant-output) elasticity of the shadow price of input i with respect to changes in the quantity of input j . Analogously, Blackorby and Russell [13] proposed the Allen elasticity of complementarity (alternatively, the dual Allen elasticity of substitution): σijDA (x, y) = =

D(x, y)Dij (x, y) Di (x, y)Dj (x, y) ijD (x, y) sjD (x, y)

,

(24)

(25)

where sjD (x, y) = ρj (x, y) · xj is the cost share of input j (assuming cost-minimizing behavior). If σijDA (p, y) < 0 (i.e., if increasing the j th quantity decreases the shadow price of input i), we say that inputs i and j are Allen-Uzawa dual substitutes; if σijDA (p, y) > 0, they are Allen-Uzawa dual complements. Similarly, if

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R. R. Russell

σijDM (p, y) < 0 (i.e., if increasing the j th quantity decreases the shadow price of input i relative to the shadow price of input j ), we say that input j is a dual Morishima substitute for input i; if σijDM (p, y) > 0, input j is a dual Morishima complement to input i. Recall from section “Two-Input Elasticity of Substitution Redux” that in the twoinput case the elasticities of substitution and complementarity are simple inverses of one another. This is clearly not the case when n > 2.30 Interestingly, since D (x, y) in (23) is non-positive, the distance function is concave in x, and hence jj the Morishima elasticity leans more toward dual complementarity than does the Allen elasticity (in sharp contrast to the primal taxonomy in section “AES and MES and the Comparative Statics of Income Shares”). Similarly, if two inputs are dual Allen-Uzawa complements, they must be dual Morishima complements, whereas two inputs can be dual Allen-Uzawa substitutes but dual Morishima complements. There exist, of course, dual comparative-static results linking factor cost shares and elasticities of complementarity.31 Consider first the effect of quantity changes on absolute shares (for i = j ): ∂ ln siD (x, y) = ijD (x, y) = σijDA (x, y) sjD (x, y), ∂ ln xj so that an increase in xj increases the absolute share of input i if and only if ijD (x, y) > 0 or, equivalently, σijDA (x, y) > 0 : that is, if and only if inputs i and j are dual Allen-Uzawa complements. Thus, the dual elasticities provide immediate qualitative comparative-static information about the effect of quantity changes on (absolute) shares. To obtain quantitative comparative-static information, one needs to know the share of the j th input as well as the Allen-Uzawa elasticity of complementarity. Of course, the (constant-output) elasticity derived from the distance function ijD (x, y) yields the same (qualitative and quantitative) comparative-static information. Comparative-static information about relative income shares in the face of quantity changes can be extracted from the Morishima elasticity. As the share functions, siD , i = 1, . . . n, are homogeneous of degree zero in quantities, we can re-write their images as s˜iD (x i , y) := siD (x, y). We then obtain      ∂ ln s˜iD x i , y /˜sjD x i , y ∂ ln(xj /xi )

DM = ijDM (x, y) − jj (x, y) − 1 = σijDM (x, y) − 1.

course, the Allen and Morishima elasticities of complementarity are identical when n = 2, as is the case with Allen and Morishima elasticities of substitution. 31 While shadow prices and dual elasticities are well defined even if the input requirement sets are not convex, the comparative statics of income shares using these elasticities requires convexity (as well, of course, as price-taking, cost-minimizing behavior), which implies concavity of the distance function in x. By way of contrast, convexity of input requirement sets is not required for the comparative statics of income shares using dual elasticities, since the cost function is necessarily concave in prices. See Russell [71] for a discussion of these issues. 30 Of

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Thus, an increase in xj increases the share of input i relative to input j if and only if σijDM (x, y) > 1 : that is, if and only if inputs i and j are sufficiently complementary in terms of the dual Morishima elasticity of complementarity. Moreover, the degree of departure from unity provides immediate quantitative information about the effect on the relative factor share. Thus, the dual Morishima elasticities provide immediate quantitative and qualitative comparative-static information about the effect of quantity changes on relative shares. As pointed out by Blackorby and Russell [13, p.153], constancy of the dual Morishima elasticities of substitution entails precisely the same restrictions on the production technology as does constancy of the MES elasticities. This is because the required structure of the cost function and the distance function is self-dual, as can be seen by inspection of (13)–(14) and (19)–(20).

Symmetric Elasticity of Complementarity Stern [75] points out that the dual Morishima elasticity does not reflect the curvature of the isoquant. While the log derivative in (21) holds output quantities y constant in assessing the effect on the shadow-price ratio of a change in the quantity ratio – changing only the j th quantity – it does not maintain D(y, x) = 1. Consequently, the direction of the differential change in the quantity ratio is not consistent with containment in the y-isoquant. Stern defines the symmetric elasticity of complementary as follows:      i   ∂ ln ρˆi x i , y ρˆj x , y   σijSEC (x, y) =  ∂ ln(x /x ) j

=

i

(26) D(x,y)=1

Ψ (x, y) (x, y)

(27)

where Ψ (x, y) = −

Djj (x, y) Dij (x, y) Dii (x, y) − +2 Di (x, y)Dj (x, y) Dj (x, y)2 Di (x, y)2

and (x, y) = 1/Di (x, y)xi + 1/Dj (x, y)xj . This elasticity is symmetric, reflecting the required (differential) movement along the isoquant. Moreover, it can be expressed as a share-weighted average of the dual Morishima elasticities [75],

280

R. R. Russell

σ SEC (x, y) =

siD (x, y) siD (x, y) + sjD (x, y)

σijDM (x, y) +

sjD (x, y) siD (x, y) + sjD (x, y)

σjDM i (x, y), (28)

or the dual Allen elasticities (Stern [76]), σ SEC (x, y) =

siD (x, y) siD (x, y) + sjD (x, y)

σijDA (x, y) +

sjD (x, y) siD (x, y) + sj (x, y)

σjDA i (x, y). (29)

The asymmetric elasticity of complementarity is dual to the shadow elasticity of substitution (McFadden [61] and Mundlak [63]), which is derived by evaluating the derivative of a quantity ratio with respect to a price ratio along a constant-cost frontier:      ∂ ln Ci p, y Cj p, y   σijSES (p, y) =  ∂ ln(p /p ) j

i

C(p,y)=y

˜ = Ψ˜ (p, y)/(p, y) where Ψ˜ (p, y) = −

Cjj (p, y) Cij (p, y) Cii (p, y) − +2 2 Ci (p, y)Cj (p, y) Dj (p, y)2 Ci (p, y)

and ˜ (p, y) = 1/Ci (p, y)pi + 1/Dj (p, y)pj . As shown by Chambers [22] and Stern [76], respectively, the shadow elasticity of substitution can be written as share-weighted averages of Morishima or Allen elasticities, as in the relationships between for dual elasticities of complementarity (28) and (29).

Gross Elasticities of Substitution The (primal and dual) Allen and Morishima elasticities of substitution are formulated in terms of constant-output demand functions. Their immediate usefulness in studies of the comparative statics of factor incomes is limited to firms that are output constrained or to firms with homothetic technologies (in which case the elasticities are independent of output quantities). This limitation prompted the formulation of elasticities of substitution that incorporate the effects of optimal output adjustments as input prices change. Following Blackorby et al. [16], I refer to these measurement

6 Elasticities of Substitution

281

concepts as gross elasticities of substitution, contrasting them to the standard Allen and Morishima elasticities, which assess net input quantity change – that is, changes that abstract from the effects of output changes. The gross analogue of the Allen elasticity, first formulated by Mundlak [63] using primal production theory methods, was formulated in the dual by Lau [56] and referred to by Bertoletti [9, 10] as the Hotelling-Lau elasticity in his resurrection of this concept. The gross analogue of the Morishima elasticity was proposed by Davis and Shumway [28] and contrasted with the Hotelling-Lau elasticity by Blackorby et al. [16] and Syrquin and Hollender [77].32 Both of these gross elasticities are most evocatively expressed in terms of the profit function (and in fact are formulated by simply substituting the profit function for the cost function in the Allen elasticity). Let r ∈ Rn++ be the vector of output prices, indexed by k,  = 1, . . . , m. The m profit function,  : Rm ++ × R++ , is defined by (p, r) = max{r · y − p · x | x, y ∈ T } x,y

= r · φ(p, r) − p · ζ (p, r),

(30) (31)

where φ and ζ are the (vector-valued) input-demand and output-supply functions, respectively. The profit function is nondecreasing in r, nonincreasing in p, and convex, jointly continuous, and homogeneous of degree one in r, p. We assume in addition that it is twice continuously differentiable in all prices. The vector of supply functions and the vector of input demand functions are derived from the profit function using the envelope theorem – often termed Hotelling’s Lemma in this context: φk (p, r) = k (p, r) ∀ k and ζi (p, r) = −i (p, r) ∀ i, where subscripts on the profit function  indicate differentiation with respect to the indicated output or input price. The extension of the Allen elasticity to encompass output-quantity changes as formulated by Lau [56] – the Hotelling-Lau elasticity (HLES) – is given, for inputs i and j , by σijH L (p, r) =

32 See

(p, r) ij (p, r) (p, r) ∂ ln φi (p, r) = . i (p, r) j (p, r) pj xj ∂ ln pj

also Hicks [44], Sato and Koizumi [72], and Stern [76] for discussions of these issues.

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R. R. Russell

Blackorby et al. [16] demonstrate that the HLES inherits the inadequacies of the AES, from which it is constructed by analogy. They summarize their evaluation as follows (page 206): The HLES “is not a logarithmic derivative of a quantity ratio with respect to a price ratio—allowing output to change, and it does not provide comparative static content about relative factor incomes. In fact it is not even a generalization of the AUES in any meaningful sense, since it does not reduce to the latter under the assumption of homotheticity.” Blackorby et al. go on to construct a gross version of the MES that rectifies the problems with the HLES. Note that the optimal input ratio can be written as 

ζi (p, r) ln ζj (p, r)





−i (p, r) = ln −j (p, r)





i (p, r) = ln , j (p, r)

(32)

where the second term is an application of Hotelling’s Lemma. To differentiate the ratio in (32) with respect to the log of pi /pj , note that, using homogeneity of degree one of  in all prices, (p, r) = pj j (p−j /pj , r/pj ),

(33)

where p−j is the (n − 1)-dimensional vector of price ratios with pj purged from p. Similarly, owing to homogeneity of degree zero of the demand functions, ζi (p, r) = φi (p−j /pj , r/pj ) ∀i. j

Application of Hotelling’s Lemma to (33) yields the Morishima gross elasticity of substitution: σ MG (p, r) =

  j j ∂ ln(ζi (p−j /pj , r/pj )/ ln ζj (p−j /pj , r/pj )

∂ ln(pi /pj )  ij (p, y) jj (p, r) = pj − . j (p, r) j (p, y)

∗ = ij∗ (p, r) − jj (p, r),

(34) (35) (36)

where ij∗ (p, r) is the (gross) cross elasticity of demand for input i with respect to ∗ (p, r) is the own (gross) price elasticity of demand for input the j th price and jj j . Thus, analogous to the MES, the MGES is simply the difference between the appropriate (gross) cross elasticity of input demand and the (gross) own elasticity of the input associated with the j th axis, along which the price ratio is being varied. By construction, the MGES is a derivative of an optimal input-quantity ratio with respect to the relevant input price ratio when outputs as well as inputs are allowed to adjust. This elasticity, moreover, provides immediate information about the comparative statics of factor income shares. Define the relative shares of inputs i and j :

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283

sij (p, r) =

pi ζi (p, r) pj ζj (p, r)

or, in terms of price ratios, sˆij (p−j /pj , r/pj ) =

pi ζˆi (p−j /pj , r/pj ) . pj ζˆj (p−j /pj , r/pj )

Some tedious but straightforward calculations yield ∂ sˆij (p−j /pj , r/pj ) = 1 − σijMG (pj , r). ∂ ln(pi /pj ) That is, an increase in the price of input i relative to the price of input j (actually, holding pj fixed) increases the share of input i relative to input j if and only if σijMG (pj , r) < 1. Thus, the Morishima gross elasticity, unlike the HotellingLau elasticity, yields immediate (qualitative and quantitative) comparative static information about the effect of changes in relative input prices on the relative factor income shares. As shown by Blackorby et al. [16], the MGES reduces to the MES when the technology is homothetic, whereas the Hicks-Lau elasticity does not collapse to the Allen elasticity under this restriction.

Elasticities of Substitution and Separability Technological separability – independence of technical rates of substitution of a subset of pairs of inputs or outputs from the quantities of inputs or outputs not belonging to this subset – is a powerful restriction rationalizing the existence of aggregate input or output quantities and the decentralization of optimization problems (e.g., output-constrained cost minimization).33 Applications of the concept to dual representations of the technology – independence of a set of dual marginal rates of substitution (or complementarity) from price levels of inputs or outputs not in the set – have dual implications for (price) aggregation and decentralization.34 As was first noticed by Berndt and Christensen [6], certain identity restrictions on the Allen elasticities of substitution are equivalent, under some strong regularity conditions (principally homotheticity), to some corresponding separability restrictions on the technology. Russell [70] and Blackorby and Russell [12] generalized these results for the AES and then extended them to encompass the Morishima elasticities in Blackorby and Russell [13]. The requisite technological restrictions

33 The

concept was independently conceived by Leontief [57, 58] and Sono [74]. See Blackorby et al. [15] for a comprehensive development of the concept and its applications and for citations to the literature extending the Leontief-Solo concept. 34 In fact, the concept is abstract: it can be applied to any (multiple variable) function.

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take the form of separability conditions for the dual cost function (or, equivalently, the distance function).

Separability and Functional Structure Inputs i, j , say, are separable from input k in the distance function if   Di (x, y) ρi (x, y) ∂ ∂ = = 0. ∂xk Dj (x, y) ∂xk ρj (x, y)

(37)

That is, the technical rate of substitution of input i for input j , given the output level y, is independent of the quantity employed of input k. In the single-output case, this condition is equivalent to35  Fi (x) ∂ = 0. ∂xk Fj (x)

(38)

Similarly, under the assumption of differentiability of the cost function, input prices i, j  are separable from input price k in C if   Ci (p, y) δi (p, y) ∂ ∂ = = 0. ∂pk Cj (p, y) ∂pk δj (p, y)

(39)

That is, the ratio of constant-output demand-function images i and j is independent of input price k for given output y. the set of input variable indices I = 1, . . . , n into subsets I =

1Now partition  I , . . . , I S . The corresponding decompositions of the vectors x and p are x = x 1 , . . . , x S  and p1 , . . . , pS . Define the set of triples, IS = {i, j, k | i, j  ∈ I r × I r ∧ k ∈ I s , r = s} and / I s ∪ I r }. IC = {i, j, k | i ∈ I r , j ∈ I s , k ∈ We say that D, F , or C is separable in the partition I if (quantity or price) variables i and j are separable from k for all i, j, k ∈ IS and completely separable in the partition I if variables i and j are separable from k for all i, j, k ∈ IC . That is, D, F , or C is separable in I if ratios of derivatives (i.e., trade-offs between) variables in any set belonging to I are independent of values of variables outside that the case where m = 1, D(x, F (x)) = 1 on the isoquant for output F (x). Differentiate this identity with respect to xi and xj and take the ratio to obtain this equivalence.

35 In

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set, and D, F , or C is completely separable in I if ratios of derivatives between any two variables in I are independent of values of variables outside the sets containing the variables in the ratio. Positing these separability conditions imposes structural restrictions on the functions representing the technology. In particular separability of F in the partition I holds if and only if the production-function image (in the single-output case) can be written as   F (x) = Fˆ F 1 (x 1 ), . . . , F S (x S ) .

(40)

The “aggregator” functions, F 1 , . . . , F S , inherit the curvature and monotonicity properties of F and are interpreted as aggregate input quantities. The “macro” function Fˆ is increasing in its arguments. The cost function is separable in the partition I if and only if   C(p, y) = Cˆ y, C 1 (p, y), . . . , C S (p, y)

(41)

and the distance function is separable in the partition I if and only if   D(x, y) = Dˆ y, D 1 (x, y), . . . , D S (x, y) .

(42)

The sectoral distance and cost functions, D 1 , . . . , D S and C 1 , . . . , C S , inherit ˆ are increasing in the the properties of D and C; the macro functions, Dˆ and C, aggregator-function images. The structures, (41) and (42), are self-dual: that is, the structure (41) holds if and only if (42) holds.36 If the technology is homothetic, the aggregator functions in (40) can be normalized to be homogeneous of degree one. Moreover, the dual representations, (41) and (42), simplify as follows:   C(p, y) = (y) C˜ 1 (p), . . . , S (p)

(43)

  D(x, y) = (y) D˜  1 (x), . . . ,  S (x) ,

(44)

and

where the aggregator functions are homogeneous of degree one and can be interpreted as sectoral price and quantity indexes,  is an increasing function, and  is a decreasing function. Finally, if the production function satisfies first-degree homogeneity, (43) and (44) simplify to

36 Proofs

of these and other results in this section can be found in Blackorby et al. [15].

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  C(p, y) = y C˜ 1 (p), . . . , S (p)

(45)

  D(x, y) = y −1 D˜  1 (x), . . . ,  S (x) .

(46)

and

Note that, on the frontier, D(x, y) = 1 and inversion of (46) in y yields the production function (40). We say that D, F , or C is completely separable in the partition I if (quantity or price) variables i and j are separable from k for all i, j, k ∈ IC and completely separable in the partition I if variables i and j are separable from k for all i, j, k ∈ IC . Assume that the partition of the price and quantity variables I contains more than two groups (S > 2).37 Then the (symmetrically dual) cost and distance functions have the following images if and only if they satisfy complete separability in the partition I 38 :   S s s ¯ C(p, y) = C y, C (p , y)

(47)

s=1

= (y) Cˆ

 S

s

s

C (p , y)

ρ(y) ˆ

1/ρ(y) ˆ 0 = ρ(y) ˆ ≤ 1,

,

(48)

s=1

or (y)

S 

C s (ps , y)βs (y) ,

βs (y) > 0 ∀ s,

s=1 S 

βs (y) = 1,

(49)

s=1

and   S D(x, y) = D¯ y,  s (x s , y)

(50)

s=1

= (y)−1 Dˆ

 S

1/ρ(y)  s (x s , y)ρ(y)

,

0 = ρ(y) ≤ 1,

(51)

s=1

37 Don’t

ask. Or if you can’t resist, I refer you to Section 4.6 of Blackorby et al. [15] on “Sono independence” and additivity in a binary partition.   38 Analogous to the case (13), ρ(y) = ρ(y)/ ˆ ρ(y) ˆ −1 .

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or (y)−1

S 

βs (y) > 0 ∀ s,

 s (x s , y)βs (y)

s=1 S 

βs (y) = 1.

(52)

s=1

Thus, complete separability results in a dual structure for the cost and distance functions that is CES in the aggregator-function images, C 1 (p1 , y), . . . , C S (pS , y) and D 1 (x 1 , y), . . . , D S (pS , y). These aggregator function images cannot be interpreted, however, as sectoral price and quantity indexes, since they depend on the value of the output vector as well as input-specific prices and quantities. If, however, we conjoin complete separability and the assumption of homotheticity of the technology, the above structure simplifies to C(p, y) = (y)

 S

s ρˆ

s

1/ρˆ

 (p )

0 = ρˆ ≤ 1,

,

(53)

s=1

or (y)

S 

s (ps )βs ,

βs > 0 ∀ s,

s=1 S 

βs = 1,

(54)

0 = ρ ≤ 1,

(55)

s=1

and D(p, y) = (y)

−1

 S

1/ρ s

s ρ

Λ (x )

,

s=1

or (y)−1

S 

Λs (x s )βs ,

βs > 0 ∀ s,

s=1 S 

βs = 1.

(56)

s=1

The functions, s (ps ) and  s (x s ), s = 1, . . . , S, satisfy the salient (monotonicity and homogeneity) properties of price and quantity indexes, respectively. If m = 1 in (55) and (56), inversion of D(x, y) = 1 yields the production function,

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y = F (x) = 

  S

1/ρ , Λ (x ) s

s ρ

0 = ρ ≤ 1,

(57)

s=1

or  S  Λs (x s )βs ,

βs > 0 ∀ s,

s=1 S 

βs = 1.

(58)

s=1

If, in addition, the production function is homogeneous of degree one, (y) = y and F (x) =

 S

1/ρ Λs (x s )ρ

,

0 = ρ ≤ 1,

(59)

s=1

or S 

Λs (x s )βs ,

βs > 0 ∀ s,

s=1 S 

βs = 1.

(60)

s=1

Elasticity Identities and Functional Structure Berndt and Christensen [6] were the first to notice a relationship between functional structure and certain restrictions on the values of (Allen) elasticities of substitution. Maintaining linear homogeneity of a single-output production function and n > 2, they showed that A (p, y) ∀i, j, k ∈ IS σkiA (p, y) = σkj

if and only if F is separable in the partition I. This result was generalized by Diewert [30], Russell [70], and Blackorby and Russell [12], and the latter results were extended to Morishima elasticities by Blackorby and Russell [13]. These results can be summarized as follows: The following conditions are equivalent (under the maintained assumption that n > 2): (i) C is separable in the partition I (structure (43)). (ii) D is separable in the partition I (structure (44)). M (p, y) ∀i, j, k ∈ I . (iii) σkiM (p, y) = σkj S

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A (p, y) ∀i, j, k ∈ I . (iv) σkiA (p, y) = σkj S DM (p, y) ∀i, j, k ∈ I . (v) σkiDM (p, y) = σkj S DA DA (vi) σki (p, y) = σkj (p, y) ∀i, j, k ∈ IS .

That is, separability of C or D is equivalent to identity of both Allen and Morishima elasticities between all variables within a separable sector and all variables outside that sector. The following conditions are equivalent (under the maintained assumption that n > 2): (i) (ii) (iii) (iv) (v) (vi)

C is completely separable in the partition I (structure (48) if S > 2). D is completely separable in the partition I (structure (51) if S > 2). M (p, y) ∀i, j, k ∈ I . σkiM (p, y) = σkj C A A σki (p, y) = σkj (p, y) ∀i, j, k ∈ IC . DM (p, y) ∀i, j, k ∈ I . σkiDM (p, y) = σkj C DA DA σki (p, y) = σkj (p, y) ∀i, j, k ∈ IC .

That is, strict separability of C or D is equivalent to identity of both Allen and Morishima elasticities between all variables in any two sectors and all variables outside those sectors. These results provide powerful tools for hypothesis testing because tests for separability – i.e., for aggregate inputs or outputs – are equivalent to tests for certain equality conditions for pairs of elasticities.

Concluding Remarks The elasticity of substitution concept grew out of the interest of prominent English economic theorists, at the time of the Great Depression, in the distribution of income between capital and labor. The concept surged to prominence with the SMAC characterization of constant elasticity of substitution production functions and the emergence of modern growth theory in the 1960s. The elasticity of substitution between labor and capital turns out to be fundamental to many theoretical aspects of economic growth, including the possibility of perpetual growth or decline, the growth of per capita income, and the speed of convergence to an equilibrium growth path. The elasticity of substitution is especially salient in the insightful analysis of the growth process by Klump and de la Grandville [51]. See Chirinko [23] for a discussion of these issues and references to the relevant literature. Most of the growth theory literature, relying originally on the historical constancy of labor and capital income shares, features the Cobb-Douglas production function.39 But beginning about 1980, the labor share began to fall, and an accumulation

39 In fact, as first pointed out by Antras

[3], the pre-1980 constancy of income shares does not imply unitary elasticity of substitution when one takes into account the empirical evidence of aggregate

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of empirical evidence has indicated that the (aggregate) elasticity of substitution is significantly less than one (see Chirinko [23] and Chirinko and Mallick [24]).40 Although the CES has only one more parameter than the Cobb-Douglas, its greater flexibility seems to be more attuned to the evidence. The first empirical estimation of elasticities of substitution for more than two inputs (to my knowledge41 ) is that of Griliches [38], who estimated Allen elasticities to assess the effect of increases in human capital (reflected in years of educational attainment) on the relative wages of skilled and unskilled labor, with capital as a third important input.42 Shortly thereafter, Parks [67] estimated a translog production function to obtain estimated Allen elasticities for five inputs (capital, labor, and three material inputs). Another early estimation of Allen elasticities, as well as implementation of the aggregation theorems in section “Elasticities of Substitution and Separability” and Berndt and Christensen [6], is the test in Berndt and Christensen [7] for the existence of an aggregate capital stock comprising equipment and structures in a production technology using labor as well as the two types of capital. Allen elasticities for technologies with more than two inputs play an important role in the research on energy economics, beginning with the classic KLEM (capital, labor, energy, and materials) paper of Berndt and Wood [8]. Their paper employs the theorems described in section “Elasticities of Substitution and Separability” to test for the existence of a value-added production function – that is, for separability of labor and capital inputs from material inputs. Thompson and Taylor [78] follow up on the analysis of these issues using Morishima elasticities. Elasticities of substitution with more than two inputs have also played an important role in the assessment of the substitutability of (multiple) monetary assets. Barnett et al. [5] estimated Allen elasticities, whereas Davis and Gauger [27] and Ewis and Fischer [33] each employed Morishima elasticities. Finally, Allen elasticities play an important role in the empirical study of the effect of immigration on the relative wages of domestic and immigrant labor (Grossman [39], Borjas [17], Borjas, Freeman, and Katz [18], Borjas, Freeman, and Katz [19] and of the effect of the increase in the number of guest workers on resident and non-resident labor (Kohli [49]). The only empirical estimation of dual elasticities of substitution of which I am aware is in the study by Mundra [64] of the substitutability of resident and nonresident (guest) labor (and other inputs). The paper compares estimates of primal and dual elasticities. labor-saving technological change, which would tend to increase the share of capital, offsetting its declining share owing to a increasing capital intensity and an elasticity of substitution below one. 40 Karabarbounis and Neiman [48] estimate an elasticity of substitution greater than one, but Acemoglu and Robinson [1] argues that their use of cross-country data makes their estimates more likely to correspond to endogenous-technology elasticities. 41 Some of this discussion is based on a working paper by Mundra and Russell [65]. 42 Follow-ups of the Griliches study can be found in Johnson [47], Kugler et al. [55], and Welch [80].

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Without doubt, many other empirical applications of elasticities of substitution have escaped my attention. The focus of this chapter has been on the elasticity of substitution between (or among) inputs, but the concept is equally relevant to output substitution. That is, for each of the elasticities defined in sections “Two-Input Elasticity of Substitution: Early Formulations and Characterizations”, “Allen and Morishima Elasticities of Substitution”, and “Dual Elasticities of Substitution”, one can formulate a corresponding (primal or dual) output elasticity of substitution – a characterization of the curvature of the output possibility curve (or surface).43 The elasticity of substitution also shows up in utility theory, where it reflects the ease of substitution between consumer goods (and characterizes the curvature of indifference surfaces). In fact, it was in utility theory that the CES function made its first appearance in the economics literature, when Burk (Bergson) [20] showed that additivity of the utility function and linearity of Engel curves implies that the utility function belongs to the CES family, referred to as the “Bergson family” in consumer theory. The elasticity of substitution in intertemporal utility functions plays an important role in macroeconomic theory (Hall [40]) and in optimal growth theory (Cass [21] and Koopmans [52]). Finally, the CES utility function has proved useful in the study of optimal product diversity in the context of monopolistic competition [32]. Consistent with the theme of this volume, the chapter has focused primarily on the theoretical development of the measurement of substitutability: primal and dual characterizations and their close relationships to separable sectors of a production or utility function. The taxonomy for n-variable elasticities implicit in the discussions in sections “Allen and Morishima Elasticities of Substitution”, “Dual Elasticities of Substitution”, and “Gross Elasticities of Substitution” dichotomizes elasticity-ofsubstitution concepts along the following lines: partial vs. ratio elasticities (Allen vs. Morishima elasticities), direct vs. dual elasticities (quantities vs. (shadow) prices as “endogenous” variables), and net vs. gross elasticities (fixed output [or technological homotheticity] vs. variable output). Other elasticity-of-substitution concepts have been proposed, but I see them as variations on these themes.44

43 See,

e.g., the analysis of the substitutability between a “good” and a “bad” output (in this case, electricity and sulfur dioxide) in Färe et al. [35]. 44 This may be an unfair oversimplification: Stern [76], building on Mundlak [63], proposes a related but somewhat different and more comprehensive taxonomy of the elasticities. (Nevertheless, I’m reminded of a (private) comment made by a prominent social choice theorist back in the heyday of research in his area: “The problem with social choice theory is that there are more axioms than there are ideas.” Well, perhaps we have reached the point where there are more elasticity-ofsubstitution concepts than there are ideas.)

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Cross-References  Distance Functions in Production Economics  Duality in Production  Functional Structure and Aggregation  Multiproduct Technologies

References 1. Acemoglu D and Robinson JA (2015) The rise and decline of general laws of capitalism. J Econ Perspect 29:3–28 2. Allen RGD (1938) Mathematical analysis for economists. Macmillan, London 3. Antras P (2004) Is the U.S. aggregate production function Cobb-Douglas? New estimates of the elasticity of substitution. Contrib Macroecon 4:1–34 4. Arrow KJ, Chenery HP, Minhaus BS, Solow RM (SMAC) (1961) Capital-labor substitution and economic efficiency. Rev Econ Stat 63:225–250 5. Barnett WM, Fisher D, Serletis A (1992) Consumer theory and the demand for money. J Econ Lit 30:2086–2119 6. Berndt ER, Christensen L (1973) The internal structure of functional relationships: separability, substitution, and aggregation. Rev Econ Stud 40:403–410 7. Berndt ER, Christensen L (1973) The translog function and the substitution of equipment, structures, and labor in U.S. manufacturing 1929–68. J Econ 1:81–114 8. Berndt ER, Wood DO (1975) Technology, prices, and the derived demand for energy. Rev Econ Stat 57:259–268 9. Bertoletti P (2001) The Allen/Uzawa elasticity of substitution as a measure of gross input substitutability. Rivista Italiana Degli Economisti 6:87–94 10. Bertoletti P (2005) Elasticities of substitution and complementarity a synthesis. J Prod Anal 24:183–196 11. Blackorby C, Russell RR (1975) The partial elasticity of substitution. Discussion Paper No. 75-1, Department of Economics, University of California, San Diego 12. Blackorby C, Russell RR (1976) Functional structure and the Allen partial elasticities of substitution: an application of duality theory. Rev Econ Stud 43:285–292 13. Blackorby C, Russell RR (1981) The Morishima elasticity of substitution: symmetry, constancy, separability, and its relationship to the Hicks and Allen elasticities. Rev Econ Stud 48:147–158 14. Blackorby C, Russell RR (1989) Will the real elasticity of substitution please stand up? A comparison of the Allen/Uzawa and Morishima elasticities. Am Econ Rev 79:882–888 15. Blackorby C, Primont D, Russell RR (1978) Duality, separability, and functional structure: theory and economic applications. North-Holland, New York 16. Blackorby C, Primont D, Russell RR (2007) The Morishima gross elasticity of substitution. J Product Anal 28:203–208 17. Borjas GJ (1994) The economics of immigration. J Econ Lit 32:1667–1717 18. Borjas GJ, Freeman RB, Katz LF (1992) On the labor market effects of immigration and trade. In: Borjas G, Freeman R (eds) Immigration and the work force. University of Chicago Press, Chicago 19. Borjas GJ, Freeman RB, Katz LF (1996) Searching for the effect of immigration on the labor market. AEA Pap Proc 8:246–251 20. Burk (Bergson) A (1936) Real income, expenditure proportionality, and Frisch’s ‘new methods of measuring marginal utility’. Rev Econ Stud 4:33–52 21. Cass D (1965) Optimum growth in an aggregative model of capital accumulation. Rev Econ Stud 32:233–240

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22. Chambers R (1988) Applied production analysis. Cambridge University Press, Cambridge 23. Chirinko RS (2008) σ : the long and short of it. J Macroecon 30:671–686 24. Chirinko RS, Mallick D (2016) The substitution elasticity, factor shares, and the low-frequency panel model. Am Econ J Macroecon 9:225–253 25. Cobb CW, Douglas PH (1928) A theory of production Am Econ Rev 18:23–34 26. Cornes R (1992) Duality and modern economics. Cambridge University Press, Cambridge 27. Davis GC, Gauger J (1996) Measuring substitution in monetary-asset demand systems. J Bus Econ Stat 14:203–209 28. Davis GC, Shumway CR (1996) To tell the truth about interpreting the Morishima elasticity of substitution. Can J Agric Econ 44:173–182 29. de la Grandville O (1997) Curvature and the elasticity of substitution: straightening it out. J Econ 66:23–34 30. Diewert WE (1974) Applications of duality theory. In: Intriligator M, Kendrick D (eds) Frontiers of quantitative economics, vol 2. North-Holland, Amsterdam 31. Diewert WE (1982) Duality approaches to microeconomic theory. In: Arrow K, Intriligator M (eds) Handbook of mathematical economics, vol II. North-Holland, New York 32. Dixit AK, Stiglitz JE (1977) Monopolistic competition and optimum product diversity. Am Econ Rev 67:297–308 33. Ewis NA, Fischer D (1984) The translog utility function and the demand for money in the United States. J Money Credit Bank 16:34–52 34. Färe R, Grosskopf S (1990) A distance function approach to price efficiency. J Pol Econ 43:123–126 35. Färe R, Grosskopf S, Noh D-W, Weber W (2005) Characteristics of a polluting technology: theory and practice. J Econ 126:469–492 36. Färe R, Primont D (1995) Multi-output production and duality theory: theory and applications. Kluwer Academic Press, Boston 37. Fuss M, McFadden D (eds) (1978) Production economics; a dual approach to theory and applications. North-Holland, Amsterdam 38. Griliches Z (1969) Capital-skill complementarity. Rev Econ Stat 51:465–468 39. Grossman J (1982) The substitutability of natives and immigrants in production. Rev Econ Stat 64:596–603 40. Hall RE (1988) Intertemporal substitution in consumption. J Polit Econ 96:339–357 41. Hardy GH, Littlewood JE, Pólya G (1934) Inequalities. Cambridge University Press, Cambridge 42. Hicks JR (1932) The theory of wages. MacMillan Press, London 43. Hicks JR (1936) Distribution and economic progress: A revised version. Rev Econ Stud 4:1–12 44. Hicks JR (1970) Elasticity of substitution again: substitutes and complements. Oxford Econ Pap 22:289–296 45. Hicks JR, Allen RGD (1934) A reconsideration of the theory of value, part II. Economica 1:196–219. N.S 46. Houthakker HS (1965) A note on self-dual preferences. Econometrica 33:797–801 47. Johnson GE (1970) The demand for labor by educational category. South Econ J 37: 190–204 48. Karabarbounis L, Neiman B (2014) The global decline of the labor share. Q J Econ 129:61–103 49. Kohli U (1999), Trade and migration: a production theory approach. In: Faini R, de Melo J, Zimmermann KF (eds) Migration: the controversies and the evidence. Cambridge University Press, Cambridge 50. Kim HY (2000) The Antonelli versus Hicks elasticity of complementarity and inverse input demand systems. Aust Econ Pap 39:245–261 51. Klump R, de la Grandville O (2000) Economic growth and the elasticity of substitution: two theorems and some suggestions. Am Econ Rev 90:282–291 52. Koopmans TC (1965) On the concept of optimal economic growth. In: The econometric approach to development planning. North Holland (for Pontificia Academic Science), Amsterdam

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53. Kuga K (1979) On the symmetry of Robinson elasticities of substitution: the general case. Rev Econ Stud 46:527–531 54. Kuga K, Murota T (1972) A note on definition of elasticity of substitution. Macroeconomica 24:285–290 55. Kugler P, Müller U, Sheldon G (1989) Non-neutral technical change, capital, white-collar and blue-collar labor. Econ Lett 31:91–94 56. Lau L (1978) Applications of profit functions. In: Fuss M, McFadden D (eds) Production economics: a dual approach to theory and applications. North-Holland, Amsterdam, pp 133– 216 57. Leontief WW (1947) A note on the interrelation of subsets of independent variables of a continuous function with continuous first derivatives. Bull Am Math Soc 53:343–350 58. Leontief WW (1947) Introduction to a theory of the internal structure of functional relationships. Econometrica 15:361–373 59. Leontief WW (1953) Domestic production and foreign trade: the American capital position re-examined. Proc Am Philos Soc 97:331–349 60. Lerner AP (1933) Notes on the elasticity of substitution II: the diagrammatical representation. Rev Econ Stud 1:68–70 61. McFadden D (1963) Constant elasticity of substitution production functions. Rev Econ Stud 30:73–83 62. Morishima M (1967) A few suggestions on the theory of elasticity (in Japanese). Keizai Hyoron (Econ Rev) 16:144–150 63. Mundlak Y (1968) Elasticities of substitution and the theory of derived demand. Rev Econ Stud 35:225–236 64. Mundra K (2013) Direct and dual elasticities of substitution under non-homogeneous technology and nonparametric distribution. Indian Growth Dev Rev 6:204–218 65. Mundra K, Russell RR (2004) Dual elasticities of substitution. Discussion Paper 01-26, Department of Economics, University of California, Riverside 66. Murota T (1977) On the symmetry of Robinson elasticities of substitution: a three-factor case. Rev Econ Stud 42:79–85 67. Parks RW (1971) Price responsiveness of factor utilization in Swedish manufacturing, 1870– 1950. Rev Econ Stud 53:129–139 68. Pigou AC (1934) The elasticity of substitution. Econ J 44:23–241 69. Robinson J (1933) Economics of imperfect competition. MacMillan, London 70. Russell RR (1975) Functional separability and partial elasticities of substitution. Rev Econ Stud 42:79–85 71. Russell RR (1997) Distance functions in consumer and producer theory. Essay 1. In: Färe R, Grosskopf S (eds) Index number theory: essays in honor of Sten Malmquist. Kluwer Academic Publishers, Boston, pp 7–90 72. Sato R, Koizumi T (1973) On the elasticities of substitution and complementarity. Oxford Econ Pap 25:44–56 73. Solow RM (1956) A contribution to the theory of economic growth. Q J Econ 65:65–94 74. Sono M (1945, 1961) The effect of price changes on the demand and supply of separable goods. Int Econ Rev 2:239–271 (Originally published in Japanese In: Kokumin Keisai Zasshi 74:1–51) 75. Stern DI (2010) Derivation of the Hicks, or direct, elasticity of substitution from the input distance function. Econ Lett 108:349–351 76. Stern DI (2011) Elasticities of substitution and complementarity. J Prod Anal 36:79–89 77. Syrquin M, Hollender G (1982) Elasticities of substitution and complementarity: the general case. Oxford Econ Pap 34:515–519 78. Thompson P, Taylor TG (1995) The capital-energy substitutability debate: a new look. Rev Econ Stat 77:565–569 79. Uzawa H (1962) Production functions with constant elasticities of substitution. Rev Econ Stud 29:291–299 80. Welch F (1970) Education in production. J Polit Econ 78:35–59

7

Distance Functions in Production Economics Robert G. Chambers and Rolf Färe

Contents Intuitive Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Basic Assumptions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Distance Functions Defined and Their Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Differential Properties of Distance Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Distance Functions at Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Duality Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Efficiency Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Index Numbers and Productivity Measurement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Empirical Implementation of Distance Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Commentary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Cross-References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Abstract

This chapter treats distance functions used in production economics and operations research. The economic intuition behind the different notions of distance functions is discussed to set the stage for a more formal analysis. A minimal set of regularity conditions needed to ensure the existence of well-behaved distance functions are presented, distance functions are defined, and the uses of distance functions in a variety of settings are surveyed. R. G. Chambers () Department of Agricultural and Resource Economics, University of Maryland, College Park, MD, USA e-mail: [email protected] R. Färe Department of Economics and Department of Agricultural and Resource Economics, Oregon State University, Corvallis, OR, USA e-mail: [email protected] © Springer Nature Singapore Pte Ltd. 2022 S. C. Ray et al. (eds.), Handbook of Production Economics, https://doi.org/10.1007/978-981-10-3455-8_14

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Keywords

Distance functions · Production · Technology · Measurement

JEL codes

D21, D22, D24 Modern production economics, following Debreu [11] and Shephard [22], is typically cast in set-theoretic terms. Even the most casual reader of that literature is confronted with concepts such as technology sets, input sets, and output sets that are far removed from the production functions and input-requirement functions familiar from intermediate microeconomics. The set-theoretic approach conveys clarity, precision, and the ability to generalize. But its abstractness often proves problematic when examining more mundane situations involving observed data. A cardinal bridge is needed to connect abstract set-theoretic analysis and actual data. In intermediate microeconomics, the production function and the inputrequirement function are the cardinal bridges to which we introduce our students. The former is defined as the maximum obtainable output for a given input bundle, and the latter as the minimal amount of an input needed to produce a given output. That’s clear enough and easily illustrated using Fig. 1. There the “lazy S-shaped” curve emanating from the origin, labelled f (x), traces out a traditional version of the graph of the production function. The vertical axis is denominated in units of output, y, the horizontal axis is denominated in units of input, x, and the technically feasible input-output combinations are those falling on or below f (x). Thus, for input use of x ∗ , the maximum output attainable is f (x ∗ ) that corresponds to the point on f (x) directly above x ∗ . On the other hand, if the goal is to produce y ∗ , only e (y ∗ ), which corresponds to the point on f (x) at the same vertical height as y ∗ , is needed. Thus, depending upon your perspective, f (x) illustrates both the production function and the input-

y

Fig. 1 Production, input-requirement, and distance functions

y*

0

f(x)

B

f ( x *) C

e ( y *)

A

x*

x

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requirement function. Figure 1 depicts x as the independent variable and y as the dependent variable reflecting the idea that the production function gives the maximum amount that can be produced given y. Switching axes and measuring x vertically and y horizontally, however, does not change the substance of the relationship. There are several things to note. First, the production function is measured as the vertical distance between, say, (x ∗ , 0) and (x ∗ , f (x ∗ )) and is denominated in output units. Second, the input-requirement function is measured as the horizontal distance (0, y ∗ ) and (e (y ∗ ) , y ∗ ) and is denominated in input units. Finally, in the case illustrated e∗ (y) = f −1 (y) and f (x) = e−1 (x), with f −1 (y) = {x : f (x) = y} , and e−1 (x) = {y : e (y) = x} . Unfortunately, real technologies are not describable in terms of a single input or a single output. In general, production technologies involve using multiple inputs to produce multiple outputs. So, despite their intuitive value, production functions and the input-requirement function are often lacking in empirical settings. Production economists and operations researchers have accommodated this difficulty by developing cardinal representations of the set-theoretic representations of production technologies. Those cardinal representations are commonly referred to as distance functions. Like the production function and the input-requirement function, they characterize the technology in terms of a measured distance (hence the name) between some point in input-output space and a frontier of the feasible set of production outcomes. Unlike the production function and the input requirement function, however, they can be adapted to both multiple input and multiple output settings. They are the subject matter of this chapter. This chapter proceeds as follows. First, the basic intuition behind the different notions of distance functions is presented. Then we present and discuss intuitively a minimal set of assumptions that are needed to ensure a meaningful representation of a production technology. A formal definition of a distance function is then offered, and its basic mathematical properties are discussed. Then we turn to a discussion of the smoothness (differential) properties of distance functions, and we show the connection between standard notions of differentiability and a slightly generalized notion needed to ensure that points are associated with potentially meaningful economic equilibria. The penultimate section discusses different uses (in a production contexts) to which distance functions have been applied. And the final section ends with a brief overview of the history of distance functions in economic analysis.

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Intuitive Background The typical economist conceptualizes the technology using a production function. Specialists in production economics are more prone to think of it in terms of a technology set, T , that is a subset of RN +M and is defined by   T = (x, y) ∈ RN +M : x can produce y , where x ∈ RN denotes an N−dimensional input vector, and y ∈ RM an M−dimensional output vector.1 If x and y are scalars, the connection between the production function and T is obvious. Measuring x on the horizontal axis and y on the vertical axis as in Fig. 1, the production function represents the upper boundary of T . One thus writes2 f (x) = max {y : (x, y) ∈ T } to define the production function f : R → R. Although production functions are usually treated as primitives by economists, viewing it from the perspective of T clarifies that deriving it formally involves solving an optimization problem. And the type of optimization problem reveals much about the way economists often view production economics problems. Defining f (x) as the largest possible output attainable from x is a natural way of looking at things if you are the individual who keeps the returns from producing y. Not all individuals, however, get to keep what they produce. Many times, contracting works the other way around. Individuals are hired to produce a given y but have no claim on y. Intuition would then suggest that they would concentrate on economizing on the use of x leading one to solve e (y) = min {x ∈ R : (x, y) ∈ T }, where e : R → R is the input-requirement function. If we were to depict this optimization problem formally using traditional mathematical conventions, then

1 Distinguishing

between inputs and outputs is unnecessary and often arbitrary because what are perceived as outputs in one context can be inputs in another. For example, milk produced from cows can be converted by the producer into cream, cheese, and other milk products. A more general representation, which is especially common in general-equilibrium models, is obtained by using the concept of a netput which allows commodities to play either role. When a commodity acts as an input in a particular process, it enters with a negative sign, and when it acts as an output, it enters with positive sign. In that case, the technology is written   T = z ∈ RN +M : z is technically feasible .

Although this level of generality is possible, throughout our presentation we maintain the artificial distinction between inputs and outputs because of its familiarity and its continuing prevalence in applied production analysis. 2 Subject to suitable regularity conditions that we will discuss below.

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unlike Fig. 1, y as the independent variable would be measured along the horizontal axis and x as the dependent variable would be measured on the vertical axis. The production function and the input-requirement function portray two different ways of looking at the same T . Residual claimants for y view T via f (x), while residual claimants for x via e (y). One way to portray the intuitive connection between them and distance functions, which are the subject of this chapter, is to think in terms of an arbitrary (x, y) and then ask if (x, y) represents the best inputoutput combination available from T ? Answering that question properly requires one to be precise about what best means. One possible sense of best would be in terms of whether that y represented the maximal output that could be had if x were committed to production. One could then compare y to f (x) if the latter were known but more directly would also solve: o (y, x) = min {β ∈ R : (x, y − β) ∈ T } = y + min {β − y ∈ R : (x, y − β) ∈ T } = y − max {y − β ∈ R : (x, y − β) ∈ T } = y − f (x) . Alternatively, another possible way to define best would be to determine whether y could be produced using something smaller than x. That would lead one to solve the following optimization problem: i (x, y) = max {α ∈ R : (x − α, y) ∈ T } = x + max {α − x ∈ R : (x − α, y) ∈ T } = x − min {x − α ∈ R : (x − α, y) ∈ T } = x − e (y) . The functions o (y, x) and i (x, y) are examples of distance functions. The first, o (y, x), measures the minimal shrinkage of y required to make the “shrunken output” producible by x.3 The second, i (x, y), measures the maximal shrinkage

3 One can also write this problem as determining the maximal expansion of y possible given x. Mathematically,     oˆ (y, x) = max βˆ : x, y + βˆ ∈ T

    = max y + βˆ : x, y + βˆ ∈ T = f (x) − y = −o (y, x) , so that apart from the sign difference, identical results will be obtained. A similar argument shows that i (x, y) can also be recast as a minimization problem without changing its true nature.

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of x that will allow the “shrunken input” to produce y. Figure 1 again illustrates. For the input-output pair (x ∗ , y ∗ ) illustrated by point A, o (y ∗ , x ∗ ) is given by (minus) the vertical distance AB, and i (x ∗ , y ∗ ) is given by the horizontal distance CA. Before we move on to a more detailed discussion, it’s important to emphasize that the different representations of the technology, even in a two-dimensional setting, result from looking at the constraints imposed by the technology from different economic perspectives. One emphasizes output considerations, and the other emphasizes input considerations. Those perspectives echo throughout even the most basic economic discussion of production problems. Visually, we often think of producers moving onto or along isoquants (an input perspective) or onto or along transformation curves (an output perspective). There are many reasons for this, and not the least is that in a multiple input-multiple output setting, graphical analysis of technologies requires that something be held constant. It’s also true that those different ways of looking at things reflect the different ways producers think about outputs and input, maximization versus minimization.

Basic Assumptions Our most basic assumption is that there exists a nonempty subset of RN +M that we denote as T and define by   T = (x, y) ∈ RN +M : x can produce y . Maintaining the conventional split between inputs and output, T represents the technically feasible bundles of inputs and outputs, (x, y). We treat T as a physical datum that is not subject to manipulation by individual producers. Individual producers view T as a constraint to which they react in designing their economic activities. The assumption that T is a subset of real space carries the implicit assumption that both inputs and outputs can be divided into arbitrarily small units. The jargon for this is input and output divisibility. Everyday experience, however, teaches us that many commodities are not so divisible. So, for example, where a tiny scoop of ice cream remains a tiny amount of ice cream, a tiny slice of the ice-cream scooper is not a scooper but a metal shard. It’s important to remember, therefore, that x and y do not correspond to everyday notions of commodities that do not distinguish between stocks and flows but that they refer exclusively to the economic flows from the commodities, and it is those flows that are arbitrarily divisible and not the stocks.

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From T we derive two correspondences. The first4 V : RM ⇒ RN , called the input correspondence, is defined   V (y) = x ∈ RN : (x, y) ∈ T , y ∈ RM , and its image V (y) is referred to as the input set for y, or more simply the input set. It represents the set of input bundles that can produce the output bundle y. The second Y : RM ⇒ RN , called the output correspondence, is defined   Y (x) = y ∈ RM : (x, y) ∈ T , x ∈ RN , and its image is referred to as the output set for x. It represents the output bundles that can be produced from the input bundle x. Y and V are lower inverses (in the sense of Berge 1963) to one another. Figure 1 represents both concepts in the two-dimensional case. There T continues to be represented by all (x, y) pairs falling below the lazy S-shaped curve emanating from the origin. Therefore, for the output y ∗ , V (y ∗ ) consists of all input quantities (measured on the horizontal axis) greater than or equal to e (y ∗ ). Conversely, Y (x ∗ ) consists of the output quantities less than or equal to f (x ∗ ). Hence, it’s natural to envision Y (x) and V (y) as generalizations of f (x) and e (y), respectively. Figure 1 also nicely summarizes regularity conditions sufficient to ensure the existence of well-defined function representations of the technology. Recall that we represented the production function as the solution to a pointwise, that is for given x, optimization problem. In terms of output correspondences, that definition can be rewritten as f (x) = max {y : y ∈ Y (x)} . As we have drawn Fig. 1, this optimization problem is well defined over x ≥ 0 because each Y (x) is bounded from above there by the lazy S-shaped curve emanating from the origin. But elsewhere, things are not so clear. For example, f is not really defined for x < 0. We assume that there exists a closed, convex subset of RN +M that we shall refer to as commodity space, and denote by Z. We require that T ⊆ Z and to be closed. To economize on notation, we assume in all that follows that vectors or sets resulting

A ⇒ B denotes a point-to-set mapping from a point in set A to a set in B. Some writers use A → 2B , where 2B is the power set of B to denote the same correspondence. It is to be distinguished from A → B that denotes a point-to-point mapping.

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from Minkowski addition or scalar multiplication of sets belong to Z. To cover what happens in cases such as Y (x) = ∅ for the production function definition, for ¯ to denote the set of real numbers formed example, we set f (x) = −∞ and use R by appending −∞ and +∞ to R. We’ve isolated some mathematically convenient assumptions that facilitate our discussion of production functions, input-requirement functions, and distance functions. Next, we consider the mathematical requirements that enable us to reverse course. That is, assumptions that allow us to generate T from knowledge of o (y, x) or i (x, y). Figure 1 again illustrates. One sees immediately that if one had complete knowledge of f then T as depicted could be recaptured from the following operation   T = (x, y) ∈ R2 : f (x) ≥ y .

(1)

In formal mathematical terms, T in this instance is the hypograph of the production function. In everyday English, T is everything falling on or below the graph of the production function. One can also recapture T as   T = (x, y) ∈ R2 : x ≥ e (y) ,

(2)

which corresponds to the epigraph of the input-requirement function, that is, everything lying or above the graph of the input-requirement function. It follows quickly from their definitions that equivalently   T = (x, y) ∈ R2 : o (y, x) ≤ 0 ,

(3)

  T = (x, y) ∈ R2 : i (y, x) ≥ 0 .

(4)

and

So, as drawn, T , f (x), e (y), o (y, x), and i (x, y) all summarize the same information. The visual reason that this occurs is that once a point on the “boundary” of T is isolated using the appropriate function representation, all that remains is to identify the remaining points by looking to one side of that boundary. The standard production economics jargon for the requisite property is disposability of outputs and/or inputs. That jargon, however, masks the role of the numeraire. The production function and o (y, x) share the common numeraire y. Viewed in these terms, the property that permits algorithms (1) and (3) to be applied is that once a boundary point, (x o , y o ), is isolated adding or subtracting units of the numeraire from y o suffices to identify T . Similarly, algorithms (2) and (4) work by isolating a boundary point, (x o , y o ), and then adding or subtracting units of the numeraire for e (y) and i (x, y), which is x, from x o .

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Thus, what’s crucial to developing the algorithms is the ability of the numeraire to identify where T lies relative to points on its frontier. We refer to this property as goodness of the numeraire. Stating this precisely requires a definition of the numeraire. Our focus in this chapter is on two types of commodities, inputs and outputs, and thus on two specific types of numeraire. We denote the output N numeraire by ω ∈ RM + \ {0} and the input numeraire by ι ∈ R+ \ {0}. Our formal 5 criteria for goodness are : Definition 1. T satisfies goodness of the output numeraire (GON) if Y (x) − εω ⊆ Y (x) , for ε > 0. Definition 2. T satisfies goodness of the input numeraire (GIN) if V (y) + θ ι ⊆ V (y) , for θ > 0. Remark 1. As traditionally defined free disposability of output (FDO) requires that y ∈ Y (x) ⇒ y o ∈ Y (x)

for y o ≤ y.

Free disposability of input (FDI) requires that x ∈ V (y) ⇒ x o ∈ V (y)

for x o ≥ x.

Weak disposability of output (WDO) requires that y ∈ Y (x) ⇒ μy ∈ Y (x)

for μ ∈ (0, 1],

and weak disposability of input (WDI) requires x ∈ V (y) ⇒ μx ∈ V (y)

5 If

for μ ≥ 1.

one operated in terms of netputs,   T = z ∈ RN +M : z is technically feasible ,

+M and a numeraire γ ∈ RN \ {0}, the parallel notion of goodness in the netput numeraire (GNN) + would require that

z ∈ T ⇒ z − εγ ∈ T for all ε > 0.

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The weaknesses and strengths of each of these disposability assumptions is addressed elsewhere in the handbook (see, e.g., the  Chap. 12, “Bad Outputs”). For our purposes, it suffices to note that each represents a strengthening of GON or M \ {0}, while FDI requires GIN. FDO requires that T satisfies GON for all ω ∈ R+ N that T satisfies GIN for all ι ∈ R+ \ {0}. WDO and WDI narrow things down a bit, but they still require, respectively, that for any x T satisfies GON for all y ∈ Y (x) and that for any y T satisfies GIN for all x ∈ V (y).

Distance Functions Defined and Their Properties Once we admit the possibility that production processes involve multiple inputs and multiple outputs, it becomes obvious that representing technologies in terms of either production functions or input-requirement functions may be unnecessarily restrictive. Thus, few economists would blink if confronted by technologies represented pictorially by either isoquants or transformation curves (production possibilities frontiers). One would understand that the axes for isoquants involve inputs (holding output constant), that the axes for transformation curves are outputs (holding inputs constant), and that these curves shift as outputs and inputs, respectively, change. Isoquants and transformation curves are usually represented in mathematical terms as level sets of a numerical function describing the technology. Many writers call that numerical function a transformation function and denote it as t (x, y). We’ll honor that tradition. The exact origins of t (x, y) often remain unspecified, even though it is often endowed with extremely strong (and convenient) mathematical properties. And in very many instances, t is used to define the technology via T ≡ {(x, y) : t (x, y) ≤ 0} . In other words, t is treated as the primitive, and T is derived from it and not the other way around. Properties of T thus flow from functional structure and restrictions placed on t. This section shows how the process can be reversed so that T is treated as the primitive and functional representations, such as this ambiguously defined t, are induced from assumptions on T . Distance functions provide the link.6 Before we offer a precise definition of a distance function, it’s important to say a word about the perspective that we take. When distance-type measures were

6 One,

but not the only, for example, one can always define what’s known as an indicator function for T as follows  0 (x, y) ∈ T . δ (x, y) = ∞ ∞ otherwise

Indicator functions are an essential part of modern variational analysis and convex analysis. However, in practical settings, they can prove quite difficult to use.

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originally introduced into economics by Konüs [15], Malmquist [18], and Shephard [21], they were defined in radial terms. By that we mean that for any arbitrary (x, y), input-based measures took that x as the numeraire and then moved toward the frontier in input space in the direction of that x. Visually, that corresponds to radially contracting x. Similarly, output-based measures used y as the numeraire. We don’t take that approach. But as we demonstrate below, the radial approach is an important, but still special, case of our general approach. In our set up, an isoquant intuitively corresponds to a lower boundary of an input set, V (y), and a transformation curve to an upper boundary of an output set, Y (x). Isoquants and transformation curves are, thus, correspondences derived by projecting a slice7 of T onto RN and RM , respectively, and then looking at its appropriate boundary. Hence, both V (y) and Y (x) offer natural ways to formalize the concept of a transformation function. We have as the following natural generalizations of our two-dimensional measures: ¯ is defined Definition 3. An output-oriented distance function, O ω : RN +M → R, O ω (y, x) ≡ min {β ∈ R : y − βω ∈ Y (x)} if there exists β ∈ R such that y − β ∈ Y (x) and +∞ otherwise. ¯ is defined Definition 4. An input-oriented distance function, I ι : RN +M → R, I ι (x, y) ≡ max {λ ∈ R : x − λι ∈ V (y)} , if there exists λ ∈ R such that x − λι ∈ V (y) and −∞ otherwise. Because these distance functions are themselves functions of the numeraire (as recognized by the appropriate superscript), the ability to think in terms of different potential numeraires ensures that many distance functions potentially exist. The output-oriented (or output for short) distance function is visualized with the aid of Fig. 2.8 Y (x) is illustrated by all the points on or below the negatively sloped curvilinear frontier. In this instance, the numeraire illustrated by the vector labelled ω contains positive amounts of both outputs. The output couple of interest, y, lies outside of Y (x) and cannot be produced using the input bundle x. Hence, y must be “shrunk” to be technically feasible. The numeraire, ω, determines how (in what

7 Slices,

in fact, have a precise mathematical definition with which we need not concern ourselves here. The intuitive idea is straightforward. Imagine T in say three-dimensional space with one dimension representing output and the other two representing inputs. Now mark off a particular level of y and imagine taking a knife and slicing through T at this point parallel to the input axes. That’s the slice that you project onto the input axes to get the isoquant. If you can represent T via a production function, the equivalent operation is to obtain its upper contour set for a particular y. 8 The visual intuition for the input-oriented distance function is similar with V (y) replacing Y (x) and x replacing y.

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Fig. 2 Output-oriented distance function

y

y* Y(x)

ω y1

direction) y is to be shrunk. The shrinkage is envisioned as moving from y “in the direction of the numeraire.” One visualizes this as sliding y along the dotted line parallel to ω until it encounters Y (x). For the case illustrated, o (y, x) is the Euclidean length of the line segment y ∗ y divided by ω . Put differently, y is projected onto the boundary of Y (x) in the direction defined by ω. The resulting projection is at point y ∗ , whence y ∗ = y − O ω (y, x) ω. The basic properties of distance functions are described by (for a proof, see Chambers [7], Chapter 3): Theorem 1. If Y (x) satisfies GON in ω: (a) O ω (y, x) ≤ 0 ⇔ y ∈ Y (x) ⇔ x ∈ V (y) ⇔ (x, y) ∈ T (Indication); (b) for b ∈ R, O ω (y + bω, x) = O ω (y, x) + b (Translation); and (c) O μω (y, x) = μ−1 O ω (y, x) μ > 0 (Normalization). If V (y) satisfies GIN in ι: (a) I i (x, y) ≥ 0 ⇔ x ∈ V (y) ⇔ y ∈ Y (x) ⇔ (x, y) ∈ T ,(Indication); (b) for l ∈ R, I ι (x + lι) = I i (x, y) + l (Translation); and (c) I μi (x, y) = μ−1 I i (x, y) μ > 0 (Normalization). The Indication property ensures that our distance functions are complete function representations of T . That is, knowledge of them is equivalent to knowledge of the technology. An immediate implication is that either one can be used as a formal means of defining t (x, y), albeit under different regularity conditions. Mathematically, Translation is an arithmetic consequence of how the distance function is defined, and it has nothing to do with the regularity properties that we have imposed upon T . One visualizes it using Fig. 2. From the illustrated point y, y + bω for b real is illustrated by sliding y along the dotted line b units in the

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direction of the numeraire. Calculating O ω for that translated point carries one back to the same y ∗ as y, and the only difference in the calculation is the arithmetic correction for the translation amount. Put another way, Translation ensures that the distance calculation between, for example, y and Y (x) is invariant to translations of the origin in the direction of ω. An immediate consequence of Translation is, for example, I ι (x + ϕι, y) − I ι (x, y) =1 ϕ→0 ϕ lim

for all (x, y).9 Thus, differentially small adjustments of x in the direction of ι are matched by equivalent changes in I ι . Where Translation ensures invariance to translations of the origin in the direction of ω, Normalization ensures that rescaling axes of the numeraire rescale the resulting distance functions by the same proportion. Remark 2. Choosing ω to be y and ι to be x, respectively, yields O y (y, x) = min {β : y − βy ∈ Y (x)} = min {β − 1 : (1 − β) y ∈ Y (x)} + 1 = 1 − max {1 − β : (1 − β) y ∈ Y (x)} = 1 − g Y (y, x)−1 with g Y (y, x) = inf {γ > 0 : y ∈ γ Y (x)} , and I x (x, y) = max {λ : x − λx ∈ V (y)} = 1 − d V (x, y)−1 with d V (x, y) = sup {θ > 0 : x ∈ θ V (y)} . g Y is often called a radial output-oriented distance function,10 and d V a radial inputoriented distance function. An immediate consequence of Theorem 1 is that if T

9 In

formal terms, this is equivalent to I ι (x, y) being Gateaûx differentiable in the direction of x for all (x, y). 10 g Y is also often called a gauge function or a Minkowski functional for the set Y .

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satisfies WDO g Y (y, x) ≤ 1 ⇔ y ∈ Y (x) , and if T satisfies WDI d V (x, y) ≥ 1 ⇔ x ∈ V (y) , so that both satisfy a version of Indication. In principle, O ω , g Y , I ι , and d V all provide a suitable means of defining t. And, despite the claims made by some authors, there seems to be no reason to prefer one over any of the others for purposes of representing the technology. The situation is not unlike that faced by a carpenter or a mechanic. Depending upon the task at hand, they will use the most appropriate tool. One can, of course, use a hammer to drive a wood screw, but a deftly applied drill bit and screwdriver (spanner) work much more efficiently. Similarly, a pliers can be used to fasten and loosen bolts, but properly sized wrenches work even better. So too for our distance functions. In some cases, good reasons exist to prefer output-oriented ones to input-oriented ones, and just the reverse in other situations. This same reasoning applies to choosing the perspective for looking at T and a numeraire, a suitable choice will depend the problem to be solved. Along these lines, we have derived the output distance functions using Y (x) and the input distance functions using V (y) . It follows trivially, however, from Theorem 1 that we could have derived either directly from T , the former from V (y), or the latter from Y (x). Our derivation reflects traditional treatment and the natural bias that issues mainly involving outputs are best examined in output space and issues involving inputs are best examined in input space because they are most easily visualized, and thus illustrated, in those terms.11

11 Using

the netput representation,   T = z ∈ RN +M : z is technically feasible ,

one can also define a netput-distance function as N γ (z) = min {ψ ∈ R : z − ψγ ∈ T } if there exists ψ ∈ R such that z − ψγ ∈ T and ∞ otherwise that satisfies appropriate versions of Indication, Translation, and Normalization under GNN in the direction γ .

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Differential Properties of Distance Functions Pictorial depictions of isoquants, transformation curves, and graphs of production functions are fundamental to production economics. Any discussion of optimal producer behavior teems with references to marginal rates of technical substitution, marginal rates of transformation, and marginal products depicted as slopes of frontiers. One way to investigate the resulting concepts is to impose smoothness on an appropriate function representation of T , use the function representation to generate the appropriate boundary of T , and then use the calculus to investigate the slope and curvature of that boundary. Our approach to such matters relies less on mechanical calculation and more on viewing matters geometrically. Generally speaking, the idea is to start with an appropriate boundary of T . Because we assume T to be closed, little or no generality is lost in assuming that the boundary is for a closed set and hence belongs to it. We then examine what different aspects of the visual representations of the boundary imply about the differential properties of the function generating the boundary. The basic ideas can be demonstrated using either O ω or I ι . Going from one to the other requires slight changes in wording. We concentrate the visual and intuitive discussion on O ω and Y (x) and follow with a relatively terse discussion focused on I ι , leaving it to the interested reader to fill in the details. We start with O ω (y, x), given GON for ω,   T = (x, y) : O ω (y, x) ≤ 0 . Our particular interest is in the outer boundary for Y (x), which we identify with   Y¯ (x) = y : O ω (y, x) = 0 , and call the transformation curve. (We’re fudging here for the sake of easy intuition. We have yet to place enough structure upon T to ensure how to characterize the boundary of Y (x) properly.) Figure 3 illustrates the transformation curve as the curve connecting points A and B and Y (x) as all points falling below the transformation curve. For most economists, identifying Y (x) with points below the transformation curve surely will seem natural. But, this is precisely where FDO usually comes into play. FDO ensures GON in all directions, so that if points on AB belong to Y (x), so too must the points falling below it. Moreover, for y  ≤ y,

 ω ω ω  y − O (y, x) ω ≤ y − O (y, x) ω ∈ Y (x) so that O y , x ≤ O ω (y, x) is nondecreasing under FDO, which with Indication is another way of saying that any point falling below the transformation curve must belong to Y (x). FDO also ensures that the slope of the transformation curve must not be a strictly positive real number. Intuitively, increasing one output prompts an increase in O ω , and to balance it, one needs another output to decrease to return production feasibility. The slope of the transformation curve that reflects this principle is

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Fig. 3 Transformation curve A

C

D Y(x)

E B

y1

traditionally called the marginal rate of transformation. A negative value ensures that outputs, when viewed as pairs, must be substitutes in production.12 Figure 3 depicts a smooth transformation curve. The marginal rate of transformation is obtained by finding a tangent to Y¯ (x) or by implicitly differentiating O ω (y, x) when the latter is smooth. Figure 3 has been drawn so that at C, D, and E, the marginal rate of transformation is illustrated by the slope of the parallel dotted tangent line segments implying the same marginal rate at C, D, and E. This has been done to illustrate the connection between a smoothly differentiable structure and a more general differential representation that determines whether points on the transformation curve are economically relevant in a sense to be made precise. From technical and mechanical perspectives, points C, D, and E appear similar. They each satisfy O ω (y, x) = 0, and they share a common marginal rate of transformation. But they represent different output mixes. And looking more closely, D is located on a portion of Y¯ (x) that is convex to the origin, while C and E are located on portions that are concave to the origin. In production economics jargon, the technology exhibits a diminishing marginal rate of transformation at D and an increasing marginal rate of transformation at C and E. If the slopes of the dotted line segments were to reflect relative output prices, D would be judged economically inferior to C and E because less revenue can be obtained from D than from the latter two. Using the same criterion, however, C is judged inferior to E. What Fig. 3 illustrates is that distinct points can be closely similar from the mechanical calculus perspective in which we frequently discuss economic concepts but drastically different economically. They can be on the frontier, and they can have equal marginal rates of transformation, and marginal rates of transformation

12 Good

reasons exist to believe that this is not always true. In other words, FDO might be too strong a restriction (hence our insistence upon the presence of GON ). Perhaps the best example is given by pollutants that have complementary relationships with many outputs. The  Chapter 12, “Bad Outputs” discusses such concerns in detail.

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increasing at the same rate, but still differ in terms of their economic efficiency.13 Differentiability is often intuitive and analytically convenient for those practiced in calculus manipulation for identifying potential optima. But, in the final analysis, something else is required. As it turns out, a one-sided differential notion that slightly generalizes the geometric association of the gradient with the hyperplane tangent to the function’s level set suffices. Point E in Fig. 3 illustrates. Letting the dotted line segments C E E 1 represent relative output prices −p p2 , point E satisfies p1 y1 + p2 y2 > p1 y1 + p2 y2C > p1 y1D + p2 y2D and moreover that p1 y1E + p2 y2E ≥ p1 y1 + p2 y2 for all (y1 , y2 ) ∈ Y (x). A simple tangency between the hyperplane with normal

(p1 , p2 ) that passes through y1E , y2E and Y¯ (x) won’t satisfy this criterion (points C and D both illustrate). What’s required in addition to tangency is that the associated hyperplane associated supports Y (x) from above so that Y (x) falls in the half-space below

the tangent hyperplane. Economically, that means for prices (p1 , p2 ), y1E , y2E must be at least as valuable as any other output combination falling in Y (x).

We can write the desired criterion for y1E , y2E as (p1 , p2 ) satisfying       O ω (y, x) − O ω y E , x ≥ p1 y1 − y1E + p2 y2 − y2E for all (y1 , y2 ) ∈ R2 .

To see why this always works, recall that O ω y E , x = 0 and that for any (y1 , y2 ) ∈ Y (x), O ω (y, x) ≤ 0, whence     0 ≥ O ω (y, x) ≥ p1 y1 − y1E + p2 y2 − y2E , or     p1 y1E − y1 + p2 y2E − y2 ≥ −O ω (y, x) ≥ 0, as desired. More generally, if y E is to satisfy such a criterion, there must exist a p ∈ RM such that14     O ω (y, x) − O ω y E , x ≥ p y − y E for all y ∈ RM .

13 This

is another way of saying that location, first-order, and second-order conditions can be met without truly identifying the true optimum. Note that at C second-order conditions for an interior optimum are satisfied, but C remains non-optimal. 14 Here and elsewhere p  for p ∈ RM denotes the transpose of an M-dimensional column vector, and p  y for p, y denotes the standard inner product.

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Fig. 4 Subdifferentiable but not smooth A

C

Y(x)

B

y1

A p ∈ RM satisfying this criterion is called a subgradient (in y) of O ω (y, x) at y E . In Fig. 3, all the parallel dotted line segments can be identified with gradients of O ω (y, x) at C, D, and E, but only the one at E defines a subgradient. When such a p exists, O ω is said to be subdifferentiable at y E .15 Because derivatives are typically defined via limits, gradients describe point-topoint mappings. Subdifferentiability is not defined in terms of limiting behavior but in terms of a global condition that must apply at a particular point. That definition permits the existence of multiple subgradients at a point. Figure 4 illustrates with point C possessing multiple supporting hyperplanes for Y (x). The subdifferential correspondence ∂O ω : RN +M ⇒ RM is defined at y E , x by    p ∈ RM : O ω (y, x) − O ω y E , x ≥ p y − y E  E . ∂O y , x = for all y ∈ RM ω

Despite the fact that gradients and subgradients are both intuitively identified with tangent hyperplanes, one does not imply the other. Numerical functions can be differentiable at point but not subdifferentiable (see C in Fig. 3) or subdifferentiable but not differentiable (see C in Fig. 4). The key distinction is that subdifferentiability directly reflects global optimality16 , while differentiability does not. This distinction is highlighted by noting that     p ∈ ∂O ω y E , x ⇔ O ω (y, x) − p y ≥ O ω y E , x − p y E for all y ∈ RM , whence 15 The “sub” terminology arises from (y, O ω (y, x)) falling in the half-space lying above the affine hyperplane       (y, O) ∈ RM+1 : O = O ω y E , x + p  y − y E for all y ∈ RM .

16 In

the sense of minimization

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Fig. 5 Convex Y(x) A

F

Y(x)

G

B

y1

    0 ∈ ∂O ω y E , x ⇔ O ω (y, x) ≥ O ω y E , x for all y ∈ RM . Smooth numerical functions are not generally subdifferentiable. On the other hand, as Fig. 4 illustrates non-smooth functions can be subdifferentiable but not differentiable. Examining Figs. 3 and 4 gives us a hint at what’s required to ensure subdifferentiability. At both points of subdifferentiability, the transformation curve is (locally) concave to the origin. But point C in Fig. 3 also demonstrates that local concavity does not ensure subdifferentiability. What’s needed is envisioned by taking the transformation curve in Fig. 3 and somehow eliminating the salient between points C and E that points toward the origin. Figure 5 depicts this operation as pushing the inward pointing salient from Fig. 3 northwest toward the dotted line segment until it is replaced by the line segment connecting points F and G. The resulting modified transformation curve AFGB is subdifferentiable everywhere on the interior of the positive orthant. The resulting transformation curve is illustrated visually by a contour that is everywhere concave toward to the origin, and the modified Y (x) is a convex set.17 As the visual argument suggests, convexity of Y (x) suffices to ensure that O ω is convex in y and subdifferentiable as required. We formalize this visual intuition with: Theorem 2. If T satisfies F DO and Y (x) is a convex set for all x ∈ RN , O ω (y, x) is nondecreasing and convex in y, ∂O ω (y, x) = ∅ for all y in the relative interior of {y : O ω (y, x) < ∞}, and O ω (y, x) is differentiable almost everywhere in the relative interior of {y : O ω (y, x) < ∞}. Figure 5, with the inner-directed salient removed, depicts a “normal”-looking transformation curve that is an “upper bound” for a nicely convex Y (x). Our discussion reveals that what’s considered “normal” is one consequence of requiring an everywhere subdifferentiable O ω . In other words, our most common visualization of an output set embeds subdifferentiability in it. This ensures that for any boundary

17 More

formally, the modified set is the convex hull of Y (x).

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points on Y¯ (x), one can find a suitable set of prices that will make it optimal. In a more general setting that does not correspond to our stylized representations, subdifferentiability enables discrimination between boundary points that are economically rational and ones that are not. In economics jargon, subdifferentiability is the differential condition required to ensure a point on the transformation curve can be identified with an output “price” vector that makes the point revenue maximal, what economists refer to as a “shadow” or “virtual” price vector. The identification of subdifferentiability with shadow prices is formalized and strengthened by the following result (see, e.g., Chambers [7], Chapter 3):  Theorem 3. Under FDO, p ∈ ∂O ω (y, x) ⇒ p ∈ RN + and p ω = 1 for all ω y ∈ {y : ∂O (y, x) = ∅}, and



po ∈ ∂O ω y o , x ,





poo ∈ ∂O ω y o , x ⇒ p oo − po y oo − y o ≥ 0.

(cyclical monotonicity)

for all y, y o ∈ {y : ∂O ω (y, x) = ∅} Theorem 3 establishes that F DO and subdifferentiability of O ω guarantee the existence of non-negative shadow price vectors that price the numeraire bundle at 1. Moreover, these shadow prices are cyclically monotone with their associated output bundles. Cyclical monotonicity of shadow prices generalizes the concept of an increasing marginal rate of transformation between outputs to higher dimensions. Cyclical monotonicity is perhaps best understood as a generalization of the univariate notion of monotonicity. For example, N = 1, cyclical monotonicity of ∂O ω (y, x) implies that the subdifferential with respect to y is nondecreasing in y, which also requires O ω (y, x) to be convex in y. Turning to the differential properties of I i (x, y), we first note that replacing “transformation curves” with “isoquants,” “marginal rate of transformation” with “marginal rate of substitution,” and “increasing rate of marginal transformation” with “decreasing marginal rate of substitution” provides the needed jargon. Then, if one uses −I i (x, y), replaces F DO with F DI , and recycles arguments made in terms of y in terms of x, the extension is straightforward. Subdifferentiability of O ω in y is replaced with subdifferentiability of −I i in x,18 maximization of revenue with minimization of cost, and concave to the origin transformation curves with convex to the origin isoquants that are consistent with F DI .

18 Subdifferentiability

of −I i in x is equivalent to superdifferentiability of I i in x.

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Distance Functions at Work Distance functions play a central role in many areas of economics including producer analysis, consumer theory, efficiency analysis, index number theory, and equilibrium computation among others. This section presents an overview of the uses of distance functions in duality theory, efficiency measurement, index number theory, and productivity measurement. As with the discussion of the differential properties of distance functions, the basic ideas can be demonstrated using outputor input-based concepts. Going from one to the other requires slight changes in wording. When relevant, we concentrate the discussion on V (y) and leave the extension to Y (x) to the reader.

Duality Theory The duality terminology is borrowed from mathematics, where the dual of a primal vector space X is identified as the space of the linear functionals on X. Using the traditional Euclidean norm, the linear functions of elements of RN , x ∈ RN , assume the form l  x with l ∈ RN so that the space of linear functionals of RN is itself, RN . Put another way, RN is self-dual. Because our exclusive focus is on real numbers, what is primal and what is dual is, therefore, often ambiguous and depends importantly upon how axes are labelled. Economists typically resolve this ambiguity by invoking the aphorism that “primal means things you can eat, and dual things you cannot.”19 Thus, RN conceived of as measuring quantities of physical commodities is typically identified with the primal space, and RN consisting of the linear functions of quantities associated with prices that value these quantities is the dual space. The core notion of duality in production economics is that technologies can be characterized either in primal or dual terms. And under appropriate restrictions upon T the primal characterization and the dual characterization are equivalent.20 To our knowledge, the first complete demonstration of a dual relationship was Shephard [21] who demonstrated that a radial input-oriented distance function and a cost function formed a dual pair. We now quickly illustrate why this is true using our set up. Letting input prices N M ¯ be denoted w ∈ RN ++ , the cost function, c : R++ × R → R for V (y) is defined:   c (w, y) = min w  x : x ∈ V (y) x

19 We

(5)

attribute this Professor Shawna Grosskopf, and even if it is bit imprecise because primal also contains items such as coal, it nicely conveys the general idea. 20 Please see Chapter 3 of Volume 1 of this Handbook for a thorough discussion of duality theory.

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if V (y) = ∅ and +∞ otherwise. As is well-known, the cost function is nondecreasing, positively homogeneous, and concave in w. Moreover, it can be recognized as the (lower) support function for V (y). Thus, if V (y) satisfies F DI and is convex as a subset of RN + , a well-known result in convex analysis is that (Rockafellar [20])    N V (y) = x ∈ RN : w x ≥ c y) for all w ∈ R (w, + ++ .

(6)

Together, expressions (5) and (6) are referred to as a dual pair. They show that knowledge of V (y) is sufficient to generate c (w, y) for given input prices and that knowledge of c (w, y) is sufficient to generate V (y). As a result, any information that is imbedded in the primal technology, V (y), can be recaptured from its dual c (w, y) provided V (y) satisfies F DI and is convex. Conversely, information imbedded in the dual c (w, y) can be recaptured from V (y). In this sense, c (w, y) and V (y) are informationally equivalent. In the words of McFadden [19, p.4], the cost function is “. . . a sufficient statistic for all the economically relevant characteristics of the underlying technology.” Input distance functions offer another perspective on this duality that can be analyzed using simple optimization arguments. If V (y) satisfies F DI , x − I ι (x, y) ι ∈ V (y) , then

N w  x − I ι (x, y) ι ≥ c (w, y) for all x ∈ RN + , w ∈ R++ . Using the positive homogeneity of c (w, y) and rearranging gives

∗ N w ∗ x ≥ I ι (x, y) + c w ∗ , y for all x ∈ RN + , w ∈ R++

(Fenchel’s inequality), (7)

where w ∗ = ww ι represents real input prices normalized by the value of the numeraire bundle. Holding x fixed, Fenchel’s inequality implies

w ∗ x − c w ∗ , y ≥ I ι (x, y) for all w ∗ ∈ RN ++ , while holding w∗ fixed gives

w ∗ x − I ι (x, y) ≥ c w ∗ , y for all x ∈ RN +. Thus, I ι (x, y) provides a lower bound for all affine functions of x describable as ∗ w ∗ x − c (w ∗ , y) for w ∗ ∈ RN ++ , and c (w , y) provides a lower bound for all ∗ ∗ affine functions of w describable as w x − I ι (x, y) for x ∈ RN + . An immediate consequence is that

7 Distance Functions in Production Economics

 inf

w∗ ∈RN ++

317



 w ∗ x − c w ∗ , y ≥ I ι (x, y) , and

 

min w ∗ x − I ι (x, y) ≥ c w ∗ , y .

x∈RN +

(8)

The left-hand sides of the inequalities in (8) are, respectively, the concave conjugates of c (w ∗ , y) and I ι (x, y) (Rockafellar [20]). When V (y) satisfies F DI and is convex as a subset of RN + , these inequalities convert to equalities to form a conjugate dual pair (Rockafellar [20]) I ι (x, y) =

 inf

w∗ ∈RN ++



 w ∗ x − c w ∗ , y , and

(9)

 

c w ∗ , y = min w ∗ x − I ι (x, y) . x∈RN +

An immediate consequence is the following version of Shephard’s Lemma:

x ∈ ∂c w ∗ , y ⇔ w ∗ ∈ ∂I ι (x, y) , where, for example, ∂c (w, y) now denotes the superdifferential of c (w, y) in w.21 In words, the efficient solution to the cost minimization problem belongs to the superdifferential of the cost function and the efficient solution to the dual formulation of the distance-function problem belongs to the superdifferential of I ι .

Efficiency Analysis Another important use of distance functions is to measure technical efficiency. Recall, for example, the radial input-oriented distance function d V (x, y) = sup {θ > 0 : x ∈ θ V (y)} . Because

−1 d V (x, y) = 1 − I x (x, y) , its effective numeraire is x. Moreover, if x ∈ V (y), d V (x, y) ≥ 1, and under WDI, the converse is true. One important interpretation of d V (x, y) is as a measure of technical efficiency for producing y (Debreu [10], Farrell [14]). If d V (x, y) = 1, x might be called technically efficient relative to V (y) because x cannot be shrunk radially and stay

21 Recall

the superdifferential of c is the subdifferential of −c.

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in V (y). On the other hand, if d V (x, y) > 1, x “might” be called technically x belongs to V (y). inefficient because the radially shrunken d V (x,y) We’ve employed the “might” qualification to emphasize that precise definitions of technical efficiency are necessarily entangled with how the “boundary” for V (y) is characterized. To illustrate, recall that we have previously referred to V (y) s boundary as an isoquant. Intuitively, its name is meant to connote the intermediate micro intuition that it represents the input bundles that can produce the same output. But that definition also describes V (y), so more precision is needed. In particular, we want to exclude points falling in V (y) but not on its boundary. That intuitive idea can be accomplished in a number of different ways. One popular alternative is to identify the isoquant correspondence, V¯ R : RM ⇒ RN , via / V (y) , λ < 1} . V¯ R (y) = {x ∈ V (y) : λx ∈ Another alternative is   / V (y) . V¯ L (y) = x ∈ V (y) : x  ≤ x ⇒ x  ∈ Because λx ≤ x for λ < 1, one can show V¯ L (y) ⊆ V¯ R (y). The converse is not true because V¯ L (y) can exclude points falling in V¯ R (y). The Leontief production function f (x) = min {x1 , x2 } illustrates. For that case, V¯ L (y) = {(y, y)} , is a singleton set. V R (y), on the other hand, is represented as an L−shaped isoquant emanating upward from (y, y) with its vertical “arm” extending parallel to the vertical axis and its horizontal “arm” extending outward and parallel to the horizontal axis. Hence, x ∈ V¯ L (y) ⇒ d V (x, y) = 1, ¯R

d (x, y) = 1 ⇔ x ∈ V (y) , V

and (10)

so that points can be declared technically efficient according to the Debreu-Farrell criterion but not fall on the “isoquant” V¯ L (y). Expressions (10) illustrate the close connection between notions of technical efficiency and different versions of GIN. Choosing V¯ R (y) to be the boundary concept associates technical efficiency with radial shrinkage of x and naturally links technical efficiency judgments to GIN for all x ∈ V (y). V¯ L (y), on the other hand, associates technical inefficiency with being able to decrease any single input and, N \ {0}. Because thus, links technical efficiency comparisons to GIN for all ι ∈ R+ L R ¯ ¯ the latter is a more stringent criterion, V (y) ⊆ V (y). More generally, different boundary notions derived from either O ω or I ι by varying either ω or ι will lead to definitions of technical efficiency that are specific to the choice of the numeraire.

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Expressions (10) also illustrate that input bundles, which could never be economically efficient, can be technically efficient according to some criteria. Again the Leontief technology, y = min {x1 , x2 }, illustrates. As long as input prices are positive, the only economically efficient point in V¯ R (y) is (y, y). Economists are usually more interested in economic efficiency than in technical efficiency. And economic efficiency is often defined in dual terms. A natural measure of cost efficiency for a given input bundle x ∈ V (y) is the ratio of minimal cost to observed cost c (w, y) . w x If x ∈ V (y) and c(w,y) = 1, x is judged cost efficient. If c(w,y) < 1, x is cost w x w x inefficient because costs can be reduced while still producing the same output. x Because it is definitionally true that d V (x,y) ∈ V (y), it follows for x ∈ RN ++ and V (y) ⊂ RN ++ that w  x ≥ c (w, y) d V (x, y) .

(Mahler’s inequality)

(11)

Mahler’s inequality is the multiplicative manifestation of the same economic phenomenon lying behind Fenchel’s inequality. Farrell [14] suggested closing the Mahler inequality by defining allocative efficiency residually as the ratio of the righthand side divided by the left. That is, a V (x, w, y) ≡

c (w, y) d V (x, y) . w x

The input bundle x is allocatively efficient for (w, y) if a V (x, w, y) = 1 and inefficient otherwise. If FDI is maintained and V (y) is assumed to be a convex set, Mahler’s inequality forms the basis for yet another dual pair: 

   w x  v w = min x : d y) ≥ 1 , (x, x x d V (x, y)     w x = min w  x : c (w, y) ≥ 1 . d V (x, y) = min w w c (w, y) c (w, y) = min

(12)

Moreover, convexity of V (y) ensures the following version of Shephard’s Lemma x ∈ ∂c (w, y) ⇔ w ∈ ∂d V (x, y) . The Farrell [14] definitions of technical, cost, and allocative efficiency depend upon the chosen numeraire. Choosing different numeraire results in different notions

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of technical, cost, and allocative efficiency (Chambers, Chung, and Färe [9]). Recalling that x − I ι (x, y) ι ∈ V (y) implies w  x − I ι (x, y) w  ι ≥ c (w, y) , whence

w  x − c (w, y) = w ∗ x − c w ∗ , y ≥ I ι (x, y) .  w ι

(13)

Following Chambers, Chung, and Färe [9], a difference-based measure of allocative efficiency is then determined residually to “balance” Fenchel’s inequality as



w ∗ x = Ai w ∗ , x, y + c w ∗ , y + I ι (x, y) . To recycle this discussion for measuring output inefficiency, replace V (y) with Y (x), “isoquants” with “transformation curves,” and x technically efficient for y with y technically efficient for x provides the needed jargon. Then, replacing W DI with W DO, F DI with F DO, w ∈ RN ++ with strictly positive output prices p ∈ RM ++ , and recycling arguments made in terms of x in terms of y will proved the needed extension.

Index Numbers and Productivity Measurement Konüs ([1924], [15])22 proposed using cost or expenditure functions to represent price indexes. Following that contribution, Sten Malmquist [18], following the basic logic of duality, proposed using distance functions to represent quantity indexes. Malmquist [18], who was working in the context of consumer theory, phrased things in slightly different terms than ours. But the translation is easy enough. Simply equate Malmquist’s commodities with our inputs and his indifference level with our output (now treated as a scalar). o For the commodity bundles, x o , x 1 ∈ RN ++ , and the indifference level, y , Malmquist [18, p. 230] defined his quantity index  o

 x y = min μ > 0 : μx 1 ∈ V y o Io1

22 Konüs

[15] is a translation of a paper originally published by Konüs in Russian in 1924.

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 −1 = d V x1, yo , with x o chosen so that d V (x o , y o ) = 1. Hence, more generally, o d V (x o , y o ) x Io1 y = V 1 o . d x ,y x x Translated into inputs and outputs, either I01 (y o ) or its reciprocal I01 (y o )−1 defines a Malmquist input index (Caves, Christensen, and Diewert [3]) that relates input x bundles x o , x 1 in terms of their ability to produce the output y o . If Io1 (y o ) < 1, x 1 x o must be radially expanded to be capable of producing output y , and if Io1 (y o ) > 1, o it can be radially shrunk. Thus, in this sense, x would be judged “larger” than x 1 in the first instance and smaller in the second. The Malmquist input index effectively ranks x o and x 1 in terms of their ability to produce y o . If one instead uses y 1 , one obtains

x Io1

  d V x o , y 1

y1 = V 1 1 . d x ,y



x y 1 = I x o In general, Io1 o1 (y ) so that the two indexes typically differ. This is a familiar problem in index number theory. Namely, indexes always depend upon the “reference” situation. Here the reference is determined by the level at which the output, y, is set. A common way23 to address this indeterminacy is to take geometric averages to arrive at the following input index   x yo, y1 = I˜o1



1

d V x o , y 1 d V (x o , y o ) 2



. d V x1, y1 d V x1, yo

Exactly parallel arguments on the output side suggest defining a Malmquist output index (Caves, Christensen, and Diewert [3]) as   y

Oo1 x o = max μ > 0 : μy 1 ∈ Y x o  −1 = gY y 1, x o with y o chosen so that g Y (y o , x o ) = 1, whence

23 Other

ways exist. For example, one could simply choose an arbitrary y that is neither y o nor y 1 as the reference. That resolves the dilemma of choosing either o or 1 as the basis, but it does not resolve its arbitrariness.

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g Y (y o , x o ) y

Oo1 x o = Y 1 o . g y ,x Again, either Oo1 (x o ) or its reciprocal, Oo1 (x o )−1 , defines an index. But now the index relates y o and y 1 in terms of x o s ability to produce them. If y Oo1 (x o ) 0 : x ∈ λK} . It has played a central role in both functional analysis and convex analysis (e.g., Rockafellar [20]). Both input-oriented and output-oriented radial distance functions are versions of Minkowski functionals. Debreu [10], Shephard [21], and Malmquist [18] seem to be independently responsible for introducing versions of the Minkowski functional into economics. Debreu [10] used it to define a coefficient of resource utilization. Shephard [21] used an input-oriented version to derive a dual relationship between a singleproduct technology and its cost function. And Malmquist [18], explicitly reflecting an older argument that he attributes to Könus [15], used it to define a quantity index. Somewhat later, Shephard [22] used a Minkowski functional to define an output-oriented radial distance function. Färe and Lovell [12] recognized the formal connection between input-oriented radial distance functions and the Farrell [14] efficiency score and showed that input-oriented and output-oriented distance functions for a technology are reciprocals if and only if the technology satisfies constant returns to scale. The output-oriented and input-oriented distance functions, which explicitly recognize the use of a numeraire, can be attributed to Blackorby and Donaldson’s [2] translation function as used in inequality measurement and to Luenberger’s [16,17] benefit function and shortage function as used in equilibrium characterization. But as Luenberger [16] points out, the benefit function generalizes Allais’s [1] even older idea of disposable surplus. Their use to characterize technological structures is due to Chambers, Chung, and Färe [8], who also explicitly recognized them as generalized versions of radial distance functions.

Cross-References  Bad Outputs  Duality in Production  Multiproduct Technologies

References 1. Allais M (1943) Traité d’Économie pure, vol 3. Imprimerie Nationale, Paris 2. Blackorby C, Donaldson D (1980) A theoretical treatment of indices of absolute inequality. Int Econ Rev 21(1):107–136 3. Caves DW, Christensen LR, Diewert WE (1982) Multilateral comparisons of output, input, and productivity using superlative indexes. Econ J 92:73–86

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4. Chambers RG (1996) Computable profit as a superlative technical change, input, output, and productivity measure. Department of Agricultural and Resource Economics, The University of Maryland, College Park, pp 96–01 5. Chambers RG (1998) Input and output indicators. In: Färe R, Grosskopf S, Russell RR (eds) Essays in honor of Sten Malmquist. Kluwer Academic Publishers, Boston 6. Chambers RG (2002) Exact nonradial input, output, and productivity measurement. Econ Theory 20:751–767 7. Chambers RG (2020, forthcoming) Competitive agents in certain and uncertain. Oxford University Press, Oxford 8. Chambers RG, Chung Y, Färe R (1996) Benefit and distance functions. J Econ Theory 70: 407–419 9. Chambers RG, Chung Y, Färe R (1998) Profit, directional distance functions, and nerlovian efficiency. J Optim Theory Appl 98:351–364 10. Debreu G (1951) The coefficient of resource utilization. Econometrica 19(3):273–292 11. Debreu G (1959) The theory of value. Yale University Press, New Haven 12. Färe R, Lovell CAK (1978) The structure of technical efficiency. J Econ Theory 19:150–162 13. Färe R, Lundberg A (2006) mimeo 14. Farrell MJ (1957) The measurement of productive efficiency. J R Stat Soc 129A:253–281 15. Konüs AA (1939) The problem of the true index of the cost of living. Econometrica 7:10–29 16. Luenberger DG (1992a) Benefit functions and duality. J Math Econ 21:461–481 17. Luenberger DG (1992b) New optimality principles for economic efficiency and equilibrium. J Optim Theory Appl 75(2):221–264 18. Malmquist S (1953) Index numbers and indifference surfaces. Trabajos De Estadistica 4: 209–242 19. McFadden D (1978) Cost, revenue, and profit functions. In: McFadden D, Fuss M (eds) Production economics: a dual approach to theory and applications. North Holland, Amsterdam 20. Rockafellar RT (1970) Convex analysis. Princeton University Press, Princeton 21. Shephard RW (1953) Cost and production functions. Princeton University Press, Princeton 22. Shephard RW (1970) Theory of cost and production functions. Princeton University Press, Princeton

8

Stochastic Frontier Analysis: Foundations and Advances I Subal C. Kumbhakar, Christopher F. Parmeter, and Valentin Zelenyuk

Contents Introduction and Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The Benchmark SFM . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The Distribution of ε . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Alternative Specifications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Estimation of Individual Inefficiency . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Do Distributional Assumptions Even Matter? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Finite Sample Identification of Inefficiency . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Handling Endogeneity in the SFM . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A Corrected Two-Stage Least Squares Approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A Likelihood Approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A Method of Moments Approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Estimation of Individual Inefficiency . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . An Economic Approach to Deal with Endogeneity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Modeling Determinants of Inefficiency . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Proper Modeling of the Determinants of Inefficiency . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Incorporating Determinants When u Is Truncated-Normal . . . . . . . . . . . . . . . . . . . . . . . . . The Scaling Property . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Estimation Without Imposing Distributional Assumptions . . . . . . . . . . . . . . . . . . . . . . . . . Estimation When Determinants of Efficiency and Endogeneity Are Present . . . . . . . . . . .

332 335 336 338 342 346 347 349 350 350 352 353 354 355 357 359 360 361 363

S. C. Kumbhakar () Department of Economics, State University of New York at Binghamton, Binghamton, NY, USA Inland Norway University of Applied Sciences, Lillehammer, Norway e-mail: [email protected] C. F. Parmeter Department of Economics, University of Miami, Miami, FL, USA e-mail: [email protected] V. Zelenyuk School of Economics and Centre for Efficiency and Productivity Analysis (CEPA), The University of Queensland, Brisbane, QLD, Australia e-mail: [email protected] © Springer Nature Singapore Pte Ltd. 2022 S. C. Ray et al. (eds.), Handbook of Production Economics, https://doi.org/10.1007/978-981-10-3455-8_9

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Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Cross-References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

365 365 366

Abstract

This chapter (as well as Chap. 11) reviews some of the most important developments in the econometric estimation of productivity and efficiency surrounding the stochastic frontier model. Keywords

Efficiency · Productivity · Panel data · Endogeneity · Nonparametric · Determinants of inefficiency · Quantile · Identification

JEL Classification

C10, C13, C14, C50

Introduction and Overview The primary goal of this and the next chapters is to introduce the wide audience of this Handbook to the range of methods, developed over the last four decades, within one of the most popular paradigms in modern productivity analysis – the approach called Stochastic Frontier Analysis, often abbreviated as SFA. The first, and one of the most important, question a reader might wonder about is why a researcher on productivity should ever care about SFA in general and especially about the enormous variety of different types of SFA models that have been proposed over the last four decades. Our goal in writing these two chapters was to provide the reader with a good answer to this important question. Here, we strive to outline the essence of major types of SFA methods, providing the minimal but most essential details and focusing on advantages and disadvantages of each method for dealing with various aspects that arise in practice. We hope that upon finishing reading these chapters, the reader who is barely or even unfamiliar with SFA obtains a general understanding of the importance and relevance of different SFA methods, along with useful/key references for further details on each method. Of course, the reader also deserves to receive a quick answer now, to decide if it is worth it for a productivity researcher to read these chapters further – we try to give such a quick answer in this section. The Nobel Laureate Paul Krugman was hardly exaggerating when he once quipped that “Productivity isn’t everything, but in the long run, it’s almost everything.” The root of this statement can be seen when looking at various theoretical models of economic growth, e.g., starting from Solow’s growth model, the related

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variations of more advanced growth theory models or empirical growth accounting approach to productivity measurement, as well as from the more sophisticated measurements of productivity. Regardless of how the productivity is measured, it is inevitably tied to measuring production relationships. Such relationships are usually modeled through the so-called production functions or, more generally, transformation functions (e.g., Shephard’s distance functions, directional distance functions), cost functions, etc. In the classical growth accounting approach [95], all the variation in growth apart from the variation of inputs is attributed to the socalled Solow’s residual, which under certain restrictions measures what is referred to as the change in total factor productivity (TFP). A well-known problem of simple growth accounting is that it piles up and hides many sources for growth, the most obvious of which is the statistical error. Standard regression methods can and are often used to, basically, estimate average relationships conditional on various factors (inputs, demographic and geographic factors, etc.) to filter out the effect of statistical noise. All the deviations from the estimated regression curves in such approaches are attributed to the statistical error, and all the decision-making units (DMUs) represented in the data as observations (e.g., firms, countries, etc.) are typically assumed to be fully efficient or on the frontier of the production relationship. Such full efficiency assumption certainly simplifies the measurement complexity, but is it really an innocent assumption? Indeed, while many economic models admit the assumption that all firms are efficient, the reality that one observes in practice usually suggests there are reasonable amounts of inefficiency in this world. Such inefficiencies could arise, for example, because of asymmetric information or, more generally, the problem of incomplete markets (e.g., see [97]), which to some extent are present almost in every aspect of our lives. Differences in inefficiencies (or in relative productivity levels)1 across firms or countries can also arise due to different managerial practices (e.g., see [19]), which could in turn be implied by the asymmetric information problem, different cultural beliefs, traditions, and expectations [17]. Does accounting for such inefficiency matter for productivity measurement? Vast literature on the subject suggests that it indeed often matters substantially, as has been documented in thousands of articles in the last four decades. The difference is in the approach – SFA, data envelopment analysis (DEA), free disposable hull (FDH), etc. – and the goal of this and the next chapters is to give a sense of a few major approaches within the SFA paradigm. In a nutshell, the main premise of the SFA approach is a recognition that whether all DMUs are efficient or not is an empirical question that can and should be statistically tested against the data, while allowing for a statistical error. To 1 Despite the variety of definitions, intuitively, production efficiency can be understood as a relative

measure of productivity. In other words, production efficiency is a productivity measure that is being normalized (e.g., to be between 0 and 1 to reflect percentages) relative to some benchmark, such as the corresponding frontier outcome, optimal with respect to some criteria: e.g., maximal output given certain level of input and technology in the case of technical efficiency or minimal cost given certain level of output and technology in the case of cost efficiency.

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enable such testing, the SFA approach provides a framework where the production relationship is estimated also as a conditional average (of outputs given inputs and other factors, in the case of production function), but the total deviation from the regression curve is decomposed into two terms – statistical noise and inefficiency. Both of these terms are unobserved by a researcher but with relatively mild assumptions the different approaches within SFA allow the analyst to estimate them for the sample as a whole (e.g., representing an industry) or for each individual DMU. Importantly, SFA approach also allows for the inefficiency term to be statistically insignificant, if the data might suggest so, thus encompassing the classical approach with a naive assumption of full efficiency as a special case and, importantly, allowing for this assumption to be tested. Moreover, the SFA approach also contains the other extreme where one assumes no statistical noise with all the deviations treated as inefficiency to the frontier. Thus, the SFA approach is a natural compromise between approaches which make two extreme assumptions, yet also encompassing them as special cases, which can still be followed if the data and the statistical tests from SFA would not recommend otherwise. If the tests support (or at least cannot reject) the full efficiency hypothesis, then one can proceed with the standard regression techniques, or even with Solow’s growth accounting, but if not then accounting for possible inefficiency could be critical for both quantitative and qualitative conclusions and, perhaps more importantly, for the resulting policy implications. Indeed, if statistical tests reject the hypothesis of full efficiency of DMUs, then it can be imperative to decompose the productivity (be it Solow’s residual or any other productivity measure) further – to estimate the inefficiency component for the sample (e.g., representing an industry) and for each individual DMU. Moreover, SFA also provides a framework to analyze the sources of production inefficiency and the variation of productivity levels, both of which can give important insights into how to reduce inefficiency and increase productivity. We discuss these interesting and important issues in this and the next chapters, while some of the stylized facts we present here can be also found in previous reviews [38, 59, 61, 70, 79], and it is impossible to give a good review without following them to some degree, here we also summarize many of (what we believe to be) the key recent developments as well as (with their help) shed some novel perspectives onto the workhorse methods. So, all in all, our belief is that there is much value added for the reader to complement what was done well in earlier reviews on this theme. The rest of the chapter is structured as follows: sections “The Benchmark SFM”, “Handling Endogeneity in the SFM”, and “Modeling Determinants of Inefficiency” focus on stochastic frontier models (SFM) for cross-sectional variation in efficiency (relative productivity), where section “The Benchmark SFM” covers the foundation laid by [4] and some closely related research, with section “Handling Endogeneity in the SFM” discussing endogeneity issues and section “Modeling Determinants of Inefficiency” focuses on modeling the determinants of inefficiency, while section “Conclusions” concludes.

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The Benchmark SFM One of the main approaches to study productivity and efficiency of a cross section of firms is the SFM, independently proposed by [4] (ALS hereafter) and [72] (MvB hereafter).2 Using conventional notation, let Yi be the single-output for observation (e.g., firm) i and let yi = ln(Yi ). The SFM can be written for a production frontier3 as yi = m(x i ; β) − ui + vi = m(x i ; β) + εi .

(1)

Here m(x i ; β) represents the production frontier of a firm (or more generally a DMU), with given input vector x i . Our use of β is to clearly signify that we are parametrically specifying our production function.4 The main difference between a standard production function setup and the SFM is the presence of two distinct error terms in the model. The ui term captures inefficiency, shortfall from maximal output dictated by the production technology, while the vi term captures stochastic shocks. The standard neoclassical production function model assumes full efficiency – so the SFM embraces it as a special case, when ui = 0, ∀i, and allows the researcher to test this statistically.5 One shortcoming of the benchmark SFM is that the appearance of inefficiency in (1) lacks any specific structural interpretation. Where is inefficiency coming from? It could stem from inputs being used sub-optimally: workers may not put forth full effort or capital may be improperly used, e.g., due to asymmetric information or other reasons hidden to the researcher or even the firm. Without a specific structural link, it is difficult to know just how to treat inefficiency in (1). Thus, to estimate the model, several assumptions need to be imposed. First, it is commonly assumed that inputs are independent of u and v, ui ⊥ x and vi ⊥ x ∀x.6 Second, u and v are assumed to be independent of one another. Next, given that ui leads directly to a shortfall in output, it must come from a one-sided distribution implying that E[εi |x] = 0. This has two effects if one was to estimate the SFM using OLS. First, the intercept of technology would not be identified, and second, without

2 Battese

and Corra [16] and Meeusen and van den Broeck [73], while appearing in the same year, are applications of the methods. 3 Our discussion in both chapters will focus on a production frontier, as it is the most popular object of study, while the framework for dual characterizations (e.g., cost, revenue, profit) or other frontiers is similar and follows with only minor changes in notation. 4 See  Chapter 9 “Stochastic Frontier Analysis: Foundations and Advances II” for a discussion on relaxing parametric restrictions on the production frontier in the SFM. 5 Prior to the development of the SFM, approaches which intended to model inefficiency typically ignored vi leading to estimators of the SFM with less desirable statistical properties: see the work of [1, 3, 29, 84, 87, 99]. 6 See section “Handling Endogeneity in the SFM” for a discussion on estimation of the SFM when some inputs are allowed to be endogenous.

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any additional information, nothing can be said about inefficiency. Additionally, if ui is an independently and identically distributed random variable, there is no policy implication behind it given that nothing can directly increase or decrease inefficiency. That is, the conclusions of such a study would be descriptive (reporting presence or absence of inefficiency) rather than prescriptive or normative.7 Denote E[u] as μu and εi∗ = vi −(ui −μu ), the benchmark SFM can be rewritten as yi = m(x i ; β) − μu − (ui − μu ) + vi ≡ m∗ (x i ; β) + εi∗

(2)

and E[εi∗ |x] = 0. The OLS estimator could be used to recover mean inefficiency adjusted technology m∗ (x i ; β) = m(x i , β) − μu in this case. However, rarely is the sole focus of an analysis of productivity on the production technology. It is more likely that both the production technology and information about inefficiency for each DMU are the targets of interest; more structure is required on the SFM in this case. ALS’ and MvB’s approach to extract information on inefficiency, while also estimating technology, was to impose distributional assumptions on ui and vi , recovering the implied distribution for εi and then estimating all of the parameters of the SFM with the maximum likelihood estimator (MLE). vi was assumed to be distributed as a normal with mean 0 and variance σv2 by both sets of researchers, while the distribution of ui differed across the papers; [4] assumed that ui was generated from a half-normal distribution, N+ (0, σu2 ), whereas MvB assumed ui was distributed exponentially, with parameter σu .8 Even though the half-normal and exponential distributions are distinct, they possess several common aspects. Both densities have modes at zero and monotonically decay (albeit at different speeds) as ui increases. The zero mode property is indicative of an industry where there is a tendency for higher efficiency for the majority of the DMUs. Both densities would be classified as single-parameter distributions, which means that the mean and variance both depend on the single parameter, and these distributions also possess the scaling property, which we will discuss in section “The Scaling Property.”

The Distribution of ε Estimation of the SFM in (1) with maximum likelihood requires that the density of ε, f (ε), is known. f (ε) can be determined through the distributional assumptions invoked for v and u. Not all pairs of distributional assumptions for v and u will

7 See

section “Modeling Determinants of Inefficiency” for models handling determinants of inefficiency. 8 ALS also briefly discussed the exponential distribution, but its use and development is mainly attributed to MvB.

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lead to a tractable density of f (ε), permitting estimation via maximum likelihood. Fortunately, the half-normal specification of [4] and the exponential specification of MvB (along with the normal assumption for v), produce a density for ε that has a closed form solution; direct application of maximum likelihood is straightforward in this setting. For brevity we report the density of the composed error for the normalhalf-normal specification. f (ε) =

2 φ(ε/σ )(−ελ/σ ), σ

(3)

where φ(·) is the standard normal probability density function (pdf), (·) is the standard  normal cumulative distribution function (cdf), with the parameterization σ = σu2 + σv2 , and λ = σu /σv . λ is commonly interpreted as the proportion of variation in ε due to inefficiency. The density of ε in (3) can be characterized as that of a skew normal random variable with location parameter 0, scale parameter σ , and skew parameter −λ.9 This connection has only recently appeared in the efficiency and productivity literature [27]. From f (ε) in (3), along with independence assumptions on ui and vi the loglikelihood function is  n  n n   1  2 ln L = ln f (εi ) = −n ln σ + ln (−εi λ/σ ) − εi , (4) 2σ 2 i=1

i=1

i=1

where εi = yi −m(x i ; β). The SFM can be estimated using the traditional maximum likelihood estimator (MLE). The benefit of this is that under the assumption of correct distributional specification of ε, the MLE is asymptotically efficient (i.e., consistent, asymptotically normal and its asymptotic variance reaches the CramerRao lower bound). A further benefit is that a range of testing options are available. For instance, tests related to β can easily be undertaken using any of the classic trilogy of tests: Wald, Lagrange multiplier, or likelihood ratio. The ability to readily and directly conduct asymptotic inference is one of the major benefits of stochastic frontier analysis over DEA.10

pdf of a skew normal random variable x is f (x) = 2φ(x)(αx). The distribution is right skewed if α > 0 and is left skewed if α < 0. We can also place the normal, truncated-normal pair of distributional assumptions  in this class. The pdf of x with location ξ , scale ω, and skew  parameter α is f (x) = ω2 φ x−ξ  α x−ξ . See [12, 76] for more details. ω ω 9 The

10 This

in no way suggests that inference cannot be undertaken when the DEA estimator is deployed; rather, the DEA estimator has an asymptotic distribution which is much more complicated that the MLE for the SFM, and so direct asymptotic inference is not available; bootstrapping techniques are required for many of the most popular DEA estimators [93, 94].

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Alternative Specifications The half-normal assumption for the one-sided inefficiency term is almost without question the most common distribution for inefficiency in practice. This stems partly from posterity, partly from the closed form solution of the likelihood function, and partly from the availability of software to estimate the model for applied researchers. However, none of these reasons are sufficient for blind application of the half-normal density for inefficiency in the SFM.

The Exponential Distribution The exponential assumption on inefficiency is also popular. The exponential density is f (u) =

1 −u/σu e , σu

u ≥ 0.

(5)

For the normal-exponential distributional pair, the density of ε is f (ε) =

1 2 2 (−ε/σv − σv /σu )eε/σu +σv /2σu , σu

(6)

with likelihood function

σv2 ln L = −n ln σu + n 2σu2

+

n  i=1

ln (−εi /σv − σv /σu ) +

n 1  εi . σu

(7)

i=1

Like the half-normal specification for u, the exponential specification monotonically decreases in u, suggesting that larger levels of inefficiency are less likely to occur than small levels of inefficiency. Both the half-normal and exponential specifications for inefficiency stem from what are known as single-parameter distributions. Single-parameter distributions are the simplest distributions, and an unfortunate (yet sometimes very convenient) property of them is that all of their moments depend on this single parameter, which can restrict the shape that the density can potentially take.11

The Truncated Normal Distribution To allow more generality into the SFM, while guarding against distribution misspecification, a variety of one-sided distributions have been proposed for modeling ui in the SFM. Stevenson [96] proposed the truncated-normal distribution as a generalization of the half-normal distribution; whereas the half-normal distribution is the truncation of the N(0, σu2 ) at 0, the truncated-normal distribution is the

11 See

[79] for a more detailed analysis of the SFM with u distributed exponentially.

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truncation of the N(μ, σu2 ) at 0. The pre-truncation mean parameter, μ, affords the SFM more flexibility in the shape of the distribution of inefficiency. The truncated-normal density is

f (u) = √

1 2π σu (μ/σu )

e

− (u−μ) 2 2σu

2

,

u ≥ 0.

(8)

This density reduces to the half-normal distribution when μ = 0 and thus provides a generalization (more specifically a nesting structure), and an opportunity for inference on μ. An intuitive appeal of deploying truncated-normal distribution in practice is that, unlike the half-normal and exponential densities, the truncatednormal density has a mode at 0 only when μ ≤ 0 but otherwise has a mode at μ. When μ > 0, the implication is that producers in a given market would tend to have inefficiency ui near μ > 0 rather than near 0. This connotation may be more realistic in some settings (e.g., the regulatory environment) than the half-normal assumption, where the probability of being less efficient is much larger than of being grossly inefficient. For the normal-truncated-normal distributional pair, the density of ε is 1 f (ε) = φ σ





μ ε+μ ελ   (μ/σu ). − σ σλ σ

(9)

The corresponding log-likelihood function is

ln L= − n ln σ −0.5

n  εi +μ 2 i=1

σ

−n ln (μ/σu )+

n  i=1

ln 

μ εi λ − . (10) σλ σ

Other Distributions Aside from the truncated-normal specification for the distribution of u, a variety of alternatives have been proposed throughout the literature. Greene [34, 35] and Stevenson [96] both proposed a gamma distribution for inefficiency. The gamma distribution generalizes the exponential distribution in much the same way that the truncated-normal distribution nests the half-normal distribution. Ritter and Simar [85] advocate against the use of the gamma specification in practice noting that large samples were required to reliably estimate the parameters of the gamma distribution due to computational identification problems with the constant of the regression. Lee [66] proposed a four parameter Pearson density for the specification of inefficiency; unfortunately, this distribution is intractable for applied work and until now has not appeared to gain popularity. Li [68] proposed the use of the uniform distribution for inefficiency noting an intriguing feature of the subsequent

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composed error density: that it could be positively skewed.12 Another specification for inefficiency appears in [22], who assumes that the distribution of u follows a binomial specification; this allows the skewness of the composed error to be positive or negative. Gagnepain and Ivaldi [32] specify inefficiency as being beta distributed when inefficiency can be defined as a percentage (scaled between 0 and 1), while [6] further generalize Stevenson’s [96] framework by assuming a doubly truncatednormal distribution for inefficiency. This distributional assumption also allows the convolved error term to be either positively or negatively skewed. A common theme of all of the papers just mentioned is that they focus exclusively on the distribution of inefficiency inside the SFM. Recent literature has shed light on the features of f (ε) for the SFM when both the density of v and the density of u are changed. Horrace and Parmeter [46] study the behavior of the composed error when v is distributed as Laplace and u is distributed as truncated Laplace. Nguyen [75] considers the Laplace-exponential distributional pair as well as the Cauchy-Half Cauchy pair for the two error terms of the composed error. While these alternative distributional pairs do provide different insights into the behavior of the composed error, it remains to be seen if they will be regularly adopted in practice and whether they provide substantially different conclusions than the most frequently adopted distributional pairs (e.g., normal-half-normal); see section “Do Distributional Assumptions Even Matter?” for more discussion on the perceived importance of distributional assumptions regarding estimation of the SFM. It is important to note that the main idea behind the SFM is that nearly any pair of distributions can be used to model u and v. The advantage of the normal-halfnormal pair that is dominant in the literature is that the likelihood function has an easy to evaluate expression. In general this should not be expected. More likely than not, for a range of distributional assumptions, the likelihood function will contain one or more intractable integrals, complicating estimation.13

Alternative Estimation Approaches of the SFM Given the focus on inefficiency in the SFM and the impact that the distributional assumption on u is likely to have on the MLE, studying the behavior of the SFM across a range of distributional assumptions is desirable. However, outside of a few specifications (half-normal, exponential, truncated-normal for ui , and normal for vi ), the composed error density will not have a likelihood function that lends itself for easy evaluation. In these cases it can be difficult to estimate all of the parameters of the SFM, but several approaches exist, ranging in complexity, to estimate the SFM when direct estimation of the likelihood function is not feasible. The simplest

12 Prior

to [68] all of the previously proposed distributions always produced a composed error density that was theoretically negatively skewed. Note that if u is distributed uniformly over the interval [0, b], inefficiency is equally likely to be either 0 or b. 13 Note that the likelihood function for the normal-half-normal pair is dependent upon the cdf of the normal distribution, (·) which contains an integral, but this can be quickly and easily evaluated across all modern software platforms.

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approach, dubbed corrected OLS (COLS) by [77],14 recognizes that OLS estimation of the SFM produces consistent estimates of the coefficients of the frontier function aside from the intercept. The intercept √ is biased downward by the expected level of industry inefficiency E [u] = 2/π σu .15 Olson et al.’s [77] insight was that for a given pair of distributional assumptions (normal-exponential, say), the central moments of the OLS residuals could be used to construct consistent estimators of the parameters of the convolved error. Once these were estimated, expected inefficiency could be estimated and the bias in the intercept corrected. The beauty of COLS from the applied perspective is that OLS can be used and difficult likelihood functions do not have to be derived or estimated.16 Several newer approaches exist as well. One that is becoming popular is maximum simulated likelihood (MSL) estimation [71]. Greene [37] used MSL estimation to estimate the parameters of the SFM for the normal-gamma convolution. The key to implementation of the SFM when the composed error does not produce a tractable likelihood is to notice that the integrals that commonly remain in the density (from integrating u out of the density) can be treated as expectations and evaluated by simulation rather than analytic optimization. Given that the distribution of u is assumed known (up to unknown parameters), for a given set of parameters, draws can be taken, and the subsequent expectation that is evaluated can replace the integral. Optimization proceeds by searching over the parameter space until a global maximum is found. An even more recent approach to evaluating intractable likelihoods is found in [102] who suggested estimation of the parameters of the SFM through the characteristic function of the composed error. The reason that this will work is that the characteristic function is a unique representation of a distribution (whether the

14 See

also [34, pp. 31–32]. Richmond [84] also proposed adjusting the intercept from OLS estimation, however, his model differs from that of [77] by assuming the presence of inefficiency (which follows a gamma distribution) but no noise. 15 An alternative approach would be to estimate a weighted average efficiency of an industry, as described theoretically in  Chapter 25 “Aggregation of Efficiency and Productivity: From Firm to Sector and Higher Levels”. 16 There exists some confusion over the terminology COLS as it relates to another method, modified OLS (MOLS). Beginning with [111] and discussed in [31] and [34, pp. 32–34], MOLS shifts the estimated OLS production function until all of the observations lie on or below the “frontier.” At issue is the appropriate name of these two techniques. Greene [38] called the bounding approach COLS, crediting [70, p. 21] with the initial nomenclature, and referred to MOLS as the method in which one bias corrects the intercept based on a specific set of distributional assumptions. Further, [59, pp. 70–71] also adopted this terminology. However, given that [77, p. 69] explicitly used the terminology COLS, in our review we will adopt COLS to imply bias correction of the OLS intercept and MOLS as a procedure that shifts up (or down) the intercept to bound all of the data. The truth is both COLS and MOLS are the same in the sense that the OLS intercept is augmented, it is just in how each method corrects, or modifies, the intercept that is important. While we are departing from the more mainstream use of COLS and MOLS currently deployed, given the original use of COLS, coupled with myriad papers written by Peter Schmidt and coauthors that we discuss here, we will use the COLS acronym to imply a bias corrected intercept.

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density does or does not exist), and following from the convolution theorem, the characteristic function of two independent random variables (here v and u) added together is the product of the individual characteristic functions. The characteristic functions for all of the densities described above are known, and so, using the Fast Fourier Transform, the estimated characteristic function can be mapped to the underlying density and, subsequently, the likelihood function. Tsionas’s [102] method is somewhat computationally complicated, but it offers another avenue to estimate the SFM under alternative distributional assumptions on both v and u.

Estimation of Individual Inefficiency Once the parameters of the SFM have been estimated, estimates of firm-level productivity and efficiency can be recovered. Observation-specific estimates of inefficiency are one of the main benefits of the SFM relative to neoclassical models of production. Firms can be ranked according to estimated efficiency; the identity of underperforming firms as well as those who are deemed best practice can also be gleaned from the SFM. All of this information is useful in helping to design more efficient public policy or subsidy programs aimed at improving the market, for example, insulating consumers from the poor performance of heavily inefficient firms. As a concrete illustration, consider firms operating electricity distribution networks that typically possess a natural local monopoly given that the construction of competing networks over the same terrain is prohibitively expensive.17 It is not uncommon for national governments to establish regulatory agencies which monitor the provision of electricity to ensure that abuse of the inherent monopoly power is not occurring. Regulators face the task of determining an acceptable price for the provision of electricity while having to balance the heterogeneity that exists across the firms (in terms of size of the firm and length of the network). Firms which are inefficient may charge too high a price to recoup a profit but at the expense of operating below capacity. However, given production and distribution shocks, not all departures from the frontier represent inefficiency. Thus, measures designed to account for noise are required to parse information from εi regarding ui . Alternatively, further investigation could reveal what it is that makes these establishments attain such high levels of performance. This could then be used to identify appropriate government policy implications and responses or identify processes and/or management practices that should be spread (or encouraged) across the less efficient, but otherwise similar, units. This is the essence of the determinants of 17 The

current literature is fairly rich on various examples of empirical values of SFA for the estimation and use of efficiency estimates in different fields of research. For example, in the context of electricity providers, see [42,54,62]; for banking efficiency, see [23] and references cited therein; for the analysis of the efficiency of national healthcare systems, see [33] and a review by [45]; for analyzing efficiency in agriculture, see [14, 15, 21, 69], to mention just a few.

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inefficiency approach which we will discuss in section “Modeling Determinants of Inefficiency.” More directly, efficiency rankings are used in regulated industries such that regulators can set tougher future cost reduction targets for the more inefficient companies, in order to ensure that customers do not pay for the inefficiency of firms. The only direct estimate coming from the normal-half-normal SFM is σu2 . This provides context regarding the shape of the half-normal distribution on ui and the industry average efficiency E[u], but not on the absolute level of inefficiency for a given firm. If we are only concerned with the average level of technical efficiency for the population, then this is all the information that is needed. Yet, if we want to know about a specific firm, then something else is required. The main approach to estimating firm-level inefficiency is the conditional mean estimator of [50], commonly known as the JLMS estimator. Their idea was to calculate the expected value of ui conditional on the realization of composed error of the model, εi ≡ vi − ui , i.e., E[ui |εi ].18 This conditional mean of ui given εi gives a point prediction of ui . The composed error contains individual-specific information, and the conditional expectation is one measure of firm-specific inefficiency. Jondrow et al. [50] show that for the normal-half-normal specification of the SFM, the conditional density function of ui given εi , f (ui |εi ), is N+ (μ∗i , σ∗2 ), where μ∗i =

−εi σu2 σ2

(11)

σ∗2 =

σv2 σu2 . σ2

(12)

and

Given results on the mean of a truncated-normal density it follows that σ∗ φ( μσ∗i∗ ) E[ui |εi ] = μ∗i +   .  μσ∗i∗

(13)

The individual estimates are then obtained by replacing the true parameters in (13) with MLE estimates from the SFM. Another measure of interest is the Afriat-type level of technical efficiency, defined as e−ui = Yi /em(x i ) evi ∈ [0, 1]. This is useful in cases where output is measured in logarithmic form. Further, technical efficiency is bounded between 0 and 1, making it somewhat easier to interpret relative to a raw inefficiency score. −ui Since

−ue  is not directly observable, the idea of [50] can be deployed here, and i E e |εi can be calculated [13, 67]. For the normal-half-normal model, we have

18 Jondrow

et al. [50] also suggested an alternative estimator based on the conditional mode.

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E e−ui |εi = e



−μ∗i + 12 σ∗2



 

 − σ∗   ,

μ∗i σ∗



μ∗i σ∗

(14)

where μ∗i and σ∗ were defined in (11) and (12), respectively. Technical efficiency estimates are obtained by replacing the true parameters in (14) with MLE estimates from the SFM. When ranking efficiency scores, one should use estimates of 1 − E [ui |εi ], which is the first-order approximation of (14). Similar expressions for the [50] and [13] efficiency scores can be derived under the assumption that u is exponential [59, p. 82], truncated-normal [59, p. 86], and Gamma [59, p. 89]; see also [61].19

Inference About the Presence of Inefficiency Having estimated the benchmark SFM, a natural hypothesis is whether inefficiency is even present. In this case the null hypothesis of interest is H0 : σu2 = 0 against H1 : σu2 > 0.20 The direct way to test the H0 is through a likelihood ratio test, keeping in mind that the unrestricted model is the assumed SFM and the restricted model is the linear regression model (or more specifically the normal regression model). There is a problem with implementation of this test however. Under H0 σu2 is restricted to lie on the boundary of the parameter space, and this precludes direct use of a likelihood ratio test. Battese and Coelli [28] demonstrates that under H0 , the likelihood ratio statistic in this setting is a 50:50 mixture of a χ12 distribution, the distribution of the ordinary likelihood ratio statistic if the parameter was not on the boundary of the parameter space, and a χ02 , known as the chi-bar-square distribution, χ¯ 2 , [28, 89]. This second piece is what captures the potential presence of the σu2 parameter to lie on the boundary of the parameter space and creates a point mass in the asymptotic distribution of the likelihood ratio statistic. Calculation of the test statistic itself is invariant to whether the parameter lies on the boundary under H0 . What does change is how one goes about calculating either the p-value or the critical value to assess the outcome of the test. In the case of the 50:50 mixture, the critical values are determined by looking at the 2α-level critical value from a χ12 distribution. For example, whereas the critical value for a 5% significance level is 3.841 for χ12 , it is 2.706 for the 50:50 mixture. More specifically, Table 1 presents the critical values of both the χ12 and the 50:50 mixture for a range of significance levels.

19 In

principle, these individual efficiency scores can then be used for estimating weighted average efficiencies of an industry or a group within it, as described theoretically in  Chapter 25 “Aggregation of Efficiency and Productivity: From Firm to Sector and Higher Levels”, which seems novel for SFA context. 20 One could test if other moments of the distribution were 0 as well, but most of the SFMs parameterize the distribution of u with σu and so this seems the most natural.

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Table 1 Right tail critical values for both a χ12 and a 50:50 mixture of a χ02 and a χ12 , denoted as χ¯ 2 Significance Level χ12 χ¯ 2

0.01 6.634 5.412

0.05 3.841 2.706

0.1 2.706 1.642

0.15 2.072 1.074

0.2 1.642 0.708

0.25 1.323 0.455

An alternative type of test for the presence of inefficiency is based on the skewness of the residuals. A variety of tests for skewness exist, notably [2, 43, 63]. Henderson and Parmeter [43] proposed a bootstrap based version of Ahmad & Li’s [2] asymptotic test, noting that in finite samples the bootstrap version is likely to have superior performance. This test involves estimating the SFM using OLS and then testing whether the distribution of the OLS residuals is symmetric. Kuosmanen and Fosgerau’s [63] test of symmetry is also based on the bootstrap, but rather than focus on the estimated distribution of the OLS residuals, their test focuses exclusively on the skewness coefficient of the residuals. Both of these tests of symmetry are appealing because they do not require parametric distributional assumptions and can be implemented after having estimated the SFM using OLS.

Inference About the Distribution of Inefficiency It is important to recognize, despite the frequent misuse of terminology, that the JLMS or Battese-Coelli (or similar types) efficiency estimators are not estimators of ui or e−ui , respectively, and do not converge to them for n → ∞. As n → ∞, the new observations represent different firms each with their own level of inefficiency and noise (upon which JLMS conditions), rather than observations from the same firm. Even more importantly, the JLMS estimator was not intended to estimate unconditional inefficiency. The JLMS estimator is, however, a consistent estimator for the expected level of inefficiency conditional on the particular realizations of ε.21 The JLMS efficiency scores can be used to provide a (limited) test of the distribution of inefficiency. The key insight to understand how a test can be constructed is that if the distributional assumptions are correct, then the distribution i |εi ] of E[ui |εi ] is completely known. Hence a comparison of the distribution of E[u to the true distribution of E[ui |εi ] will shed light into the statistical validity of the assumed distributions for u and v. Wang and Schmidt [108] derived the distribution of E[ui |εi ] for the normal-half-normal SFM, while [107] proposed χ 2 and Kolmogorov-Smirnov-type test statistics against this distribution.22 We caution readers regarding a rejection with use of this test. A rejection does not necessarily imply that the distributional assumption on u is incorrect, it could be that the normality distributional assumption on v or some other assumptions about the SFM (e.g., the parametric form of m) is violated, and this is leading to 21 The

JLMS efficiency estimator is known as a shrinkage estimator; on average, it understates the efficiency level of a firm with small ui while it overstates efficiency for a firm with large ui . 22 See also [66] for a different test based on the Pearson distributional assumption for u.

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the rejection. Similarly, one must be careful in interpreting tests on the distribution of ε (or functionals of ε) when the distribution of v is also assumed to be normal. Alternative tests similar to [107] could be formulated using the Laplace-exponential SFM of [46].

Predicting Inefficiency Aside from testing for the appropriate distribution of inefficiency, one should also test, or present uncertainty, as it pertains to an individual efficiency score. Each JLMS efficiency score is a prediction of inefficiency, and it is possible to calculate prediction intervals. Interestingly, few applied papers cover in depth uncertainty of estimated efficiency scores. A prediction interval for E[ui |εi ] was first derived by [98] and also appeared in [18, 44, 47] (see the discussion of this in [92]). The prediction interval is based on f (ui |εi ). The lower (Li ) and upper (Ui ) bounds for a (1 − α)100% prediction interval are



  α μ∗i Li =μ∗i +  1− 1− σ∗ , 1− − 2 σ∗



 α μ∗i σ∗ , 1− − Ui =μ∗i + −1 1 − 2 σ∗ −1

(15) (16)

where μ∗i and σ∗ are defined in (11) and (12), respectively, and replacing them with their MLE estimates will give estimated prediction intervals for E[ui |εi ]. Wheat et al. [109] derived minimum width prediction intervals noting that the confidence interval studied in [47] was based on a symmetric two-sided interval. Given that the distribution of ui conditional on εi is truncated (at 0) normal and asymmetric, this form of interval is not of minimum width. Parmeter and Kumbhakar [79] showed that depending upon the ratio of σu to σv , the difference in relative widths of Horrace & Schmidt’s [47] and Wheat et al.’s [109] prediction intervals can be quite substantial. It is thus recommended to use the intervals provided by [109] as these are not based on symmetry. Note that although we could predict u and construct a prediction interval, this information is not that useful for policy purposes unless there are some variables that affect inefficiency and such variables can be changed by a specific policy.

Do Distributional Assumptions Even Matter? An important empirical concern when using the SFM is the choice of distributional assumptions made for v and u. The distribution of v has almost universally been accepted as being normal in both applied and theoretical work (a recent exception is [46]); the distribution of u is more commonly debated, but relatively little work has been devoted to discerning the impact that alternative shapes of the distribution can have. Moreover, choice of u is often driven through available statistical software to

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implement the method rather than an underlying theoretical link between a model of productive inefficiency and the exact shape of the corresponding distribution. A majority of applied papers studying productivity do not rigorously check differences in estimates, or perform inference, across different distributional assumptions. Greene [36] is often cited as one of the first analyses to compare average inefficiency levels across several distributional specifications (half-normal, truncated-normal, exponential, and gamma), and he finds little difference in average inefficiency across 123 US electric generation firms. Following Greene’s [36] investigation into the choice of distribution, [59] calculated the rank correlations among the JLMS scores from these same four models, producing rank correlations as low as 0.75 and as high as 0.98.23 The intuition underlying these findings is that one’s understanding of inefficiency, as measured through the JLMS score, is robust to distributional choices, at least from a ranking perspective. The reason for this can be found in the work of [78, p. 438] who have shown that the JLMS efficiency scores are monotonic in ε provided that the distribution of v is log-concave (which the normal distribution is). The implication here is that firm rankings can be obtained via the OLS residuals without the need of distributional assumptions whatsoever [18]. Thus, in light of these insights, the important aspect of distributional choice for u is the impact that it has on the corresponding estimates of the production function; when these estimates are robust to distributional choice, so too will be the inefficiency rankings. Thus, if interest hinges on features of the frontier, then so long as inefficiency does not depend on conditional variables (see section “Modeling Determinants of Inefficiency”), one can effectively ignore the choice of distribution, as this only affects (usually but not substantially) the level of the estimated technology, but not its shape – which is what influences measures such as returns to scale and elasticities of substitution.

Finite Sample Identification of Inefficiency An early analysis of the finite sample performance of the normal-half-normal SFM by [77] uncovered an interesting phenomena, quite regularly the corrected OLS estimator would produce an estimate of σu2 ≤ 0. This was deemed a “Type I” failure of the SFM; further [77, p. 70] noted that “It is also true that, in every case of Type I failure we encountered, the MLE estimate of [σu2 ] also turned out to equal zero. (This makes some sense, though we cannot prove analytically that it should happen.)” Waldman [104] provided the analytic foundation behind this

23 In

a limited Monte Carlo analysis, [86] compared rank correlations of stochastic frontier estimates assuming that inefficiency was either half-normal (which was the true distribution) or exponential (a misspecified distribution) and found very little evidence that misspecification impacted the rank correlations in any meaningful fashion; [46] conducted a similar set of experiments and found essentially the same results.

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result, demonstrating that a stationary point of the log-likelihood function exists, and this stationary point is a local maximum when the sign of the skewness of residuals stemming from OLS estimation of the SFM is positive. This is broadly viewed as a deficiency of the SFM as an estimate of σu2 of 0 is literally interpreted as a finding of no inefficiency. However, this is an unfortunate interpretation because it is purely a finite sample issue. If in fact u is distributed half-normal, then as shown in [28, 92, 103], as n → ∞ the likelihood of drawing a random sample which will have positive skewness decreases, and the rate of this decrease is directly related to σu2 /σv2 ; the larger this ratio, the faster the decrease in the probability of observing a random sample with positive skew.24 The observance of OLS residuals with positive skew is, by and of itself, of no concern. What is concerning is that for an applied researcher whose focus is to study the efficiency level of firms, analysis of a sample where the residuals from the SFM have positive skewness leads to the conclusion of all firms being efficient, and this finding might be incongruent with either preconceptions about the industry or perceived publication standards when applying these methods. This has often led to various forms of respecification: using a different data set, trying an alternative functional form for the production function or, most likely, deploying different distributional assumptions regarding inefficiency. As noted by [92], none of these respecification approaches are appropriate or warranted. Again, Table 1 in [92] evinces that even when everything about the SFM is correctly specified, positively skewed OLS residuals are still a regular occurrence. Their suggestion is to use special resampling techniques based on bootstrapping to conduct inference on either overall inefficiency of the industry under study or specific firms. The finding of OLS residuals with positive skewness is commonly denoted the “wrong skew problem,” though it is not clear where this term initially originated. It is unfortunate that this term has crept into the lexicon of productivity analysis as there really is no problem at all, except for the problem of misinterpretation and mistreatment. One reason why respecification is troubling is that classical statistical inference assumes that model specification is selected independently of estimation. When specification searches are conducted, this introduces biases into the final parameter estimates. Further, there is the concern in published research that if the researcher did encounter positive skew, this information is not provided to the reader. It is worth mentioning that not all SFMs are plagued by this issue. In fact, some distributional combinations will lead to identification of inefficiency regardless of the sign of the skewness of the OLS residuals. Examples include the normal-uniform SFM of [68], the normal-Weibull SFM of [101], the normal-binomial SFM of [22], and the normal-doubly truncated SFM of [6]. Even more recently, [46] demonstrated, in the style of [104], that the log-likelihood function of the Laplace-exponential SFM is not dependent upon the sign of the skewness of the OLS residuals. This mainly

24 Note that the estimator of the skewness coefficient is distributed asymptotically standard normal,

so it is feasible to have either negative or positive skewness in any finite sample.

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stems from the fact that as σu2 → 0, this model converges to a regression model with error term distributed Laplace, for which the MLE is the least absolute deviations (LAD) estimator. Despite the history behind the impact of the sign of the skewness of the OLS residuals on the SFM, interest still abounds surrounding this issue. Recently, [40] presented a generalized method which always ensures that the SFM can be identified and that this model will converge to the traditional SFM model as n → ∞ if the traditional SFM is correctly specified. Bonanno et al. [20] introduced a generalized SFM which allows v to be distributed as a Type 1 generalized logistic which introduces asymmetry in v, coupled with allowing dependence between u and v. These two additional assumptions, similarly to [40], allow the parameters of the SFM to be identified regardless of the sign of the OLS residuals. Feng et al. [30] describe a constrained MLE that uses the traditional normal-half-normal distributional pair but imposes a penalty in estimation to combat the potential for positive skewness of the OLS residuals to lead to an estimate of σu2 of 0. Finally, [48] generalize the theory of [104] by studying the SFM without explicit distributional assumptions. All told, this issue is one that still generates a substantial amount of interest in the academic community, and it is one that is not likely to fade any time soon (see the discussion in [7]).

Handling Endogeneity in the SFM A common assumption in the SFM is that x is either exogenous or independent of both ui and vi . If either of these conditions are violated, then the MLE will be biased and most likely inconsistent. Yet, it is not difficult to think of settings where endogeneity is likely to exist. For example, if shocks are observed before inputs are chosen, then producers may respond to good or bad shocks by adjusting inputs, leading to correlation between x and v. Alternatively, if managers know they are inefficient, they may use this information to guide their level of inputs, again, producing endogeneity. In a regression model, dealing with endogeneity is well understood. However, in the composed error setting, these methods cannot be simply transferred over but require care in how they are implemented [10]. To incorporate endogeneity into the SFM in (1), we set m(x i ; β) = β0 + x 1i β 1 +x 2i β 2 where x 1 are our exogenous inputs and x 2 are the endogenous inputs, where endogeneity may arise through correlation of x 2 with u, v, or both. To deal with endogeneity, we require instruments, w, and identification necessitates that the dimension of w is at least as large as the dimension of x 2 . The natural assumption for valid instrumentation is that w is independent of both u and v. Our following discussion here will center on the distributional assumptions of ALS. Why worry about endogeneity? Economic endogeneity means that the inputs in question are choice variables and chosen to optimize some objective function such as cost minimization or profit maximization. Statistical endogeneity arises from simultaneity, omitted variables, and measurement errors. For example, if the

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omitted variable is managerial ability, which is part of inefficiency, inefficiency is likely to be correlated with inputs because managerial ability affects inputs. This is the Mundlak’s argument for why omitting a management quality variable (for us inefficiency) will cause biased parameter estimates. Endogeneity can also be caused by simultaneity meaning that more than one variable in the model are jointly determined. One way to address the problem is to look at it from a purely statistical angle and use instrumental variables. The other solution is economic, that is, address the economic issue that is causing endogeneity. We consider first the statistical solution and then the economic solution. In many applied settings, it is not clear what researchers mean when they attempt to handle endogeneity inside the SFM. An excellent introduction into the myriad of influences that endogeneity can have on the estimates stemming from the SFM can be found in [74]. Mutter et al. [74] used simulations designed around data based on the California nursing home industry to understand the impact of endogeneity of nursing home quality on inefficiency measurement.

A Corrected Two-Stage Least Squares Approach The simplest approach to accounting for endogeneity is to use a corrected two-stage least squares (C2SLS) approach, similar to the common COLS approach that has been used to estimate the SFM. This method estimates the SFM using standard 2SLS with instruments w. This produces consistent estimators for β 1 and β 2 but not β0 , as this is obscured by the presence of E[u] (to ensure that the residuals have mean zero). The second and third moments of the 2SLS residuals are then used to recover estimatorsof σv2 and σu2 . Once σu2 is determined, the intercept can be

corrected by adding π2 σˆ u . This represents a simple avenue to account for endogeneity, and it does not require specifying how endogeneity enters the model, i.e., through correlation with v, with u, or both. However, as with other corrected procedures based on calculations of the second and third (or even higher) moments of the residuals, from [77] and [104], if the initial 2SLS residuals have positive skew (instead of negative), then σu2 cannot be identified, and its estimator is 0. Further, the standard errors from this approach need to be modified for the estimator of the intercept to account for the stepwise nature of the estimation.

A Likelihood Approach The SFM with endogeneity has recently been studied by [10, 52, 64, 100]. Here we describe maximum likelihood estimation of the SFM under endogeneity. Our discussion here follows [10] as their derivation of the likelihood relies on a simple conditioning argument as opposed to the earlier work relying on the Cholesky

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decomposition. While both approaches lead to the same likelihood function, the conditioning idea of [10] is simpler and more intuitive. Consider the stochastic frontier system: yi =x i β + εi

(17)

x 2i =wi  + ηi

(18)

where x i = (x 1i , x 2i ), β = (β 1 , β 2 ), w i = (x 1i , q i ) is the vector of instruments, ηi is uncorrelated with wi , and endogeneity of x 2i arises through cov(εi , ηi ) = 0. Here simultaneity bias (and the resulting inconsistency) exists because ηi is correlated with either vi , ui or both. The following assumptions are used by [10]: ui ∼N+ (0, σu2 ), m(x 1i , x 2i ; β 1 , β 2 ) =β0 + x 1i β 1 + x 2i β 2 , and conditional on w i , ψi = (vi , ηi ) ∼ N(0, ), where 

 σv2 Σvη = . Σηv Σηη  vi . ηi To derive the likelihood function, [10] condition on the instruments, w. Doing this yields f (y, x 2 |w) = f (y|x 2 , w) · f (x 2 |w). With the density in this form, the log-likelihood follows suite: ln L = ln L1 + ln L2 , where ln L1 corresponds to f (y|x 2 , w) and ln L2 corresponds to f (x 2 |w). These two components can be written as 

Amsler et al. [10] focused on the setting where ui is independent of ψi =

ln L1 = − (n/2) ln σ 2 −

n n  1  2  ε ˜ + ln  (−λc ε˜ i /σ ) i 2 2σ i=1

ln L2 = − (n/2) ln |Σηη | − 0.5

n 

i=1

−1 η i Σηη ηi ,

i=1 −1 η , σ 2 = σ 2 + σ 2 , λ = σ /σ , and where ε˜ i = yi − β0 − x i β − μci , μci = Σvη Σηη c u c i v u −1 Σ . The subtraction of μ in ln L is an endogeneity correction σc2 = σv2 −Σvη Σηη ηv ci 1 while it should be noted that ln L2 is nothing more than the standard likelihood function of a multivariate normal regression model (as in (17)). Estimates of the model parameters (β, σv2 , σu2 , , Σvη ) and Σηη can be obtained by maximizing the likelihood function ln L. While direct estimation of the likelihood function is possible, a two-step approach is also available [64]. However, as pointed out by both [64] and [10], this two-step approach will have incorrect standard errors. Even though the twostep approach might be computationally simpler, it is, in general, different from full optimization of the likelihood function of [10]. This is due to the fact that the twostep approach ignores the information provided by  and Σηη in ln L1 . In general

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full optimization of the likelihood function is recommended as the standard errors (obtained in a usual manner from the inverse of the Fisher information matrix) are valid.25

A Method of Moments Approach An insightful avenue to deal with endogeneity in the SFM that differs from the traditional corrected methods or maximum likelihood is proposed by [10], who used the work of [41]. The idea is to use the first-order conditions for maximization of the likelihood function under exogeneity:   E ε22 /σ 2 − 1 = 0   εi φi =0 E 1 − Ψi   φi = 0, E x i εi /σ + λx i 1 − i

(19) (20) (21)

where φi = φ( λεσ i ) and i = ( λεσ i ). Note that these expectations are taken over x i and yi (and by default, εi ) and solved for the parameters of the SFM. The key here is that these first-order conditions (one for σ 2 , one for λ and the vector for β) are valid under exogeneity, and this implies that the maximum likelihood estimator is the generalized methods of moments estimator. Under endogeneity however, this relationship does not hold directly. But the seminal idea of [10] is that the first-order conditions (19) and (20) are based on the distributional assumptions on v and u, not on the relationship of x with v and/or u. Thus, these moment conditions are valid whether x contains endogenous components or not. The only moment condition that needs to be adjusted is (21). In this case the firstorder needs to be taken with respect to w, the exogenous variable, not x. Doing so results in the following amended first-order condition:

25 Typically

the standard errors can be obtained either through use of the outer product of gradients (OPG) or direct estimation of the Hessian matrix of the log-likelihood function. Given the nascency of these methods, it has yet to be determined which of these two methods is more reliable in practice, though in other settings both tend to work well. One caveat for promoting the use of the OPG is that since this only requires calculation of the first derivatives, it can be more stable (and more likely to be invertible) than calculation of the Hessian. Also note that in finite samples, the different estimators of covariance of MLE estimator can give different numerical estimates, even suggesting different implications on the inference (reject or do not reject the null hypothesis). So, for small samples, it is often advised to check all feasible estimates whenever there is suspicion of ambiguity in the conclusions (e.g., when a hypothesis is rejected only at say around the 10% of significance level).

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φi E wi εi /σ + λwi 1 − i

353

 = 0,

(22)

where φi and i are identical to those in (21). It is important to acknowledge that this moment condition is valid when εi and w i are independent. This is a more stringent requirement than the typical regression setup with E[εi |wi ] = 0. As with the C2SLS approach, the source of endogeneity for x 2 does not need to be specified (through v and/or u).

Estimation of Individual Inefficiency An interesting and important finding from [10] is that when there is endogeneity, one can potentially improve estimation of inefficiency through the JLMS estimator. The traditional predictor of [50] is E(ui |εi ). However, more information is available when endogeneity is present, namely, via ηi . This calls for a modified JLMS estimator, E(ui |εi , ηi ). Note that even though it is assumed that ui is independent from ηi (as in [10]), because ηi is correlated with vi , there is information that can be used to help predict ui even after conditioning on εi . Amsler et al. [10] showed that ηi is independent of (ui , ε˜ i ): E(ui |εi , ηi ) = E(ui |˜εi , ηi ) = E(ui |˜εi ). and that the distribution of ui conditional on ε˜ i = yi − β0 − x i β − μci is N+ (μ∗ , σ∗2 ) with μ∗ = −σu2 ε˜ i /σ 2 and σ∗2 = σu2 σc2 /σ 2 , which is identical to the original JLMS estimator, except that σv2 is replaced with σc2 and ε˜ i taking the place of εi . The modified JLMSestimator in the presence of endogeneity becomes  φ(ξi ) E(ui |εi , ηi ) = σ∗ 1−(ξi ) − ξi with ξi = λ˜εi /σ . Note that E(ui |εi , ηi ) is a better predictor than E(ui |εi ) because σc2 < σv2 . The improvement in prediction follows from the textbook identity for variances, where for any random vector (X, Z), where X and Z are random sub-vectors, we have var(X) = var[E(X|Z)] + E(var[X|Z]) .       Explained

U nexplained

In this case, by conditioning on both εi and ηi , the conditioning set is larger than simply conditioning on εi , and so it must hold that the unexplained portion of E(ui |εi , ηi ) is smaller than that of E(ui |εi ). It then holds that there is less variation in E(ui |εi , ηi ) as a predictor than E(ui |εi ), which is a good thing. While it is not obvious at first glance, one benefit of endogeneity is that researchers may be able to more accurately predict firm-level inefficiency, though it comes at the expense of having to deal with endogeneity. This improvement in prediction may also be accompanied by narrower prediction intervals; however, this is not known as [10] did not study the prediction intervals.

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An Economic Approach to Deal with Endogeneity An alternative to developing valid instruments and correcting for endogeneity is to use what is known as a primal system approach, when inputs are endogenous [61, Chap. 8]. This setup estimates the traditional SFM but appends the first-order conditions stemming from cost minimization (one could alternatively attach profit maximization or return to the outlay conditions instead if this was a more representative behavior for the industry under study). That is, if a producer minimizes costs26 min p x, s.t. y = m(x; β) + v − u,

(23)

for input prices p, the first-order conditions in this case are pj mj (x; β) = , m1 (x; β) p1

j = 2, . . . , J,

(24)

where mj (x; β) is the partial derivative of m(x; β) with respect to xj . These first-order conditions are exact, which usually does not arise in practice; rather, a stochastic term is added, which is designed to capture allocative inefficiency. That m (x;β) p is, our empirical first-order conditions are mj1 (x;β) = pj1 eξj for j = 2, . . . , J where eξj captures allocative inefficiency for the j th input relative to input 1 (the choice of input to compare to is without loss of generality). The idea behind allocative inefficiency is that firms could be fully technically efficient and still have room for improvement due to over or under use of inputs, relative to another input, given the price ratio. In general if firms are cost minimizers and one estimates a production function, the inputs will be endogenous as these are choice variables to the firm. Hence, a different approach is needed. The primal system approach estimates the SFM as in (1) but also incorporates the information in the J − 1 conditions in (24) with allocative inefficiency built in. Shephard’s lemma in microeconomics dictates that the first-order conditions are actually cost share information; when the logarithm of the production function is taken, the first derivatives represent the cost shares of the corresponding inputs, mj (x; β) = m1 (x; β)

∂ ln m ∂ ln xj ∂ ln m ∂ ln x1

=

sj /xj . s1 /x1

When these are equated to the ratio of input prices, one obtains which can be rearranged to yield

sj s1

=

pj xj ξj p1 x1 e .

(25) sj /xj s1 /x1

=

pj ξj p1 e ,

Taking logarithms produces

is possible to treat a subset of x as endogenous; i.e., x = (x 1 , x 2 ), where x 1 is endogenous and x 2 is exogenous.

26 It

8 Stochastic Frontier Analysis: Foundations and Advances I

ln(sj ) − ln(s1 ) − ln(pj xj ) + ln(p1 x1 ) = ξj .

355

(26)

If distributional assumptions are imposed on v, u, and ξ , the parameters of the production function can be estimated along with technical and allocative efficiency. An unfortunate consequence of the primal system approach is that the input demand and cost functions are analytically tractable only for quite specific assumptions on the production function (Cobb-Douglas being one). In these cases a more complicated process is required to determine the impact of technical and allocative inefficiency on costs [60]. See [56, 57] for more detailed discussion of these types of primal system approaches to handle economic endogeneity across a range of settings.

Modeling Determinants of Inefficiency The use of the SFM is exciting for productivity analysis because a prediction of firm-level efficiency can be obtained. However, in the benchmark SFM, ui is treated as completely random, and so nothing connects the level of inefficiency to variables which might serve as an explanation for the existence and the level of inefficiency. As the SFM has gained popularity in applied productivity analysis, it has become common to introduce variables outside the main production structure which influence output through their effect on inefficiency.27 As a concrete example, consider the study of productivity within the banking industry. A researcher may want to know whether a bank’s level of efficiency is affected by the use of information technology, the amount of assets the bank has access to, the type of bank, or the type of ownership structure in place, corporate governance practices, etc. Similarly, the government might be interested in whether regulations (such as allowing banks to merge) improve banks’ performance. To answer these questions, the relationship between efficiency and its potential determinants needs to be modeled and estimated. Consider estimating what influences firm-level inefficiency in the benchmark SFM. This model assumes that both vi and ui are homoskedastic. In a traditional linear regression, heteroskedasticity has no impact on the bias/consistency of the OLS estimator. However, if we were to allow σu2 to depend on determinants of inefficiency, z, then ignoring this will lead to, except in special settings, a biased and inconsistent estimator of the parameters of the SFM. Both [59, Section 3.4] and [106] provide detailed accounts of the consequences of ignoring the presence of determinants of inefficiency in the SFM.

27 Reifschneider

and Stevenson [83] used the term “inefficiency explanatory variables,” while others call them “environmental variables,” but it is now common to refer to these variables as “determinants of inefficiency.” A variety of approaches have been proposed to model the determinants of inefficiency with the first pertaining to panel data models [14, 55] (see chapter 11).

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√ Recall from section “The Benchmark SFM” that E[u] = 2/π σu . Now imagine ignoring the composed structure of ε and estimating the SFM via OLS. If it is the case that determinants of inefficiency are present, so that σu2 = σu2 (z), this omission leads to biased parameter estimates of the SFM given that the assumed model is yi = m(x i ; β) + with εi∗ = εi −

√



2/π σu + εi∗ ,

2/π σu , whereas the true model is

yi = m(x i ; β) +

 2/π σu (zi ) + εi∗ ≡ m(x ˜ i , zi ; β, δ) + εi∗ .

The estimates of m(x i ; β) are conflated with σu (zi ), unless x and z are uncorrelated. The reason that this issue presents itself is the fact that the mean of u, due to the truncation at 0, must depend on the variance. Thus, it is not possible to allow u to be heteroskedastic without the mean of u being a function of z as well. Notice here that we have specifically separated the impacts of x and z on output, with x capturing pure production and z capturing inefficiency. This is commonly known as the separability assumption. In some settings this assumption does not have to be made, but in other settings, it is a necessity for identification. See [80] for a more detailed discussion of the separability assumption. Our use of it here is more for expositional clarity. Exactly how to model the influence of z on inefficiency is unknown, and at various points in time, practitioners have deployed a simpler, two-step analysis to account for the presence of determinants of inefficiency. This approach constructs JLMS predictions in the first step and then regresses these inefficiency estimates on z in the second step. Pitt and Lee [82] were the first to implement this type of approach (in a panel data setting), and many others followed this two-step approach blindly [5, 21, 51]. However, this route to modeling determinants of inefficiency has been met with criticism repeatedly and for good reason. As explained in [15], the first-stage model is misspecified if z is ignored. Further, [106] note that if x and z are correlated, then an omitted variable bias exists in the first step rendering the second step ineffectual. Even in the special case where x and z are uncorrelated, ignoring the dependence of u on z will lead to the estimated JLMS predictions in the first stage to have too little variation (see also [88]), and, subsequently, the estimator in the second-stage regression will be biased downward. Caudill and Ford [24] provide Monte Carlo evidence on the effects that ignoring the impact of z on u has on the estimator of the parameters of the SFM, while [106] provide a detailed analysis of the bias of the second-stage parameter estimators. As should be clear, the two-stage approach to account for determinants of inefficiency in the SFM has no statistical foundation and is widely agreed upon to yield poor insights on the actual behavior of inefficiency, as such this approach should be strictly avoided; even with these criticisms of the two-step approach, one will occasionally happen across research that adopts this flawed two-step methodology.

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While the two-stage approach has undesirable statistical properties, this does not mean that determinants of inefficiency cannot be accounted for. Quite the contrary, the preferred approach to studying the exogenous influences on efficiency is a single-step procedure that explicitly accounts for z.

Proper Modeling of the Determinants of Inefficiency The first proper proposals to model z in the SFM are [58, 83], who used the normal truncated-normal SFM as the basis for estimation.28 While their focus was on the normal truncated-normal SFM, the key insights hold for the normal-half-normal SFM, which is what we will base our discussion on here. The main idea is to specify σu2 as a parametric function of z.29 Formally, their parameterization of σu2 is

σu2 = ez δ ,

(27)

The log-likelihood function of the heteroskedastic model is the same as in (4), except that we replace σu2 with (27).30 Here all of the model parameters are estimated simultaneously, and once they are found, technical inefficiency can be computed using (13) or (14) with the appropriate form of σu2 substituted into the expressions. If u follows a half-normal distribution, with the σu2 function depending upon z, then the mean of ui is E[ui |zi ] =

 1

2/π ezi δ = e 2 ln(2/π )+zi δ .

(28)

Note that the 12 ln(2/π ) term can be absorbed by the constant term in z i δ. Therefore, by parameterizing σu2 , we allow z to affect the expected value of inefficiency. More importantly, however, is that the parameterization (27) produces maximum likelihood estimates of δ which may not be very informative. This is because E[ui |zi ] is nonlinear in z, and therefore the slope coefficients δ are not the marginal effects of z. For instance, assume the j th variable in z has an estimated coefficient of 0.5. This number itself tells us very little about the magnitude of the j th variable’s (marginal) effect on the inefficiency, though it does tell us the direction of the effect on inefficiency. Also, the nonlinearity of the conditional mean of u implies that for different levels of z, there will be different expected levels of u. In these instances the marginal effect of z may be useful for empirical purposes.

28 Caudill

and Ford [24], Huang and Liu [49], Battese and Coelli [15], Caudill et al. [25], Hadri [39], and Wang [105] present alternative specifications as well. 29 It is also possible to model σ 2 as a function of variables, but this poses fewer problems, and we v omit the details here. See [79] and [91] for more discussion. 30 Actually, given the reparameterization of the log-likelihood function, the specification for σ u implies a particular specification for both λ and σ .

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For the given parameterization of the normal-half-normal SFM, the marginal effect of the j th variable of zi , zj i on E[ui |zi ] is  ∂E[ui |zi ] = δj 2/π σu,i ∂zj where

√

(29)

2/π is approximately 0.80. It is clear that (29) also implies

∂E[ui |zi ] sign ∂zj

= sign(δk )

(30)

so that the sign of the coefficient reveals the direction of impact of zj i on E[ui |zi ]. This property does not always hold across distributional assumptions, for example, in the normal-truncated-normal SFM, the sign of the coefficient cannot be interpreted directly [79]. In general, only in one parameter families for the pdf of u (exponential, half-normal, etc.) does this correspondence hold; this suggests caution in directly interpreting the impact that a particular variable zj has on inefficiency based purely on the sign of δj . The nonlinear nature of the relationship of E[u|z] with z implies that for a sample of n observations, we have n marginal effects for each variable. A concise statistic to present is the average partial effect (APE) on inefficiency or the partial effect of the average (PEA):

AP E(zj )

=(δju



 −1

2/π ) n



P EA(zj ) =δju 2/π ez¯ u δ .

n 

 σu,i

(31)

i=1

(32)

Either of these measures can be used to provide an overall sense for the impact of a given variable on the level of inefficiency. However, these statistics should also be interpreted with care. Neither necessarily reflects the impact of a given covariate for a given firm but rather on average and ceteris paribus, i.e., holding other covariates fixed; for example, it could be that half of the sample has a very negative effect that is balanced by positive effects in the other half of the sample, thus getting nearly zero on average, which might misrepresent the phenomenon. It is√ also possible to standardize further by using elasticities, which will cancel out the 2/π term; this occurs when the variables are measured in logarithms. It could also prove useful to present the estimates of these at either quartiles or at particular points of interest suggested by a particular empirical context (for example a specific regulation output target).

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Incorporating Determinants When u Is Truncated-Normal As we have discussed earlier, the truncated-normal distribution offers greater flexibility to model an array of shapes of the true, but unknown, distribution of u. When determinants of inefficiency are present and one elects to assume the truncated-normal distribution, several additional modeling choices become available to the researcher. These additional choices are important because, as with the choice of distributional assumption, there is typically little guidance on how best to incorporate the determinants. What do we mean? Consider again the truncated-normal density that would be assumed for u, when determinants of inefficiency are present: f (u) = √

1 2π σu (z; δ 1 ) (μ(z; δ 2 )/σu (z; δ 1 ))

e

−

(u−μ(z;δ 2 ))2 2σu (z;δ 1 )2

,

u ≥ 0.

(33)

In this case the impact of z on u can be modeled through the pre-truncation mean, μ, and the pre-truncation standard deviation, σu . The issue with where to assume that z influences u is that modeling either parameter as a function of z impacts all of the moments of u, due to the truncation. Consider the conditional (on z) mean of a truncated normal random variable   2) φ( σμ(z;δ ) μ(z; δ 2 ) (z;δ ) u 1 + E[u|z] = σu (z; δ 1 ) . (34) σu (z; δ 1 ) ( μ(z;δ 2 ) ) σu (z;δ 1 )

Regardless of whether σu or μ is constant, z still influences the mean of inefficiency unless both are constant. This is what makes the choice of where to incorporate z abstruse when using the truncated normal distribution. Parametric specification of either σu or μ will allow for z to influence expected inefficiency, but in different manners and in nonlinear fashion. Given that μ can be positive or negative, it is common to model it in a linear fashion, i.e., μ(z; δ 2 ) = z δ 2

and to model σu (z; δ 1 ) as ez δ 1 , to ensure positivity of the pre-truncation standard deviation. When we assume that u has the half-normal distribution, our choice is easy because only a single parameter exists and it is clear where z enters. However, in the truncated-normal setup, we could elect to have z enter only through the pretruncation mean, only through the pre-truncation standard deviation, or both. In fact, various applied papers have used any of these three approaches. Kumbhakar et al. [58] and Reifschneider and Stevenson [83] modeled the impact of determinants of inefficiency through μ,31 while [24] incorporated determinants through σu .32 Lastly, [105] modeled the determinants through both μ and σu . The benefit of

31 See

also [49] and [15] for early approaches following this strategy. early approaches that followed this route include [25] and [39].

32 Other

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S. C. Kumbhakar et al.

modeling both pre-truncation parameters jointly as functions of z is that this leaves little room for ambiguity and makes inference of where z belongs a viable option. The costs are that the model is more complex to estimate and may lead to identification problems, as raised in [85]. An alternative approach, which we discuss next, is to invoke a special assumption on the distribution that makes it more amenable to modeling the influence of determinants of inefficiency in the SFM.

The Scaling Property Many of the main proposals to incorporate determinants of inefficiency did so through the normal truncated-normal SFM. The two-parameter nature of the truncated-normal distribution implies that determinants could influence the pretruncation mean, μ, the pre-truncation variance, σu2 , or both. Further still, different variables could influence each parameter. A popular simplification [90, 106], which encapsulates the normal-half-normal SFM, is to assume that inefficiency behaves as ui ∼ g(zi ; δ) u∗i ,

(35)

where g(·) ≥ 0 is a function of the exogenous variables, while u∗i ≥ 0 is a random variable. This behavior is known as the scaling property. Single-parameter distributions, such as the half-normal and the exponential, automatically possess this property, but more flexible distributions, such as truncated-normal or gamma, can have this property imposed. The key feature of the scaling property is that u∗i does not depend on zi in any fashion; u∗i is known as base inefficiency [8, 106]. When a distribution possesses the scaling property, the shape of the distribution of ui is the same for all firms, which can be viewed as an attractive feature. The scaling function, g(·), expands or contracts the horizontal axis so that the scale of the distribution of ui changes while preserving the underlying shape of the distribution. In comparison, the normal truncated-normal SFM models allow different scalings for each ui , so that for some firms the distribution of inefficiency is close to a normal (if the pre-truncation mean is large), while for other firms, the distribution of inefficiency is the extreme right tail of a normal with a mode of zero (if the pretruncation mean is negative). In comparison, for a model with the scaling property, the mean and the standard deviation of u change with zi , but the shape of the distribution is fixed. Another advantage of the scaling property specification is the ease of interpreta tion of δ when g(zi , δ) = ezi δ , ∂ ln E[ui |z] = δj . ∂zj

(36)

That is, δj is the semi-elasticity (or elasticity if z is already measured on the logarithmic scale) of expected inefficiency with respect to the j th element of z, and

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more importantly, this interpretation is distinct from any distributional assumption placed on u∗ . An interpretation of this ilk is generally not available in other model specifications. Further, the sign of the elements of δ can be directly interpreted. The scaling property provides an attractive economic interpretation as well. u∗ can be interpreted as a benchmark level of inefficiency of the firm [8]. The scaling function then allows a firm to exploit (or fail to exploit) these talents through other variables, z, which might include experience of the plant manager, the operating environment of the firm, or regulatory restrictions. The scaling property is not a fundamental feature; rather, as with the choice of distribution on u, it is an assumption on the features of the inefficiency distribution. As such it can be tested against models that do not possess this property for the inefficiency distribution. As it currently stands, all tests of the scaling property hinge on a given distributional assumption, for example, estimating the normal truncated-normal SFM and then estimating a restricted version of the same model but imposing the scaling property. An important avenue for future research is the development of a test (or tests) that does not require specific distributional assumptions.

Estimation Without Imposing Distributional Assumptions In settings where the researcher is comfortable with imposing the scaling property on the distribution of inefficiency, the SFM can be estimated without parametric distributional assumptions. This is perhaps the key benefit of invoking the scaling property. To understand how it is possible to estimate the SFM without distributional assumptions, we expound on the discussion of [8,90,106]. The SFM with the scaling property can be written as33

yi = m(x i ; β) + vi − ezi δ u∗i .

(37)

The conditional mean of y given x and z is

E[y|x, z] = x β − ez δ μ∗

(38)

where μ∗ = E[u∗ ] and E[v|x, z] = 0. The SFM is then



yi = m(x i ; β) − ezi δ μ∗ + vi − ezi δ (ui − μ∗ ) = m(x i ; β) − ezi δ μ∗ + εi∗ , (39)

with εi∗ = vi − ezi δ (ui − μ∗ ), which, for a given parameterization of m(x i ; β), can be estimated using nonlinear least squares (NLS) as

33 Note

here that we are making the implicit assumption that z is different from x. The nonlinearity of the scaling function does allow z and x to overlap however.

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S. C. Kumbhakar et al. n  2   

yi − m(x i ; β) + μ∗ ezi δ . β, δ, μ∗ = min∗ n−1 β,δ,μ

(40)

i=1

The elegance of invoking the scaling property is that the SFM can be estimated in a distribution-free manner via NLS; the need for NLS stems from the fact that the scaling function must be positive, and if it was specified as linear, this would be inconsistent with theoretical requirements on the variance of the distribution. Direct NLS will produce a consistent estimator of all of the terms of the SFM. However, the error term εi∗ is heteroskedastic,

var(εi∗ |x i , zi ) = σv2 + σu2∗ e2zi δ , where σv2 = var(vi ) and σu2∗ = var(u∗i ). As such, a generalized NLS estimator would be called for to produce an efficient estimator (as similar to the MLE). Unfortunately, a generalized NLS algorithm hinges on distributional assumptions to appropriately separate σv2 and σu2∗ . An alternative, which allows valid inference to be undertaken, is to compute heteroskedasticity robust standard errors for β and δ [110]. An interesting extension of this idea was recently proposed by [81] for the setting where ui has already been converted into technical efficiency. In this case the level of inefficiency must be bound between 0 and 1. To account for this [81] model, the impact of z on the level of inefficiency through a probit function. Again, given the nonlinear nature of the probit function, this necessitates the use of NLS if one wishes to eschew distributional assumptions. With the wide range of statistical software that can quickly implement an NLS problem, it is perhaps surprising that this avenue has not been exploited in applied research. Certainly the scaling property is an assumption that requires judicious justification, but not more so than distributional assumptions imposed on the composed error structure of the SFM. It is also possible in this nonlinear setup that the calculation of expected firm efficiency can be done without requiring distributional assumptions, leading to the potential for more robust conclusions regarding observation-specific inefficiency. It is also possible to estimate the SFM in (39) without imposing assumptions on the scaling function. Currently no test of the scaling property exists without enforcing distributional assumptions. Alvarez et al. [8] proposed standard tests of the scaling property by using the nesting structure of the normal-truncated-normal distributional pair against the normal-half-normal distributional pair. Unfortunately this testing facility requires distributional assumptions on both vi and ui . A test of the statistical significance of the determinants of inefficiency, using the NLS framework just described, is available [53]. Under H0 : δ = 0 it follows that



yi = m(x i ; β) − μ∗ ezi δ + εi∗ = m∗ (x i ; β) − μ∗ (1 − ezi δ ) + εi∗ ,

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where m∗ (x i ; β) = m(x i ; β) + c for a constant c. That is, one can only identify μ

if at least one element of δ is nonzero; note that under H0 : 1 − ezi δ = 0, μ∗ cannot be separately identified. This lack of identification creates issues for inference under the null hypothesis and invalidates the common asymptotic behavior of Wald and likelihood ratio tests. The solution, which [53] proposed to avoid this problem, is to use the Lagrange multiplier (LM) test which involves estimation imposing the null hypothesis. A novel insight of [53] is that the LM test they proposed has power in directions where the scaling property does not hold. This is due to the fact that the model being tested H0 is indifferent to “how” inefficiency enters the model. Thus, while an explicit test of the scaling property without requiring distributional assumptions would be a useful tool, the [53] LM test is likely to be sufficient. The LM test is based on the derivative of the NLS criterion function in (40) with respect to δ, evaluated at the restricted estimates (δ = 0):   2  yi − m(x i ; β) + μ∗ μ∗ zi . n n

(41)

i=1

The test statistic is designed to determine how close the derivative of the NLS objection function (with respect to the parameters under the null hypothesis) is to 0. If the parameter restrictions are true, then this should be close to 0. The reason that distributional assumptions are not needed for this test to work properly is that this test is identical to an F -test, and F -tests are invariant to the scale of the covariates [53]. Thus, one can simply set μ∗ = 1 and use NLS to regress y on (x, z) and test the significance of δ.

Estimation When Determinants of Efficiency and Endogeneity Are Present Quite recently, attention has focused on estimation of the SFM when some of the determinants of inefficiency may be endogenous [11, 65]. These models can be estimated using traditional instrumental variables methods. However, given that the determinants of inefficiency enter the model nonlinearly, nonlinear methods are required. To begin, we consider the model of [11],

yi = x i β + vi − ui = x i β + vi − u∗i ezi δ ,

(42)

where the scaling property has been invoked. The covariates x i and zi are partitioned as  x 1i xi = , x 2i 

 z1i zi = , z2i 

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where x 1i and z1i are exogenous and x 2i and z2i are endogenous. The set of instruments used to combat endogeneity are defined as ⎡

⎤ x 1i w i = ⎣ z1i ⎦ , qi where q i are the traditional outside instruments. Identification of all the parameters requires that the dimension of q be at least as large as the dimension of x 2 plus the dimension of z2 (the rank condition). In the model of [11], endogeneity arises through correlation between a variable in the model (x 2 and/or z2 ) and noise, v. That is, both x and z are assumed to be independent of basic inefficiency u∗ . Given that E[ui ] is not constant, the COLS approach to deal with endogeneity proposed by [10] cannot be used here. To develop an appropriate estimator, add and subtract the mean of inefficiency to produce a composed error term that has mean 0,



yi = x i β − μ∗ ezi δ + vi − (u∗i − μ∗ )ezi δ .

(43)

Proper estimation through instrumental variables requires that the moment condition  

E vi − (u∗i − μ∗ )ezi δ |wi = 0.

(44)

The nonlinearity of these moment conditions would necessitate use of nonlinear two-stage least squares (NL2SLS) [9]. Latruffe et al. [65] have a similar setup as [11], using the model in (42), but develop a four step estimator for the parameters; additionally, only x 2 is treated as endogenous. Latruffe et al.’s [65] approach is based on [26] using the construction of efficient moment conditions. The vector of instruments proposed in [65] is defined as ⎡

⎤ x 1i wi (γ , δ) = ⎣ q i γ ⎦ ,

zi e zi δ

(45)

where q i γ captures the linear projection of x 2 on the external instruments q. The four-stage estimator is defined as Step 1 Regress x 2 on q to estimate γ . Denote the OLS estimator of γ as γ. Step 2 Use NLS to estimate the SFM in (42). Denote the NLS estimates of (β, δ) ¨ δ). ¨ Use the NLS estimate of δ and the OLS estimate of γ in Step 1 to as (β, ¨ construct the instruments wi ( γ , δ).

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¨ calculate the NL2SLS Step 3 Using the estimated instrument vector w i ( γ , δ), estimator of (β, δ) as (β, δ). Use the NL2SLS estimate of δ and the OLS estimate of γ in Step 1 to construct the instruments wi ( γ , δ). γ , δ), calculate the NL2SLS Step 4 Using the estimated instrument vector w i ( estimator of (β, δ) as ( β, δ). This multistep estimator is necessary in the context of efficient moments because the actual set of instruments is not used directly; rather w i (γ , δ) is used, and this instrument vector requires estimates of γ and δ. The first two steps of the algorithm are designed to construct estimates of these two unknown parameter vectors. The third step then is designed to construct a consistent estimator of w i (γ , δ), which is not done in Step 2 given that the endogeneity of x 2 is ignored (note that NLS is used as opposed to NL2SLS). The iteration from Step 2 to Step 3 does produce a consistent estimator of wi (γ , δ), and as such, Step 4 produces consistent estimators for β and δ. While [65] proposed a set of efficient moment conditions to handle endogeneity, the model of [11] is more general because it can handle endogeneity in the determinants of inefficiency as well.

Conclusions In this chapter we covered the workhorse SFM and discussed avenues to include determinants of inefficiency and productivity and how to deal with potential endogeneity issues. This material must give a good stepping-stone for a general reader to proceed to the next chapter, where we will discuss about various ways to analyze stochastic frontier problems with panel data, how to adapt quantile estimation to SFA, how to use robust methods involving nonparametric regression and local-likelihood, as well as what software can be used to implement the various methods of stochastic frontier analysis. We will defer further concluding remarks till the end of  Chapter 9 “Stochastic Frontier Analysis: Foundations and Advances II”.

Cross-References  Activity Analysis in Production Economics  Conceptualization and Measurement of Productivity Growth and Technical

Change: A Nonparametric Approach  Cost, Revenue, and Profit Function Estimates  Data Envelopment Analysis: A Nonparametric Method of Production Analysis  Distance Functions in Production Economics  Duality in Production  Elasticities of Substitution  Modeling Technical Change: Theory and Practice  Multiproduct Technologies

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 Reminiscences of “Returns to Scale in Electricity Supply”  Scale Elasticity and Returns to Scale

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9

Stochastic Frontier Analysis: Foundations and Advances II Subal C. Kumbhakar, Christopher F. Parmeter, and Valentin Zelenyuk

Contents Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Panel Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Time-Invariant Technical Inefficiency Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Time-Varying Technical Inefficiency Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Models That Separate Firm Heterogeneity from Inefficiency . . . . . . . . . . . . . . . . . . . . . . . . Models That Separate Persistent and Time-Varying Inefficiency . . . . . . . . . . . . . . . . . . . . . Models That Separate Firm Effects, Persistent Inefficiency, and Time-Varying Inefficiency . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The Four-Component Panel Data SFM with Determinants of Inefficiency . . . . . . . . . . . . . Inference Across the Panel Data SFM . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Nonparametric Estimation of the SFM . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Early Attempts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Local Likelihood Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Local Least-Squares Approaches . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Avoiding Distributional and (Some) Parametric Assumptions When Determinants of Inefficiency Are Present . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Future Directions in Semi- and Nonparametric Estimation and Inference of the SFM . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Quantile Estimation of the SFM . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

372 372 374 376 378 380 383 385 386 386 386 388 391 395 398 399

S. C. Kumbhakar () Department of Economics, State University of New York at Binghamton, Binghamton, NY, USA Inland Norway University of Applied Sciences, Lillehammer, Norway e-mail: [email protected] C. F. Parmeter Department of Economics, University of Miami, Miami, FL, USA e-mail: [email protected] V. Zelenyuk School of Economics and Centre for Efficiency and Productivity Analysis (CEPA), The University of Queensland, Brisbane, QLD, Australia e-mail: [email protected] © Springer Nature Singapore Pte Ltd. 2022 S. C. Ray et al. (eds.), Handbook of Production Economics, https://doi.org/10.1007/978-981-10-3455-8_11

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Additional Approaches/Extensions of the SFM . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Available Software to Estimate SFMs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Cross-References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

403 403 404 404 405

Abstract

This chapter continues to review some of the most important developments in the econometric estimation of productivity and efficiency surrounding the stochastic frontier model. As in the previous chapter, we continue to place an emphasis on highlighting recent research and providing broad coverage, while details are left for further reading in the rich (although not exhaustive) references at the end of this chapter. Keywords

Efficiency · Productivity · Panel data · Endogeneity · Nonparametric · Determinants of inefficiency · Quantile · Identification JEL codes

C10, C13, C14, C50

Introduction In this chapter, we will continue our discussion of one of the most popular paradigms in modern productivity analysis – stochastic frontier analysis, or SFA. In particular, section “Panel Data” focuses on various SFA models for analyzing variation of efficiency (or relative productivity) not only across firms but also over time, i.e., in the panel data context. Section “Nonparametric Estimation of the SFM” reviews several prominent semi- and nonparametric approaches to SFA. Section “Quantile Estimation of the SFM” briefly discusses a recent vein of literature focusing on quantile estimation of the SFM. Section “Additional Approaches/Extensions of the SFM” presents some further extensions of the SFM, while section “Available Software to Estimate SFMs” briefly summarizes some of the available software to estimate SFMs in practice. Section “Conclusions” concludes. We will use the same notation as in the previous chapter, with some modifications given the new contexts we discuss here.

Panel Data Our current discussion of the SFM has focused on having access to crosssectional data. When repeated observations of firms are available, then more useful information about inefficiency (and often with more flexibility) can be extracted,

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and a range of panel data SFMs are available to the applied researcher. Here we highlight some of the most prominent models. The advantage of panel data is that more information on inefficiency and productivity can be parsed, and in particular, shed light on changes in efficiency or productivity, which differs from a crosssectional setting, which can only provide a static portrayal of inefficiency. While [87] were the first to consider extending the cross-sectional SFM to the panel data setting, it was [92] who brought prominence to the use of models tailored exclusively to panel data. They raise three problems with cross-sectional models that are used to measure inefficiency and productivity: First, if the MLE is used to estimate the parameters of the SFM and inefficiency through JLMS, everything is contingent on distributional assumptions for both noise and inefficiency; second, technical inefficiency is assumed to be independent of the regressor(s)1 ; and third, the JLMS estimator is not a consistent estimator of u, as E[u|ε] never approaches u as the number of cross-sectional units approaches infinity (n → ∞). Access to panel data can, to varying degrees, mitigate all of these issues. However, with panel data comes a range of additional assumptions that the researcher needs to carefully consider before proceeding. To begin, consider the benchmark linear panel data regression model: yit = m(x it ; β) + ci + vit .

(1)

Aside from the indexing of our data by individual, i, and time, t, we have the presence of firm-specific heterogeneity, ci . The common dilemma facing application of the linear panel data regression model is how to treat the relationship between ci and x it . Under the fixed-effects (FE) framework [103], x it is allowed to be correlated with ci , and the parameters of the model can be estimated consistently using the within transformation [7]. Under the random-effects (RE) framework, x it and ci are required to be uncorrelated, leading to OLS being a consistent estimator, but is ultimately inefficient given that the variance-covariance matrix of the composed error term c + v is no longer diagonal. A feasible generalized leastsquares approach is available to obtain asymptotically efficient estimators of the parameters of the regression model in this case. Now, to think about where inefficiency enters the model in (1), we must characterize the nature of inefficiency. If inefficiency is assumed to be constant over time, then it is likely that ci might be augmented to also capture inefficiency. If inefficiency is time-varying, then we could include a second, one-sided error term to be convolved with vit in (1), in much the same way we did in the benchmark SFM. Or, it could be that inefficiency is composed of both a time-invariant component and a time-varying component. All told, the general panel data SFM is yit = m(x it ; β) + ci − ηi + vit − uit = m(x it ; β) + αi + εit ,

1 If

(2)

firms maximize profit, and inefficiency is known to the firm, then this assumption is unlikely to be true as firms may adjust their inputs to account for inefficiency (e.g., see [75]).

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where αi = ci − ηi with ci capturing time-invariant heterogeneity and ηi encapsulating time-invariant inefficiency, while εit = vit − uit with uit representing time-varying inefficiency. The panel data SFM looks identical to the panel data regression model in (1), except that, due to uit > 0, εit no longer has mean zero, and αi no longer solely captures individual specific heterogeneity. Early approaches that studied inefficiency in panel data settings placed restrictions on how inefficiency entered the panel data SFM. As time progressed, fewer assumptions were made, especially as more advanced econometric techniques were exploited.

Time-Invariant Technical Inefficiency Models When inefficiency in the panel data SFM is assumed to be time-invariant, it is possible to estimate the model without the need for distributional assumptions. To begin, we assume that uit does not exist in (2) and all time-invariant unobserved heterogeneity is inefficiency, αi = ηi . With these restrictions, the panel data SFM is written as yit = m(x it ; β) − ηi + vit ; i = 1, . . . , n; t = 1, . . . , T .

(3)

This model is termed the time-invariant SFM. Aside from the one-sided nature of ηi , this model can be estimated with standard panel data regression techniques, once an assumption on the underlying statistical relationship (either the FE or RE framework) between x it and ηi is made. Which framework to deploy depends upon the relationship that one assumes exists between the covariates of the model and firm-level inefficiency. Under the FE framework, correlation is allowed between x it and ηi , whereas under the RE framework, no correlation is permitted between x it and ηi . Regardless of which framework is deemed appropriate, neither requires distributional assumptions for η or v. This freedom from imposing a parametric assumption on the distribution of ηi (i.e., we have some statistical requirements on the distribution but do not require a precise parametric form) has led to the timeinvariant SFM being referred to as a distribution-free approach [92]. To estimate the time-invariant SFM respecting the one-sided nature of ηi , a simple transformation is needed to interpret the individual effect as time-invariant inefficiency as opposed to pure firm heterogeneity. One major limitation of the time-invariant SFM is that separate identification of inefficiency and individual heterogeneity is not considered. Additionally, the production technology is assumed to be time constant, which may be a further limitation depending upon the time dimension one has access to. We briefly discuss how to estimate the time-invariant SFM under the FE framework, which was first proposed by [92]. For ease of exposition, we assume m(·) is linear in x it . The time-invariant SFM is yit =β0 + x it β + vit − ηi

(4)

=(β0 − ηi ) + x it β =ci + x it β + vit

(5)

+ vit

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where ci ≡ β0 − ηi . Under the FE framework, ηi and thus αi , i = 1, . . . , n are allowed to have arbitrary correlation with x it . Given the similarity of the time-invariant SFM and a traditional panel data regression model, [92] used standard estimation methods to estimate the parameters of the model, namely, within estimation. The within transformation subtracts crosssectional means of the data from each  cross section (e.g., replacing yit by yit − y¯i· and xit by xit − x¯i· , where y¯i· = (1/T ) t yit , etc.), thereby eliminating ci . OLS can then be used to estimate the transformed model, essentially regressing transformed y on transformed x. The OLS estimator with the transformed data,  β, is a consistent estimator for β. An estimator of ci , cˆi , is constructed from the mean of the residuals for each cross-sectional unit, i.e., cˆi = y¯i· − x i· β, but it is biased, because ηi > 0 ∀i. A simple transformation will produce a consistent estimator of ηi . Once cˆi is determined, ηˆ i is estimated as [92]: ηˆ i = max{cˆi } − cˆi ≥ 0, i

i = 1, . . . , n.

(6)

This formulation implicitly assumes that the most efficient firm/DMU in the sample is 100% efficient. In other words, estimated inefficiency in the fixed-effects model is relative to the best firm/DMU in the sample. If one is interested in estimating firm-specific technical efficiency, it can be obtained from T E i = e−ηˆ i ,

i = 1, . . . , n.

(7)

Operating under the FE framework may be more appropriate for empirical applications in which inefficiency is believed to be correlated with the inputs used. However, a disadvantage of using the time-invariant SFM under the FE framework is that no other time-invariant variables can be included, for example, the gender of a plant manager or ownership status of the firm (which may not change over a short time). Effectively, the influence of time-constant variables will be accumulated in (and distort) the estimates of inefficiency. In settings where time-invariant variables are expected to be relevant regressors in the production model, an alternative is to operate under the RE framework. Estimation of the model still does not require distributional assumptions on v or η, but OLS on the transformed model no longer represents an efficient estimator given that the composed error term, vit − ηi , no longer has a diagonal variancecovariance matrix [besides the requirement of no correlation between inefficiency and inputs]. Schmidt and Sickles [92] discuss estimation under the RE framework through generalized least-squares as well. Another alternative, if one was uncomfortable with the implications stemming from RE framework, would be to make distributional assumptions and estimate the model via maximum likelihood. This avenue was suggested by [87] and can allow time-invariant covariates to enter the model while still identifying time-invariant inefficiency. The cost is the use of distributional assumptions so that the likelihood function can be constructed. Following [1, 87] assume that ηi follows a half-normal distribution and vit follows a normal distribution. Kumbhakar [53] discussed estimation of inefficiency in such a model by extending the JLMS formulation.

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Time-Varying Technical Inefficiency Models The time-invariant SFM allows inefficiency to differ across individuals but restricts any change over time. The implication of this is that an inefficient firm could not improve productivity over time by lessening inefficiency. This may be unrealistic in a variety of applied settings or where T is large. We must consider models that allow both technology and inefficiency to change over time to accommodate the idea of productivity and efficiency improvement at the firm level. A nice feature of time-varying SFMs is that the time-invariant SFM is a special case and, correspondingly, the time-invariant specification can be tested, opening up a variety of inferential opportunities for empirical analyses. To introduce the timevarying SFM, recall the model in (5): yit = ci + x it β + vit .

(8)

To allow ci to be time-varying, one may impose some reasonable and tractable structure, e.g., [26] suggested replacing ci by cit where cit = c0i + c1i t + c2i t 2 ,

(9)

where t is the time trend variable. The parameterization in (9) allows the parameters to be firm-specific. If the number of cross-sectional units (n) is not large, one can define n firm dummies and interact these dummies with time and time squared. These variables along with the regressors (i.e., the x variables) are then used in a standard OLS regression. The coefficients associated with the firm dummies and their interactions are the estimates of c0i , c1i , and c2i . These estimated coefficients can be used to obtain estimates of cit , c˜it . Again, the within estimator can be used to consistently estimate β along with the 3n parameters from the parameterization of cit . Finally, cˆit (the estimator of relative inefficiency) is obtained from cˆit = cˆt − c˜it

where

cˆt = max(c˜j t ) j

∀t.

(10)

In this model, efficiency is calculated relative to the best firm in each year. Since the firm with the maximum c˜j t is likely to change over time, different firms may be fully efficient (or inefficient at different levels) in different years. An alternative would be to calculate cˆj t = maxj t (c˜j t ), the maximum over all j and t, and replace cˆt with this definition in (10), and then efficiency is relative to the firm that was the most efficient over the entire sample period. The [26] estimation procedure is easy to implement. It relies on the standard panel data estimator with the FE framework. Note that since t appears in the inefficiency function, it cannot also appear as a regressor in x it , which would be required if one were to capture technical change, i.e., a shift in the production frontier, m(x). In other words, the above model cannot separate inefficiency from technical change, which is an obvious drawback of this approach. In general, if

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one wants to have both time-varying inefficiency and technical change, then the distribution-free route of [26] will not work. In this case, distributional assumptions will be necessary to allow time (and higher powers of it) to enter the model in various places. In a model with large n and small T , the model will have too many parameters (3n parameters in the cit function alone). A somewhat parsimonious time-varying inefficiency model was proposed by [69]: yit = m(x it ; β) + vit − uit = m(x it ; β) + εit .

(11)

where uit = ui t and t represent time-specific effects to be estimated. This model is quite flexible in its ability to model time-varying inefficiency. However, the temporal pattern of inefficiency is assumed to be exactly the same for all firms (t ). Under the FE framework, this specification can be viewed as an interactive effects panel data model, and estimation can be undertaken by introducing both firm and time dummies. Though no distributional assumptions are required by [69], the structure of inefficiency is similar to that assuming the scaling property discussed above. Again, given that inefficiency depends directly upon time, it is difficult to model both time-varying inefficiency and technical change in (11). A similar idea was used prior to [69] in [54] and [11], who proposed time-varying SFMs, but made distributional assumptions on both vit and uit and estimated the corresponding likelihood functions. Lee & Schmidt’s [69] model is more general than either the [54] or [11] models as both can be derived as special cases with appropriate parametric restrictions on t . Further still, the time-invariant SFM is also a special case: t = 1 ∀ t. Once t and ui are estimated, inefficiency can be estimated from uˆ it = max{uˆ j ˆt } − uˆ i ˆt . j

(12)

So far, the time-varying models that we have discussed treat inefficiency in a fully deterministic fashion, i.e., no distributional assumptions are required. In the [69] time-varying SFM, both ui and t are deterministic. This model can also be estimated treating the time component as deterministic, but the individual component as stochastic (through a distributional assumption). The deviation from the [69] time-varying SFM in (11) is that uit = G(t)ui with G(t) being a deterministic function of time and ui ∼ N+ (μ, σu2 ) [11, 54]. The ideas discussed pertaining to the scaling property appear here, where firms have a base level of inefficiency, and then, through time, become more or less efficient. The stochastic component, ui , utilizes the panel structure of the data in this model. The G(t) component is common across individuals (as in, but not limited to, [69]). Given ui ≥ 0, uit ≥ 0 is ensured by having G(t) > 0. Undoubtedly, the most popular form of G(t) is that proposed by [11] G(t) = exp [γ (t − T )] ,

(13)

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where T is the terminal period of the sample. The specification for G(t) is a simplification of the first attempt to introduce stochasticity into the time-varying SFM by [54] that assumes a more general specification of G(t) given by  −1 G(t) = 1 + exp(γ1 t + γ2 t 2 ) .

(14)

The [11] specification essentially enforces γ2 = 0 in the [54] time-varying SFM. The popularity of the [11] time-varying SFM has been aided by the freely available statistical package Frontier V4.1 which implements this model at the push of a button (see section “Available Software to Estimate SFMs” as well). Other specifications for G(t) have also been proposed; see [27] and [65] for more recent examples. Little research has been done on comparing a variety of forms of G(t). Lastly, modeling technical change in the [54] or [11] framework is trivial because the imposition of distributional assumptions allows inclusion of t (as a deterministic time-trend, e.g., linear, quadratic, etc.) as a component of x it .

Models That Separate Firm Heterogeneity from Inefficiency While the time-invariant SFM is a standard panel data model where ci is the unobservable individual effect, a notable drawback of this approach is that inefficiency is indistinguishable from individual heterogeneity. All time-invariant heterogeneity is confounded with inefficiency, and therefore cˆi will capture heterogeneity in addition to, or even instead of, inefficiency [37]. An important question for practitioners using the time-invariant SFM is how to view the time-invariant component. Should it be thought of as persistent inefficiency (as per [55–58]) or is it more appropriate to think of it as individual heterogeneity, capturing the effects of unobserved timeinvariant covariates? If it is the latter, then the insights from the time-invariant panel data SFMs are incorrect. A less rigid perspective is that the truth lies somewhere in the middle; inefficiency may be decomposed into two components: one that is persistent over time and one that varies over time. Unless persistent inefficiency is disentangled from time-invariant individual heterogeneity, practitioners need to choose between either the case in which ci represents persistent inefficiency or ci represents an individual-specific effect (heterogeneity). Here, we will discuss both specifications. In particular, we will consider models in which inefficiency is time-varying irrespective of whether the time-invariant component is treated as inefficiency or not. Thus, the model we will describe is yit = ci + x it β + vit − uit .

(15)

Compared to a standard panel data model, we have the additional time-varying inefficiency term, −uit , in (15). If one treats ci , i = 1, . . . , n as a random variable that may be correlated with x it but does not capture inefficiency, then the above

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model becomes what has been termed the “true fixed-effects” panel SFM [36]. The model is labeled as the “true random-effects” SFM when ci is treated as uncorrelated with x it . Note that these specifications are of the same nature as the models proposed by [55–58]. The difference is in the interpretation of the “time-invariant term,” ci . Estimation of the model in (15) is not straightforward. When ci , i = 1, . . . , n, are embedded in the FE framework, the model encounters the incidental parameters problem [76]. The incidental parameters problem arises when the number of parameters to be estimated increases with the number of cross-sectional units in the data, which is the case with the ci in (15). In this situation, consistency of the parameter estimates is not guaranteed even if n → ∞ because the number of ci increases with n. Therefore, usual asymptotic results may not apply. In addition to this specific statistical problem, another technical issue in estimating (15) is that the number of parameters to be estimated can be prohibitively large for large nT . For a standard linear panel data model (i.e., one that does not have −uit in (15)), the literature has developed estimation methods to deal with this problem. These methods involve transforming the model so that ci is removed before estimation. Without ci in the transformed model, the incidental parameters problem no longer exists, and the number of parameters to be estimated no longer increases with the number of individuals. Methods of transformation include conditioning the model on ci ’s sufficient statistic2 to obtain the conditional MLE, and the withintransformation model or the first-difference transformation model to construct the marginal MLE (e.g., [25]). For the basic panel data SFM, this could be done by transforming the error term if assumptions on vit and uit are such that the composed error term’s distribution is closed-skew normal (i.e., the normal-halfnormal distributional pair). These standard methods, however, are usually not applicable to (15). For the conditional MLE of (15), [37] showed that there is no sufficient statistic for ci . For the marginal MLE, the resulting model after the within or first-difference transformation usually does not have a closed-form likelihood function, if one uses standard procedures.3 In general this would not pose an issue as regression methods can be easily applied. However, given the precise interest in recovering estimates of the parameters of the distribution of inefficiency, maximum likelihood or specific moments of the distribution of the transformed error component are needed. This precipitates methods that can recover information regarding uit . Greene [37] proposed a tentative solution. He assumed uit follows a simple i.i.d. half-normal distribution and suggested including n dummy variables in the model for ci , i = 1, . . . , n and then estimating the model by MLE without any transformation. He found that the incidental parameters problem does not cause significant bias to the model parameters when T is relatively large (e.g., T ≥10). The

2 A sufficient statistic contains all the information needed to compute any estimate of the parameter. 3 Colombi

et al. [24] showed that the likelihood function has a closed-form expression. Chen et al. [20] considered a special case of [24] and derived a closed-form expression.

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problem of having to estimate more than n parameters is dealt with by employing an advanced numerical algorithm. There are some recent econometric developments on this issue. First, [20] proposed a solution in the FE framework. They showed that the likelihood function of the within-transformed and the first-difference model have closed-form expressions using results in [30]. The same theorem in [30] is used by [23] to derive the loglikelihood function in the RE framework. Using a different approach, [102] solve the problem classified in [37] by proposing a class of SFMs in which the within and first-difference transformations on the model can be carried out while also providing a closed-form likelihood function. The main advantage of such a model is that because the ci s are removed from the model in (15), the incidental parameters problem is avoided entirely. As such, consistency of the estimates is obtained for either n → ∞ or T → ∞, which is invaluable for applied settings. A further computational benefit is that the elimination of ci s reduces the number of parameters to be estimated to a manageable number. The catch is in the specification of inefficiency which is the product of an i.i.d non-negative random component and a deterministic function of zit (determinants of inefficiency). Formally, the [102] model is yit = ci + x it β + εit ,

(16)

where εit = vit − uit with vit ∼ N(0, σv2 ) and uit = git u∗i with u∗i ∼ N+ (μ, σu2 ), which is the now familiar scaling property model with a truncatednormal distribution for the basic distribution of inefficiency. For the scaling function [102] set git = g(zit δ). What allows the model transformation to be applied is the scaling property; the within and first-difference transformations leave this stochastic term intact as u∗i does not change with time. As [102] showed that the within-transformed and the first-differenced models are algebraically identical we have only provided discussion on the first-differenced model. However, a limitation of their model is that it does not completely separate persistent and time-varying inefficiency, a subject which we now turn our attention to. Lastly, as with the models of [54] or [11], the use of distributional assumptions allows both time-varying inefficiency and technical change to be modeled in (16).

Models That Separate Persistent and Time-Varying Inefficiency Although several models discussed earlier can separate firm heterogeneity from time-varying inefficiency (which is either modeled as the product of a time-invariant random variable and a deterministic function of covariates or distributed i.i.d. across firms and over time), none of these models consider persistent technical inefficiency. It is important to quantify persistent inefficiency, especially in short panels, as it captures the effects of inputs like management quality [75]. Unless there is a change in something that affects management practices at the firm (e.g., new government

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regulations or a change in ownership), it is unlikely that persistent inefficiency will change. The importance of persistent inefficiency contrasts with time-varying as this can change over time without requiring structural changes which impact the firm. This distinction between the time-varying and persistent components is important from a policy perspective as each yields different implications. Colombi et al. [23] refer to time-varying inefficiency as short-run inefficiency and mention that it can arise due to failure in allocating resources properly in the short run. They argued that, for example, a hospital with excess capacity may increase its efficiency in the short run by reallocating the work force across different activities. Thus, some of the physicians’ and nurses’ daily working hours might be changed to include other hospital activities such as acute discharges. This is a short-run improvement in efficiency that may be independent of short-run inefficiency levels in the previous period, which can justify the assumption that uit is i.i.d. However, this does not impact the overall management of the hospital and so is independent from timeinvariant inefficiency. To help formalize this issue more clearly, we consider the model4 yit = β0 + x it β + εit = β0 + x it β + vit − (ηi + uit )

(17)

Technical inefficiency is represented as ηi + uit where ηi is the persistent, firmspecific component (e.g., time-invariant ownership or geographic location) and uit is the time-varying component of technical inefficiency which is firm- and timespecific. Model (17) generalizes the previously discussed models because it allows for firm heterogeneity and time-invariant and time-varying inefficiency all at once. Such a decomposition is desirable because, since ηi does not change over time, for a firm to improve efficiency, a structural change in policy or management would need to arise. Additionally, ηi does not fully capture firm-level inefficiency because it does not account for learning over time since it is time-invariant; the time-varying component, uit can capture this aspect. In (17) the level of overall firm inefficiency, as well as the components, is important to know because they convey different types of information. Thus, for example, it may be argued that if residual inefficiency for a firm is relatively large in a particular year, this is due to an event which is unlikely to occur in the following year. Alternatively, if persistent inefficiency is large, then a firm is expected to operate with a relatively high level of inefficiency over time, unless some changes in policy and/or management occur. Therefore, a large value of ηi is more concerning in the long run given its persistent nature than is a high value of uit . The specification in (17) offers that advantage of testing for the presence of the persistent nature of technical inefficiency without the imposition of a specific parametric form of time dependence. Furthermore, by including time in the x it vector, (17) has the ability to separate exogenous technical change from technical inefficiency.

4 This

is the model proposed by [56–58], among others.

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To estimate the model, we rewrite (17) as yit = αi + x it β + ωit = (β0 − ηi − E[uit ]) + x it β + vit − (uit − E[uit ]). (18) The error, ωit , has zero mean and constant variance. Model (18) is a standard panel data model with firm-specific heterogeneity (one-way error component model) and can be estimated either by the within transformation (under the FE framework) or by generalized least-squares (under the RE framework). The SFM in (18) can be estimated under the FE framework using a step-wise procedure. Step 1 The standard within transformation can be performed on (18) to remove αi before estimation. Since both the components of ωit are zero mean and constant variance random variables, the within transformed ωit will generate a random variable that has zero mean and constant variance. OLS can be used on the within-transformed version of (18) to obtain consistent estimates of β. ˆ from Step 1, construct the pseudo-residuals Step 2 Given the estimate of β, β,  ˆ rit = yit − x it β, which contain information on αi + ωit . Using these, we first estimate αi from the mean of rit for each i. Then, we can estimate αi from maxi αˆ i − αˆ i = maxi {¯ri } − r¯i where r¯i is the mean (over time) of rit for firm i. Note that the intercept, β0 , and ωit are eliminated by taking the mean of rit over time for a firm. The above formula gives an estimate of αi relative to the best firm in the sample. Step 3 With our estimates of β and ηi , we calculate residuals eit = yit − x it βˆ + ηˆi , which contains information on β0 + vit − uit . At this stage, additional distributional assumptions are required to separate vit from uit . Here we follow convention and assume vit ∼ i.i.d. N(0, σv2 ) and uit ∼ i.i.d. N+ (0, στ2 ). MLE can be deployed here, treating eit as the dependent variable, to estimate β0 and the parameters associated with vit and uit . The log-likelihood for this setup is, letting N = nT , ln L = −N ln σ +

T n   i=1 t=1

ln Φ(−eit λ/σ ) −

n T 1  2 eit 2σ 2

(19)

i=1 t=1

Note that the parameters to be estimated here are β0 , σν2 , and στ2 . Once these parameters have been estimated, a JLMS conditional mean or median technique can be used to estimate uit for each observation. To summarize estimation under the FE framework, we estimate (18) using standard FE panel data tools to obtain consistent estimates of β in Step 1. Step 2 estimates persistent technical inefficiency, ηi . Lastly, Step 3 involves estimation of β0 and the parameters associated with the distributional assumptions imposed on the random components, vit and uit . One can then use the JLMS formula to estimate the time-varying (residual) component of inefficiency, uit . Note that no distributional

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assumptions are used in the first two steps. Without further assumptions, residual inefficiency cannot be identified, and hence, distributional assumptions are needed in the last step. This model can also be estimated under the RE framework (see also [23]).

Models That Separate Firm Effects, Persistent Inefficiency, and Time-Varying Inefficiency All of the panel data SFMs introduced so far have departed from the general model introduced in (2) in some aspect pertaining to the four separate error components. This is due to the fact that, until recently, it was not clear how to estimate the full panel data SFM represented by (2). The models of [23,59] overcome the limitations of the previous models by embracing the nature of the four-component structure inherent in the general panel data SFM. In the SFM represented in (2), the four components take into account different factors affecting output, given the inputs. As in [36, 37], the first component captures firms’ latent heterogeneity, which needs to be extricated from inefficiency; the second component captures time-varying inefficiency, the third component captures time-invariant inefficiency as in [56–58] while the fourth component captures stochastic shocks beyond control of the firm. The ability to estimate model (2) allows improvement over the previous models in several ways. To begin, while some of the time-varying inefficiency models just described can accommodate firm effects, these models fail to acknowledge the potential for factors that might have time-invariant effects on firm inefficiency. Second, SFMs which allow time-varying inefficiency commonly assume that the inefficiency level of the firm at time t is independent of its previous level of inefficiency; it is more reasonable to assume that a firm may eliminate some of its inefficiency by mitigating short-run rigidities, while other sources of inefficiency may remain over time. The former is captured by the time-invariant component, ηi , and the latter by the time-varying component, uit . Finally, many panel SFMs do consider time-invariant inefficiency, but do not simultaneously account for the presence of unobserved firm heterogeneity. In doing so, these models confound time-invariant inefficiency with firm effects (heterogeneity). The models proposed by [20,36,37,65,102] decompose the error term in the production function into three components: a firm-specific time-varying inefficiency term, a firm-specific effect capturing latent heterogeneity, and a time- and firm-varying random error term. However, these models consider any producer-specific, time-invariant component as unobserved heterogeneity. Thus, although firm heterogeneity is now accounted for, it comes at the cost of ignoring long-term inefficiency. As before, latent heterogeneity is confounded with long-run inefficiency. Estimation of the panel data SFM in (2) can be undertaken in a single-stage MLE method based on distributional assumptions on the four components [24]. We first describe a simpler, multi-step procedure [59]. For this, we rewrite the model in (2) as

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yit = β0∗ + x it β + αi + εit ,

(20)

where β0∗ = β0 − E[ηi ] − E[uit ]; αi = ci − ηi + E[ηi ]; and εit = vit − uit + E[uit ]. With this specification, both αi and εit are zero mean and constant variance random variables. (20) is estimated in three steps. Step 1 Standard random-effect panel regression is used to estimate βˆ (since (20) is a common panel data model). Predicted values of αi and εit , denoted by αˆ i and εˆ it , are also available after estimating (20). Step 2 Time-varying technical inefficiency, uit , is estimated using εˆ it from Step 1. Since εit = vit − uit + E[uit ],

(21)

2 2 by assuming √ vit is i.i.d. N(0, σv ) and uit is i.i.d. N+ (0, σu ), which yields E[uit ] = 2/π σu , and ignoring the difference between the true and predicted values5 of εit , we can estimate (21) using standard SFA techniques. Doing so produces predictions of the time-varying technical inefficiency component uit ,  E e−uit |εit , (i.e., [10]), which we call relenting technical efficiency (RTE). Step 3 Estimate ηi following a similar strategy as in Step 2. For this we use αˆ i from Step 1. Since

αi = ci − ηi + E[ηi ],

(22)

√ by assuming ci is i.i.d. N(0, σμ2 ), ηi is i.i.d. N+ (0, ση2 ), where E[ηi ] = 2/π ση , estimate (22) using the standard normal-half-normal cross-sectional SFM and obtain estimates of the persistent technical inefficiency component, ηi , following JLMS. Persistent technical efficiency (PTE) can then be estimated as PTE = e−ηi , where ηˆi is the JLMS estimator of ηi . Overall technical efficiency (OTE) is then constructed as the product of PTE and RTE, i.e., OTE = PTE×RTE. It is possible to extend this model (in steps 2 and 3) to include PTE and RTE that is distributed as truncated-normal or exponential as opposed to half-normal. While the multi-step approach of [59] is straightforward to implement, it is inefficient relative to full MLE. However, given the structure of the four separate errors, deriving the likelihood function was previously seen as infeasible. However, using insights related to the closed-skew normal distribution, as in [23], a tractable likelihood function turned out to be easily obtainable. Colombi et al. [23] made skew normal distributional assumptions for both ci −ηi and vit − uit in (20).6 Assuming vit is i.i.d normal and uit is i.i.d half-normal, the composed error vit − uit has a skew normal distribution. The same set of 5 Which

is the standard practice in any two- or multi-step procedure.

6 The skew normal distribution is a more general distribution than the normal distribution, allowing

for asymmetry [5].

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assumptions can be used for ci and ηi . Thus, model (2)’s likelihood can be derived. Even though the log-likelihood for (2) can be determined based on skew normal assumptions for the time-varying and time-invariant error components, it can be daunting to implement. Greene and Fillipini [39] recently proposed a simulationbased optimization routine which circumvents many of the challenges associated with direct optimization. They used a trick suggested by [17], conditioning on ci and ηi . This conditioning eliminates many of the computational hurdles that direct optimization of the likelihood function presents.

The Four-Component Panel Data SFM with Determinants of Inefficiency A further generalization of the four-component model in (2) involves the inclusion of determinants of inefficiency, either for the time-varying or the time-invariant components. An estimator for this model was recently proposed in [6], yit = m(x it ; β) + ci − ηi + vit − uit ,

(23)

2 ), u ∼ N (0, σ 2 ), c ∼ N(0, σ 2 ), and v ∼ N(0, σ 2 ). where ηi ∼ N+ (0, ση,i it + i it u,it c,i v,it These distributional assumptions are imposed so that the time-invariant composed error and the time-varying composed error both follow the closed-skew normal distribution. Each of the variance parameters of the four components is dependent  2 = σ 2 ezη,i δ η , upon a set of covariates and specified as an exponential function: ση,i η 





2 = σ 2 ezc,i δ c , σ 2 2 zu,it δ u , and σ 2 2 zv,it δ v . The time-constant and σc,i c u,it = σu e v,it = σv e time-varying z vectors can overlap due to the assumed distributional assumptions, that is, zc,i can share elements with zη,i and zu,it can share elements with zv,it . To estimate this four-component model, [6] used the insights of [39] and deployed simulated maximum likelihood techniques. The benefit of this approach is that rather than having T integrals to evaluate, by conditioning on ci − ηi , the likelihood function can be written as the product of T univariate integrals. Simulation methods are required to construct draws of ci −ηi inside the convolution density. The final log-likelihood function is

L=

n  i=1

log R

−1

T  R  2 εitr  εitr λit  Φ φ , σit σit σit r=1

(24)

t=1



     ezu,it δ u + ezv,it δ v , λit = ezu,it δ u −zv,it δ v , εitr = it − where σit = 

    ezc,i δ c Vir − ezη,i δ η |Uir | and it = yit − m(x it ; β). R is the number of draws over which to numerically evaluate the integral (larger R increases accuracy but slows down the routine; smaller R leads to faster computation but decreases accuracy). Lastly, both Vir and Uir are random draws from a standard normal

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distribution. Implementation of this routine is straightforward if one has access to a standard normal random number generator (which is typically available in any general statistical software). Once draws for Vir and Uir have been constructed, the likelihood is evaluated for the current set of parameters (β, δ u , δ v , δ η , δ c ). This process is then iterated over different sets of parameter values. Naturally, one can impose constancy at various parts of the error components by restricting δ  = 0 for  ∈ {u, v, c, η}.

Inference Across the Panel Data SFM The most general SFM in the panel context is the model which allows for firm-specific heterogeneity, persistent technical efficiency, relenting technical inefficiency, and individual time-specific idiosyncratic shocks. Colombi et al. [23] denote this model as TTT (True for having firm-specific heterogeneity, True for having time-constant inefficiency, and True for having time-varying inefficiency). The majority of all panel data models that have appeared in the literature are special cases of TTT. For example, the widely used true RE model of [37] is a special case of the TTT model. The same holds for all of the models we have discussed above. Naturally, inference is necessary to determine the model which best fits the data at hand. One benefit of nearly all of the panel data SFM discussed here is that standard panel data type tests (coefficient significance, fixed-versus random-effects framework, serial correlation, etc.) are easily implemented. This is similar to the benefits of the cross-sectional SFM that we discussed earlier. What is less straightforward is to test the most general TTT model against more restricted versions. Testing any of the previous models against the most general TTT model is a nonstandard problem because, under the null hypothesis, one or more of the parameters of interest lie on the boundary of the parameter space. Under reasonable assumptions, the asymptotic distribution of the log-likelihood ratio test statistic is χ¯ 2 , as discussed in chapter 9. For example, the model of [87] could be tested against the TTT model with the log-likelihood ratio test statistic but using χ¯ 2 to determine the p-value; see Table 1 in chapter 9. Future research focusing on adapting testing procedures to the TTT framework is important moving forward. As discussed earlier, the presence of both time-varying and time-invariant efficiencies yields different policy recommendations, and so working with models that document their presence, or lack of one, is important for proper analysis.

Nonparametric Estimation of the SFM Early Attempts In a nutshell, the semiparametric and nonparametric approaches to SFA typically use the benchmark SFM of [1] as the stepping-stone, generalizing it in different ways by relaxing all or some parametric assumptions by utilizing existing semiparametric

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and nonparametric statistical methods, such as the Nadaraya-Watson estimator, the local-polynomial estimator, or the likelihood (pseudo or local) estimators. To facilitate further and more precise discussion, recall that the benchmark SFM for a sample of n DMUs is given by yi = m(x i ) + vi − ui = m(x i ) + εi ,

i = 1, . . . , n,

(25)

where m(·) is the frontier of the production technology that can be used to transform q vector of inputs x ∈ R+ into scalar output yi , perturbed by some statistical noise vi and adjusted by technical inefficiency ui . As we discussed in Section 2in the previous chapter, traditional parametric estimation of the model begins by assuming a particular functional form for the production technology, most commonly a CobbDouglas or a Translog, besides making distributional assumptions on both vi and ui , which help to identify and estimate the unknown parameters via, say, the maximum likelihood approach. All the asymptotic results (consistency, asymptotic normality) are conditional on these assumptions, and if they happen to be incorrect, then, strictly speaking, all these results may be invalid. In such cases, the parametric MLE will be inconsistent or converging in probability not to the truth (e.g., true elasticities) but to some other numbers, which can even be very far from the truth if the parametric assumption made on a function is far from the true one. The early attempts to estimating SFM nonparametrically or semiparametrically go back to at least [8,34,47]. Specifically, [8] proposed a nonparametric approach in the spirit of the DEA estimator but embedded in a maximum likelihood framework, similar to parametric SFA, and thus allow for modeling both the noise and the inefficiency. A few years later, [34] proposed estimating the production frontier in another flexible manner, using nonparametric kernel regression methods embedded into the parametric maximum likelihood. About the same time, [47] suggested using the kernel regression estimator (Nadaraya-Watson in particular) for the panel data SFM. Importantly, note that the estimated conditional mean E[yi |x] of the production frontier is a biased estimator when ignoring the inefficiency term. Indeed, a critical assumption for consistent estimation of the production frontier in a regression setting is E[εi |x] = 0 and due to the one-sided nature of ui , this assumption is not satisfied, because E[εi |x] = μu = 0 in the simplest case when inefficiency is independent of the inputs, or more generally, E[εi |x] = μu (x) = 0 ∀x. Therefore, the production frontier cannot be identified in the regression setup, where one would estimate yi = m(x i ) + εi = m(x i ) + μu + (εi − μu ) ≡ m∗ (x i ) + εi∗ .

(26)

Realizing this, [34] proposed correcting the estimation bias of m(x) via a threestage semiparametric pseudo-likelihood estimation of the SFM. In this approach,

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at the first stage, one estimates (26) nonparametrically.7 Results from this first stage are then fed into the second stage, involving parametric MLE with particular assumptions on the distribution of the noise and inefficiency that help identifying and disentangling the two.8 Once the parameters of this symbiosis of MLE and kernel regression are estimated, the estimated conditional mean can then, in the third stage, be corrected for the bias by the estimated mean of inefficiency (as in COLS), μˆ u (x i ) to get a consistent estimator m(x i ) given by ∗ (x i ) − μˆ u (x i ), m (x i ) = m

(27)

[47] also proposed a similar strategy for correcting the bias occurring in estimating (26) nonparametrically, but avoided using MLE due to the possibility of disentangling the noise from inefficiency without distributional assumptions, by utilizing the panel-data SFA framework. The approaches of [34,47] provided a useful framework and formed a foundation on which many other approaches have been built.9 For example, more recent approaches of [68, 84] share some essence of [34] except that they required the estimated production frontier to obey traditional axioms of production, such as monotonicity and concavity, something that [34] did not accommodate in their approach. Specifically, [84] employ the framework of [34] but combine it with constraint weighted bootstrapping [31, 40] to ensure that monotonicity and concavity are enforced during estimation. More recently, [77] made improvements to the approach of [84], which resulted in small sample performance gains. On the other hand, [68] used an entirely different estimation approach, concave nonparametric least-squares (CNLS), to impose monotonicity and concavity. Lastly, [73] showed that while the estimator of [34] is consistent, the parametric estimator for the parameters of √ the density of the convolved error yields an asymptotic bias (when normalized by n) and proposed an alternative estimator that estimates the distributional parameters and the unknown frontier jointly.

Local Likelihood Methods The local likelihood approach [99] is known to be a natural alternative to the semiparametric pseudo-likelihood, and it was first proposed in the SFA context by

7 They

used a local constant (Nadaraya-Watson) regression, although other consistent nonparametric estimators can be used there too. 8 In their work, the normal-half-normal assumption was used, but other assumptions as discussed above can be used there too. Note, however, that for some alternative distributional assumptions on u, for example, exponential or truncated-normal, a concentrated version of the log-likelihood function may not exist, causing identification problems. 9 See [86] for a more comprehensive review of this topic.

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[60]. This approach closely resembles the parametric likelihood approach with the only (yet fundamental) difference being the kernel-based weights (instead of the equal weights) used to weigh each individual contribution to the likelihood, which help in localizing the estimation in the direction of each continuous variable through the bandwidths. Specifically, for a given regression error density, fε (ε, θ ), we have the local log-likelihood function Lˇ n (θ (x), mx ) = (n|h|)−1

n 

ln fε (yi − m(x i ); θ (x))Kix ,

(28)

i=1

where mx captures the conditional mean of y given x (a q × 1 vector of covariates)   q  x is −x s and θ is the vector of remaining parameters of fε , Kix = h−1 s k hs s=1

is the standard product kernel where k(·) is any second order univariate kernel (Epanechnikov, Gaussian, e.g.), hs is the smoothing parameter for the sth covariate (and is the sth element of vector h), while |h| = h1 h2 · · · hq . Kumbhakar et al. [60] used a local-linear approximation for the unknown production function m(x i ) combined with the assumption of a normal, half-normal convolved error term, where parameters are also modeled as unknown functions of the covariates: Lˇ n =(n|h|)−1

n   2 −0.5σ¨ x2 (x i ) − 0.5¨εi2 e−σ¨ x (x i ) i=1

  2 ¨ + ln  −¨εi eλx (x i )−0.5σ¨ x (x i ) Kix

(29)

¨ x (x i ), m ¨ x (x i ) = m ¨0 −m ¨ 1 (x i − x), σ¨ x2 (x i ) = σ¨ 02 + σ¨ 12 (x i − x), where ε¨i = yi − m  10 and λ¨ x (x i ) = λ¨ 0 + λ¨ 1 (x i − x). Noting that often the main focus of interest is related to σu ; [82] suggested directly parameterizing the local likelihood function in terms of ln σv2 and ln σu2 which also impose positivity of σv2 and σu2 throughout the estimation, making it more stable computationally. Park et al. [82] also outlined an asymptotic theory for modeling discrete variables in the context of the local likelihood approach, which can be imperative for many applications, which many covariates that researchers have access to are categorical in nature (regulated vs. non-regulated firms or industries, private vs. publicly owned companies, male vs. female managers, etc.). The local likelihood function in this case would be

10 One

could also use a quadratic approximation, but note that even in this local-linear case, there are already 3 + 3q parameters to estimate (i.e., optimize over) at each point of interest x: these are the three functional estimates, m ¨ 0 , σ¨ 02 and λ¨ 0 and the 3q derivative estimates of the functions, m ¨ 1, 2 ¨ σ¨ 1 and λ1 .

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ˇ (x c , x d ), mx c ,x d ) = (n|h|)−1 L(θ

n   2 c d   2 c d −0.5 ln eσ¨ v (x i ,x i ) + eσ¨ u (x i ,x i ) i=1

 2 c d 2 c d −0.5¨εi2 / eσ¨ v (x i ,x i ) + eσ¨ u (x i ,x i )  

σ¨ u2 (x ci ,x di )/2−σ¨ v2 (x ci ,x di )/2 σ¨ v2 (x ci ,x di ) σ¨ u2 (x ci ,x di ) Kix c W i (x di ). + ln  −¨εi e / e +e 

(30) where x ci is a vector of continuous regressors, while x di is a vector of discrete regressors, and W i (x di ) is an appropriate discrete kernel, e.g., the one proposed by [2] or its variations. The theory in [82] is derived for the case of kernel from I (xijd =xjd ) k , which is a standardized version of [90], given by W i (x d ) = j =1 ωj the Aitchison-Aitken kernel, standardized so that the bandwidths for a j th discrete variable, here denoted as ωj , are always between 0 and 1, regardless of the number of categories. However, this theory also extends (with some modifications) to cases with other discrete kernels. For example, one might prefer the so-called discrete Epanechnikov kernels, which are particularly useful and can be superior to others in case of sparse data (e.g., see [21] and the references cited therein). One can also use more adaptive bandwidths, e.g., allow for bandwidths of some or all continuous regressors to differ across categories of some or all discrete variables (e.g., see [70] for related discussion). Standard optimization algorithms can be used here, but as with any nonlinear optimization, careful choice of starting values is imperative, especially in selecting the bandwidths. For example, [60] suggested starting with the local-linear leastsquares estimates for m ¨ 0 and m ¨ 1 and the global, parametric maximum likelihood estimates for σ 2 and λ (from [1]) so that m ¨ 0 is properly corrected (as in [34]). Selection of the bandwidths is a very important step here (as is true in general for kernel-based methods), and many interesting general selection methods can be adapted to the current context. One of the most popular approaches is cross-validation.11 Kumbhakar et al. [60] outlined how to use least-squares cross-validation (LSCV) for their approach. Meanwhile, [82] suggested using maximum likelihood cross-validation (MLCV), which is more natural for the local likelihood approach, although it may be more demanding in computation. For the starting values in numerical optimization of LSCV or MLCV for selecting optimal bandwidths, one could use the so-called rules-of-thumb bandwidths that reflect the rates of convergence required for the asymptotic theory, e.g., for a continuous variable xsc , use h0 (xsc ) = 1.06 × n−1/(4+q) σˆ xsc , where σˆ xsc is estimated standard deviation of xsc , and ω0 = n−2/(q+4) for the discrete bandwidths. Kneip et al. [48] provide an update of the [60] estimator whereby the distributional assumption on the inefficiency term can be dropped. The only 11 For

[42].

more discussions on the pros and cons, as well as references on this approach in general, see

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parametric assumption required in [48] is that the two-sided error term is normal, which allows them to rely on penalized likelihood, where the unknown density is constructed nonparametrically via a histogram over the support of the covariate space and the penalty term is included to ensure appropriate smoothness of the resulting density. Both the theory and simulated evidence appearing in [48] suggest that this estimator works quite well in a variety of settings. To date, no application of this method has appeared to our knowledge, and so it represents an exciting opportunity moving forward.

Local Least-Squares Approaches In spite of the appealing theoretical advantages of the likelihood-based approaches, they involve numerical optimization of the local likelihood function over many parameters at each point of interest, which can be computationally complex, especially if bootstrap methods are needed to conduct inference. An attractive alternative that is much simpler to compute is provided by adopting the local leastsquares methods; because these methods do not require nonlinear optimization (given closed-form solutions), only basic matrix operations are required, marking dramatic improvements in computation time. Recently, [94] (SVKZ hereafter) proposed what can be viewed as a semi- or nonparametric generalization of COLS [79],12 which also allows for modeling determinants of inefficiency. Specifically, they considered a generalization of (25) given by yi = m(x i , zi ) + vi − ui = m(x i , zi ) + εi .

(31)

where m(x i , zi ) is the production frontier evaluated at x i , the realizations of inputs for observation i, and at zi , the realization of the so-called environmental factors faced by the observation i and disturbed by the realizations of statistical noise vi and inefficiency ui . In general, they required fairly general and mild conditions on the model, e.g., (ui |x i = x, zi = z) ∼ D + (μ(x, z), σu2 (x, z)) with D + (·, ·) being a non-negative random variable with mean μ(·, ·) and finite positive variance σu2 (·, ·), while (vi |x i = x, zi = z) ∼ D(0, σv2 (x, z)) with D(0, ·) being a random variable with mean zero and finite positive variance σv2 (·, ·). They also assumed that, conditional on (x i , zi ), ui and vi are independent random variables. Further, given that vi has a symmetric distribution around zero, while ui is a positive random variable from a skewed distribution E[εi |x, z] = −E[ui |x, z] = 0. Therefore, after recentering, we have yi = m(x i , zi ) + vi − ui + E[ui |x, z] − E[ui |x, z] = m∗ (x i , zi ) + εi∗

(32)

12 As with our earlier discussion, SVKZ referred to this approach as nonparametric MOLS, but cite

[79], who used the term COLS and so we refer to it as COLS here.

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where m∗ (x i , zi ) = m(x i , zi ) − E[ui |x, z] and εi∗ = εi + E[ui |x, z]. Adapting the strategy of COLS from [79], SVKZ proposed in the first stage the estimator of m∗ (x, z), m ∗ (x, z) using local-polynomial least-squares, noting that under mild regularity conditions and appropriate choice of bandwidths, such estimators have desirable statistical properties (consistency, asymptotic normality, etc.; see [32,42,71]). Then, in the second stage, they utilized the moment conditions implied by the assumptions on ui and vi , namely, E[ε∗ |x, z] = 0, E[(ε∗ )2 |x, z] = σu2 (x, z) + σv2 (x, z),   E[(ε∗ )3 |x, z] = −E (u − E[u|x, z])3 |x, z , and estimate the second and third moments of ε∗ using local-polynomial methods with the residuals  εi∗ = yi − m ∗ (x i , zi ) from the first stage, i.e., m 2 (x, z) =

n 

Ai (x, z) εi2

(33)

Ai (x, z) εi3 ,

(34)

i=1

and m 3 (x, z) =

n  i=1

where Aj (x, z) would vary depending upon the local smoothing method used. If one desires to estimate the level of the frontier in SVKZ’s setup, then a (local) parametric distributional assumption for ui is needed, although the ranking of output would be independent of this distributional choice. Importantly, note that if the moments of ui depend on x or z, then the frontier correction will also depend on x and z implying that any features of the production frontiers, such as returns to scale, may depend on the distribution of ui . One therefore needs to either make some type of distributional assumption or to assume a type of separability assumption, such as E[u|x, z] = E[u|z]. With the normal-half-normal framework, SVKZ showed (adapting [79]) that    1/3  π π m 3 (x, z)  σu (x, z) = max 0, 2 π −4 

π −2 , 2 (x, z) −  σu2 (x, z)  σv2 (x, z) = m π

(35) (36)

These estimates can then be used to obtain the estimates of the efficiency scores for each observation, in the spirit of [46], generalized to the heteroskedastic

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case involving E[ui |εi , x i , zi ] instead of E[ui |εi ]. However, as mentioned in the parametric context above, one should be careful interpreting these estimates of efficiency scores, as they are “predicted values” conditional on unobserved εi , replaced with its estimate for the specific realization i, and as such the prediction intervals tend to be quite wide (see [96] for related discussion). In turn, the conditional mean of inefficiency can be consistently estimated as   μu (x, z) =

2  σu (x, z). π

(37)

and then use it at any point of interest (x, z) to form a consistent estimate of the level of frontier, m(x, z), using μu (x, z). m (x, z) = m ∗ (x, z) + 

(38)

SVKZ also derived the asymptotic properties of these estimators, generalizing earlier results from [33] and [19]. Finally, and perhaps most interestingly, SVKZ pointed out that if one is only interested in the influence of z or x on the (conditional mean) efficiency, or as a special case to test if E[u|x, z] is a constant, then no parametric distributional specification is required for ui , only a condition that it belongs to the oneparameter scale family of distributions. Specifically, they showed that an elasticity of E[u|x, z], ψ , defined as ξψ (x, z) =

∂μu (x, z) ψ ∂ψ μu (x, z)

(39)

assuming that μu (x, z) = 0, can be estimated as  ξψ (x, z) =

 3 (x, z) ψ 1 ∂m 3 ∂ψ m 3 (x, z)

(40)

 3 (x, z)/∂ψ are the estimates from the local-polynomial where m 3 (x, z) and ∂m estimator and provided that m 3 (x, z) = 0 for the particular combination of interest (x, z). Importantly, SVKZ also derived the asymptotic law for this elasticity estimator, showing that   ξψ (x, z) − ξψ (x, z) −→ N(0, sξ2 (x, z)), (nhp+d+2 )1/2 

(41)

In turn, these asymptotic results can be used for statistical testing about influence of elements in (x, z) onto expected inefficiency. A practical limitation of SVKZ is that the estimated production technology may not satisfy axioms of production. One might be tempted to follow [68] or [84],  imposing the desired constraints first and then recovering E[u|x, z]. However, as we noted earlier, the methods of [68] and [84] work when the distribution of inefficiency

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is independent of x and z, i.e., when u is homoskedastic. The issue the applied researcher faces here is much more subtle. When heteroskedasticity is present in u, one must recognize that what is being estimated in the first stage is a conditional mean, and not a production frontier. Thus, it is not necessarily the case that the axioms of production should be expected to hold when estimating the conditional mean. Consider the case of a monotonic production function. The conditional mean of  output could be non-monotonic if E[u|x, z] was non-monotonic, even though the production function is monotonic. Further, it is well-known that adding two concave  functions might not produce a concave function, so even if E[u|x, z] was concave, adding it to the production frontier may not produce a concave production function. And therein lies the danger of imposing constraints when estimating the conditional mean, it is not necessarily the case that they should be satisfied. This might seem innocuous except for the fact that imposing constraints on a conditional mean that are incorrect will not produce a consistent estimator, and typically, consistent estimates in the first stage are needed for the second stage (recovering inefficiency) to produce valid estimates. Take, for example, the discussion in [67, pp. 233], who consider estimation of a production frontier nonparametrically, while also allowing u to depend on x. In this case, they stated (in our notation) “. . . Note that the shape of function g can differ from that of frontier m because E(ui |x i ) is a function of inputs x . . . It is also worth noting that function g is not necessarily monotonic increasing and concave even if the production function m satisfies these axioms because −E(ui |x i ) can be a non-monotonic and non-concave function of inputs . . . To apply CNLS in step 1, we need to assume that the curvature of the production function m dominates and that function g is monotonic increasing and concave (at least by approximation).” Unless the conditional mean of output satisfies the axioms of production, it is recommended the axiomatic restrictions be enforced after consistent, unrestricted estimation of the conditional mean as this will ensure that the first-stage estimator of the conditional mean is consistent. How exactly to do this is a relatively unexplored area in stochastic frontier analysis and is a fruitful avenue for future research. Figures 1, 2, and 3 illustrate the pitfalls of enforcing constraints ex ante on the conditional mean of y (given x). We have a single input, x, and our production frontier is logarithmic, which is naturally monotonic and concave. When inefficiency is homoskedastic, we see that the conditional mean is just a shift down of the production frontier and remains both monotone and concave. However, if we allow heteroskedasticity of inefficiency, e.g., through a quadratic relationship, then, depending on the nature of heteroskedasticity, we can violate monotonicity, Fig. 2, or concavity, Fig. 3, of E[y|x]. This quadratic relationship is not beyond the pale, even in the parametric setting.13

13 Wang

[101] documents non-monotonic efficiency effects in a panel of Philippine rice farmers based on the age of the farmer.

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Conditional Mean is Concave and Monotonic

0.55

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Fig. 1 Concave and Monotonic Conditional Mean and Production Frontier Under Homoskedastic Inefficiency. The solid line is the production frontier, while the dashed line is the conditional mean of output

Avoiding Distributional and (Some) Parametric Assumptions When Determinants of Inefficiency Are Present Here we discuss the approach of [85, 100]. Let the SFM be yi = m(x i ) + vi − ui = m(x i ) + vi − ui + E[ui |zi ] − E[ui |zi ] = m∗ (x i , zi ) + εi∗ .

(42)

where m∗ (x i , zi ) = m(x i ) + g(zi ), (ui |zi = z) ∼ D + (μ(x, z), σu2 (x, z)), while (vi |x i , zi ) ∼ D(0, σv2 ). This model is a special case of SVKZ’s model. Now, if we specify our production technology as m(x i ) = x i β and E[ui |zi ] = g(zi ), then if β were known, g(zi ) could be identified as the conditional mean of ε˜ i = yi − x i β given zi . However, β is unknown and must be estimated. It can be estimated as follows. Conditioning only on zi in equation (42), we have

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Fig. 2 Concave but Non-monotonic Conditional Mean and Production Frontier Under Heteroskedastic Inefficiency. The solid line is the production frontier, while the dashed line is the conditional mean of output

E[yi |zi ] = E[x i |zi ] β − g(zi ).

(43)

Subtracting (43) from (42) yields yi − E[yi |zi ] = (x i − E[x i |zi ]) β + εi .

(44)

If E[yi |zi ] and E[x i |zi ] were known, β could be estimated via OLS from (44). The idea is to replace the unknown conditional means with their nonparametric estimates [91]. To estimate both β and g(zi ), we replace E[yi |zi ] and E[x i |zi ] in (44) with ˆ E[y|z i] =

n  j =1

Aj (zi )yj

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Conditional Mean is Non−Concave l

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Fig. 3 Monotonic but Non-concave Conditional Mean and Production Frontier Under Heteroskedastic Inefficiency. The solid line is the production frontier, while the dashed line is the conditional mean of output

ˆ s |zi ] = E[x

n 

Aj (zi )x sj ,

j =1

For a given bandwidth, the conditional expectations for y and each element of x can be estimated, and OLS can then be used to obtain a consistent estimator of β. That is, instead of the usual regression of y on x, one performs the modified OLS ˆ ˜ where we have used the notation w˜ = w − E[w|z] regression of y˜ on x, to denote a random variable that has been conditionally demeaned. The estimates for β can then be used to obtain a consistent estimator of the conditional mean of inefficiency via standard nonparametric regression techniques. ˜ Let εˇ i = yi − x i  β, where  β is our estimate from the OLS regression of y˜ on x. We then estimate g(zi ) nonparametrically via local-polynomial least-squares as  g (zi ) =

n  j =1

Aj (zi )ˇεj .

(45)

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In the cross-sectional regression setting, without assuming some structure on the distributions of the error components, it is not possible to identify the impact that any given variable has on output directly, i.e., through the frontier, indirectly through inefficiency, or both.14 One way to achieve identification is through invocation of the separability assumption. This assumption, described in exceeding detail in [95], essentially requires two distinct sets of variables: those which influence the frontier and those which solely influence inefficiency. In the context of a model for which two-sided noise does not exist (the standard DEA framework), when this assumption is satisfied, a two-step approach is available which can produce consistent estimators of both the frontier function and the inefficiency of a firm [9, 95, 97]. In general, it is recommended that if variables which influence inefficiency exist, this information should be used directly, with a single-stage estimator, such as maximum likelihood. When the separability assumption holds, then the partly linear model of [85, 100] could be deployed (albeit with some parametric assumptions imposed) or the additive model previously described can be used.15 Importantly, the separability assumption can be tested in the stochastic frontier context, including the fully nonparametric or semiparametric frameworks. We can compare the estimates from the additively separable SFM, with that from a fully nonparametric model to determine if there are statistical differences. Fortunately, this type of setup is conducive to inference through either a residual sum of squares test or a conditional moment test. See the discussion in Chapter 6 of [42].

Future Directions in Semi- and Nonparametric Estimation and Inference of the SFM One of the future directions of research within non- and semiparametric SFA is, naturally, related to statistical inference. The asymptotic results developed in the above mentioned papers as well as various testing procedures developed in the

14 Hall and Simar

[41] discussed nonparametric identification of the mean of inefficiency subject to the variance of the noise distribution diminishing as n → ∞. Horrace and Parmeter [43] showed how to nonparametrically identify the full distribution of inefficiency if one assumes that v is distributed normal. 15 The approach of SVKZ allows for both x and z to influence both the frontier and inefficiency, and as such the separability assumption is not required. Yet, one may say that there is also a kind of “separability” structure involved implicitly: (x, z) is assumed to influence the frontier via the first moment, while for the inefficiency term, u, the same (x, z) is modeled through the skedastic function defining the second moment. Besides helping with statistical identification, such structure can be viewed as quite natural to the context of measurement. Indeed, one often thinks of the frontier as the level, and so using the (conditional) first moment, measuring the (conditional) average level of outputs, would be very natural. Meanwhile, the inefficiency is often understood as the deviation from the frontier, so it would be a more natural way to model it with the second moment. In addition, one could also think of the inefficiency as a reflection of the uncertainty and related “risk” to produce less than the potential and beyond the usual (and symmetric) noise, and it is very common to model risk through the second moment.

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general statistics community make a solid foundation for this to happen, with careful adaptation and extensive Monte Carlo evidence supporting the theory. Additionally, few of the methods discussed here have been fully developed in the panel data setting. It is worth noting that neither [60], nor [73], nor [82], nor SVKZ imposed any axioms of production on the frontier, e.g., monotonicity (i.e., require ∇mx ≥ 0 ∀x), although some of them have brief discussions about possible extensions to do so. Specifically, to impose the desired constraints, one could adapt ideas from [29] and [28], or use DEA or FDH on the fitted values from these methods (thus using the stochastic DEA or stochastic FDH approaches of [98]), or to employ the constraint weighted bootstrapping [31, 40], as was adapted to the baseline SFM by [84].

Quantile Estimation of the SFM A recent development in the estimation of the SFM has been to embrace the use of quantile methods [15, 16, 49, 72]. Quantile regression is known to provide a more complete picture of a conditional distribution [50, 52] and provides a robust alternative to ordinary least squares. Whereas the ordinary least-squares estimator stems from minimization of the sum of squared errors, the conditional quantile estimator is determined through minimization of the “check” function [51] defined for a particular quantile, the median say. The conditional quantile function Qy (τ |x) for a random variable y with conditional CDF F (y|x) is defined as F −1 (τ |x) = inf {y : F (y|x) ≥ τ } where τ is the τ th conditional quantile of the random variable y. Rather than directly inverting of the conditional distribution function, the conditional quantile can be determined through the loss function: ρτ () =  (τ − 1{ < 0}) .

(46)

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frontier with the conditional quantile estimator using τ = 0.5, 0.9 and 0.975. τ = 0.5 corresponds to the median and is equivalent to the conditional mean in the case that σu2 = 0 (see the discussion in [44]). Know et al. [49, p. 79] estimate conditional quantiles for τ = 0.85, 0.9 and 0.95, while [72, p. 1080] consider τ = 0.5 and 0.8. Lastly, [15, p. 572] recommended the use of τ = 0.95 for estimation of production frontiers and τ = 0.05 for estimation of cost frontiers. What is lost in the recommendations of this earlier research is how one estimates (or predicts) individual efficiency once the frontier has been estimated. Currently the standard practice is to treat any firm whose output lies above the frontier as fully efficient, and any firm whose output is below the frontier as inefficient, with inefficiency defined as the difference between the estimated frontier and observed output. However, both of these recommendations ignore the fact that the composed error term represents inefficiency and noise. There does not exist at present an approach that separates inefficiency from noise in a manner similar to [46]. One idea could be to use the conditional mode as proposed in [74]. This estimator can be interpreted as a maximum likelihood estimator for the distribution of the joint density of v and u, and more importantly, for positive residuals, it is always 0, which is akin to how inefficiency is currently calculated using conditional quantile estimation. Unfortunately, as with the conditional mean, the conditional mode estimator requires distributional assumptions for it to be operational. Lastly, we mention two important caveats with quantile estimation of frontiers. First, heteroskedasticity in either v or u has, until now, not been accounted for. This is a severe limitation as heteroskedasticity is commonly seen as present in v in applied efficiency studies, and researchers typically have access to an array of determinants of inefficiency, which induce heteroskedasticity in the inefficiency term. Moreover, unlike estimation of a conditional mean, when conditional heteroskedasticity is present, this can affect consistent estimation of the conditional quantile. Second, estimation of the conditional quantile for a specific value of τ is an implicit assumption on the ratio of signal to noise between σu2 and σv2 . To see this, more clearly, Figs. 4, 5, 6, and 7 present the results of quantile estimation for τ = 0.5, 0.8, 0.85, 0.9, and 0.95 for 1,000 observations drawn from the model: yi = xi0.4 evi −ui ,

(48)

with vi ∼ N(0, 1) and ui ∼ N+ (0, σu2 ). In Fig. 4 the inefficiency draws are taken with σu2 = 0.01, in Fig. 5 we have σu2 = 0.25, in Fig. 6 σu2 = 1, and in Fig. 7 σu2 = 4. In the case where σu2 = 4, this corresponds to a λ = σu /σv = 2 which is of a decent size for an applied efficiency study. In this case, the true frontier is approximately equal to the 85th quantile. It is clear that interpreting the frontier for a given quantile as the benchmark for a firm being efficient or inefficient is implicitly a statement on the ratio between the variance of the noise and the inefficiency for the sample. In Fig. 4, where λ = 0.01, the situation where there is almost no inefficiency, the frontier is nearly equivalent to the median, which is the least absolute deviation estimator that [44] discussed.

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While the quantile estimator marks an interesting and robust alternative to traditional stochastic frontier analysis, it should be clear that more work needs to be done. We direct the reader to the earlier referenced papers for more details and additional insights on how best to use conditional quantile methods at present for conducting efficiency analysis. Furthermore, panel estimation of quantiles, as well

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as semi- and nonparametric estimation of quantiles, is still in its infancy in this area, and extensions to the SFM have as yet to appear in the literature.

Additional Approaches/Extensions of the SFM As with any review or summary article, there is never enough space to cover all topics equally or broadly enough. The SFM has been studied and used for 40 years now, and even though we have covered a range of approaches and insights, there are still many topics which we did not cover. These include finite mixture models [18, 37, 80], the zero-inefficiency SFM [63], the meta-frontier [13, 14], total factor productivity change and its individual components [45], the two-tier frontier [88, 89], sample selection in the SFM [38, 64], and directional distance function estimation [4]. Parmeter and Kumbhakar [83] cover broadly estimation and inference of finite mixture models, the zero-inefficiency SFM, and issues pertaining to sample selection. Full details on the measurement of total factor productivity and separation into distinct components can be found in [66, chap. 11]. Both the two-tier frontier [61, 62, 81] and meta-frontier [3, 78] have started to receive more attention recently, but as of yet, no broad review of either exists. Regarding the estimation of directional distance functions, we refer interested readers to [35] for a thorough treatment.

Available Software to Estimate SFMs Despite the popularity of the SFM, only the most basic implementations of it are available across a wide array of statistical platforms. For example, in the R programming environment, the frontier [22] package allows for cross-sectional estimation of the SFM assuming either the half-normal or truncated-normal distribution for ui , and the [11, 12] panel data estimators of the SFM are implemented.16 There are similar estimators available in LIMDEP through the NLOGIT module, but these also include the normal-gamma specification as well as the true fixed- and true random-effects estimators along with the latent class stochastic frontier estimator. There are also several modules in the STATA software as described in [66] which implement several other panel data estimators as described earlier. Additionally, many authors provide their own personal codes. For example, Young Hoon Lee provides GAUSS code for a variety of cross-sectional and panel data stochastic frontier estimators on his webpage complete with several datasets (https://sites. google.com/site/yhnlee3/SFM-code). Federico Belotti provides integrated STATA

16 The frontier package accesses the Frontier V4.1 Fortran codes originally developed by Tim Coelli, which is also freely available (at http://www.uq.edu.au/economics/cepa/frontier.php), although fairly outdated by now (see also https://cran.r-project.org/web/packages/frontier/frontier. pdf).

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code which works with the basic frontier capabilities. These new codes are sfcross and sfpanel and can be obtained through his blog http://www. econometrics.it.17 However, there does not yet exist a singular software that implements all of the available estimators described here. This should not be surprising. As with any applied field, as statistical improvements are made, there is a lag with available software, and the array of options makes it infeasible to include all discussed models in a singular package. Researchers interested in the newest methods can invest in programming these methods and disseminating them to the field, or can collaborate with the authors of the original models to develop software that can be made widely available, and we strongly encourage researchers to do so.

Conclusions The review that we made in this and the previous chapters was meant to highlight some of the most important econometric developments over the past 40 years which improve the estimation of measurements of productivity and efficiency. While, in the previous chapter, we covered the workhorse SFM and how to include determinants of inefficiency and productivity and how to deal with endogeneity, in this chapter, we focused on the panel data, quantile estimation, and robust methods involving nonparametric regression and local likelihood. All told, a variety of methods and models exist for the practitioner, and our hope is that this review will encourage applied researchers to move away from some of the basic SFMs in search of more robust and insightful conclusions. While much has been covered, much remains unsaid. Important areas that are still being developed include modeling dependence between statistical noise and inefficiency, selection of firm technology, handling heterogeneous technology in a sample of firms, and how to allow a subset of firms to be fully efficient. While our discussion was couched in terms of the single equation stochastic production frontier, system-based approaches surrounding cost, profit, or revenue frontiers are also available, and, similar to the other methods mentioned earlier without any details, they deserve attention and separate reviews.

Cross-References  Activity Analysis in Production Economics  Data Envelopment Analysis: A Nonparametric Method of Production Analysis

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can install these commands via net install sfcross, all from (http://www. econometrics.it/stata) net install sfpanel, all from (http://www.econometrics.it/ stata), see also https://sites.google.com/site/productivityefficiency/home1 and for details refer to Chapter 17 of [93]

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Data Envelopment Analysis: A Nonparametric Method of Production Analysis

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Contents Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The Production Technology and Technical Efficiency . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Shephard Distance Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Input and Output Sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Nonparametric Construction of the Technology and Measurement of Technical Efficiency . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Assumptions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . DEA Models for Measuring Output and Output-Oriented Technical Efficiency . . . . . . . . . Technology and Efficiency Under Constant Returns to Scale . . . . . . . . . . . . . . . . . . . . . . . Multiplier Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Scale Efficiency . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Ray Average Productivity and Returns to Scale . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Most Productive Scale Size . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Identifying the Nature of Local Returns to Scale . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Identifying Returns to Scale for Inefficient Unit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The Case of Multiple MPSS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Choice Between Input- and Output-Oriented Projections . . . . . . . . . . . . . . . . . . . . . . . . . . . Graph Efficiency Measures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Graph Hyperbolic Distance Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Directional Distance Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Non-radial Measures of Efficiency . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Non-radial Russell Output Efficiency . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Non-radial Russell Input Efficiency . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Pareto-Koopmans Measures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Efficiency Measurement with Market Prices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Cost Efficiency . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Fixed Inputs and Short Run Cost Minimization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Using Total Cost as an Aggregate Input . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

411 413 414 415 416 416 417 420 422 424 425 425 426 429 431 433 434 434 436 438 439 440 440 442 443 445 446

S. C. Ray () Department of Economics, University of Connecticut, Storrs, CT, USA e-mail: [email protected] © Springer Nature Singapore Pte Ltd. 2022 S. C. Ray et al. (eds.), Handbook of Production Economics, https://doi.org/10.1007/978-981-10-3455-8_24

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Multi-location Cost Minimization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Revenue Efficiency . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Profit Efficiency . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Capacity Utilization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A Physical Measure of Short Run Capacity Utilization . . . . . . . . . . . . . . . . . . . . . . . . . . . . Long Run Capacity Output . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Economic Scale Efficiency . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Efficiency Measurement with Bad Outputs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Bad Output as Input . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Good and Bad Outputs as Joint Products . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Bad Output as a By-Product . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Joint Disposability and Material Balance Principle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Contextual Variables in DEA . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . All-Inclusive DEA . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A Second Stage Regression . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A Three-Stage Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

448 449 450 451 452 454 457 457 458 458 459 461 462 463 464 465 466 467

Abstract

In the Operations Research/Management Science literature, the nonparametric method of Data Envelopment Analysis (DEA) has gained wide popularity as a valid analytical format for efficiency evaluation. In economics, however, its reception has been far less enthusiastic. Yet, the intellectual roots of DEA go back to the seminal contributions to nonparametric analysis of production by Debreu, Shephard, Farrell, Afriat, and others. Over the past four decades, DEA has matured into a full blown non-parametric methodology for measuring productive efficiency that serves as an alternative to parametric Stochastic Frontier Analysis (SFA). Both grounded into the neoclassical theory of production, DEA and SFA provide the researcher alternative ways to calibrate testable relations between inputs, outputs, costs, revenue, and profit. Staring from the central concept of the Production Possibility set, this chapter provides a broad overview of the literature on DEA methodology for radial and non-radial measurement of technical efficiency from input and output quantity data under alternative returns to scale assumptions. This is followed by models for performance evaluation in the presence of market prices through cost, revenue, and overall profit efficiency- both in the long run when all inputs are variable and in the short run, when some inputs are fixed. DEA models for physical measures of the capacity output in the short run and economic measures of capacity in the long run are discussed. Alternative ways to incorporate the production of ‘bad’ or undesirable outputs collaterally with the ‘good’ or intended output in DEA models for efficiency measurement are presented. Finally, the role of contextual or environmental variables that affect efficiency is also discussed.

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Keywords

Neoclassical production theory · Production efficiency · Returns to scale · Cost efficiency · Capacity utilization · Bad outputs

Introduction Charnes, Cooper, and Rhodes (CCR) [19] introduced data envelopment analysis (DEA) as a nonparametric method of measuring technical efficiency in a paper published in the European Journal of Operational Research in 1978. At that time, the journal was only in its second year of publication. In the Operations Research (OR)/Management Science (MS) literature, DEA was enthusiastically accepted as a path-breaking contribution and within a few years found its place in the analytical tool kit of empirical researchers. By contrast, its reception in economics was lukewarm at best and outright skeptical, in general. Apparently, CCR first tried The American Economic Review, the iconic economics journal, as the outlet for their new model and later went to EJOR only after they were frustrated by rejections by top economics journals. In general, economists do not feel quite comfortable dealing with mathematical programming problems incorporating multiple inequality constraints, which cannot be easily assumed to hold as equalities simultaneously, thereby reducing it to a standard Lagrange multiplier problem. Moreover, DEA does not readily accommodate random noise in the data. Finally, in the absence of an explicit functional form of the underlying production, cost, or profit function, one cannot extract ready-to-use elasticities for policy evaluation. All this may account (at least partially) for the general reluctance to accept DEA as a valid empirical method in mainstream economics. Nonetheless, in economics the intellectual roots of DEA go all the way back to Debreu [28], Koopmans [44], Shephard [72], and Farrell [37] continuing further in the works of Afriat [1] and Hanoch and Rothschild [42] and beyond. At present, among the academics, there are two different views of DEA. In the OR/MS field, DEA is considered to be an extension of the method of linear programming (LP) introduced by Dantzig [27] and Charnes, Cooper, and Mellon [18] and others. The novelty in DEA is that unlike in earlier formulations of LP problems, the input requirements per unit of the output (sometimes described as the activity vector) are unknown. The primary emphasis there is on the algebraic formulation of the optimization problems and their solution algorithms. In economics, by contrast, DEA is an extension and refinement of the nonparametric approach to production analysis and is firmly grounded in neoclassical production economics. This is evident from the fact that originally designed to measure technical efficiency of production units, the DEA methodology has been extended to address general questions about the technology ranging from capacity utilization to technical change.

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The objective of this chapter is to provide a broad overview of DEA as a nonparametric method of empirical analysis of the production technology and producers behavior. The sequence of topics discussed in this chapter is not necessarily in the chronological order of their development in the DEA literature. Rather, it starts with the most general characterization of the technology (exhibiting variable returns to scale (VRS)) and gradually imposes additional restrictions (like constant returns to scale (CRS)) and/or makes behavioral assumptions like cost minimization, revenue maximization, or profit maximization.1 Our discussion of the different DEA LP models is prefaced by the underlying economic theory so that each of these models can be clearly interpreted economically. At the same time, this chapter does not provide an overall review of nonparametric analysis of production. The chapter is organized as follows. Section “The Production Technology and Technical Efficiency” introduces the production technology represented either by the production possibility set or by the families of input and output sets as the reference for measuring technical efficiency in production and/or Shephard distance functions. Section “Nonparametric Construction of the Technology and Measurement of Technical Efficiency” spells out the underlying assumptions for empirically constructing a nonparametric production possibility set and formulates the DEA models for measuring technical efficiency under variable and constant returns to scale. Section “Scale Efficiency” deals with scale efficiency and the most productive scale size (MPSS) and shows how to identify the nature of (local) returns to scale at any point on the frontier. Section “Graph Efficiency Measures” considers the graph hyperbolic and the directional distance functions for measurement of graph efficiency. Section “Non-radial Measures of Efficiency” deals with non-radial output and output-oriented Russell measures and the overall Pareto-Koopmans measure (also known as the slack-based measure (SBM)). Section “Efficiency Measurement with Market Prices” covers different measures of economic efficiency including DEA models for measurement of cost efficiency (both in the long run and in the short run) along with multi-location cost efficiency, revenue efficiency, and profit efficiency. Measurements of capacity utilization – both physical measures of capacity utilization (with fixed inputs) in the short run and economic measures in the long run (without fixed inputs), are considered in section “Capacity Utilization.” Section “Efficiency Measurement with Bad Outputs” deals with measurement of efficiency in the presence of undesirable outputs. Section “Contextual Variables in DEA” deals with non-discretionary (or contextual variables) that influence productivity but are not within the control of the decision-maker. Section “Summing Up” wraps up the chapter.

1 By

contrast, the earlier models in DEA and activity analysis assumed CRS (e.g., CCR [19]) and generalization to the VRS technology was a subsequent development [8].

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The Production Technology and Technical Efficiency The conceptual foundation of the neoclassical production economics is the production possibility set. Consider an industry where the individual production decision-making units (generically described as firms in economics and as DMUs in DEA) produce bundles of m outputs y = (y1 , y2 , ..yr , .., ym ) using bundles of n inputs, x = (x1 , x2 , . . . , xi , . . . , xn ). An input-output pair (x0 , y0 )constitutes a feasible production plan if, and only if, output y0 can be produced from input x0 .2 The production possibility set, T, consists of all feasible production plans and can be defined as   n m , y ∈ R+ ; y can be produced from x T = (x, y) : x ∈ R+

(1)

m+n . The production It is assumed that T is a closed and bounded subset of R+ technology of the industry is completely defined by the set T. In the single output case, one defines a production function:

y ∗ = f (x1 , x2 , .., xi , .., xn ) = f (x)

(2)

where y* is the maximum quantity of the (scalar) output y that can be produced from the input bundle x = (x1 , x2 , .., xi , .., xn ). Under most circumstances, one may assume that inputs may be left idle or otherwise wasted. As a result, the output produced is less than what is maximally producible. In this case, the production possibility set can be defined as T = {(x, y) : y ≤ f (x)} .

(3)

The graph of the production function (also known as the frontier of the production possibility set) is G = {(x, y) : y = f (x)} .

(4)

For any firm with the observed input-output bundle (x0 , y0 ) ∈ T, y0 ≤ f (x0 ). Its output-oriented technical efficiency can be measured as   y y0 0 τy x 0 , y0 = ∗ =  0  . y0 f x

(5)

  The input-output bundle x 0 , y0∗ lies on the graph of the technology and serves as the benchmark for measuring the output-oriented technical efficiency of the firm.

2 In

this chapter, vectors are denoted by superscripts and scalars by subscripts.

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An alternative way to measure technical efficiency would be to look for the maximum reduction in inputs feasible without lowering the output below the observed level, y0 . In other words, we want an input bundle x∗ such that f (x∗ ) = y0 . Because there will be many such bundles, one possible way to get a unique projection is to require that the target input bundle x∗ is proportional to the actual bundle x0 . Let x∗ = θ x0 . Then f (x∗ ) = f (θ x0 ) = y0 and the output-oriented technical efficiency of the firm is   τx x 0 , y0 = θ.

(6)

When the technology involves multiple outputs and multiple inputs, one cannot define the production possibility set T in terms of a production function and, instead, has to consider a production correspondence mapping from the input space into the output space through the transformation function F (x, y) = k.

(7)

An input-output combination (x0 , y0 ) is a feasible production plan if and only if F(x0 , y0 ) ≤ 0. Hence, the production possibility set can be defined as    T = (x, y) : F x, y ≤ 0 .

(8)

For any specific input-output pair (x0 , y0 ), the output-oriented technical efficiency is     1 τy x 0 , y 0 = ∗ where ϕ ∗ = max ϕ : x 0 , ϕy 0 ∈ T . ϕ

(9)

The corresponding output-oriented technical efficiency is     τx x 0 , y 0 = min θ : θ x 0 , y 0 ∈ T .

(10)

Shephard Distance Functions Shephard [72, 73] defined the (output) distance function evaluated at any arbitrary input-output bundle (x0 , y0 ) as 

  0 0 0 1 0 D x , y = min λ : x , y ∈ T . λ y

(11)

Note that λ > 1 implies that (x0 , y0 ) is infeasible whereas if λ < 1, (x0 , y0 ) is technically inefficient and the output bundle can be scaled upward without using any more input. Finally, λ = 1 implies that the bundle is technically efficient. Thus,

10 Data Envelopment Analysis: A Nonparametric Method of Production Analysis

415

the distance function relates to the transformation function as 

  1 D y x 0 , y 0 = min λ : F x 0 , y 0 ≤ 0 λ

(12)

and an alternative characterization of the production possibility set is    T = (x, y) : D y x, y ≤ 1 .

(13)

Also, it is evident from a comparison of (12) with (9)     D y x 0 , y 0 = τy x 0 , y 0 .

(14)

Analogous to the output distance function is the Shephard input distance function 

  1 0 0 x ,y ∈ T. D x x 0 , y 0 = max β : β

(15)

Note that β in (15) is the inverse of θ in (10). Hence,   Dx x 0, y 0 =

1  . τx x 0 , y 0

(16)

Input and Output Sets An alternative characterization of the production possibility set T is possible in terms of a family of input sets. For any output bundle y, the input (requirement) set consists of all input bundles that can produce that output bundle and can be expressed as V (y) = {x : (x, y) ∈ T } .

(17)

The frontier of the input requirement set is the input isoquant        / V y0 . V y 0 = x : x ∈ V y 0 and α < 1 ⇒ αx ∈

(18)

    It is clear that τx x 0 , y 0 = 1 ⇒ x 0 ∈ V y 0 . In a comparable manner, one can define the output set of any input bundle x as P (x) = {y : (x, y) ∈ T } and the output isoquant

(19)

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   P (x) = y : (x, y) ∈ T and α > 1 ⇒ x, αy ∈ /T .

(20)

    Further, τy x 0 , y 0 = 1 ⇒ y 0 ∈ P x 0 .

Nonparametric Construction of the Technology and Measurement of Technical Efficiency In practice, the production technology is unknown, and one needs to construct an approximation to the production possibility set T from data. The nonparametric method of data envelopment analysis (DEA) enables one to construct from the observed input-output data a piecewise linear approximation to the frontier of the production possibility set.3 In DEA one relies on a number of fairly weak assumptions about the production technology but leaves the exact functional form of the frontier unspecified. The result is a conservative approximation to the frontier.

Assumptions In particular, it is assumed that: (A1) All observed input-output bundles are feasible (A2) The production possibility set is convex (A3) Inputs are freely disposable (A4) Outputs are freely disposable Consider the data set consisting of the input-output bundles of N firms in the sample D=



 x j , y j ; j = 1, 2, . . . , N

(21)

  • Assumption (A1) implies that each (xj , yj )∈ D ⇒ x j , y j ∈ T . • The convexity assumption in (A2) implies (x1 , y1 ) ∈ T ∧ (x2 , y2 ) ∈ T ⇒ (λx1 + (1 − λ)x2 , λy1 + (1 − λ)y2 ) ∈ T ∀ λ ∈ (0, 1). Thus, (A1–A2) together imply that every convex combination of the observed bundles in D will also be in T. • Free disposability of inputs in (A3) implies that if (x0 , y0 ) ∈ T, then for every x ≥ x0 , (x, y0 ) ∈ T.

3 The

more popular alternative is Stochastic frontier analysis (SFA) introduced by Aigner, Lovell, and Schmidt [2] where one uses the maximum likelihood procedure to estimate a parametrically specified frontier production function incorporating a one-sided error term representing inefficiency and another two-sided error term representing random noise. See Kumbhakar and Lovell [46] for a detailed discussion of Stochastic frontier analysis (SFA).

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• Free disposability of outputs in (A4) implies that if (x0 , y0 ) ∈ T, then for every y ≤ y0 , (x0 , y) ∈ T.4 Under the assumptions (A1)–(A4), an empirical approximation of the production possibility set T is ⎧ ⎨

Tˆ = (x, y) : x ≥ ⎩

N 

λj x j ; y ≤

j =1

N  j =1

λj y j ;

N 



λj = 1j λj ≥ 0; j = 1, 2, . . . , N

⎫ ⎬

j =1

⎭

(22) The set Tˆ is often described as the free disposal convex hull of the set D of the observed input-output bundles and is the smallest set satisfying assumptions (A1)– (A4). The frontier of the set Tˆ provides the tightest envelop that covers the data from above and is an under approximation of the true production possibility set T.5

DEA Models for Measuring Output and Output-Oriented Technical Efficiency The output-oriented technical efficiency of a firm using input x0 and producing output y0 may be evaluated as   1 τy x 0 , y 0 = ∗ ϕ where ϕ ∗ = max ϕ N  j λj yr ≥ ϕyr0 (r = 1, 2, . . . , m) ; s.t. j =1

N 

j =1 N  j =1

j

λj xi ≤ xi0 (i = 1, 2, . . . , n) ;

(23)

λj = 1;

λj ≥ 0, (j = 1, 2, . . . , N) ; ϕ unrestricted. 4 If the transformation function is differentiable, weak disposability of inputs and outputs will imply ∂F ∂F ∂xi ≤ 0 for each input i and ∂yr ≥ 0 for each output r. In a later section in this chapter and in

much greater details in the  Chapter 12, “Bad Outputs” by Murty and Russell in this volume of the Handbook, free or strong disposability is contrasted with weak disposability where an output cannot be decreased (or an input increased) unilaterally but simultaneous reduction in multiple outputs or increase in multiple inputs may be feasible. 5 In the DEA literature, this is often described as minimum extrapolation. It should be noted that this is a criterion for estimation rather than a property of the technology.

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Even though ϕ is unrestricted, when (x0 , y0 ) is one of the bundles in D (say the bundle of firm k), (λk = 1, λj = 0 (j = k), ϕ = 1) is a feasible solution and hence 1 would be a lower bound for ϕ.6   N N   ∗ ∗ 0 0 ∗ j ∗ j The benchmark input-output bundle for (x , y ) is x = λj x , y = λj y j =1

j =1

constructed from the optimal solution of the problem. For any output r, the N  j output slack sr+ = λ∗j yr -ϕ ∗ yr0 represents additional expansion of the output j =1

feasible beyond the common expansion by the scalar ϕ* . Similarly, the input slack N  j λ∗j xi is the potential reduction in input i. The scalar ϕ∗ shows the si− = xi0 − j =1

factor by which all outputs can be expanded without requiring any additional input. In the single output case, the optimal value of the objective function in the outputoriented DEA problem (ϕ* ) yields an estimate of the maximum output7 producible from the input bundle x0 as   fˆ x 0 = ϕ ∗ y0 .

(24)

Based only on assumptions (A1)–(A4),   it is the most conservative estimate of the frontier output, and, hence, τy x 0 , y0 = ϕ1∗ is an upper bound on the outputoriented technical efficiency of the firm. This model was formulated in Banker, Charnes, and Cooper [9] and is commonly known as the output-oriented BCC DEA model. It should be emphasized that although the credit for this formulation of Tˆ is given to BCC, it was already formulated in Afriat ([1] Theorem 1.2, p. 571) for the single output case. The corresponding output-oriented technical efficiency of the firm using input x0 and producing output y0 may be evaluated as   τx x 0 , y 0 = θ ∗ , where

6 When

(x0 , y0 ) is not one of the observed bundles, non-negativity of the λs and the outputs will ensure that ϕ will never be negative even if it is lower than 1. However, if any individual input in the bundle x0 is smaller than the smallest value of the corresponding input across all observations in the data set D, (23) will not have a feasible solution. 7 The true maximum may actually be considerably higher than ϕ∗ y . But we cannot infer that on 0 the basis of the observed input-output bundles without making additional assumptions about the technology. However, it cannot be any smaller than ϕ∗ y0 if the assumptions (A1)–(A4) hold.

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Fig. 1 (a) Output-oriented technical efficiency. (b) Input-oriented technical efficiency

θ ∗ = min θ N  j λj yr ≥ yr0 (r = 1, 2, . . . , m) ; s.t. j =1

N 

j =1 N  j =1

j

λj xi ≤ θ xi0 (i = 1, 2, . . . , n) ;

(25)

λj = 1;

λj ≥ 0, (j = 1, 2, . . . , N) ; θ unrestricted. 

N  It should be noted that the benchmark input-output bundle x ∗ = λ∗j x j , j =1  N  λ∗j y j on the frontier for the output-oriented DEA problem will generally y∗ = j =1

be different from what was obtained as the efficient output-oriented projection. Output and output-oriented measurement of technical efficiency are shown graphically for the 1-output 1-input case in Fig. 1a and b, respectively. The curve y∗ = f (x) is the production frontier. Points A and B show two observed inputoutput bundles (xA , yA ) and (xB , yB ). The  output-oriented   efficient  projections of the two points shown in Fig. 1a are A∗ xA , yA∗ and B ∗ xB , yB∗ , respectively. The A output-oriented technical efficiencies of the two bundles are τy (A) = Oy Oy ∗ and A

B τy (B) = Oy ∗ . In Fig. 1b, their input-oriented efficient projections are the point  ∗ OyB   C xA , yA and D xB∗ , yB . Their input-oriented technical efficiency measures are

Ox ∗

Ox ∗

τx (A) = OxAA and τx (B) = OxBB . For the output bundle y0 , the input set can be empirically constructed as

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S. C. Ray

⎧ ⎫ N N N ⎬   ⎨    λj x j ; λj y j ≥y 0 ; λj =1; λj ≥0, (j =1, 2, . . . , N) . Vˆ y 0 = x : x≥ ⎩ ⎭ j =1

j =1

j =1

(26) The input-oriented technical efficiency τ x (x0 , y0 ) can alternatively be measured empirically as   θ ∗ = min θ : θ x 0 ∈ Vˆ y 0 .

(27)

For the input bundle x0 , the output set can be empirically constructed as ⎧ ⎫ N N N ⎬   ⎨    Pˆ x 0 = y : y≥ λj y j ; λj x j ≤x 0 ; λj =1; λj ≥0, (j = 1, 2, . . . , N ) . ⎩ ⎭ j =1

j =1

j =1

(28) The output-oriented technical efficiency can alternatively be measured empirically as   1 τy x 0 , y 0 = ∗ ϕ where   ϕ ∗ = max ϕ : ϕy 0 ∈ Pˆ x 0 .

(29)

Technology and Efficiency Under Constant Returns to Scale So far no assumption has been made about returns to scale. In some cases, it would be reasonable to assume constant returns to scale. The technology exhibits constant returns to scale (CRS) globally if8 (x, y) ∈ T ⇒ (kx, ky) ∈ T ∀k ≥ 0.

(30)

Under the CRS assumption, an empirical estimate of the production possibility set is

8 The

case of k = 0 corresponds to inaction when no input is used and no output is produced.

10 Data Envelopment Analysis: A Nonparametric Method of Production Analysis

Tˆ C

421

⎧ ⎫ N N ⎨  ⎬   = (x, y) : x ≥ λj x j ; y ≤ λj y j ; λj ≥ 0; j = 1, 2, . . . , N . ⎩ ⎭ j =1

j =1

(31) The set Tˆ C is sometimes described as the free disposal conical hull of D 9 . The output-oriented CRS technical efficiency is   1 τyC x 0 , y 0 = ∗ ϕC

(32)

where ϕC∗ = max ϕ N  λj y j ≥ ϕy 0 0 ; s.t. j =1

N 

j =1

(33)

λj x j ≤ x 0 ;

λj ≥ 0; (j = 1, 2, . . . , N) ; ϕ unrestricted. Similarly, the input-oriented CRS technical efficiency is   τxC x 0 , y 0 = min θ N  λj y j ≥ y 0 ; s.t. j =1

N 

j =1

(34)

λj x j ≤ θ x 0 ;

λj ≥ 0; (j = 1, 2, . . . , N) ; θ unrestricted. It may be noted under CRS, the output distance function is  that correspondingly,  y DC (x, y) = τyC x 0 , y 0 = ϕ1∗ , while the input distance function is DCx (x, y) = τyC

1 (x 0 ,y 0 )

=

1 θC∗ .

C

It is easy to verify that under the CRS assumption, input- and

output-oriented measures of technical efficiency are identical. In Fig. 2, the line f (x) = kx is the CRS production frontier. For the input-output bundle shown by

9 that the λs add up to unity. Under convexity only, Note the absence of the restriction  N N N    j j is feasible so long as x= λj x , y = λj y λj = 1 and no λj is negative. With j =1

j =1

j =1

the added assumption  of CRS, (kx, ky)  is also feasible for any k ≥ 0. CRS implies that for N N N    j j μj = kλj , k ≥ 0, μj x , μj y is feasible. But μj = k need not be equal to 1. j =1

j =1

j =1

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S. C. Ray

Fig. 2 Technical efficiency under CRS

the point PA , the output-oriented projection onto the frontier is QA , and the input∗ ∗ RA x A OxA PA x A oriented projection is RA . It can be seen that τyC (PA ) = Q = = = x Q x Ox A A A A A τxC (PA ) . Similarly, τyC (PB ) =

PB x B QB xB

=

RB xB∗ QB xB

=

OxB∗ OxB

= τxC (PB ) .

Multiplier Models It is worthwhile at this point to look at the following LP dual of the minimization problem in (34) above:

max s.t.

m 

r=1 m 

j

ur yr −

r=1 n  i=1

ur yr0 n  i=1

j

vi xi ≤ 0; (j = 1, 2, . . . , N )

vi xi0 = 1; ur , vi ≥ 0; (i = 1, 2, . . . , n; r = 1, 2, . . . , m) .

(35)

10 Data Envelopment Analysis: A Nonparametric Method of Production Analysis

Utilizing the normalization constraint

n  i=1

m 

h = max r=1 n  m 

s.t. r=1 n  i=1

i=1 j ur yr j

423

vi xi0 = 1, one can rewrite (35) as

ur yr0 vi xi0

(36)

≤ 1; (j = 1, 2, . . . , N)

vi xi

ur , vi ≥ 0; (i = 1, 2, . . . , n; r = 1, 2, . . . , m) . This is the original ratio form of technical efficiency proposed by CCR [19]. The multipliers u = (u1 , u2 , . . . , um ) are the shadow prices of outputs, while v = (v1 , v2 , . . . , vn ) are the shadow prices of inputs.10 The objective function is a measure of the shadow return on outlay of the unit under evaluation. For the VRS technology, consider the following dual of the output-oriented BCC model in (23) ψ = min v0 +

n 

vi xi0 i=1 n m   j j s.t.v0 + vi xi − ur yr i=1 r=1 m  0 ur yr = 1; r=1

≥ 0; (j = 1, 2, . . . , N)

(37)

ur , vi ≥ 0; (i = 1, 2, . . . , n; r = 1, 2, . . . , m) ; v0 unrestricted.

By standard duality results, ψ ∗ in (37) is equal to ϕ∗ in (23). Hence, the ratio measure of the output-oriented technical efficiency under VRS will be   τy x 0 , y 0 =

n 

i=1

v0 +

= max

r=1

v0 +

ur yr0

n 

vi xi0

j

ur yr

r=1

v0 +

1 ψ

i=1

m 

s.t.

m 

j

≤ 1; (j = 1, 2, . . . , N)

vi xi

n 

i=1

vi xi0 = 1;

ur , vi ≥ 0; (i = 1, 2, . . . , n; r = 1, 2, . . . , m) ; v0 unrestricted.

10 These

shadow prices or multipliers are uniquely designed for the unit under evaluation.

(38)

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S. C. Ray

Scale Efficiency11 While full technical efficiency requires a firm to produce the maximum output from its observed input bundle, in order to be considered scale efficient, the firm needs to operate at the scale of the input where average productivity reaches a maximum. In the 1-output 1-input case, the average productivity of a firm with input-output (x0 , y0 ) is yx00 . Now suppose that the production function is y∗ = f (x). Along the production function, y = f (x) so that AP (x) = yx = f (x) x . Thus, if 0) y = f (x0 ) , AP (x0 ) = f (x . Under VRS, average productivity varies across x0 * different levels of the input. Let x be the input level where average productivity

(x) (x) = xf (x)−f = 0 at the input level x* . attains a maximum. In that case, dAP dx x2 Frisch [40] described the input level where average productivity is maximum as the technical optimal production scale (TOPS). At the technically optimal input level (x) (x* ), dAP = 0. That is, within a small neighborhood of x* , average productivity dx remains unchanged as x changes. Hence, locally constant returns to scale holds. Also, at the input level (x* ), marginal productivity and average productivity are

f x∗ equal. Thus, f (x ∗ ) = (x ∗ ) . This implies that f (x∗ ) = f (x∗ )x∗ . Hence, SE (x0 ) =

f (x0 ) x0 f (x ∗ ) x∗

=

f (x0 ) . x0 f (x ∗ )

(39)



Now define δ ≡ f (x∗ ) and consider an artificial CRS production function y ∗∗ = r(x) = δx.

(40)

Then, the denominator in (39) becomes δx0 = r(x0 ). Therefore, an alternative measure of scale efficiency is f (x0 ) = SE (x0 ) = r (x0 )

y0 r(x0 ) y0 f (x0 )

τyC (x0 , y0 )

y

D (x0 , y0 ) = Cy . = τy (x0 , y0 ) D (x0 , y0 )

(41)

It should be noted that the expression in (41) measures the output-oriented scale efficiency of the input level x0 . In a perfectly analogous manner, one can take the output level y0 as given and measure the input-oriented scale efficiency SE (y0 ) =

D x (x0 , y0 ) τxC (x0 , y0 ) = x . τx (x0 , y0 ) DC (x0 , y0 )

(42)

11 For a more detailed discussion of this topic, refer to the  Chap. 17, “Scale Elasticity and Returns

to Scale”, by Podinovski and Førsund in this volume of the Handbook.

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425

As in the case of technical efficiency, the scale efficiency measure also will generally be different for input and output orientation. In Fig. 3 the point A shows the input-output bundle (x0 , y0 ), and the point B on the production function y∗ = f (x) is the output-oriented technically efficient projection. The most productive input scale f x∗ Dx0 Cx ∗ is x∗ and AP (x ∗ ) = (x ∗ ) = Ox ∗ = Ox . Also, the tangent to the production 0 function at the point C can be treated as a counterfactual CRS production function

y∗∗ = r(x) = δx; δ ≡ f (x∗ ). Thus, SE (x0 ) =

f (x0 ) x0 f (x ∗ ) x∗

Bx0 f (x0 ) = = = Dx0 r (x0 )

y0 r(x0 ) y0 f (x0 )

y

D (x0 , y0 ) . = Cy D (x0 , y0 )

(43)

Ray Average Productivity and Returns to Scale The single input single output case was useful for illustrative purposes but is of little relevance in real life because seldom, if ever, any output is produced from one input alone. Now consider a multiple input single output technology. The production function now shows the maximum scalar output producible from a vector of inputs. Consider an input-output combination (x0 , y0 ) that lies in the graph of the technology, G. That is, y0 = f (x0 ). Now consider another bundle (x1 , y1 ) also in the graph such that x1 = βx0 . The two input bundles differ only in scale but not in input proportions. The vectors x0 and x1 lie on the same ray through the origin in the input space. If the bundle x0 is considered to be 1 unit of a composite input, then x1 represents β units of the same input. If β is greater than 1, then x1 is a radial expansion of the x0 bundle. Now suppose that y1 = αy0 . The ray average productivity measured by output per unit of the composite input at (x0 , y0 ) is y0 and at (x1 , y1 ) is αyβ 0 . Now, if α > β > 1, then ray average productivity is increasing at (x0 , y0 ), and one can conclude that locally increasing returns to scale (IRS) holds at this point on the graph. On the other hand, 1 < α < β signifies locally diminishing returns to scale (DRS). Finally, α = β implies constant returns to scale (CRS). Note that these are all local characteristics of the technology and are evaluated as β → 1 from above. The technology may exhibit increasing, constant, or diminishing returns to scale at different points on the graph. This is why it is described as variable returns to scale (VRS).

Most Productive Scale Size Banker [5] generalized Frisch’s concept of the technically optimal production scale  output multiple input case. A feasible input-output bundle  0 to nthe 0multiple m is a most productive scale size (MPSS) if for all non-negative x ∈ R+ , y ∈ R+ scalars (α, β) for which (βx0 , αy0 ) is a feasible input-output combination, βα ≤ 1. In other words, (x0 , y0 ) is an MPSS only if there is no other feasible input-output bundle

426

S. C. Ray

Fig. 3 Scale efficiency

with the same mix of inputs and outputs but a higher ray average productivity. It is obvious that no feasible input-output bundle can be an MPSS unless it is in the graph.12 Banker and Thrall [8] and Ray [60] have shown that when the production possibility set is convex, IRS holds at all scales smaller than the smallest MPSS. Similarly, DRS holds at all scales larger than the largest MPSS.

Identifying the Nature of Local Returns to Scale There are three alternative ways to identify the nature of returns to scale at a specific input-output bundle: (a) a primal approach [5], (b) a dual approach [8], and (c) a nesting approach due to Färe, Grosskopf, and Lovell (FGL) [31]. This chapter, considers the primal approach in details both because it is the most

(x0 , y0 ) ∈ T but ∈G, then there will exist either some β < 1 such that (βx0 , y0 ) ∈ T or some α > 1 such that (x0 , αy0 ) ∈ T. In the former case, one gets βα > 1 for α = 1. In the latter case, α β > 1 for β = 1.

12 If

10 Data Envelopment Analysis: A Nonparametric Method of Production Analysis

427

popular alternative and also because it serves to identify the MPSS. The other two approaches are mentioned only briefly13 .

Banker’s Primal Approach Banker [5] developed the following important theorem that serves as a basis for identifying the nature of local returns to scale at the input-output bundle (x0 , y0 ) if it is on the VRS frontier and at its efficient projection if it is an interior point. Theorem 1 An input-output bundle (x0 , y0 ) is an MPSS if and only if the optimal value of the objective function of a CCR-DEA model equals unity for this inputoutput combination. Proof See Banker ([5], p. 40). This theorem only determines whether (x0 , y0 ) is an MPSS or not. It does not say anything directly about the nature of local returns to scale when it is not an MPSS. However, three important corollaries follow from the theorem: 1. If k =

N  j =1

λ∗j = 1 at the optimal solution of the DEA LP problem (34) above,

(x0 , y0 ) is an MPSS and CRS holds locally. N  λ∗j < 1 at the optimal solution of the DEA LP problem (34) above, 2. If k = j =1

IRS holds locally at (x0 , y0 ) or at its input-oriented efficient projection on to the VRS frontier if it is technically inefficient. N  3. If k = λ∗j > 1 at the optimal solution of the DEA LP problem (34) above, j =1

DRS holds locally at (x0 , y0 ) or at its input-oriented efficient projection on to the VRS frontier if it is technically inefficient. The intuition behind these corollaries is quite simple. When k = 1, the optimal solution from the output-oriented CRS problem in (34) is an optimal solution for the corresponding VRS problem. Because the CRS and VRS technical efficiency measures are identical, scale efficiency equals unity and (x0 , y0 ) is an MPSS. Moreover, by virtue of part (a) of the theorem, θ * equals unity and (x0 , y0 ) is on 0 ∗ 0 the frontier. If k = 1, the CRS input-oriented projection  ∗ (θ0 x0 , y ) is not a feasible 1 solution for the corresponding VRS problem. But k θ x , y is on both the CRS and the VRS frontiers. If k < 1, the input-oriented projection is to be scaled up to attain an MPSS, and it lies in the IRS region. On the other hand, if k > 1, it is to the right of the MPSS, and the input-oriented projection falls in the DRS region on the VRS frontier. 13 The more interested reader may see Ray [57] for detailed treatment of all these three approaches.

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S. C. Ray

A potential problem with this method of returns to scale characterization is that there may be multiple optimal solutions to the DEA problem in (34) with the sum of λs greater than 1 in some and less than 1 in others. In that situation conflicting conclusions would be drawn depending on which optimal solution was reached. This requires a modification of corollaries (2) and (3) as follows: (2a) Locally increasing returns to scale holds if k =

N  j =1

λ∗j < 1 at all optimal

solutions of the CRS DEA problem in (34). N  λ∗j > 1 at all optimal solutions of (3a) Locally diminishing returns holds if k = j =1

the CRS DEA problem in (34). This can be implemented in two steps. In step 1, the DEA problem in (34) is solved, and the optimal value θ * is determined. For (2a) above, in step 2, the following problem is solved:

max s.t.

N 

j =1 N 

j =1 N  j =1

λj

λj x j ≤ θ ∗ x 0 ;

(44)

λj y j ≥ y 0 ;

λj ≥ 0, (j = 1, 2, . . . , N) . If the maximum value of the objective function is less than 1, it can be concluded N  that k = λ∗j < 1 at all optimal solutions of (34). Similarly, in order to check for j =1

(3a), one minimizes the sum of λs in (44), and if the minimum is greater than 1, one can conclude that DRS holds locally.

A Dual Approach Banker, Charnes, and Cooper (BCC) [9] offer an alternative method of identifying local returns to scale from the following dual of the input-oriented VRS DEA problem: max u y 0 − u0 s.t. u y j − u0 ≤ v x j , (j = 1, 2, . . . , N ) ; v x 0 = 1; u, v 0 ≥ 0; u0 unrestricted.

(45)

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429

Note that at the optimal shadow prices (u∗ , v∗ ), the shadow profit from any



observed input-output bundle (xj , yj ) is u∗ yj − v∗ xj . Thus the constraints in (45) imply that u∗0 is the upper bound on the shadow profit from the bundle (xj , yj ). BCC have shown that: (i) CRS holds at (x0 , y0 ) if at the optimal solution of (45) u0 is zero (ii) IRS holds at (x0 , y0 ) if at the optimal solution of (45) u0 is 0 As in the case of Banker’s approach, multiple optimal solutions pose a problem, and the conditions (ii) and (iii) have to be appropriately modified.

A Nesting Approach Färe, Grosskopf, and Lovell (FGL) [31] consider a technology that lies in between CRS and the VRS technologies. They call it a non-increasing returns to scale (NIRS) technology. Under the assumption of NIRS     x 0 , y 0 ∈ T ⇒ kx 0 , ky 0 ∈ T for any k ∈ (0, 1) . The DEA approximation to an NIRS production possibility set is ⎧ ⎫ N N N ⎨  ⎬    Tˆ NI RS= (x, y) :x≥ λj x j ;y≤ λj y j ; λj ≤1j λj ≥0; j =1, 2, . . . , N ⎩ ⎭ j =1

j =1

j =1

(46) It may be noted that the frontiers of the CRS and NIRS production possibility sets coincide in the region of IRS. Similarly, the VRS and NIRS frontiers are identical in the DRS region. Therefore, when IRS holds at (x0 , y0 ), in an input-oriented model θ∗C = θ∗ N I RS < θ∗V where the superscripts C, N, and V refer to CRS, NIRS, and VRS. Similarly, θ∗C < θ∗ NI RS = θ∗V implies DRS. Of course, in the case of CRS, all three estimates of technical efficiency equal unity.

Identifying Returns to Scale for Inefficient Unit The concept of returns to scale is meaningful only when the relevant input-output bundle lies on the frontier of the production possibility set. For an inefficient bundle, one must consider its efficient projection – either input- or output-oriented. Unless similar returns to scale are found at both projections, one cannot conclusively determine the returns to scale at the observed input-output bundle. The following minor modification of Banker [5] considered by Cooper, Thompson, and Thrall [22] can be used not only to determine whether an input-output  bundle (x0 , y0 ) is an MPSS but also to identify the bundle x∗0 , y∗0 which is an

430

S. C. Ray

MPSS for (x0 , y0 ): max

s.t.

α β N 

λj y j ≥ αy 0 ;

j =1 N 

λj x j ≤ βx 0 ;

(47)

j =1 N 

λj = 1;

j =1

α, β, λj ≥ 0, (j = 1, 2, . . . , N) . As such, the objective function is nonlinear. However, it can be easily transformed into a linear programming problem. Define t = β1 , ρ = βα , and μj = tλj (j = 1, 2, . . . ,N). Note that non-negativity of β and λj s ensures that t and μj s are also non-negative. Problem (47) can, therefore, be reformulated as the following linear programming problem: max ρ s.t.

N 

μj y j ≥ ρy 0 ;

j =1 N 

μj x j ≤ βx 0 ;

(48)

j =1 N 

μj = t;

j =1

t, μj ≥ 0, (j = 1, 2, . . . , N ) . ∗

From the optimal solution of this problem, we can derive β ∗ = t1∗ and α ∗ = ρt ∗ . One can then infer the nature of returns to scale from these values of α* and β* . It may be pointed out here that because the only restriction on t is non-negativity, (48) is simply the output-oriented CCR DEA problem and ρ1∗ is the same as the   output-oriented CRS technical efficiency τyC x 0 , y 0 . Because (x0 , y0 ) is assumed to be a feasible input-output bundle, (α = β = ρ = 1) is a feasible solution for this problem. Hence, the optimal value ρ ∗ is always greater

10 Data Envelopment Analysis: A Nonparametric Method of Production Analysis

431

∗

than or equal to 1.When ρ ∗ = βα ∗ exceeds unity, we know that (x0 , y0 ) is not an MPSS. But we can also conclude that (β ∗ x0 , α ∗ y0 ) is an MPSS. When the bundle (x0 , y0 ) is not itself an MPSS, ρ ∗ > 1 so that α ∗ > β ∗ . If the MPSS is unique, there are five different possibilities: (i) 1 < β ∗ < α ∗ ; (ii) β ∗ < α ∗ < 1; (iii) β ∗ = 1 < α ∗ ; (iv) β ∗ < 1 = α ∗ ; and (v) β ∗ < 1 < α ∗ . When the MPSS is unique, if 1 < β ∗ < α ∗ , both input- and output-oriented projections of the bundle (x0 , y0 ) fall in the region of IRS. In this case, the unit is conclusively too small relative to its MPSS. Similarly, if β ∗ < α ∗ < 1, both input- and output-oriented projections fall in the region of DRS. The implication is that the unit is too large. When β ∗ = 1 < α ∗ , the input scale corresponds to the MPSS but the output scale is too small. This is only due to output-oriented technical inefficiency but there is no scale inefficiency. Similarly when β ∗ < 1 = α ∗ , there is output-oriented technical inefficiency but no scale inefficiency. Finally, in the intermediate case, where β ∗ < 1 < α ∗ , the input scale is bigger than the MPSS, and the output-oriented projection falls in the region of DRS. At the same time, the input scale is smaller than the MPSS, and the inputoriented projection falls in the region of IRS. When β ∗ = 1 < α ∗ , the (α = 1, β = 1) point lies directly below the MPSS. Similarly, for β ∗ < α ∗ = 1, the (α = 1, β = 1) point lies on the horizontal line through the MPSS. Figure 4a through c graphically illustrate the three cases mentioned above. In each diagram, the horizontal axis measures the input scale, and the vertical axis measures the output scale of a specific input-output bundle shown by the point (α = 1, β = 1) in each of these three diagrams. The broken line shown as the frontier is the fixed-mix graph of the technology      G x 0 , y 0 = (α, β) : F βx 0 , αy 0 = 0

(49)

The MPSS is shown by the point (α ∗ , β ∗ ). In Fig. 4a, 1 < β ∗ < α ∗ and both the input and output-oriented projections of (x0 , y0 ) will be in the region of increasing returns to scale. Figure 4b illustrates the case where β ∗ < α ∗ < 1 and both projections will be in the region of diminishing returns to scale. The intermediate case, where β ∗ < 1 < α ∗ , is shown in Fig. 4c. In this case, the output scale is smaller than the MPSS and the output-oriented projection would be in the region of increasing returns but the input scale is bigger than the MPSS and the outputoriented projection will be in the region of diminishing returns. When β ∗ = 1 < α ∗ , the (α = 1, β = 1) point lies directly below the MPSS. Similarly, for β ∗ < α ∗ = 1, the (α = 1, β = 1) point lies on the horizontal line through the MPSS.

The Case of Multiple MPSS Next consider the possibility of multiple MPSS. It is obvious that when (48) has a unique optimal solution (in particular, t* is unique), there cannot be multiple MPSS. If different values of t are obtained in multiple optimal solutions (of course,

432

S. C. Ray

a

b

c

Fig. 4 Unique and ambiguous RTS characterization of inefficient units

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433



μ∗j = t1 across all optimal   ∗ solutions of (48) corresponds to the smallest MPSS α1∗ = ρt1 , β1∗ = t11 .Similarly,  ∗ the smallest t ∗ = μj = t2 at an optimal solution yields the largest MPSS j   ∗ α2∗ = ρt2 , β2∗ = t12 .

with the same optimal value ρ ∗ ), the largest t ∗ =

j

Now the returns to scale classification of (x0 , y0 ) needs to be revised appropriately. In particular: (i) β1∗ < α1∗ < 1 corresponds to IRS. (ii) β1∗ < α1∗ < 1 < β2∗ < α2∗ corresponds to CRS. (iii) 1 < β2∗ < α2∗ corresponds to DRS. For other feasible combinations of (α ∗ , β ∗ ), the returns to scale classification will depend on the direction of projection (whether input- or output-oriented). Zhu [77] uses a single input single output example to partition the interior of the production possibility set into six different regions for returns to scale classification of inefficient production units14 . In three out of these six regions, both input- and output-oriented efficient projections exhibit the same returns to scale: increasing, constant, or diminishing. In the remaining three, increasing returns at the input-oriented projection combines with constant or diminishing returns at the output-oriented projection, or constant returns at the input-oriented projection is associated with diminishing returns at the output-oriented. In order to correctly locate an inefficient unit in the appropriate region, one has to ascertain returns to scale at both projections.

Choice Between Input- and Output-Oriented Projections Except in the case of globally constant returns to scale, output- and input-oriented technical efficiency measures would differ for the same firm. An important question is how to decide which measure is preferable. As a general rule, the answer depends on whether output augmentation is more important that input conservation in a specific context. In many situations, however, there is no clear-cut priority. A rule of thumb would then be to select the orientation that yields a lower measure of efficiency under the VRS assumption. The logic behind this criterion is that the corresponding efficient projection would have a higher level of scale efficiency. This can be explained by a simple 1-input 1-output example. Consider a technically inefficient input-output combination (x0 , y0 ). Now suppose that the output-oriented efficient projection is (x0 , ϕ∗ yo ) while the input-oriented projection is (θ ∗ x0 , yo ). Thus the corresponding technical efficiency measures are τy = ϕ1∗ and τ x = θ ∗ .

14 See

also the earlier paper by Seiford and Zhu [71].

434

S. C. Ray

Assume, arbitrarily, that τ y < τ x . This implies y0 θ ∗ x0

∗

1 ϕ∗

< θ ∗ or

1 θ∗

< ϕ ∗ . Therefore,

< ϕx0y0 . This shows that average productivity is higher at the output-oriented efficient projection than at the input-oriented projection of (x0 , y0 ).

Graph Efficiency Measures For the output-oriented radial measure of technical efficiency of an inefficient unit, one projects the observed input-output bundle vertically upward onto the frontier. Similarly, for the output-oriented measure, it is projected horizontally leftward. One may, however, prefer to project to a point on the frontier which is strictly to the northwest of the observed point. Movement to such a point involves some increase in output simultaneously with some decrease in input. A comparison of this efficient projection with the observed bundle yields a graph efficiency measure that takes account of potential increase in output and decrease in input simultaneously. However, there is an ambiguity about the optimal projection in this case. In a 1-output 1-input diagram with output measured along the vertical and input along the horizontal axis, the optimal output-oriented projection is directly above the actual point. Similarly, for the input-oriented model, it is horizontally toward the left. However, for the graph efficiency, one can, in principle, select any point in between these two limits. The choice of a specific point in this segment of the frontier must be based on some other reasonable criterion. While there are several alternative models, the more popular ones are the graph hyperbolic and the directional distance function.

Graph Hyperbolic Distance Function The graph hyperbolic distance function (GHDF) is defined as

δ

GH



  1 0 0 0 0 x , δy ∈ T . x , y = max δ : δ

(50)

It derives its name from the fact that in the 1-output 1-input case, the optimal projection would be located at the point where a rectangular hyperbola through the observed point intersects the production frontier in the input-output space. In Fig. 5 the point A shows the input-output bundle (x0 , y0 ). The curve xy = k is a rectangular hyperbola through A. The point B where this curve intersects the production frontier y∗ = f (x) is the graph hyperbolic efficient projection of A. At B, x ∗ = 1δ x0 , y ∗ = δx0 . The DEA LP problem for (50) above is

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435

Fig. 5 Graph hyperbolic distance function

max δ  s.t. λj yrj ≥ δyr0 ; (r = 1, 2, . . . , m) j λj xij ≤ 1δ xi0 ; (i = 1, 2, . . . , n) j  λj = 1; λj ≥ 0; (j = 1, 2, . . . , N)

(51)

j

Under CRS, the constraint on the sum of the λs is deleted. This is a nonlinear programming problem. However, one can define β = δ 2 and μj = δλj (j = 1, 2, . . . , N), and the model can be rewritten as15 max β  s.t. μj yrj ≥ βyr0 ; (r = 1, 2, . . . , m) j μj xij ≤ xi0 ; (i = 1, 2, . . . , n)

(52)

j

μj ≥ 0; (j = 1, 2, . . . , N) For VRS, one can use the linear approximation as

15 See

Färe, Grosskopf, Lovell, and Pasurka [33].

1 δ

≈ 2 − δ at δ = 1 to revise (51)

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S. C. Ray

max δ  s.t. λj yrj ≥ δyr0 ; (r = 1, 2, . . . , m) j λj xij + δxi0 ≤ 2xi0 ; (i = 1, 2, . . . , n) j  λj = 1; λj ≥ 0; (j = 1, 2, . . . , N )

(53)

j

Directional Distance Function Chambers, Chung, and Färe (CCF) [15, 16] introduced the directional distance function (DDF) based on Luenberger’s benefit (or shortage) function [49] as a measure of inefficiency. While in the GHDF the direction of projection of an inefficient bundle on to the frontier is endogenously determined, in the case of the DDF, the direction is prespecified by the analyst. Consider a bundle g = (gx , gy ) which may or may not be feasible. However, the DDF measures the potential for radial movement from the observed bundle along that direction as it is projected onto the frontier. The DDF is measured as   − →  0 0 x y (54) D x , y ; g , g = max β : x 0 − βg x , y 0 + βg y ∈ T . The CRS DEA LP problem for (54) for given pair of direction vectors (gx , gy ) is max β s.t. N  λj y j − βg y ≥ y 0 ; j =1 N  j =1

(55)

λj x j + βg x ≤ x 0 ;

λj ≥ 0; j = 1, 2, . . . , N. The dual LP problem for (55) is min v x 0 − u y 0 s.t. v x j − u y j ≥ 0, (j = 1, 2, . . . , N ) v g x + u g y = 1, u, v ≥ 0.

(56)

The numeraire in the DDF model (56) shows that when the same direction vectors (gx , gy ) are chosen for evaluating efficiency of all units, the numeraire is the sum of the shadow values of the same input-output bundle for all units, and the efficiency measures are more directly comparable across units. By contrast in

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437

a radial output-oriented model like (35), the numeraire is the shadow value of the output bundle of the unit under evaluation and varies across units making a direct comparison of the efficiency ratios  somewhat problematic.   j y  j x Leleu and Briec [47] used g = X = x , g = Y = y , the industry j

j

aggregate of input-output bundle, for the direction of projection. Ray [58] offered a measure of “overall technical inefficiency” as   ξ = max (ϕ − θ ) : θ x 0 , ϕy 0 ∈ T .

(57)

Because ϕ − θ = (ϕ − 1) + (1 − θ ), one can think of the overall inefficiency as the sum of output-oriented and input-oriented inefficiencies. Aparicio, Pastor, and Ray (APR) [3] extended the model in (57) as   − →  0 0 x y D x , y ; g , g = max β x + β y : x 0 − β x g x , y 0 + β y g y ∈ T .



x

APR have shown that β , β

y



 =

1 N

 j

βjx , N1

 j

(58)

 y βj

solves the problem in

− → (58) for D (X, Y ; g x , g y ) when g x = X, g y = Y . Another direction vector often used is (gx = − ι, gy = ι). In this case, β is the maximum (absolute) amount by which each output can be expanded and every input be contracted simultaneously. Unlike in (58), this time β does not have a simple intuitive interpretation. A popular choice of the direction is (gx = x0 , gy = y0 ). In that case   − →  0 0 x y D x , y ; g , g = max β : (1 − β) x 0 , (1 + β) y 0 ∈ T .

(59)

Here β can be interpreted as the maximum proportion by which the output vector can be expanded and the input vector be contracted simultaneously. Note that for (gx = 0, gy = y0 ) one gets the familiar output-oriented projection, whereas (gx = x0 , gy = 0) yields the output-oriented projection. In Fig. 6, A is the observed bundle (x0 , y0 ). The point B defines the direction of movement (−x0 , y0 ). The point C on the production frontier shows the maximum feasible movement within the production possibility set in the direction parallel to AC OB. In this case, the directional distance function is β = OB = OD OB . The DEA model for (59) under VRS will be

S. C. Ray

Output ( )

438

∗

=

( )

C 0

B

A

D −

0

0

0

Input ( )

Fig. 6 Directional distance function

max β  λj yrj − βyr0 ≥ yr0 ; (r = 1, 2, . . . , m) s.t. j



λj xij + βxi0 ≤ xi0 ; (i = 1, 2, . . . , n)

(60)

j



λj = 1; λj ≥ 0; (j = 1, 2, . . . , N)

j

Non-radial Measures of Efficiency It should be emphasized that all of the DEA efficiency measures considered so far are radial in the sense that technical efficiency is determined by the maximum proportional expansion of all outputs or contraction of all inputs. The measured efficiency does not reflect the potential for expanding any individual output beyond the common rate of expansion or decreasing individual inputs on top of the common rate of contraction. For example, in a 2-output case, a firm is radially efficient if no increase is possible in one output even if the other output could be doubled! In fact, CCR were bothered by this problem right from the beginning and soon after their original 1978 paper introduced the so-called non-Archimedian number (ε) as a penalty for the presence of a slack in any of the input or output constraints in their note [20] revising the DEA model as

10 Data Envelopment Analysis: A Nonparametric Method of Production Analysis

max ϕ + ε

 m  r=1

s.t.

N 

sr+

+

n 

 sii

i=1

λj yr − sr+ = ϕyr0 ; (r = 1, 2, . . . , m) ; j

j =1 N 

439

(61) λj xi + si− = xi0 ; (i = 1, 2, . . . , n) j

j =1

sr+ , si− ≥ 0; (r = 1, 2, . . . , m; i = 1, 2, . . . , n) ; λj ≥ 0; (j = 1, 2, . . . , N) ; ϕ unrestricted. Now, even when ϕ∗ equals 1 in the optimal solution, the unit under evaluation will not be considered efficient unless all slacks are also 0. It can be seen from the dual of this LP problem in (61) that all the shadow prices of outputs and inputs have a lower bound of ε and cannot be strictly equal to 0. However, its practical usefulness is virtually nil because in order to solve the problem in (61) with data one cannot assign a positive real numeric value to ε and at the same time treat this as non-Archimedian (smaller than any Archimedian number).16 The non-radial Russell measure of efficiency introduced by Färe and Lovell [30] avoids the presence of slacks (in outputs in output-oriented models and in inputs in output-oriented models) by allowing outputs to expand (or inputs to contract) at different rates.

Non-radial Russell Output Efficiency The input-oriented Russell efficiency measure is   RM y x 0 , y 0 =

ρy



1  x0, y0

m   1  ρy x 0 , y 0 = max ϕr m r=1

16 Rumor

has it that when Färe raised this point at a conference in Austin TX, Charnes was so irritated that he excluded the former from his guest list to a barbecue!

440

S. C. Ray

s.t.

N 

j

λj yr ≥ ϕr yr0 ; (r = 1, 2, . . . , m) ;

j =1 N 

j

λj xi ≤ xi0 ; (i = 1, 2, . . . , n)

j =1 N 

(62) λj = 1;

j =1

ϕr ≥ 1; (r = 1, 2, . . . , m) ; λj ≥ 0; (j = 1, 2, . . . , N) ;

r

unrestricted.

It may be noted that although the output constraints are in the form of inequalities, because the ϕ s appear in the objective function in (62), there will be no output slacks in the optimal solution.

Non-radial Russell Input Efficiency Analogous to (62) above is the input-oriented Russell efficiency measure n     1  RM x x 0 , y 0 = ρx x 0 , y 0 = min θi n i=1

s.t.

N 

j

λj yr ≥ yr0 ; (r = 1, 2, . . . , m) ;

j =1 N 

j

λj xi ≤ θi xi0 ; (i = 1, 2, . . . , n)

(63)

j =1 N 

λj = 1;

j =1

θi ≤ 1; (i = 1, 2, . . . , n) ; λj ≥ 0; (j = 1, 2, . . . , N) ;

r

unrestricted.

Pareto-Koopmans Measures An input-output bundle is Pareto-Koopmans efficient [44, 45] if and only if no output can be increased without reducing some other output or increasing some

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441

input, and at the same time, no input can be reduced without increasing some other input or reducing some output. In other words, for any Pareto-Koopmans efficient bundle, there cannot be any slack in any input or in any output. As has been shown above for the non-radial Russell measure, there cannot be any output slack in an efficient non-radial output-oriented projection, but slacks in inputs are not ruled out. Conversely, the input-oriented non-radial projection may allow output slacks, but there cannot be any slack in any input. Thus, Pareto-Koopmans (PK) efficiency combines both input- and output-oriented Russell efficiency. There are different variants of this PK efficiency, but the most popular of them is the product of the Russell output and input efficiencies.17 It is called enhanced Russell measure by Pastor, Louis, and Sirvent (PRS) [52], slack-based measure (SBM) by Tone [74], and simply Pareto-Koopmans efficiency by Ray [57] and can be measured as

τ

PK

s.t.



0

x ,y



0



1 n

= min

1 m



θi

i



ϕr

r

λj yrj ≥ ϕr yr0 ; (r = 1, 2, . . . , m)

j



λj xij ≤ θi xi0 ; (i = 1, 2, . . . , n)

(64)

j

ϕr ≥ 1; (r = 1, 2, . . . , m) ; θi ≤ 1, ; (i = 1, 2, . . . , n)  λj = 1; λj ≥ 0; (j = 1, 2, . . . , N) j

Note that in (64) every input and output will be strictly binding.  constraint λ∗j xij = θi∗ xi0 (i = 1, 2, . . . , n) . Therefore at the optimal projection xi∗ = j

Define the total reduction in input i as for each input xi0 − xi∗ = si− ≥ 0. This leads to θi∗ =

xi∗ s− =1− i . xio xi0

(65)

Similarly by defining sr+ = yr∗ − yro , we can derive

17 Portela

function.

1

and Thanassoulis [54] use the measure

(θi ) n 1 (ϕ r ) m

and called it the geometric distance

442

S. C. Ray

ϕr∗ =

xi∗ s+ = 1 + r , (r = 1.2 . . . , m) xio yr0

(66)

Hence the objective function in (64) becomes 1 n 1 m

 i

 r

1−

θi ϕr

=

1+

1 n 1 m

 i

 r

si− xi0

(67)

+ sir yr0

SBM.18

which is the Both (PRS) and Tone use the expression in (67) for the objective function and resort to a normalization to convert the linear fractional functional programming problem into an LP (following Charnes and Cooper [17]). Ray [56], Ray and Jeon [63], and Ray and Ghose [62], on the other hand, used a linear approximation of the objective function at (θ i = 1, ϕr = 1) (i = 1, 2, . . . , n; r = 1, 2, . . . , m) to get 1 n 1 m

and used min

 i

θi −





θi

i

 r

ϕr

≈2+

 i

θi −



ϕr

(68)

r

ϕr as the objective function.

r

Efficiency Measurement with Market Prices There is a commonly held belief that DEA should be used only to measure efficiency of non-profit organizations like government departments, non-governmental organizations (NGOs), and other non-profit institutions like schools. For decisionmaking units operating in the market, it is believed, one should use econometric methods like SFA to estimate a parametric cost, revenue, or profit function. This is a result of confusing conceptualization with calibration. The appropriate criterion of measuring efficiency – cost, revenue, or profit – is determined by the scope of decision-making. With outputs exogenously assigned efficiency lies in minimizing cost. Similarly, for a given input bundle, the objective is to maximize revenue. Lastly, when both outputs and inputs can be freely chosen, profit maximization is the appropriate criterion of efficiency. The nonparametric method of DEA and the parametric method of SFA are merely two alternative methods of computing the appropriate measure of efficiency.

18 See

also the range-adjusted measure (RAM) introduced by Cooper, Park, and Pastor [23].

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443

Cost Efficiency The different kinds of efficiency considered so far – radial, non-radial, outputoriented, input-oriented, and graph – are based entirely on input and output quantity data only. In all of these cases, improvement in efficiency came from reduction in inputs or increase in outputs or some combination of both. In measuring the radial output-oriented technical efficiency, proportional reduction in each input is given the same importance, and the objective is to scale the entire input bundle down as much as possible without changing the input mix. In practice, however, different inputs have different market prices and account for different proportions of the total cost of the firm. When the technical efficiency is below unity, say 0.90, the firm can, obviously, reduce all inputs by 10% and reduce its total cost by 10%. It may be possible, however, to achieve an even greater reduction in cost by changing different inputs by different proportions. In fact, sometimes cost minimization involves increasing some inputs while reducing some others. Cost minimization is an important objective for a variety of production decisionmaking units. Even when a firm is maximizing profit and selects its optimal input and output bundles together, cost minimization remains an objective embedded within the overall objective of profit maximization because profit is not maximized unless the selected output bundle is produced at the lowest cost. In a non-profit organization, the output may be exogenously given, but accountability to the stakeholders makes cost minimization an important objective. Consider a firm facing a vector of input prices w0 that produces the output bundle

0 y using the input bundle x0 . Thus its actual cost is C0 = w0 x0 . The minimum cost of producing the output bundle y0 at input prices w0 is   C0∗ = min w 0 x : x, y 0 ∈ T .

(69)

  C0∗ = min w 0 x : x ∈ V y 0 .

(70)

Alternatively,

Given that x0 ∈ V(y0 ), obviously C0∗ ≤ C0 . The cost efficiency of the firm is measured as γ =

C0∗ . C0

(71)

The relevant DEA LP problem for cost minimization under the VRS assumption is

444

S. C. Ray

C0∗ = min w 0 x s.t.

N 

λj x j ≤ x;

j =1 N 

λj y j ≥ y 0 ;

(72)

j =1 N 

λj = 1;

j =1

x ≥ 0; λj ≥ 0, (j = 1, 2, . . . , N ) . As usual, when CRS is assumed, the restriction

N  j =1

λj = 1 is dropped. It is

important to note that like the λj s, the elements of the optimal input vector x are also choice variables. Moreover, there cannot be any input slack at the optimal solution of (70). Following Farrell [37] one can decompose the (overall) cost efficiency of the firm into two multiplicative components separately representing technical and allocative efficiencies. If the input-oriented technical efficiency of the firm is τ x (x0 , y0 ) = θ , it would be possible to scale down its input bundle from x0 to x0T = θ x 0 . As a result its cost would be lowered to   C0T = θ. w 0 x 0 = θ C0 . (73)   The scaled down input bundle x0T is in the isoquant V y 0 , and there remains no further scope for reducing cost  through proportionate reduction in all inputs. Now, all input bundles in V y 0 will produce the output bundle y0 . But they do not cost the same amount of money. There remains the possibility of further cost reduction by moving from the bundle x0T to another bundle on the isoquant.   Because all bundles in V y 0 are technically efficient, reduction in any input has to be counterbalanced by increase in some other input(s).19 The potential for cost reduction through input substitution along the isoquant will depend on both the degree of substitution possible and the relative prices of the inputs. Note that input substation will alter the input mix. The amount of cost reduction through a change in the input mix as a proportion of the technically efficient cost of the firm is a measure of its allocative efficiency. Thus,      C∗  C0T C0∗ 0 0 0 0 = CE x |w , y = . . (74) C0 C0 C0T

19 An

exception is when there is any slack in the technically efficient input bundle.

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445

2 0 1

2

( 0

2

1

1, 2)

=

0

0

1

Fig. 7 Technical, allocative, and cost efficiency

As is apparent from (73) above,

C0T C0

= θ . Now define

C0∗ C0T

= α. Then we

have Farrell’s decomposition of cost efficiency in (71) into technical and allocative efficiency as γ = (θ ) . (α) .

(75)

Figure 7 shows the measurement and decomposition of cost efficiency for the 2-input case. Suppose that the point P represents the input bundle x0 and

the expenditure line A0 B0 through P shows the cost C0 = w0 x0 . Point Q on the 0 isoquant is the bundle xT , the technically efficient projection of x0 . The cost of this bundle is CT0 = w 0 xT0 shown by the expenditure line A1 B1 through Q. The minimum expenditure C0∗ = w 0 x ∗ is shown by the line A2 B2 through R which is the cost minimizing input bundle. We therefore have a measure of cost efficiency C∗ 2 γ = C00 = OA OA0 . This is further broken up into θ = efficiency and α =

C0∗ CT0

=

OA2 OA1

=

OS OQ .

CT0 C0

=

OA1 OA0

=

OQ OP

representing technical

representing allocative efficiency.

Fixed Inputs and Short Run Cost Minimization In the short run, one distinguishes between fixed and variable inputs. Suppose that the input vectors x is partitioned as x = (v, K) where v is the vector of variable

446

S. C. Ray

inputs and K is a single fixed input. The vector of variable input prices is wv , while the rental rate of the fixed input is rK . The firm has a target (scalar) output y0 and has to select a bundle of variable inputs (v) that can produce y0 when combined with the given quantity of fixed input K0 . Note that like y0 , K0 is also given in the short

run. Hence, minimizing the short run total cost SRTC = wv v + rK K0 involves

minimization of only the variable cost wv v. The firm’s optimization problem in this case is min V C = wv v  λj v j ≤ v; s.t. j



λj Kj ≤ K0 ; (76)

j



λj yj ≥ y0 ;

j



λj = 1; v ≥ 0; λj ≥ 0; (j = 1, 2, . . . , N)

j

The optimal solution of (76) along with the fixed cost (rK K0 ) leads to the optimal

short run total cost SRTC = wv v∗ + rK0 . It should be noted that when the fixed ∗ input constraint is binding, the corresponding dual variable ∂V∂KC ≡ zk measuring the reduction in the variable cost as the quantity of the fixed input is increased will be negative. In production economics,  = − zk is treated as the shadow value of the fixed input. If  > rK , the firm can lower the SRTC for its output y0 by increasing ∗ K if possible. A measure of the short run marginal cost is MC = ∂V∂yC0 which is the dual variable associated with the output constraint.

Using Total Cost as an Aggregate Input In many situations in order to evaluate the output-oriented technical efficiency of a firm, one has to use the total expenditure or cost as an aggregate measure of input. As argued by Banker, Chang, and Natarajan (BCN) [10], because the total cost involves both quantities and prices of inputs, the measure of inefficiency obtained using a single aggregated input variable equals the aggregate (technical and allocative) efficiency of the firm. For this to be true, one has to assume that all firms face the same input prices even though these prices may not be known. Assume that all firms face the same strictly positive price vector w. Thus, the total cost of the input bundle xj of firm j is Xj = w x j . Now construct the set

10 Data Envelopment Analysis: A Nonparametric Method of Production Analysis

T˜ (w) =

⎧ ⎨ ⎩

(X, y) : X ≥

N 

λj Xj ; y ≤

j =1

N  j =1

λj y j ;

N  j =1

⎫ ⎬ λj = 1; λj ≥ 0 ⎭

447

(77)

derived from the estimated production possibility set Tˆ define earlier.   Consider the single input technical efficiency, θ X = min θ : θ X0 , y 0 ∈ T˜ . The relevant DEA problem is θ z = min θ s.t.

N 

λj Xj ≤ θ X0 ;

j =1 N 

λj y j ≥ y 0 ;

(78)

j =1 N 

λj = 1; λj ≥ 0; (j = 1, 2, . . . , N ) .

j =1

Contrast this with the cost minimization problem in (72). It is clear that the N N       constraint λj x j ≤ x in (72) implies λj w x j ≤ w x . If we define j =1 j =1   w x = θ. w x 0 , it is obvious that the single input DEA technical efficiency problem in (78) is the same as the multiple input cost minimization problem in (72).   In other words, τ x (X0 , y0 ) = θ z from (72) is the same as γ x 0 |y 0 , w . 20 We have already seen that cost efficiency coincides with technical efficiency if and only if allocative efficiency is unity. An implication of this result is that technical efficiency obtained from using total cost as the single input will in general underestimate the technical efficiency of a multiple input firm. It is very important to realize that in the cost minimization problem, the input price vector, w0 , is assumed to be a parameter like the target output bundle y0 . No matter whether the product market is competitive or monopolistic, the input markets are assumed to be competitive, and the firm is a price taker in the factor markets. It is assumed that the firm can purchase any input bundle it wants at the given vector of input prices. Failure to recognize this competitive assumption leads to the erroneous perception that between two units producing the same output and using the same input bundle, one is more cost-efficient because it faces lower input prices in the factor market and has a lower cost.21

20 See 21 See,

the discussion in Ray and Mukherjee [64]. for an example, Tone [75].

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S. C. Ray

Multi-location Cost Minimization Ray, Chen, and Mukherjee (RCM) [65] introduced some measure of flexibility in the choice of input prices on the part of the firm by considering multiple production locations across which input prices vary. In the RCM multi-location cost minimization problem, the firm still has a target output vector y0 to produce. But it has the option to produce all or parts of y0 in any one or several out of a given set of S sites. The vector of input prices, ws , that the firm has to pay at any specific location s is fixed. But the firm can choose to pay a different vector of prices, wq , by selecting a different site, q. There is an important constraint, however. Any output produced at any specific site, s, must be produced from inputs procured at the local market at the locally applicable prices, ws . The firm is not allowed to “cherry pick” by purchasing various inputs at favorable prices from different markets. Suppose that the firm has a choice of sites at three locations: A, B, and C. The input prices at these locations are wA , wB , and wC , respectively. The firm decides to produce the output bundles yA , yB , and yC at these locations. Of course, it may decide not to produce anything at location s, in which case ys = 0 (s = A, B, C). The input bundles used at these locations are (xA , xB , xC ), and the corresponding



costs of production are CA = wA xA , CB = wB xB , andCC = wC xC . The firm has to choose its multi-location production plan such that its total output target is met at the minimum cost. Of course, the production plan (xs , ys ) at each location s must be feasible. As formulated in RCM, the multi-location minimum cost is22 min C = w A x A + w B x B + w C x C s.t.

N 

λsj x j ≤ x s , (s = A, B, C) ;

j =1 N 

λsj y j ≥ y s , (s = A, B, C) ;

j =1



ys ≥ y0;

s=A,B,C N 

λsj = Bs , (s = A, B, C) ;

j =1

Bs ∈ {0, 1} , (s = A, B, C) ; λsj ≥ 0 (s = A, B, C; j = 1, 2, . . . , N ) . (79)

22 This

problem assumes homogeneous technology across all locations. RCM also consider the case where the technology varies across locations.

10 Data Envelopment Analysis: A Nonparametric Method of Production Analysis

449

In the above LP problem, Bs is an indicator variable. When Bs equals zero, every λsj also equals zero. Thus, both xs and ys are null vectors. That is, the firm does not select location s for producing any output. On the other hand, when Bs equals unity, (xs , ys ) is some convex combination of observed input-output bundles and is a strictly positive input-output bundle. That is, the firm does produce a positive output bundle at location s.

Revenue Efficiency In some cases, the objective of a firm is to select the optimal output bundle that can be produced from a given endowment of inputs in order to generate the maximum revenue. This time, the output prices (like the input quantities) are treated as given parameters. Suppose that a firm produces the output bundle y0 from the input bundle

x0 and at the output price vector p0 earns the revenue R0 = p0 y0 . Now the maximum possible revenue that can be generated from this input bundle and at these output prices is   R ∗ = max p0 y : x 0 , y ∈ T .

(80)

  x0 .

(81)

Alternatively, R ∗ = max p0 y : y ∈ P The VRS DEA formulation of this problem is max p0 y s.t. 

λj y j ≥ y;

j



(82) λj x j ≤ x 0 ;

j



λj = 1; λj ≥ 0, (j = 1, 2, . . . , N) .

j

It may be noted in passing that the revenue maximization problem is the nonparametric version of the familiar product mix problem. Here the resource and return vectors are specified but the technology matrix is not. The optimal value of

the objective function in (82) yields the maximum revenue R∗ = p0 y∗ , and the revenue efficiency of the firm is measured as

450

S. C. Ray

 p0 y 0  R0 η y 0 ; x 0 , p0 = 0 ∗ = ∗ R . p y

(83)

As in the case of cost efficiency, revenue efficiency also can be multiplicatively decomposed into technical and allocative efficiency.

Profit Efficiency In the case of cost minimization, the output quantities and input prices were treated as exogenously determined, and only the input quantities were the choice variables. Similarly, for revenue maximization, input quantities and output prices were parameters, and the output quantities were the choice variables. For profit maximization by a competitive firm, both input and output quantities are choice variables, while the input and output prices are parameters. In order to maximize its profit, the firm is free to select any input-output bundle so long as it represents a feasible production plan in the sense that the selected output bundle can be produced from the corresponding input bundle. Thus, the profit maximization problem can be expressed as   π p 0 , w 0 = max p0 y − w 0 x : (x, y) ∈ T .

(84)

The DEA problem for profit maximization is max p 0 y − w 0 x s.t. 

λj y j ≥ y;

j



(85) λj x j ≤ x;

j



λj = 1; λj ≥ 0, (j = 1, 2, . . . , N) .

j

It is important to remember that the optimal profit cannot assume a non-zero yet finite value if CRS holds.



The optimal solution of (85) yields the maximum profit π ∗ = p0 y∗ − w0 x∗ , and the profit efficiency of the firm can be measured as υ=

π0 p0 y 0 − w 0 x 0 = . π∗ p0 y ∗ − w 0 x ∗

(86)

10 Data Envelopment Analysis: A Nonparametric Method of Production Analysis

451

A potential problem with this measure of efficiency is that if the actual profit is negative, one gets υ < 0. Note, however, that from the standpoint of economic theory, this is really a question of misspecification of the problem. Because all inputs are being freely chosen, (85) represents an analysis of the behavior of the firm in the long run. If the firm is making losses even in the long run, it would have the option to quit in which case there would be 0 profit. The fact that the firm is earning negative profit but still is in business would suggest that there are fixed costs and not all inputs are freely variable. In that case (85) is not an appropriate analytical framework for evaluating efficiency. Varian [76] proposed a quick test of profit maximizing behavior simple enough to be carried out on the back of an envelope.23 Consider the observed input-output bundles (xj , yj ), (j = 1, 2, . . . , N). Suppose that the output and input price vectors of



firm i are (pi , wi ). Thus its actual profit is π i = pi yi − wi xi . Firm i is not profit





maximizing if π i = pi yi − wi xi < pi yj − wi xj for some j = i. Varian calls this the weak axiom of profit maximization (WAPM). The intuition behind this is quite simple. Every observed input-output bundle is feasible by assumption. Hence, if a different bundle yields higher profit at the prices applicable for firm i, then this firm cannot have maximized profit. In that sense, WAPM is a necessary condition for consistency of the observed bundle with profit maximizing choice. It can also be shown that under the standard assumptions of convexity and free disposability, it is also a sufficient condition. (See Ray [56], p. 261.)

Capacity Utilization24 While there is considerable interest in measuring capacity utilization, there is no general agreement on the definition of the capacity output. Indeed, there are several definitions of the capacity output – each is valid in a particular context. By far, the simplest of them is the maximum level of output that can be produced from a given level of quasi-fixed inputs (like plant and machinery) even when variable inputs (like labor or materials) are available without restriction. By definition, the actual output produced cannot exceed this maximum quantity. This is a physical measure of capacity that is technologically determined. First proposed by Johansen [43], it has been subsequently popularized in empirical applications by Färe, Grosskopf, and Kokkelenberg (FGK) [32]. Note that this capacity is a short run concept because at least one variable is held fixed. Moreover, the level of the capacity output would depend on which input is being held fixed and at what level. In contrast

23 Also see Afriat [1] and Hanoch and Rothschild [42]. Banker and Maindiratta [6] further extended Varian [76] to construct upper and lower bounds on technical, allocative efficiency, and overall profit efficiencies. 24 See the  Chap. 24, “Capacity and Capacity Utilization in Production Economics” in this volume of the Handbook for a detailed discussion of capacity utilization in production economics in general.

452

S. C. Ray

to this physical measure, the textbook definition of the economic capacity output corresponds to the minimum of the long run average cost curve of a competitive firm. Because this is a long run measure, there is no fixed input. In the case of the long run average cost curve, its U-shape derives from economies of scale followed by diseconomies of scale and is not due to any fixed input. An economic measure of the capacity output differs from the physical measure even in the short run. The presence of fixed costs associated with the quasi-fixed inputs of the firm justifies the U-shaped average cost curve, and the output level where the short run average (total) cost reaches the minimum is the capacity level for the given bundle of the (quasi)-fixed inputs. In fact, Cassell [13] argued that, “since the absolute technical upper limit of output obtainable from the fixed factors is likely to lie far beyond the realm of practical economic operations, their capacity output should be taken as that at which the average full costs of production are at their minimum.” It is impoo emphasize at the outset that the average cost curve – whether in the short run or in the long run – is unequivocally defined only in the context of a single product technology. In the multiple output case, there is no natural definition of average cost. One approach to deal with this problem is to obtain an aggregate measure of output (like total revenue) and to treat it as single product problem. Alternatively, following Baumol, Panzer, and Willig [11], one may consider variations in only the scale of an output bundle but keep the output mix unchanged and measure the ray average cost. The economic capacity scale will then correspond to the minimum point of the ray average cost curve for the given output mix. In this section only the single output case is considered. In a recent paper, Ray, Walden, and Chen [67] developed DEA models to determine the rate of capacity utilization in the multiple output case both in the short run and in the long run under alternative returns to scale assumptions and accommodating multiple fixed inputs.

A Physical Measure of Short Run Capacity Utilization To obtain the physical measure of the short run capacity output, partition the input vector as x = (v, f ), where v is the sub-vector of variable inputs and f is the subvector of fixed inputs. Then for any given bundle of fixed inputs f0 , capacity output is defined by FGK as     y ∗∗ f 0 = maxv y : v, f 0 , y ∈ T . In empirical applications, the FGK capacity output is measured as y ∗∗ = ϕ ∗∗ y0 where

(87)

10 Data Envelopment Analysis: A Nonparametric Method of Production Analysis

453

ϕ ∗∗ = max ϕ  s.t. λj yj ≥ ϕy0 ;   j

j

λj v j ≤ v;

j

(88)

λj f j ≤ f 0 ;



λj = 1;

j

v ≥ 0; λj ≥ 0 (j = 1, 2, . . . , N) . Note that apart from non-negativity the variable inputs are unconstrained and, in consequence, play no role in the optimization problem in (88). Recall that the technically efficient output producible from the given levels of variable and fixed inputs (v0 , f0 ) is y ∗ = ϕ ∗ y0 where ϕ ∗ = max ϕ  s.t. λj yj ≥ ϕy0 ; 

j

λj v j ≤ v 0 ;

j



λj f j ≤ f 0 ;

(89)

j



λj = 1;

j

v ≥ 0; λj ≥ 0 (j = 1, 2, . . . , N ) . FGK defined capacity utilization as CU ∗ =

y∗ ϕ∗ = ∗∗ . ∗∗ y ϕ

(90)

It is clear from above that ϕ∗∗ ≥ ϕ∗ and, therefore, their measure of capacity utilization can never exceed unity. One might prefer to measure capacity utilization

454

S. C. Ray

by the ratio of the actual output y0 to the capacity output y∗∗ . The corresponding rate of capacity utilization is y0 1 CU = ∗∗ = ∗∗ = y ϕ



0

1 ϕ∗

 ∗ ϕ . . ϕ ∗∗

(91)

The first factor on the right measures wasted capacity due to inefficiency, while the second relates to excess capacity.

Long Run Capacity Output The short run physical capacity output is not economically appealing because it does not pay any attention to the cost of adjusting the variable inputs that would be needed to produce the capacity output and it is not clear why a firm would like to produce at that level. By contrast, the minimum point of the long run average cost curve is arguably the only sustainable output level for a firm in the long run in a competitive market with free entry and exit. The minimum point of the long run average cost (LAC) curve corresponds to the output level where economies of scale have been exhausted but diseconomies of scale have not yet set in. Of course, the LAC will have usual U-shape only when the technology exhibits variable returns to scale. There is a subtle difference between the MPSS and the cost-efficient scale size. The former maximizes the average productivity AP (x) = f (x) x in the single input single output case and ray in the multiple input single output case. On average productivity RAP (t, x) = f (tx) t the other hand, the cost-efficient output scale minimizes the average cost AC(y) = C(w,y) in the single output case and the ray average cost RAC(t, y) = C(ty,w) in y t the multiple output case. Ray [61] addressed the problem of finding the capacity output in the single output multiple input case. In order to find the output level where the LAC under VRS reaches a minimum, one needs to solve the problem w 0 x y  s.t. λj x j ≤ x; min



j

λj yj ≥ y;

j



λj = 1;

j

λj ≥ 0; (j = 1, 2, . . . , N) .

(92)

10 Data Envelopment Analysis: A Nonparametric Method of Production Analysis

455

The optimal input-output bundle x∗ from (92) above yields the minimum average cost AC V RS (w0 , y∗ ) = C V RS

(w0 , y∗ )(w 0 , y∗ ) w 0 x ∗ = y∗ y∗

(93)

along with the efficient output level (y* ). Although the problem in (92) is nonlinear, one can utilize the following two lemmas from production economics to determine both the output level y* and the minimum average cost ACVRS (w0 , y∗ ) by solving a simple DEA LP problem. Lemma 1 Locally constant returns to scale holds at the input-output bundle (x* , y* ) where the average cost reaches a minimum. Lemma 2 If the technology exhibits constant returns to scale globally, average cost is a constant at all output levels. Proof See Ray [61]. Suppose that firm k faces the input prices wkandproduces the output quantity yk . The objective is to find the output levelyk∗ = y w k that minimizes ACVRS (wk , y). For this, first consider the CRS cost minimization problem min w k x  s.t. λj x j ≤ x; 

j

(94)

λj yj ≥ yk ;

j

λj ≥ 0. Suppose the optimal solution  ∗ that at λj = q and λ∗j x j = x ∗ . j

j



This implies that wk x∗ = CCRS (w0 , yk ) is the minimum cost of producing yk at input prices wk under the CRS assumption. Then, by Lemma 2, q1 w k x ∗ =     k 1 0 , 1 y . Define the output level y = 1 y and the = CCRS w CCRS w , y k k q q q k  ∗ 1 ∗ ∗ ∗ weights μj = q λj . Clearly, μj = 1 and each μj ≥ 0. Further, define the input j  vector x˜ = q1 x ∗ = μ∗j x j . j

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S. C. Ray

Now consider the VRS cost minimization problem   C V RS w k , y = min w k x  λj x j ≤ x; s.t. 

j

λj yj ≥ y;

(95)

j



λj = 1;

j

λj ≥ 0. Suppose that the optimal solution of (95) yields a minimum cost w k x. It can be seen from above that the μ∗j s and x˜ above constructed from (94) constitute a feasible solution for (95). Hence, ˜ w k x ≤ w k x.

(96)

  At the same time, wk x˜ = q1 w k x ∗ = C CRS w k , y is the optimal solution of the less restrictive CRS problem min w k x  s.t. λj x j ≤ x; 

j

(97)

λj yj ≥ y;

j

λj ≥ 0. Hence, w k x˜ ≤ w k x.

(98)

The inequalities (96) and (98) together imply w k x˜ = w k x.

(99)

Therefore, at the output level y, the total and average costs are the same under both CRS and VRS assumptions. That is, at the output level y, the average cost ACVRS (wk , y) reaches a minimum. In other words, y is an efficient production scale.

10 Data Envelopment Analysis: A Nonparametric Method of Production Analysis

Recall, however, that y = utilization is

1 q yk .

457

Hence, an economic measure of capacity

CU e =

yk = q. y

(100)

In order to obtain the capacity output, one only needs to solve the CRS cost minimization problem and from the optimal values of the λs compute y=

yk yk =  ∗. q λj

(101)

j

Economic Scale Efficiency The scale efficiency of an input-output bundle (x0 , y0) is measured by the inverse of  the ray average productivity at its MPSS. Thus, SE x 0 , y 0 = βα when (βx0 , αy0 ) is its MPSS. By contrast, the economic scale efficiency of the output (bundle) y0 is     RAC ty 0 ; w C t ∗y0, w   =   ESE y ; w = RAC y 0 ; w t ∗C y0, w 

0



(102)

where t∗ is the output scale that minimizes the ray average cost. In the scalar output case,  AC w 0 , y0   0   ESE y0 |w = AC w 0 , y ∗

(103)

where y∗ is the output level where average cost reaches a minimum.

Efficiency Measurement with Bad Outputs25 In many cases, the production of the desired or intended output results simultaneously in the production of some undesirable or “bad” outputs as well. The most common example is one of electricity generation in thermal power plants resulting in air pollution as well. It is recognized that just as increased production of the desirable output from a given bundle of inputs implies higher efficiency, increased production of the undesirable should be considered lower efficiency. While “good/bad” classification of outputs is a matter of preference, production

a detailed treatment of bad outputs, see  Chap. 12, “Bad Outputs” by Murty and Russell in this Handbook.

25 For

458

S. C. Ray

of an output in spite of it being considered bad by the producer calls into question the assumption of free disposability of such output. Clearly, it is not possible to eliminate totally or at least to reduce significantly the bad output unilaterally without increasing some input(s) and/or reducing some desired or good output as well. While there is no general agreement about how to reconceptualize the production technology to accommodate bad outputs, there are three principal approaches found in the relevant literature.

Bad Output as Input The first approach is to treat the bad output like a conventional input (e.g., in Baumol and Oates [12] or Cropper and Oates [25]). In light of the positive correlation of the good output (power) and the bad output (smoke emission), it is intuitively appealing to treat the bad output like an input (say coal). But even though observationally equivalent, the bad output is conceptually quite different from an input. First, an input exists even before the production process starts. There was no smoke in the air before production started. Second, an input is depleted in stock as production is carried out. In this case, there is more smoke in the atmosphere as more power is generated.

Good and Bad Outputs as Joint Products In the second, and by far the most widely used, approach introduced by Färe and Grosskopf and applied in numerous papers with their coauthors,26 the good and the bad outputs are treated as weakly disposable and null joint. Two outputs are weakly disposable if a unilateral decrease in one output holding the other output and inputs unchanged is not feasible but they can be reduced together. They are null joint if production of one output can be stopped only if the other output is not produced as well. For example, consider an industry producing one good output (g) along with a single bad output (b) using two inputs x1 and x2 . Under the assumption of free disposability of outputs,         x10 , x20 , g 0 , b0 ∈ T ∧ g 1 , b1 ≤ g 0 , b0 ⇒ x10 , x20 , g 1 , b1 ∈ T .

(104)

But under weak disposability 

       x10 , x20 , g 0 , b0 ∈ T ∧ g 1 , b1 = αg 0 , αb0 ; 0 ≤ α ≤ 1⇒ x10 , x20 , g 1 , b1 ∈ T . (105)

26 See, for example, Färe, Grosskopf, Lovell, and Pasurka [33]; Färe, Grosskopf, Lovell, and Yaisawarng [34]; and Färe, Grosskopf, Noh, and Weber [36] among many others.

10 Data Envelopment Analysis: A Nonparametric Method of Production Analysis

459

Following Färe and Grosskopf [29], assuming (i) CRS and (ii) weak disposability of the bad output but free disposability of the good output, the production possibility set under joint production can be approximated as

SJWPD =

⎧ N  ⎪ j ⎪ ⎪ , x ; g, b) : x ≥ μj xi ; (i = 1, 2) ; (x 1 2 i ⎪ ⎪ ⎪ j =1 ⎨

⎫ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎬

⎪ ⎪ ⎪ ⎪ N N ⎪ ⎪   ⎪ ⎪ ⎪ ⎪ g ≤ μ g ; b = μ b ; μ ≥ 0 = 1, 2, . . . , N) . (j ⎪ ⎪ j j j j j ⎭ ⎩ j =1

(106)

j =1

Bad Output as a By-Product Murty and Russell [50] and Murty, Russell, and Levkoff (MRL) [51] regard the bad output as an unintended by-product of producing the good output.27 Both the neutral input x1 and the polluting input x2 are used for the production of the good output, g, while only the polluting input produces the bad output b as an undesired side effect.28 In this conceptualization, the production possibility set consists of two sub-technologies       ∂F g ∂F g > 0 (107) T g = (x1 , x2 ; g) : F g x1 , x2 ; g ≤ 0; < 0 i = 1, 2 ; ∂xi ∂g and     ∂F b (kx 2 , kb) b T = (x2 ; b) : F x2 ; b ≥ 0; >0 . ∂k b

(108)

MRL [51] define the corresponding nonparametric construction of the production possibility set for the by-production technology under CRS as

27 See

their Chap. 12 on  “Bad Outputs” in this volume of the Handbook.

28 MRL [51] also consider another model including pollution abatement as a separate desired output

produced by diverting resources from the production of the desired output g. An example would be treatment of polluted waste water before discharging into the stream.

460

S. C. Ray

g

SBP =

⎧ ⎨ ⎩

(x1 , x2 , g) : x1 ≥



j

λj x1 ; x2 ≥

j

b = SBP

⎩

(x2 , b) : x2 ≥



j

λj x2 ; g ≤



j

  ≥ 0; j = 1, 2, . . . , N ⎧ ⎨



j

μj x2 ; b ≥



j

λj gj ; λj

j



λj bj ; μj ≥ 0; j = 1, 2, . . . , N

j

⎫ ⎬ ⎭

     g b SBP = (x1 , x2 , g, b) : x1 , x2 , g ∈ SBP ∧ x2 , b ∈ SBP (109) MRL [51] formulate two separate optimization problems one for maximizing   g ϕg : x10 , x20 , ϕg g ∈ SBP

(110)

  b θb : x20 , θb b0 ∈ SBP

(111)

and another for minimizing

and measure overall efficiency as  ∗

ψ =α

1 ϕg∗

 + (1 − α) θb∗

(112)

where 0 < α < 1 is a preselected weight. There are two problems with treating (110) and (111) as separate optimization problems. First, the optimal value of the common input x2 can be different across the two problems. As argued in Lozano [48] and Ray, Mukherjee, and Venkatesh (RMV) [66], the quantity of the polluting input, x2∗ , in (110) that is used to produce the good output g∗ should be exactly the x2∗ in (111) that produces the optimal bad output b∗ . There is no constraint in the MRL formulation that ties the two problems together. Another problem (although much less serious than the other) is that unless the optimal λs in (110) and the optimal μs in (111) are the same, the peer group from the good output benchmark would be different from the one for the bad output benchmark. This implicitly assumes that there are two decision-making units within the firm: one maximizing the good output and the other minimizing the bad output.

10 Data Envelopment Analysis: A Nonparametric Method of Production Analysis

461

Joint Disposability and Material Balance Principle RMV [66] modified the weak disposability assumption into what they call joint disposability in respect of the pollution generating technology. In the joint production approach, the two outputs (good and bad) are considered weakly disposable. This is a relation between a pair of outputs. RMV [66] connect the bad output and the polluting input through joint disposability in the sense that the bad output can be reduced only if the polluting input is also reduced.29 Their formulation of Tb is b = SBP

⎧ ⎨ ⎩

(x2 , b) : x2 = α 



j

λj x2 ; b = α

j

≥ 0; λj ≥ 0; j = 1, 2, . . . , N





λj bj ; 1 ≥ α (113)

j

Førsund ([38], [39]) points out that there is a physical law that requires that any part of the (material) input that is not incorporated into the output must take the form of a residual, which is treated as a bad output in specific situations. Accordingly, he argues strongly for imposing a materials balance condition30 relating the good and the bad outputs and the polluting input. In the simple case considered above, such materials balance condition would be of the form a1 b = a2 x2 − a3 g

(114)

For the by-production technology, he recommends an additional restriction of the form ⎛ ⎞ ⎛ ⎞ ⎛ ⎞    j λ∗j bj ⎠ = a2 ⎝ λ∗j x2 ⎠ − a3 ⎝ λ∗j gj ⎠ . (115) a1 ⎝ j

j

j

Note that RMV [66] in their “unified” models require that the same λs must be used for creating the benchmark bundles for the good and the bad output production technologies. That should ensure that the material balance condition in (115) is automatically satisfied. Because (114) is assumed to hold as a physical necessity for j every (bj , gj , x2j ) combination that is observed, a1 bj = a2 x2 −a3 gj holds for every j.

29 The

joint disposability of the bad output and the polluting input is comparable to the two materials balance postulates MB1 and MB2 in Dakpo et al. ([26], p. 352). 30 Ayres and Kneese [4] introduced the question of materials balance in economics. In a number of subsequent papers, it has been extensively discussed in the context of production efficiency by a number of authors including Pethig [53]; Coelli, Lauwers, and Van Huylenbroeck [21]; Chambers and Melkonyan [14]; Hampf [41]; Rodseth [68, 69]; and Førsund [39] among others. See, in particular, Dakpo, Jenneauxe, and Latruffe [26].

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j λ∗j a2 x2



 



 

 j λ∗j x2

Hence, λ∗j a1 bj = − λ∗j a3 gj ⇒ a1 λ∗j bj = a2 − j j j j j    ∗ λj gj . This would make imposition of (115) as an additional restriction a3 j

redundant. It is important to remember that environmental pollution is not the only case where a bad output is produced alongside the good output and there is not a single model that applies in every context. Consider, for example, the case of nonperforming loans in banking. Such loans in default are best modeled as joint products with the good output (loans in good standing). Because the probability of loan default can never be reduced to 0, the only way to eliminate them is to stop lending altogether. Thus, they are indeed null joint. The joint production model is a more appropriate analytical format in this case.

Contextual Variables in DEA The production possibility set consists of all feasible input-output bundles and is thus defined in the input-output space. However, production takes place in a specific physical, social, and cultural environment. Differences in environmental conditions can play a decisive role in defining the feasibility of a particular input-output bundle. In measuring the efficiency of a decision-making unit, one assumes that it can choose the input bundle it uses or the output bundle it produces. Unlike inputs or outputs, the environmental factors cannot be chosen by the firm and has to be treated as “non-discretionary.” An obvious example of an environmental factor is rainfall in the context of agricultural production. The maximum output producible from a given bundle of inputs (say labor, fertilizer, and land) depends on the amount of rainfall. In that sense, rainfall contributes to the output much the same way as irrigation. However, while the farmer can choose the level of irrigation, the amount of rainfall is not within his control. Here, rainfall acts as a non-discretionary input. In defining the feasible set for a DEA LP problem, one has to include a constraint for the amount of rainfall, but while measuring the radial input-oriented technical efficiency, the proportional scaling factor should be applied only to the discretionary inputs (like labor, fertilizer, and land) but not to rainfall. For another example, consider a secondary school where the average performance of its pupils in a standardized test in mathematics is one of the outputs and hours of classroom instruction in math is one of the inputs. An increase in this input is expected to improve the average test score in math. Now consider another variable – the median family income of the town where the school is located. There is ample evidence to conclude that students from more affluent families where the parents are professionals are better motivated and spend more time on homework and perform better in tests. In that sense, the economic status of the pupil acts

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like class time spent on math. However, the former is an input, while the latter is a contextual variable. They are also referred to as “non-discretionary inputs.” Another example of a contextual variable is the marital status of parents of a pupil. A child from a single parent family (irrespective of income) is unlikely to get the same level of parental attention as when both parents are present. Thus, an increase the proportion of pupils from single parent families will lower the average math score even when all other variables are unchanged. Two things emerge out of this illustrative example. First, unlike an input, a contextual variable may be either favorable (like family income) or detrimental for production (like single parent families). Second, a decision-maker at an appropriate level of authority (the school superintendent or the Board of Education) can select the input bundle used. This is not true for the contextual variables. For yet another example, consider the efficiency of a water utility. The outputs are the number of customers served and the gallons of water distributed. The inputs are pumps, length of pipelines, and hours of labor. Note that in an urban area the higher density of population implies that the same number of customers can be served and the same volume of water dispensed with a smaller network of pipelines than what is required in a rural area. Moreover, when many customers are located in the same building (as is the case in an urban community), the labor hours needed for meter reading will be lower than in a rural area where customers are located at distant points. In this case, density of population is an environmental variable. Where contextual variables are considered to be significant determinants of performance, an appropriate way to conceptualize the production technology is to define the production possibility set conditional on a specific vector of contextual variables z0 . With explicit inclusion of the contextual variables in the transformation function, the production possibility set becomesT = {(x, y, z) : F(x, y, z) ≤ 0}, and the conditional production possibility set is      T z0 = (x, y) : F x, y, z0 ≤ 0 .

(116)

Efficiency is still evaluated at the inputs used and outputs produced. But the appropriate benchmark bundle depends on the applicable vector of contextual variables. The disposability and convexity assumptions about the technology apply to the input-output set but are not necessarily extended to the contextual variables z0 . There are mainly two different ways to formulate the DEA problem depending on how the revised transformation function is conceptualized.

All-Inclusive DEA In one approach, following Banker and Morey [7], one imposes additional constraints for the non-discretionary  variable. Assume that the vector of non discretionary inputs is z0 = z10 , z20 where z1 is a favorable and z2 an unfavorable factor. Then the input-oriented VRS technical efficiency of a firm producing output

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y0 from input x0 while facing the external conditions z0 is obtained as θ ∗ = min θ  λj y j ≥ y 0 ; s.t. j



λj x j ≤ θ x 0 ;

j



j

λj z1 ≤ z10 ;

(117)

j



j

λj z2 ≥ z20 ;

j



λj = 1; λj ≥ 0 (j = 1, 2, . . . , N) .

j

Several points are to be noted. In the problem above, the scaling factor θ applies to the discretionary inputs only. Also, the actual value of the favorable nondiscretionary input is treated as an upper bound, and that of the unfavorable one is set as a lower bound for the corresponding values in the benchmark. This all-inclusive DEA model is intuitively appealing because it does not allow the hypothetical benchmark unit to have a lower level of the unfavorable factor or a higher level of the favorable factor than what is faced by the unit under evaluation. However, extending the convexity and disposability assumptions to the non-discretionary inputs is open to question. Also, all of the non-discretionary factors have to be unequivocally identified beforehand as favorable or unfavorable. Lastly, the DEA LP scores have to be recomputed whenever any non-discretionary factor is added or removed.

A Second Stage Regression The other approach introduced by Ray [56] includes only the discretionary inputs and outputs in the DEA LP problem in the first stage and estimates a regression of the DEA efficiency score on the non-discretionary variables. The conceptual foundation of the second stage regression is provided in Ray [55, 56]. Consider the single output case and assume that the production function is multiplicatively separable as y ∗ = f (x, z) = g(x).h(z). The actual output is

(118)

10 Data Envelopment Analysis: A Nonparametric Method of Production Analysis

y = y ∗ e−η ; η ≥ 0

465

(119)

where yy∗ = e−η is the technical efficiency of the firm. In this case, any change in the contextual variables causes a neutral shift in the production frontier that does not alter the marginal rates of substitution between inputs or marginal rates of transformation between outputs. Assuming CRS and that h(z) is naturally bounded between 0 and 1 for all values of z, Ray [55] has shown that the DEA efficiency score τy = ϕ1∗ is an estimate of h(z)e−η . One can specify    j αp zp j ln h zj = α0 +

(120)

p

and estimate an OLS regression ln τy = α0∗ + j



j

αp zp j + εj ;     where εj = E ηj − ηj ; α0∗ = α0 − E ηj . p

(121)

Ray [55] proposed a corrected OLS (COLS) procedure to adjust the intercept by the largest OLS residual and to use the adjusted residuals to measure technical efficiency. Apart from its simplicity, the second stage regression is quite appealing because the regression coefficients measure the marginal effects of changes in different non-discretionary variables on the DEA efficiency score. However, the statistical properties of the second stage regression estimators have been questioned in the literature. It is important to remember that the second stage regression will be invalid unless the non-discretionary variables are all uncorrelated with the inputs.

A Three-Stage Analysis Ruggiero [70] developed a three-stage procedure that uses the second stage regression and then reverts to the all-inclusive DEA of Banker and Morey with a modification. In this procedure, the predicted value of the dependent variable from the second stage regression is treated as a composite measure of “environmental harshness” (Ej ) for each decision-making unit in the sample. In the subsequent third stage of the analysis, this composite variable is used for data screening so that the benchmark frontier for any unit with input-output (x0 , y0 ) is constructed from the (xj , yj ) data for only those units with Ej ≥ E0 . The Ruggiero [70] three-stage VRS input-oriented DEA problem is

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θ ∗ = min θ  s.t. λj y j ≥ y 0 ; j



λj x j ≤ θ x 0 ;

(122)

j

λj = 0 if Ej < E0 ;  λj = 1; λj ≥ 0 (j = 1, 2, . . . , N) . j

As already noted, the convexity assumption about the input-output bundles in the all-inclusive problem (117) may not be applicable for some contextual variables. This is particularly true for categorical variables. Suppose that in the water utility example, the service areas are classified as rural, urban, and metropolitan but the exact measure of population density is not available for each observation. In this case, all one knows is that water delivery is most difficult in the rural areas and the least difficult in the metropolitan areas. Creating convex combinations of a categorical variable representing population density is not meaningful in this context. One may handle this by treating the conditional production possibility sets as nested in the sense that all input-output bundles that are feasible in a less favorable condition are also feasible in a more favorable condition but not the other way around. In this case, one includes only the rural observations to construct the frontier for evaluating utilities serving rural areas but all observations to construct the frontier that is to be used for evaluating the utilities serving the most densely populated areas. It should be noted though that for multiple contextual variables that are categorical, such cross-classification may severely restrict the number of observations available for constructing the frontier for the less favored groups. The LP problem in (122) by focusing on an aggregated measure of overall disadvantage helps significantly to mitigate the problem of data attenuation.

Conclusion This chapter has covered only some of the major topics from DEA. Given the limited scope of a book chapter (as compared to a full-length book), many other topics which are themselves quite important had to be excluded. Among them are such important areas of research as productivity growth and technical change31 , network DEA, centralized resource allocation, non-convexity (including free disposal hull 31  Chapters

20, “Conceptualization and Measurement of Productivity Growth and Technical Change: A Nonparametric Approach” by Ray and  21, “Modeling Technical Change: Theory and Practice” by Kumbhakar in this volume of the Handbook cover non-parametric DEA and parametric SFA approaches to measuring productivity growth and technical change.

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analysis),32 and others. The criterion for selecting topics was how closely they were related to core neoclassical production theory.33

References 1. Afriat S (1972) Efficiency estimation of production functions. Int Econ Rev 13(3):568–598 2. Aigner DJ, Lovell CAK, Schmidt P (1977) Formulation and estimation of stochastic frontier production function models. J Econ 6(1):21–37 3. Aparicio J, Pastor JT, Ray SC (2013) An overall measure of technical inefficiency at the firm and at the industry level: the ‘lost profit on outlay’. Eur J Oper Res 226(1):154–162 4. Ayres RU, Kneese AV (1969) Production, consumption, and externalities. Am Econ Rev 59:282–297 5. Banker RD (1984) Estimating the most productive scale size using data envelopment analysis. Eur J Oper Res 17(1):35–44 6. Banker RD, Maindiratta A (1988) Nonparametric analysis of technical and allocative efficiencies in production. Econometrica 56(5):1315–1332 7. Banker RD, Morey RC (1986) Efficiency analysis for exogenously fixed inputs and outputs. Oper Res 34(4):513–521 8. Banker RD, Thrall RM (1992) Estimating most productive scale size using data envelopment analysis. Eur J Oper Res 62:74–84 9. Banker RD, Charnes A, Cooper WW (1984) Some models for estimating technical and scale inefficiencies in data envelopment analysis. Manage Sci 30(9):1078–1092 10. Banker RD, Chang H, Natarajan R (2007) Estimating DEA technical and allocative inefficiency using aggregate cost or revenue data. J Prod Anal 27:115–121 11. Baumol WJ, Panzar JC, Willig RD (1982) Contestable Markets and the Theory of Industry Structure. New York: Harcourt, Brace, Jovanovich. 12. Baumol WJ, Oates WE (1988) The theory of environmental policy, 2nd edn. Cambridge University Press, Cambridge 13. Cassell JM (1937) Excess capacity and monopolistic competition. Q J Econ 51(3):426–443 14. Chambers RG, Melkonyan T (2012) Production technologies, material balance, and the income-environmental quality trade-off. University of Exeter working paper 15. Chambers RG, Chung Y, Färe R (1996) Benefit and distance functions. J Econ Theory 70: 407–419 16. Chambers RG, Chung Y, Färe R (1998) Profit, directional distance functions, and nerlovian efficiency. J Optim Theory Appl 98:351–364 17. Charnes A, Cooper WW (1968) Programming with linear fractional functionals. Nav Res Logist Q 15:517–522 18. Charnes AC, Cooper WW, Mellon B (1952) Blending aviation gasolines – a study in programming interdependent activities in an integrated oil company. Econometrica 20(2): 135–159 19. Charnes A, Cooper WW, Rhodes E (1978) Measuring the efficiency of decision making units. Eur J Oper Res 2(6):429–444

a detailed discussion of non-convexity in general, refer to the  Chap. 18, “Nonconvexity in Production and Cost Functions: An Exploratory and Selective Review” by Briec, Kerstens, and Van de Woestyne in this volume of the Handbook. 33 For detailed discussion of DEA from an OR/MS perspective, the reader should refer to Zhu [77] and Cooper, Seiford, and Tone [24]. Fare, Grosskopf, and Lovell [35] and Ray [56] explain the economic theory behind DEA. 32 For

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20. Charnes A, Cooper WW, Rhodes E (1979) Short communication: measuring the efficiency of decision making units. Eur J Oper Res 3(4):339 21. Coelli T, Lauwers L, Van Huylenbroeck GV (2007) Environmental efficiency measurement and the materials balance condition. J Prod Anal 28:3–12 22. Cooper WW, Thompson RG, Thrall RM (1996) Introduction: extensions and new developments in DEA. Ann Oper Res 66:3–45 23. Cooper WW, Park SK, Pastor JT (1999) RAM: a range adjusted measure of inefficiency for use with additive models, and relations to other models and measures in DEA. J Prod Anal 11:5–42 24. Cooper WW, Seiford L, Tone K (2002) Data envelopment analysis: a comprehensive text with uses, example applications, references and DEA-solver software. Kluwer, Norwell 25. Cropper ML, Oates WE (1992) Environmental economics: a survey. J Econ Lit 30:675–740 26. Dakpo KH, Jeanneauxe P, Latruffe L (2016) Modeling pollution generating technologies in performance benchmarking: recent developments, limits, and future prospects in the nonparametric framework. Eur J Oper Res 250:347–359 27. Dantzig GB (1951) Maximization of a linear function of variables subject to linear inequalities. In: Koopmans TC (ed) Activity analysis of production and allocation. Wiley, New York, pp 339–347 28. Debreu G (1951) The coefficient of resource utilization. Econometrica 19(3):273–292 29. Färe R, Grosskopf S (2003) Nonparametric productivity analysis with undesirable outputs: comment. Am J Agric Econ 85:1070–1074 30. Färe R, Lovell CAK (1978) Measuring the technical efficiency of production. J Econ Theory 19(1):150–162 31. Färe R, Grosskopf S, Lovell CAK (1985) The measurement of efficiency of production. Kluwer-Nijhoff, Boston 32. Färe R, Grosskopf S, Kokkelenberg EC (1989) Measuring plant capacity, utilization and technical change: a nonparametric approach. Int Econ Rev 30(3):655–666 33. Färe R, Grosskopf S, Lovell CAK, Pasurka C (1989) Multilateral productivity comparisons when some outputs are undesirable: a non-parametric approach. Rev Econ Stat 71(1):90–98 34. Färe R, Grosskopf S, Lovell CAK, Yaisawarng S (1993) Derivation of shadow prices for undesirable outputs: a distance function approach. Rev Econ Stat 75:374–380 35. Färe R, Grosskopf S, Lovell CAK (1994) Production frontiers. Cambridge University Press, Cambridge 36. Färe R, Grosskopf S, Noh DW, Weber W (2005) Characteristics of a polluting technology: theory and practice. J Econ 126:469–492 37. Farrell MJ (1957) The measurement of technical efficiency. J R Stat Soc Ser A Gen 120(Part 3):253–281 38. Førsund F (2009) Good modelling of bad outputs: pollution and multiple-output production. Int Rev Environ Resour Econ 3(1):1–38 39. Førsund F (2018) Multi-equation modeling of desirable and undesirable outputs satisfying the material balance. Empir Econ, online 54(1):67–99 40. Frisch R (1965) Theory of production. Rand McNally and Company, Chicago 41. Hampf B (2014) Separating environmental efficiency into production and abatement efficiency: a nonparametric model with application to US power plants. J Prod Anal 41:457–473 42. Hanoch G, Rothschild M (1972) Testing the assumptions of production theory: a nonparametric approach. J Polit Econ 80(2):256–275 43. Johansen L (1968) Production functions and the concept of capacity. Reprinted in Førsund FR (ed) Collected works of Leif Johansen, vol 1. North Holland, Amsterdam 44. Koopmans TJ (1951) Analysis of production as an efficient combination of activities. In: Koopmans TJ (ed) Activity analysis of production and allocation. Wiley, New York, pp 33–97 45. Koopmans TJ (1957) Three essays on the state of economic science. McGraw Hill, New York 46. Kumbhakar S, Lovell CAK (2000) Stochastic frontier analysis. Cambridge University Press, New York

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47. Leleu H, Briec W (2009) A DEA estimation of a lower bound for firms’ allocative efficiency without information on price data. Int J Prod Econ 121:203–211 48. Lozano SC (2015) A joint-inputs network DEA approach to production and pollutiongenerating technologies. Expert Syst Appl 42:7960–7968 49. Luenberger DG (1992) Benefit functions and duality. J Math Econ 21:115–145 50. Murty S, Russell RR (2016) Modeling emission-generating technologies: reconciliation of axiomatic and by-production approaches. Empir Econ 54(1):7–30 51. Murty S, Russell R, Levkoff SB (2012) On modeling pollution-generating technologies. J Environ Econ Manag 64:117–135 52. Pastor JT, Louis JL, Sirvent I (1999) An enhanced DEA Russell-graph efficiency measure. Eur J Oper Res 115:596–607 53. Pethig R (2006) Non-linear production, abatement, pollution and materials balance reconsidered. J Environ Econ Manag 51:185–204 54. Portela MCAS, Thanassoulis E (2005) Profitability of a sample of Portuguese bank branches and its decomposition into technical and allocative components. Eur J Oper Res 162(3): 850–866 55. Ray SC (1988) Data envelopment analysis, non-discretionary inputs and efficiency: an alternative interpretation. Socio Econ Plan Sci 22(4):167–176 56. Ray SC (1991) Resource-use efficiency in public schools: a study of Connecticut data. Manag Sci 37(12):1620–1628 57. Ray SC (2004) Data envelopment analysis: theory and techniques for economics and operations research. Cambridge University Press, New York 58. Ray SC (2007) Shadow profit maximization and a measure of overall inefficiency. J Prod Anal 27:231–236 59. Ray SC (2009) Are Indian firms too small? A nonparametric analysis of cost efficiency and the optimal organization of industry in Indian manufacturing. Indian Econ Rev XXXXVI(1):49–67 60. Ray SC (2010) A one-step procedure for returns to scale classification of decision making units in data envelopment analysis. University of Connecticut Economics working paper 2010-07 61. Ray SC (2015) Nonparametric measures of scale economies and capacity utilization: an application to U.S. manufacturing. Eur J Oper Res 245:602–611 62. Ray SC, Ghose A (2014) Production efficiency in Indian agriculture: an assessment of the post green revolution years. Omega 44:58–69 63. Ray SC, Jeon Y (2009) Reputation and efficiency: a non-parametric assessment of America’s top-rated MBA programs. Eur J Oper Res 189(2008):245–268 64. Ray SC, Mukherjee K (2016) Data envelopment analysis with aggregated inputs and a test of allocative efficiency when input prices vary across firms. Data Envel Anal J 2(2):141–161 65. Ray SC, Chen L, Mukherjee K (2008) Input price variation across locations and a generalized measure of cost efficiency. Int J Prod Econ 116:208–218 66. Ray SC, Mukherjee K, Venkatesh A (2018) Nonparametric measures of efficiency in the presence of undesirable outputs: a by-production approach with weak disposability. Empir Econ 54(1):31–65 67. Ray SC, Walden J, Chen L (2018) Economic Measures of Capacity Utilization: A Nonparametric Cost Function Analysis. Working Paper 2018–02, University of Connecticut, Department of Economics 68. Rodseth KL (2015) Axioms of a polluting technology: a materials balance approach. Environ Res Econ 67(1):1–22. Online October 2015 69. Rodseth KL (2016) Environmental efficiency measurement and the materials balance condition reconsidered. Eur J Oper Res 250:342–346 70. Ruggiero J (1998) Non-discretionary inputs in data envelopment analysis. Eur J Oper Res 111:461–469 71. Seiford L, Zhu J (1999) An investigation of returns to scale in data envelopment analysis. Omega Int J Manag Sci 27:1–11 72. Shephard RW (1953) Cost and production functions. Princeton University Press, Princeton

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73. Shephard RW (1970) Theory of cost and production functions. Princeton University Press, Princeton 74. Tone K (2001) A slacks-based measure of efficiency in data envelopment analysis. Eur J Oper Res 130:498–509 75. Tone K (2002) A strange case of the cost and allocative efficiencies in DEA. J Oper Res Soc 53:1225–1231 76. Varian HR (1984) The nonparametric approach to production analysis. Econometrica 52(3):579–597 77. Zhu J (2003) Quantitative models for performance evaluation and benchmarking: data envelopment analysis with spreadsheets and DEA excel solver. Kluwer Academic, Boston

Activity Analysis in Production Economics

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Thijs ten Raa

Contents Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The Origin of Activity Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Substitution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Activity Foundation of the Production Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Variants of Houthakker’s Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Activity Foundation of Input-Output Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Efficiency . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Cross-References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

472 472 473 475 477 478 479 481 481 481

Abstract

This chapter opens with the historical root of activity analysis. The framework of activity analysis admits multiple techniques to produce a commodity. Substitution theorems investigate when the market mechanism singles out a best technique for each product and if the best techniques vary with the data of an economy, such as resource availabilities. Houthakker’s Theorem initiated a literature on the relationship between the distribution of activities and the form of the aggregate production function. Activity analysis is connected to modern input-output analysis, where the numbers of products and industries differ, which facilitates the measurement of the efficiency of the production units of an economy and of the economy.

T. ten Raa () Utrecht School of Economics, Utrecht University, Utrecht, The Netherlands e-mail: [email protected] © Springer Nature Singapore Pte Ltd. 2022 S. C. Ray et al. (eds.), Handbook of Production Economics, https://doi.org/10.1007/978-981-10-3455-8_25

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Keywords

Activity analysis · Efficiency · Houthakker’s theorem · Input-output analysis · Substitution theorem

Introduction This chapter shows how activity analysis is a basis of production functions. One of the advantages of activity analysis is that it offers a framework for the measurement of efficiency. Given the ensemble of all activities, one may determine the frontier and then measure how close activities are to the frontier. There are two problems: First, in which direction should one go to measure the gap between an activity and a best practice frontier element? Second, how should one aggregate the efficiencies of the individual activities? Both issues are solved in the recent literature, and these results neatly complement neoclassical production analysis with its explicit or implicit assumption of cost minimization. In the section “The Origin of Activity Analysis,” I discuss the historical root of activity analysis. The framework of activity analysis admits multiple techniques to produce a commodity. An important economic question is: Does the market mechanism single out a best technique for each product? If so, does the best technique vary with the data of an economy, such as resource availabilities? Substitution theorems are discussed in the section “Substitution.” Section “Activity Foundation of the Production Function” discusses in detail the relationship between the distribution of activities and the form of the production function. The classic result is Houthakker’s Theorem. Section “Variants of Houthakker’s Theorem” discusses the further literature on this relationship. Section “Activity Foundation of Input-Output Analysis” connects activity analysis to modern input-output analysis, where the numbers of products and industries differ. Section “Efficiency” analyzes the measurement of the efficiency of the production units of an economy and of the economy.

The Origin of Activity Analysis Activity analysis originates from Tjalling Koopmans’ [10] Activity Analysis of Production and Allocation, a conference volume. The conference was on linear programming, and indeed, activity analysis is still considered a practical tool for the analysis of production and allocation. The conference volume is an icon in the history of economic thought. In 1975, his research on this topic earned Dutch American Koopmans, jointly with Russian Leonid Kantorovich, the Nobel Prize in economics. Incidentally, Koopmans was named by the then young William Baumol, who sadly died during the writing of this chapter. Baumol’s [1] Economic Theory and Operations Analysis popularized the use of operations research in economics and was a leading microeconomic textbook until well into the 1980s.

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Koopmans’ Activity Analysis of Production and Allocation consists of four parts: theory (ten papers), applications (six papers), convex analysis (four papers), and algorithms (five papers). The third and fourth parts are now standard fare in applied mathematics and software, greatly facilitated by recent computer power. The second part is a collection of assorted papers. The first part is the most important one, at least for our objective, an overview of activity analysis in the context of production economics. The first part of Activity Analysis of Production and Allocation begins with Dantzig’s mathematical programming, and Koopmans’ “Analysis of Production as an Efficient Combination of Activities,” continues with Von Neumann and Leontief’s dynamic models and completes with five (!) papers on the static Leontief model. Indeed, there is a close connection between input-output analysis and activity analysis, both at the level of concept and of application. To begin with the latter, mind that activity analysis blossomed shortly after World War II. Planning was an important policy tool, not only in Russia, but also in the United States. Warfare prompted a change in demand (toward aircraft and other equipment), and this had to be reconciled with resource scarcity.

Substitution At the conceptual level, an activity is a pair of an input vector and an output vector. Leontief, who won the Nobel Prize 2 years earlier, was ahead of Koopmans and activity analysis, but the latter, not surprisingly, offers a more general framework for economic analysis. By the same token, input-output analysis is a special, albeit important, case of activity analysis. This is particularly true of static input-output analysis. Here an activity is a pair of an input vector and a pure output vector, where “pure” means that only one component is nonzero. Joint production is ruled out. Dynamic input-output analysis is more general. It features two types of inputs, namely, absorbed inputs and capital, and the output is basically still pure, but accompanied by the same capital. There may be depreciation, but that is modeled by including it in the absorbed inputs. The main simplification of input-output analysis is the implicit assumption that for every output there is only one activity that produces it. Each product has a unique input structure. Economists, particularly students of Leontief, starting with Paul Samuelson, were intrigued by this simplifying, implicit assumption and theorized about it. Samuelson, Koopmans, and Arrow each contribute a paper on this subject to Koopmans [10]. The result of their analysis is the substitution theorem. If activities are pure, but different ones coexist for the production of commodities, and if there is only one nonproduced input, called “labor,” then there exists a collection of activities, one for each product, that minimizes the labor input of the net output vector produced, irrespective of the proportions of the net output vector. Except for labor, commodities feature in output vectors and input vectors, and the difference between the two is net output, available for household consumption. Thus, the substitution theorem states that there is a dominant technology to produce net output, whatever

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its commodity composition. An activity is also called a “technique,” and a collection of available activities or techniques is also called a “technology.” For an obvious reason, the substitution theorem – industries can substitute techniques but do not do so when minimizing the resource use – is also called the nonsubstitution theorem. Activity analysis sticks to the input-output assumption of constant returns to scale. Consequently, the minimization of the resource input for a given level of net output is equivalent to the maximization of the level of net output given a resource input. This observation is obvious in the context of the substitution theorem (a single nonproduced input and pure output vectors) and remains valid when net outputs are not pure but feature multiple nonzero components. For example, if a given net output vector is producible with only 4/5th or 80% of the actual use of the observed amount of resource, then, under constant returns to scale, the actual amount of resource could produce 5/4th or 125% of the given net output vector. We say the efficiency of the economy is 80%. Equivalently, potential output is 125% of actual output. In determining these performance measures, the collection of available activities is considered to be given, but the intensities with which each activity is run are to be determined by the mathematical program that minimizes the resource requirement or maximizes the level of net output. There are two constraints in either program. Commodity balances require that the activity intensities are large enough so that the supply of net output is at least equal to household demand, but small enough so that production demand for the resource does not exceed the available stock. The constraints pick up Lagrange multipliers: a factor reward for the resource and also shadow prices for the produced commodities. The shadow prices, including the one for the factor input, fulfill the so-called dual constraints. The dual constraints are such that all activities have nonpositive profit and the activities running with positive intensity break even. Therefore, the shadow prices are competitive prices. Competitive prices would signal to entrepreneurs which activities to undertake. The competitive prices are also a useful analytical tool. An advantage is that in the general activity model, there are more activities than commodities, thus allowing substitution. Therefore, the dual variables (the prices) have lower dimension than the primal variables (the activity intensities). Johansen [8] approached the substitution theorem using the competitive prices, and ten Raa [21] filled the gaps. The substitution theorem has the striking result that prices are independent of demand, hence determined by supply, more precisely, by technology. This is classical economics. It rests on the classical assumptions of a single nonproduced input and no joint production. The classical assumptions have a built-in tension. The assumption of a single nonproduced input, labor, suggests that capital is a produced commodity. We have no difficulty with this view. Capital is buildings, machinery, equipment, and infrastructure, and these are produced commodities indeed. However, the essence of productive capital is that at least some of it (after correcting for depreciation) remains when an activity has been completed. But then, the activity has at least two outputs: the commodity produced and the remaining capital. In other words, there is joint production. There are two approaches to deal with this issue. The first, going back to Von Neumann [25], is to accommodate multiple positive output components in the activity vectors. However, Von Neumann

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trades off this generality on the supply side against more specificity on the demand side, assuming that labor services are the output of the household activity with fixed consumption coefficients, which essentially is a classical, Marxian assumption. The second is to accommodate joint production and to analyze to which extent substitution emerges [15]. A more blatant violation of the substitution theory assumptions occurs when there are multiple nonproduced inputs, for example, labor and land, or, as in many neoclassical economic models, labor and capital, where at any point of time the latter is considered to be given by the past. Then the choice of techniques will depend on the composition of the resources. A relatively more labor-endowed economy will employ more labor-intensive techniques when maximizing the level of output. The argument is simple, particularly when the relative factor intensities range from very low to very high values compared to the endowment ratio. Then both endowments can be fully employed, and the average factor intensity in production will be equal to the endowment ratio. When East Germany was absorbed, labor became less scarce and capital more so. The shadow price of labor became smaller, supporting more efficient labor-intensive production. This reasoning pertains to the potential output of the economy. The actual economy may have followed a different, less efficient path.

Activity Foundation of the Production Function The economy has a supply side, populated with producers, and a demand side, populated with consumers. Center pieces of the supply side are production functions. Production functions have different functional forms. Implicit are alternative degrees of substitution and scale economies [5]. In activity analysis, however, the situation is more basic. An activity is like a recipe. There are input requirements per unit of output. Output may be multidimensional as well, for example, juice may be a byproduct when cooking. Differences in output are accompanied by differences in inputs. And even when there are no differences in outputs, such as in an industry producing a homogeneous product, there may be differences in inputs. Moreover, alternative activities may produce the same output. Local conditions vary, alternative production techniques compete, and some production units are simply less efficient than others, a phenomenon which shows in a different (higher) input structure. Activities replace each other. For example, when a new supply of some resource is discovered, activities which make relatively intensive use of this resource will expand. The economy will use a different mix of inputs. This, indeed, may be described by a production function, but it is an interesting question how differences in activities translate into alternative functional forms of production. In activity analysis, alternative techniques to produce commodities coexist, and the efficient ones are determined using a mathematical program in which intensities, one for each activity, that is, technique, are the variables. How can we reconcile this framework with a neoclassical production function, such as the CobbDouglas function, Y = AK α L β? Here K and L are inputs, Y is output, and A, α,

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and β are parameters. There are decreasing/constant/increasing returns to scale if α + β < / = / > 1, respectively. A simplistic link would be as follows, for the case of constant returns to scale. An activity would be a pair of variable inputs and an output, which can be normalized, (k, l; 1), with Akα l β = 1. The activities can be parameterized by one input, for example, k. Then l = (Ak α)−1/ β, and therefore the technology set of activities is {(k, (Ak α)−1/β; 1)| k > 0}. Each activity can be run with intensity s k. Total output will be s k dk, where the integral is taken over the positive numbers. The  constraints are s k kdk ≤ K and s k ldl ≤ L, where K and L are the factor endowments. Because of the convexity of the technology set and the assumption of constant returns to scale, α + β = 1, running different activities with positive intensity can be improved upon, in terms of output, by replacement of the activities by their intensity-weighted average. Hence, output is maximized by running the single activity with the right factor intensity that matches the endowments ratio, k/l = k(Ak α)1/ β = K/L. Solving, using α + β = 1, k∗ = (K/L)β/A. The intensity is determined by s k∗ k∗ = K, hence s k∗ = K(K/L)− β A = AK α L β. All other intensities s k are zero. Since activities were normalized by output and returns to scale are constant, output equals the activity intensity, Y = AK α L β. In other words, the aggregate production function is the same as the underlying microtechnology. In this simple activity analytic underpinning of the aggregate production function, all production units are free to choose from a continuum of activities, from labor to capital intensive. And all production units would select the same activity. This extreme flexibility, with its concentrated optimal activity pattern, is not very realistic. Capital intensities of production units are fixed once installed and vary across production units. Individual production units cannot access the full menu of activities, technology. In activity analysis, it is customary to assume that production units have given techniques and are represented by their activities. The implicit assumption is that production units cannot substitute inputs. However, at the macrolevel, substitution may take place. When a factor input becomes abundant, such as labor in the time of German unification, its price will go down, making activities with intensive use of the abundant factor input financially feasible. The subpopulation of active production units will shift to the more intensive users of the abundant endowment. In this framework, we better do not assume that production intensities can vary freely from zero to infinity. If so, then a single production unit with the right factor intensity, which matches the endowments ratio, would pick up all activity. The result would be the same as in the simplistic world were all production units to have access to the full technology. In line with the factor specificity of an activity, it is assumed there is a capacity constraint for each activity. A fixed input causes the capacity constraint. The fixed input is other than the variable inputs, capital, and labor. Houthakker [7] suggests entrepreneurial resources. The distribution of the capacity constraint (of entrepreneurial resources) over activities (k, l; 1) is considered to be given, y(k, l). This distribution need not be concentrated on a frontier like {(k, l)| Ak α l β = 1}. Some activities may dominate others, with both components of (k, l) smaller. Yet a dominated activity may be run, because the superior activity, like all activities, has a

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capacity constraint. Activities 0 ≤ s(k, l) ≤ y(k, l). Subject  can be run with intensities  to the factor constraints s(k, l)kdkdl ≤ K and s(k, l)ldkdl ≤ K, we maximize  output s(k, l)dkdl. This is a linear program with a continuum of variables s(k, l). Denote the shadow prices of the two factor constraints by r and w, respectively. By the phenomenon of complementary slackness, unprofitable activities, with unit cost rk + wl > 1, are not run, s(k, l) = 0. By the same argument, profitable activities, with unit cost rk + wl < 1, are run at full capacity, s(k, l) = y(k, l). Activities which break even, rk + wl = 1, have activity 0 ≤ s(k, l) ≤ y(k, l), but since the set of such activities has measure zero we may set s(k, l) = y(k, l). It follows that  inputsand output are K = rk+ wl≤1 y(k, l)kdkdl, L = rk+ wl≤1 y(k, l)ldkdl, and Y= rk+ wl≤1 y(k, l)kdkdl, respectively. The implicit assumption is that all factor input can be fully employed. There must be activities with factor intensity k/l below endowment ratio K/L and activities with factor intensity above the endowment ratio. The three expressions, for inputs K and L and output Y, are interrelated by the two shadow prices r and w. The idea of Houthakker [7] is to use the first to expressions to solve for r and w in terms of K and L. Substitution of the results in the third expression yields output as function of the inputs. Houthakker [7] carries out this calculation for the capacity distribution with Pareto density function, y(k, l) = μk κ−1 l λ−1 , where μ, κ, and λ are positive constants. The result is Y = AK α L β with α = κ(κ + λ + 1), β = λ(κ + λ + 1), and A a positive constant depending on μ, κ, and λ. In other words, a Pareto capacity distribution yields a Cobb-Douglas production function. This is Houthakker’s Theorem. At the microlevel, activities have fixed input-output ratios – it takes given amounts of labor to operate given machinery and equipment – but a change in resources, such as the inclusion of the East German labor force, is accommodated by the activation of new activities and the deactivation of some incumbent activities. Reallocations of resources across activities manifest as substitutions. The capacity distribution is not concentrated on a single isoquant in input space. Both k and l can be bigger. In solving the output maximization, smaller input combinations are activated, but only to full capacity. Residual inputs are employed by more input-intensive activities. The capacity constraints thus yield decreasing returns to scale. Indeed, the Cobb-Douglas function has exponents summing to a number less than one. Houthakker’s activity foundation of neoclassical production functions works only if returns to scale are decreasing.

Variants of Houthakker’s Theorem Clearly, different capacity distributions for the activity levels will generate different production functions. Houthakker [7] has generated a stream of theoretical and applied research, to date. The bulk of this literature features a lower dimension, with only one variable input, namely labor, and again one fixed output, which is now capital. In this one fixed-one variable input framework, Levhari [14] found the capital distribution for which total output is a CES function of the total fixed input (capital) and the total variable input (labor) and showed it encompasses the

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Cobb-Douglas function. Muysken [16] has consolidated the Cobb-Douglas, CES, and VES functions by showing they are all generated by beta distributions, with alternative parametrizations. Two books on the distribution approach to production are Johansen [9] and Sato [19]. In this literature, activities have fixed input-output proportions, and capacity constraints explain the existence of inefficient activities. Increases in levels of inputs prompt the activation of less efficient activities, in Ricardian style. The law of one price yields rents to the more efficient activities. The activation of different activities prompts different proportions between the input totals and the output. Substitution is considered a symptom of the change in the range of active activities (run with positive intensity). The interrelation of total output to two inputs is a shortcut with a strong macroeconomic flavor (e.g., Lagos [12]). One way to reconcile economy-wide analysis with activities is to aggregate in stages, from activities (production units) via conglomerates (industries) to the economy. In the second stage, one has to aggregate production functions more general than fixed proportions functions, also called Leontief functions. In the one variable-one fixed output framework [18] analyzed how micro-CES functions and an appropriate inefficiency distribution (reflecting capacity constraints) generate a macro-CES function, with a greater elasticity of substitution (for the same reason as capacity, constraints create substitutability when the microfunctions are Leontief). Growiec [6] generalizes the capacity distribution with Pareto density function, y(k, l) = μkκ−1 lλ−1 . He keeps the multiplicative structure, in other words the independence of the unit factor productivities, F a and F b. For each (K, L), firms maximize CES output A[ψ(bK)θ + (1 – ψ)(aL)θ ]1/ θ with respect to unit factor productivities a and b, subject to Fa(a)Fb(b) = N, where N indexes the technology frontier, 0 < N < 1. If Fa(a) = c a a γ , F b(b) = c b b γ α /(1 − α) , then maximum output is AK α L1 − α . This is the case where a Pareto distribution of unit factor productivities and free choice of technology yields Cobb-Douglas output. However, the mechanism is very different than in Houthakker [7]. In Growiec [6], firms freely choose from a menu which is parametrized by a distribution. The formal similarity – a Pareto distribution translates into a Cobb-Douglas function – is coincidental. In Growiec, the Weibull distribution translates into a CES function, while this is not the case in Levhari’s [14] CES analysis of the Houthakker [7] model.

Activity Foundation of Input-Output Analysis The two-stage aggregation, from activities via industries to the economy, is a useful framework to accommodate the output differences between industries and to relate their inefficiencies. We return to the definition of an activity: a pair of an input vector and an output vector. Unlike traditional input-output analysis, modern activity analysis allows for multiple outputs and even different numbers of outputs and production units or industries. The advantage of input-output analysis, the accommodation of intermediate inputs, is preserved though. An input vector

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consists of m commodities and l factor inputs. An output vector consists of m commodities (the same as in the input vector). There are n production units, that is, activities. The first production unit has produced inputs (u11 , . . . , u m1 ), factor inputs (f11 , . . . , f l1 ), and outputs (v11 , . . . , v m1 ). Here f stands for factor, u stands for use, and v, the next letter in the alphabet, stands for output. Writing these vectors as column vectors (as the index notation suggests), and stacking the column vectors representing the other production units next to them, the n activities are represented by the triplet of l × n-dimensional factor input matrix F, m × ndimensional intermediate input matrix U, and m × n-dimensional output matrix V. If all activities are included, the economy is represented by the triplet (F, U, V) and the nonnegative l-dimensional available resource vector ω. An allocation is a nonnegative n-dimensional activity vector, s, where the i-th component is the scale of production of unit i. For example, if s i = 1.1, all inputs and outputs of activity i are 10% greater than observed. An allocation is feasible if Fs ≤ ω. Intermediate demand is Us. Gross output is Vs. Net output is the difference, (V − U) s. This is final consumption. In traditional input-output analysis, the number of activities equals the number of commodities, i.e., m = n. In this literature, gross output Vs is denoted x and net output (V – U) s is denoted y. It is reasonable to assume that output matrix V has a dominant diagonal. Then V is invertible, and the choice between the allocation variable s of activity analysis and the gross output variable x of inputoutput analysis is a matter of a change of variable, x = Vs and s = V−1 x. The material balance y = (V – U) s can be rewritten as Leontief’s [13] basic equation, y = x − Ax. Here A is the matrix of input-output equations determined by A = UV−1 . The latter specification is the so-called commodity technology model, which has superior balance and invariance properties [11]. The upshot is that activity analysis encompasses input-output analysis. In the System of National Accounts [2], the number of commodities is greater than the number of industries: m > n. Standard input-output analysis is problematic, but activity analysis remains doable [24]. When input and output data are used in raw form, at the level of reporting production units, without aggregation to industries, the number of activities is greater than the number of commodities, m < n. In this case, there is a wealth of data, and activity analysis facilitates stochastic input-output analysis. The commodity technology model, A = UV−1 , does not exist when output matrix V is not square, but input-output coefficients matrix A can be estimated by regressing inputs U on outputs V: U = AV + ε, including an error term [17].

Efficiency An allocation is efficient if no other allocation is better to one consumer without being worse to the other consumers. Observed allocations tend to be inefficient. The efficiency of an economy is measured by Debreu’s [4] coefficient of resource

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utilization, ρ, a number between 0 and 1. ρ is the lowest number such that all consumers can be equally well off if the endowment is reduced from ω to ρω. The coefficient of resource allocation depends on the preferences of the consumers. If there is much substitutability, there is much scope for reallocations and, therefore, for potential efficiency gains. In this case, the coefficient of resource utilization will be low. Conversely, ten Raa [22] has shown that if the consumers have Leontief preferences (consumption with fixed proportions), the coefficient of resource utilization attains its upper bound. In other words, the assumption of Leontief preferences yields a conservative inefficiency measure. ten Raa [22] coins this measure the Debreu-Diewert coefficient of resource allocation. The Debreu-Diewert coefficient of resource allocation, by its assumption of fixed consumption bundles, rules out efficiency gains due to consumers’ exchanges. If Debreu’s coefficient of resource allocation is 0.7 and the Debreu-Diewert coefficient is 0.8, then the difference represents consumer inefficiency. In this example, overall inefficiency is 30%, production inefficiency 20%, and consumer inefficiency 10%. ten Raa [22] shows that microdata of final consumption are not needed to calculate the Debreu-Diewert coefficient of resource utilization. The calculation of the Debreu-Diewert coefficient of resource allocation is simple in the activity framework of the economy. ten Raa [22] shows that the better set of Pareto noninferior allocations is {s ≥ 0| (V – U)s ≥ (V – U)e}, where e is the unit or summation vector with all components equal to one and the inequalities are commodity constraints. Over this set, one must minimize ρ subject to feasibility condition Fs ≤ ρω. This is a linear program. The Lagrange multipliers of the commodity and factor constraints are competitive prices. By the phenomenon of complementary slackness, activities with positive activity level break even, and unprofitable activities are shut down. In other words, the principle of profit maximization selects the activities that minimize resource use. The competitive commodity prices can be used to evaluate the net output growth, competitive factor rewards are used to evaluate the factor input growth, and the difference is total factor productivity growth (TFP). A classical result is that for perfectly competitive economies, TFP equals the shift in the production function or technical change (TC; see Solow [20]). In general, TFP equals the sum of TC and the change in the Debreu-Diewert coefficient of resource utilization or, briefly, efficiency change (EC; see ten Raa [22]). Both components can be decomposed further in numerous ways. The decomposition of efficiency involves a bias issue. The efficiency of a system of production units is less than the average efficiency of the production units. The reason is that the allocation of resources may be inefficient. This bias issue was first analyzed by Blackorby and Russell [3] who demonstrated that only when production is linear in the sense that marginal rates of substitution and marginal rates of transformation are constant and these constants are common to the production units, there is no bias issue. ten Raa [23] showed that the bias measures the inefficiency of the industrial organization of the production units.

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Conclusion Activity analysis bridges the gap between input-output analysis with its fixed input proportions and neoclassical production theory with abounding substitutability, using the former as a foundation of the latter and showing that the latter encompasses the former. Substitutability of factor inputs is a manifestation of reallocations between activities with different factor intensities. Activity analysis accommodates efficiency analysis by measuring and decomposing inefficiencies.

Cross-References  Multiproduct Technologies  Neoclassical Production Economics: An Introduction

References 1. Baumol WJ (1961) Economic theory and operations analysis. Prentice-Hall, New York 2. Beutel J (2017) Chapter 3. The supply and use framework of national accounts. In: ten Raa T (ed) Handbook of input–output analysis. Edward Elgar, Cheltenham 3. Blackorby C, Russell RR (1999) Aggregation of efficiency indices. J Prod Anal 12(1):5–20 4. Debreu G (1951) The coefficient of resource utilization. Econometrica 19(3):273–292 5. Diewert WE, Fox KJ (2008) On the estimation of returns to scale, technical progress and monopolistic markups. J Econ 145(1):174–193 6. Growiec J (2008) Production functions and distributions of unit factor productivities: uncovering the link. Econ Lett 101(1):87–90 7. Houthakker HS (1955) The Pareto distribution and the Cobb-Douglas production function in activity analysis. Rev Econ Stud 23(1):27–31 8. Johansen L (1972) Simple and general nonsubstitution theorems for input–output models. J Econ Theory 5(3):383–394 9. Johansen L (1972) Production functions: an integration of micro and macro, short run and long run aspects. North-Holland, Amsterdam 10. Koopmans TC (1951) Activity analysis of production and allocation. Wiley, New York 11. Kop Jansen P, ten Raa T (1990) The choice of model in the construction of input–output coefficients matrices. Int Econ Rev 31(1):213–227 12. Lagos R (2006) A model of TFP. Rev Econ Stud 73(4):983–1007 13. Leontief WW (1936) Quantitative input and output relations in the economic system of the United States. Rev Econ Stat 18(3):105–125 14. Levhari D (1968) A note on Houthakker’s aggregate production function in a multifirm industry. Econometrica 36(1):151–154 15. Mirrlees JA (1969) The dynamic nonsubstitution theorem. Rev Econ Stud 36(1):67–76 16. Muysken J (1983) Transformed beta-capacity distributions of production units. Econ Lett 11(3):217–221 17. Rueda-Cantuche J (2017) Chapter 4. The construction of input–output coefficients. In: ten Raa T (ed) Handbook of input–output analysis. Edward Elgar, Cheltenham 18. Sato K (1969) Micro and macro constant-elasticity-of-substitution production functions in a multifirm industry. J Econ Theory 1(4):438–453

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19. Sato K (1975) Production functions and aggregation. North-Holland, Amsterdam 20. Solow RM (1957) Technical change and the aggregate production function. Rev Econ Stat 39(3):312–320 21. ten Raa T (1995) The substitution theorem. J Econ Theory 66(2):632–636 22. ten Raa T (2008) Debreu’s coefficient of resource utilization, the Solow residual, and TFP: the connection by Leontief preferences. J Prod Anal 30(3):191–199 23. ten Raa T (2011) Benchmarking and industry performance. J Prod Anal 36(3):285–292 24. ten Raa T, Shestalova V (2015) Supply-use framework for international environmental policy analysis. Econ Syst Res 27(1):77–94 25. von Neumann J (1945) A model of general economic equilibrium. Rev Econ Stud 13(1):1–9

Bad Outputs

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Sushama Murty and R. Robert Russell

Contents Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Single-Equation Modeling of the Technology Under Standard Disposability Assumptions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Treating Pollution as a Conventional Production Output . . . . . . . . . . . . . . . . . . . . . . . . . . . Treating Pollution as a Conventional Production Input . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Weakly Disposable Technologies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Multiple-Equation Modeling of Pollution-Generating Technologies . . . . . . . . . . . . . . . . . . . . Rival vs. Joint Production of Multiple Outputs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Multi-equation Modeling: The Case of Factorially Determined Multi-output Production . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Multi-equation Modeling: The Case of Rival and Joint Production . . . . . . . . . . . . . . . . . . . Multi-equation Modeling of Emission-Generating Technologies with Abatement Activities and Multiple Emissions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Rival Production of Abatement and the Economic Output . . . . . . . . . . . . . . . . . . . . . . . . . . Modeling the Generation of Multiple Emissions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The Overall By-Production Technology with Abatement and Multiple Emissions . . . . . . Axiomatic Approach to Modeling Emission-Generating Technologies . . . . . . . . . . . . . . . . . . Efficiency Measurement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Properties of Environmental Efficiency Indexes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Hyperbolic and Directional Distance Indexes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The “Färe-Grosskopf-Lovell” Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Extension of the FGL Index to Graph Space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Critiques and Suggested Modifications of the By-Production Structure . . . . . . . . . . . . . . .

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S. Murty () Centre for International Trade and Development, School of International Studies, Jawaharlal Nehru University, New Delhi, India e-mail: [email protected] R. R. Russell Department of Economics, University of California, Riverside, Riverside, CA, USA e-mail: [email protected] © Springer Nature Singapore Pte Ltd. 2022 S. C. Ray et al. (eds.), Handbook of Production Economics, https://doi.org/10.1007/978-981-10-3455-8_3

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Concluding Remarks: The Material Balance Condition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Cross-References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Abstract

Inadequacies of the traditional, single-equation representations of models of emission-generating technologies prominently associated with the classic book by Baumol and Oates (The theory of environmental policy, 1st and 2nd edn. Cambridge University Press, Cambridge, 1975, 1988) are first laid out. In particular, these models lack the “monotonicity degrees of freedom” to capture adequately the complex trade-offs in the production of unintended as well as intended outputs using emission-generating inputs. Reprising ideas in the classic 1965 book on Theory of Production by Ragnar Frisch, it is shown that the use of multiple functional restrictions, a phenomenon referred to as by-production in Murty, Russell, and Levkoff (J Environ Econ Manag 64:117–135, 2012), facilitates the modeling of pollution-generating technologies. In particular, a by-production technology is obtained as the intersection of an intended-output sub-technology and an unintended-output sub-technology. These principles are illustrated by sketching a model of coal-fired electrical power generation. A data envelopment analysis (DEA) methodology for measuring technical efficiency under the by-production approach is also discussed.

Introduction The modeling of production technologies has a long history. Early conceptualization of relations among inputs and outputs, based on stylized facts and empirical observations, were manifested in the law of diminishing marginal productivity (or increasing marginal cost) and various types of returns to scale. These modeling efforts culminated in a rigorous axiomatization of production technologies and their representations by production functions in the middle of the twentieth century. Prominent among the main features of a technology recognized by this literature were the free disposal properties of the inputs and outputs. Together they imply the empirically observed positive relationship between inputs and outputs. Combined with the assumption of convexity, this axiomatization of the technology laid a foundation for many pathbreaking theoretical results, including the existence of a general competitive equilibrium, the formalization of the two fundamental theorems of welfare, and the duality between technological constraints and optimizing behavior (e.g., profit maximization and cost minimization). These results facilitated a plethora of applied work with significant consequences for economic policy in areas like public economics, measurement of efficiency and productivity, economic growth, and industrial organization. Operations of many technologies lead to the production of not only desirable economic outputs but also incidental outputs that may have undesirable consequences for the rest of the economy. Rigorous study of the relations between inputs and outputs that are satisfied by such technologies was generally lacking for a long while.

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It was only in the latter part of the twentieth century, as the field of environmental and natural resource economics established its roots and gained momentum, that the researchers faced the challenge of modeling the generation of bad (unintended) outputs as by-products of the production of the desired (intended) economic outputs. The technology of an emission-generating producer was recognized as an important primitive in the theoretical analysis of market externalities and the formulation of policies aimed at efficient mitigation of the inimical effects on social welfare of the production of the bad outputs. The initial treatment of bad outputs in the modeling of production technologies was very simple. The standard approach, adopted in the classic Baumol-Oates [7] book and persisting to this day, is simply to include in the production function an emission variable, assumed to satisfy the same (free disposability) conditions as a conventional input. An early exception to the standard approach can be found in Färe, Grosskopf, and Pasurka [22] and Färe, Grosskopf, Lovell, and Pasurka [19], where emissions are modeled as (bad) outputs satisfying a weak disposability assumption: bad and good outputs can only be disposed of in tandem (proportionately). The main idea behind these “input” or “output” approaches to modeling bad outputs is to capture the empirically observed positive relation between the production of good and bad outputs: as the production of good outputs increases, the technology also generates more of the bad outputs. Under such approaches, it became possible to represent the technology set of an emissiongenerating production unit by a single production function/equation. As first pointed out by Førsund [27] and Murty and Russell [42], each of these approaches to modeling pollution-generating technologies entails implausible properties of the technology. Most egregiously, the Baumol-Oates formulation implies that, ceteris paribus, increases in the use of a pollution-generating input lowers the levels of emissions. The weak disposability approach of Färe, Grosskopf, Lovell, and Pasurka [19] entails free disposability of emission-generating inputs, implying for example that coal input can be increased without bound and without generating additional pollution. Building on ideas of Frisch [30], Førsund [27] and Murty and Russell [42] argued analytically that the perverse trade-offs engendered by the single-equation representations of pollution-generating technologies can be avoided by using multiple functional restrictions to describe the technology, a construction that Murty and Russell call by-production. These ideas have been further developed in Førsund [28], Murty [40], and Murty, Russell, and Levkoff [44]. The chapter unfolds as follows. The single-equation (Baumol-Oates) and weak disposability approaches are presented and critiqued in sections “Single-Equation Modeling of the Technology Under Standard Disposability Assumptions” and “Weakly Disposable Technologies.” The multiple constraint approach without abatement activities is developed in section “Multiple-Equation Modeling of Pollution-Generating Technologies.” To study the required form of the multifunctional restrictions, a distinction is made between rival and joint production of multiple outputs. It is argued that the production of multiple economic (desirable) outputs can be rival or joint but that there is jointness in the production of good and bad outputs.

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While the analysis in section “Multiple-Equation Modeling of Pollution-Generating Technologies” is restricted to the case where only one emission is generated, section “Multi-equation Modeling of Emission-Generating Technologies with Abatement Activities and Multiple Emissions” extends the multi-equation modeling approach to allow for the generation of multiple emissions by a producing unit and to incorporate abatement activities that it can undertake to mitigate its emissions. Ayres and Kneese [3] and Pethig [46] argue that abatement activities mitigate harmful emissions by transforming them into less harmful matter. When multiple emissions are generated by a production unit, some may be jointly produced, while others may be rival in nature; e.g., Levkoff [35], Kumbhakar and Tsionas [33], and Murty and Russell [43] distinguish between complementarity and substitutability in the generation of emissions. Section “Axiomatic Approach to Modeling Emission– Generating Technologies” adopts an axiomatic approach, proving that any model of pollution-generating technologies satisfies a set of desirable axioms if and only if it is a by-production technology. Section “Efficiency Measurement” studies the implications of the multiple production-relations approach to modeling an emissiongenerating technology for the measurement of technical inefficiency of a producing unit. The multi-equation models of emission-generating technologies that are developed in this chapter are motivated by both the first and the second law of thermodynamics.1 Together these laws explain why emission generation is an inevitable consequence of economic production. Of the two, the first law of thermodynamics, also called the material balance or the mass balance condition, is especially popular in the literature, where it has often been employed to measure the extent of emission generation. Intuitively, it states that matter cannot be destroyed and hence that the mass of all material inputs must equal the mass of all outputs produced. Section “Concluding Remarks: The Material Balance Condition” concludes with some comments on the consequences of this condition for the economic modeling of production technology. A couple of caveats about the content of the chapter are in order. First, in keeping with the theme of this volume, we consider only theoretical characterizations of a pollution-generating technology and only non-stochastic notions of efficiency measurement. Second, the chapter is not a standard survey. Rather, its primary objective is to develop a consistent framework for modeling technologies that generate by-products, drawing on the relevant literature as necessary.

1 See

[5].

Ayres and Kneese [3], Ayres [2], Baumgärtner and de Swaan Arons, [6] and Baumgärtner

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Single-Equation Modeling of the Technology Under Standard Disposability Assumptions Consider a very parsimonious model in which two inputs are employed to produce a single intended (economic) output, with a single unintended (bad) output as a by-product. Denote the quantities of the two inputs by x1 and x2 and the quantities of the intended and unintended outputs, respectively, by y and z. Finally, denote the underlying technology set by T and the production vector by x1 , x2 , y, z ∈ R4+ . Assume that the technology satisfies standard free disposability with respect to both the inputs and the intended output2 : x1 , x2 , y, z ∈ T ∧ x¯1 ≥ x1 ∧ x¯2 ≥ x2 ∧ y¯ ≤ y =⇒ x¯1 , x¯2 , y, ¯ z ∈ T .

(1)

In particular, output free disposability (implied by (1)), ¯ z ∈ T , x1 , x2 , y, z ∈ T ∧ y¯ < y =⇒ x1 , x2 , y, states that, for fixed quantities of inputs (and emissions), the economic output can be arbitrarily reduced. Thus, reduction of the economic output is costless: it need not entail use of additional inputs (or reduction of other economic outputs if y were a vector of several economic outputs). Similarly, input free disposability (implied by (1)), x1 , x2 , y, z ∈ T ∧ x¯1 ≥ x1 ∧ x¯2 ≥ x2 =⇒ x¯1 , x¯2 , y, z ∈ T . states that, holding the economic output (and emissions) fixed, input quantities can be arbitrarily increased. Thus, the use of additional amounts of inputs is costless: it need not entail reductions in the production of outputs (both intended and unintended). If we also assume that T is a closed set and that there are upper bounds on production of the economic output when inputs are held fixed, T can then be represented by a single explicit production function, F : R3+ −→ R+ , with image F (x1 , x2 , z) := max{y | x1 , x2 , y, z ∈ T }.

(2)

Given that the economic output is freely disposable, it is clear that x1 , x2 , y, z ∈ T ⇐⇒ y ≤ F (x1 , x2 , z).

(3)

2 Vector notation: x¯ ≥ x if x¯ ≥ x for all i; x¯ > x if x¯ ≥ x for all i and x¯ = x; and x¯ x if i i i i x¯i > xi for all i. The conjunction symbol ∧ stands for “and”.

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The frontier of the technology is defined to be the set of production vectors x1 , x2 , y, z ∈ T such that y = F (x1 , x2 , z). Free disposability of inputs implies that the function F is nondecreasing in each of the inputs. To see this, suppose y = F (x1 , x2 , z) and x¯1 ≥ x1 . Then x1 , x2 , y, z ∈ T and free input disposability imply that x¯1 , x2 , y, z ∈ T . Hence, (3) implies that F (x1 , x2 , z) = y ≤ F (x¯1 , x2 , z). Thus, F is nondecreasing in inputs. Note that (1) imposes no disposability restriction on the unintended output. The alternative (standard) assumptions are to treat the unintended output either as a conventional output or as a conventional input.3 As is demonstrated below, either of these assumptions ensures that the technology has a single-equation functional representation (albeit these modeling assumptions both lead to counterintuitive properties of the technology).

Treating Pollution as a Conventional Production Output Suppose first that emission is treated as a standard output, so that T also satisfies standard output free disposability with respect to this variable: x1 , x2 , y, z ∈ T ∧ z¯ ≤ z =⇒ x1 , x2 , y, z¯  ∈ T .

(4)

The implications of assuming emission is a freely disposable output are counterintuitive. This assumption implies that the technology permits arbitrary reductions in the emission, holding all other inputs and economic output quantities fixed. This implies in turn that there is no cost associated with reducing the emission– emission can be reduced without affecting the production of the economic output, an implication that is refuted by simple empirical observation in many situations. In real-life situations, decreases in emissions like greenhouse gases come at the cost of decreasing the economic output. In particular, assuming that emission is also a standard output implies that the function F is nonincreasing in emission, i.e., the trade-off along the frontier of the technology between the maximum-producible economic output and the emission is nonpositive.4 This perverse trade-off between the intended and unintended outputs implies that there is no trade-off between growth and environment: ceteris-paribus, a reduction in emission (i.e., an improvement in the environmental quality) increases the production of the economic output. The negative relation between emission and the economic output when emission is treated as a standard output of the technology can also be interpreted to imply that emission has a detrimental effect on the production of the economic output. Førsund

3A

nonstandard disposability assumption is explored in section “Weakly Disposable Technologies”. 4 Sketch of proof: Suppose y = F (x , x , z) and z¯ ≤ z. Free output disposability of emission (4) 1 2 implies that x1 , x2 , y, z¯  ∈ T . Hence, y ≤ F (x1 , x2 , z¯ ), and from (3), it follows that y = F (x1 , x2 , z) ≤ F (x1 , x2 , z¯ ). 

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[27] demonstrates that, if emission also has a detrimental effect on social welfare, maximization of social welfare subject to such a technological constraint results in a solution where no emission is generated, while a positive amount of the economic output is produced and consumed. This solution, as Førsund argues, contradicts the inevitability of emission generation when economic outputs are produced.

Treating Pollution as a Conventional Production Input The arguments presented above have been well understood in the literature, which has consistently refrained from assuming that emission is a freely disposable output. Rather, it has aimed at developing models of technology that yield a positive relation between the generation of emissions and the production of economic outputs. One strand of this literature,5 going back to Baumol and Oates [7] and Cropper and Oates [14], models emissions as freely disposable inputs. In the context of the parsimonious model presented in the previous section, this modeling strategy entails replacing (4) with x1 , x2 , y, z ∈ T ∧ z¯ ≥ z =⇒ x1 , x2 , y, z¯  ∈ T ,

(5)

while maintaining standard disposability (1) with respect to all other goods and all other hypotheses made about the technology T in the previous section. The input approach has some appeal: it relates emissions to the waste disposal capacity of the environment, which is interpreted as an input in production, just as other economics inputs. Since emission is now treated as a standard input and satisfies standard input free disposability, the resulting trade-off between the emission and the economic output obtained under this approach is nonnegative. To see this, suppose y = F (x1 , x2 , z) and z¯ > z. Free (input) disposability of the emission (5) implies that x1 , x2 , y, z¯  ∈ T . Hence, y ≤ F (x1 , x2 , z¯ ), and it follows ¯ That is, the function F is from (3) that y = F (x1 , x2 , z) ≤ F (x1 , x2 , z¯ ) =: y. nondecreasing in the emission, so that the emission and the economic output are positively related. This relationship is consistent with empirical observation: in real life, emission generation and economic output production usually go hand in hand. The proponents of the input approach6 also justify the positive trade-off between emission generation and intended-output production under this approach by invoking abatement activities. Economic resources are shared between the production of abatement and the economic output, so that the more the resources of a producing unit are diverted to abatement activities, they less are the available for production of the economic outputs; thus, the lower are the amounts produced of both economic outputs and emissions.

5 See,

for example, Njuki and Bravo-Ureta [45] and the references therein. e.g., Baumol and Oates [7], Laffont [34, Ch. 2], Cropper and Oates [14], Reinhard, Lovell, and Thijssen [48], and Ball, Lovell, Luu, and Nehring [4]. 6 See,

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Taking a very different approach, Førsund [27, 28] argues that, although the solution to a standard social welfare maximization problem subject to a technological constraint that assumes emission is a freely disposable input and where the emission is detrimental to social welfare is well-defined, the input approach to modeling emission-generating technologies is unsatisfactory, as it is not revealing of the underlying purification (abatement) activities. Abatement activities are only implicitly assumed, and this approach therefore fails to show how abatement is produced from the given inputs. While the input approach seems to generate the correct trade-off between emission generation and economic output production, Murty and Russell [42] and Murty, Russell, and Levkoff [44] (hereafter MRL) show that it also generates two unacceptable implications for production trade-offs. To discuss the first of these unacceptable implications, let us first differentiate inputs according to whether they are emission-causing (such as fossil fuels) or nonemission-causing (such as labor and capital). Emission-causing inputs are composed of substances that generate emissions. For example, coal contains sulfur and carbon content, so that when it is combusted in the process of generating energy, it liberates CO2 and SO2 into the atmosphere. In the context of our parsimonious model, assume that the second input is emission causing, while the first is not. MRL demonstrate that treatment of the emission as a standard input results in a nonpositive trade-off between emission and any emission-causing input. For example, the input approach implies that an emission like CO2 decreases with an increase in an emission-causing input like coal, a finding that is inconsistent with common sense. Below we provide an alternative (non-differential) proof of this counterintuitive implication of free disposability of emissions in a single-equation representation of the technology. The function,  : R3+ −→ R+ , with image (x1 , y, z) := min{x2 | x1 , x2 , y, z ∈ T }, identifies the minimal amount of the emission-causing input that is required to produce economic output y and emission z when the non-emission-causing input use is x1 . Since T satisfies input free disposability, it can also be represented functionally as7

7 The set T can have more than one functional representation. The function F , defined in (2), offers one, the function  offers another, and later in this section, we define a function ð that offers yet another. Along the strictly efficient frontier of T , we have y = F (x1 , x2 , z) ⇐⇒ x2 = (x1 , y, z) ⇐⇒ z = ð(x1 , x2 , y). A production vector in T is a strictly efficient point of T if there exists no other point in T with no greater amounts of emission and inputs and no smaller amount of the good output. The set of all strictly efficient points of T forms the strictly efficient frontier of T . (See section “Axiomatic Approach to Modeling Emission-Generating Technologies” for a formal definition of the strictly efficient frontier of technology T .)

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x1 , x2 , y, z ∈ T ⇐⇒ x2 ≥ (x1 , y, z). A production vector x1 , x2 , y, z is a frontier point of T if x2 = (x1 , y, z). Suppose x2 = (x1 , y, z) and z¯ ≥ z. Since the emission is treated as a standard input, T satisfies free disposability of the emission. Consequently, x1 , x2 , y, z¯  ∈ T , and the definition of the function  implies that x2 ≥ (x1 , y, z¯ ). Hence, (x1 , y, z) = x2 ≥ (x1 , y, z¯ ) =: x¯2 . Thus, the function  is nonincreasing in the emission; i.e., when the amount of the non-emission-causing input is held fixed at x1 , the minimal amount of the emission-causing input that is required to produce y amount of the economic output and z¯ amount of the emission is less than the minimal amount required to produce the same amount of the economic output but a lower amount z of emission. Hence, the input approach implies, contrary to common sense, a nonpositive relation between the emission and the emission-causing input. MRL and Murty [41] demonstrate a second paradox associated with the input approach: if we assume, as is realistic, that emission generation is positively related to the use of, emission-causing input, then the technology violates free input disposability of the emission-causing input. We demonstrate this violation below. The function, ð : R3+ −→ R+ , defined by ð(x1 , x2 , y) := min{z | x1 , x2 , y, z ∈ T }, identifies the minimal emission level under technology T when the economic output quantity is y and the input use is x1 , x2 . Thus, we have x1 , x2 , y, z ∈ T ⇐⇒ z ≥ ð(x1 , x2 , y). Since the second input is emission causing, its use should not decrease the minimal level of emission that can be generated, so that ð should be nondecreasing in x2 . Suppose, in conformance with our intuition, that ð is strictly increasing in the use of the second (emission causing) input. Let z = ð(x1 , x2 , y). Then x1 , x2 , y, z ∈ T , and z is the minimal emission generated by quantity x2 of the emission-causing input. Suppose, ceteris paribus, there is an increase in the use of this input, x¯2 > x2 . Define the minimal emission that can now be generated as z¯ := ð(x1 , x¯2 , y). Since ð is increasing in the emission-causing input, we have z¯ > z. This clearly implies that x1 , x¯2 , y, z ∈ / T , because otherwise z¯ would not have been the minimal emission generated by quantity x¯2 of the emissioncausing input. Thus, to summarize, we have x1 , x2 , y, z ∈ T and x¯2 > x2 but x1 , x¯2 , y, z ∈ / T . Clearly, this is a violation of free disposability of the emissioncausing input. Thus, the input approach is not consistent with the empirically observed positive relation between the emission and an emission-causing input. To see a final critique of the input approach, define the production possibility set corresponding to a given vector of input, say x, as the set of emission and intended output configurations that can be produced by input vector x under technology T :

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P (x) = {y, z ∈ R2+ | x, y, z ∈ T }. If the production function F represents the technology, then output disposability of y implies that P (x) = {y, z ∈ R2+ | y ≤ F (x)}. The (weakly efficient) frontier of P (x) is the set    W P (x1 , x2 ) := y, z ∈ R2+ | y = F (x1 , x2 , z) . Thus, along the frontier of the technology, when inputs are held fixed, there is a rich menu of combinations of the quantities of the emission and the economic output. As noted earlier, under free input disposability of the emission and free output disposability of the economic output, the relation between emission and economic output along the frontier when all inputs are held fixed is nonnegative. A combination of good and bad outputs in P (x) is a strictly efficient point of P (x) if there exists no other combination in P (x) with no larger amount of the bad output and no smaller amount of the good output. The set of all strictly efficient points of P (x) – its strictly efficient frontier – is denoted Pˆ (x). Panel (a) of Fig. 1 gives an example of a production possibility set P (x) satisfying free disposability assumptions (1) and (5), where (5) in particular implies that emission is treated as a freely disposable input. The strictly efficient frontier, the bold part of the weakly efficient frontier, reflects a positive relationship between the good and the bad outputs. The overall frontier reflects a nonnegative relation between the good and the bad outputs, but this is counterintuitive: if we hold the quanitity of the emission-causing input fixed, there exists a unique minimal level of emission, i.e., the minimal level is independent of the output quantity. For example, the minimal amount of smoke that can be produced by one ton of coal containing a fixed amount of carbon is unique.

Fig. 1 Production possibility sets satisfying (a) free disposability and (b) weak disposability

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Weakly Disposable Technologies Over the years, the principal alternative to the Baumol-Oates (single equation) method of modeling emission-generating technologies has been the set-theoretic approach inaugurated by Färe, Grosskopf, Lovell, and Pasurka [19] (hereafter referred to as FGLP). This approach, which is generally oriented toward data envelopment analysis (mathematical programming) methods of estimating or constructing technologies, characterizes technologies by sets of inequality conditions for the inputs and outputs (rather than by use of explicit or implicit production functions).8 The technologies constructed by this method satisfy conditions (1) on the free disposability of economic output and standard inputs but not condition (5) regarding free input disposability of the emission. Instead, the authors propose the weak disposability condition, x, y, z ∈ T ∧ λ ∈ [0, 1] =⇒ x, λy, λz ∈ T , and the null-jointness condition, x, y, z ∈ T ∧ z = 0 =⇒ y = 0. Panel (b) of Fig. 1 illustrates a production possibility set P (x) for a weakly disposable technology given a fixed vector of inputs x. The bold region of its boundary, denoted by Pˆ (x), is its strictly efficient frontier, showing a positive relation between the good and the bad outputs. By not treating emission as a conventional output, the FGLP approach eliminates the “global” possibility of the perverse negative trade-off between emission and the economic output demonstrated in section “Treating Pollution as a Conventional Production Output.” Under the weak disposability condition, pollution cannot be freely disposed of as a standard output but can instead be reduced only in tandem (proportionally) with intended output. As is well documented (see, e.g., Førsund [27, 28]), however, the FGLP approach does not altogether eliminate the negative trade-off between emission and the economic output: local regions of the production space can exist where this trade-off is negative. See panel (b) of Fig. 1, in which the boundary of P (x) has a negatively sloped region. Moreover, the MRL and Murty [41] critique of emission-generating technologies satisfying free input disposability of emission-causing inputs, which was discussed in the previous section, continues to apply even in the weak disposability approach,

8 See, e.g., Färe, Grosskopf, Lovell, and Yaisawarng [20], Coggins and Swinton [13], Murty and Kumar [38], Murty and Kumar [39], Färe, Grosskopf, Noh, and Weber [21], and Boyd and McClelland [8]. See Zhou, Ang, and Poh [56] for a comprehensive survey of a number of papers employing this approach.

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as this approach also maintains free disposability of all inputs. The strictly efficient frontier Pˆ (x) of the production possibility set P (x) for the weakly disposable technology in panel (b) of Fig. 1 contains multiple points. As in the rationalization of the input approach to modeling emissions, the proponents of the FGLP approach justify the positive relation between economic output production and emission generation in terms of abatement activities that can be undertaken by the production unit. However, since such activities are not explicitly modeled, what is modeled can only be interpreted as a reduced form of the technology in the space of all intended and unintended outputs and all inputs.9 MRL demonstrate that even this reduced form of the technology violates free disposability of the emission-causing input. Moreover, MRL argue that, when abatement activities are produced by a producing unit along with the economic outputs, an emission-generating technology can violate the null-jointness assumption in the weak disposability approach. Although the use of emission-causing inputs results in the generation of emissions alongside the generation of the economic output, it is possible that abatement activities so produced can totally eliminate the emissions. Thus, generation of zero net emissions alongside positive levels of economic outputs is a theoretical possibility.

Multiple-Equation Modeling of Pollution-Generating Technologies The output and input approaches to modeling emission-generating technologies, critiqued in sections “Treating Pollution as a Conventional Production Output” and “Treating Pollution as a Conventional Production Input,” impose disposability conditions on the technology that make possible its representation by a single functional relation F (see, e.g., Eq. (3)) or, equivalently, by the function  or ð. These sections demonstrated that some of these disposal properties do not conform to our intuitive understanding and empirical observations of the features of emission-generating technologies and, more particularly, that a single functional relation is not sufficient to capture all the complex trade-offs among inputs and outputs involved in the production of economic outputs and the generation of emissions. Redress of the problems with the single-equation modeling has focused on using multiple functional restrictions to implement richer and more plausible disposability conditions on the representation of the technology. The conceptual framework for multiple function specifications of technologies was laid out long ago in a book by Ragnar Frisch [30]. Although inadequately appreciated by the profession for years, Frisch’s ideas have been reprised for the special case of modeling pollutiongenerating technologies in a series of papers by Finn Førsund ( [26–28]; 2017). Based on the ideas of Frisch, Førsund proposes the use of multiple functional relations to represent emission-generating technologies. But identification of the

9 See

Sect. 3.2 of MRL for further explanation of this restrictive interpretation

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precise functional relations that correctly capture the trade-offs among goods in production processes that generate emissions leads to dual questions about the realistic disposal properties satisfied by such technologies. Such questions led to the development of an axiomatic framework for modeling such technologies in a series of papers by Murty and Russell [42, 43] [hereafter MR], Murty, Russell, and Levkoff [44], and Murty [40, 41]. To provide a rationale for the introduction of multiple functional relations in the modeling of such technologies, we first distinguish below between rival production and joint production. We argue that the production of emissions and economic outputs is not rival in production; rather, this is a special case of joint production that is discussed in Frisch. While Førsund proposes a model where all goods (including abatement activities) are jointly produced, in the model proposed by MR and MRL, the independent production of economic outputs is rival, but the production of economic outputs and emission is collectively joint, a phenomenon they call byproduction. MR and MRL consider the case of a single emission in their theoretical model.10 Later in this chapter, we show that, in the case of multiple emissions, independent production of emissions can also be joint or rival.11 Moreover, intuition suggests that the production of economic outputs and explicit abatement activities (such as mitigation of emissions by treatment plants) by a single producing unit is rival in nature. The proposed framework can be extended to the case where some economic outputs are also jointly produced.

Rival vs. Joint Production of Multiple Outputs Let T ⊂ Rn+m be a general technology set producing m outputs using n inputs.12 + Thus, x, y ∈ Rn+m is a production vector, where x ∈ Rn+ denotes an input vector + m and y ∈ R+ denotes an output vector. Outputs are indexed by j , while inputs are indexed by i.

Rival Production of Outputs The following definition of rival production is equivalent to the definition of input non-jointness in an earlier literature (e.g., Kohli [32] and Chambers [9], p. 287). Definition. T exhibits rivalry in the production of (all) outputs if there exist production functions, f j : Rn+ −→ R+ , one for each output, such that

10 However,

in their data envelopment analysis (DEA) model and its empirical application to the measurement of efficiency of a production unit, they adopt a multi-emission framework. 11 See section “Modeling the Generation of Multiple Emissions” of this chapter. 12 In this section, we do not distinguish between economic outputs and emissions. Both are considered as outputs of the technology.

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x, y ∈ T ⊂ Rn+m ⇐⇒ ∃ x Yj ∈ Rn+ for all j = 1, . . . , m, satisfying + m 

x Yj = x and yj ≤ f j (x Yj ) ∀j.

j =1

Thus, rivalry in production means that a given vector of input quantities x employed by the production unit is allocated to (divided among) the production of its m outputs as x Y1 , . . . , x Ym . So, if more of any input is diverted to the production of a particular output, less of that input is available for the production of the remaining outputs.13 The multiple production functions, f j for j = 1, . . . , m (each representing the production of a single output), can be combined into a single production function representing the overall technology. For example, when m = 2, we can define F(x, y1 ) := max {f 2 (x Y2 ) | y1 ≤ f 1 (x Y1 ), x Y1 + x Y2 ≤ x}. x Y1 ,x Y2

(6)

Given an input vector x and a level of production y1 of the first output, this problem finds the optimal split of the input vector between the production of the two outputs. The optimal split is one that maximizes the production of the second output without reducing the amount of the first output below y1 . Clearly, T can equivalently be represented by the function F as follows: x, y1 , y2  ∈ T ⇐⇒ y2 ≤ F(x, y1 ). If the production functions, f j , j = 1, . . . , m, are nondecreasing, the inputs and outputs are freely disposable under this representation of the technology T . If f j is also differentiable for all j = 1, . . . , m, then it can be shown, employing the envelope theorem on problem ((6)), that holding the input vector x fixed, an increase in the production of the first economic output comes at the cost of a decrease in the production of the second economic output. This is because, given the input vector, an increase in the first economic output involves diversion of inputs to its production, which implies that lesser amounts of inputs are available for producing the second economic output. Thus, if F is differentiable, we have (in a slight abuse of notation)

13 See

also Kohli [32]. A related literature on network DEA (e.g., Färe and Grosskopf [17], Färe, Grosskopf and Pasurka [23], and Hampf [31]) features various subprocesses of production among which inputs are shared (divided). One strand of this literature (see, e.g., Lozano [36] and references therein) distinguishes between joint and non-joint inputs. While non-joint inputs are associated with rival production, joint inputs are not shared (or divided) among production processes and lead to the joint production of outputs, a concept that is defined in the next subsection.

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dy2 ∂F(x, y1 ) = ≤ 0. dy1 ∂y1 In the general case of m outputs, the technology can be represented by a single output distance function, given the individual production functions f j for j = 1, . . . , m, as follows:  yj DO (x, y):= λ>0 | ≤f j (x Yj ) min Y Y λ λ,x 1 ,...,x m

∀ j =1, . . . , m

∧

m 

 x ≤x , Yj

j =1

so that x, y ∈ T ⇐⇒ DO (x, y) ≤ 1. It can be shown that DO is nondecreasing in the outputs. Given an x ∈ Rn+ , the set of weakly efficient output vectors is14    W P (x) = y ∈ Rm + | DO (x, y) = 1 . An output vector is strictly efficient given an input vector x if it can be produced with input vector x, and there exists no other output vector that can be produced with x containing a larger amount of at least one output and no smaller amount of any other output. Let Pˆ (x) denote the set of all strictly efficient output vectors given input vector x. If f j is a differentiable (hence continuous) and increasing function for all j = 1, . . . , m, then DO is increasing and differentiable in the outputs (∂DO (x, y)/∂yj >  0 for all j ) and W P (x) = Pˆ (x), that is, the sets of weakly and strictly efficient output vectors contingent on an input vector coincide. Further, it follows from the n+m−1 −→ implicit function theorem that there exists a continuous function, Fˆ : R+ R+ , such that the implicit production function DO can be solved to express the level of the j th output as an explicit function of the levels of all the remaining goods; i.e., DO (x, y) = 1

⇐⇒

 yj = Fˆ x, y−j ,

where y−j is the vector of all outputs other than the j th. Thus, the efficient set of outputs Pˆ (x) has a continuum of points, i.e., given an input vector x, there exists a rich menu of efficient output combinations that can be produced. Further, the implicit function theorem also implies that the trade-off between the outputs along the strictly efficient frontier Pˆ (x) is negative and is given by

14 Given

input vector x, y is a weakly efficient vector of outputs if there exists no other output vector that can be produced by input vector x with larger amounts of all outputs.

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Fig. 2 Production possibility frontier: rival outputs

∂DO (x,y)

∂yj  ∂yj = − ∂D (x,y) < 0, O ∂yj 

∀ j = j  .

∂yj

The famous guns and butter example in the classic textbook by Paul Samuelson and William Nordhaus [52] is an example of rival production: the more guns produced, the lesser are the resources available for producing the other good, butter. Figure 2 illustrates the set of strictly efficient output vectors when the input vector is held fixed at x¯ for the case when m = 2. This is given by the set   Pˆ (x) ¯ = y ∈ R2+ | F(x, ¯ y1 ) = y2 , where ∂F(x, ¯ y1 )/∂y1 < 0. The diagram shows that there are many ways of allocating input vector x¯ efficiently between the production of the two outputs. The negative slope of the strictly efficient frontier implies that, as greater amounts of inputs are allocated to production of good 1, less and less are available for production of good 2. Thus, when the production of outputs is rival, it is possible to represent the technology by a single production function. Holding input levels fixed, there is a continuum of efficient output combinations, and the trade-off between any two outputs along the efficient frontier of the technology is nonpositive.

Joint Production of Outputs The following definition of joint production of outputs is equivalent to the concept of input price non-jointness defined in Kohli [32]. See also Chambers [9], p. 289. Definition. T jointly produces outputs 1, . . . , m if there exist production functions f j : Rn+ −→ R+ , one for every output, such that x, y ∈ T ⇐⇒ yj ≤ f j (x), j = 1, . . . , m. Intuitively, if a production unit employs a given vector of inputs, then the same vector of inputs is available for the production of each of its economic outputs.

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Fig. 3 Production possibility frontier: joint outputs

Thus, in contrast to rival production of outputs, the amounts of inputs are not shared/ divided among the various lines of production of the unit; rather, they are equally available to all lines of production. Frisch [30] and Førsund [27,28] provide real-life examples of joint production. A sheep as an input jointly produces milk, wool, and mutton. A chicken jointly yields both eggs and poultry meat. If, for j = 1, . . . , m, the production function f j is increasing in the inputs, the set of strictly efficient output vectors for a given input vector is a singleton15 :   Pˆ (x) = f 1 (x), . . . , f m (x) . Thus, in contrast to the case of rival production, there is no trade-off in the production of the outputs along an efficient frontier with fixed input quantities. Rather, there is a positive correlation in the production of various outputs: if f j is increasing in inputs for all j , then as input amounts increase, the unique efficiently produced vector of outputs becomes larger, i.e., x¯ > x

∧

 y = f 1 (x) . . . , f m (x)

∧

 y¯ = f 1 (x), ¯ . . . , f m (x) ¯ =⇒ y¯ > y.

Figure 3 illustrates a case of joint production of two outputs. Shown in the diagram are production possibility sets for three input vectors satisfying x¯ ≥ x  ≥ x. The unique strictly efficient points of P (x), P (x  ), and P (x) ¯ are y1 , y2  = ¯ f 2 (x), ¯ (f 1 (x), f 2 (x), y1 , y2  = (f 1 (x  ), f 2 (x  ), and y¯1 , y¯2  = (f 1 (x), respectively.

15 For

example, there is a unique efficient combination of milk and wool that a single sheep can produce, for it can produce only a certain maximal amount of milk and a certain maximal amount of wool. In general, it seems realistic to assume that there is no trade-off in the production of milk and wool by a sheep.

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Multi-equation Modeling: The Case of Factorially Determined Multi-output Production Extending the framework in sections “Treating Pollution as a Conventional Production Output,” “Treating Pollution as a Conventional Production Input,” and “Weakly Disposable Technologies,” we henceforth assume that there are n inputs of which nz are emission causing, while the remaining n − nz =: no are non-emission n causing. The input vector x ∈ Rn+ is partitioned as xz , xo , where xz ∈ R+z no is the vector of use of emission-generating inputs, while xo ∈ R+ is the vector of use of non-emission-generating inputs. Inputs continue to be indexed by i or, alternatively, when the partition into emission-causing and non-emission-causing inputs is relevant, by zi or oi .16 We assume that there are m economic outputs (indexed by j ) and m emissions (indexed by k); the respective quantity vectors m  are denoted by y ∈ Rm + and z ∈ R+ . Let t := n + m + m Viewing the production of emissions and economic outputs as a clear case of joint production, Førsund [27, 28] argues that the particular multi-equation model of Frisch [30] that is best suited for modeling emission-generating technologies is the case Frisch called “factorially determined multi-output production,” where there is joint production of all economic outputs and emissions. He specifically suggests the following multi-equation system: yj = Fj (x1 , . . . , xn ), j = 1, . . . , m,

and

zk = Gk (x1 , . . . , xn ), k = 1, . . . , m ,

(7)

where, for all j and all k, Fj and Gk are differentiable functions with derivatives j satisfying Fxi (x1 , . . . , xn ) ≥ 0 for all i = 1, . . . , n, Gkxz (x1 , . . . , xn ) ≥ 0 for all i

i = 1, . . . , nz , and Gkxo (x1 , . . . , xn ) ≤ 0 for all i = 1, . . . , no . That is, the signs of i

the derivatives of Fj with respect to inputs imply that the marginal products of all inputs in the production of the economic outputs are nonnegative. The signs of the derivatives of Gk imply that emission-causing inputs (weakly) increase emissions, while non-emission-causing inputs (called service inputs by Førsund) (weakly) decrease emissions. As discussed above (in section “Treating Pollution as a Conventional Production Input”), Førsund [27, 28] argues that the single-equation input approach of Baumol and Oates to modeling an emission-generating technology does not reveal the underlying purification/abatement activities that explain the positive relation between emissions and economic activities. He goes on to argue that purification activities can be inbedded in the technology when it is modeled by equation system (7). In

xzi refers to the amount of ith emission-causing input for i = 1, . . . , nz , and xoi refers to the amount of ith non-emission-causing input for i = 1, . . . , no .

16 Thus,

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particular, he assumes that some or all service inputs such as labor and capital can be employed to mitigate emissions, an assumption reflected in the nonpositive signs of the derivatives of the functions Gk , k = 1, . . . , m , with respect to service inputs. But the problem with adopting a full-fledged joint production approach to multiequation modeling of a technology producing multiple economic outputs and also engaging in abatement activities is that it fails to recognize that not only are many economic outputs (such as guns and butter in the classic example of Paul Samuelson) rival in production but that the production of abatement activities and economic outputs are also rival. If the economic unit employs a vector x of inputs, it may have to share these resources in the production of many of its economic outputs, so that if inputs are diverted to the production of some economic output, a lesser amount of the input vector is available for the production of its other economic outputs. But the formulation (7) assumes that the input vector x is jointly and equally available across all lines of production. Similarly, when service inputs are employed by the economic unit for mitigating its emissions, lesser amounts of these inputs are available for the production of its economic outputs. This explains why a cost minimizing/profit maximizing producing unit diverts no resources to abatement activities when it is unregulated. The purpose of regulation is to force a production unit to internalize abatement activities in its operational calculus. Profit maximization, which implies minimization of abatement expenditure, requires it not only to choose the aggregate levels of inputs to purchase and use but also to simultaneously choose the optimal split of the purchased input quantities between economic output production and abatement activities. Contrast this description with Førsund’s (2017, p.18), approach, in which inputs going into abatement do not come from a common pool of resources of the producing unit. Rather, it is recommended that abatement and economic production be treated as separate “profit centres.” Given the arguments above, however, this may not be realistic when a producing unit engages in both economic output production and abatement activities. For example, scrubbing activities form an integral part of several regulated thermal plants, where SO2 emissions produced are instantly subjected to treatment. If thermal power plants were unregulated, they would fail to undertake scrubbing, as it eats into their profits. Under regulation, profit maximization internalizes scrubbing costs as scrubbing activities are vertically integrated into (i.e., become a part of) the production structure.

Multi-equation Modeling: The Case of Rival and Joint Production In contrast to the pure case of joint production (or equivalently, the factorially determined multi-output production) discussed above, MR and MRL propose a multi-equation model that allows rival production of economic outputs on the one hand and joint production of economic outputs and emissions on the other. They

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call this approach to modeling emission-generating technologies the by-production approach.17 Murty [41] and Murty and Russell [43] argue that there is no unique model that can encompass all emission-generating technologies. Models must vary depending upon case-specific characteristics of emission generation and economic output production. They argue further, however, that the by-production approach encompasses production relations that can characterize all cases. These will generally be of two types: (i) those that describe the production of the economic and abatement outputs and (ii) those based mainly on considerations such as the mass balance conditions that (a) relate generation of emissions to emission-causing inputs used in the production of the economic and abatement outputs and (b) describe mitigation of emissions by abatement activities. Each of these production relations describes a sub-technology with its own disposability features. The overall emission-generating technology is obtained as an intersection of these sub-technologies; i.e., it contains production vectors that satisfy all the production relations in (i) and (ii). Disposability properties of the overall technology are engendered by the disposability properties of its sub-technologies. In the simple by-production technology studied in MR and MRL, only one type of emission is produced (i.e., m = 1), and it is generated because the production unit uses a particular input that is known to be a natural cause of this emission. Denote the quantity of this input by xz . There are only two inputs, and the other input is non-emission causing. Denote the quantity of this input by xo . In this section, “Multi-equation Modeling: The Case of Factorially Determined Multi-output Production”, for simplicity of exposition, we retain these assumptions and assume, in addition, that more than one type of economic output is produced (i.e., m > 1) and that there is rivalry in the production of economic outputs.18 This model also assumes that the production unit does not engage in explicit abatement activities.19 Yet, MR and MRL show that this simplified model yields a positive relation between the emission and the economic outputs. This relation is based purely on the fact that the use of the emission-causing input affects both economic output production and emission generation, resulting in a positive correlation between these two types of outputs.

The Technology Producing Economic Outputs The first sub-technology is a standard technology restricting the allowable combinations of economic outputs and conventional inputs. It represents the production 17 This

model is also extended by MR and MRL to include abatement activities, production of which is rival to the production of economic outputs. We study this model in section “Multi-equation Modeling of Emission-Generating Technologies with Abatement Activities and Multiple Emissions”. 18 This model can be generalized to encompass the case where some economic outputs are jointly produced. 19 Extensions of the model to include such activities are studied in section “Multi-equation Modeling of Emission-Generating Technologies with Abatement Activities and Multiple Emissions”.

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relation of human engineering design. The formulation below assumes that the emission by the unit does not affect the production of its economic outputs20 :   T1 = xz , xo , y, z ∈ Rt+ f (xz , xo , y) ≤ 0 ,

(8)

where the implicit production function f is differentiable and satisfies fxi (xz , xo , y) ≤ 0 for i = z, o, and fyj (xz , xo , y) ≥ 0 for all j = 1, . . . , m. These monotonicity conditions, together with the sign of the inequality constraint in ((8)), imply the following standard neoclassical disposability conditions for inputs and economic outputs: xz , xo , y, z ∈ T1 ∧ x¯z ≥ xz =⇒ x¯z , xo , y, z ∈ T1 xz , xo , y, z ∈ T1 ∧ x¯o ≥ xo =⇒ xz , x¯o , y, z ∈ T1

(9)

xz , xo , y, z ∈ T1 ∧ y¯ ≤ y =⇒ xo , xz , y, ¯ z ∈ T1 . The signs of the derivatives of f imply that, along the frontier of sub-technology T1 (i.e., the set of production vectors satisfying f (xz , xo , y, z) = 0), standard tradeoffs between goods hold. In particular, if fyj (xz , xo , y) > 0 for some j = 1, . . . , m, the implicit function theorem implies that, holding inputs fixed, there is nonpositive trade-off among economic outputs: fyj  (xz , xo , y) ∂yj =− ≤0 ∂yj  fyj (xz ,xo ,y)

∀ j  = 1, . . . , m.

The implicit production function f is similar to the output distance function DO derived in the discussion on rival production in section “Rival vs. Joint Production of Multiple Outputs.” It is clear that the above trade-offs among economic outputs imply that, under the maintained assumptions, sub-technology T1 exhibits rival production of economic outputs. Holding inputs fixed, the greater the production of some economic outputs, the lesser will be the production of the remaining economic outputs along the frontier of the technology set.

The Emission-Generating Mechanism The second sub-technology, T2 ⊂ Rt+ , links the emission generation to its various causes in nature. Emissions are generated because many processes producing marketable outputs necessarily require the use of emission-causing inputs,21 and many components of these inputs are not fully transferred to the good outputs during the process of production. Rather, some amounts of these components are

20 This

feature is generalized in Murty [41], where emissions of a unit can affect its economic output production detrimentally or beneficially. 21 This follows from the second (entropy) law of thermodynamics. See, for instance, Baumgärtner and Arons [6] and Baumgärtner [5].

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transformed into other outputs (wastes), many of which are harmful to society.22 The exact amounts of emissions produced depend also on the physical conditions and parameters under which the production takes place, some of which may be unobservable to the researcher. Thus, the set T2 embodies nature’s emissiongenerating mechanism. In general, one expects that the material balance condition would imply a positive relation between the use of emission-causing input and the generation of emission along the frontier of the sub-technology T2 . To obtain additional insights into the structure of this sub-technology, we begin by describing the disposability properties of this set. We show that, given these disposability properties, the function that best represents this sub-technology implies a positive relation between the emissioncausing input and the emission along the frontier. The following disposal properties are assumed by MR and MRL for subtechnology T2 : xz , xo , y, z ∈ T2 ∧ x¯z ≤ xz =⇒ x¯z , xo , y, z ∈ T2 xz , xo , y, z ∈ T2 ∧ z¯ ≥ z =⇒ xz , xo , y, z¯  ∈ T2

(10)

xz , xo , y, z ∈ T2 ∧ y¯ = y ∧ x¯o = xo =⇒ xz , x¯o , y, ¯ z ∈ T2 . The last assumption in ((10)) restricts the generation of emission to the use of emission-causing inputs, as it implies that, ceteris paribus, arbitrary changes in the levels of economic outputs and non-emission-causing inputs have no effect on the generation of emission. Thus, the by-production technology described here is not applicable to cases where the emissions are generated by outputs rather than inputs.23 As discussed in section “Treating Pollution as a Conventional Production Input,” the emission-causing input is not freely disposable: the quantity generated of the emission might not remain unchanged if the use of this input increases. This feature of the technology is reflected in the first condition in ((10)). This restriction, the polar opposite of standard free disposability of inputs, is called costly disposability of the emission-causing input. In contrast to free input disposability, it says that if quantity xz of the emission-causing input produces amount z of emission, a lower use of this input can also continue producing this amount of emission. This reflects inefficiencies in the functioning of the emission-generating mechanism.24 This will be true, for example, if production takes place under physical conditions (or other unobservable parameters) that are not conducive to minimizing emission generation.

22 This follows from the first law of thermodynamics – equivalently, the material-balance condition. 23 See

Murty [41] for the case where emissions can also be also generated by the economic output, once it has been produced. 24 In contrast, when this mechanism works efficiently, lowering use of the emission-generating input will lower the emission level. For example, if coal is burnt in an efficient manner, a lower use of coal implies a lower emission of CO2 .

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In section “Single-Equation Modeling of the Technology Under Standard Disposability Assumptions,” we noted that emission is an output that does not satisfy standard free output disposability: ceteris paribus, reductions in the emission comes at the cost of reductions in the production of economic outputs. The second condition in ((10)) is the polar opposite of standard free disposability of outputs and is therefore called costly disposability of emission. More intuition on this assumption will be provided in section “Axiomatic Approach to Modeling Emission-Generating Technologies” of this chapter.25 But we note here that this assumption permits inefficiencies in emission generation: if a given amount of this input generates a certain amount of emission, then owing to inefficiencies caused by unfavorable physical and other unobservable conditions, this input quantity could also generate more emission. Now define the function gˆ : Rt−1 + −→ R+ with image g(x ˆ z , xo , y) := min{z ≥ 0 | xz , xo , y, z ∈ T2 }.

(11)

Since ((10)) implies that emission generation is not caused by and hence is unaffected by changes in the economic outputs and the non-emission-generating input, the image of the minimum emission function gˆ can be redefined as g(x ˆ z , xo , y) =: g(xz ). The second costly disposability assumption in ((10)) implies that ˆ z , xo , y) ≡ g(xz ). xz , xo , y, z ∈ T2 ⇐⇒ z ≥ g(x Hence, T2 can be functionally represented as   T2 = xz , xo , y, z ∈ Rt+ z ≥ g(xz ) ,

(12)

We now show that under the first costly disposal condition in ((10)), g is nondecreasing in the use of the emission-causing input. Sketch of proof. Suppose z = g(x ˆ z , xo , y) = g(xz ) and x¯z ≤ xz . Hence, xz , xo , y, z ∈ T2 and costly disposability of the emission-causing input in ((10)) imply that x¯z , xo , y, z ∈ T2 . Thus, ((12)) implies that z ≥ g(x¯z ). But this implies g(xz ) = z ≥ g(x¯z ).  Thus, the costly disposability assumptions in ((10)) imply that the efficient frontier of the emission-generating set can be represented functionally by employing the function g and that, along this frontier, emission is positively related to its natural cause.

25 See

also Murty [41] and MR.

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An alternative formulation of the set T2 can be found in Ray, Mukherjee, and Venkatesh [47]. To capture the positive relation between emission and emissioncausing inputs along the frontier of the technology T2 , they assume that this set satisfies weak disposability of emissions and emission-causing inputs; i.e., emissions can be reduced in tandem with emission-causing inputs. With no further disposability assumptions on emissions, however, this formulation could lead to cases where the frontier of T2 has local regions with negative slopes.26 This problem, however, can be solved if costly disposability of emissions is assumed in addition to this weak disposability assumption.

The Overall Emission-Generating Technology The overall by-production technology is the intersection of the two subtechnologies:   TB := T1 ∩ T2 ≡ xz , xo , y, z ∈ Rt+ f (xz , xo , y) ≤ 0 ∧ z ≥ g(xz ) . (13) The efficient frontier of this set comprises all production vectors xz , xo , y, z ∈ Rt+ that simultaneously satisfy equations f (xz , xo , y) = 0

∧

z = g(xz ).

(14)

Since all inputs are (potentially27 ) shared in the production of the economic outputs, technology TB exhibits rivalry in the production of these goods, implying that, when all inputs (including emission-causing inputs) are held fixed, there is a menu of efficient combinations of economic outputs. At the same time, there also exists a unique minimal level of emission. This is because the emission-causing input independently influences economic output production and emission generation. It is shared in the production of economic outputs but results in a unique minimal level of emission. Thus, TB also exhibits jointness in economic output production and emission generation. The panels of Fig. 4 illustrate the structure of a by-production technology. Panels (a), (b), and (c) are drawn under the assumption that there is one good and one bad output and only one emission-causing input. Since sub-technology T1 is independent of emission generation and sub-technology T2 is independent of the production of the good output, the former is depicted in the restricted space of the input and the good output in panel (a), while the latter is depicted in the restricted space of the input and the bad output in panel (b). As drawn, it is clear that T1 satisfies input and output free disposability as defined in ((9)), while T2 satisfies costly disposability of the emission-causing input and the emission as defined in ((10)). Hence, the production point x, ¯ y  , z  is in technology T = T1 ∩ T2 . However, the

26 This

problem is similar to that encountered in the output approach to emission modeling, which assumes weak disposability of emissions and good outputs. 27 Obviously, this specification could be specialized to restrict the use of some inputs.

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Fig. 4 By-production technology (n = m = m = 1)

maximum producible amount of the good output and the minimum emission that can be generated when the input level are held fixed at x¯ are y¯ and z¯ , respectively. In panel (c), ¯z, y ¯ is the sole strictly efficient point of the production possibility set P (x). ¯ Compare this with Fig. 1, where emission is treated as a freely disposable input and the set P (x) ¯ has many strictly efficient points. Panels (a) and (b) show that, as the level of the input increases from x¯ to x ∗ , the maximum level of the good output and the minimum level of the bad output that are producible increase to y ∗ and z∗ , respectively. The shift in the production possibility set resulting from the increase in the level of the input is seen in panel (c) of Fig. 4. The sole strictly efficient point of the set P (x ∗ ) is z∗ , y ∗ . Panel (a) of Fig. 5 illustrates the strictly efficient frontier of the production possibility set P (x) ¯ for the case of two good outputs and a single bad output. The strictly efficient frontier, denoted Pˆ (x), ¯ also shows the rivalness in the production of the two good outputs as well as the by-production of the good and bad outputs. Given input vector x, ¯ the figure shows that there are a number of possible combinations of the two good outputs–the greater is the good output y1 produced given a fixed vector of inputs x, ¯ the lesser is the amount of the good output y2 produced. The maximum amount of the first (respectively, second) good output that

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Fig. 5 strictly efficient frontier with two good outputs and one bad output and (b) input possibility set with two inputs and one bad output

can be produced from x¯ is y¯1 (respectively, y¯2 ). However, given the input vector x, ¯ there is only one feasible level of the bad output z¯ that is generated. Panel (b) of Fig. 5 illustrates the input possibility sets for emission generation for the case where there are one bad output and two inputs, both of which are emissioncausing; i.e., m = 1 and n = nz = 2. The diagram illustrates that both inputs satisfy costly disposability in the production of the emission and that the input possibility sets expand as the level of the emission increases. The disposability properties of TB are derived from those of the sub-technologies. Since T1 satisfies standard free disposability with respect to the economic outputs and non-emission-causing inputs and the constraint defining T2 is independent of quantities of these goods (see the third condition in ((10))), TB also satisfies standard free disposability with respect to these goods. But because T1 satisfies free input disposability with respect to the emission-causing input, while T2 violates this condition, instead satisfying costly disposability, TB does not satisfy free disposability with respect to this input. Recall that this is predicted in the latter part of section “Treating Pollution as a Conventional Production Input.” The trade-offs among goods along the efficient frontier of TB can be obtained by applying the implicit function theorem to ((14)). As the number of outputs including the emission is m + 1, the degree of assortment (using Frisch’s terminology) in equation system ((14)) is m − 1 (the number of outputs minus the number of equations). Thus, if fyj (xz , xo , y) > 0 for j = 1, . . . , m, there exists an explicit m+1 −→ R+ such that F(xz , xo , y−j ) = yj ⇐⇒ f (xz , xo , y) = 0, function F : R+ and equation system ((14)) can be written as:

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yj = F(xz , xo , y−j )

∧

z = g(xz ).

(15)

Moreover, if the derivative of g is positive, we can invert to solve for xz as a function of the emission level z: z = g(xz ) ⇐⇒ xz = h(z). Substitution into the first equation in ((15)) then yields  yj = F h(z), xo , y−j . Thus, the trade-off between the j th economic output and the emission along the frontier of TB is positive (under our maintained sign convention for derivatives of functions f and g):    fxz h(z), xo , y−j  ∂yj  h (z) > 0, = Fxz h(z), xo , y−j h (z) = − ∂z fyj h(z), xo , y−j as suggested by our intuition. The substitution of the non-emission-causing input for the emission-causing input affects both economic output production and emission generation. Suppose the differential changes in the emission-causing and non-emission-causing inputs are dxz < 0 and dxo > 0 and the effect of these differential changes on economic production is zero:   dyj = Fxz xz , xo , y−j dxz + Fxo xz , xo , y−j dxo = 0. This implies  Fxo xz , xo , y−j  dxo < 0, dxz = − Fxz xz , xo , y−j indicating a substitution of the non-emission-causing input for the emission-causing input in the production of the j th economic output. The effect that this substitution has on emission generation is negative:  Fxo xz , xo , y−j  dxo < 0. dz = g (xz )dxz = −g (xz ) Fxz xz , xo , y−j 



Thus, the above specification of an emission-causing technology using the byproduction approach, where the cause of emission in nature is attributed solely to the good used as an input in intended production, yields the correct effect on the

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emission when another input that does not cause the emission is substituted for the emission-causing input.28 In order to capture this input substitutability in the factorially determined multi-output production system (7), Førsund [29] [pp. 10–13] includes service inputs as arguments in the emission-generating functions, Gk , k = 1, . . . , m . The above calculations, however, show that the by-production model captures this substitutability in the overall technology without the need to include these services as arguments of the emission-generation function g in ((15)).

Multi-equation Modeling of Emission-Generating Technologies with Abatement Activities and Multiple Emissions As argued in section “Multiple-Equation Modeling of Pollution-Generating Technologies,” the production of economic outputs and abatement of emissions in treatment plants is rival in nature. Resources diverted toward either of these ends reduce resources available to meet the other. Moreover, the law of conservation of mass implies that abatement activities merely transform targeted (usually harmful) emissions into other forms of “less harmful” or even “useful” matter. These abatement activities might also use inputs that generate additional harmful emissions. Pethig [46] makes these points and develops a model that includes these aspects. It is important to note, however, that many of these less harmful emissions generated during the abatement process are outside the purview of economic policy analysis and hence often not modeled by the researcher. Further, generation of multiple emissions can itself be joint or rival. Two emissions are jointly produced when there is no trade-off in their production for a given vector of emission-causing inputs. On the other hand, the production of two emissions is rival if an increase in the generation of one type of emission implies a decrease in the generation of the other type for a given vector of the emissioncausing inputs. To illustrate these points, consider the operations of a thermal power plant that uses coal along with other inputs such as labor and capital to generate electricity as its economic output. The coal employed generates CO, CO2 , and SO2 as the

28 Bäumgartner

[5] refers to thermodynamic inefficiencies in the use of fossil fuels. These inefficiencies arise when the heat generated by the combustion of fossil fuels is not fully (100%) converted into the desired form of energy (such as electricity) that is required to produce the economic output. Some of this heat can be lost. Increased use of the service inputs or improvements in the quality of these inputs, such as large-scaled plants or better capital equipment, can reduce thermodynamic inefficiencies, so that a given amount of fossil fuel can generate a greater amount of the desired form of energy. A reduction in thermodynamic inefficiencies attributable to better quality or more use of service inputs hence implies that the same amounts of the economic outputs can be produced with lower amounts of fossil fuels. At the same time, lower use of fossil fuels, together with the production relations characterizing the sub-technology T2 , implies lower amounts of emission generation.

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three emissions owing to its carbon and sulfur content. Of these three emissions, CO and CO2 are rival, since the total carbon content of a given amount of coal is limited, and, depending upon the availability of oxygen, the greater the production of CO2 , the lesser is the production of CO.29 On the other hand, assuming that coal contains carbon and sulfur in fixed proportions, SO2 is jointly produced with the two carbon-based emissions. Suppose, in addition, that the plant has a scrubbing unit that employs lime or limestone as sorbents to mitigate its sulfur emission. The use of lime in scrubbing converts a part of the sulfur emission into gypsum, which is either treated as a marketable by-product by the producing unit or is treated as a relatively less harmful emission by the researcher. The extent of conversion of SO2 into gypsum depends on the amount of lime employed and the efficiency of the scrubbing unit. In the spirit of this real-world example, we devote this section of the chapter to the development of another parsimonious model, one entailing two emissioncausing inputs, four types of emissions, and one economic output; i.e., nz = 2, m = 1, and m = 4. An abatement activity helps in mitigating one type (say the third type) of emission (e.g., scrubbing mitigates SO2 emission), while it is solely responsible for generating the fourth type of emission because of its use of the second emission-causing input (e.g., scrubbing leads to production of gypsum, which we treat as another–the fourth – emission). Economic output production employs the non-emission-causing inputs (e.g., labor and capital) in conjunction with the first emission-causing input (say coal) to produce thermal electricity. The use of the first emission-causing input leads to the generation of the first three types of emissions (say CO2 , CO, and SO2 ). Of these, the first two types of emissions are rival in nature, while the third type is jointly produced with the other two types of emissions. Thus, the space of all goods under study has dimension t = nz + no + m + m + 1 = no + 8. We first model the rival production of abatement and the economic output. The sub-technology that produces these is the intended-production technology. We then develop the structure of the sub-technology generating multiple emissions from emission-causing inputs. The overall technology that produces the economic output, multiple emissions, and abatement from all inputs is obtained as the set of production vectors that lie simultaneously in both of these sub-technologies.

Rival Production of Abatement and the Economic Output Individual Technologies Producing Economic Output and Abatement The technology that produces the desired economic outputs is represented by the set n+1 30 Y Y Y TY1 ⊂ Rn+1 + and consists of production vectors xz , xo , y =: x , y ∈ R+ . 29 If the concentration of oxygen

in the air is high, relatively more CO2 is produced, and if it is low, relatively more CO is produced. 30 Since the second emission-causing input is not employed in the production of the economic output, xzY2 = 0 whenever xzY , xoY , y ∈ T1 .

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Let us pause to describe the nature of the output of an abatement technology aimed at the reduction of a particular type of emission. The net output of a pollution treatment technology–e.g., a scrubber technology in the case of SO2 emission–is often measured in terms of the resultant reduction in the “gross” emission level.31 The gross emission of SO2 generated by the combustion of sulfur contained in coal, g say z3 , is reduced by the end of the scrubbing procedure. Denote this reduction in the gross amount of SO2 by a ∈ R+ , so that the “net” emission of SO2 generated g by the producing unit is z3 = z3 − a. The abatement technology employs inputs such as labor, capital, and lime or limestone to produce reductions in the emission.32 It is clear that there are bounds on emission reductions given fixed amounts of these inputs.33 For example, the amount of SO2 reduction from the flue gas depends upon the amount of lime or limestone used as a sorbent during flue gas desulfurization (FGD).34 Any given quantity of lime or limestone, along with fixed amounts of the service inputs used by the abatement technology, fixes the maximal amount of SO2 reduction. Thus, the abatement technology is defined by relations among all the inputs used by it and the extent of reduction that is made possible by the use of these inputs.35 n Denote the production technology that captures these relations by TA 1 ⊂ R+ . This A A A technology contains production vectors of the form xz , xo , a = x , a ∈ TA 1. are standard neoclassical technologies We assume that technologies TY1 and TA 1 satisfying the following assumptions: (T1 C) TY1 and TA 1 are non-empty and closed. (T1 B) The sets {y ∈ R+ | x Y , y ∈ TY1 } and {a ∈ R+ | x A , a ∈ TA 1 } are bounded for all x Y ∈ Rn+ and for all x A ∈ Rn+ . ¯ y ¯ ∈ TY1 . (T1 F D) x, y ∈ TY1 ∧ x¯ ≥ x ∧ y¯ ≤ y =⇒ x, A A ¯ a ¯ ∈ T1 . x, a ∈ T1 ∧ x¯ ≥ x ∧ a¯ ≤ a =⇒ x, . (T1 SD) 0n , 0 ∈ TY1 and 0n , 0 ∈ TA 1

31 For

example, in the case of the scrubber technology, reductions are usually measured as percentages of the gross emission. 32 As will be seen in section “Modeling the Generation of Multiple Emissions,” the reduction in the third emission, say SO2 , is accompanied by an increase in the fourth emission, say gypsum. This is because, depending on the quantity of the second input (say lime) used, the third emission is converted into the fourth emission during the abatement process (say scrubbing). 33 As an analogy, a pound of a cleaning powder can only clean a finite amount of dirty surface area. If used inefficiently, it cleans less than in its potential. 34 See, e.g., Srivastava and Jozewicz [54]. 35 Hampf [31] provides a network DEA formulation of technology that includes rival production of abatement. The inputs employed by the abatement technology include standard inputs and gross emissions, while its output is measured in terms of reductions in emission levels. While net emissions are observable, gross emissions are computed employing the material balance condition as the difference between the mass of the emission-generating input used and the mass of these inputs transferred to the marketable output during production. The difference in the gross and net emissions is defined as the reduction in the emission levels attributable to abatement.

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While Assumption (T1 B) implies that the outputs of the economic production technology TY1 and abatement producing technology TA 1 are bounded when the respective vectors of inputs used by these technologies are fixed at x Y and x A , Assumption (T1 F D) implies that these technologies satisfy standard free disposability conditions with respect to their respective outputs and inputs. Assumption (T1 SD) says that it is possible to shut down operations of the two technologies. Under these assumptions, the technologies TY1 and TA 1 have functional representations. Define the functions,  : Rn+ → R+ and  : Rn+ → R+ , by (x Y ) = max{y ≥ 0 | x Y , y ∈ TY1 } and (x A ) = max{a ≥ 0 | x A , a ∈ TA 1 }.

(16)

Under the maintained assumptions, technologies TY1 and TA 1 can be functionally represented as x Y , y ∈ TY1

⇐⇒

y ≤ (x Y )

and

x A , a ∈ TA 1

⇐⇒

a ≤ (x A ).

The Overall Intended-Production Technology T1 We define the intended production of the economic unit as its production of both the economic and the abatement outputs. The intended-production technology, denoted by T1 ⊂ Rt+ , combines the two technologies, TY1 and TA 1 , as follows:

 

n +n n +n T1 := xz , xo , a, y, z ∈ Rt+ ∃ xzY , xoY ∈ R+z o and xzA , xoA ∈ R+z o such that xzY + xzA = xz , xoY + xoA = xo , 

Y Y 

 xz , xo , y ∈ TY1 , and xzA , xoA , a ∈ TA 1 .

(17)

Thus, T1 is a set of all production vectors x, a, y, z such that the production vectors y and a are possible with some allocation of the aggregate input vector x between the two technologies TY and TA . Thus, if the vector xzY , xoY  of the two types of inputs are employed in the production of the economic output, only the remaining amounts of inputs xzA , xoA  := xz , xo  − xzY , xoY  are available for abatement. The technology defined by (17) explicitly incorporates the resource cost of cleaningup: the diversion of resources toward scrubbing reduces the resources available for electricity generation. The proposition below, which directly follows from the restrictions imposed on TY1 and TA 1 , states the properties of the intended-production technology T1 : it is a closed set that permits shutting down; the set of combinations of economic and abatement outputs that are feasible under T1 with finite amounts of inputs is bounded; it satisfies free disposabiity in all inputs and the economic and abatement

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outputs, and it is independent of the level of net emissions (net emissions do not affect intended production).36 Proposition 1. Under Assumptions (T1 C), (T1 B), (T1 F D), and (T1 SD), the following conditions are satisfied: 

m and 0n , 0s , 0m , z ∈ T1 for all (i) T1 is closed

 z ∈ R+ . m+s (ii) the set a, y ∈ R+ x, a, y, z ∈ T1 is bounded for all x, z ∈ Rn+4 + . (iii) x, a, y, z ∈ T1 , x ≤ x, ¯ y ≥ y, ¯ and a ≥ a¯ implies x, ¯ a, ¯ y, ¯ z¯  ∈ T1 .

Note that (iii) holds for z = z¯ as well as z = z¯ , a reflection of the fact that the set T1 simply constrains the production of intended outputs for given quantities of the inputs, independently of the pollution (by-product) levels. Employing the functions  and , we can obtain an implicit distance function representation of the overall intended-production technology T1 . Define F (x, y, a, z) = max

λ, x Y ,x A



λ ≥ 0 λy ≤ (x Y ), x Y + x A ≤ x,

λa ≤ (x A ),

x Y ∈ Rn+ ,

and

 x A ∈ Rn+ .

(18)

Then the set T1 can be functionally represented by x, y, a, z ∈ T1 ⇐⇒ F (x, y, a, z) ≥ 1. Remark 1. If the functions  and  are differentiable, production efficiency implies that the input vector x is split between the production of the economic output and the abatement output such that (on the interior of Rt+ )37 the marginal rates of technical substitution between any two inputs in economic output and abatement output production are equalized.38

Modeling the Generation of Multiple Emissions Levkoff [35], Kumbhakar and Tsionas [33], and Murty and Russell [43]) have argued that, while a single restriction on emissions and emission-causing inputs suffices to capture the generation of emissions that are rival (or substitutable) in production, multiple restrictions, one for each type of emission, are required when emissions are jointly produced (complementary).

36 See

Murty [41] for the case where emissions also affect production of the economic outputs. For example, smoke from a factory can have detrimental effects on the productivity of its labor. 37 And at the boundaries for appropriately defined directional derivatives. 38 This follows from considering the first-order conditions of problem (18).

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Continuing our example where coal is used to produce electricity, we first model the sub-technology that generates emissions as the by-product of economic output production. These emissions include CO2 , CO, and SO2 . We then study the sub-technology that generates the new emission (gypsum) during the abatement (scrubbing) process. The overall technology that captures generation of all emissions from emission-causing inputs and their mitigation by abatement activities is obtained by combining these two sub-technologies.

Modeling Generation of Carbon and Sulfur Emissions Attributable to Combustion of Coal We capture the rivalry in the production of CO and CO2 and the jointness in the production of SO2 and the carbon emissions when coal is combusted to generate thermal electricity by first defining the set,

TY2 := xz1 , a, z1 , z2 , z3  ∈ R5+ ðC (xz1 , z1 , z2 ) ≥ 0 ∧  z3 ≥ max{ðS (xz1 ) − a, 0} , where z1 , z2 , and z3 denote the net emissions of CO, CO2 , and SO2 , respectively. g The function ðS : R+ −→ R+ with image z3 = ðS (xz1 ) gives the minimal amount of gross emission of SO2 associated with xz1 level of coal. Given an arbitrary level of abatement a ≥ 0, the minimal “net” emission generated is z3 = g S (xz1 ) − a if a ≤ g S (xz1 ). If, however, a > g S (xz1 ), the minimal net emission is z3 = 0.39 Hence, the minimal net emission of SO2 is given by z3 = max{ðS (xz1 ) − a, 0}. The actual level of net emission, z3 , can be more than this if there are inefficiencies in emission generation. The implicit production function ðC captures the rival production of CO2 and CO emissions owing to the use of coal. Thus, ðC (xz1 , z1 , z2 ) ≥ 0 implies that z1 and z2 levels of the two carbon emissions are feasible given combustion of xz1 amount of coal. We assume that the functions, ðC and ðS , are differentiable and that their C derivatives have the following signs: ðC xz (xz1 , z1 , z2 ) ≥ 0; ðzk (xz1 , z1 , z2 ) < 0 for 1

k = 1, 2; and dg S (xz1 )/dxz1 > 0. From the implicit function theorem, it follows that the carbon and sulfur emission are increasing in the use of coal and that there is rivalry in the production of the two carbon emissions. The latter follows because, holding the quantity of coal fixed, there is a negative trade-off between these two emissions: ðC ∂z2 z (xz1 , z1 , z2 ) < 0. = − C1 ∂z1 ðz2 (xz1 , z1 , z2 ) 39 The

potential level of abatement a can be greater than the gross emission level g S (xz1 ) if the inputs used in the abatement technology are capable of reducing more than g S (xz1 ) of SO2 .

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We assume in addition that, when no coal is used, none of the carbon or sulfur-based emissions are produced: 05 ∈ TY2 .

Modeling the Production of Gypsum During Scrubbing Limestone used by the abatement technology transforms the SO2 emission into gypsum. Thus, the scrubber technology jointly produces the abatement output (a reduction in SO2 ) and a new emission (gypsum). It is clear that, given an amount of limestone used for scrubbing, there is an upper bound on the amount of SO2 that the scrubbing can abate. Since the abated SO2 is converted into gypsum, this upper bound also defines the maximum amount of gypsum that can be produced by the given amount of limestone. Inefficiency in abatement implies that less than the maximum reduction of SO2 by the given amount of limestone takes place, resulting in a lower amount of gypsum production. In the extreme case of inefficiency, no reduction of SO2 takes place, and so no gypsum is produced by the scrubber. Define a function ðG : R+ −→ R+ , with image,  z4 = ðG xz2 specifying the (maximal) amount of gypsum that can be produced when xz2 amount of limestone is used efficiently in the scrubber to reduce the SO2 emission. Assume that this function is differentiable. Since it is increasing in the amount of limestone used, its derivative is positive. The following sub-technology captures the production of gypsum during the scrubbing process:

 2 G TA 2 := xz2 , z4  ∈ R+ | z4 ≤ ð (xz2 ) . In addition, we assume that when no limestone is used, no gypsum is produced: ðG (0) = 0.

Combining Sub-technologies Generating Carbon and Sulfur Emissions and Gypsum In the space Rt+ of all goods, the set depicting the net generation of all emissions by emission-causing inputs is obtained from the individual net emission generating sub-technologies, TY2 and TA 2 , as 

T2 = xz , xo , a, y, z ∈ Rt+ xz1 , a, z1 , z2 , z3  ∈ TY2 ∧ xz2 , z4  ∈ TA 2 . (19) The proposition below states the properties of set T2 . Proposition 2. Under the maintained assumptions, the following are true: no (i) T2 is closed and 0nz , xo , a, y, 04  ∈ T2 for all y ∈ Rm + , a ∈ R+ , and xo ∈ R+ . ¯ zk ≤ z¯ k for k = 1, 2, 3, and z4 ≥ z¯ 4 , (ii) x, a, y, z ∈ T2 , xz ≥ x¯z , a ≤ a, imply x¯z , x¯o , a, ¯ y, ¯ z¯  ∈ T2 for arbritary x¯o , y¯o .

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Thus, T2 satisfies joint essentiality of coal and limestone in generating emissions. It also satisfies costly disposability of the emission-causing inputs, sulfur and carbonbased emissions, and the abatement activity. However, it satisfies free disposability of gypsum. Thus, ceteris paribus, T2 permits arbitrary increases in carbon and sulfurbased emissions, decreases in gypsum, decreases in the emission-causing inputs, and increases in abatement activity. Moreover, emission generation is independent of the levels of the non-emission-causing inputs and the economic output–these goods do not influence the amounts generated of emissions.

The Overall By-Production Technology with Abatement and Multiple Emissions Given the intended-production technology T1 defined in (17) and the set T2 depicting emission generation defined in (19), a by-production technology, denoted by T B ⊂ Rt+ , is defined as in MRL, Murty [41], and MR as the intersection of these two sets: T B = T1 ∩ T 2 .

(20)

Once again, as seen in section “Multi-equation Modeling: The Case of Rival and Joint Production,” the disposability properties of set T B with respect to the emission-causing inputs are not obvious. The disposability property of T B with respect to the abatement activity is also unclear. This is because, while set T1 satisfies free input disposability of emission-causing inputs and free output disposability of abatement, it satisfies costly disposability with respect to the emission-causing input and abatement. We next study the disposability properties of such an overall technology with respect to all goods.

Axiomatic Approach to Modeling Emission-Generating Technologies The primitive concepts in the modeling of emission-generating technologies in section “Multi-equation Modeling of Emission-Generating Technologies with Abatement Activities and Multiple Emissions” are two sub-technologies, one for characterizing the intended production (set T1 in the previous section) and the other for characterizing emission generation in nature (set T2 in the previous section). The first sub-technology is an engineering construct, while the latter captures natural laws that link emissions to their basic sources in nature and the mitigation of these emissions through human abatement activities. The intersection of these sub-technologies yields the by-production technology (BPT). As seen in the previous section, these sub-technologies have well-defined disposability properties that conform to our intuitive understanding of the processes and are consistent

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with empirical observation. The properties of the overall BPT, however, remained undetermined. In the analysis that follows, with a view to understanding the basic disposability properties of the overall by-production technology, we adopt a reverse approach, which is based on Murty [41] and Murty and Russell [43]. We perceive observable data to have been generated by a technology T that engages simultaneously in economic output production and emission generation. We postulate its disposability properties in the form of some axioms. Murty and Russell show that, if a technology T satisfies these axioms, it can be decomposed into an intended production subtechnology and a set that describes residual generation in nature. Moreover, a BPT, as defined in the section Modeling the Generation of Multiple Emissions satisfies these axioms. We generalize our model to include s types of abatement activities. A quantity vector of abatement outputs is denoted by a ∈ Rs+ . Redefine the number of commodities as t = n + s + m + m . An emission-generating technology comprises a set of technologically feasible production vectors x, a, y, z ∈ Rt+ and is denoted by  ⊂ Rt+ . This technology should capture all relations that describe how the use of inputs in production generates the economic and abatement outputs and the emissions as well as the mechanism by which abatement/cleaning-up activities help in mitigating emissions. From the material balance conditions of nature, one can infer that there are both upper and lower limits to production of emissions once the levels of emissioncausing inputs and abatement are fixed; e.g., because of its carbon content, the combustion of coal must generate a nonnegative amount of CO2 , but the amount of CO2 emitted depends on the oxygen supply in the air. As economists, we are usually concerned with emissions that are harmful and economic policies that aim, ceteris paribus, to minimize the generation of such emissions. At the same time, economic policies aim, ceteris paribus, to maximize the production of economic outputs from inputs. Hence, the relevant economic frontier of a technology generating harmful emissions combines the lower limits of emission generation attributable to the use of emission-causing inputs and abatement activities with the upper limits of intended output production from all inputs. In this chapter, we assume that abatement activities transform harmful emissions generated by the producing unit into less harmful emissions that are outside the purview of economic policy and hence not modeled by the researcher. Thus, the strictly efficient frontier of  contains only those production vectors in  for which there do not exist other production vectors, also in , with no larger amounts of inputs or emissions and no smaller amounts of economic and cleaningup outputs. Thus, x, a, y, z in  is a strictly efficient point of  if −x, ¯ a, ¯ y, ¯ −¯z > −x, a, y, −z implies that x, ¯ a, ¯ y, ¯ z¯  is not contained in . Murty and Russell [43] show that, to study the properties of the true emissiongenerating technology  relative to its frontier, it suffices to study the properties of its costly disposal hull in the direction of emissions, which is defined as the set,

  T := x, a, y, z +  ∈ Rt+ x, a, y, z ∈  and  ∈ Rm + .

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The set T includes any production vector v = x, a, y, z ∈  as well as production vectors of type x, a, y, z +  ∈ Rt+ that, ceteris paribus (i.e., holding levels of all other goods unchanged), produce arbitrarily larger amounts of emissions than z. This approach is adopted because the economically relevant frontiers of the two technologies, T and , are identical, and the set T is analytically more tractable than the true technology set . In what follows, we therefore adopt the costly disposal hull T as the relevant emission-generating technology. It is helpful to define some subspaces of the set T in Rt+ . These include the intended-output possibility set, T y (x, a, z) = {y ∈ Rm + | x, a, y, z ∈ T }, the pollution-generation set, 

T z (x, a, y) = {z ∈ Rm + | x, a, y, z ∈ T }, and the set of vectors of economic outputs and emissions that are feasible under T , 

T y,z (x, a) = {y, z ∈ Rm+m | x, a, y, z ∈ T }. + For example, T y (x, a, z) is the set of all economic outputs that are feasible under technology T with the fixed vectors of inputs, cleaning-up activities, and emissions x, a, z. It is possible for such a subspace to be empty: for example, T y (x, a, z) could be empty if the amount of some component of the emission vector z is smaller than the minimal amount of net emissions that can be generated by input and cleaning-up vectors xz and a. (This will be true, for example, if the given amounts of fossil fuels in vector xz , combusted under the most favorable of atmospheric conditions, generate far more CO2 than the amount indicated by the relevant component of z.) In this case, there is no economic output vector y such that x, a, y, z is technologically feasible, because generation of the emission vector z is infeasible under physical laws of nature given the vector xz of emission-causing inputs and the cleaning-up vector a. Similarly, the set T z (x, a, y) can be empty if the levels of inputs in vector x are too small or too large a part of the inputs is siphoned into producing abatement vector a to ensure production of intended-output vector y. (E.g., a given amount of coal, when burnt, may be too small to produce the amount of heat required to generate the given amount of electricity.) In such a case, there is no emission vector z such that x, a, y, z is technologically feasible, because y is infeasible in intended production given the combination x, a of inputs and abatement levels. It is useful, therefore, to define the sets,     = x, a, z ∈ Rn+s+m | T y (x, a, z) = ∅ and +   n+s+m | T z (x, a, y) = ∅ .  = x, a, y ∈ R+

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Murty and Russell [43] impose the following assumptions on the set T 40 : (EG0) T is closed and contains 0t . (EG1) T y (x, a, z) is bounded, satisfies free disposability of non-emission-causing inputs and outputs, y ∈ T y (x, a, z), x¯o ≥ xo , and y¯ ≤ y =⇒ y¯ ∈ T y (xz , x¯o , a, z),

(21)

conditional free disposability of emission-causing inputs and cleaning-up activities, y ∈ T y (x, a, z), x¯z ≥ xz , a¯ ≤ a, and x¯z , xo , a, ¯ z ∈  =⇒ y ∈ T y (x¯z , xo , a, ¯ z),

(22)

and independence of emissions, y ∈ T y (x, a, z) =⇒ y ∈ T y (xz , xo , a, z¯ )

∀ z¯ = z.

(23)

(EG2) T z (x, a, y) satisfies joint essentiality of emission-causing inputs for emission generation: 

xz = 0(nz ) =⇒ 0(m ) ∈ T z (x, a, y),

(24)

conditional costly disposability of emission-causing inputs, cleaning-up activities, and emissions: ¯ y ∈  z ∈ T z (x, a, y), x¯z ≤ xz , a¯ ≥ a, z¯ ≥ z, and x¯z , xo , a, =⇒ z¯ ∈ T z (x¯z , xo , a, ¯ y),

(25)

and independence of intended outputs and non-emission-causing inputs: ¯ ∈ z ∈ T z (x, a, y) and xz , x¯o , a, y =⇒ z ∈ T z (xz , x¯o , a, y) ¯ ∀ x¯o , y. ¯

(26)

To understand these axioms, recall that the set T contains only those production vectors that simultaneously satisfy constraints on intended production and emission generation. While the intended-production technology satisfies free disposability of all inputs, emission-causing inputs are not freely disposable in nature’s emission generation mechanism. For example, ceteris paribus, increasing coal combustion

40 These

assumptions attribute emission generation to the use of emission-causing inputs only. See Murty [41] for the case where emissions can also be generated by the economic output once it has been produced.

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is not free: it comes at the cost of increasing the emission levels. Similarly, we can argue that, while the intended-production technology satisfies free output disposability in the direction of cleaning-up outputs, these goods are not freely disposable in the generation of emissions: decreasing the level of the scrubbing activity comes at a cost of decreasing the mitigation of the SO2 emission. Hence, the overall technology T –a composition of the intended production technology and the laws that govern emission generation–is not freely disposable in the direction of emission-causing inputs and cleaning-up activities. The (EG0) assumption is standard in production theory, while (EG1) captures properties that the technology T inherits from the intended-production technology of human engineering design, and (EG2) reflects relations between goods that describe emission generation and remain relevant for the technology T. More specifically, (21) in (EG1) states that the set T permits standard free disposability of the economic outputs and standard free disposability of non-emissioncausing inputs, while (22) states that emission-causing inputs and cleaning-up activities are only conditionally freely disposable: if the production of intendedoutput vector y is permitted given the quantities of inputs, cleaning-up levels, and emissions, then y is also permitted by T under a larger vector of emissioncausing inputs x¯z and a smaller vector of cleaning-up activities a, ¯ provided that the vector x¯z , a ¯ can continue generating z amounts of the emissions–i.e., provided that x¯z , xo , a, ¯ z ∈ . In addition, condition (23) in (EG1) states that changes in the levels of emissions do not affect production of the intended outputs. Implicit in this independence is an assumption that emissions produced by a producing unit are not detrimental to its production of intended outputs.41 Condition (25) in (EG2) states that, if the emission vector z is permitted by technology T , given the vector x, a, y of inputs, cleaning-up levels, and intended outputs, then technical inefficiencies in emission generation can imply that z is also permitted by T under a smaller vector of emission-causing inputs x¯z and a larger vector of cleaning-up activities a, ¯ provided the vector xo , x¯z , a ¯ can still produce amount y of intended outputs–i.e., provided x¯z , xo , a, ¯ y ∈ . Thus, emission-causing inputs and cleaning-up activities satisfy only conditional costly disposability. In addition, condition (26) states that emission generation is not affected by changes in the production levels of intended outputs or changes in the use of non-emission-causing inputs. Definition. The set T ⊂ Rt+ is an emission-generating technology (EGT) if it satisfies (EG0), (EG1), and (EG2). Let T be an EGT. To recover the intended-production technology and the emission-generation set underlying T and to obtain its functional representation, Murty and Russell [43] propose the use of distance functions. Define D1EG : Rt+ → R+ and D2EG : Rt+ → R+ by 41 For

generalization to the case where the emissions produced by a producing unit also affect the production of its economic output, see Murty [41].

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inf λ ∈ R++ | y/λ ∈ T y (x, a, z)

 D1EG (x, a, y, z) =

∞

 D2EG (x, a, y, z) =

if T y (x, a, z) = ∅ if T y (x, a, z) = ∅



min λ ∈ R+ | λz ∈ T z (x, a, y)} if T z (x, a, y) = ∅ ∞

if T z (x, a, y) = ∅.

Thus, D1EG is the inverse of the maximum technologically permissible amount by which we can expand the intended output vector y holding input, abatement, and emission levels fixed, while D2EG is the inverse of the maximum technologically permissible amount by which we can contract the emission vector z holding input, abatement, and intended output levels fixed. The function D1EG provides an implicit functional representation of the intendedproduction technology, which can be recovered as Tˆ1 := {x, a, y, z ∈ Rt+ D1EG (x, a, y, z) ≤ 1}, while the function D2EG provides a functional representation of the underlying emission-generation set, which can be recovered as Tˆ2 := {x, a, y, z ∈ Rt+ D2EG (x, a, y, z) ≤ 1}. The set of production vectors x, a, y, z satisfying D1EG (x, a, y, z) = 1 forms the upper frontier of the set Tˆ1 , indicating the upper bounds to the production of intended outputs. The set of production vectors x, a, y, z satisfying D2EG (x, a, y, z) = 1 forms the lower frontier of the set Tˆ1 , indicating the lower bounds of emission generation. The following theorem in Murty and Russell [43] shows that all the intuitive trade-offs between goods in intended production and emission generation hold along the frontiers defined by the functions D1EG and D2EG , respectively. In particular, along the frontier of the underlying intended production technology defined by D1EG , the trade-offs between standard economic outputs and inputs are nonnegative, and those between two inputs or two economic outputs are nonpositive. On the other hand, along the frontier of the emission-generating set defined by D2EG , the tradeoffs between emission-causing inputs and emissions are non-negative, and those between cleaning-up activities and emissions are nonpositive. Theorem 1. Suppose T is an EGT. D1EG is independent of z and linearly homogeneous in y, and D2EG is independent of y and homogeneous of degree minus one in z. D1EG is nonincreasing in x and nondecreasing in y on the set , while D2EG is non-decreasing in x and non-increasing in z and a on the set .

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The distance functions, D1EG and D2EG , provide a functional representation of the EGT, which we denote T EG : x, y, a, z ∈ T EG ⇐⇒ D1EG (x, a, y, z) ≤ 1

and

D2EG (x, a, y, z) ≤ 1.

The set of frontier points of T EG are the production vectors x, a, y, z that satisfy D1EG (x, a, y, z) = 1 or D2EG (x, a, y, z) = 1. As a conclusion to this section, we draw attention to two important points: First, recall that an EGT is defined as a technology set that satisfies axioms EG0, EG1, and EG2, properties that hold, we have argued, for realistic/empirically observed emission-generating technologies. We have demonstrated above that, in contrast to the technologies derived under the input or output approaches to emission modeling discussed in sections “Single-Equation Modeling of the Technology Under Standard Disposability Assumptions and Weakly Disposable Technologies,” an EGT has a multiple-equation representation. We have earlier argued that the input and output approaches result in many counterintuitive consequences for technology modeling. Second, Murty and Russell [43] show that if T is a BPT – i.e., T = T1 ∩T2 , where T1 is defined in (17), T2 is defined in (19), and T1 and T2 satisfy the properties in Propositions 1 and 2 – then T is also an EGT. Hence, the disposability properties of a by-production technology are fully specified by the properties of an EGT; that is, a BPT satisfies properties EG0, EG1, and EG2.

Efficiency Measurement The emission-generating technologies throughout this chapter have been characterized in terms of production and emission sets, encompassing the possibility of firms producing off the frontier because of technological or managerial inefficiencies. This approach therefore facilitates discussion of the measurement of technical (in)efficiency – calculation of a scalar measure of the “distance” from the point of operation of the firm to the technological frontier. Formally, an environmental technological efficiency index is a mapping, E : Rt ∩ T → (0, 1], where T is the set of allowable technologies42 , and, informally, larger image values are interpreted as higher levels of efficiency. A technological inefficiency index is a mapping, I : Rt+ ∪ T → [0, ∞), where higher image values are interpreted as greater levels of inefficiency. Clearly, any inefficiency index can be converted into an efficiency index, and vice versa, by a simple renormalization: I (x, y, z, T ) = [1/E(x, y, z, T )] − 1. The production vector x, y, z is a frontier point of T if and only if E(x, y, z, T ) = 1 or I (x, y, z, T ) = 0.

42 Dakpo,

Jeanneaux, and Latruffe [16] provide a comprehensive survey of recent developments in DEA modeling of pollution-generating technologies.

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Properties of Environmental Efficiency Indexes These definitions have no interesting content without a rigorous definition of efficiency and the stipulation of properties satisfied by the indexes. To that end, we first define the notion of technical efficiency, which has as its basis (i) a normative criterion of discouraging generation of harmful emissions and (ii) a criterion of minimizing wastage of scarce and productive economic inputs. A production vector x, y, z ∈ T is (technologically) efficient if x, ˆ −y, ˆ zˆ  < x, y, z implies x, ˆ −y, ˆ zˆ  ∈ / T and weakly efficient if x, ˆ −y, ˆ zˆ   x, y, z implies x, ˆ −y, ˆ zˆ  ∈ / T . Intuitively, a production vector is efficient if there does not exist another production vector with no smaller amounts of the good outputs and no larger amounts of emissions and inputs. Next, we stipulate additional properties that efficiency indexes are required to satisfy. The most important possibilities are • identification of (weakly) efficient points: E(x, y, z, T ) = 1 (or I (x, y, z, T ) = 0) if and only if x, y, z is (weakly) efficient for all T ∈ T, and • monotonicity: x, ˆ −y, ˆ zˆ  > x, −y, z) =⇒ E(x, ˆ y, ˆ zˆ , T ) < E(x, y, z, T ) for all T ∈ T or • weak monotonicity: x, ˆ −y, ˆ zˆ  x, −y, z) =⇒ E(x, ˆ y, ˆ zˆ , T ) < E(x, y, z, T ) for all T ∈ T. Note that the satisfaction or violation of these properties depends on the maintained set of admissable technologies T as well as the specific formulation of the index E or I .

Hyperbolic and Directional Distance Indexes A large number of specific (in)efficiency indexes have been proposed in the literature.43 The first application of efficiency measurement to emission-generating technologies was carried out by Färe, Grosskopf, and Pasurka [22]. They proposed the (output oriented) hyperbolic efficiency index (HYP),44 defined by EH (x, y, z, T ) = min {β ∈ (0, ∞) | x, y/β, βz ∈ T } . The inverse of this index provides the maximal, technologically feasible (scalar) amount by which the vector of intended-output quantities can be scaled up and

43 See

Russell and Schworm [51] for an analysis of these indexes and their properties. because it measures the distance from the stipulated production/emission quantity vector to the frontier along a hyperbolic path.

44 So-called

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the vector of unintended-output quantities can be scaled down, holding all input quantities fixed.45 In recent years, the more widely employed environmental efficiency index is the (output-oriented) directional distance inefficiency index (DD), proposed by Chung, Färe, and Grosskopf [11]46 and defined by   

IDD (x, y, z, T , g) = max β ∈ (0, ∞) | x, y + βgy , z − βgz ∈ T , 

where g = gy , gz  ∈ Rm+m is the arbitrary (output) “direction vector.” This index + provides the maximal technologically feasible (scalar) amount by which the vector of intended outputs can be increased in the direction gy , and, concomitantly, the vector of unintended outputs can be decreased in the direction gz while holding all the inputs fixed.



 The vectors x d , y d /β ∗ , β ∗ zd and x d , y d + β ∗ gy , zd − β ∗ gz , where β ∗ is the solution value in each case, are referred to as “reference points”; they are comparison vectors for assessing the efficiency of a particular production vector. The hyperbolic and directional distance indexes work well when applied to the weak disposability technologies advanced by the authors. MRL, however, have noted a fundamental problem with the conventional measures of efficiency when using the by-production (BP) approach for constructing the technology: the efficiency score for a firm may take the value 1 for HYP measures or 0 for the DD measure even though the firm is not weakly efficient in both environmental and intended-output directions. In addition, the reference point, itself, with which the firm is compared may not be weakly efficient in both these dimensions, resulting in an understatement of overall inefficiency (overstatement of efficiency). In the BP approach, the emission-generating technology is an intersection of one or more subtechnologies, each possessing distinct disposability properties that capture different types of production relations among the inputs and outputs.47 MRL argue that the DD index is particularly unsuitable for use as an inefficiency index for a BP technology. It is well-known that the inefficiency scores obtained from the DD measure can be very sensitive to the choice of the direction vector g.48 This sensitivity, moreover, seems to be more salient in the BP approach, since the choice of g in this context is typically tantamount to predetermining a choice

45 The

index is called “output oriented” because it measures efficiency in output space (as opposed to the entire space). We return to this point later in this section. 46 Based on the notion of the directional distance function formulated by Luenberger [37] in his novel approach to duality analysis. For recent applications of the DD to the measurement of environmental efficiency, see Aparicio, Barbero, Kapelko, Pastor, and Zofio [1] and the papers cited therein. 47 For example, Serra, Chambers, and Lansink [53] specify a rich model of a BP technology that takes into account the stochastic nature of agricultural production and incorporates several subtechnologies governing not only the production of the good (marketable) outputs and the bad outputs but also the damage to human health. 48 See, e.g., Vardanyan and Noh [55] and Färe, Grosskopf, and Pasurka [24].

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between the selection of the environmental and the intended production inefficiency components as the measure of overall inefficiency.

The “Färe-Grosskopf-Lovell” Index Because of these problems with the employment of the HYP or DD efficiency measure on BP technologies, MRL propose an alternative index motivated by the input-oriented index proposed by Färe and Lovell [25] and extended to the full input, output space for standard technologies (with no unintended outputs) by Färe, Grosskopf, and Lovell ( [18], pp. 153–154). The key feature of this index is that the reference points it uses to assign efficiency scores to production vectors are strictly efficient, in contrast to the HYP and DD indexes for which the reference points are weakly efficient. In particular, this measure deems a production vector to be efficient if and only if it is both environmentally efficient and efficient in intended production.49 As the MRL modification is minor, they continue to refer to the measure as the (output oriented) Färe-Grosskopf-Lovell (FGL) index and define it as follows: EF GL (x, y, z, TBP ; α) :=     j θj k γk + (1 − α) min α x, y  θ, γ ⊗ z ∈ TBP ,  m m θ,γ ∈(0,1]m+m where y  θ = y1 /θ1 , . . . , ym /θm , γ ⊗ z = γ1 z1 , . . . , γm zm , and α ∈ (0, 1) is an arbitrary weighting factor (which could depend on analytical or policy considerations). This index maps into the (0,1] interval and is equal to 1 if and only if the output vectors are strictly efficient given the input vector. Moreover, in the case of BP technologies, the index decomposes as follows: EF GL (x, y, z, TBP ; α)  

j θj



k γk m

x, y  θ, γ ⊗ z ∈ TBP



+ (1 − α) m 

 θj j k γk = min + (1 − α) α x, y  θ, z ∈ T1 ∧  m m θ,γ ∈(0,1]m+m

=

min

θ,γ ∈(0,1]m+m



α

x, y, γ ⊗ z ∈ T2

49 This



feature is attributable to the fact that the Färe-Grosskopf-Lovell index involves a maximal contraction/expansion of all inputs/outputs in coordinate-wise directions (rather than in a maximal radial or hyperbolic direction). Hence, all the slack in inputs and outputs is removed. (Of course, the output-oriented version of the MRL index takes up all slack only in the output space, leaving the possibility of residual slack in inputs. More on this in section “Extension of the FGL Index to Graph Space” below.)

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 = α min m θ∈(0,1]

j θj

m

x, y  θ, z ∈ T1 

+(1 − α)

min

γ ∈(0,1]m



k γk m



 x, y, γ ⊗ z ∈ T2

=: αEF1 GL (x, y, z, T1 ) + (1 − α)EF2 GL (x, y, z, T2 ) =: αβ1 + (1 − α)β2 = β, where the second identity follows from independence of T1 from z and independence of T2 from y. This index is a weighted average of the sum of the average maximal coordinate-wise expansions of economic output quantities and the average maximal coordinate-wise contractions of unintended-output quantities subject to the constraint that the expanded/contracted output quantity vector remain in the production possibility set for a given input vector. Under the independence assumptions, the index decomposes into the sum of a standard intended-outputoriented index defined on T1 and an environmental index defined on T2 (β1 and β2 , respectively). The FGL environmental efficiency index and the underlying MRL by-production structure have come under criticism recently. A number of technical criticisms and proposed corrections appear in Dakpo [15], Dakpo et al. [16], and Ray et al. [47]. We respond to these critiques in section “Critiques and Suggested Modifications of the By-Production Structure” below but first address a more fundamental issue raised by both Dakpo [15] and Lozano [36].

Extension of the FGL Index to Graph Space Each of the efficiency indexes is described in sections “Hyperbolic and Directional Distance Indexes” and “The “Färe-Grosskopf-Lovell” Index” limits the measurement of efficiency to (intended and unintended) output space (for given input quantities), leaving open the possibility of remaining slack in input space. This feature–also adopted by Färe Grosskopf, and Pasurka [22] and by Färe, Grosskopf, Noh, and Weber [21] in their specifications of environmental efficiency indexes–has been criticized recently by Dakpo [15] and Lozano [36] for ignoring slack in input quantities. The FGL environmental index, however, can be readily extended to the full (or graph) space. This extension requires the incorporation of additional contraction factors for inputs, δ = δz , δo  ∈ (0, 1]n , as follows: m

 EFGGL (x, y, z, TBP ) =

min



θ,γ ,δ∈(0,1]m+m +n

α1

j =1 θj

m

m + α2

k=1 γk + α3 m

n



i=1 δi

n

 δ ⊗ x, y  θ, γ ⊗ z ∈ TBP

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m

 =

min



θ,γ ,δ∈(0,1]m+m +n

α1

j =1 θj

m

m

k=1 γk + α3 m

+ α2

n



i=1 δi

n

 δ ⊗ x, y  θ, z ∈ T1 ∧ δ ⊗ x, y, γ ⊗ z ∈ T2 , (27)

 where αν ∈ (0, 1] for ν = 1, 2, 3 and ν αν = 1. The minimization problem in this formulation takes up the slack in all inputs as well as all intended and unintended outputs, assuring that the reference point is strictly efficient.50,51 Although well-defined, it appears that decomposition of the efficiency index EFGGL into a conventional (economic output) production index and an environmental index is not possible in the full input, output space (owing to the interaction between contractions of input vector x with contractions of both the economic output and the unintended-output vectors y and z). Thus, while adaption of the FGL index to account for slack in all directions is straightforward, the cost of this adaptation is the loss of the decomposition of overall efficiency measurement into production and environmental components. This loss will, of course, be consequential in some contexts and not in others.

Critiques and Suggested Modifications of the By-Production Structure Some recent critiques of the by-production structure are presented in the context of data envelopment analysis. Under the assumption of constant returns to scale and convexity of the technology, EF GL (x, y, z, TBP ; α) can be rewritten in DEA parlance as 52 EDEA (x, y, z, X, Y, Z; α) = α min

θ ∈(0,1],λ

θ Xo λ ≤ xo ;

+ (1 − α)

50 Contraction

min

γ ∈(0,1],μ

 y ; λ ≥ 0U θ  Zμ ≤ γ z; μ ≥ 0U ,

Xz λ ≤ xz ;

γ Xz μ ≥ xz ;

Yλ ≥

of effluent-generating input quantities (lowering the components of the vector δz ) paradoxically moves xz , z away from the frontier in its ambient subspace, but under reasonable assumptions on the technology, reductions in these quantities will be bounded from below by the constraints in the x, y subspace. Thus, the effective constraints in the xz , z subspace are the lower bounds on the pollution variables. 51 Note that, in this program, removal of slacks in the input direction leads to production vectors with the same amounts of all inputs in the two sub-technologies T1 and T2 . This important point has been made by Ray, Mukherjee, and Venkatesh [47]. 52 See p. 133 in MRL.

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where Xo , Xz , Y , and Z are no × U , nz × U , 1 × U , and 1 × U data matrices of non-emission-causing input quantities, emission-causing input quantities, intended output quantities, and emission levels, and U is the number of decision-making units (DMUs) in the dataset. Here, we have assumed that m = m = 1. This optimization problem is clearly equivalent to EDEA (x, y, z, X, Y, Z; α) =

min

θ,γ ∈(0,1]2 ,λ,μ

αθ + (1 − α)γ

subject to Xo λ ≤ xo ;

Xz λ ≤ xz ;

Xz μ ≥ xz ;

Zμ ≤ γ z;

y ; θ μ ≥ 0U . Yλ ≥

λ ≥ 0U (28)

A Missing Constraint? Dakpo et al. [16] add an additional constraint, Xz λ = Xz μ, to the optimization problem (28). They argue that the absence of such a restriction can lead to inconsistencies in the emission-causing input levels in the benchmark/reference points computed by the program for the two sub-technologies. We argue below that this is not so. Let θ BP , γ BP , λBP , μBP  solve problem (28). Then convexity and constant return to scale of the constructed DEA technology, independence of sub-technology T1 from emission, and independence of T2 from intended production and use of non-emission-causing inputs together imply Xo λBP , Xz λBP ,

y , γ BP z ∈ T1 θ BP

and Xo λBP , Xz μBP ,

y , γ BP z ∈ T2 . θ BP

It is possible (as Dakpo et al. argue) that the former production vector does not lie in sub-technology T2 and/or the latter production vector does not lie in sub-technology T1 . But since the solution satisfies constraints of problem (28), we also have Xz λBP ≤ xz

and

XzBP μ ≥ xz .

Free disposability of T1 in all inputs and costly disposability of T2 in emissiony causing inputs then imply that the production vector xo , xz , θ BP , γ BP z belongs to both sub-technologies T1 and T2 . Thus, although it is possible that Xz λBP = Xz μBP at the optimum of problem (28), we nevertheless always have xo , xz ,

y , γ BP z ∈ T1 ∩ T2 = TBP . θ BP

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The case Xz λBP = Xz μBP is simply indicative of slacks in the use of emissioncausing inputs at the optimum: Xz λBP < xz (indicating slack in T 1 ) or Xz μBP > xz (indicating slack in T 2 ).

Conflicting Efficiency Improvements in T 1 and T 2 ? Dakpo [15] raises some issues regarding the removal of slack in the input direction in the particular context of a BP technology (as in ((27))). Given that the intendedoutput technology T1 satisfies standard free disposability with respect to the inputs and the economic outputs, input efficiency improvements in intended-output production entail reductions in the quantities of inputs with no reductions in the amounts of good outputs produced. On the other hand, features of the emission generation set T2 –namely, costly disposability in the directions of both emissions and emission-causing inputs – suggest that, starting from an inefficient (e.g., an interior) point of T2 , it is possible to increase the use of the emission-causing inputs without increasing the emission levels. Hence, according to Dakpo, efficiency improvements in the direction of inputs with respect to the sub-technologies T1 and T2 have conflicting implications, as they involve decreasing the use of inputs with respect to T1 and increasing the use of inputs in the context of T2 . It is also for such a reason that Lozano [36] restricts efficiency improvements in the input direction to only non-emission-causing inputs. The expressed rational for this restriction is that the set T2 is assumed to be independent of non-emission-causing inputs so that, holding the use of emission-causing inputs fixed, efficiency improvements boil down to standard reductions in the use of non-emission-causing inputs, increases in the production of the good outputs, and reductions in generations of emissions. Despite the concerns raised by Dakpo, the general definition of economic efficiency, as spelled out in ((27)), is unambiguous about the input reductions that imply efficiency improvement. Scarcity of all productive inputs (both emissiongenerating and non-emission generating) implies that efficiency improvements entail reducing wastage (removing slacks) in any input direction. Thus, in the context of the overall BP technology–an intersection of the sets T1 and T2 –efficiency improvements involve reduced use of any input without decreasing the production of the good outputs or increasing the generation of emissions. An Overall Intensity Factor? Ray, Mukherjee, and Venkatesh [47] distinguish between a unified and a decentralized (by-production) approach in their DEA constructions of pollution-generating technologies. The latter approach involves construction of two sub-technologies, T1 and T2 , from data on DMUs using two distinct sets of intensity vectors, one for each sub-technology. The objective is to capture the distinct sets of production relations satisfied by sub-technologies T1 and T2 . The unified DEA approach, on the other hand, uses only a single intensity vector to construct an overall technology satisfying (i) standard free disposability of good outputs and non-emission-causing inputs and (ii) weak disposability of emissions and emission-causing inputs:

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xz , xz , y, z ∈ T ∧ λ ∈ [0, 1] =⇒ xo , λxz , y, λz ∈ T . Specifically, the Ray et al. technology is U 

TRMV = xo , xz , y, z ∈ Rt+ λj yj ≥ y; j =1

α

U 

λj zj = z;

j =1



λj = 1;

U 

j

λj xo ≤ xo ;

j =1

α

U 

j

λj xz = xz ;

j =1

0 ≤ α ≤ 1;

 λj ≥ 0 ∀ j = 1, . . . , U .

(29)

j

As pointed out in section “Axiomatic Approach to Modeling Emission-Generating Technologies,” a BP technology is equivalent to an overall emission-generating technology satisfying axioms (EG0), (EG1), and (EG2). In particular, (EG1) and (EG2) imply conditional free disposability and conditional costly disposability of emission-causing inputs. It is possible that the unified approach of Ray et al. could violate these axioms. Specifically, weak disposability of emissions and emission-causing inputs could be in conflict with conditional costly disposability of the pollution-generating inputs. Intuitively, weak disposability of emissions and emission-causing inputs implies that, holding the quantities of all good outputs and non-emission-causing inputs fixed, radial contractions of emissions and emissioncausing inputs are points in the technology. Realistically, however, this may not be possible, as reductions in emission-causing inputs, such as fossil fuels, may require a decrease in the good outputs produced when all other inputs are held fixed. The Ray et al. DEA technology in (29) is predicated on the use of “two different intensity vectors” λ and μ (both in RU ), in MRL’s DEA construction of the byproduction technology. They argue that, “[w]hen separate intensity vectors are used, the peer group for good output production may (and in many cases will) be different from the one for bad output production.” In fact, the programs (6.1) and (6.2) on page 130 of MRL generate a single reference point, y θ  , β  ⊗z, where θ  solves (6.1), and β  solves (6.2). (Note that the program (6.1) is independent of z, and the program (6.2) is independent of y.) Alternatively, in the notation of this chapter (see Eq. (28)), the programs generate a unique reference point, y/θ BP , γ BP z, held fixed at xo , xz . Thus, the reference production vector is

when inputsBPare BP xo , xz , y/θ , γ z , which lies in TBP = T1 ∩ T2 . This construction allows for the possibility that a firm using xo , xz  amounts of inputs is efficient with respect to the intended production sub-technology T1 but not with respect to the environmental sub-technology T2 , or vice versa, or is inefficient with respect to both sub-technologies.

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Concluding Remarks: The Material Balance Condition Throughout this chapter, we have made frequent reference to the material balance condition, a physical relationship that must hold for all production processes. This condition, embedded in the first law of thermodynamics, intuitively states that matter cannot be destroyed and hence that the mass of all material inputs must equal the mass of all material outputs produced. This law on the preservation of mass energy was introduced into economics in the seminal work of Ayres and Kneese [3]. They employed this principle to account for wastes generated at the macroeconomic level based on the knowledge of the masses of material inputs employed by the economy and the economic outputs produced. The roles played by both the first and the second (entropy) laws of thermodynamics in the generation of emissions as an inevitable consequence of production activities have been more comprehensively discussed in subsequent papers by Baumgärtner and de Swaan Aron [6] and Baumgärtner [5]. We have endeavored to present in this chapter multi-relation models of pollutiongenerating technologies that are consistent with both of the physical laws of thermodynamics. There exists, however, a microeconomic literature that aims at explicit incorporation of the first law into the specification of the technology. Especially noteworthy are the papers by Pethig [46], Coelli, Lawers and Van Huylenbroeck [12] Chambers and Melkonyan [10], Hampf [31], and Rodseth [49, 50]. Essentially, these papers introduce a material balance identity along the following lines into the model of the production process: α · xz = β · y + γ · z,

α, β, γ > 0,

(30)

where the coefficient vectors α, β,, and γ convert input and (intended and unintended) output flows into common mass units. Many of these works also demonstrate that material balance conditions are generally violated in the conventional input and output approaches to modeling emission-generating technologies. While the material balance condition–a physical law–must hold at both the macro and the micro level, MRL and Försund (2017) discuss some concerns that may arise when it is directly employed to quantify generation of emissions at a micro level. In particular, the accounting nature of this condition accurately measures the amounts of wastes generated only if the researcher has full information about all the inputs (economic and noneconomic) used and the full set of outputs (good and bad) produced. This seems possible only if the production process is a completely closed system, which in turn requires no leakage of some unaccounted-for effluents or inputs. In the space of observable and deducible variables, we cannot expect the material balance condition to hold as an equality when unobservable variables are not available to complete the balance. For example, one of the most important forms of matter in the universe is oxygen, which is an input in many industrial processes and is difficult to account for in the specification of the technology. As another example, if only some of the wastes generated during production are policy relevant,

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the researcher may not find it worthwhile to collect data on the remaining wastes (or noneconomic outputs), especially if these wastes are not directly observable. For these and other reasons, we think the research on the incorporation of the material balance condition into models of pollution-generating technologies is only in its formative stage and not yet ready for synthesizing. This inchoate feature of the research suggests ample opportunities for researchers interested in contributing to the development of models featuring this condition. We recommend that they start with careful review of the works cited above.

Cross-References  Data Envelopment Analysis: A Nonparametric Method of Production Analysis  Distance Functions in Production Economics  Economics of Externalities: An Overview  Multiproduct Technologies

References 1. Aparicio J, Barbero J, Kapelko M, Pastor JT, Zofio JL (2017) Testing the consistency and feasibility of the standard Malmquist–Luenberger index: environmental productivity in world air emissions. J Environ Manag 196:148–160 2. Ayres RU (1996) Eco-thermodynamics: economics and the second law. Ecol Econ 26:189–209 3. Ayres RU, Kneese AV (1969) Production, consumption and externalities. Am Econ Rev 59:282–297 4. Ball VE, Lovell CAK, Luu H, Nehring R (2004) Incorporating environmental impacts in the measurement of agricultural productivity growth. J Agric Resour Econ 29:436–460 5. Baumgärtner S (2012) Ambivalent joint production and the natural environment: an economic and thermodynamic analysis. Springer Science & Business Media, Berlin 6. Baumgärtner S, de Swaan Arons J (2003) Necessity and inefficiency in the generation of waste. J Ind Ecol 7:113–123 7. Baumol WJ, Oates WE (1975, 1988) The theory of environmental policy, 1st and 2nd edns. Cambridge University Press, Cambridge 8. Boyd GA, McClelland JD (1999) The impact of environmental constraints on productivity improvement in integrated paper plants. J Environ Econ Manag 38:121–142 9. Chambers RG (1988) Applied production analysis: a dual approach. Cambridge University Press, Cambridge 10. Chambers RG, Melkonyan T (2012) Production technologies, material balance, and the income-environmental quality trade-off. University of Exeter Working Paper 11. Chung YH, Färe R, Grosskopf S (1997) Productivity and undesirable outputs: a directional distance function approach. J Environ Manag 51:229–240 12. Coelli T, Lauwers L, Van Huylenbroeck GV (2007) Environmental efficiency measurement and the materials balance condition. J Prod Anal 28:3–12 13. Coggins JS, Swinton JR (1996) The price of pollution: a dual approach to valuing SO2 allowances. J Environ Econ Manag 30:58–72 14. Cropper ML, Oates WE (1992) Environmental economics: a survey. J Econ Lit 30:675–740 15. Dakpo KH (2015) On modeling pollution-generating technologies: a new formulation of the by-production approach. Paper presented at the 6th EAAE PhD Workshop, Rome Co-organized by AIEAA (Italian Association of Agricultural and Applied Economics) and the Department

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of Economics of Roma Tre University. http://prodinra.inra.fr/ft?id=1DE1A17F-F41C-41F9A853-305563DA93A9 16. Dakpo KH, Jeanneaux P, Latruffe L (2016) Modelling pollution-generating technologies in performance benchmarking: recent developments, limits and future prospects in the nonparametric framework. Eur J Oper Res 250:347–359 17. Färe R, Grosskopf S (2000) Network DEA. Socio Econ Plan Sci 34:35–49 18. Färe R, Grosskopf S, Lovell CAK (1985) The measurement of efficiency of production. Kluwer-Nijhoff, Boston 19. Färe R, Grosskopf S, Lovell CAK, Pasurka CA (1989) Multilateral productivity comparisons when some outputs are undesirable: a nonparametric approach. Rev Econ Stat 71:90–98 20. Färe R, Grosskopf S, Lovell CAK, Yaisawarng S (1993) Derivation of shadow prices for undesirable outputs: a distance function approach. Rev Econ Stat 75:374–380 21. Färe R, Grosskopf S, Noh D-W, Weber W (2005) Characteristics of a polluting technology: theory and practice. J Econ 126:469–492 22. Färe R, Grosskopf S, Pasurka C (1986) Effects of relative efficiency in electric power generation due to environmental controls. Resour Energy 8:167–184 23. Färe R, Grosskopf S, Pasurka C (2013) Joint production of good and bad outputs with a network application. In: Shogren J (ed) Encyclopedia of energy, natural resources, and environmental economics, vol 2. Elsevier, Amsterdam, pp 109–118 24. Färe R, Grosskopf S, Pasurka C (2007) Environmental production functions and environmental directional distance functions. Energy 32:1055–1066 25. Färe R, Lovell CAK (1978) Measuring the technical efficiency of production. J Econ Theory 91:150–162 26. Førsund F (1972) Allocation in space and environmental pollution. Swed J Econ 74:19–34 27. Førsund F (1998) Pollution modeling and multiple-output production theory. Discussion Paper #D-37/1998, Department of Economics and Social Sciences, Agricultural University of Norway 28. Førsund F (2009) Good modelling of bad outputs: pollution and multiple-output production. Int Rev Environ Resour Econ 3:1–38 29. Førsund F (2018) Multi-equation modeling of desirable and undesirable outputs satisfying the material balance. Empir Econ 54:67–99 30. Frisch R (1965) Theory of production. D. Reidel Publishing Company, Dordrecht 31. Hampf B (2014) Separating environmental efficiency into production and abatement efficiency: a nonparametric model with application to US power plants. J Prod Anal 41:457–473 32. Kohli U (1983) Non-joint technologies. Rev Econ Stud 50:209–219 33. Kumbhakar SC, Tsionas EG (2016) The good, the bad and the technology: endogeneity in environmental production models. J Econ 190:315–327 34. Laffont J-J (1998) Ch. 1 in Fundamentals of public economics, translated by Bonin JP and Bonin H, MIT Press, Cambridge, Massachusetts; London, England 35. Levkoff SB (2013) Efficiency trends in U.S. coal-fired energy production & the 1990 clean air act amendment: a nonparametric approach. Working paper, UC San Diego. Online version: http://stevelevkoff.com/uploads/Clean_Air_Act.pdf 36. Lozano SC (2015) A joint-inputs network DEA approach to production and pollutiongenerating technologies. Expert Syst Appl 42:7960–7968 37. Luenberger DG (1992) New optimality principles for economic efficiency and equilibrium. J Optim Appl 75:221–264 38. Murty MN, Kumar S (2002) Measuring cost of environmentally sustainable industrial development in India: a distance function approach. Environ Dev Econ 7:467–486 39. Murty MN, Kumar S (2003) Win-win opportunities and environmental regulation: testing of porter hypothesis for indian manufacturing industries. J Environ Manag 67:139–144 40. Murty S (2010) Externalities and fundamental nonconvexities: a reconciliation of approaches to general equilibrium externality modeling and implications for decentralization. J Econ Theory 145:331–353

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41. Murty S (2015) On the properties of an emission-generating technology and its parametric representation. Econ Theory 60:243–282 42. Murty S, Russell RR (2002) On modeling pollution-generating technologies. Department of Economics, University of California, Riverside, Discussion Papers Series, No 02-14 43. Murty S, Russell RR (2018) Modeling emission-generating technologies: reconciliation of axiomatic and by-production approaches. Empir Econ 54:7–30 44. Murty S, Russell RR, Levkoff SB (2012) On modeling pollution-generating technologies. J Environ Econ Manag 64:117–135 45. Njuki E, Bravo-Ureta BE (2015) The economic costs of environmental regulation in U.S. dairy farming: a directional distance function approach. Am J Agric Econ 97:1087–1106 46. Pethig R (2006) Non-linear production, abatement, pollution and materials balance reconsidered. J Environ Econ Manag 51:185–204 47. Ray SC, Mukherjee K, Venkatesh A (2018) Nonparametric measures of efficiency in the presence of undesirable outputs: a by-production approach with weak disposability. Empir Econ 54:31–65 48. Reinhard S, Lovell CAK, Thijssen GJ (1999) Econometric estimation of technical and environmental efficiency: an application to Dutch dairy farms. Am J Agric Econ 81:44–60 49. Rodseth KL (2016) Environmental efficiency measurement and the materials balance condition reconsidered. Eur J Oper Res 250:342–346 50. Rodseth KL (2017) Axioms of a polluting technology: a materials balance approach. Environ Resour Econ 67:1–22 51. Russell RR, Schworm W (2011) Properties of inefficiency indexes on input, output space. J Prod Anal 36:143–156 52. Samuelson P (1948) Economics. McGraw-Hill [19th Edition (2009) by Samuelson P, Nordhaus WD. McGraw-Hill, Irwin 53. Serra T, Chambers RG, Lansink AO (2016) Measuring technical and environmental efficiency in a state-contingent technology. Eur J Oper Res 236:706–717 54. Srivastava RK, Jozewicz W (2001) Flue gas desulphurization: the state of the art. J Waste Air Manag Assoc 51:1676–1688 55. Vardanyan M, Noh D-W (2006) Approximating pollution abatement costs via alternative specifications of a multi-output production technology: a case of us electric utility industry. J Environ Manag 80:177–190 56. Zhou P, Ang B, Poh K-L (2008) A survey of data envelopment analysis in energy and environmental studies. Eur J Oper Res 189:1–18

Market Structures in Production Economics

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Devin Garcia, Levent Kutlu, and Robin C. Sickles

Contents Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The Structure-Conduct-Performance Paradigm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The Bounds Approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Commonly Used Basic Market Structures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Conduct Parameter Approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Dynamic Market Structures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Market Structures with Differentiated Products . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Market Structure and Market Power . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Market Structure and Innovation Studies with No Explicit Treatment for Distorted Production Decisions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Theoretical Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Empirical Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Cross-References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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D. Garcia Ernst and Young, LLP, Houston, TX, USA e-mail: [email protected] L. Kutlu Department of Economics and Finance, University of Texas Rio Grande Valley, Edinburg, TX, USA e-mail: [email protected] R. C. Sickles () Department of Economics, Rice University, Houston, TX, USA e-mail: [email protected] © Springer Nature Singapore Pte Ltd. 2022 S. C. Ray et al. (eds.), Handbook of Production Economics, https://doi.org/10.1007/978-981-10-3455-8_4

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Abstract

Our chapter begins by discussing the structure-conduct-performance (SCP) paradigm, which is an early descriptive literature that provided many of the stylized facts about market behaviors. This is followed by a discussion of the bounds approach, which concentrates on making predictions that can hold across a broad range of industries and is achieved by aiming conclusions based on minimal assumptions. We then briefly talk about commonly used fundamental market structures and illustrate how different combinations of various standard concepts are combined to describe market structures. As dynamic en and markets with product differentiation play important roles in defining market structures, we provide additional information about models with dynamic environments and models with product differentiation in separate sections. We finalize our review by discussing in depth the literature on market structure on innovation, productivity slowdowns and related problems of income inequality. Keywords

Structure-conduct-performance paradigm · Structural models · Dynamics · Innovation

Introduction One of the important purposes of industrial organization is to understand the environments, i.e., the market structures, in which firms interact with each other, their customers, and potential entrants as well as the implications of these environments for market outcomes. Market structure, among other factors, is characterized by the number of firms in the market, the concentration of the market, the technology, the nature of the products, and the presence of information asymmetry between firms and customers.1 These factors help describing the market structure and examining how it relates to firm conduct, firm efficiency, and market performance. Issues that are taken up by this literature involve whether collusion and higher prices are facilitated in more concentrated markets and whether market concentration affects firms’ research and development (R&D) or other types of investments. Among other benefits, an understanding of such relationships guides the policy makers to determine whether regulatory actions are needed and, if needed, how particular regulations may be implemented. Often, economic policies are implemented at a level that affects a wide range of industries, which requires economists to develop tools that apply in a broad range of settings and industries. However, it is also essential to understand industry-specific differences and industry- or even

1 Some

of these factors may be endogenous, which we will talk about later in the chapter.

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firm-specific predictions as in merger cases. Therefore, not surprisingly, economists have developed tools that can be useful in both these settings. The structure-conduct-performance (SCP) paradigm is an early descriptive literature that provided many of the stylized facts about market behaviors. The bounds approach concentrates on making predictions that can hold across a broad range of industries. This is achieved by aiming conclusions based on minimal assumptions. On the other hand, structural methods rely on game theoretic modeling of structure and generally focus on specific industries. Since the focus is a specific industry, stronger assumptions that are compatible with the industry-specific properties are acceptable. This chapter begins with a discussion of the SCP paradigm and related literature, which is followed by a description and literature review on the bounds approach. Then, the chapter briefly talks about commonly used fundamental market structures. The section illustrates how different combinations of various standard concepts are combined to describe market structures. At the end of this section, additional potential weaknesses of the SCP approach is briefly discussed. This section is followed with a description and brief literature review of the conduct parameter approach, which is mainly used in the market power literature. As dynamic environments and markets with product differentiation play important roles in defining market structures, additional information about models with dynamic environments and models with product differentiation are presented in separate sections. The review is finalized by discussing in depth the literature on market structure on innovation, borrowing from perspectives of Schumpeter. Finally, as innovation plays a major role in the ongoing public policy debates about productivity slowdowns and the related problem of income inequality, this seems particularly relevant to a Handbook Chapter that addresses market structure in a Handbook of Production Economics. The last section concludes.

The Structure-Conduct-Performance Paradigm The structure-conduct-performance paradigm proposes a one-way causation path that starts with market structure’s effect on firm conduct, followed by the effect of firm conduct on market performance. As mentioned in the introduction, market structure is a function of number of factors, such as the number of firms in a market, the concentration of a market, technology, the nature of products, and the presence of information asymmetry between firms and customers. A general approach taken by SCP studies is describing market structure through market concentration. It is argued by SCP that in highly concentrated markets, the firm’s conduct is likely to be more collusive, which would lead to higher prices and thus higher profits. A potential reason for high concentration is entry barriers. One particular entry barrier is the presence of scale economies. Bain [7] argues that the need of a firm to be large in order to obtain productive efficiency creates an entry barrier. Bain [7] showed that there is a significant correlation between scale economies and market concentration. This result is consistent with the idea that scale economies

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facilitate entry barriers. The follow-up studies by Bain and others showed that scale economies are not always the only driving force for high market concentration. For example, in markets with intensive investment in advertisement and/or R&D, it is possible to observe high market concentrations. In the SCP literature, the hypothesis that increased concentration leads to higher prices is strongly supported by both empirical and theoretical studies. Hence, it is argued that concentration is bad for consumers, which historically paved the way for further antitrust legislation. However, a similar strong relation could not be deduced for market concentration and profitability. In particular, the empirical results for the relationship between market concentration and profitability are mixed. One difficulty in examining this relationship stems from measurement and interpretation issues [31]. Another difficulty is that cross-sectional data used in these studies frequently come from different industries. This is problematic due to the fact that demand and supply conditions in these industries may be substantially different. The final criticism for SCP studies is that market structure is assumed to be exogenous. Demsetz [27] argues that positive correlation between market concentration and profits may be due to efficiency differences of the firms. That is, those firms with superior efficiencies can gain market share and this may lead to higher market concentration levels. Weiss [113] argues that the studies that focused on concentration and price do not face all the criticisms that market concentration and profit studies receive. For example, many of the price-concentration studies use data across specific local markets within an industry. However, as pointed out by both Bresnahan [17] and Schmalensee [91], many of the price-concentration studies suffer from serious endogeneity issues. More precisely, the unobserved demand and supply shocks in a market may not only affect prices but also the market structure (e.g., entry barriers may be affected through changes in costs).

The Bounds Approach The underlying idea of bounds approach, developed by Sutton2 [104, 105], is identifying strong mechanisms that can characterize the market outcomes across a broad range of environments. Hence, the approach aims to make as few assumptions as possible about market structure to generate general testable predictions. Ideally, these assumptions should be based on strongly confirmed empirical regulations to reach a reasonable conclusion. This approach deviates from the notion of a fully specified model, which leads to an (unique) equilibrium outcome. Different models that have been proposed in the literature lead to different equilibrium outcomes, and the bounds approach aims to find the bounds for these outcomes. Therefore, rather than providing a single equilibrium outcome, the approach provides a set of equilibrium outcomes that are reasonable. Below we provide the details of this

2 These

papers are inspired by an idea introduced by Shaked and Sutton [98].

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approach. Sutton [106] and Ellickson [30] are other studies that provide summaries of this approach. In order to give some idea about the bounds approach, we give a simple example. We borrow this simple version of bounds approach from Sutton [106], which involves a two-stage game. There are N0 (≥2) firms with constant marginal cost, c, and they produce a homogenous good. In the first stage, the strategic and forward-looking firms decide about entry and in the second stage they compete for customers. The payoff of a firm equals the profit in the second stage minus the sunk cost of entry, e. All consumers have the same utility given by U = xα z1−α where x is the focus of the study and z is the outside good. Assume that there is a (high) price p0 so that if p > p0 the consumers do not by x. The demand schedule is given by X = S/p where p is the market price, S is the total number of expenditures on x, X is the total quantity sold by all firms. Sutton [106] characterizes the equilibrium as a perfect Nash equilibrium of the game. Let N denote the number of entrants in the first stage, and we solve the symmetric Cournot game in the second stage. As expected, the price falls as N increases and eventually converges to c. Also, the equilibrium profit in the second stage is π = S/N2 . This enables us to solve the number of entrants by setting π ≥ e. It turns out that as S increases, the 1-firm concentration ratio, C1 = 1/N monotonically converges to zero and output per firm increases. Finally, Sutton [106] notes that the number of firms increases less than proportionally with S.3 One of the key differences in this setting from SCP is that entry is endogenous. Two fundamental assumptions for the equilibria are that they must satisfy stability and viability. Stability assumes that any firm that chooses to not enter expects negative profits if they enter the market. Viability assumes that any firm that chooses to enter the market expects nonnegative profits if they enter the market. In general, the bounds approach leads to different conclusions depending on whether advertising and R&D plays an important role in the market. Sutton shows that when advertising and R&D are relatively less important for an industry, the market size negatively affects the minimum level of concentration. In other words, the lower bound for concentration decreases as the market grows larger. This result, however, does not imply a functional relationship between concentration and market size. That is, the result does not rule out the possibility of a positive relationship between market size and concentration. What Sutton finds is a relationship between the lower bound of concentration and market size. The vagueness of this result is an artifact of general assumptions applied. Another related question that is answered by Sutton is the relationship between the strength of competition and the lower bound for concentration. He finds that, for a given market size, an intensification of price competition will shift the lower bound of concentration upwards. Sutton [106] argues that there are two candidate mechanisms that may lead to this increase in concentration. Some firms may exit and/or consolidate via mergers and acquisitions. An important policy implication

3 For

additional examples, see Sutton [106].

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of this finding is that regulations which aim to increase competition may indeed lead to market structures that are more concentrated. Of course, as explained earlier, the bounds approach does not provide functional relationships and thus this result does not necessarily imply that an increase in competition would lead to more concentrated market structures. For industries where advertising and R&D are relatively more essential, Sutton finds that the share of the largest firm remains positive as the market size grows. Hence, the concentration is bounded below. This result contrasts with the findings for industries where advertising and R&D are relatively less crucial. In these industries, larger markets are still more profitable than small ones, but also a larger market size increases the incentives to increase the amount of fixed investments. This in turn prevents market concentration from falling indefinitely. These results explain why in some industries firms have incentives that result in a growing amount of unprofitable sunk investments. One interesting yet difficult question is whether the results of standard multistage models are carried out to the dynamic games framework. Sutton [105] (Chap. 13, “Technology and Market Structure Revisited”) presents a dynamic game in this context. Sutton [106] argues that the multistage game framework excludes certain kinds of equilibria that may arise in a dynamic framework, which is not surprising given the richer essence of dynamic games. Sutton explains that this leads to the appearance of what he refers to as “underinvestment equilibria.” In a dynamic setting, firms may underspend on R&D. This is because of strategies that result in a firm increasing R&D spending at time t + 1 after their rival has increased its R&D spending at time t. Nocke [76] shows that this kind of equilibrium may happen in dynamic games in which the firms can react to rivals arbitrarily quickly. As a consequence, the lower bound to concentration reduces relative to the multistage counterpart of a dynamic game. Symeonidis [108, 109] tests some of the important predictions of Sutton via systematic tests that take advantage of a natural experiment, namely a change in British competition law that took place in the 1960s. It turns out that as the laws strengthened, market concentrations generally increased in manufacturing industries. Ellickson [29] provides empirical support from the US supermarket industry. The US grocery industry is not very concentrated at the national level and even less so at the city level (generally dominated by 3 or 4 chain firms). Ellickson [29] argues that while R&D and advertising play relatively small roles, supermarkets invest competitively in information technology to increase product variety. As a result of these investments, the number of local firms is limited. Marin and Siotis [70] examine Sutton’s results in the chemicals industry. The advantage of this study is that the chemicals industry is comprised of many distinct markets in R&D intensity. Moreover, except for pharmaceuticals, advertising expenditures are low. Hence, this allowed them to concentrate on R&D aspects. As in many market structure studies, this study encountered the difficulty of defining the markets and products. Their results support the findings in Sutton [105] that the lower bound on market concentration is higher and increasing in product concentration where markets have relatively high R&D intensities.

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We finalize this section by pointing out that although the bounds approach of Sutton is based on minimal assumptions, the key predictions that are described above depend critically on some of these assumptions. In particular, the assumption that the game form is exogenous to the model may be challenged. That is, the assumption that the form of the game remains unchanged since the market size changes may not be plausible in every context.

Commonly Used Basic Market Structures The structure-conduct-performance paradigm generally uses market concentration measures to describe market structure. However, some of the commonly used economic models provide more precise descriptions about market structures. In this section, we present a very brief summary of these fundamental market structures that constitute the core of more detailed and potentially more technical market structure representations. The market environment may depend on the nature of products (e.g., homogenous vs. differentiated products), the choice of strategy variables (e.g., quantity vs. price), timing of the strategic decisions, presence of conditions that allow price discrimination or not, etc. The simplest market structure models are given in the context of homogenous goods, single price, and static environments. One widely used market structure type is perfect competition, which is used as a benchmark model in many settings. In this market structure type, a firm is said to be competitive if it believes that it cannot affect the market price. The importance of this market structure type is that the first-best welfare outcome is achieved. Under this framework, firms do not find a room for strategic interactions. Therefore, the firm’s residual demand curve is flat. At the other extreme, another benchmark market structure type is monopoly where there is only one producer/seller for a product(s). It is possible to add some more structure to this market structure type by providing more details about the costs and/or technology, whether the monopolist can and is willing to price discriminate by some specific price discrimination mechanism or not, the nature of the product(s), etc. The base version of monopoly sets one price and sells a perishable good. If average costs are declining over all meaningful quantity ranges, the most efficient outcome would be a single firm to produce all output (e.g., public utilities). This version of monopoly is said to be a natural monopoly. In general, this happens in markets where the fixed costs are very high and marginal cost is relatively low. In some market environments, the monopolist may find ways to implement some form of price discrimination . The essential ingredient for price discrimination is the ability to identify the customer’s types (e.g., their valuations) and ability to prevent arbitrage opportunities. Broadly, there are three types of price discrimination. Firstdegree price discrimination (personalized pricing) happens when the firm has perfect knowledge of the valuations of the customers. In this framework, the firm charges different prices for each customer based on their valuations. Total welfare in this setting is the same as in perfect competition. The distinction is that all the

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welfare goes to the monopolist. While initially this type of price discrimination seemed to be an abstract concept that is hard to find in real world scenarios, recent advancements in machine learning algorithms and big data make this type of price discrimination in some market structures closer to reality than fiction. With second-degree price discrimination (menu pricing), the firm creates slightly different products (e.g., different amounts of the same product) for the purpose of price differentiation. Finally, third-degree price discrimination (group pricing) divides the market in segments and charges everyone the same price according to segment. The reason for such segmentation is that the firm may not have the tools to identify individual valuations of the customers. Hence, the firm rather identifies groups in an attempt to get a heuristic approximation that simplifies the identification problem. Two obvious examples of third-degree price discrimination are student discounts and geographical price discrimination. Armstrong [5] and Stole [102] survey this literature and provide details of advancements in the price discrimination literature. If a product is a durable good rather than perishable good, this would have serious implications for the monopolist. One important implication of market structures with durable goods is that the monopolist needs to consider the future. In a typical market with perishable goods, it is possible to use static models; however, this is inappropriate with durable goods. Coase’s conjecture claims that if consumers do not discount time heavily and expect the price to fall in the future then this would affect the current demand negatively. This in turn forces the monopolist to charge a lower price compared to what it would charge for a perishable good. Some examples that may solve consumers’ lower future price expectation problem include introducing capacity constraints, announcing future prices, or renting. Whether this conjecture holds or not is discussed by many researchers and has led to a large literature [32]. So far the market structure examples that we provided were either perfect competition or monopoly. Cases that are more interesting can be explored in market structures with imperfect competition. One important distinction of monopoly and imperfect competition models is that for the latter, in general, the choice variable plays a more important role. That is, in imperfect competition models the outcomes may change substantially depending on whether the firms choose quantity or price. Similarly, the timing of a firm’s actions plays an important role, e.g., simultaneous versus sequential. We first start with the imperfectly competitive market structures where the firms choose quantities. Cournot competition assumes that there are multiple firms in a homogenous product market that choose quantities to maximize profit. The equilibrium output is determined such that no firm can increase its profits by changing its output level if other firms produce Cournot output levels. That is, firms maximize their profits in accordance with a Nash equilibrium. A characteristic of this market structure is that an increase in another firm’s output leads the firm to produce less. In other words, the best response functions are downwards sloping. When the number of firms is one, this market structure coincides with the (based)

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monopoly structure. When the number of firms goes to infinity, the outcome of this market structure converges to that of perfect competition. In the Cournot competition model, firms act simultaneously. If one firm has some advantage to enable it to move first, then this would change the market outcomes and lead to another market structure known as Stackelberg competition. The simplest version of this model assumes that there are two firms in the market that compete on quantity: the leader and the follower. The leader moves first and after observing the leaders action the follower moves. It is assumed that the leader knows ex ante that the followers observe their actions. Moreover, the follower must not have the ability to commit to a future non-Stackelberg follower action, and this has to be known by the leader. One reason why a leader might have a first-mover advantage is that it may be the incumbent monopolist in the industry, and the follower is a potential new entrant. Hence, the Stackelberg competition model and its variations play an important role in describing markets that face potential entrants.4 Similar to monopoly setting, it is possible to add more details about the market structure to the quantity-competition-based models. For instance, Stole [102] gives an example of third-degree price discrimination in a quantity choice setting. Similarly, Hazledine [47, 48] and Kutlu [62] incorporate a form of second-degree price discrimination to the Cournot competition framework. They show that, under linear and some nonlinear demand functions, the quantity-weighted average price does not depend on the extent of price discrimination. Kutlu [61] incorporates the same price discrimination type to the Stackelberg competition framework where the demand is linear and costs are symmetric. He shows that the leader does not use price discrimination but the follower does. The leader directs all of its first mover advantage to attract the highest value consumers. Whether firms actually set quantities or not depends on the particularities of relevant markets. Bertrand competition assumes that the firms set prices rather than quantities. In a homogenous goods market with Bertrand competition, the consumer is assumed to buy from the firm with the lowest price. When the firms charge the same price, we need a sharing rule. A sensible sharing rule is equally dividing the demand among those firms that charge the smallest price. Under these conditions, if the marginal costs are the same for the firms, the market price reduces to the marginal cost. This result is paradoxical because even with two firms the market price becomes the competitive price. Based on the empirical evidence in the markets with small number of firms the price–cost markup is positive in general. In the literature, this result is referred to as the Bertrand Paradox. Hence, when firms choose prices, the market outcome is materially different compared to Cournot competition. When capacity constraints are introduced to the price choice framework, it is possible to achieve outcomes like Cournot competition [59]. In particular, Kreps

4 In other contexts, a seminal work on static entry models is Bresnahan and Reiss [18], which examines the strategic entry decisions of small retail firms.

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and Scheinkman [59] show that it is possible to get Cournot-like outcomes if the firms first choose capacity and then prices. This suggests a potential solution to the Bertrand Paradox. However, their result critically depends on the rationing rule that is used. A technical aspect of Bertrand competition is that the equilibrium may not exist if the marginal costs are not constant. This leads to alternative models such as the supply function equilibrium of Klemperer and Meyer [57]. One advantage of the supply function equilibrium approach compared to Bertrand and Cournot games is that in this setting a firm adjusts to the uncertainty in an optimal way given the other firms’ behavior. Until now, we have concentrated on market structures with homogenous goods. However, there are many markets where such an assumption may not be realistic. That is, with rare exceptions, the goods vary in features, quality, location, etc. Also, although Kreps and Scheinkman [59] provide a plausible solution to the Bertrand Paradox, a more applicable solution is to introduce differentiated products. Product differentiation allows the consumers to choose the product variations close to their tastes. The greater the variety of the products sold in the market, the more likely consumers would find a better match to their ideal preferences. Hence, product differentiation helps the firm to increase the potential amount of goods that they can sell. However, product differentiation comes with the potential cost that firm(s) may have to worry about determining the optimal degree of product differentiation. There are two ways to achieve product differentiation. In vertical differentiation, all consumers have the same preference ranking of the products given that they are charged the same price. In horizontal differentiation, consumers do not have the same preference rankings of the products even when the products are charged the same price. In the vertical differentiation setting, consumers may rank the products, say, based on quality. With horizontal differentiation, differences of the products may be due to factors other than quality such as the color, etc. In the literature, markets with vertical differentiation are generally examined by models that account for quality differences of the product variations (e.g., [33, 34, 96, 97]). A common way to model horizontal differentiation is using the variations of linear city model of Hotelling [52] and circular city model of Salop [88]. In these models, the distinct preferences of consumers for the products are modelled through a parameter called transportation cost. If a variation of the product is not close to a consumer’s taste, then the transportation cost of this individual would be high for this product variation. Many of the current empirical studies choose to model horizontal product differentiation via discrete choice models. In these models, it is important that consumers have heterogeneous preferences and choose only one product (out of a set of finitely many products) to ensure a smooth aggregate demand curve, which seems to be a requirement that is naturally satisfied in most cases. Later, we will provide a more detailed discussion for these types of models. As we discussed in the SCP section, market structures may be described by market concentration measures such as four-firm concentration measures or Herfindahl–Hirschman Index (HHI) . For example, the Department of Justice Merger Guidelines uses HHI in their merger evaluation analysis. One problem

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using concentration measures is that it may be difficult to come up with a proper market definition where the market shares are calculated. Another difficulty is that market shares are highly imperfect at describing market dynamics. Thus, they do not give us a complete or precise description of the market structure. For example, two homogenous product industries with the same market concentration measures may have very different characteristics. In particular, in one industry the market structure may be described with Cournot competition and in the other industry it may be Bertrand competition. Similarly, in one industry the market structure may be described by uniform-pricing firms, and in the other industry the market structure may be described by price discriminating firms. Hence, using HHI as a measure of competition in a cross-industry study may not always be the best way to proceed. In such cases, the models that describe the market by more precise structural information can be used as alternative options. In the next section, we consider the conduct parameter (or conjectural variations) approach that enables researchers to model market structures by “generalizing” some of the equilibrium concepts that are used in the commonly used models described above.

Conduct Parameter Approach An exhaustive survey of the conduct parameter approach along with technical details and some empirical examples is provided by Perloff et al. [78]. Bresnahan [17] is another study that provides a detailed summary of this approach. Some of the earlier works that use this approach are Gollop and Roberts [40], Iwata [53], Appelbaum [4], Porter [81], and Spiller and Favaro [100]. As mentioned earlier, the focus of the SCP literature is the cross-sectional study of many industries. The conduct parameter approach, on the other hand, concentrates on a single industry in an attempt to estimate a conduct parameter that characterizes firm behavior. Hence, conduct parameter models use economic theory to guide the empirical model specification while concentrating on a single industry. These empirical models rely on the theory of conjectural variations to estimate conduct parameters, which is mainly used as a measure of market power. Based on the conjectural variations interpretation, the conduct parameter measures the market power of firms in a market in a fairly general way by allowing equilibrium outcomes that may not be supported by the standard equilibrium concepts such as Nash equilibrium. For example, in the standard Cournot model, the conjecture is that the firms will have zero reaction; yet conjectural variations theory allows more general types of reactions. Basically, the conjectural variations of the firms determine the slopes of their reaction functions. Hence, similar to the common models that we described, the researchers may add some structure (e.g., capacity constraints, dynamic factors, price discrimination, etc.) to the model that describes the market structure in a market; but at the same time, the researcher may also be agnostic about the firms’ competitive behavior, i.e., the firm conduct, and estimate it using the available economic data.

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For example, Puller [83] incorporates capacity constraints; Puller [82] and Kutlu and Sickles [63] incorporate dynamic strategic factors; Graddy [42] and Kutlu and Sickles [64] incorporate price discrimination in their conduct parameter models. Corts [25] and Kutlu and Sickles [64] state that in the language of conjectural variations theory, the conduct is described in terms of firms’ conjectural variations, which are their expectations about other firms’ reactions. Conduct parameter models use the conduct parameters to represent the conjectural variations of the firms. Based on this interpretation, the conduct parameter can take a continuum of values. It is important to note that the conjectures do not refer to what firms believe will happen if they change their quantity levels. Rather, what is being estimated is what firms would do because of their expectations. Hence, as argued by Corts [25], the conduct parameter can be estimated “as if” the firms are playing a conjectural variations game, which would reveal the price-cost margins. However, some researchers may not be comfortable with the idea of a conjectural variation that allows nonstandard equilibrium outcomes, e.g., equilibria other than the Bertrand, Cournot, collusion, etc. Although some other researchers have argued that the folk theorem allows a range of conduct parameter values that are consistent with Nash equilibrium, they could not make a strong case against critics that question the validity of using a static model to represent a dynamic game. Hence, some researchers prefer to view conduct as a parameter that can take values consistent with existing theories, which would be estimated in the conduct parameter model. That is, the estimated conduct parameter value can be used to categorize the competitive behavior of firms by using statistical tests. In many cases, the researcher would face more than two alternative models to pick from. For example, the researcher may need to test whether the market outcome is consistent with perfect competition, Cournot competition, or monopoly. Therefore, nonnested hypothesis tests (e.g., [112]) can be used as in Gasmi et al. [37]. Due to its simplicity, some researchers may prefer to choose a compatible model by making pairwise comparisons using the standard statistical tests similar to pairwise model comparisons done in Bresnahan [16]. Besides the interpretation-related criticisms, the standard conduct parameter models that do not incorporate the dynamics specifically are intrinsically static. Corts [25] argues and illustrates by an example that this may lead to severely mismeasured market power estimates. Puller [83] and Kutlu and Sickles [63] present conduct parameter models that are robust to the criticism of Corts [25]. That is, they offer general empirical models that allow the consistent estimation of the parameters of the model (including the conduct parameter) that is robust to efficient collusion. As argued by Puller [83], while these models are robust to efficient tacit collusion, they may not be robust to other forms of dynamic solutions. Nevertheless, these models nest the static scenario in a testable way, and it would be extremely difficult if not impossible to design economic models that are robust to every, or even most, dynamic market behaviors. More details about dynamic market structures in other contexts are provided in the next section. One of the distinctions of conduct parameter models from the standard economic models is that the identification requires an extra effort. In particular, when

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estimating a demand-supply system of equations, the researcher needs to be more careful compared to standard demand-supply models. The source of this problem is that not every functional form choice for demand and marginal cost functions enables separate identification of the marginal cost and conduct parameter. If the functional form choices are not carefully done, it would be possible to confuse competitive markets with high marginal cost and collusive markets with low marginal cost. Lau [68] and Bresnahan [15] provide some conditions for identification in the conduct parameter setting. Bresnahan [15] suggests that this identification issue can be solved by using more general demand functions so that the exogenous variables do more than parallel shifts, i.e., change the demand slope by rotations. Hence, the rotations around the equilibrium point would identify the conduct parameter. This can be achieved by including an interaction term with the quantity variable. However, Perloff and Shen [79] show that such rotations may cause some multicollinearity issues. Another, potentially more realistic, identification approach would be the nonparametric structural identification approach in Brown [19], Roehrig [86], and Brown and Matzkin [20]. Recently, Orea and Steinbucks [77] and Karakaplan and Kutlu [56] proposed conduct parameter models that can be estimated using stochastic frontier approaches. The advantage of these methods is that they model the firm and time-specific conduct parameters as random draws from a doubly truncated normal distribution. Hence, in contrast to existing conduct parameter models, they use skewness of the distribution of conduct parameter in order to identity marginal cost and conduct parameters separately without requiring some of the strong functional form restrictions on the demand and marginal cost functions. In another context, Kumbhakar et al. [60] also use the stochastic frontier approach to estimate market powers (i.e., markups) of firms. Their approach allows estimation of market power even when the input price data are not available. Moreover, their method can reliably estimate market power with or without constant returns to scale assumption.

Dynamic Market Structures Perloff et al. [78] distinguish two types of properties that affect the dynamics of market structure of a market: fundamental and strategic. If the dynamics of the market structure is due to a stock variable that affects future profits, they call this type of reason fundamental. If the dynamic interactions of firms stem from the beliefs that the rivals will respond to current actions, they call this type of reason strategic. In a dynamic setting, the optimization condition should be modified so that price equals full marginal cost, where the full marginal cost equals the sum of marginal cost and a term that is a function of shadow value of the constraints that the firms face. If the reasons are strategic, this can be a function of the shadow value of the incentive compatibility constraints due to cooperation (e.g., [63, 83]). If the reasons are fundamental, this can be a function of the shadow value of a stock (e.g., [80, 82]).

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The stock variable can be amount of natural resources, knowledge, or a quasi-fixed input. A common example of a quasi-fixed input is capital, as in many applications it is more expensive to make quicker adjustments in the capital. This is an example of production-related fundamental reasons (e.g., [28, 85]). Any market structure that involves a quasi-fixed input requires that firms solve a dynamic optimization problem. The reason is that the quasi-fixed input affects the current profits and future levels of quasi-fixed input, which in turn affects future profits. Therefore, the optimal level of investment path depends on the current period’s quasi-fixed input amount and the firm’s belief about future factors such as input prices. Similarly, in markets where advertising is a relatively more important tool that can change demand, the firms face a dynamic optimization problem. This is an example of demand-related fundamental reasons. Perloff et al. [78] argues that advertising may create a stock effect by increasing the firms’ customers today and in the future. If the firm has a small portion of the potential customers, the value of additional advertisement would be large. If this firm invests a large amount in the current time period, it may boost the current demand by capturing potentially a large portion of the market demand. This behavior, however, affects the need for advertisement in the next period because the firm already has a high demand due to large investment on advertisement in the current period; and thus the value of additional advertisement for the next period may be small. The intertemporal connection of advertisement decisions consequently makes the firm’s optimization problem a dynamic one. For market structures where dynamics play an important role, open-loop equilibria and Markov perfect equilibria are among the most commonly used solution concepts. With Markov perfect equilibria, firms know that rivals will respond to a change in the state variable. On the other hand, in open-loop equilibria, the firm will assume that the rivals will not respond to these changes. Therefore, it seems that Markov perfect equilibria more closely reflect what many would call rational firms’ behavior. Perloff et al. [78] provide detailed examples of those models that use these equilibrium concepts.

Market Structures with Differentiated Products In this section, we concentrate more on the approaches that are used when modelling markets with differentiated products and their estimation. The first approach estimates residual demands for close substitutes. Since the degree of market power of a firm depends on the residual demand elasticity, the residual demand approach may be useful in studies that analyze market power. Some earlier examples that use this approach are Bresnahan [14, 16] and Spiller and Favaro [100]. The main difficulty of this approach is that it is not clear how one estimates all own- and cross-price elasticities. The second approach estimates a neoclassical demand system. The general idea is simultaneously estimating a demand system of goods along with marginal cost functions for each good. Hence, for n goods this approach estimates a system

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of 2n equations (n for demand and n for supply). However, when the number of goods is relatively large, estimation of such a system of equations may be impossible due to data requirements. A solution to this problem is estimating a model with a multilevel demand specification (e.g., Hausman et al. [46]; Hausman and Leonard [45]. This approach imposes some implicit restrictions on the crossprice elasticities. In particular, changes in the prices of one category do not affect the demand for another category. Another solution is estimating a so-called almost ideal demand system (AIDS). A relatively simple version of this method is the linear approximation of AIDS that is proposed by [26]). This approach uses Stone’s [103] geometric approximation to the price index. One overlooked issue in the literature is the fact that the estimated demand system actually may not be a complete one by ignoring other goods. It is, however, possible to estimate an incomplete demand system in a way that is consistent with utility maximization (e.g., [66, 67]). The third and relatively more widely used approach estimates a random parameter utility model. This method solves the potential data requirement problems mentioned above. Moreover, Nevo [73] argues that the assumptions for consumers to have preferences so that an aggregate consumer exists and has a demand function that satisfies conditions assumed by economic theory are strong, and many times these assumptions are empirically falsifiable. Hence, using an aggregate model may lead to different conclusions compared to a model that explicitly models individual heterogeneity. One potential solution is to use discrete choice models, which solve the dimensionality problem by projecting the products into a characteristics space (e.g., [8, 73–75]). These models allow the researchers to model the market structure at a very detailed microlevel. For example, Nevo [75] estimates brand-level demand for the ready-to-eat cereal industry and uses the estimates along with the pricing rules to recover price-cost markups without observing the costs. In contrast to the conduct parameters method, which estimates the firm’s conduct along with other parameters, Nevo’s approach assumes that the firms compete under NashBertrand setting. Hence, the conduct is exogenously given. He instead uses three hypothetical ownership structures to determine the extent of market power: singleproduct firms; the current structure observed in the data; and a multibrand monopoly producing all brands. Based on these different ownership scenarios, he calculates the counterfactual price–cost markups that correspond to each of these scenarios. By using crude measures of actual price–cost markups, he can determine which ownership structure fits better to the observed data. Using the same approach, Nevo [74] provides an analysis of merger impact prior to its consummation. This provides an important tool for policy-makers for merger analysis.

Market Structure and Market Power There is a strong connection between market structure and market power. At a high level, market structure creates the environment that forms the base of market power. Although they actually measure market concentration, HHI and, to a lesser extent,

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concentration ratios are commonly used proxies for market power. The appeal of HHI is that it not only gives more weight to larger firms, but it also increases as the number of firms decreases. However, HHI does not consider the particularities of markets. For example, although not very likely, it is possible to imagine relatively competitive markets even with two symmetric firms. Application of the HHI in practice is also problematic since it requires that goods be homogenous. Hence, in markets with differentiated products, the usage of HHI requires additional assumptions, e.g., calculating the market shares based on sales rather than actual quantities. Another common measure of market power is the Lerner index, defined as the ratio of price-marginal cost markup to price. In a standard static setting, when price equals marginal cost, this measure equals zero, which indicates a perfectly competitive market. One issue with the Lerner index is that efficiency improvements could be mistaken as increments in market power. Koetter et al. [58] propose efficiency adjusted measures of Lerner index to overcome this issue. Moreover, if the market structure involves dynamics (due to either strategic or fundamental reasons), the relevant marginal cost concept for optimality conditions is the full marginal cost. This suggests that the Lerner index also must use the full marginal cost in its calculation. Kutlu and Sickles [63] suggest using efficiency adjusted full marginal cost, which is robust to efficiency- and dynamics-related concerns mentioned above. Kutlu and Wang [65] provide a variety of conduct-parameter-based models in this context. The Lerner index may not always be easy to estimate due to available data issues. More precisely, the calculation of Lerner index requires knowledge of marginal cost, which is not directly observable and thus requires to be estimated. One way to do so is estimating a cost function and calculating the marginal cost from the parameter estimates of this cost function. This, however, requires total cost data, which is not always easy to obtain. The conduct parameter approach enables the estimation of marginal cost implicitly without using the total cost data. Also, this approach provides an alternative measure of market power, i.e., the conduct parameter. At least some variations of conduct parameters are shown to be highly correlated with the Lerner index. In particular, under some assumptions, the conduct parameter equals price elasticity adjusted Lerner index. Boone [12] proposes a new way to measure market power. He motivates his analysis by demonstrating (e.g., [87, 101]; and [21]) the theoretical possibility of more intense competition leading to higher price–cost margins. Hence, Boone [12] aims to develop a competition measure that is theoretically robust yet requires similar data sets with price–cost margin estimation. Boone [12] calls his measure relative profit differences (RPD). This measure is defined as follows: Let π (n) be the profit level of a firm with efficiency level n where higher n denotes higher efficiency.   Consider three firms with different efficiency levels, n < n < n and calculate   the following variable (π (n ) − π (n))/(π (n ) − π (n)). Boone [12] argues that an increase in competition raises this variable in models where a rise in competition reallocates output from less efficient to more efficient firms, which covers a broad range of models. Another related study with a very similar measure is Boone [13],

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which proposes a measure called relative profits. This time Boone considers the variable for two firms with efficiency levels. Boone [13] argues that an increase in competition reallocates profits from the less efficient firm to the more efficient firm and thus increases the relative profits measure.

Market Structure and Innovation Studies with No Explicit Treatment for Distorted Production Decisions Schumpeter [93] argued that innovation plays the principal role in advancing economic prosperity. This is supported by the historical record of innovations that have come and gone since the industrial revolution. The steam engine, light bulb, automobile, airplane, radio, and integrated circuit immediately come to mind – each of these innovations have pushed the economic frontier forward, noticeably improving consumer welfare. Understanding what innovation is and how competition policy could affect it is consequently important. The chief goal of this section is to explore the latter, but we first provide a short summary of the origin of the literature, what innovation is and some of the issues surrounding measurement. Innovation is defined generally as a new or materially improved product, process, service or business method. What drives it is the profit motive. Economic policy, therefore, can only be effective at fostering innovation if it helps to safeguard or promote the incentive to innovate. Broadly speaking, one may view the profit motive from two perspectives: (1) the objective and expectation of realizing an economic gain solely on the merits of the innovation, with or without existing competition; or (2) the objective to maintain profitability5 at a minimum or prevent an economic loss when competition is present. The first approach to innovation is purely entrepreneurial in that the innovator’s investment decision is independent of its competitive environment. In contrast, the second approach to innovation is a mixture of entrepreneurship and an effort to survive; i.e., the innovator also takes into account its competitive environment to optimize its investment strategy. This section limits discussion around the second approach to innovation. Whether or not competition spurs innovation has been a question of significant interest in the industrial organization literature. One may attribute this to Schumpeter [93], who argues that in a capitalist system, perfect competition is both incompatible with economic progress and inferior to large scale enterprise. Taken at face value, his conclusion might indicate that a more competitive industry stifles innovation, but this is not Schumpeter’s intention.6 Schumpeter’s claim that “big business” is superior to atomistic competition in a capitalist system is secondary to his main concern – that capitalism is superior to

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profitability is high enough to fully cover economic costs. example, Aghion et al. [3] define the “Schumpeterian effect” as the effect of increased competition lowering post-innovation rents, thereby reducing the incentive to innovate. 6 For

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socialism because perfectly competitive markets maximize social welfare. Schumpeter advanced the idea that capitalism is superior not because of price competition or any particular market structure, but rather because free enterprise begets the introduction of new products, processes, services, and methods of production. If one therefore accepts the premise that economic prosperity is the product of innovation, then perfect competition cannot be the reason for capitalism’s success as an economic system. Schumpeter demonstrates this point with the following observations. The first of these observations is that capitalism is a system of constant and disruptive structural change. Accordingly, it is not appropriate to assess capitalism’s performance based on an outcome of static conditions. In particular, while the static outcome of perfect competition is socially desirable at any given point in time, it implies that goods and production methods never change. The static efficiency of perfect competition thus comes at the cost of precluding innovation’s promise of dynamic and long-run efficiency. Schumpeter also juxtaposed the assumption of free entry in a perfectly competitive market against an innovator’s expected return on investment. That is, even though free entry is a necessary condition for maximizing social welfare, quick and costless entry into a new market is at odds with an innovator’s incentive to invest.7 In particular, if free entry implies costless and nearly instantaneous replication of the innovator’s idea, then the innovator’s expected return will not exceed its opportunity cost of investment. Even ignoring the assumption of homogeneous goods and static production technologies, it follows that a firm in a perfectly competitive market has no incentive to innovate. The final observation is that large scale and anticipated market power facilitate investment in innovative ideas. More specifically, the financial position of a profitable, large firm grants it the opportunity to take on riskier innovative activity than a small firm. And anticipated market power, by way of an intellectual property right, promotes innovative activity by allowing the innovator to temporarily charge a price in excess of what the market would bear. The latter observation, while not novel, is further emphasized by Schumpeter that free and prompt entry is problematic for innovation. Schumpeter additionally advocates for intellectual property rights as it permits a firm to better plan for the future. On the other hand, Schumpeter’s view that large scale is a boon rather than a detriment to innovation is meant to highlight the limitations of a perfectly competitive firm, specifically limitations for growth due to relatively little capital to finance risky innovative activity, and the relative inability to weather the constant flow of outside innovations.

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idea that free entry will quickly erode an innovator’s supranormal profit assumes that the innovator’s competition is effective; that the idea embedded in the innovation is copied well and easily; and that being a first mover does not confer a material advantage. While this is not always the case (see, for example, Boldrin and Levin [11]), the topic of intellectual property rights is outside the scope of this chapter.

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It bears repeating that Schumpeter’s criticism of perfect competition as a model of efficiency is not a statement about the relationship between innovation and ex ante market structure, market power or level of competition. Rather, Schumpeter’s intention is to demonstrate that the assumptions of perfect competition are wholly in conflict with innovation, and therefore economic progress. The idea that ex ante market power fosters innovation is nevertheless commonly ascribed to Schumpeter. This is borne out by the multitude of studies that have tested some form of Schumpeter’s proposition. Before going into some detail about the literature alluded to above, we briefly touch on the issue of measurement. A common measure of competition is the Lerner index. The ubiquity of this measure is traced to its relationship with the price elasticity of demand, the source of a firm’s market power when price is the only choice variable. Indeed, when a single, profit maximizing firm takes its demand as given, its equilibrium Lerner index will be inversely proportional to its price elasticity of demand. Thus, a larger Lerner index is taken to indicate greater market power or less competition. There are several reasons why this measure could fail to indicate changes in market power. One possibility is if a firm engages in cost-minimizing behavior. In this case, a temporal increase in the Lerner index would reflect nothing more than the competitive process at work. The life cycle of a product will also affect the margin we observe in the data. For many goods, as a product matures, focus shifts from differentiation and quality improvement to cost reduction ([111]). That is, commoditization will take effect in the later stages of a product’s life cycle, and margins will generally fall as a result. A firm’s margin will also increase if it is able to generate a competitive advantage. For example, in a hypothetical industry of two firms selling the same good, successful product innovation by one of the firms will allow the innovator to command a higher price for its product. The average margin in the industry will increase, but if the competitive advantage possessed by the innovator pushes the noninnovator to invest in developing a better product, then rivalry will have been preserved if not strengthened. Thus, an increase or decrease in margins does not fundamentally indicate a fall or rise in competitive intensity, respectively. The Herfindahl–Hirschman index is another commonly used measure of competition. The main attraction of this measure is that it resonates with the intuition that more concentrated industries are less competitive (i.e., industries with few firms and/or firms with substantial market share are viewed as relatively less competitive). Also attractive is the Herfindahl–Hirschman index’s theoretical link to market structure. For example, the Herfindahl–Hirschman index corresponds to monopoly when the market is captured entirely by one firm and perfect competition when there are infinitely many firms with equal shares. The prevalence of the above measures is based on the presumption that changes in market structure or market power will identify changes in competitive intensity. This presumption is questionable, however, if the defining feature of a more competitive market is greater rivalry to deliver superior products and services. In particular, it does not follow that a more profitable or concentrated industry will

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be less inclined to compete vigorously (or vice versa), especially when changes in market structure or market power are measurably small. A more competitive market will arise when, on average, a firm faces a greater risk of falling behind (resp., a greater prospect of moving ahead) its competitors if it offers inferior (superior) value to consumers. That is, a more contestable market will result in greater rivalry. Shapiro [99] calls this the “contestability” principle, and it is adopted by Garcia [36] to examine the relationship between competition and innovation. Application of the “contestability” principle is consistent with antitrust policy.8 The 2010 Horizontal Merger Guidelines (HMG) of United States Department of Justice and the Federal Trade Commission state the following9 : “The unifying theme of these Guidelines is that mergers should not be permitted to create, enhance, or entrench market power or to facilitate its exercise . . . [where by definition] a merger enhances market power if it is likely to encourage one or more firms to raise price, reduce output, diminish innovation, or otherwise harm customers as a result of diminished competitive constraints or incentives.”

The Guidelines identify the root of enhanced market power as any market characteristics that would lessen rivalry or the incentive to compete. While it is possible that enhanced market power could manifest as greater profitability or market concentration, the converse is not necessarily true. In contrast, any policy that would serve to make a market more contestable, all else equal, will intensify competition. The measurement of innovation is equally important. The four defining features of innovation are: investment; expansion of revenue and/or profit not attributable to routine labor and capital; relatively rare occurrence; and growth in economic output that exceeds growth in inputs. The three most common measures of innovative activity are R&D intensity, patent counts, and productivity growth. We briefly argue below that productivity growth is the most appropriate measure for capturing the above features. R&D intensity, defined as the ratio of R&D expenditure to sales, is perhaps less used today than it was in the nascent stages of the literature. Intuitively, R&D intensity captures the “innovative effort” of a firm, but it is not a measure of innovative output. The latter ultimately makes R&D intensity an inappropriate measure of innovation. But even as a measure of effort, it suffers from a variety of problems. Notably, there is no requirement for R&D expenditure to be reported, and the peculiarities of firm-level accounting methods make it an unsuitable as an estimate of relative innovative activity [10].

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Bingaman, a former Assistant Attorney General for the Antitrust Division in the U.S. Department of Justice during 1993–1996, said the following about rivalry and innovation: “The fundamental thesis of strong antitrust enforcement is that rivalry, not market power, fosters innovation and efficiency over the long run . . . Antitrust has an important role in preserving the rivalry that spurs innovation.” See Bingaman [9]. 9 See HMG [49] for the complete set of guidelines governing antitrust policy in the United States.

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Patent statistics are often used as a measure of innovation because patents represent a novel idea. A citation-weighted patent count, in particular, is the number of forward citations a patent has; its use is based on the idea that more citations are indicative of greater economic value. There are several problems with patents as a measure of innovation, however. One, patents represent only a fraction of innovative output [41]. Two, the incentive to patent encompasses more than just the intent to protect a novel and valuable idea; companies also seek out patents to defend against litigation or to litigate themselves [43]. Three, evidence suggests that companies view patents as relatively weak mechanisms for protecting intellectual assets; instead, trade secrecy and lead time to market are viewed as more effective [22]. Four, the presumption that a greater number of citations reflect a more valuable idea has been challenged. Empirical evidence indicates that the relationship between citations and economic value is an inverted-U [1]. Intuitively, the most valuable patents are the ones firms actively try to protect the most, resulting in fewer citations. For the reasons above, it has been argued that productivity growth is the most appropriate measure of innovation. Productivity growth, as has been argued by Dale Jorgenson and others at the NBER (see, for example, [54]), is the key economic indicator of innovation: . . . Productivity growth is the key economic indicator of innovation. Economic growth can take place without innovation through replication of established technologies. Investment increases the availability of these technologies, while the labor force expands as population grows. With only replication and without innovation, output will increase in proportion to capital and labor inputs, as suggested by Schultz ([94], [95]). By contrast the successful introduction of new products and new or altered processes, organization structures, systems, and business models generates growth of output that exceeds the growth of capital and labor inputs. This results in growth in multifactor productivity or output per unit of input . . . .

Thus, not only does productivity growth as a measure of innovation avoid the economic complications underlying patents, it is consistent with long-run economic growth – precisely why Schumpeter was such an advocate for anything that would serve to promote innovation. The main concern of early empirical work in the literature was the effect of firm size and market concentration on innovative activity, the latter being some measure of R&D expenditure. Little evidence was found that supported a statistically robust or economically significant cross-sectional size effect. More recent empirical IO studies have turned their attention to the relationship between ex ante market power and innovation. Typical measures of competition and innovation in this literature are the Lerner index and patent statistics, respectively. The results are mixed despite the use of panel data, similar methods and more sophisticated econometric techniques. Some studies find a negative relationship, some find a positive relationship and some find an inverted-U relationship. A related but distinct empirical literature presents strong evidence of a positive relationship between competition and innovation. This literature examines changes in productivity at the microlevel in response to discrete changes in the competitive environment. In particular, this literature uses natural experiments to identify changes in competitive intensity. For example, one study finds that a group of US

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iron ore manufacturers nearly doubled their labor productivity in the 1980s after Brazilian manufacturers entered their market. Economic theory also examines the relationship between competition and innovation. Broadly speaking, two approaches have been taken. One approach compares how much an incumbent firm would be willing to invest in R&D compared to a potential entrant, and the other approach examines the relationship between a parameter that affects market structure and R&D effort. Like the empirical IO literature, the results are mixed. The incumbent-entrant class of models predicts both a positive and negative competition-innovation relationship, while predictions from the parametric class of models range from a positive relationship to an inverted-U. Overall, both empirical evidence and theory paint an unclear picture of the competition-innovation relationship. The nebulous qualities and complex interaction of these variables is doubtless a major reason why. Our endeavor in this section is to provide a synthesis of the competition-innovation debate and the implications it has for economic growth.

Theoretical Work Theoretical Models of Market Structure, Incumbency, and the Incentive to Innovate Arrow [6] was the first to rigorously examine the relationship between market structure and the incentive to innovate, the latter defined as the difference between post- and pre-innovation profit. Under the crucial assumption that property rights are perfectly exclusive and infinitely lived, Arrow analyzes the relationship by comparing a monopolist’s incentive to reduce marginal cost to that of a perfectly competitive firm. Two types of process innovation are possible in his model: drastic and nondrastic. With drastic innovation , the new technology achieves a profit maximizing price that is less than the marginal cost of the old technology. Consequently, the monopolist remains a monopolist with drastic innovation, and the perfectly competitive firm becomes one.10 It follows that the incentive to innovate will be higher for a competitive firm since its post-innovation profit is the same as the monopolist’s, while its pre-innovation profit of zero is lower. In the case of nondrastic innovation (i.e., the new profit maximizing price is higher than the marginal cost of the old technology), the post-innovation profit for a competitive firm will be less than a monopolist’s. This is a consequence of the fact that a competitive firm cannot profitably charge a price that exceeds the prevailing competitive price. Instead, a competitive firm’s post-innovation profit

10 To

see the latter, note that the marginal cost of the old technology is equal to the prevailing price in a perfectly competitive market. Thus, with drastic innovation and exclusive property rights, a perfectly competitive firm will drive its competitors out of the market with the new price and subsequently become a monopolist.

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will be limited to the unit royalty it charges (i.e., the difference between the old and new marginal costs) times the number of units sold in the market at the old price. Notwithstanding, Arrow shows that a monopolist’s incentive to innovate will still be less than a competitive firm’s, precisely because the monopolist applies the cost reduction to relatively less units of output in equilibrium. Gilbert and Newbery [39] build on Arrow’s model by allowing for the possibility of entry into the monopolist’s market. Specifically, an incumbent (i.e., monopolist) bids for a patent on a new, substitute technology to preempt entry by a challenger, and a challenger bids for a chance to compete with the incumbent. In this setup, Gilbert and Newbery find that preemption is a rational strategy – in fact, a Nash equilibrium – for the incumbent if monopoly profits with the new technology exceed the costs of preemption. This will attain if post-entry industry profit is less than pre-emptive monopoly profit. That is, letting e and m denote entrant and monopolist, respectively, if π m (pm 1 , pm 2 ) > π m (pm 1 , pe 2 ) + π e (pm 1 , pe 2 ) attains, where π i (·) represents the firm i’s profit function, i ∈ {m, e}, and pi j represents the price of product j, j ∈ {1, 2}, then the incumbent will have the incentive to submit a larger bid than the challenger.11 Gilbert and Newbery show that this condition holds under fairly weak assumptions, implying that the incumbent’s incentive to innovate will be relatively greater if it has more to lose from entry than the challenger has to gain. Reinganum [84] extends the model of Gilbert and Newbery [39] by introducing uncertainty to the innovation process. Specifically, she assumes that innovation follows a Poisson process with an exponentially distributed date of successful innovation. In turn, the incumbent and challenger choose investment levels that increase their chance of innovating first. Her model has three possible outcomes. If the incumbent succeeds in reducing its current marginal cost from c to c < c and secures a patent before the challenger, then it will earn flow profits (c). If the challenger succeeds before the incumbent, then the incumbent and challenger will earn Cournot flow profits π I (c) < (c) and π C (c), respectively. If neither succeeds, then the incumbent and challenger maintain their pre-innovation profit flows at R and zero, respectively.12 In the spirit of Arrow [6], Reinganum defines drastic innovation as an innovation that lowers marginal cost to a level c ≤ c0 , where c0 is the largest value of c such that π I (c) = 0 Coupled with the assumption of constant returns to scale, this condition implies that the incumbent produces zero output. The incumbent is thus knocked out of the market by the challenger when innovation is drastic, allowing the latter to earn profit flows π C (c) = (c). The Nash equilibrium of this game implies that the challenger will unambiguously spend more on R&D than the incumbent when innovation is drastic. And to

j = 2 represents the new, substitute technology. assumes that (c) and π C (c) are nonincreasing and that π I (c) is nondecreasing in c, the intuition being that the successful (unsuccessful) innovator’s flow profits are higher (lower) the greater is the reduction in cost.

11 Product

12 Reinganum

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the extent that the equilibrium solution can be analyzed under nondrastic innovation, Reinganum shows that there exists a nontrivial set of nondrastic innovations where the challenger’s incentive to innovate is relatively greater.13 The assumption of stochastic innovation is critical to the innovation-market structure relationship, as demonstrated by the different results reached by Reinganum [84] and Gilbert and Newbery [39]. Why the difference occurs may be explained by how investment is modeled. Specifically, whereas in the model of Gilbert and Newbery there is no investment decision (the incumbent and challenger innovate with probability one), Reinganum assumes that further investment can only marginally increase the probability of successful innovation; i.e., the assumptions of Reinganum’s model imply diminishing returns to investment. The marginal increase in expected post-innovation flow profits (via the marginal increase in probability of relatively early innovation) is accordingly offset by the marginal cost of further investment at some point. And this point occurs earlier for the incumbent precisely because its pre-existing profit flows must be replaced.

Theoretical Models on the Degree of Competition and the Incentive to Innovate Theoretical models in the vein of Arrow [6] may be viewed as “discrete” in the sense that they compare the incentive to innovate across two types of firms: one with market power and one without. More recent theory, however, examines the competition-innovation relationship with competition measured on a continuum. Kamien and Schwartz [55] were the first to take this approach. In their model, a firm chooses a date to introduce innovation based on the degree of rivalry it faces, where greater rivalry is modeled as a parameter that accelerates the expected date of rival innovation. Specifically, the decision problem facing the firm is to choose an innovation arrival time that maximizes its present value of expected innovation cash flows, conditional on the probability of a rival innovating first.14 The firm formally chooses its development date T ∗ to solve ∞ maxT

e−(r−g)t [P0 (1 − F (t)) + P1 (F (t) − F (T )) + P2 F (T )] dt − C(T ),

T

where r > g is the discount rate; g is market growth; F(τ ) is the probability of rival introduction by time τ ; P0 (1 − F(t)) is the expected payoff to the firm conditional

13 This

is for nondrastic innovations where the reduction in cost is sufficiently close to the drastic level c0. 14 Kamien and Schwartz [55] examine two versions of this model, one with patent protection and one without. We omit the model with patent protection as the results are not affected by the differences in appropriability.

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on no rival entry by time t; P1 (F(t) − F(T)) is the expected payoff to the firm conditional on it innovating before a rival and the rival appearing between time T and t; P2 is the expected payoff to the firm conditional on a rival innovating before time T; and C(T) is the minimum present value of the cost of completing   development by time T. It is assumed that C (T) < 0 and C (T) > 0, and P0 ≥ P1 and P0 ≥ P2 . In addition to other parameters, the optimal date T ∗ is a function of the hazard  rate of successful innovation by a rival, h ≡ F (t)/(1 − F(t)). Kamien and Schwartz assume for simplicity that the probability of innovation is memoryless, implying a constant hazard rate with cumulative density function F(τ ) = 1 − ehτ , τ ∈[0, T]. The significance of h is its tie to the degree of rivalry. Namely, because the inverse of h is the expected date of innovation by a rival, an increase in h heightens the probability of a rival innovating first. Kamien and Schwartz accordingly interpret an increase in h as an intensification of rivalry. As Kamien and Schwartz demonstrate, the relationship between h and T∗ depends on the size of the innovation (defined as the amount of profit rewarded for innovating). When the size of innovation is sufficiently large, the relationship takes a U shape. That is, for relatively small values of h, the marginal cost saving of postponement is less than the marginal loss of delay, pushing the firm to accelerate its development date. But for sufficiently high values of h, the effort required to preempt a rival’s development date becomes excessively costly, driving the firm to postpone. In the case of small innovation size, h and T∗ are strictly positively related. Intuitively, an already small reward for innovation makes any additional effort to pre-empt rival innovation less attractive. Loury [69] extends the model of Kamien and Schwartz [55] to a game-theoretic setting. In particular, he examines how R&D incentives are affected by rivalry when a finite number of symmetric firms, n, choose their investment strategies simultaneously. As in Kamien and Schwartz [55], Loury assumes that the date of successful innovation is memoryless. Thus, the probability of successful innovation by time t is given by P(τ (x) ≤ t) = 1 − e−h(x)t , where τ (·) is the random date of successful innovation, x is R&D expenditure and h(·) is the hazard rate of    innovation. It is assumed that h (x) > 0, h (x) ≥ 0 for x∈[0,x] and h (x) < 0 for x∈(x, ∞). Each firm chooses its R&D expenditure to maximize its present value of discounted cash flows, taking as given the collective R&D expenditure of its rivals. The symmetry of the problem implicitly defines the optimal R&D expenditure function x ∗ = x(a, ˆ r, V ), where a = (n − 1)h(x∗ ) and V and r denote the flow of revenues from successful innovation and the interest rate, respectively. In the special case of no strategic effects, Loury shows that a firm’s R&D expenditure will either first increase and then decrease with rival R&D expenditure (i.e., the parameter a), or monotonically decrease. This mirrors the results of Kamien and Schwartz [55]. When strategic effects are allowed (and n ≥ 2), however, individual R&D expenditure and the expected date of innovation strictly decrease

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with n.15 Hence, while greater rivalry (as measured by an increase in n) lowers individual R&D expenditures, it also accelerates the introduction of innovation. Intuitively, an increase in the number of firms in a symmetric industry will lower expected profits, thus deterring firms to invest in R&D; concurrently, more firms will simultaneously invest in R&D, which increases the chance of earlier innovation. Aghion et al. [2] develop a dynamic macroeconomic model of “step-by-step” innovation whereby a leading (laggard) firm in its sector can widen (narrow) its technological lead (lag) if it successfully innovates. The continuum of sectors that makeup the economy are each comprised of two cost-asymmetric firms that ultimately compete in price with differentiated goods. In the first stage of a two-stage game, each firm takes their second-stage profit function as given and makes an R&D investment decision to maximize expected future discounted profits, conditional on the technological gap in their industry and the level of competition. In particular, the firm makes a cost-reducing investment that increases its probability of favorably changing the technological gap in its industry, where the technological gap is a function of the relative production cost between the two firms. In the second stage, firms compete on price alone, taking as given their relative cost. The equilibrium profit in this stage is an implicit function of relative marginal cost16 and the degree of product substitutability. The latter is Aghion et al.’s [2] measure of competition; however, it is more precisely a reflection of consumer tastes, which affects the interpretation of an industry’s market structure.17 Aghion et al. [2] analyze the effect of competition on innovation under the following assumptions: (1) there is a steady state composition of n-gap (unleveled) and 0-gap (leveled) industries; (2) the laggard can immediately catch-up to the leader if the laggard is the sole innovator; (3) there is at-most a one-step increase in the gap if a leader or neck-and-neck firm innovates and; (4) changes in product substitutability affect the whole economy. When innovation is large, they find that an increase in competition can either foster or retard innovation. On one hand, large innovation implies that a one-step lead will raise the would-be leader’s profit to the maximal level, so a leader will not innovate further. On the other hand, large innovation effectively fixes the postinnovation rent for a neck-and-neck firm and the pre-innovation rent for a laggard. The effect of competition on innovation thus only operates through the pre- and

15 Loury

shows that the expected arrival date of innovation is strictly decreasing with respect to n when a unit increase in R&D investment by any single firm causes every other firm to invest a smaller amount into R&D. 16 Defined as the ratio of firm i’s marginal cost to firm j’s marginal cost. 17 Aghion et al. [2] concede that product substitutability is a taste parameter. By construction, product substitutability will affect the structure of a firm’s demand. When product substitutability is at its lowest, a firm has no competitors, so the firm will behave like a monopolist. On the other hand, a firm will behave like a perfectly competitive firm when product substitutability is at its highest. Thus, a higher level of product substitutability may be interpreted as a “less monopolistic” market structure.

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post-innovation rents of neck-and-neck and laggard firms, respectively. In fact, these rents are the same: the pre-innovation rent for a neck-and-neck firm is the postinnovation rent for a laggard. Thus, since greater competition lowers neck-and-neck rents, competition will have a nonlinear effect on economy-wide innovation. The shape of the relationship found by Aghion et al. [2] under large innovation is an inverted-U. This arises because the steady state distribution of leveled and unleveled industries is itself a function of the level of competition. More specifically, when competition is already relatively high, neck-and-neck firms will have a greater incentive to innovate than laggards as they seek to “escape competition.” This will push the economy into a state where there are more unleveled industries than leveled ones. And because this transition does not affect the level of competition (i.e., the level of neck-and-neck profits), the incentive to innovate across the economy will diminish – laggards have little incentive to innovate when competition is already high, and leaders do not innovate at all. Aghion et al. [2] call this the “Schumpeterian” effect of competition. Contrast this to the case of relatively low competition. When competition is relatively low, laggard firms will be more inclined than neck-and-neck firms to innovate. This will push the economy into a state with more leveled than unleveled industries where the “escape competition” effect dominates. Thus, innovation initially increases with competition but then declines. In the case of small innovation, innovation is found to monotonically increase with competition. Intuitively, the increment in profit from innovating is approximately the same for leaders, laggards and neck-and-neck firms; and since the increment in profit for a neck-and-neck firm increases with competition (due to the “escape competition” effect), it follows that economy-wide innovation will also increase. Finally, Aghion et al. [2] examine the general case numerically and confirm their analytical results. That is, they find innovation to increase with competition for intermediate values of innovation size, but eventually an inverted-U shape arises when innovation is large and the probability of imitation is low. Aghion et al. [3] reexamine the “step-by-step” innovation model developed in Aghion et al. [2] under two modifications. First, a firm can only advance its technological position by one step through successful innovation. If, for example, the current state of a sector is m – a nonnegative integer that indexes the efficiency gap between two firms – and the leader (laggard) successfully innovates while the laggard (leader) does not, then the state of the sector will change from m to m + 1 (respectively, from m to m − 1). Thus, unlike Aghion et al. [2], this model assumes that the laggard cannot immediately catch up to the leader. Second, instead of explicitly using product substitutability as a proxy for competition, greater competition is measured as the degree to which neck-and-neck firms cannot collude. That is, letting  ∈ [0, 1/2] denote a neck-and-neck firm’s profit as a fraction of a leader’s profit, Aghion et al. [3] treat a smaller value of  as greater competition. To operationalize  in their model, Aghion et al. [3] use  ≡ 1 −  = (π 1 − π 0 )/π 1 , where π 0 and π 1 are the profit levels of a neck-andneck and leader firm, respectively.

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With the above modifications, Aghion et al. [3] restrict attention to the case where m ∈ {0, 1}, so that a leader (laggard) can be at most one step ahead (behind). This mirrors the “large innovation” scenario analyzed in Aghion et al. [2], and implies that a leader will not innovate. The inverted-U competition-innovation relationship found in Aghion et al. [2] (under large innovation) remains intact in Aghion et al. [3]. Hashmi [44] argues that the theoretical model of Aghion et al. [3] is not wellsuited for industry level analysis, particularly because Aghion et al. [3] model the interaction of competition and innovation at the economy level. Hashmi notes, however, that an industry-level analysis is possible with Aghion et al.’s [3] duopoly model of competition, so he accordingly adopts it. Namely, Hashmi considers a setting where two cost-asymmetric firms price compete with differentiated goods. The demand for firm i, i = 1, 2, is given by: 1/(α−1)

qi =

pi

α/(α−1)

pi

α/(α−1)

+ pj

,

where p is price, and α ∈ [0, 1] is the degree of product substitutability. Hashmi follows Aghion et al. [3] and uses α to measure competition. Letting ci = wγ −ki denote firm i’s constant marginal cost of production, where w is the wage rate, γ is the size of innovation and ki is the technology level of firm i, the equilibrium profit function for firm i is πi (n) =

(1 − α) Ri (n) , 1 − αRi (n)

where n ≡ ki − kj is the technology gap between firm i and firm j, and Ri (·) is firm i’s market share. Finally, Hashmi defines the probability of successful innovation as     P = 1 − e−ax + max 0, 1 − eη(n−nˆ ) , where x is R&D investment. The first term is common to both firms; it represents the baseline probability of success. The second term is specific to an unleveled industry, and it allows the laggard to more easily catch up to the leader as the technological gap grows. This term is introduced by Hashmi to ensure that zero investment is not chosen by both the laggard and leader when the gap is large. Given the above, both firms choose their R&D investment level to maximize their expected future discounted profits. The relationship between optimal R&D investment and competition depends on the technology gap. When the technology gap is small, the relationship is approximately monotonically increasing; when it is intermediate, the relationship is an inverted-U; and when it is large, the relationship is monotonically decreasing. Thus, Aghion et al. [3]’s result is not robust to variations in the technology gap.

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Hashmi’s results are interesting from a policy perspective, particularly if one views the technology gap as a better proxy, or at least a characteristic, of market structure. Intuitively, a technological lead confers a competitive advantage (in this case, the ability to charge a more competitive price due to a cost-reducing technology), and as the lead gets larger, the industry will move closer to monopoly.18 Perhaps even more interesting is the result that innovation decreases with the technological gap. Ostensibly, a large gap lulls the leader and discourages the laggard from innovating, whereas a small gap keeps rivals from becoming complacent with their competitive positions.

Empirical Work Empirical Results on Market Structure, Incumbency, and the Incentive to Innovate The literature on firm size, market structure, and innovation is voluminous [38, 107]. Scherer [90], an extension of Scherer’s earlier work, Scherer [89], is a seminal paper in the literature. There are two material differences between Scherer [90] and Scherer [89]. One, Scherer [90] uses a more comprehensive dataset of 56 industries, as opposed to the 48 used in Scherer [89]. And two, whereas Scherer [89] estimates the impact of firm size on patents granted, Scherer [90] estimates the effect of market concentration on the total number of engineers and scientists employed as a proportion of total employees. Thus, he moves from an output-based to an inputbased measure of innovation. Two findings stand out in Scherer’s cross-sectional study. One is that the explanatory power of market concentration drops substantially when industryspecific dummies are added to the regression. In other words, industry differences in technological opportunity account for most of the variation in R&D investment. Another notable finding is that innovation effort exhibits an inverted-U relationship with concentration. Thus, R&D investment increases with market concentration when market concentration is relatively low, and decreases when it is relatively high. Scherer interprets this as are a rejection of the Schumpeterian hypothesis that market power fosters innovation. The drop in market concentration’s explanatory power after technological opportunity is accounted for is a common theme in the empirical literature related to market structure, incumbency, and the incentive to innovate [107]. However, it is unclear from these studies if the elimination of market concentration’s effect is theoretically driven, or if it is a statistical artifact of the limitations of cross-

18 In

fact, when an industry consists only of two firms, a larger technological gap will necessarily lead to greater concentration. This relationship does not necessarily hold for an industry with more than two firms, however. When there are three firms with different efficiency levels, for example, the laggard firm could narrow the technological gap with the second-place and leader firm, but the reallocation of sales among the three firms could lead to lower or higher concentration.

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sectional methods. The latter is addressed by more recent empirical studies, which use panel data methods to identify the effect of market power on innovation. More significant from a policy perspective is the inverted-U relationship brought forward by Scherer, which implies that too much competitive rivalry will retard innovation. The possibility of a nonlinear competition-innovation relationship has been and continues to be investigated in the IO literature. We discuss this more recent literature shortly.

Empirical Results on the Degree of Competition and the Incentive to Innovate Relatively early empirical work was concerned with the hypothesis that large scale stimulates innovation. More recent empirical IO research is expressly concerned with the effect of competition on innovation. In addition to this different, albeit somewhat related policy question, the newer literature differentiates itself via more sophisticated econometric techniques, measures of competition and innovation, and breadth of data. The literature is broadly split into two categories; models that test competition’s effect on innovation under the assumption that the relationship is linear, and models that test the effect assuming the relationship is nonlinear. In particular, Nickell [72] and Blundell et al. [10] estimate linear specifications, while Aghion et al. [3], Correa [23], Correa and Ornaghi [24], and Hashmi [44] estimate nonlinear specifications. Both Nickell [72] and Blundell et al. [10] estimate a linear, dynamic panel-data model and find a positive competition-innovation relationship based on a panel of publicly traded firms in the UK. Nickell uses a firm’s average operating margin as a proxy for competition and total factor productivity to measure of innovation, whereas Blundell et al. [10] measure competition in terms of individual market share, market concentration and import penetration, and innovation in terms of survey-based innovation counts. The nonlinear class of empirical IO models, all based on Aghion et al. [3], generate a range of qualitative results. Aghion et al. [3] use Poisson regression and a 1973–1994 panel of publicly traded firms in the UK to estimate the competitioninnovation relationship at the industry-year level. They use citation-weighted patent counts to measure innovation and operating margins to measure competition. They find that competition’s effect on innovation is positive for relatively low and intermediate levels of competition, but negative when competition is sufficiently high. Using the same data and econometric techniques as Aghion et al. [3], Correa [23] instead finds a monotonically increasing relationship after accounting for structural changes in the data, namely a policy that facilitated patent grants. Correa and Ornaghi [24] also find a monotonically increasing relationship, but with panel data on US manufacturing firms spanning the period 1974–2001. Their finding is robust to different measures of innovation, including citation-weighted patent counts and productivity growth. Finally, Hashmi [44] finds a weak, monotonically negative relationship using a panel of US publicly traded firms over the period 1976–2001. His econometric model is largely the same as Aghion et al.’s [3], except he uses negative binomial regression to estimate the competition-innovation relationship.

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The range of conclusions for the nonlinear class of models is striking given the strong similarity in data and methods. This bears further discussion, which we pursue below. As noted above, the nonlinear empirical models that test the competitioninnovation relationship are closely related to the Aghion et al. [3] model. Accordingly, this warrants some discussion of the Aghion et al. [3] model and the models that followed. Specifically, Aghion et al. [3] estimate following conditional mean function: E[p|c,x] = eg(c) + xβ , where p is the number of citation-weighted patents, c is competition, x is a set of industry and time dummy variables and g(·) is some function to be estimated. It is assumed that p follows a Poisson process. Aghion et al. [3] estimate the model with data on publicly traded manufacturing firms in the UK over the period 1973–1994. Correa [23] revisits the Aghion et al. [3] model using the same sample of data and empirical formulation, but allows for structural breaks in the data. Correa reasons that this is appropriate because the establishment of the United States Court of Appeals for the Federal Circuit (henceforth CAFC) in 1982 made it effectively easier to have a patent granted. Correa takes two approaches to test the structural break hypothesis. The first approach is a Chow test. The base model estimated is the same as in Aghion et al. [3]:  pj t = exp β0 + β1 cj t + β2 cj t 2 + ϕ vˆj t + δ1 Dτ cj t +δ2 Dτ cj t 2 +

 j

αj Dj +

 t

 γt Dt + uj t ,

where Dτ = 1 for all t ≥ π (π denotes a pre-defined structural break), 0 otherwise; cjt is the level of competition for industry j at time t, measured as one minus the industry average price-cost margin; vˆ jt is the residual for industry j at time t from regressing the competition index on policy and foreign-industry instruments (i.e., endogeneity is accounted for with a control function approach); and the last two terms are industry- and time-fixed effects. The null hypothesis of time stability at t = 1983 is rejected by the Chow test at the 5% level of significance. Correa also carries out a Wald-type test for structural breaks, finding only one structural break at year 1981. Correa gives several reasons why this year, instead of 1982, was detected. One of these reasons is that the political discussion to establish the CAFC began in 1979. Thus, in anticipation of the CAFC being established, a structural change in patent incentives may have manifested before the CAFC’s official introduction. The indication of a structural break in the data by both tests prompted Correa to test the joint statistical significance of the competition coefficients. Correa concluded from this exercise that before the establishment of the CAFC, the competition-innovation relationship is statistically significant; but after the CAFC’s introduction, the relationship is not statistically significant. Correa then estimated

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the relationship under the two regimes, i.e., with the two identified structural breaks. In both cases, he found that the relationship between competition and innovation is monotonically increasing before the structural break. Hashmi [44] and Correa and Ornaghi [24] also revisit the empirical Aghion et al. [3] model. Both use a negative binomial instead of a Poisson specification for the conditional mean function (to account for over-dispersion in the data), data on publicly traded manufacturing firms in the US and the Lerner index as a proxy for competition. Nevertheless, Correa and Ornaghi [24] find a monotonically positive relationship using total factor productivity growth, labor productivity and citationweighted patent counts as a proxy for innovation, while Hashmi [44]|, using only citation-weighted patent counts as a proxy for innovation, finds a monotonically “mild,” but negative relationship. There remains no clear consensus of empirical evidence that points to the direction of the effect of competition on innovation or on the productivity growth that is presumably engendered by innovation.

Empirical Results Using Natural Experiments to Identify the Effect of Competition The empirical IO literature on competition and innovation has exclusively focused on concentration and profitability to infer the level of competitiveness of an industry, and largely patents and R&D to capture the level of innovation. Another literature investigates the link between structural changes to the competitive environment and productivity. Following Holmes and Schmitz [50], this section presents a brief review of that literature. We would first like to draw attention to some comments made by Holmes and Schmitz [50] regarding the measurement of competition and how competition affects productivity. Holmes and Schmitz [50] claim that concentration and profitability are inadequate at identifying structural changes in a competitive environment and, in fact, have the potential to mislead. To illustrate, they consider an industry that has a strong trade barrier and is made up of small, unproductive firms. The government then lifts the trade barrier, subsequently drawing the attention of large, highly productive to enter the market. From the perspective of the researcher who observes only market shares and profitability, they might conclude that the industry became less competitive due to a significant increase in concentration and profitability. This, however, is at odds with the conventional thinking that less entry barriers stimulate competition. They also note the selection effect of competition, whereby relatively unproductive firms are “selected out” of an industry because they cannot compete effectively. Taken altogether, market concentration may increase in a more competitive state. Moreover, to the extent that productivity is positively correlated with profitability, average profitability will also increase in a more competitive state. Notwithstanding the above, Holmes and Schmitz [50] note that there is no model that can comprehensively explain why or how competitive pressure induces firms to be more productive. The body of evidence strongly suggests that it does, however. Matsa [71], for example, found that incumbent supermarket retailers significantly upgraded their inventory systems (to maximize product availability) after Wal-Mart

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entered their local markets. Importantly, the increase in productivity cannot be attributed to an increase in market or average firm sizeas demand did not all of a sudden increase for the existing firms, nor did the existing firms substantially increase in scale. The observed gain in productivity was, therefore, largely a response to increased competitive pressure. Competition can also boost productivity by lowering the opportunity cost of investment. For example, Schmitz [92] found that plant managers were reluctant to adopt new managerial practices because they feared losing profits to a job strike. From this, he argues that the competitive process – which in the absence of innovation tends to shrink margins over time – will reduce forgone profits and thereby spur investment into new, more efficient forms of management. Major shifts in the competitive landscape are perhaps the best way to identify the effect of competition on innovation. Holmes and Schmitz [51] examine the effect of railroad transportation on water shipping in the 19th and 20th centuries. Before transportation by railroad was economically feasible in the US (1850s), freight transportation by water was effectively the only way to ship cargo across the nation. This meant that ports not only had tremendous market power, but also had the incentive to keep it. Their market power was heavily weakened, however, when railroads became a viable alternative for transportation. Railroads undermined the market power of ports in two ways. On one hand, railroads gave consumers easier accessibility to other ports. So, if a consumer was not happy with the price or service of a port, it could use a train to ship its cargo to another port. On the other hand, railroads could in some cases entirely replace the function provided by water services. Ultimately, the threat that railroads presented to ports manifested as an effort by the latter to increase productivity. In the same vein, Galdon-Sanchez and Schmitz [35] and Schmitz [92] examine the effect of Brazil’s entry into the lower Great Lakes iron ore market during the 1980s. They note that before Brazil’s entry markets were characterized by few and distant producing locations and high transportation costs, which fostered significant market power. Market power was in fact evident since unions and local government exercised their power to extract as much of the surplus from iron ore producers as possible. The market power of iron ore producers around the Great Lakes eventually eroded, however. Due to a substantial decrease in transportation costs, Brazil entered the iron market around the Great Lakes in the 1980s. This put tremendous price pressure on the domestic iron ore producers and, in turn, pressure to improve labor productivity. Labor productivity actually doubled in the mid-1980s, and GaldonSanchez and Schmitz [35] demonstrate that the source of productivity growth was not due to the closing of inefficient mines or increases in scale, but rather surviving mines that made investments to lower costs. Finally, Syverson [110] investigates the effect of spatially dense competition on the distribution of productivity in the US ready-mix concrete industry. He finds that more densely clustered markets exhibit higher average productivity and lower productivity dispersion. The reason for this is that more densely clustered markets lower switching costs for consumers, and thus an inefficient firm is more likely to

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exit an industry that is highly dense. In turn, average productivity and productivity dispersion will increase and decrease, respectively, in more competitive markets.

Conclusion In this chapter, we have demonstrated what market structure is, how it is measured and how it can affect productivity and innovation. Such a chapter of course leaves much room for other studies and perspectives and we do not expect that this handbook chapter will satisfy all of those in this rather dense literature. We do trust, however, that our treatments and perspectives on important contributions to this literature are balanced and that they provide a relatively complete perspective on such an important issue in regard to a broad array of assumptions and methodological approaches.

Cross-References  Capacity and Capacity Utilization in Production Economics  Cost, Revenue, and Profit Function Estimates  Multiproduct Technologies

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Production Under Uncertainty

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Contents Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Uncertainty and Risk . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The Stochastic Technology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Incorporating Randomness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Some Common Assumptions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The Structure of Stochastic Technologies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Stochastic Production Decisions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Producer Preferences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Cost Minimization, Duality, Risk-Neutral Probabilities, Fisher Separation, and More Duality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Revenue Cost and Graphical Illustration of Producer Equilibrium . . . . . . . . . . . . . . . . . . . Concluding Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Cross-References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Abstract

This chapter describes a formal model of a stochastic production technology. Alternative axioms and different structural restrictions are presented, and producer decision-making under uncertainty is examined. The presentation emphasizes the formal similarities between the stochastic production environment and

My thanks to Spiro Stefanou for comments that considerably improved the presentation. R. G. Chambers () Department of Agricultural and Resource Economics, University of Maryland, College Park, MD, USA e-mail: [email protected] © Springer Nature Singapore Pte Ltd. 2022 S. C. Ray et al. (eds.), Handbook of Production Economics, https://doi.org/10.1007/978-981-10-3455-8_6

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more traditional models of a nonstochastic technology and producer behavior under certainty. The nonstochastic multiple-output technology is shown to be special case of the more general stochastic production structure. Keywords

Production · Uncertainty · Risk · Producers · Cost · Decision-making under uncertainty

JEL codes

D21, D24, D25, D81, D84

Introduction Virtually all economic decisions involve uncertainty. Even such mundane transactions as buying a fast-food hamburger carry the potential for foreseeable, but uncertain, unpleasant outcomes. In such settings, the uncertainty extends over a relatively short time horizon, perhaps a day, and thus may seem relatively inconsequential. Producers, on the other hand, routinely make important economic decisions well in advance of exact knowledge of the conditions that will prevail when their products are ready for market. Traditionally, microeconomic analyses of producer behavior swept this awkwardness under an analytically convenient rug woven from the fibers of a nonstochastic decision environment. Of the reasons offered for this abstraction, perhaps the most basic was the most forceful. Nonstochastic decision environments are simpler than stochastic ones. And despite their relative simplicity, they still presented sufficient challenges to fully employ entire generations of production economists. Better, perhaps, to resolve the challenges posed by the simpler setting before attempting to resolve those posed by the more realistic, but more challenging, setting. Still, important contributions were made. Knight [26], by arguing that profit was the entrepreneurial reward for bearing uncertainty, made uncertainty the centerpiece of his theory of the firm. Arrow (1953, later translated as Arrow [3]), Savage [39], and Debreu [17] provided the basic analytic foundations for analyzing uncertain decision-making. Both Arrow [2] and Debreu [17] treated generalequilibrium settings. Arrow’s [2] analysis focused on a pure-exchange economy, but he suggested that “. . . the introduction of production would not be difficult.” And Debreu [17] later extended the basic model to incorporate uncertain production possibilities. Nevertheless, the bulk of the formal production economics literature remains focused on nonstochastic decision settings. This chapter’s goal is to provide a capsular depiction of a formal model of stochastic production technologies and then to use it to discuss briefly the fundamentals of producer decision under making uncertainty. The emphasis, following Magill and Quinzii [31] and Chambers and

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Quiggin [10], is on identifying formal similarities between models developed for nonstochastic production structures. The view taken is that nonstochastic models should be special cases of a more general stochastic production structure. Thus, readers of other chapters in this Handbook should encounter many familiar concepts, albeit slightly recycled to accommodate the more general decision setting. One thing this chapter will not attempt is a review of the stochastic production function literature (e.g., [24] and [25], Antle [1], Lapan and Moschini [28], Pope and Chavas [34], Chavas and Holt [14], and Pope and Just [35, 36]). In its place, this chapter shows why this approach is a restrictive special case of the more general Arrow-Debreu framework. Readers interested in more details on this approach are referred to the excellent overview presented in Moschini and Hennessy [33]. The chapter is organized as follows. The first section discusses the conceptual framework for analyzing uncertain decisions. A formal model of uncertain production is then developed. The first step is to recall some basic concepts from nonstochastic production analysis. These concepts are then extended to an uncertain setting. The result is a general model that contains traditional nonstochastic models as a special case. Alternative axioms for that technology are then discussed. After that, different structural representations of stochastic technologies are examined. The primary emphasis is on discriminating between an econometrically framed version of that technology and one that flows more naturally from the Arrow and Debreu contributions and is thus better grounded in axiomatic production theory. The penultimate section contains a brief treatment of producer decision-making in a stochastic environment, and then the chapter concludes.

Uncertainty and Risk Following Knight [26], economists distinguish between risk and uncertainty. Randomness that is “susceptible of measurement” by statistical calculation, historical experience, or deduction is called risk. Uncertainty is defined residually as randomness that is not “susceptible of measurement” by statistical calculation, historical experience, or deduction. Practically speaking, risk involves randomness to which objective probability assessments can be attached, and uncertainty involves randomness to which objective probability assessments cannot be attached. With the obvious exceptions of formal and informal gambling fora, such as card games, casinos, and publicly supported lotteries, most economic decisions over random outcomes involve uncertainty and not risk. A bit of introspection reveals why. Consider a simple coin-flip game where two individuals wager on the outcome of tossing a coin. If the coin is known to be fair, it’s a 50–50 proposition and the game involves risk. On the other hand, we’re all aware of the phenomenon of unfair coins. And so unless the coin is subjected to testing before the wagers are made, any assessment of its fairness is purely subjective. Thus, the game involves uncertainty. More generally, many of our daily endeavors are influenced by interacting random factors that are beyond our control and which we only understand imprecisely. Creating laboratory-like settings for evaluating the likelihood of different outcomes

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in such settings is not possible. Any probability assessments we make to aid our decision-making in such settings are inherently subjective. In discussing individual attitudes toward randomness, and in providing an axiomatic basis for making decisions about potentially random outcomes, distinguishing between risk and uncertainty is vital. In modelling random production structures, it is not. A complete and probabilistically-free theory of random production is available. The claim is not that probabilities are unimportant for decision-makers confronted by stochastic technologies. Whether that is true or not is the subject of decision theory and not production economics. Instead, the claim following Debreu [17] is that probabilities are not needed to describe a physical stochastic technology and that a complete theory of producer decision-making under uncertainty can be developed without any reference to the notion of probabilities. The formal structure, traceable to Arrow [2] and Savage [39], is common to both decision theory and financial analysis. There are two periods, 0 and 1. In period 0, which predates 1 and is called the decision period, no uncertainty exists. In period 1, uncertainty exists. Decisions made in period 0 are not uncertain, but they can have uncertain consequences (outcomes) in period 1. Uncertainty is modelled by a set of states, S, that is exogenous to producers and for which one, and only one, realization, s ∈ S, called a state or a state of Nature actually occurs in period 1. Uncertainty is resolved by a neutral player, Nature, making a unique choice from S. Nature’s choice is made after period 0 and cannot be affected by producers. In general, S can contain either a finite number of elements or an infinite number of elements. In this chapter attention is restricted to the case where S contains a finite number of elements. The elements of S are indexed, with an abuse of notation, as S = {1, 2, . . . , S}. Hence, S is used to denote both the set of states and its dimensionality. Subsets of S are referred to as events. From the perspective of a production economist, the main practical difference between the case of finite-dimensional S and infinite-dimensional S is that the latter results in random variables being defined as infinite-dimensional vectors. Conceptually, this causes no real problems, but mathematically it does require notions of differentiability that depart from those usually used by economists. Finite-dimensional S, on the other hand, allows us to operate in terms of commonly understood gradients and partial derivatives. Outcomes realized in period 1 are always expressible as real numbers. Thus, the choices made in period 0 that have uncertain outcomes in period 1, commonly referred to as acts, random variables, or state-contingent choices, are representable as vectors in RS . For an uncertain choice f ∈ RS , its realized value if Nature chooses s ∈ S is denoted by fs ∈ R. An example illustrates. At planting time, a farmer is uncertain how much rain will occur during the growing season. There are two possibilities, drought (D) and adequate rainfall (A). The farmer can either plant a drought-resistant seed variety (R) or a non-resistant seed variety (N ). If the non-resistant seed variety is planted, yield will be yAN if adequate rainfall occurs and yDN if drought occurs with yAN >

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yDN . If the drought-resistant variety is planted, yield will be yAR if adequate rainfall occurs and yDR if drought occurs with yAR > yDR . In this example, period 0 is “planting time,” and period 1 is “end of the growing season.” Abusing notation, S = {A, D} and the decision to be made is “non-resistant variety” or “drought-resistant variety,” that is, choose either yN = (yAN , yDN ) or yR = (yAR , yDR ) . (The notation  denotes the transpose of the vector.) The period 0 choice reduces to choosing between two real vectors, each with the characteristic that its sth element corresponds to the realized value for that choice if Nature chooses s from S. The farmer’s available choices can be represented in several ways. One is in tabular format: StateChoice Resistant Non-resistant . Adequate yAR yAN Drought yDR yDN Another is the corresponding matrix format: 

yAR , yAN Y = yDR , yDN

 ∈ R2×2 .

Our notational convention is that the mth column of an S × M matrix corresponds to the mth random variable. Each row, s ∈ S, on the other hand represents the corresponding outcome for each of the random variables in realized state s.

The Stochastic Technology When talking production, economists typically rely upon three interrelated concepts. Each characterizes the technology but views it from a different visual perspective. The concepts are a technology set, an input set, and an output set. The technology set, when depicted in R2 , consists of input-output combinations that lie on or below a technical frontier typically conceived as the graph of a production function. The input set, when depicted in R2 , consists of input combinations capable of producing a given output. These input combinations fall on or above another technical frontier, the isoquant. The output set, when depicted in R2 , portrays output bundles producible using a fixed input bundle. These output bundles fall on or below a technical frontier known, variously, as the transformation frontier, the production possibilities frontier, or the production transformation frontier. Figure 1, which measures a single output, z, on the vertical axis and a single input, x, on the horizontal axis, depicts the technology set as everything falling on or below the familiar “lazy-S” shaped production function. Figure 2 illustrates the input set for the case of two inputs, x1 and x2 . The input set is obtained by holding

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Fig. 1 The technology set

Fig. 2 The input set

output, z, constant and orthogonally projecting the locus of input bundles that can produce z onto R2 . Its lower boundary is the isoquant. More formally, identify input bundles as N -dimensional real vectors, x ∈ RN , and output bundles as M-dimensional real vectors, z ∈ RM . The technology set, T , is the subset of input-output space, RN × RM + , satisfying   T ≡ (x, z) ∈ RN × RM : x and z are technically feasible . + The input set is obtained by applying a point-to-set mapping (correspondence) to T . The notation P : A ⇒ B denotes a point-to-set mapping from the real subspace A to the real subspace B. That is, P takes points, a ∈ A, and maps them into subsets of B, B o ⊂ B. The input correspondence maps points in output space, z ∈ RM , into the subset of the input space, RN , containing the input bundles that can produce z:   X (z) ≡ x ∈ RN : (x, z) ∈ T . The image of the correspondence evaluated at z, X (z) , is the input set for z.

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Fig. 3 The output correspondence

Fig. 4 The output set

The output correspondence transmutes the process and maps input space into output space, Z : RN ⇒ RM , with   Z (x) ≡ z ∈ RM : (x, z) ∈ T . Figure 3 illustrates the output correspondence visually for the case N = 1 and M = 2, and Fig. 4 illustrates the output set Z (x0 ) obtained by projecting the surface in Fig. 3 onto RM for x0 . Its outer boundary is the production possibilities frontier. Formally, there is no need to discriminate between inputs and outputs. One can always think in terms of netputs letting positive realizations denote outputs and negative realizations denote inputs. This easily handles the case of intermediate inputs where some outputs from one production process are inputs in another. We discriminate between inputs and outputs because it allows us to develop our ideas using non-specialist language and widely familiar concepts. The input correspondence and the output correspondence are lower inverses of one another. That is,

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Z (x) = X− (x) ≡ {z : x ∈ X (z)} , and X (z) = Z − (z) ≡ {x : z ∈ Z (x)} . From the definitions, it is apparent that the following equivalence holds x ∈ X (z) ⇔ (x, z) ∈ T ⇔ z ∈ Z (x) .

(1)

The different perspectives from which to view production problems are equivalent. Different perspectives are used because different economic problems may require focusing on different aspects of the technology. For example, for problems involving the interplay between a single input and a single output, T is the natural choice. If the interplay between different inputs is the primary concern, X (z) is the natural focus, and Z (x) when the interplay between different outputs is the focus.

Incorporating Randomness Despite the prominent role that nonstochastic technologies play in economic theory, very few real technologies are nonstochastic. Everyone is familiar with very simple circumstances where seemingly identical efforts produce different outcomes. Most need go no further than attempting to draw two uniform circles freehand on a single sheet of paper to grasp this phenomenon. Try as we might, slightly different images will likely emerge each time the process is repeated. Just what causes such phenomena is not uncontroversial. And because our purpose is not to explain why such things happen, we attempt no concrete explanation. Instead, we outline a formal framework that permits incorporating such possibilities into models of producer behavior. To that end, assume that inputs, x, are committed in period 0 and that outputs, although also chosen in period 0, are only realized in period 1. After the producer makes that input-output choice, Nature makes a selection, s, from S that resolves uncertainty and determines what level of outputs are actually realized. Just what S incorporates and what constitutes a state is often vague and typically depends upon the problem setting. So, for an agricultural producer, Nature’s choice might involve picking such things as moisture and radiation that characterize the physical conditions under which plant growth actually occurs. Moisture and radiation are physical inputs that the producer knows can affect production but which he or she cannot feasibly know prior to planting. Hence, in this case, each state of the world may reasonably be interpreted as specifying distinct moistureradiation combinations, and S would include all such possible combinations. Of course, in other decision settings, producers may not care about moisture and radiation because those random inputs may not affect production outcomes. Then,

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S would likely be conceptualized differently to accommodate the relevant sources of uncertainty. Regardless of the setting, however, the states and S must satisfy several criteria. S and its elements must be exogenous to the producer. S must exhaustively characterize the uncertainty faced by the producer. Put in language used by Donald Rumsfeld,1 all unknowns are known, Nature’s choice must always be an element of S, and one element of S must be chosen. S must cover all random outcomes that can possibly affect production. And, finally, the states must be mutually exclusive so that no two states can occur simultaneously. For example, drought and adequate rainfall cannot both occur. As a practical matter, the exhaustive nature of S in the Savage set up is often violated. In describing criteria that states need satisfy, Savage (1954, p. 8) only refers to “. . . descriptions that might thinkably apply.” Despite this vagueness, it is critical that the producer knows each possible state and what it means. What the producer does not know in period 0 is which state will actually be chosen. But once Nature chooses an s from S, the production process becomes entirely deterministic. However, as has become increasingly apparent, individuals often face choices for which the range of alternative outcomes is unknowable or imperfectly foreseen, what Rumsfeld called unknown unknowns. There is a burgeoning literature on modeling and describing decision-making over unknown unknowns, but that remains beyond the scope of the current chapter. The conceptual approach to accommodating this uncertainty in modelling production decisions, traceable to Arrow [2], recognizes that just as commodities are differentiated according to their physical characteristics, location, and time provided, they also should be differentiated according to the state of Nature, s ∈ S, that eventuates. For example, when it doesn’t rain, those galoshes on your feet are superfluous attire. But when it does rain, the same galoshes handily keep your feet dry. Therefore, outputs chosen by the producer, z, before Nature’s choice is revealed should not only be subscripted according to their physical characteristics (e.g., wheat of a particular variety) but also according to the state of Nature that occurs. Thus, multiple stochastic outputs are not represented by vectors but as matrices, z ∈ RS×M (and not z ∈ RM ) with typical element zsm denoting the amount of the mth commodity produced if state of Nature s ∈ S is realized. And so if in period 0, the producer chooses the input and stochastic output mix, (x, z) ∈ RN × RS×M and then Nature picks s ∈ S, the realized period-1 output bundle is zs· = (zs1 , zs2 , ..., zsM ) , that corresponds to the sth row of z. 1 Donald

Rumsfeld famously said: “‘. . . there are known knowns; there are things we know we know. We also know there are known unknowns; that is to say we know there are some things we do not know. But there are also unknown unknowns – the ones we don’t know we don’t know” (US Department of Defense, 2002).

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With these changes in notation and interpretation, the definition of the stochastic technology becomes   T = (x, z) ∈ RN × RS×M : x and z are technically feasible ,

(2)

which is virtually indistinguishable from that for a nonstochastic technology. The main difference is the domain for z. The formal similarity in definitions is especially attractive because it implies that concepts used to analyze nonstochastic production decisions can be productively recycled to analyze stochastic ones. Of course, because more forces are at play, actual decisions are now more complicated. But fundamentally, one is still interested in input substitutability, output transformability, and the interaction between inputs under the control of producers and outputs realized. What’s new is Nature’s role in this process. Nature’s role necessarily complicates things, but no reason exists to suggest that it invalidates ideas developed in nonstochastic models. Indeed, in a formal setting, one would hope that familiar nonstochastic models would emerge as the special case of the more general stochastic technology for S = {1} . Mathematically, when one wishes to focus on certain aspects of a Q-dimensional set, one often orthogonally projects the set onto a lower dimensional space holding one or several elements of the set constant. (This is the basic idea behind input sets and output sets.) Using similar logic, nonstochastic technologies can be visualized as projections of the stochastic technologies from RN × RS×M onto, say, RN × RM for a particular s ∈ S. However, this projection is only economically relevant if it is known for sure in period 0 that Nature’s choice will be s. Some examples will help illustrate. First, take the case where M = 1. The producer produces a single stochastic output. In our formalism, in period 0 the producer chooses x ∈ RN and z ∈ RS . The sth element of z, zs , is the amount of the output produced if Nature picks s. Something akin to Figs. 3 and 4 remains a natural way to depict the output choice. But despite the similarity of the visual representations, the intuition differs subtly. For now the horizontal axis in Fig. 4 measures output realized from x if Nature’s choice is 1 ∈ S = {1, 2} , and the vertical axis measures output produced if 2 ∈ S = {1, 2} is chosen. As drawn, T permits transforming z1 into z2 holding x constant. This would be done, for example, by diverting efforts to prepare for state 1 to preparing for state 2. In this instance, the visual depiction for X (z) remains essentially unchanged from Fig. 2. And Fig. 1, as drawn, corresponds to how much of one state-specific output, say z1 , would be produced as x varies now holding the other state-specific output, z2 , constant. It emerges from projecting the three-dimensional surface in Fig. 3 onto the (x, z1 ) coordinate plane holding z2 constant. Now suppose that M = 2 so that two stochastic outputs are produced. Then instead of just one two-dimensional transformation frontier for a fixed x, as in Fig. 4, there are now six: between z11 and z21 (same output, different states), z11

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and z12 (same state, different outputs), z21 and z22 (same state, different outputs), z11 and z22 (different outputs and different states), and z12 and z21 (different outputs and different states). Presumably, one might represent these distinct output sets as similar to Fig. 4. For our current purposes, however, the specific shape is not crucial. And, in any case, it likely varies considerably across problem settings. What is important is that trade-offs occur both across products and across states of Nature. In the following, we shall always speak in terms of outputs as being random and inputs as being deterministic. Again this is a naming convention and is pursued to promote an intuitive grasp of the basic ideas while avoiding an excessively complicated notation. The case of stochastic inputs is easily handled within our formalism. To illustrate, consider the example of a farmer who plants a crop in early spring knowing that it might be subject to a pest infestation closer to harvest time that can only be treated upon emergence. Then at planting time, when the farmer picks z ∈ RS , he or she also picks a stochastic treatment plan t ∈ RS to accommodate the potential pest outbreak.

Some Common Assumptions Economic analysis always requires assumptions. This subsection discusses the technical details that facilitate analysis of stochastic technologies. As a general rule, the mathematics relevant for stochastic technologies are the same as for nonstochastic ones. Where differences emerge, they typically result from the subtle difference between a stochastic commodity, z ∈ RS and an S-dimensional bundle of distinct, but nonstochastic, outputs. The discussion focuses on those differences, and readers requiring a more complete discussion of technical issues are referred to Chambers and Quiggin [10]. And because the focus is stochastic production and not multiple outputs, the remaining sections exclusively treat the case of a single stochastic output so that T ⊂ RN × RS instead of RN × RS×M . The assumptions relevant to modelling stochastic production can be classified as feasibility assumptions, continuity assumptions, disposability assumptions, and curvature or smoothness assumptions. Each will be discussed in turn. Feasibility assumptions ensure that T , defined above, is non-empty. That’s simple enough and needs relatively little motivation. After all, if T = ∅, it would be hard to justify economists studying it intensively. Continuity assumptions describe the type of set that we want T to be. In the main, these are chosen for mathematical convenience, and depending upon context they can assume different forms. For example, if our interest lies in studying input behavior, one might only need restrictions on the input correspondence, X. To be sure, because X and Z are lower inverses of one another, this results in restrictions on Z. But those induced restrictions on Z may not be the same invoked if one were only studying output behavior. For our purposes, however, simplicity is best even if it results in a loss of generality. Therefore, we require T to be compact, that is, as a subset of RN × RS , it is both closed and bounded. In certain contexts, this can prove overly strong. For

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example, it rules out technologies satisfying global constant returns to scale. But the gain in simplification in presenting basic results more than compensates for any lost generality. As the chapters “Distance Functions” and “Bad Outputs” demonstrate, poorly chosen disposability assumptions can be quite limiting even when technologies involve no uncertainty. Similar issues occur here. For example, imposing free disposability of certain inputs and outputs can violate the laws of Nature as we now understand them. On the other hand, disposability properties frequently convey the most important analytic consequence of any assumptions – the ability to write down a “function representation of T .” Where, for example, would firm theory or empirical production analysis be if one could not describe the technology in functional terms, be it a production function, a transformation function, or an input requirement function? The disposability assumption we invoke, while acknowledging its potential limitations, is free disposability of z ∈ RS . Formally, T ⊂ RN × RS satisfies z ∈ Z (x) ⇒ z ∈ Z (x) for z ≤ z. (Note, attention is here restricted to M = 1.) Free disposability of z has a number of consequences, but for our immediate purposes, two are particularly important. This first is that Z (x) = ∅ ⇒ 0S ∈ Z (x) . (Notation: For any r ∈ R, r S denotes the element of RS where r always occurs regardless of Nature’s choice. Alternatively, one can write r S = r1S . 0S , thus, denotes the origin associated with the usual orthonormal basis for RS .) In words, if something can be produced from a given x, one can always choose to produce nothing from that same input bundle. Put yet another way, complete output inefficiency is possible. The second consequence is that z ∈ Z (x) ⇔ D (z, x) ≤ 0,

(Indication)

(3)

where   D (z, x) ≡ min β ∈ R : z − β S ∈ Z (x) if there exists β ∈ R such that z − β S ∈ Z (x) and ∞ otherwise. Indication ensures that D (z, x) fully characterizes Z (x) and hence, by (1) T and X (z) . Thus, knowledge of D (z, x) is equivalent to knowledge of the stochastic technology.

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Fig. 5 D(z, x) illustrated for S = {1, 2}

The extended-real-valued function, D, is called either a directional output distance function or a shortage function.2 It gives the smallest amount of 1S ∈ RS that can be subtracted from z and leave the result within the output set Z (x) . It is a directional notion because geometrically it involves translating z in the direction 1S . The choice of 1S as the direction is motivated by the fact that it allows interpreting distance as being measured in units of the sure thing, which is formally equivalent to a degenerate random variable (see below). Figure 5 illustrates the determination of D (z, x) geometrically. As drawn, z lies outside of Z (x) . The direction 1S is illustrated by the vector (1, 1). D (z, x) is then derived by translating the point z in the direction parallel to (1, 1) toward the boundary of Z (x) until the translated point, z−D (z, x) 1S , just lies on the boundary of Z (x) . For the case drawn, a positive amount of 1S has been subtracted from z to make it feasible and thus D (z, x) > 0 signalling that x could not have produced that particular z. Because they focus on trade-offs, economists routinely use calculus-based arguments. In a production context, these are usually justified by curvature assumptions placed directly upon X (z) , Z (x) , or T or by smoothness assumptions placed upon a function representation of the technology such as D (z, x) . Depending upon the context, required assumptions can differ subtly. For example, assuming that Z (x) ⊂ RS is a strictly convex set ensures that D (z, x) is differentiable in z. It does not ensure, however, that D (z, x) is differentiable in x. Figure 6 illustrates. At x0 , the boundary of Z (x0 ) is strictly convex and thus nicely smooth in (z1 , z2 ) space. For points on the boundary of Z (x0 ) , D (z, x) will

2 Luenberger [30] coined the “shortage” terminology. Later building upon Luenberger [30], Chambers, Chung, and Färe [7] introduced the “directional distance” terminology to emphasize its similarities to and differences from distance function of the type studied by Shephard [? ]. See the chapter entitled “Distance Functions” by Chambers and Färe in this volume.

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Fig. 6 Smooth output set

be differentiable in z. But, as drawn, the correspondence possesses a kink as one varies x away from x0 . Similar reasoning shows that requiring X (z) ⊂ RN to be a strictly convex set ensures differentiability of D (z, x) in x, but not in z. On the other hand, one can easily imagine smooth boundaries for nonconvex Z (x) that will generate differentiable D (z, x). Broadly speaking, whether differentiability assumptions are crucial to analyzing production decisions under uncertainty is, as in most other areas of economic analysis, context-dependent. In the main, non-differentiability if it occurs will only occur at isolated points (sets of measure zero) and can be safely ignored. In such instances, differentiability assumptions are akin to continuity assumptions discussed above, analytically convenient but not terribly substantive. But in other cases, those sets of measure zero can be acutely interesting and can convey important economic information. In fact, several long-standing puzzles or paradoxes in economics (e.g., the Allais Paradox, the Ellsberg Paradox, the endowment effect) have been accommodated analytically by admitting the potential for non-differentiabilities. From the perspective of a production economist, non-differentiabilities signal a lack of substitutability or transformability. Economic analysis of alternative production schemes often hinges on a presumed ability to substitute either away from or toward economically attractive alternatives. For example, tax policies directed at reducing the use of an environmentally damaging input are likely ineffective if no substitutes exist for the damaging input. That lack of substitutability is often manifested analytically as a lack of differentiability. Curvature assumptions also often reflect economic regularities in which economists typically believe. The laws of the diminishing marginal productivity of inputs (traceable at least to von Thünen’s (1826) early nineteenth-century studies), the diminishing marginal rate of technical substitution, and the increasing rate of marginal transformation are common production examples. Each is formalized

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by imposing convexity on one of the set-based representations of the stochastic technology. Our focus is on the latter two, the diminishing marginal rate of technical substitution and the increasing marginal rate of transformation. Assuming that X (z) is a convex set yields input sets as depicted in Fig. 2. The marginal rate of technical substitution between any two inputs at a particular point on the isoquant is given by the slope of the tangent hyperplane to that point. We compute it using D (z, x) as follows. For any point, x o , belonging to X¯ (z) = {x : D (z, x) = 0} , use an appropriate version of the implicit function theorem to obtain by partial differentiation the small variation in xk that will exactly balance a small variation in xj (thus keeping one on the isoquant) as ∂D (z, x o ) /∂xj ∂xk =− , ∂xj ∂D (z, x o ) /∂xk which (depending upon naming conventions) is the marginal rate of technical substitution of xk for xj at (z, x o ) . When the input set is nicely smooth as in Fig. 2, the marginal rate of substitution is well defined, is well understood, and exhibits a diminishing marginal rate (input substitution becomes increasingly difficult). On the other hand, if X (z) is convex but possesses a kinked boundary, this standard procedure will not work. The presence of a kink implies the existence of an infinity of supporting hyperplanes, each with different slope, and signals a lack of smooth substitutability between inputs. The economic consequence is that local factor demands will exhibit “stickiness” to a continuum of relative factor-price changes. Assuming that Z (x) is convex allows one to depict output sets as in Fig. 4. But one must now remember that this familiar shape’s interpretation is subtly different than in the nonstochastic multi-output case production case. In the latter, the smooth shape implies that redirecting inputs from the production of one output to the production of another effectively permits transforming one into the other. You get more corn, from a given bundle of inputs, by sacrificing some wheat. Here, however, the message is that to get more corn in one state of Nature, say 1 ∈ {1, 2} , you have to sacrifice corn in state of Nature 2. Output transformation occurs not across dissimilar products but across different states of Nature for otherwise identical products. Mechanically, one obtains the marginal rate of transformation between statecontingent or state-specific outputs exactly as one obtains the marginal rate of substitution. Define the stochastic or state-contingent production possibilities frontier by   Z¯ (x) = z ∈ RS : D (z, x) = 0 .

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Holding all inputs and state-contingent outputs except s and m constant, implicit differentiation yields the marginal rate of transformation as ∂zs ∂D (z, x) /∂zm =− . ∂zm ∂D (z, x) /∂zs When Z (x) takes the shape in Fig. 4, this marginal rate of transformation is always well defined. However, if Z¯ (x) contains a kink, the marginal rate of transformation is not well defined. Again a kink in Z¯ (x) implies a lack of substitutability or, perhaps, transformability between outputs in different states of Nature. Thus, if state 1 were identified with ideal growing conditions for corn and state 2 with less than ideal growing conditions for corn, a kink in the stochastic production possibilities frontier for corn would signal that farmers experience stickiness in transforming corn in the ideal state of Nature into corn in the less than ideal state of Nature. This, in turn, would signal physical difficulties in redirecting inputs to different uses in preparing for different states of Nature. As later developments reveal, a broadly studied class of stochastic technologies always possesses such kinks. Other subtle differences can emerge between how one interprets curvature or smoothness properties for nonstochastic technologies and how one interprets similar properties in studying stochastic technologies. We illustrate with Fig. 7. There each point on the ray passing through (1, 1) that is labelled the “Bisector” is characterized by z1 = z2 . The terminology reminds us that the ray passing through (1, 1) splits R2 in half. In the nonstochastic case, points on this ray contain equal amounts of two different outputs and are not particularly meaningful. But in the stochastic context, points on that ray have the same output in both states of Nature. In other words, these are the non-random (or degenerately random) production outcomes. In finance, elements on that ray have certain (nonstochastic) returns in period 1. And, 1S is often referred to as the riskless asset. (Remember random variables are vectors in RS . Thus, financial assets with random period 1 payouts are representable as vectors in RS .) Fig. 7 Stochastic production possibilities

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Depicting radial contractions or expansions of 1S as intersecting Z (x) , thus, represents the nontrivial assumption that the input bundle x can be used to produce some output level nonstochastically. That is, all uncertainty can be controlled. Whether this is true depends upon the technology, but the important economic implication is that whether the producer exposes himself or herself to uncertainty now represents a conscious choice. For example, as noted, free disposability of output requires Z (x) = ∅ ⇒ 0S ∈ Z (x) , so that all uncertainty can be avoided by the simple device of choosing not to produce. The riskless asset is not the only geometric surface in RS to possess a subtly different interpretation when discussing random variables. Recall that an affine hyperplane, as a subset of RS , is defined as the set of points satisfying   H¯ (α, p) = z ∈ RS : p z = α, α ∈ R . Visually, an affine hyperplane is represented as a linear surface that is parallel to the hyperplane passing through the origin with normal p ∈ RS . In the definition, the location parameter, α, represents the amount that the hyperplane through the origin is translated in the direction of 1S to obtain the surface H¯ (α, p) . When p ∈ RS+ / {0} , hyperplanes in RS depict sets of random variables sharing the same expected value, pα1S , as defined relative to the probability measure p p 1S

.

To see why, consider two facts. First, the homogeneity properties of affine hyperplanes ensure that H¯



α p ,  S  S p 1 p 1



= H¯ (α, p) .

Dividing both the normal and the location parameter by the same constant leaves the hyperplane unchanged. And second 

p  p 1S



1s = 1.

When p ∈ RS+ / {0} , elements of the normal to H¯ p p  1S

RS+ / {0}



α , p p  1S p  1S

are non-negative

and sum to one. Hence, ∈ is interpretable as a probability measure,   z, with random variables, z ∈ RS , are interpretable and its inner products, pp1S as the expected values of those random variables.

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The Structure of Stochastic Technologies Generally speaking, economists possess little exact knowledge of what characterizes physical production technologies. As a practical matter, analytic tractability routinely trumps realism in choosing our models. Thus, an economist’s representation of the technology is often better viewed more as a parable about production than as a representation of the physical world. Hopefully, these parables don’t do too much damage to reality. But when all is said and done, our primary interest remains in how technologies constrain or condition economic behavior. And thus, if models can be designed that do not controvert the physical laws of Nature and that do not trivialize the economic responses of producers, one is likely ahead of the game. Unfortunately, economists rather persistently specify technical models that do controvert the laws of Nature (see, e.g., the chapter entitled “Bad Outputs” by Murty and Russell in this volume) and that do trivialize the economic responses of producers. In examining the structure of stochastic technologies, we must first decide which assumptions to impose on T . To preserve generality, they are kept to a minimum: a) T = ∅ (feasibility); b) T is compact (continuity); and c) z ∈ Z (x) ⇒ z ∈ Z (x) for z ≤ z (free disposability of stochastic output). As noted, a) and b) are relatively harmless and do relatively little damage to reality. Free disposability of z, on the other hand, is often violated but is maintained because it simplifies the presentation. These assumptions ensure that D (z, x) characterizes T . But together, they are not sufficient to identify a specific form for D (z, x). How to proceed? If one were to poll economists on a functional form for D (z, x), the smart money would bet that something similar to D (z, x) = A (x)

S

zsαs ,

(4)

s=1

would be among the favored answers. The Cobb-Douglas form is surely among the most popular in economics. Its popularity is traced to several roots. It admits nonlinearity while remaining relatively simple. Its economically interesting characteristics are captured by relatively few parameters. Its separability properties (see the chapter by Primont on “Functional Structure” in this volume) ensure that the interactions between those characteristics are relatively simple. And its supermodular structure ensures behavioral responses are “nice.”3

3 As it turns out, the Cobb-Douglas specification is actually not a viable candidate for characterizing

a directional distance function. It fails to satisfy a key regularity property such functions must possess. This regularity condition, which is referred to as the translation property, is important from a technical perspective. But it is not crucial to our discussion of stochastic technologies and thus has been ignored.

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Any choice of form for D (z, x) imposes structure on T . Judging that structure’s plausibility remains a craft that requires understanding the problem and the basic nature of the technology. What truly distinguishes nonstochastic technologies from stochastic ones is the latter’s treatment of output as random rather than as deterministic. Consequently, our structural focus is on D (z, x) s treatment of z. For (4), the marginal rate of transformation between two stochastic outputs is am zs ∂zs =− . ∂zm αs zm The associated partial elasticity of transformation (please see the chapter in this volume by Russell entitled “Elasticities of Substitution”) between these two outputs is thus  ∂ ln zzms = 1,  m ∂ ln ∂D/∂z ∂D/∂zs for all pairs of realized outputs. A priori, little reason exists to suggest that the elasticity of transformation should parametrically equal one (or any other value). The message this conveys is that, along the stochastic production possibilities frontier, percentage changes in ratios of stochastic outputs are exactly matched by equal percentage changes in marginal rates of transformation. This behavior mirrors the behavior in input space of Cobb-Douglas production functions. Thus, even though it may possess no strong factual basis, it is conceptually familiar. By appropriate choice of the parameters, this specification also admits stochastic production possibilities frontiers approximating that drawn in Fig. 4. Before the late 1990s, however, the most common economic specification of stochastic technologies did not resemble Fig. 4. Instead, something akin to the rectangular Z (x) in Fig. 8 was the most popular choice.4 The marginal rate of transformation between stochastic outputs is only well defined along a portion of the boundary of Z (x) in Fig. 8. Depending upon your viewpoint, the slope of the flat surfaces is either zero or infinity (in the limit). And, so, if one is zero, the other is infinity. Moreover, at the “kink” the marginal rate of transformation is not well defined. In its place, we find the continuum of potential marginal rates of transformation, (−∞, 0) .5 That kink, of course, is a set of measure zero on which differentiability of D (z, x) fails. It’s economically interesting because it represents the most efficient z that can be produced using x. Any other z producible using that

4 As

far as I am aware, Chambers and Quiggin [8] were the first to recognize that Z (x) for this particular specification took this shape. 5 Formally, gradients are replaced by subgradients at the kink. Visually, subgradients are represented by the infinity of hyperplanes tangent to Z (x) at the kink.

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Fig. 8 Cubical production possibilities

x results in a lower output for at least one state. In probabilistic terms, the outer vertex of Z (x) first-order stochastically dominates all the other elements of Z (x) . Figure 8 exhibits either “output-price-nonjointness” or “perfect output complementarity.” The former terminology is due to Kohli (1983). The latter is analogous to the consumer case where Leontief preferences identify perfect complements. The economic consequences of both are easily grasped by considering what Fig. 8 implies when the axes correspond to two distinct outputs for a nonstochastic technology. In that case, if x were held constant, altering relative output prices would elicit no production response from the producer. Thus, even though one output becomes economically more attractive as its price rises, the producer does not respond by producing more of it at the expense of producing less of the now relatively less attractive output. Because there is no substitution effect, the only way for supply to respond is by adjusting x. Imagine, in a stochastic setting, a farmer with only one tractor who has already contracted to employ a fixed amount of labor. Suppose also that the farmer must decide how to allocate that labor and that tractor without knowing what final growing conditions will be. In that setting, consider two decision scenarios. In the first, a government program guarantees that the farmer receives a fixed price for her crop. In the second, no government programs exist, and the farmer must take whatever price the market offers. The first scenario provides the farmer with perfect price insurance; the second provides none. Do you believe that the farmer produces in the same fashion under both so that the provision of perfect price insurance has no substitution effect? If your answer is no, your intuition aligns with a Z (x) resembling Fig. 4 because that technology allows one to adjust the mix of stochastic outputs holding x constant. A yes answer, on the other hand, aligns with Fig. 8 because the only available output adjustment involves throwing away output in one or both of the states. The Z (x) depicted in Fig. 8 can be identified with the stochastic production function model of the technology defined as

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  Z pf (x) = z ∈ RS : zs ≤ fs (x) , for all s ∈ S

{zs : zs ≤ fs (x)} , =

(5)

s∈S

where each fs : RN → R is a state-specific production function and the random variable f (x) = (f1 (x) , ..., fS (x)) ∈ RS is the stochastic production function. This technology works as follows. When input choices are made, the producer only knows that one of S different ways of using the chosen x will occur. But she does not know exactly which one. It’s as though she picks x and then spins a roulette wheel to determine the technology that will prevail. The producer cannot allocate the chosen x to perform different actions in response to realizations of s ∈ S. Instead, the only true choice is the maximal output that occurs in each state. The D (z, x) for this technology is derived as   D (z, x) = min β : z − β S ∈ Z (x) = min {β : zs − fs (x) ≤ β, s ∈ S} = max {zs − fs (x)} .

(6)

s∈S

Thus, assuming that s ∈ S corresponds to the state with maximal zs − fs (x) , ∇z D (z, x) = es where ∇ denotes the gradient with respect to the subscripted argument and es ∈ RS is the sth element of the usual orthonormal basis. Taking pairwise ratios of elements of ∇z D (z, x) yields either zeroes or undefined terms. The associated kink occurs at z = f (x) ∈ RS . The stochastic production function model is a natural consequence of two distinct analytic traditions. The first is the single-product production function familiar from intermediate micro theory. And the second is Haavelmo’s (1943) classic identification of the “econometric error” with the equation as a whole and not any single variable in the set of dependent and independent variables. The most common version of the model melds these ideas into the generic form zs = fs (x) ≡ g (x, εs ) ,

s∈S

(7)

where ε ≡ (ε1 , ε2 , ..., εs ) ∈ RS is an exogenously determined random variable and g : RN +1 → R. The assumption is that, prior to choosing x, the producer

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knows g and ε but not its specific realization. Hence, choosing x immediately results in the unique choice of rational z as (g (x, ε1 ) , ..., g (x, εS )) . How ε is determined is rarely broached and even less rarely explained. Typically, ε is chosen to satisfy analytically convenient statistical properties rather than with regard to the actual decision setting. This is a decidedly econometric view of stochastic production. In econometric terms, ε provides the “sample space” for the stochastic factors affecting production.6 As such, the stochastic production function specification naturally possesses important advantages in formulating statistically estimable versions of g. That is most easily seen by considering its simplest version, zs = g o (x) + εs ,

s ∈ S.

Realized values of the random variable z, in this specification, decompose into two components. One, g o (x) , is deterministic with its parameters amenable to parametric estimation using distributional assumptions on the other, zs − E [ε] . Studies using some version of (7) are varied and include theoretical and empirical analyses (see, e.g., Fuller [19], Feldstein [18], de Janvry [16], Stiglitz [? ], Batra [4], Just and Pope [24] and [25], Moscardi and de Janvry [32], Holström [22], Antle [1], Lapan and Moschini [28], Pope and Chavas [34], Chavas and Holt [14], and Pope and Just [35] and [36]). The cost of econometric tractability, however, may be economic plausibility. Chambers and Quiggin [10] provide a detailed critique of the economic implications of the stochastic production function model, which they refer to as output-cubical, and an axiomatic derivation of it in a decision-theoretic setting. Not the least of their critiques is the lack of analytic tractability that the “kinky” nature of this specification imposes on function representations of T . But there are also other conceptual problems. To give a flavor of some of these, yet another agricultural example is useful. Let z correspond to a stochastic crop output, x to the vector of inputs controlled by the producer, and ε to a composite of inputs beyond the producer’s control such   as natural moisture (m ∈ RS ), radiation r ∈ RS , and pest infestation b ∈ RS . Specifically, set εs = ϕ (ms , rs , bs ) ,

s∈S

so that zs = g (x, ϕ (ms , rs , bs )) ,

6 Most

s ∈ S.

applications treat the case where S is infinite-dimensional and thus take ε to be an interval of the real line.

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This specification requires the state-specific inputs, (ms , rs , bs ) , to be weakly separable from x. So, for example, if x contains pesticides, the marginal rate of substitution between pest infestation and moisture must be independent of the amount of pesticides applied. Similarly, if x contains chemical fertilizer, the marginal rate of substitution between natural moisture and radiation must be independent of fertilizer. Both are restrictive if not implausible. The familiar Edgeworth box and Lerner-Pearce diagrams, which provide the backdrop for the classic derivations of the Stolper-Samuelson theorem, the Rybczynski theorem, and the factor-price equalization theorem, suggest a natural way to move from the representation in Fig. 8 to that in Fig. 4. In a nonstochastic setting, the formal framework is the input-nonjoint production model (see, e.g., Chambers [5], pp. 286–8). Translated to a stochastic setting, the input-nonjoint framework models Z (x) as

Z N (x) = z ∈ RS : zs ≤ rs (xs ) , for all s ∈ S, x =



 xs .

s

Here each rs : RN → R is a state-specific production function that maps the input bundle allocated to s, xs ∈ RN , into a state-specific output. The associated random output is r (x) = (r1 (x1 ) , ..., rS (xS )) . The main difference from the output-cubical model is that the total bundle of inputs, x, is now allocable across state-specific tasks. Thus, the random variable r (x) changes as the allocation, (x1 , x2 , ..., xS ), changes even if x is held constant. Chambers and Quiggin [10] call Z N the state-allocable model. Figure 9 shows how this ability to allocate inputs to different s-specific tasks affects the stochastic production possibilities frontier. Suppose that only one input, call it effort, exists. Suppose further that the production process is agricultural and that the crucial factor determining S is the level of natural moisture. It’s easy to imagine that too much moisture is a bad thing because it will drown the plants or animals involved. Conversely, too little moisture is also a bad thing. Thus, in choosing how to apply her or his effort, the farmer must decide on how to allocate that effort to prepare for different levels of moisture. Suppose that the effort allocation choice is reflected by x1 + x2 = x in Fig. 9. For that allocation, maximal production is determined by (r1 (x1 ) , r2 (x2 )) as illustrated in the figure. A rectangular stochastic production possibilities frontier emerges for that allocation. Now suppose the same farmer almost simultaneously receives a flash of insight (or a long-term forecast) that the likelihood of bad drought conditions had diminished while the likelihood of massive flooding had dramatically increased. If given the chance to change the effort allocation, changing  without total x, the farmer responds by changing the allocation to x1∗ , x2∗ , the state allocable model permits the maximum output bundle to move to r1 x1∗ , r2 x2∗ , representing a different stochastic output mix and a different random variable. The output-cubical model would permit no such reallocation. Extending the thought experiment to all possible moisture levels, the effort reallocations define a contin-

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uum of new rectangular production possibilities frontiers, with the outer vertexes of these rectangles tracing out a representation approximating Fig. 4. For real-world settings, neither the output-cubical model nor the state-allocable model is likely correct. The same is probably true for any specific functional structure that one chooses. Paraphrasing W.M. Gorman [20], choosing a functional structure is about choosing what part of the model to analyze in detail and what to leave to later, and hopefully more thorough, analysis. Several points bear emphasis. First, choosing a specific structure limits the issues that can be examined. For example, the output-cubical structure predetermines the degree of output transformability. Thus, the rate of output transformability cannot be informatively investigated using that structure. Second, states of Nature are neither ex ante “good” nor “bad.” It can and often will depend upon how one prepares for them. Insurance provides an example. Purchasing insurance, formally speaking, is a decision to gamble, just as is purchasing a lottery ticket. You pay a premium, and if a particular state occurs, you get an indemnity back. Otherwise, you get nothing. If you receive no indemnity, ex post you have less money than if you had not taken the insurance gamble and, in that sense, you have lost the gamble. Most likely view not collecting on insurance as a good thing. Nevertheless, you have clearly lost the gamble that something bad would happen. Reversing the analogy, purchasing a lottery ticket, usually perceived as gambling, is formally equivalent to buying insurance for the unlikely event that your chosen number gets picked. So, too, for bingo and shaking hands with one-armed bandits. Formally, they can be viewed as insurance activities. The same phenomenon applies for stochastic production systems. States are not fundamentally bad or good. Instead, they reflect Nature’s role in the production process. How one prepares for them determines whether a good or a bad outcome emerges. Figure 9 again illustrates. Continuing our production parable, suppose that the farmer responds to his or her insight that flooding was more likely by reallocating effort. Output in the reallocated situation (associated with ∗) is higher

Fig. 9 State-allocable production possibilities

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in state 2 than in state 1. Thus, if that allocation were chosen, one expects the farmer to be hoping that 2 occurs because it involves a higher output. Contrast that with the original allocation. There output in 1 is higher than in 2. Then one naturally expects the farmer to be hoping that 1 occurs.7 Bad conditions, when properly prepared for, can lead to good outcomes just as gambling on a bad outcome (insurance) occurring can ameliorate losses. The output-cubical specification flips this logic on its head. Rather than the producer being able to undertake preventative actions that affect which state is “good” or “bad,” Nature alone decides. The result is a technical specification that, by its very nature, underestimates (if not eliminates) any potential for technically efficient self-protective activities.

Stochastic Production Decisions Our next step is to integrate the stochastic production model into producer decisionmaking. In this section, we first present an objective function for the producer. It includes as special cases virtually all producer objective functions that have been studied in uncertain decision settings. Important examples include risk-neutral, subjective expected-utility preferences, and mean-variance structures consistent with first-order stochastic dominance. We then discuss the primitives of the producer’s decision set, demonstrate a fundamental result underlying optimal producer behavior, and relate that behavior to net present value decision rules. The final subsection develops an equivalent, but probability-free, version of the decision model and relates it to fundamental results for competitive, but nonstochastic, markets.

Producer Preferences Period 0 wealth (income) of the producer is exogenously determined and denoted i 0 . Producer preferences over uncertain period 1 consumption, y 1 ∈ RS , and S+1 → R nonstochastic period 0 consumption, y 0 , are complete and given by W : R 1 0 that is strictly increasing and continuous in y , y . For intuitive simplicity, we also frequently assume W is quasi-concave in y 1 . Quasi-concavity imposes a generic form of risk aversion (Debreu [17]) that contains more familiar notions as special case. Although one can operate in terms of W, its existence ensures the existence of an even-more convenient cardinal representation, known as its certainty equivalent, and defined by

7 At

this juncture, it would be a good exercise for you to revisit Fig. 9 and the associated intuitive discussion in an attempt to determine whether 1 is associated with more or less moisture than 2.

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     e y 1 , y 0 = min β ∈ R : W β S , y 0 ≥ W y 1 , y 0   if there exists β ∈ R : W β S , y 0 ≥ W y 1 , y 0 and ∞ otherwise. The certainty equivalent gives the amount of the riskless asset, 1 S , that when combined with a period 0 consumption of y 0 indifferent to y 1 , y 0 . It is strictly increasing and  continuous in y 1 , y 0 and satisfies     e y 1∗ , y 0∗ ≥e y 1 , y 0 ⇔W y 1∗ , y 0∗ ≥W y 1 , y 0

(Preference Indication),

so that it is functionally equivalent to W. When W is quasi-concave in y 1 , e is concave in y 1 and thus differentiable almost everywhere.

Cost Minimization, Duality, Risk-Neutral Probabilities, Fisher Separation, and More Duality We assume the producer faces competitive period 0 markets for the inputs. Their market-determined prices are denoted w ∈ RN ++ . Period 1 price for the stochastic output z is stochastic but determined exogenously to the producer and denoted p ∈ RS++ . The producer thus views herself or himself as a price taker both for w and for p. The producer’s task is to determine how to allocate i 0 to producing stochastic output to be sold in period 1 at the stochastic spot price of p, how much to consume in period 0, and how much to consume in period 1. The producer faces two periodspecific budget constraints i 0 ≥ y 0 + w  x, p1 · z ≥ y 1 . (For two random variables m and n, the notation m · n ∈ RS denotes the random variable formed as the element-wise product of the two random variables, so that m·n = (m1 n1 , ..., mS nS ) .) The first inequality requires that period 0 consumption not exceed income available for consumption, i 0 − w  x. And the second ensures that stochastic period 1 income always is at least as large as stochastic period 1 consumption. Put another way, period 1 income first-order stochastically dominates period 1 consumption. Hence, the producer’s problem is to solve    max e y 1 , y 0 : i 0 ≥ y 0 + w  x, p1 · z ≥ y 1 .

(x,z,y)

The following result is immediate (Chambers and Quiggin [5, 10]):

(8)

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Theorem 1. (Chambers and Quiggin) The producer’s optimal production decisions satisfy max {e (p · z, i − c (w, z))} , z

where:   c (w, z) = min w  x : x ∈ X (z) x   = min w  x : z ∈ Z (x) x   = min w  x : D (z, x) ≤ 0 x

if X (z) = ∅ and ∞ otherwise. Theorem 1 shows that solving the producer’s problem decomposes, as usual, into two components. In the first, producers solve the cost-minimizing problem for z. And, once that problem is solved, producers pick their optimal stochastic output to maximize their preferences. The producer cost function, c (w, z) , is nondecreasing, closed, concave, and positively homogeneous in w. It also satisfies Shephard’s Lemma. Namely, its superdifferential in w contains the cost-minimizing solutions, and when that superdifferential is a singleton set (implying differentiability in w), the cost-minimizing solution is unique and equals the corresponding gradient. Conversely, if the cost-minimizing solution is unique, the cost function is differentiable in w, and the gradient equals that solution. An immediate consequence of the properties of c (w, z) is that the correspondence, Xˆ : RS ⇒ RN , generated by c (w, z) as   Xˆ (z) = x ∈ RN : w  x ≥ c (w, z) for all w ∈ RN ++ ˆ possesses images that are closed, convex sets satisfying Xˆ (z) + RN + ⊂ X (z) (free ˆ disposability of inputs), with X (z) ⊂ X (z) . Moreover, if X (z) satisfies these same properties, then Xˆ (z) = X (z) . In other words, standard duality relationships apply for input correspondences associated with stochastic technologies. A theorem stating that competitive producers minimize cost and that dual relations apply might strike some as pedantic. After all, the producer cost function is one of the first concepts to which we introduce our students. But the fact remains that the relevance of the cost minimization and dual cost structures for producers facing stochastic production possibilities was still being questioned until the middle of the last decade of the twentieth century [23, 34, 35]. Theorem 1 establishes that producers do minimize cost when production is uncertain. Moreover, these cost functions are fully dual to input correspondences

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satisfying familiar properties. The confusion that did exist on this issue seems to have emerged from an exclusive focus on the stochastic production function model without a formal specification of its measure-theoretic underpinnings in terms of a state space, its associated measurable events, and the definition of a random variables as mappings from the state space to the reals. Suppose, to the contrary, that a producer decided not to minimize the cost of producing z. Instead of incurring c (w, z) to produce z, he or she would incur some other strictly greater cost level, c, ˆ to produce that same output, z. That, however, would require the producer to forego period 0 consumption in the amount of cˆ − c (w, z) > 0 while generating an unchanged stochastic income of p · z! In language borrowed from financial economics, the producer ignores an arbitrage opportunity. Such behavior is always inconsistent with the choices of a rational economic individual regardless of their risk preference or the structure of W . Example 1. To illustrate the derivation of the producer’s cost function, we use the specific example of a stochastic production function. The input correspondence for that technology is given by the lower inverse to Z pf   Z pf − (z) = x ∈ RN : zs ≤ fs (x) , for all s ∈ S   = ∩s∈S x ∈ RN : zs ≤ fs (x) , which corresponds to the intersection of the input sets for the S state-specific technologies. It follows immediately that the minimal cost associated with this technology can be no lower than the minimal cost associated with the most expensive of the S state-specific technologies. That is, the cost function obeys the maximin criterion    c (w, z) ≥ max min w  x : zs ≤ fs (x) , s∈S

that implies an isocost structure of the same generic form as Z pf (x) . For the sake of a familiar argument, our discussion of the producer’s problem assumes that e y 1 , y 0 and c (w, z) are smooth and that all solutions are interior.8 The producer’s first-order conditions then require 

∇1 e y , y

8 These

1

0



 ∂e y 1 , y 0 ·p = ∇z c (w, z) ∂y 0

(9)

conditions are easily relaxed. See, for example, Chambers and Quiggin ( [10–12], 2010), Chambers [6], and Chambers and Voica [13].

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 where ∇1 e y 1 , y 0 denotes the gradient of e in y 1 and ∇z c (w, z) denotes the gradient of c in z. Some familiar transformations borrowed from the finance literature (e.g., Cochrane [15] or LeRoy and Werner [29]) permit translating these conditions into net present value terms. Define the producer’s subjective intertemporal discount  factor, denoted as δ y 1 , y 0 ∈ R, as the marginal riskless variation in period 1 consumption needed to balance a small change in period 0 consumption, ε,   S   1 1 0 0 e y + δ y ,y , y − ε = e y1, y0 . Letting ε → 0 and using the implicit function theorem gives    ∂e δ y 1 , y 0 = 0 /∇1 e y 1 , y 0 1S , ∂y  where the right-hand side reveals that δ y 1 , y 0 is also interpretable as the marginal rate of substitution between one unit of period 0 consumption and one unit of the riskless asset in period 1. Next define the producer’s risk-neutral probability  measure, denoted by π y 1 , y 0 ∈ RS , as  π y1, y0 =

 ∇1 e y 1 , y 0  .  ∇1 e y 1 , y 0 1S

  As the notation indicates, both δ y 1 , y 0 and π y 1 , y 0 are local measures  1 0 and depend upon e y , y . Therefore, both are subjectively determined. Two  producers with the same consumption profile y 1 , y 0 can easily possess different subjective intertemporal discount factors and/or risk-neutral probability measures. Second, both are intuitive devices that make the producer’s decision-making more intuitively accessibleto individuals accustomed to using net present value methods. In that context, δ y 1 , y 0 represents the return on the riskless asset that would leave the producer between a dollar today and a dollar tomorrow. just  indifferent And πs y 1 , y 0 /πs y 1 , y 0 gives the relative odds of states s and s that would form an expected net present value producer’s marginal rate of substitution between period 1 incomes in states s and s . Using these transformations and (9) gives  π y1, y0 · p = ∇z c (w, z) .  δ y1, y0

(10)

Expression (10) forms the basis of:  Theorem 2. (Fisher separation) The producer’s optimal consumption y 1 , y 0 and production decisions satisfy

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⎫ ⎧     ⎬ ⎨ π y1, y0  π y1, y0  · z) − c z) . w,  1 0 · p ≡ sup (p (w, ⎭ δ y ,y z ⎩ δ y1, y0

 c∗

   π y 1 ,y 0 w, δ y 1 ,y 0 · p , which is the convex conjugate of c (w, z) , is a net Here ( ) present value profit function defined in terms of period 0 input prices, w, and c∗

π y 1 ,y 0

discounted period 1 state-specific output prices, δ y 1 ,y 0 · p. It is positively homo( ) geneous and convex in its arguments and satisfies versions of Hotelling’s Lemma so that optimal derived demands and state-specific supplies   can be recaptured as either subdifferentials or gradients of c∗ in w,

 π y 1 ,y 0 δ (y 1 ,y 0 )

· p . Thus, when unique optimal

derived demands and state-specific supplies exist, they can be written, respectively, as       π y1, y0 π y1, y0 ∗ · p = −∇w c w, ·p   x w, 1 + δ y1, y0 1 + δ y1, y0     π y1, y0 ·p  = −∇w c w, z w, , (11) 1 + δ y1, y0 and 

     π y1, y0 π y1, y0 ∗ · p = ∇ π (y 1 ,y 0 ) c w, ·p .   z w, ·p 1 + δ y1, y0 1 + δ y1, y0 1+δ (y 1 ,y 0 )

(12)

   π y 1 ,y 0 The convexity properties of c∗ ensure that the Hessian of c∗ in w, 1+δ y 1 ,y 0 · p ( ) is positive semi-definite, thus yielding familiar comparative-static results for optimal    π y 1 ,y 0 (y 1 ,y 0 )

demands and state-specific supplies in w, 1+δ

· p . It also follows imme-

diately from (11) that optimal derived demand behavior, following Sakai [38], can be decomposed into pure substitution effects and pure expansion effects.9 The convex conjugate of c∗ defined by   c∗∗ (w, z) = sup q  z − c∗ (w, q) q∈RS

9 Although

it is not discussed here, one can alternatively express the Fisher separation theorem in terms of a revenue function defined over discounted period 1 prices and x. That derivation, in turn, allows for a decomposition of present value maximizing supplies in terms of substitution and scale effects.

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is a closed, convex function of z that is closed, concave, nondecreasing, and positively homogeneous in w (Rockafellar [37]). Moreover, if c (w, z) itself is closed, convex in z, c (w, z) = c∗∗ (w, z) , so that standard duality results also apply for net present value profit functions. The intuition behind Fisher separation is not that producers maximize the expected net present value of producing their stochastic output. Rather, Theorem 2 asserts that producers choose their stochastic output in the same manner as would an expected net present value maximizer using the subjectively determined stochastic  π y 1 ,y 0

pricing kernel, δ y 1 ,y 0 ∈ RS , to discount period 1 stochastic incomes. ( ) Terminologies regarding pricing kernels for stochastic assets can differ. The general idea is that a pricing kernel should discount the period 1 stochastic return to equal its period 0. For example, if A ∈ RS is the period 1 stochastic return from an asset with a period 0 acquisition price of 1 and the stochastic pricing kernel is denoted k ∈ RS , then k  A = 1. An alternative definition works in terms of expectations inner products and often invokes the terminology stochastic discount factor. In this setting if m ∈ RS is a stochastic discount factor, then Eπ [mA] = (π · m) A = 1. That is the expected discounted value of the asset equals its period 0 acquisition price. Thus, if k is a stochastic pricing kernel,   kS k1 ∈ RS is a stochastic discount factor for the probability measure π. π1 , ..., πS  π y 1 ,y 0

In the current context, if one knows δ y 1 ,y 0 , the optimal stochastic pro( ) duction choice is  the one that maximizes expected net present value. Because   π y 1 ,y 0 c∗ w, δ y 1 ,y 0 · p inherits all of the properties of a standard profit function, ( ) Fisher separation is both analytically and  empirically convenient. Moreover, the  existence of a technology fully dual to c∗ w,

 π y 1 ,y 0 δ (y 1 ,y 0 )

·p

ensures that no true

∗ generality is lost by operating in terms  1 of c (rather than c,1 D,0 or any other 0 representation of T ) provided that e y , y is increasing in y , y . A producer optimizing e with respect to c and then with respect c∗ makes the same economic choices for both. Thus, just as in standard producer decision-making under certainty, rational producers effectively “skip over” nonconvexities in technologies, and the “true” technology and the dual technology are observationally equivalent. Traditional shadow-pricing arguments now reveal that the producer’s appropriate shadow values for z are determined by its marginal cost, ∇z c (w, z) ∈ RS . A closely related observation is obtained by rewriting (10) as

 π y1, y0 = ∇z c (w, z) / · p,  δ y1, y0 where /· denotes element-wise division. The left-hand side of this expression  is the appropriate stochastic pricing kernel for an individual with preferences e y 1 , y 0 . But, in equilibrium, that realization reveals that ∇z c (w, z) / · p ∈ RS will represent an appropriate virtual (shadow) stochastic pricing kernel.

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Revenue Cost and Graphical Illustration of Producer Equilibrium The concepts of risk-neutral probabilities and Fisher separation are intuitively handy, but they are by no means necessary to discussing producer decision-making under uncertainty. This section shows how a probability-free approach to analyzing producer behavior is developed. A direct consequence of Theorem 1 is:   Corollary 1. The producer’s optimal consumption decisions satisfy: maxy 1 e y 1 ,   i − c w, y 1 / · p .  c w, y 1 / · p is a special case of the revenue-cost function introduced by Chambers and Quiggin [10]. It gives the minimal producer cost of obtaining a period 1 consumption y 1 given p and w.10 In the smooth case, the associated first-order conditions for the producer’s optimization problem are   ∇1 e y 1 , y 0 1  = ∇ c w, y / · p / · p, z ∂e y 1 , y 0 /∂y 0

(13)

which is fully equivalent to (9). (Substitute y 1 / · p for z in the latter to obtain the former.) Despite their equivalence, (13) motivates a subtly different interpretation of optimality conditions that does not rely on probabilities. The expression on the left-hand side of (13) equals the S-vector of marginal rates of substitution between period 1 consumption in state s and period 0 consumption. The expression on the right-hand side is the S-vector of period 0 marginal costs of raising consumption in period 1 by one unit. Optimality, not surprisingly, requires equating the two. The ratio of any two elements of the left-hand side of (13), say s and s , corresponds to  ∂e y 1 , y 0 /∂ys1  , ∂e y 1 , y 0 /∂ys1 the producer’s marginal rate of substitution between consumption in states s and s . The corresponding ratio on the right-hand side is  ps ∂c w, y 1 / · p /∂zs  , ps ∂c w, y 1 / · p /∂zs the producer’s marginal rate of transformation between income in states s and s . Thus, equilibrium corresponds to a tangency between the producer’s indifference curve between period 1 consumption in different states and the stochastic production

10 The

reader is referred to Chambers and Quiggin [10] for a more detailed and thorough treatment of revenue-cost functions.

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Fig. 10 Producer equilibrium

possibilities frontier (as  expressed in revenue terms). Figure 10 illustrates that equilibrium as y1∗ , y2∗ . The equilibrium depicted in Fig. 10 is reminiscent of basic representation results for general equilibria with representative producers and representative consumers. The difference is that the equilibrium there depicted represents an “internal” equilibrium for the producer that involves her or his assessment of optimal trade-offs and arbitrage opportunities between her or his preference structure and the physical, but stochastic, production technology. The slope of the hyperplane defined by the tangency between the stochastic production possibilities frontier and the indifference curve (labelled “Relative StateClaim Prices”) is also expressible as  πs y 1 , y 0 −  1 0 , πs y , y the relative risk-neutral probabilities (the relative risk-neutral odds) between states s and s . The visual interpretation of the Fisher separation theorem is simply that the producer chooses the stochastic output to ensure that the slope of the illustrated hyperplane in Fig. 10 equals the rate of transformation between statespecific outputs. When those relative risk-neutral odds are reinterpreted as relative prices, the visual criterion for optimality is the same as for multiple-output profit maximization. By an exactly parallel logic, the solution to the “consumer” side of the producer’s problem can be characterized as maximizing the utility from y 1 given a budget constraint defined by the hyperplane in Fig. 10. In that context, the relative prices are interpretable as subjective Arrow [2] state-claim prices – the period 0 prices of options on 1 unit of period 1 income in each of states. Thus, given these Arrow state-claim prices or the parallel risk-neutral probabilities, an individual’s behavior as a producer and a consumer can be “separated” and analyzed independently.

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R. G. Chambers  π y 1 ,y 0

The obvious catch is that one must know δ y 1 ,y 0 to operationalize these ( ) separation results. The market and price-taking behavior ensure separation between producer and consumer behavior for the corresponding analysis for nonstochastic technologies. But here the “market” is one internal to the producer, and unless competitive markets exist for Arrow securities or options (so-called complete markets), the relevant state-claim prices depend upon the solution to the producer’s problem and commingle producer preferences and the technology. Thus, individuals with different preference structures or with different technologies have different stochastic pricing kernels. Except for relatively trivial versions of S, producers do not face complete markets for period 1 state-claims. More generally, markets will be incomplete. But incompleteness is not the same as non-extant, and as Magill and Quinzii [31], Chambers and Quiggin [9,12], and Chambers and Voica [13] demonstrate, different circumstances can arise where incomplete financial markets yield situations that are effectively complete from the producer’s perspective, and price discipline over stateclaims arises from competitive behavior. In those instances, separation is complete, and producer and consumer behavior can be analyzed separately. Fully analyzing producer behavior in the presence of such incomplete markets requires a complete specification of the financial market structure that is beyond the scope of this chapter and thus is not considered. Nevertheless, a straightforward consequence of these results is that comparative-static analyses developed for dual profit structures and indirect preference structures translate directly to a stochastic setting.

Concluding Remarks This chapter presents a capsular depiction of a formal model of a stochastic production technology. The basic model is that developed by Arrow [2] and Debreu [17] as later extended by Chambers and Quiggin [10]. It has as an important special case the standard nonstochastic multiple-output production model. Alternative axioms, functional representations, and structural restrictions for the stochastic technology are examined. The model is then used to characterize optimal producer decision-making in a stochastic environment.

Cross-References  Bad Outputs  Distance Functions in Production Economics  Functional Structure and Aggregation

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References 1. Antle JM (1987) Econometric estimation of producers’ risk attitudes. American Journal of Agricultural Economics 69:509–22 2. Arrow KJ (1953) Le Role des Valeurs Boursiers pour la Repartition la Meilleur des Risques. Cahiers du Seminair d’Economie. CNRS, Paris 3. Arrow KJ (1964) The role of securities in the optimal allocation of risk bearing. Rev Econ Stud 31:91–96 4. Batra RN (1974) Resource allocation in a general equilibrium model of production uncertainty. J Econ Theory 8:50–63 5. Chambers RG (1988) Applied production analysis: a dual approach. Cambridge University Press, Cambridge 6. Chambers RG (2007) Valuing agricultural insurance. Am J Agric Econ 89:596–606 7. Chambers RG, Chung Y, Färe R (1996) Benefit and distance functions. J Econ Theory 70:407– 419 8. Chambers RG, Quiggin J (1992) A state-contingent approach to production under uncertainty.mimeo 9. Chambers RG, Quiggin J (1997) Separation and hedging results with state-contingent production. Economica 64:187–209 10. Chambers RG, Quiggin J (2000) Uncertainty, production, choice, and agency: the statecontingent approach. Cambridge University Press, New York 11. Chambers RG, Quiggin J (2008) Narrowing the no-arbitrage bounds. J Math Econ 44(1):1–14 12. Chambers RG, Quiggin J (2009) Separability of stochastic production decisions from producer risk preferences in the presence of financial markets. J Math Econ 45:730–737 13. Chambers RG, Voica D (2017) “Decoupled” farm program payments are really decoupled: the theory. Am J Agric Econ 99:773–782 14. Chavas J-P, Holt M (1996) Economic behavior under uncertainty: a joint analysis of risk preferences and the technology. Rev Econ Stat 78:329–335 15. Cochrane, J. H. (2001) Asset pricing. Princeton University Press, Princeton 16. de Janvry A (1972) The Generalized Power Production Function. Am J Agric Econ 54:234–237 17. Debreu G (1959) The theory of value. Yale University Press, New Haven 18. Feldstein M (1971) Production uncertainty with uncertain technology: some economic and econometric implications. Int Econ Rev 12:27–36 19. Fuller W (1965) Stochastic fertilizer production functions for continuous corn. J Farm Econ 47:105–119 20. Gorman WM (1976) Tricks with utility functions. In: Artis MJ, Nobay AR (eds) Essays in economic analysis. Cambridge University Press, New York 21. Haavelmo T (1943) The structural implications of simultaneous equations systems. Econometrica 11:1–12 22. Holmström B (1979) Moral hazard and observability. Bell J Econ 10:74–91 23. Just RE (1993) Discovering production and supply relationships: present status and future opportunities. Rev Mark Agric Econ 61:11–40 24. Just RE, Pope RD (1978) Stochastic specification of production functions and economic implications. J Econ 7:67–86 25. Just RE, Pope RD (1979) Production Function Estimation and Related Risk Considerations. Am J Agric Econ 61:277–84 26. Knight FH (1921) Risk, uncertainty, and profit. Augustus M. Kelley, New York 27. Kohli U (1983) Nonjoint technologies. Rev Econ Stud 50:209–219 28. Lapan H, Moschini G (1994) Futures heding under price, basis, and production risk. Am J Agric Econ 76:465–477 29. LeRoy SF, Werner J (2014) Principles of financial economics. Cambridge University Press, Cambridge 30. Luenberger DG (1994) Dual Pareto efficiency. J Econ Theory 62:70–84

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31. Magill M, Quinzii M (1996) Theory of incomplete markets. MIT Press, Cambridge 32. Moscardi E, de Janvry A (1977) Attitudes towards risk among peasants: an econometric approach. Am J Agric Econ 59:710–716 33. Moschini G, Hennessey D (2001) Uncertainty, Risk Aversion and Risk Management for Agricultural Producers. BL Gardner and GC Rausser (eds.). Handbook of Agric Econo Elseveir 1:87–115 34. Pope RD, Chavas J-P (1994) Cost functions under production uncertainty. Am J Agric Econ 76:196–204 35. Pope RD, Just RE (1996) Empirical implementation of ex ante cost functions. J Econ 72:231– 249 36. Pope RD, Just RE (1998) Cost function estimation under risk aversion. Am J Agric Econ 80:296–302 37. Rockafellar RT (1970) Convex analysis. Princeton University Press, Princeton 38. Sakai Y (1974) Substitution and expansion effects in production economics: the case of joint products. J Econ Theory 9:255–274 39. Savage LJ (1954) Foundations of statistics. Wiley, New York 40. United States Department of Defense (2002) DoD news briefing: secretary Rumsfeld and Gen. Myers Feburary 12 2002. https://archive.defense.gov/Transcripts/Transcript.aspx? TranscriptID=2636 41. von Thünen JH (1826) Der Isolierte Staat. Pergamon Press, New York

Dynamic Analysis of Production

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Contents Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The Setting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Dynamic Optimization Frameworks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Adjustment Cost Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Long History and Evolution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Incorporating Adjustment Costs into a Technology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Primal-Dual Theory Opportunities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Econometric Approaches . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Nonparametric Approaches . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Dynamic Generalizations of Modern Production Theory Concepts: Scale and Scope, Efficiency, Capacity, and Productivity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Efficiency . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Capacity Utilization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Productivity Change . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Non-convex Production Relationships . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Network Approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Cross-References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

612 614 614 616 616 616 618 619 621 622 625 626 628 629 630 631 632 633 633

S. E. Stefanou () Food and Resource Economics Department, University of Florida, Gainesville, FL, USA Wageningen University, Wageningen, Netherlands e-mail: [email protected] © Springer Nature Singapore Pte Ltd. 2022 S. C. Ray et al. (eds.), Handbook of Production Economics, https://doi.org/10.1007/978-981-10-3455-8_17

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Abstract

A parable of economic life is that some factors can adjust rapidly while others adjust slowly in a given time scale. Focusing on production analysis in the dynamic setting leads us to emphasize the technology specification that permits the theoretical construction that can be translated and amenable to empirical implementation. A historical perspective of the framing the dynamic decisionmaking is reviewed. The adjustment cost model of the investment is the key conceptual feature as it can be incorporated into the formal structure of a production technology, which offers the opportunity to exploit primal-dual theory in both analysis and empirical implementation. An overview of empirical formulations in both econometric (parametric) and nonparametric settings is discussed. Dynamic production decision environment allows explicitly for the evolution of assets implying firms may not be in long-run equilibrium at a given point in time. The dynamic generalizations of modern production theory concepts measuring economic performance are reviewed given the need to properly account and value the factors that are out of equilibrium. Empirical nonparametric and parametric approaches are addressed at length. While these cases can be addressed relatively easily within a nonparametric, dynamic data envelopment analysis setting, econometric formulations are a greater challenge. Keywords

Adjustment cost · Dynamic production technology · Dynamic data envelopment analysis · Econometric approaches · Economies of scope · Economies of scale · Capacity utilization · Productivity change · Dynamic distance function · Quasi-fixed factors · Capital adjustment

Introduction . . . if you don’t do the best you can with what you happen to have got, you’ll never do the best you might have done with what you should have had . . . (Aris [7], p. 27)

This insightful statement captures the essence of production decision making over time. Unpacking the elements of Aris’ single sentence is a driving force of this review chapter. The dynamic decision environment necessarily involves the linkage of current decisions to future opportunities, leading us back to Aris’ quote at which is a restatement of Bellman’s Principle of Optimality.1 Optimal solutions are linked 1 Bellman’s

principle of optimality states: “An optimal policy has the property that whatever the initial state and initial decision are, the remaining decisions must constitute an optimal policy with regard to the state of resulting from the first decision” ([10], p. 83).

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forward “ . . . to the best we might have done . . . ” and backward “ . . . with what you should have had . . . .” We take a step back from the basic economic setting to address the case where the time scale of decisions and the presence of factors of production that are still productive beyond the data-reporting period add another layer impacting how we characterize decision making. For the analyst, economic data arrive at different time scales than the decisions. Nature of inputs that contribute to a dynamic context for production is an economic decision making fact of life. Durable factors of production that typically include structures and machinery have a productive life beyond the reported decision period, and extracting the full value of these factors is the decision maker’s challenge. This leads us to reconsidering the technology characterization to articulate how decisions are linked over time and the mechanisms influencing these linkages. Technologies and behavior need to be specified. A dynamic production environment can be characterized as one where current production decisions impact future production possibilities. Consequently, the dynamic perspective of production relationships necessarily involves the close interplay between stock and flow elements in the input-output transformation process, and how current decisions impact the changes in future stocks. Stock elements in the production transformation process can involve physical elements, such as the volume of capital (buildings, machinery, soil nutrient endowment, etc.) that can be effectively employed in the input-output transformation process as well as the stock of technical knowledge and expertise available to the decision maker during the decision period. This chapter is organized as follows. The next section reviews the historical background on how the literature addresses dynamic aspects of production decision making and the interplay between the short and long run. This is followed by an extended presentation of the adjustment cost model of the investment and focus on how it can be incorporated into the formal structure of a production technology, which offers the opportunity to exploit primal-dual theory in both analysis and empirical implementation. An overview of empirical formulations in both econometric (parametric) and nonparametric settings is discussed. Next is the review of dynamic generalizations of modern production theory concepts measuring economic performance. Notions of scale, efficiency, capacity, and productivity now need to properly account and value the factors that are out of equilibrium. The important special case in studying the micro-oriented cases follows, which can present non-convexity adjustment process which can impact economic performance measures and the analysis of productivity change. While these cases can be addressed relatively easily within a nonparametric DEA setting, econometric formulations are a greater challenge. This is followed by a review of the Network DEA approach which offers a computational decision procession framework for production systems as outputs in one stage feed into a future stage. The final section address two directions, in particular, that are worthy of attention and discussion.

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Background The Setting The characterization of dynamics in production decision making is a theme of long interest. Samuelson ([124], Chapter IX) provides an overview of the earliest efforts to introduce dynamic notions in economics and identifies the distinction between “dynamic and causal (non-historical)” and “dynamic and historical” ([124], p. 315). “Dynamic and causal (non-historical)” refers to a subset of variables leading other variables to move, but the parameters of the system remain unchanged. “Dynamic and historical” allows for a parameter of the system (such as the state of the art) to lead to changes in the system or in its behavior over time. Frisch ([59], Part V, Chapters 16–19) focuses on the dynamic theory of production by initially commenting on the time shape (his italics), or trajectory, of input and output decisions. These early efforts focus on the investment decisions and focus particular attention on the depreciation and replacement of equipment and structures. Frisch, as others of this era, are taking on an engineering economics framework to the intertemporal case, focusing on a replacement theory approach where maintenance investment is the emphasis. The focus on replacement and maintenance leads to the investment dynamics influencing the installation of additional capital, which can lead to overcapacity [28]. The nature of factors of production is a driving force of dynamic production decision making. Viner’s [152] distinction between some factors being “freely adjusted” while others are “necessarily fixed” necessitates the distinction between the short- and long-run production technology. Capital inputs are characterized as having productive value beyond the current period and at the same time present a degree of inflexibility that freely adjusted (variable) inputs do not present. Alchian [5], Smith [137], and De Alessi [36] advanced the stock-flow production function that the notion of intertemporal investment and production decision making is formalized within the long-run cost minimization framework. Three major components of production decisions have the potential to drive a dynamic decision process. The first is driven by economic forces related largely to adjustment processes, which arise from the dichotomy between the short and long runs. The distinction between the short and long run becomes a prime consideration in determining the appropriate time scale of economic decisionmaking strategies. These strategies focus on the choice of production factors assumed to be fixed when factor allocation decisions are to be made. All economic activity occurs in the short run to the extent a factor (or factors) of production are taken as fixed [137, 152]. The long run refers to the firm planning to select a future short-run production situation. The problem with the classical description of the short and long run is that the story of the envelope curve is not entirely consistent with the story motivating the distinction between the short and long run.

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The long run consists of a range of possible short-run opportunities available to the firm. As such, the firm always operates in the short run but plans for the long run ([55], p. 198). A more complete description of producer behavior in the long-run theory of cost concentrates on the planning problem involving the minimization of the discounted stream of costs. Such a characterization focuses on long-run costs as a stock rather than a flow concept. The classical approach characterizes both short- and long-run cost functions as flows. The long run is merely the case where the fixed factor is now variable – presumably, because the time span under consideration is now long enough to view the problem as a short-run planning problem. This could entail describing the long run to last 5 or 10 years given capital adjustment rates estimated in the empirical literature. Viner’s [152] idea of some factors being “freely adjusted” while others are “necessarily fixed” is sufficiently vague to allow long-run costs to be considered a flow. Freely adjusted implies that altering the input levels of these factors does not impose a penalty on the firm other than a constant acquisition cost. The application of non-freely adjusted inputs presumably occurs because the firm must absorb some additional costs beyond the acquisition cost. Introducing the concept of adjustment costs can capture this phenomenon. Some factors are considered “fixed” in the short run, not because the operator is physically prevented from removing or introducing more of the factor, but because the economic environment places a high cost on adjusting the factor level. The full slate of costs that can lead to gradual adjustment are rarely, if ever, observed but can be proxied by a relationship (or function) that is driven by changes in the quasi-fixed factors. To those who may see this as a shortcoming, we take a step back and acknowledge that many of the concepts we have in economics are theoretical constructions proposed as the mechanisms to rationalize economic choices to observed behavior. We never observe a production function, but it serves as a mechanism to relate input choices into output realizations; we never observe a consumer’s utility function, but it serves to translate how choices of goods and services lead to the consumer’s overall satisfaction. Fundamentally, economic analysis in a static (or timeless) context entails addressing how a change in an economic variable of interest (e.g., price, tax, the level of capital stock) impacts the firm’s decision (e.g., how much to invest). This change is assumed to happen instantaneously, or more appropriately, over a time period that happens to be so long that the full impact of the stimulus for change has not taken full effect. In fact, this time period may span several years. This is akin to asking the question: How long is the long run? For a firm manufacturing automobiles, the answer can vary from 12 hours to 12 weeks; a new manufacturing facility may take 2–3 years to construct, debug, and gear up into full operational status. But for a politician, the answer is quite simple: The long run is until the next election. Understanding the decision time path associated with a policy stimulus is as important as understanding the final impact of that stimulus.

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Dynamic Optimization Frameworks The presentation of dynamic optimization approaches has evolved over the years. Many books, including Intriligator [69], Kamien and Schwartz [71], Léonard and Long [83], Takayama [139], Seierstad and Sydsaeter [126], and Caputo [19], offer complete presentations of the calculus of variations and optimal control theory, with applications to economics. Dorfman [41] is a classic contribution on the economic interpretations of optimal control theory. Sufficient conditions in optimal control can be found in Kamien and Schwartz [70] and Seierstad and Sydsaeter [126] for both the standard control problems and the case of corner solutions for the controls. Benveniste and Scheinkman [11] identify the sufficient conditions to guarantee the value function is differentiable. With enhanced computational capabilities, come treatments that have a distinctly empirical orientation exploiting the dynamic programming approach. Bellman [10] introduces the theory of dynamic programming, which predates the optimal control theory approaches and offers the foundation for computational algorithms as well as a fundamental functional equation of optimization that many applications exploit to connect with primal dual theory of dynamic production. Comprehensive discussion of the computational methods suitable to estimate structural dynamic models of investment is found in Adda and Cooper [2] and Bond and Van Reenen [17].

Adjustment Cost Model Long History and Evolution Holt et al. [67] and Eisner and Strotz [42] build the notion of a relationship that accounts for some factors of production to adjust gradually while others adjust instantaneously. The classic microeconomic theory of variable and fixed factors is now amended to refer to these gradually adjusting, or quasi-fixed, factors of production. The mechanism to capture the inertia of some factors is the cost of adjustment function that defines the relationship between cost (in either physical output or value terms) and the magnitude of the adjustment. As a result, the relative speed of adjustment can be used to characterize the degree to which a factor is variable or quasi-fixed. These models are motivated from the following propositions: (i) there are costs associated with adjusting the capital stock at a rapid rate per unit of time; and (ii) these costs increase rapidly with the absolute rate of investment; presumably, so rapid that the firm never attempts to achieve a jump in its capital stock at any given moment. The factors that are not adjusted instantaneously are referred to as quasi-fixed. The stock of quasi-fixed factors is gradually accumulated since it costs more to adjust the stock rapidly rather than slowly.

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A number of studies followed up on this basic concept by building on the parable about production2 by focusing on the properties of the adjustment cost mechanism and its relationship to investment behavior. Early contributions to this stream of thought that are exploring the properties of the adjustment cost model are Lucas [87], Rothschild [120], Treadway [145, 146], and Mortensen [97]. General overviews of the theory of the firm facing adjustment costs are Brechling [18] and Nickel [101], with Hamermesh and Pfann [62] offering an extensive history of both the theoretical and empirical applications that focus on the convex (smooth) and non-convex (kinked) quasi-fixed factor adjustment functions. These costs are the result of a reduction in output, which occurs when the quasifixed factor is absorbed (or released) too quickly. A firm may have personnel and training departments, which are adequately budgeted and staffed for the normal replacement of quits and retirements. If the firm seeks to expand its work force, more capital and labor must be devoted to the personnel and training departments. With total inputs fixed, the level of output must fall. Many such internal costs of adjustment can be viewed as learning. A manager seeking to expand the operation must spend more time learning how to manage more resources effectively. As some of this time may be devoted to more formal training (e.g., studying manuals, attending workshops), a portion of this training time can result in a loss in physical output due to the manager’s learning by production experience. With the manager’s total time available fixed, less time is available to manage the operation as a significant block of time is diverted to learning. Consequently, the level of output must fall. External adjustment costs arise from market forces or contractual obligations. With capital as a quasi-fixed factor, examples of external adjustment costs are expansion planning fees (e.g., architects, legal costs associated with zoning issues, design consultants) and imperfect capital goods markets. A firm intending to expand its capital base rapidly may be able to obtain more capital at a steeply increasing marginal cost because the rate of production of new capital goods may be insufficient. A firm may have to go to alternative credit sources to raise capital beyond the level traditional lending institutions may permit due to collateralization requirements. With labor as a quasi-fixed factor, severance pay, job advertisement, and other labor recruiting (search) costs are examples of actual costs incurred by the firm. For a production function specification for output (y), variable inputs (x), capital stock (K), and gross investment (I), =f (x, K, I), output is non-increasing in I and concave, reflecting the lost output associated with incurring investment. In a cost function setting with variable input prices (w), C(w, y, K, I), optimal short-run 2 Economists

are prone to refer to the “black box” as the unseen mechanism translating action and decisions into accomplishments. Our production functions are examples of such mysterious mechanisms. I defer here to Prof. Chambers who aptly characterizes the economist’s conceptualization of the mechanism as a parable (a simple story) in his  Chap. 14, “Production Under Uncertainty” in this volume.

618 Fig. 1 Isoquant map (I, x)

S. E. Stefanou

xt Slope = –

fI fx

It costs are increasing in investment at an increasing rate, reflecting a diseconomies of investment. Illustrating the isoquant for (I, x) in Fig. 1, additional variable input will require additional investment, which exhibits the classic congestion input behavior. The current period cost of investment is balanced against the flow of future gains associated with the embodiment of additional capital.

Incorporating Adjustment Costs into a Technology Several primal representations of the production technology are defined and characterized axiomatically in the static theory of production.3 The production function has been used, in general, as the primal representation of the adjustment-cost production technology (e.g., [45, 82, 108]). Recently, other primal representations of the adjustment-cost production technology have emerged in the literature allowing for the possibility of multiple outputs. Sengupta [127] addresses adjustment costs in an optimal control framework with a specification leading to a closed form solution of controls. The nonparametric approaches to the theory of production led to a proliferation of empirical work. These contributions start with Farrell [54] and advanced importantly by Afriat [3] and Hanoch and Rothschild [63]. These contributions provide the foundation for the more complete primal/dual nonparametric characterizations of production behavior by developing a set of inequalities that must be satisfied by observed prices, input decisions, and output realizations that would be generated by optimizing (cost minimizing and profit maximizing) behavior. Computationally

interested reader is referred to the Professors Chambers and Färe’s  Chap. 7, “Distance Functions in Production Economics” in this volume for a presentation of the production sets, and the radial and directional distance functions.

3 The

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convenient tests developed by Diewert and Parkan [39] and Varian [149] broadened the power of the revealed preference approach to production analysis. Silva and Stefanou [130] take on the dynamic production case by defining an adjustment-cost technology through a family of input requirement sets that is negative monotonic and convex in I(t) and reverse nested in k(t). Negativity monotonicity in I(t) and reverse nestedness in k(t) implies current additions to the capital stock are output decreasing in the current period but increase output in the future by increasing the future stock of capital. Convexity in I(t) implies the more rapidly the quasi-fixed factors are adjusted the greater the cost, leading to sluggish adjustment in the quasi-fixed factors. For other regularity conditions of this technology, see Silva and Stefanou [130]. Consider the data series Sc = {(yi (t), xi (t), Ii (t), ki (t), wi (t), ci (t)); i = 1, . . . , n; t = 1, . . . , T} representing the observed behavior of each production unit i at each time t, and the market input price vectors (wi (t) for variable inputs and ci (t) for quasi-fixed factors). Focusing on the input requirement set, V(y(t): k(t)), Theorem 2 in Silva and Stefanou [130] identifies VI (y(t): k(t)) as the tightest inner bound on V(y(t): k(t)) as the convex monotonic hull of (xi (t), Ii (t)) constructed as VI (y(t):k(t)) =

    λj (t)x j (t); I (t)≤ λj (t)I j (t); y(t) x(t), I (t) :x(t)≥ j

≤



λj (t)y j (t); k(t) ≥ k j (t);

j

j



λj (t) = 1; λj (t) ∈ + , ∀j



j

(1) where λ(t) is the intensity vector at time t. The distance function perspective on defining a production technology is focused on the entire opportunity set, including the boundary. For a given input-output bundle, the distance to the boundary is selected given the technology set and this distance is a measure of technical efficiency. Figure 2 presets an input requirement set in (I, x) for output identifies an interior bundle, Z, and illustrates the hyperbolic input distance function is defined in Silva and Stefanou [131] to represent a production technology with adjustment costs.

Primal-Dual Theory Opportunities At any point of time t, the firm is presumed to minimize the discounted flow of costs from time t forward as follows:   ∞ W (w, c, y, kt ) = min t e−rs w ´ xs + c´ Ks ds, x,I

s.t. K˙ = I − δK, Kt = kt (xs , Is ) ∈ V (ys : Ks ) , s ∈ (t, ∞)

(2)

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Fig. 2 Technical efficiency of variable and quasi-fixed factors

x

z z* z´

I where w is the vector of current variable input prices, c is the current rental price vector of quasi-fixed factors,4 ys is the production target at time s,5 r is the constant discount rate, K˙ = dK/dt is the vector of net investment, and δ is a diagonal (o x o) matrix of the depreciation rates δ h , h = 1, . . . ,o. The optimal current value function W(w,c,y,kt ) associated with problem in (2) focuses on long-run costs as a stock concept. The flow version of the long-run cost function is the dynamic programming equation or Hamilton-Jacobi-Bellman (HJB) equation     rW (w, c, y, kt ) = min w ´ xt + c´ kt + Wk ´ (It − δkt ) : xt , It ∈ V (yt : Kt ) , xt ,It

(3) where rW(·) is a flow version of the intertemporal cost and Wk (w, c, y, kt ) is the vector of the shadow value of capital. By definition, the shadow value of the quasifixed factor h, Wkh (·), measures the impact on the value function due to a small change in the initial capital stock, kh . Consequently, the shadow value of capital is an endogenous price and influenced by input prices (w, c), the production target, y,

4 Static

price expectations are assumed in model (21). This assumption means the firm considers that current prices contain all relevant information about future prices. The firm revises its price expectations as the initial period changes. Chambers and Lopez [21] discuss the reasons a firm that is aware of the cost of obtaining information may choose rationally to generate expectations in this way and update decisions continuously as new information appears. 5 A sequence of production targets is specified over the planning horizon in the dynamic cost model (Stefanou [138] and Epstein and Denny [47]). Consequently, the value function depends on current and future production targets. However, static output expectations are assumed in (2). The firm revises its expectations and production plans as the initial period changes and the new output targets are developed. In this way, the firm is solving an open loop optimization problem that allows but does not anticipate revisions in expectations.

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and initial capital stocks, kt . The economic interpretation of the optimized version of (2) is straightforward. The left-hand side, rW(·), is the opportunity of the production plan using the starting capital stock, kt and production target, y. The right-hand side presents the instantaneous flow of variable costs plus the user cost of capital and the imputed value of the net investment to the long-run cost function. The attractive feature of characterizing adjustment costs within a primal technology is that it creates opportunities to exploit the dual characterizations. Intertemporal (dynamic) duality focuses on the dual relation between the production function and the optimal value function of an intertemporal optimization problem (e.g., [45, 82, 108]), and duality between the optimal value function and the instantaneous variable profit function [93]. In particular, McLaren and Cooper [93] and Epstein [45] introduce the duality relationship between the value and production functions, publishing their contributions nearly simultaneously (appearing 2 months apart). McLaren and Cooper [93] exploit the optimal control approach, while Epstein [45] focuses on the HJB equation to develop the theory and econometric specification of dynamic dual models.

Econometric Approaches The emergence of focused econometric approaches to estimating dynamic adjustment econometrically estimate dynamic factor demands and test for the presence of instantaneous adjustment by determining if the marginal cost of adjustment is constant and demonstrate how the demands for quasi-fixed factors are generated as an approximate solution to the multivariate linear accelerator. Early econometric implementation of this approach is found in Denny et al. [37]. Mahmud et al. [91] examine the implications of functional form specification on the invariance of estimation, followed by Pindyck and Rotemberg [117] and Shapiro [129] and Chirinko [29] focusing on the implicit equations approach. These approaches are a mixture of dual short-run functional specifications and primal adjustment cost function specifications. Epstein and Denny [47] estimate the dynamic dual model for the US manufacturing sector and present a discussion of homotheticity and aggregation restriction for the intertemporal cost minimization. Blackorby and Schworm [16] and Epstein [46] focus on aggregation in dynamic models. With this early start, a number of applications ensue in agriculture (Taylor and Monson [141], Vasavada and Chambers [151], Howard and Shumway [68], Vasavada and Ball [150], Luh and Stefanou [88], Sckokai and Moro [125], Serra et al. [128], Rungsuriyawiboon and Hockmann [121], Yang and Shumway [153]), utilities [122], and manufacturing [15]. Several approaches to formulating and estimating dynamic decision rules under non-static expectations have been suggested in the literature. The first is the one used in Hansen and Sargent [64] and Epstein and Yatchew [48]. This approach is based on an explicit analytic solution to the Euler equation for the firm’s intertemporal optimization problem in the classic calculus of variations formulation. To yield

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an analytic solution, this approach requires a full specification of the expectation model. Moreover, due to the difficulty in solving the Euler equation, most often this approach is restricted to linear-quadratic technologies. The second approach due to Kennan [78] assumes expectations to be rational. Applying an instrumental variable estimation technique, Pindyck and Rotemberg [117] implemented this approach in their study. Unlike the first approach, this approach does not require an explicit solution of the Euler equations; therefore, it allows greater flexibility in the specification of the production technology, but this ignores the information contained in the transversality conditions. The third approach considers certainty equivalence feedback control policies and is suggested by Prucha and Nadiri [118]. Although the approach also utilizes the full solution to the intertemporal optimization problem, the algorithm suggested by Prucha and Nadiri avoids the need for explicit analytic solution. Taylor [140], LaFrance and Barney [80], and Lasserre and Ouellette [82] engage in theoretical explorations into the duality properties in stochastic environments. Empirical investigations are found in Luh and Stefanou [90] using aggregate data, and micro-level investigation is found in Pietola and Myers [116]. These dualbased formulations to address price expectations yielded to the structural investment models such as Cooper and Haltiwanger [32] for the US manufacturing plants and Roberts and Vuong [119] addressing the relationship of R&D an investment and productivity for German manufacturing firms.

Nonparametric Approaches With dynamic decision making focused on the HJB equation, the focus is on making current decisions on variable and dynamic factors, while looking to the future. This context allows for specifying nonparametric technologies for the investment decision making. The HJB equation in (3) can specify a technology nonparametrically for firm i over J firms, with M outputs, N inputs, and F capital factors i ym ≤

J 

j

γ j ym ,

m = 1, . . . , M;

j =1 J 

j

γ j xn ≤ xni − β i gxi ,

n = 1, . . . , N ;

j =1

Ifi + β i gIf − δf Kfi ≤

J 

  j j γ j If − δf Kf , f = 1, . . . , F ;

j =1

γ j ≥ 0, j = 1, . . . , J.

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The objective function in (3) is linear in the choice of the variable factor, x, and investment, I, given the shadow value of capital, Wk (·), which is the dual variable to the primal stated in (3). The solution to this class of problems must allow for the presence of both primal and dual variables in the specification of the optimization problem in such a way to guarantee the dual variables are not treated as primal variables in optimization. This can be framed as a linear complementarity problem6 as in Silva and Stefanou [131] or exploit the primal-dual structure as in Silva et al. [132]. The revealed preference approach to production analysis is one that focuses on allowing the data to reveal the technology without the constraints of a functional form. Building on Varian [149], the dynamic version leads to two boundaries for the input requirement set. The tightest inner bound presented in (2) based on the observed input-output data. When prices are available as well, the theoretical foundation of cost minimization can be used to assess if the data rationalize the dynamic cost minimizing behavior yielding the inner and outer bounds of the production technology. Theorem 3 in Silva and Stefanou [130] construct the tightest outer bound by the intersection of half-spaces that is created by the isocost planes and defined as

VO (y(t) : k(t))  (x(t), I (t)) : w i (t) x(t) + Wki (t) I (t) ≥ w i (t) x(t) + Wki (t) I (t)i ; (4) = y(t) ≥ y i (t); k(t) ≤ k i (t)

where Wki (t) is the vector of the shadow value of capital for observation i at time t. Silva and Stefanou [130] present a detailed empirical investigation of this approach with firm-level data (Fig. 3). More recent efforts emphasize the directional distance function approach to specifying the dynamic production technology. Building on the pioneering work of Chambers et al. [22, 24] and Chambers [20], Silva et al. [132] establish the primal-dual foundations for production under dynamic adjustment using the dynamic version of the directional distance function and develop decomposition of the dynamic version of the Luenberger productivity indicator in Chambers [20]. The directional distance function approach offers the flexibility of choosing the direction, rather than forces to the hyperbolic path to the frontier. The dynamics are introduced in the production technology specification as an adjustment cost in the form of the properties of the directional input distance function with respect to the dynamic factors (or the change in the quasi-fixed factors) as

6 The linear complementarity problem and solution approaches are found in Mangasarian [92] and applied in economics in Paris [111].

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Fig. 3 Identifying technically efficient reference location

X2 X0

X* XT XE 0

X1

− → D (y(t), K(t), x(t), I (t); gx , gI )    = max β ∈  : x(t) − βgx , I (t) + βgI )∈ V ( y(t)|K(t) ,

(5)

F The vector (gx , gI ) ∈ N ++ × ++ is a nonzero vector determining the − → direction in which D is defined. This function measures the distance of (x(t), I(t)) to the boundary of V(y(t)|K(t)) in a predefined direction (gx , gI ) = 0N + F . Figure 4 illustrates the dynamic directional input distance function assuming one variable input and one dynamic factor. The input vector (x(t),I(t)) is projected  − → − → onto the isoquant of V(y(t):K(t)) at a point x(t) − D (.)gx , I (t) + D (.)gI ∈ V (y(t) |K(t) ), (gx , gI ) = 0N + F . This figure presents three possible projections of the input vector (x(t),I(t)) associated with three directions: g0 , g1 , and g2 . As in the static case, the empirical implementation of the dynamic version for the directional distance function takes both parametric and nonparametric approaches. For the dynamic input distance function case, the nonparametric problem for firm i is to maximize the distance, β i , or level of inefficiency, stated as

 − → i i i i D y , K , x , I ; gx , gI = max β i β i ,γ j

s.t i ≤ ym J j =1

J j =1

j

γ j ym , m = 1, . . . , M;

j

γ j xn ≤ xni − β i gxn , n = 1, . . . , N ;

Ifi + β i gIf − δf Kfi ≤ γ j ≥ 0, j = 1, . . . , J.

J j =1

  j j γ j If − δf Kf , f = 1, . . . , F ;

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x V(y(t)|K(t)) (I(t),x(t))

(I2(t),x2(t))

(I1(t),x1(t)) (I0(t),x0(t))

g0 I g2

g0

Fig. 4 The dynamic input distance function

Dynamic Generalizations of Modern Production Theory Concepts: Scale and Scope, Efficiency, Capacity, and Productivity With the dynamic duality concepts established, rW(·) is the value function in flow terms in (3) and provides the starting point to commence the dynamic generalization of economic performance measures in modern production theory. The concepts of economic performance are muddled by the problem-specific nature of determining the length of run. As apparent in (3), the shadow value of the quasi-fixed factor, Wk , enters as an endogenous valuation of the capital asset. Conditioned on current value parameters (e.g., prices, output, and capital), it is predetermined in the sense that it is contemporaneously fixed and intertemporally endogenous. When the firm is not at a long-run equilibrium position, the under- or over-utilization of quasi-fixed factors imposes an internal cost. The time scale and the inertia of quasi-fixed factor change have two effects on our measure of economic performance. As the long-run cost function allows quasi-fixed factor stocks to evolve over time at an endogenous rate, there is not just a long run and a short run but a continuum of runs. Consequently, we can generate measures such as of returns to scale and scope economies, typically reported as a static metric, at each point in time that is driven by the fact that K˙ = 0 and an intertemporal objective is in effect. Elasticities can now be viewed as short run (with K(t) fixed at kt ), intermediate run (with K(t) variable according to the optimized investment policy), and long-run equilibrium (with optimal K(t) fixed ˙ ∗ = 0 and I ∗ = δK ∗ and K ∗ is the long-run equilibrium capital such that K(t) stock and is a function of (w, c, y)). When the technology can shift over time due to technological progress, we typically denote this progress with the time trend and the technology, (xt , It ) ∈ V(yt , Kt , t), implying the value function, W(w, c, Kt , yt , t), also shifts over time with technological progress. The measure of productivity growth in

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the dynamic production case can be generated the same way as the static case. In the classic static case, total factor productivity growth is decomposed into two components: (i) a scale effect reflecting input growth along the existing technology frontier, and (ii) a technological progress effect. If real prices are held fixed over time, the components driving productivity change are associated with those variables that change over time (inputs and technology, in this case). In the dynamic case, market input prices can be assumed to be fixed; the shadow value of capital, WK (w, c, Kt , yt , t), in the HJB equation in (3) serves as an internal price that does change over time with changes in the quasi-fixed factor stock. For the case under dynamic adjustment, Luh and Stefanou [88] find that there will be a technological progress effect but the input change effect now has several components. The first input change effect address the change in the flow variables; i.e., variable inputs and gross investment (the flow in the quasi-fixed factors stock). As investment take place along with the depreciation of the quasi-fixed factor stock, the quasi-fixed factor stock changes over time as it is driven by production choices. The last force driving productivity change emerges from the shadow value of capital, which is an endogenous value (price);  namely, the impact of an internal price

1 k ˙ + WKt · 1 . change, dW · = W K KK dt Wk Wk Measures of economic performance that are moving over time such as productivity, capacity utilization, efficiency, among others, must also now reflect the impact of disequilibrium that includes both the changing capital stock as well as the endogenously shifting shadow value of capital. Extensions to the case of returns to scale under dynamic adjustment are developed in Stefanou [138], Morrison [94, 96], and Paul and Siegel [113]. For the case of scope economies, Fernandez-Cornejo et al. [56] present an application to West German dairy farms, Oude Lansink and Stefanou [106] focusing on farm-level Dutch cash crops, and Helfat and Eisenhardt [65] presenting a business case study. A related direction arises in the strategic management literature focusing on the dynamic capabilities of firms and how resources are reallocated to build competitive advantage as firms experience rapid change in the face of competitive pressures [142]. While not based on structural production technologies, the theme and interest is related in terms of looking at the organizational capabilities, which strikes at the heart of the scope economies motivation. Silva et al. ([133], Chapter 5) constructs measures of scale, scope, capacity utilization, and productivity change under the dynamic adjustment with the directional distance function.

Efficiency Fallah-Fini et al. [49] offer an overview of the current nonparametric dynamic efficiency models. They organize the landscape to attribute dynamic behavior to one or a combination of factors associated with the dynamic aspects of production that include dynamic adjustment, cognitive capacity of decision makers, long lags in installing infrastructure, among others. The paradox of efficiency under dynamic adjustment is illustrated in Fig. 3.

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With a firm starting at X0 and the cost minimizing input bundle is X∗ , how does the decision guide her firm to move from X0 to X∗ ? If there were no friction or inertia in the decision making system, the move would be instantaneous and we would attribute the initial inefficient bundle as a lapse in managerial ability and the study of efficiency analysis would be fairly trivial. The point is that we do empirically observe firms operating off the frontier, and their presence off the frontier can be persistent. The dynamics of input reallocation may be significant. Each point on an isoquant is a technique of a production process (or technology) and the smooth isoquant results in the presence of an infinite number of techniques to achieve a given output level [31, 98]. Changing the input bundle as the firm reorients the techniques of production to enhance efficiencies can cause the firm to incur monitoring costs associated with reorganizing the production process. This can be revealed by the presence of transition costs associated with reallocating inputs, which would lead to a trajectory taking her to the optimal input bundle [30]. But as the decision-maker is following a path toward the optimal input allocation, a snapshot in time would reveal her to be inefficient; i.e., the firm may be statically inefficient but dynamically efficient (as she follows the optimal path). This approach has been applied to a range of applications in agriculture and food manufacturing (Skevas and Oude Lansink [134], Kapelko et al. [77], Kapelko and Oude Lansink [74], Kapelko [73]) and in construction [75]. As in the static version of the directional distance function, econometric applications are challenging. The need to impose the translation property restricts the range of functional form specifications, with the quadratic case remaining a tractable operational choice in the dynamic case as well, as is applied in Oude Lansink et al. [107]. The emphasis on the next discussion is driven by the error specification of a parametric frontier production function to address the production dynamics. The literature discussed here focuses on the production technology, not a cost function (although there is no restriction, other than access to price data, precluding the implementation to cost function frontiers). These econometric approaches seek to capture the sluggish movement of quasi-fixed factors by proposing that the dynamic stochastic frontier models tend to estimate firms’ long-run technical inefficiency level. Early attempts to take into account the time dependence of inefficiency in the stochastic frontier literature include studies by Battese and Coelli [9] and Kumbhakar [79]. These models allow the efficiency scores to follow deterministic functions of time. The disadvantage of this modeling approach is that it restricts the time path of efficiency to have the same structure for all firms under consideration, on average. In this sense, the results are interpretable with respect to the direction of evolution of efficiency over time for the industry. Cornwell et al. [34] develop a specification using firm-specific parameters that does not impose this restriction by proposing estimation in a generalized least squares framework. Cuesta [35] extends the Battese and Coelli [9] model by allowing the parameter in the function that describes the evolution of efficiency scores to be firm specific. All four models disregard the persistence of a positive or negative shock in firm efficiency and are,

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Table 1 Summary of studies estimating long-run persistence of efficiency Study Tsionas [147] Emvalomatis et al. [44] Emvalomatis [43] Lambarraa et al. [81] Skevas et al. [135]

Application US Banking (1989–2000) Dairy farms (1995–2005) US electric utilities firms (1986–1997) Spanish olive farms (2005–2012) German Dairy & Livestock (1999–2009)

Long-run efficiency score 0.955 Netherlands: 1.00 Germany: 0.778 0.813 0.727 0.700

therefore, not consistent with a dynamic model of firm behavior. Alvarez et al. [6] note that most of the parameters of a stochastic frontier model remain consistent even if the correlation between the efficiency scores among periods is ignored. Ahn and Sickles [4] and Tsionas [147] have specified inefficiency in a true autoregressive form. The primary focus of these two studies is the estimation of the autocorrelation parameter(s). Both studies find very strong autocorrelation, or persistence, in the efficiency scores. Although dynamic firm behavior is not modeled explicitly, the implications of such a structural model are revealed in the form of the parameter measuring the persistence of inefficiency. These approaches investigate how technical inefficiency can be present and persist, while at the same time this inefficiency is the consequence of factors that are under the firm’s control. The assumption is that these factors cannot be adjusted without entailing costs; i.e., the efficiency improvement will necessarily depend on the costs of adjustment. If such costs are high, we expect to find persistent technical inefficiency. This is a reduced form perspective in that observed relations lead to the revelation of behavior underlying the choices. Emvalomatis [43] builds on this line of work to investigate the impact of unobserved heterogeneity in the estimation of the long-run persistence of inefficiency. Table 1 summarizes empirical estimations from several studies of long run, persistent efficiency using the autoregressive frontier specification.

Capacity Utilization With capacity utilization being directly connected to the firm’s assets (or physical infrastructure), the connection to the dynamic production decision making is a natural concept to address briefly in this chapter.7 Various notions of capacity utilization have evolved over the decades. These measures are cyclical economic indicators with long standing use in public policy formation and analysis of business decisions. The derivation and calculation of capacity utilization rates has been

7 The interested reader is referred to the Professors Hayes and Jorgenson’s  Chap. 24, “Capacity and Capacity Utilization in Production Economics” in this volume.

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historically more on data analysis, both statistical and judgmental, with more recent efforts focusing on theoretical foundations. These more rigorous efforts focus on technology- and economic-based formulations. The economic characterization of capacity utilization is surveyed in the 1986 annals issue of the Journal of Econometrics (e.g., see Berndt and Fuss [12] and Slade [136]). Berndt and Fuss [13] generalize the notion of capacity utilization to multiple outputs and multiple quasi-fixed factors within a static setting. Fousekis and Stefanou [58] address the primal and dual measures of capacity utilizations under dynamic adjustment for the case of the US food processing and distribution sector. Morrison [95] explores the case for the US automobile industry, and Paul ([112], Chapter 3) disentangles capacity utilization from productivity measures under dynamic adjustment.

Productivity Change It is commonly accepted that measures of productivity growth should involve the use of multiple inputs and multiple outputs. Luh and Stefanou [88] explicitly address the change in the value of the shadow value of capital while incorporating dynamic adjustment in measuring productivity growth in a study of US agricultural productivity, finding that most productivity gains in production agriculture are from technical change. Further investigations accounting for the change in the shadow value of capital are found in Luh and Stefanou [89] accounting for learning-by-doing in productivity growth, Rungsuriyawiboon and Stefanou [122, 123], and Oude Lansink et al. [107]. Building on the dynamic analogue to the contributions of Chambers [20] and Färe et al. [53], Oude Lansink et al. [107] present the Luenberger-based measures of productivity change. The primal Luenberger productivity growth indicator is decomposed to identify the contributions of efficiency growth and technical change, while the dual Luenberger productivity growth indicator offers a further decomposition to identify the impact of quasi-fixed factor disequilibrium and allocative efficiency change. The decomposition adds context to the policy discussions. When policies promoting growth are assessed, the productivity impact is certainly important as a summary measure. But how this growth is being distributed among the components is driving the policy discussion. Is the sector gaining by the reallocation of gains by better, more efficient firms being formed (e.g., by merger and acquisition, entry vs. exits) and/or reallocation of factors within a firm? Petrin and Levinsohn [115], Dhyne et al. [38], and Nishida et al. [103] addresses the distribution of the reallocation of productivity drivers by specifying the production function controlling for simultaneity and selection problems. These production function estimation approaches controlling for the productivity shocks [1, 85, 104]. The interest in accommodating the potential endogeneity of productivity lends itself to a range of policy considerations.

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Non-convex Production Relationships The context for non-convex adjustment arises in the context of micro-level investigations. The smooth adjustment models can be justified with the aggregation across firms and quasi-fixed factors. Empirically, one finds there are frequent periods on investment inactivity at the plant level, referred to as lumpy investment or investment spikes. Non-convex adjustment can lead to investment bursts and the jumps in productive capacity can follow. Extensive evidence exists for these erratic investment patterns Cooper et al. [33], Geylani and Stefanou [60]. The findings of Letterie et al. [84] for German firms indicate that most investment spikes reflect an expansionary type of investment that have no direct relationship with improved productivity, while episodes of large investments in new technology that enhance productivity are very rare. Nielsen and Schiantarelli [102] find only very small changes in labor productivity associated with investment spikes, suggesting that productivity improvements are not related to technological change through investment spikes. At the micro-level decision-making unit, Cooper and Haltiwanger [32] review the empirical literature and find that non-convexity is likely the rule rather than the exception. Once we aggregate the data beyond the micro decision units, convexity is likely to be a tolerable abstraction. In the end, the adjustment cost function is a mechanism created by the analyst to rationalize the behavior that generates the observed data. As such, it is another of these parables of production that rationalizes all the forces that can lead to the gradual adjustment of some factors. From an empirical perspective, it is desirable to specify models that can test for the degree of inflexibility in adjusting a given factor. The relationship between investment spikes and productivity gains is an empirical relationship that has received scant attention to date. It is an empirical issue to establish the link between productivity growth and large investments, and examine how productivity growth changes in the presence of lumpy investments and the potential impact of learning-by-doing, which relates to the story of efficiency, change. Another direction related to non-convex adjustment is to explore the prospect of asymmetry in adjustment as a motivation for lumpy investment, where expansions can occur at a different rate than contractions. Chang and Stefanou [25] and Oude Lansink and Stefanou [105] take a dynamic dual modeling perspective and find the presence of adjustment asymmetry at the farm-level, and Palm and Pfann [110] explore the asymmetry of adjustment at a macro level focusing on in UK manufacturing building on a Koyck-lag mechanism driving adjustment. An alternative approach mixing dynamic DEA and econometrics is taken by Kapelko et al. [76] which assesses the impact of dynamic inefficiency, and productivity more broadly, measures over several years using the impulse response estimation postinvestment spike for the case of firm level Spanish meat processing firms finding the impact largely depends on firms’ size.

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Network Approach Network DEA is a computational decision process framework that is appropriate to modeling dynamic economic decision making [51]. It is particularly well suited to production systems when an output in one stage feeds into the next stage as an input. A network consists of sub-technologies entailing time-specific input decisions and the production of time-specific (intermediate) outputs. The constraints from one stage (or decision period) to the next can be input constrained just as capitallike factors can be in the dynamic problems. This is a mathematical programming problem where the optimization takes place over all stages (periods) to generate the optimal trajectories for the control variables. A DEA model typically describes a technology to a level of abstraction necessary for the analyst’s purpose, but leaves out a description of the sub-technologies that make up the internal functions of the technology. Essentially, the objective is to model the internal structure of firms with intermediate products or carry-over activities across multiple periods. Färe et al. [52] provide a comprehensive review of Network DEA perspective. This framework has been extended by, among others, Nemoto and Goto [99], Chen [26], Ouellette and Yan [109], Chen and Dalen [27], Tone and Tsutsui [143, 144], and Kao [72]. There are two clear points of difference between the Network DEA and the HJB equation framework presented in Section “Adjustment Cost Model” in this chapter. The HJB approach solves for the current period choice variables while looking to the impact of the flow of profits or costs into the future and has the ability to exploit the value function relations that lead to formulating the dynamic generalizations of modern production theory. The value function can be implemented computationally through DEA or can be parameterized and estimated econometrically. In addition, the HJB approach is more akin to decision-making process where firms make decisions in the short run as they look to the future. Since both approaches are consistent with dynamic optimization theory, they yield the same answers. As the Network DEA framework solves for the current period forward, a trajectory is generated for the future periods, thus, clearly, linking current decisions to the future flow of profits or costs. Chambers et al. [23] use this model to study the inefficiency of APEC countries due to dynamic misallocation of resources. Nemoto and Goto [99] extend the DEA to a dynamic framework consistent with the adjustment-cost theory of investment and obtain measures of dynamic inefficiency based on the HJB equation. Nemoto and Goto [100] applied this dynamic network model to study Japanese electricity production over time using a cost minimization criterion. Fallah-Fini et al. [50] use the network DEA approach to evaluate dynamic efficiency measurement in the performance of highway maintenance policies allowing for the inter-temporal dependencies between consumption of inputs and realization of outputs. Herrera-Restrepo et al. [66] build on Tone and Tsutsui

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[143, 144] for highway evacuation and highway-ramp closure strategies. Both of these studies combine the dynamic network DEA approach with traffic engineering and socio-behavioral theory of protective action. An initial exploration of the proposed approach allows for the discovery of efficiency interdependencies among perspectives, which in turn provides useful information and insights for the future design of holistic evacuation traffic management strategies.

Conclusion This chapter takes clear aim on the specification of a technology that can accommodate yet another parable of economic life that some factors can adjust rapidly while others adjust slowly in a given time scale. Focusing on production analysis in the dynamic setting leads us to emphasize the technology specification that permits the theoretical construction that can be translated and amenable to empirical implementation. While the decision to keep this chapter focused on the structure of the technology that support dynamic decision making, there are several directions that embrace more fully the complexity that the dynamic production parable struggles to speak. It can be a tolerable abstraction to state that we know the current state of nature better than the future states; hence, why not fully admit the case of uncertainty in the course of dynamic decision making? This is certainly a fair criticism, but my defense is that we must under that the kernel of the technology leading to the linkage of decisions over time. Grasping this starting point firmly provides us with some structure to venture out to the address future states of nature where real prices and other forces influencing decisions have elements of randomness. Several empirical econometric-oriented directions addressing unknown future prices are addressed in this chapter. Alternatively, uncertain relations have been modeled with fuzzy logic as an alternative to modeling stochastic aspects within a DEA framework and having more precise data, which comes at a cost [86]. As this chapter concludes, I turn to two directions that are worthy of attention and discussion, but outside the scope of this chapter. Both directions implicitly admit a degree of uncertainty as a key component to developing a more complete characterization. While the preoccupation in motivating application is with physical quasi-fixed factors of production, I have alluded to the case where quasi-fixed factor stocks can also be forms of human and knowledge capital [14]. This direction connotes a link to activities actively encouraging infusion (or investment) that can arise from activities starting with R&D and going beyond to the commercialization of such infusions to innovations. Hall et al. [61] provide a review of this direction to include the dynamic factors. There is a rich literature and interest in this direction and these explorations by their nature imply a degree of uncertainty in the dynamic setting. Aw et al. [8], Doraszelski and Jaumandreu [40], and Peters et al. [114] model the firm’s endogenous decisions to engage in R&D investment. These models explicitly address the concept of productivity in the context of knowledge capital while allowing for stochastic shocks to the current productivity. These

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considerations include addressing the role of research and development (R&D) in promoting productivity growth and eventually addressing the demand for R&D. The second is to pull ourselves out of the weeds of a particular technology and how the decision interacts in that setting, to a broader view that the nodes comprising the system embodies several behavioral forces. The system dynamics paradigm has its roots in Jay Forrester’s [57] pioneering work with Vaneman and Triantis [148] offering a contribution on how to embed system dynamic elements in the Network DEA framework. However, it is important to emphasize the role of managing a system that involves manufacturing activities, service delivery, human oversight, and policy decisions regarding the performance of this system.

Cross-References  Capacity and Capacity Utilization in Production Economics  Distance Functions in Production Economics  Production Under Uncertainty

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Levent Kutlu, Shasha Liu, and Robin C. Sickles

Contents Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Duality of the Technology and Characterizations of the Technology Using the Cost, Revenue, and Profit Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Cost Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Cost Function Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Functional Forms for Cost Function Estimation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Stochastic Frontier Models for Cost Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Endogeneity in Cost Function Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Marginal Cost Estimation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Revenue Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Revenue Function Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Functional Forms for Revenue Function Estimation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Stochastic Frontier Models for Revenue Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Profit Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Profit Function Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Functional Forms for Profit Function Estimation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Profit Function with Allocative and Technical Distortions . . . . . . . . . . . . . . . . . . . . . . . . . . Stochastic Frontier Models for Profit Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Alternative Profit Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Multi-output Functional Forms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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L. Kutlu Department of Economics and Finance, University of Texas Rio Grande Valley, Edinburg, TX, USA e-mail: [email protected] R. C. Sickles () Department of Economics, Rice University, Houston, TX, USA e-mail: [email protected] S. Liu Enterprise Model Risk, Freddie Mac, McLean, VA, USA e-mail: [email protected] © Springer Nature Singapore Pte Ltd. 2022 S. C. Ray et al. (eds.), Handbook of Production Economics, https://doi.org/10.1007/978-981-10-3455-8_12

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Non-parametric Estimation (and Shape Restrictions) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Concluding Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Cross-References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Abstract

This chapter reviews the ways in which cost, revenue, and profit functions are used to identify and characterize an underlying technology. It concentrates on the more widely used functional forms to motivate various issues in the flexibility of various parametric functions, in the imposition of regularity conditions, in the use of non-parametric estimation of models, and in standard econometric models used to estimate the parameters of these different functional characterizations of an underlying technology. The modeling scenarios we consider also allow allocative and technical distortions and address how such distortions may be modeled empirically in the specification and estimation of the dual functional representations of the underlying primal technology. Keywords

Duality · Flexible functional forms · Non-parametric production models · Endogeneity

Introduction The purpose of this chapter is to provide a review of how cost, revenue, and profit functions are used to identify and characterize an underlying technology. Such an undertaking for a Handbook will undoubtedly leave out certain topics. The chapter will provide a relative cursory discussion of duality theory and the links between cost, revenue, and profit functions and the underlying technology they characterize under certain testable regularity conditions. A more extensive recent treatment and summary can be found in Sickles and Zelenyuk [110]. Moreover, as the functional forms and estimation setup for the cost, revenue, and profit functions have many generic commonalities, the chapter will concentrate on the more widely used functions to motivate various issues in the flexibility of various parametric functions, in the imposition of regularity conditions, in the use of non-parametric estimation of models, and in standard econometric models used to estimate the parameters of these different functional characterizations of an underlying technology. The chapter also discusses briefly modeling settings in which allocative and technical distortions may exist and how such distortions may be addressed empirically in the specification and estimation of the dual functional representations of the underlying primal technology.

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Duality of the Technology and Characterizations of the Technology Using the Cost, Revenue, and Profit Functions 1 Very

often researchers either do not have information that allows them to identify the underlying technology and thus its characterization in terms of marginal products, substitution possibilities, and other technical aspects of the production process, or have problems estimating such a relationship due to statistical problems such as endogeneity of inputs. This situation was one of the motivations for work in the area of duality by various legendary economists. Among the masterminds, Ronald Shephard revolutionized the neoclassical production theory by developing his duality theory, a foundation for many practical results later on. The chapter will summarize and highlight some important results of this theory that will utilize discussion of various estimating relationships that rest on this theory, such as the cost, profit, and revenue functions that are the topics of this chapter. M The starting point is a firm that produces My outputs ∈ R+ y , using Mx inputs x ∈   Mx x RM + with exogenous prices = w1 , . . . . , wMx ∈ R++, using some technology M

y x T, where the technology set Tis defined as T ≡ (x, y) ∈ N + × + : y  is producible from x . The input requirement set L(y) completely characterizes the technology and is defined as

  M x L(y) ≡ x ∈ RM : y is producible from x , y ∈ R+ y . +

(1)

Moreover, the Shephard’s input distance function, defined as Di (y, x) ≡ sup {θ > 0 : x/θ ∈ L(y)}

(2)

completely characterizes the input requirement set in the sense x ∈ L(y)

⇐⇒

Di (y, x) ≥ 1.

(3)

A firm faced with a cost constraint chooses its level of inputs given the price and output level. Such a cost (we are considering here long-run costs) and its functional representation can be shown to be C (y, w) ≡ min {wx : x ∈ L(y)} . x

1 For

(4)

more details on the issues discussed in this section, see Chapter 2 (Production Theory: Dual Approach) of Sickles and Zelenyuk [110] whose notation we adopt here.

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Given cost minimizing decision by the firm in the employment of its resources, we can express the input demand functions as x (y, w) ≡ arg min { wx : x ∈ L(y)}, x

(5)

which, of course, are conditional on the level of output produced. If the input requirement sets are convex and there is free disposability of inputs, then it can be shown that the technology underlying the cost function can be identified. Thus, under these conditions the cost function is dual to the primal technology. If a firm’s behavioral objective is to maximize revenues instead of minimizing costs, then a duality can be shown to exist between the revenue function and the underlying primal technology under certain regularity conditions. First, define the technological possibilities (output set) as P (x) ≡   My x y ∈ R+ : y is producible from x , x ∈ RM + , and let the output prices for the   M M outputs be p1 , . . . , pMy ∈ R++y . The output set P(x) completely characterizes the technology. The Shephard’s output distance function is used to completely characterize P(x) as y ∈ P (x)

⇐⇒

Do (x, y) ≤ 1,

(6)

where Do (x, y) ≡ inf {θ > 0 : y/θ ∈ P (x)}.

(7)

M

y x The revenue function R: RM + × R++ → R+ ∪ {+∞}, is then defined as

R (x, p) ≡ max {py : y ∈ P (x)}, y

(8)

which leads to a set of output supply functions y (x, p) ≡ arg max {py : y ∈ P (x)}. y

(9)

M

y x Finally, the profit function π : RM ++ × R++ → R+ ∪ {+∞} is defined as

π (w, p) ≡ sup {py − wx : (x, y) ∈ T },

(10)

x,y

and the corresponding output supply and input demand equations are given by (x (w, p), y (w, p)) ≡ arg sup {py − wx : (x, y) ∈ T }, x,y

assuming profit maximizing behaviors.

(11)

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Next, the chapter turns to explicit functional forms and assumptions for cost functions and factor demand equations, revenue functions and output supply equations, and profit functions and the corresponding output supply and input demand equations. The chapter also considers their shadow prices when allocative distortions exist in the optimal relative output mix and input mix selected by the firm.

Cost Functions Simple inflexible cost functions, thanks to their parametric forms, often satisfy the regularity conditions required in the production theory and dual forms such as the cost functions. However, these simple but inflexible forms have serious limitations. Aside from the strong and often unrealistic restrictions they impose on the technology being modeled, they suffer from other shortcomings as well. A multioutput Cobb-Douglas distance function, for example, does not satisfy the concavity condition because it has a convex production possibility frontier. Since a majority of firms produce more than one output, distinguishing each output by using a different production function is empirically infeasible and theoretically dubious. Given the fact that substitution possibilities do not vary across many inputs using inflexible forms, the multi-output version of technology using inflexible functions, in general, does not have varying substitution possibilities either. More flexible functional representations of production are needed to satisfy the regularity conditions and to resolve the issues of using inflexible forms in the multi-output production. Flexible functional forms allow non-increasing marginal rates of substitution, which is a property all well-defined production functions possess. One important motivation for using flexible functional forms is that they do not impose any prior restrictions on the Allen-Uzawa elasticities of substitution. Given any arbitrary function, the flexible forms can approximate the function as well as the first two derivatives at a point with precision [19, 117]. The flexible functional forms are not completely new knowledge. In fact, they can be derived by adding secondorder terms to a wide range of functions used in the production studies. Therefore, the flexible functional forms can be considered as non-parametric versions of the commonly used functional forms such as the linear, the Leontief, and the CobbDouglas functions. The next section focuses will focus on a set of cost functions widely used in the literature: the translog, the quadratic, the generalized Cobb-Douglas, the generalized Leontief, the CES-translog, and the symmetric generalized McFadden cost functions. This section presents present some of the important features of the dual cost function and issues related to its estimation. Since many of these concepts apply to revenue and profit estimations, those sections are relatively brief.

Cost Function Properties The cost function gives the minimal amount of cost for a certain level of outputs M x y ∈ R+ y with given technological possibilities and fixed input prices w ∈ RM ++ ,

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where My and Mx are number of outputs and inputs, respectively. The duality theory shows that the cost function of a productive unit contains all the information of its technology. An immediate example is that the input distance function-based scale elasticity coincides with the cost-based measure of scale elasticity measure. Hence, understanding the cost function is essential for understanding the technology of production. This section first summarizes the properties of a cost function below as these properties play a central role in estimating a cost function [110]. 1. 2. 3. 4. 5.

C(y, w) ≥ 0 (non-negativity) C(y, w) is continuous in (y, w) (continuity)2 C(y, kw) = kC(y, w), ∀ k > 0(linear homogeneity in w) C (y, w ) ≥ C (y, w) , ∀w˜ ≥ w(monotonicity in w) C(y, w) is concave in w (concavity in w) M

x where y ∈ R+ y and w ∈ RM ++ are vectors of outputs and input prices, respectively. In practice, Conditions 1 and 2 are automatically satisfied by a proper functional choice for the cost function. Condition 1 may be violated for some functional form choices but, generally, it is satisfied at sample data points. Imposition of Condition 3 is not problematic as well. However, imposing Conditions 4 and 5 on a cost function is a relatively more difficult, yet possible, task. The difficulty stems from the fact that, for flexible functional forms, the restrictions would be observation specific. In practice, monotonicity condition is the least concern since estimated factor demands are positive, and cost is increasing in output with no parametric restrictions imposed. However, curvature conditions pose a somewhat difficult problem when estimating a flexible functional form.

Functional Forms for Cost Function Estimation This section briefly discusses briefly discuss some of the most widely used functional forms for cost function estimation and how regularity conditions are treated in this context. Although this section concentrates only on single-output cost functions, the generalizations to multi-output cases are available and straightforward.3

Translog Cost Function The translog (TL) cost function is the most widely used flexible functional form for cost function estimation and is:

2A

weaker continuity condition is that C(y, w) is continuous in w and lower semi-continuous in y.

3 See Caves et al. [19] for a discussion multi-output cost functions. See also Röller [100] for another

study that consider multi-output cost functions.

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ln C (y, w) = β0 + βy ln y + +

647



 1 βj ln wj + βyy (ln y)2 + βyj ln y ln wj j j 2

1 βj k ln wj ln wk j,k 2

(12)    where β jk = β kj (symmetry), j β j = 1, j β yj = 0, and k β jk = 0 (linear homogeneity). A standard way to impose linear homogeneity restriction is by normalizing C(y, w) and input prices using one of the input prices. It is common to estimate the cost-input share system in order to add degrees of freedom and boost the precision of the estimates. This, of course, may not be appropriate if input allocations are distorted and thus the cost minimizing input shares derived from the TL are not given by: sj (y, w) =

 ∂ ln C (y, w) = βj + βyj ln y + βj k ln wk . k ∂ ln wj

(13)

These input share equations (as opposed to the input demand equations in the level form) are linear in parameters. Regularity conditions can be tested using the cost function’s estimates. For example, the monotonicity condition is satisfied if sj (y, w) ≥ 0. Linear homogeneity in y is met whenβ yy = β yj = 0, while the less restrictive property of homotheticity only requires thatβ yj = 0. When the TL secondorder terms β yy , β yj , β jk are zero, it becomes the Cobb-Douglas (CD) cost function.

Translog Cost Functions with Allocative and Technical Distortions Kumbhakar [63] discusses inefficiencies with a focus on multiple outputs in the frameworks of cost minimizing and profit maximizing using translog functions to represent technology. A firm minimizes shadow cost given inefficient output, and its optimization problem is defined as    ∗  ∗  wj xj w , yeu c∗ w ∗ , yeu =

(14)

j

where wi∗ is the shadow price of the optimal input level, y is the actual output, and u ≥ 0 is technical inefficiency such that yeu is the maximum possible output. Since shadow costs are unobservable, actual costs are related with shadow costs by using input demand function and are derived as ln cA = ln c∗ + ln

 j

Sj∗ θj−1 ,

(15)

where Sj∗ is the shadow cost share and θ j = 1 is the allocative inefficiency. Actual cost shares can be related to the shadow cost shares by

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 SjA = Sj∗ θj−1 / Sk∗ θk−1 .

(16)

k

A translog shadow cost function for the shadow cost function is utilized with homogeneity of degree one in w∗ and is written as 

c∗ ln w1

= α0 +



αj ln w j∗ +

j

+

 m

+



1 aj k ln w j∗ ln w k∗ 2 j,k

 1      βm ln ym eu + βml ln ym eu ln yl eu 2

(17)

m,l

  γj m ln wj∗ ln ym eu ,

j,m w∗

j∗ = wj1 . where α jk = α kj , β ml = β lm , and w Then shadow cost shares can be obtained as     Sj∗ = ∂ ln c∗ /∂ ln wj∗ = αj + αj k ln w k∗ + γj m ln ym eu k

m

(18)

Technical inefficiency does not only appear additively but also interact with input prices and outputs, which results in heteroscedasticity. In the presence of input inefficiency, the shadow cost function incorporating technical inefficiency is    ∗ e ∗  c˜ w ∗ , y = wj xj w , y , j

(19)

and input demand functions are derived from Shephard’s lemma   ∂ c˜ (w ∗ , y) xje w ∗ , y = . ∂wj∗

(20)

For the translog cost function, actual cost and shadow cost can be related by

   ln cA = ln c˜ w ∗ , y + ln S˜j θj−1 + τ, j

(21)

where S˜j is the shadow cost share in the case of input inefficiency. Similar to the derivation in the output inefficiency case, actual cost shares can be derived as SjA =

 wj xj = S˜j θj−1 / S˜k θk−1 . A k c

(22)

The cost function lncA is then complete after using the translog form for ln c˜ (w ∗ , y), and S˜j is derived from the translog form.

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Sickles and Streitwieser [109] focus on distortions in the pipeline transmission of natural gas by employing a restricted cost function captured by a shadow price and estimate various aspects of a production. Assuming exogenous output and input prices, a firm minimizes its short run cost as follows: min

 i

wi xi subject to G (y, x; t) = 0,

(23)

where G is the function that transforms the technology t, and x include labor, energy, and two quasi-fixed capital inputs. The solution to this is the short-run variable cost function V C = C (y, w, x; t),

(24)

where C is homogenous of degree one, non-decreasing, and concave in factor prices w, non-increasing and convex in the quasi-fixed factors x, and non-negative and nondecreasing in output y. A non-homothetic translog function is used to approximate C. Given exogenous wi , they derive the variable cost share utilizing Shephard’s Lemma as Mi = αi + i βij ln wj + βyi ln y + k βik ln xk .

(25)

zk xk ln C The shadow share equation – ∂∂ ln xk = CV is incorporated in the model, where zk , the shadow price, can be obtained by taking the difference between revenues and variable costs. The shadow cost share in the restricted translog cost function is



Mk = − αk + i βik ln wi + βyk ln y + h βhk ln xhk .

(26)

Good, Nadiri, and Sickles [38] develop several modeling scenarios in the airline industry, which allow input price distortions incorporated in a translog variable cost function that captures the linkage between observed cost and assumed minimized cost. Airlines are assumed to use inputs x = x (xJ , xN − J ) > 0 to produce outputs y = y (yK , yM − K ), where the last N-J inputs are assumed to be fixed and the last M-K outputs are non-physical output characteristics. Consider a virtual technology and virtual input and output decisions, labeled with a “∗ ,” that are consistent with the standard assumptions of duality theory. The observed prices deviate from the virtual prices by θ = (θ 1 , . . . , θ N ) such that wi∗ = wi + θi for input i. Based on Shephard’s lemma, factor demands derived from the firm’s minimum virtual cost function are



x∗J y, wj∗ ; xN −J = ∇wj∗ C ∗ y, wj∗ ; xN −J . The observed cost function and associated short-run factor shares are

(27)

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C y, w∗j , wj ; xN −J = wj xj∗ y, wj∗ ; xN −J j

(28)

and Mi =

wi xi

, i = 1, . . . , J. C y, wj∗ , wj ; xN −J

(29)

Since M∗i = wi∗ xi /C ∗ , observed input use can be written as xi = Mi∗ C ∗ /wi∗ . Then, observed costs can be expressed as   ∗  Mi w i C=C , i wi∗ ∗

(30)

and observed factor shares expressed as

Mi =

Mi∗ wi wi∗



 j

Mj∗ wj wj∗

.

(31)

The equations above provide linkages between an observable cost function and the virtual technology when the application of the technology is distorted. Atkinson and Halvorsen [10] incorporate regulatory constraints into the cost function framework in which they assume shadow prices to be simply proportional to market prices. Later Getachew and Sickles [36] utilizes the same approach to study the impact of policy constraints on relative prices and structure of production. By imposing additional constraints R(w, x; φ), the firm minimizes the production cost as follows: minx C = w  x s.t.f (x) ≤ Q and R (w, x; φ) ≤ 0

(32)

where f(x) is a production function and Q is a certain level of output. Taking Lagrangian, the constrained cost minimization of the firm becomes L = w  x − v (f (x) − Q) −

 r

λr Rr (w, x; φ),

(33)

where λr are the Lagrangian multipliers for each of the Rr constraints. The unobserved shadow prices are approximated by using a first-order Taylor series wei = ki wi ,

(34)

where ki is a factor proportional to an input price. Derived from the shadow cost function, the updated demand function can be obtained utilizing Shepard’s Lemma. The updated demand function gives an actual cost function

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lnCA = lnC∗ + ln

 M∗ i , i ki

(35)

where M∗i is the shadow share of factor i. The actual share equation MA i is derived to be MA i

M∗i ki  Mi∗ i ki

=

(36)

.

The shadow cost function lnC∗ can be rewritten in the translog form as follows:  1 lnC∗ = α0 + αQ ln Q + γQQ (ln Q)2 + αi ln (ki wi ) i 2    1 + γiQ ln Q ln (ki wi ) + γij ln (ki wi ) ln kj wj + δt t, i i,j 2

(37)

where t is the time trend that represents technological change over time. Then, the expression for the shadow share M∗i can be obtained from the logarithmic differentiation. Substituting into the actual cost function gives lnCA = lnC∗ + ln

  i

αi + γiQ ln Q +

 j

   γij ln kj wj /ki .

(38)

Then, the actual cost share of input i can be derived as    1    MA αi + γiQ ln Q γij ln kj wj / i = αi + γiQ ln Q + j i ki    1 + γij ln kj wj . j ki

(39)

The actual cost function is then complete.

Generalized Leontief Cost Function The generalized Leontief (GL) cost function [25] is homogenous by construction and is given by: C (y, w) =

 j

βj wj + y



1/2

j,k

1/2

βj k wj wk

+ y2

 k

βyj wj

(40)

where β jk = β kj (symmetry). Input demand equations are given by: xj (y, w) = βj + y



 k

βj k

wk wk

1/2 + βyj y 2 .

(41)

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The monotonicity condition is satisfied if xj (y, w) ≥ 0. The GL cost function is non-homothetic unless β yy = β yj = 0 and incapable of distinguishing between homotheticity and linear homogeneity. When β jk = 0 for j = k, GL cost function collapses to the Leontief fix proportions cost function.

The Symmetric Generalized McFadden Cost Function The symmetric generalized McFadden (SGM) cost function [25] is given by: C (y, w) = g(w)y +

 j

βj wj + y

 j

βyj wj

(42)



where g(w) = 12 wθ Sw w , S is a symmetric non-negative semidefinite parameter matrix, and θ is a non-negative vector (not all zero). In order to achieve identification of all parameters, it’s necessary to have S w  = 0 for some w  with strictly positive components, e.g., a vector of ones. Input demand equations are given by the vector: x(w) =

1 w  Sw Sw − θ.  θ w 2 (θ  w)2

(43)

By construction, SGM cost function is linear homogenous in w. The monotonicity condition is satisfied if the components of x(w) are non-negative. It turns out that C(y, w) is globally concave in w if S is negative semidefinite. If the estimate of S  is not negative semidefinite, one can reparametrize S as S = − LL , where L is a lower triangular matrix so that L w  = 0, which would assure global concavity of C(y, w). Kumbhakar [62] gives a generalization of SGM cost function to the multioutput case that makes it relatively easy to estimate different aspects of a production technology. He applies SGM to a panel data of 12 Finnish foundry plants to estimate technical progress, economies of scale, and economies of scope. Rask [97] proposes a modified version of SGM to allow fixed factors of production so that the cost function can be applied to the processes when there are fixed costs. He estimates the modified SGM cost function for sugarcane in Brazil, which takes up over two-thirds of total costs in ethanol production and thus is important to study the technology of sugarcane production.

Imposing Regularity Conditions for Cost Functions As Barnett [11] points out, if both monotonicity and curvature conditions are not satisfied, the second-order conditions for optimization and duality theory fail. While some empirical researchers do not state these conditions, many others are careful about the regularity conditions. Guilkey and Lovell [46] and Guilkey et al. [47] exemplify some studies that provide evidence for potential poor global behavior of multi-output cost functions.4

4 See

Wales [117] for another example in the utility function context.

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If the percentage of violations for monotonicity and curvature conditions is small (e.g., smaller than 5%), some researchers attribute this to the stochastic nature of the estimations and find the violations acceptable. When the percentage of violations is high, some researchers modify the model to get an acceptable violation percentage. For example, when estimating a TL variable cost function of US airports, Kutlu and McCarthy [73] include an additional term to reduce the violation percentages for monotonicity and concavity conditions. The percentage of violations decreases from 4.2% to 0.5% after including this term. They argue that some airports have particularly higher capital levels relative to the median airport, and the additional term that they include captures this pattern. Another approach is simply imposing regularity conditions. Serletis and Feng [107] and references therein provide good discussions on how this can be done. Hence, the rest of this subsection closely follows their arguments. Serletis and Feng [107] categorize these methods as local regularity (at some data point in the sample), regional regularity (over a neighborhood of data points in the sample), pointwise regularity (at every data point in the sample), or global regularity (at all possible data points). Cholesky decomposition methods for imposing regularity conditions were first used by Wiley et al. [120]. This method is based on the Cholesky decomposition of a Hessian matrix into the product of a lower triangular matrix and its conjugate  transpose. For imposing concavity, one can reparametrize a matrix S as S = − LL where L is a lower triangular matrix. As stated by Serletis and Feng [107], this approach can be used not only for imposing the curvature but also for the monotonicity conditions. While this approach is capable of imposing local and global curvature conditions, it cannot impose regional or pointwise curvature conditions. For monotonicity, the approach can be used to impose local monotonicity condition. As an illustration, consider the TL cost function given in section “Translog Cost Function.” The concavity in input prices is satisfied if the Hessian matrix H (y, w) =

∂ 2 C (y, w) ∂w∂w 

(44)

is negative semidefinite. Diewert and Wales [25] prove that H is negative semidefinite if and only if the following matrix is negative semidefinite: G (y, w) = B − Diag (s (y, w)) + s (y, w) s  (y, w),

(45)

where B = [β ij ] is the matrix with element ij being equal to β ij , s (y, w) =   s1 (y, w) , s2 (y, w) , · · · , sMx (y, w) is the input share vector, and Diag(s(y, w)) is the Mx × Mx diagonal matrix with diagonal elements being equal to input share vector s(y, w). Since G(y, w) is observation specific, it may not be easy to impose concavity for all data points in the sample. However, as in Ryan and Wales [102] and Feng and Serletis [30], concavity can be easily imposed on G(y, w) at a reference point in the sample. Usually once the concavity is satisfied at a single reference point, it is satisfied at most of the other sample points (if not all). If the

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percentage of violations is still high, one can simply try other reference points and find the reference point that gives minimum number of violations. The TL cost function would satisfy global concavity in input prices if s(y, w) > 0 and B is negative semidefinite [25]. However, Lau [78] and Diewert and Wales [25] argue that imposing negative semidefiniteness on B destroys the flexibility of TL cost function and reduces it to the Cobb-Douglas form. The imposition of monotonicity by the Cholesky decomposition is not difficult and is explained by Serletis and Feng [107]. The non-linear optimization method for imposing regularity conditions is first used by Geman and Geman [35]. In order to reduce computational difficulties and time, Serletis and Feng [107] impose linear homogeneity by normalizing the cost and input prices by the last input price wMx .5 They impose negative semidefiniteness on G(y, w), i.e., concavity in input prices, by restricting its eigenvalues to be non-positive. They also impose non-negativity on the cost function and nonnegativity of input shares (monotonicity). This approach can impose curvature and monotonicity conditions locally, regionally, and pointwise. It is possible to impose global concavity by restricting the eigenvalues for B to be non-positive. However, the global monotonicity and non-negativity cannot be imposed if one wants to keep concavity assumption. Serletis and Feng [107] argue that the Bayesian method is a convenient way for imposing regularity conditions due to Gibbs sampling methods introduced by Geman and Geman [35] and the Metropolis-Hastings algorithm [49, 90]. Terrell [113], Koop et al. [60], and Griffiths et al. [43] exemplify some important contributions on this area that allow incorporation of non-negativity, monotonicity, and concavity conditions.6 Serletis and Feng [107] examine the performance of all three methods for imposing non-negativity, monotonicity, and concavity conditions for TL cost function. They find that, irrespective of the method, imposing global curvature conditions forces the elements of the B matrix to be close to zero as the TL cost function reduces to the Cobb-Douglas cost function in this case. Hence, they rather recommend imposing pointwise regularity using either constraint optimization or Bayesian approach. However, the Bayesian approach may be preferred on the grounds that it is easy to obtain statistical inferences for the parameters and relevant measures (e.g., elasticities and productivity), which can be expressed as functions of parameters.

Stochastic Frontier Models for Cost Functions The stochastic frontier analysis literature relaxes the neoclassical full efficiency assumption by allowing the productive units to be inefficient. Aigner et al. [5] and

5 For

another application of constrained optimization method to a flexible (i.e., globally flexible Fourier) cost function, see Feng and Serletis [31]. 6 See Kleit and Terrell [58] as an application of Bayesian approach for flexible cost functions.

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Meeusen and van den Broeck [89] exemplify earlier studies of stochastic frontier models that aim to measure efficiencies of productive units. A common feature of stochastic frontier models (SFMs) is that they assume a composed error term where the first component is the usual two-sided error and the second component is a one-sided (non-negative) error term, which represents inefficiency. A variety of distributions is proposed for the one-sided error component including the half normal [5], the exponential [89], the truncated normal, the gamma, and doubly truncated normal distributions. A stochastic cost frontier model is given by: ln C = α + x1 β + u + v

(46)

where C is the cost of the productive unit; α is the constant term; x1 is a vector of frontier variables, which does not contain the constant; u ≥ 0 is the one-sided term that captures the cost inefficiency; v is the usual two-sided error term. It is common to model the inefficiency term as u = h x2 γ u∗ , where u∗ ≥ 0 is a onesided random variable and h > 0 is a function of so-called environmental variables x2 that affect inefficiency. The smaller values of u indicate that the productive unit is cost efficient, and u = 0 means that the productive unit becomes fully efficient.  The standard stochastic frontier models assume that u∗ , v, and x1 , x2 are all independent from each other. Cost efficiency is estimated by predicting7 : Eff = exp (−u).

(47)

The earlier stochastic models (e.g., [5, 89]) are in the cross-sectional framework. Panel data can potentially give more reliable inefficiency estimates. Pitt and Lee [95] and Schmidt and Sickles [106] propose random and fixed effects models for estimating unit specific inefficiencies. These models assume time-invariant inefficiency, which may not be a reasonable assumption for relatively longer panel data. Cornwell et al. [21], Kumbhakar [61], Battese and Coelli [12], and Lee and Schmidt [79] exemplify earlier time-varying inefficiency models. Ahn et al. [4], Desli et al. [22], Tsionas [115], Huang and Chen [51], Assaf et al. [9], and Duygun et al. [28] provide dynamic efficiency models. Greene [40, 41] argues that if there is productive unit specific heterogeneity in the frontier and this is controlled, the heterogeneity may be confused with inefficiency. Greene [40, 41] proposes fixed and random effects models to control for heterogeneity, which are called true fixed effects and true random effects, respectively. The advantage of fixed effects models is that the heterogeneity can be correlated with the regressors. However, it is subject to incidental parameters problem. In particular, while the frontier parameters are consistent, the inefficiency estimates may not be accurate. Wang and Ho [118] solve this problem by introducing first difference and within transformations to eliminate the fixed effects term. Although the fixed effects models of Greene [40, 41], and

7 See

Kumbhakar and Lovell [65] for details.

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Wang and Ho [118] allow inefficiency to vary over time, the heterogeneity is timeinvariant. Kutlu, Tran, and Tsionas [76] illustrate using Monte Carlo simulations that ignoring time-varying heterogeneity may lead to biased parameter estimates and seriously distorted efficiency estimates. The individual effects model of Kutlu, Tran, and Tsionas [76] solve this issue by allowing both heterogeneity and inefficiency to vary over time without being subject to incidental parameters problem. Similar to the conventional cost function estimation, the most widely used functional form in stochastic cost frontier studies is the translog functional form. As stated earlier, in a conventional cost function model if the monotonicity and/or curvature conditions are violated, the second-order conditions for optimization and duality theory fail. The issue is even more serious for stochastic frontier models. Sauer et al. [103] illustrate that when the monotonicity and curvature conditions are not satisfied, the efficiency estimates may be seriously distorted. Many stochastic frontier studies either do not state whether the regularity conditions are satisfied or simply check these conditions at the mean or median of the sample data points. Hence, the regularity conditions may still be violated at many other sample points, indicating that the cost efficiency estimates for these sample points (and potentially other sample points) are not reliable. All these stochastic frontier studies mentioned so far can be applied to stochastic cost, production, profit, and revenue frontier model estimations with minor modifications. In particular, for production, profit, and revenue estimations, the inefficiency component u is replaced by −u to estimate efficiency. Allocative inefficiency results in utilization of inputs in wrong proportions given input prices, i.e., misallocation of inputs. A production function can be used to estimate technical inefficiency, which happens when the firm fails to produce maximum output from a given input bundle, but it cannot be used to estimate allocative inefficiency. Under the Cobb-Douglass production function assumption, Schmidt and Lovell [104] present a stochastic cost frontier model where both costs of allocative and technical inefficiency can be estimated. However, they assume that allocative and technical inefficiency are not correlated. Under the same production technology, Schmidt and Lovell [105] relax this assumption by allowing allocative and technical inefficiencies to be correlated. Modeling allocative inefficiency under translog cost function assumption is less trivial. Greene [39] models allocative and technical inefficiency in a translog cost function by assuming that allocative inefficiency departures from the cost shares. However, he does not derive cost of allocative inefficiency due to such departures. Rather, he assumes that allocative inefficiency and cost of allocative inefficiency are independent. Bauer [13] calls this “Greene problem.” Kumbhakar and Wang [68] and Kutlu [72] examine the consequences of lumping allocative inefficiency together with technical inefficiency when estimating a cost frontier, i.e., the assumption that the one-sided error term in the cost function captures the overall cost of inefficiency. They both start with the cost minimization problem for the translog cost function. Then, they calculate the exact allocative inefficiency and the corresponding cost of allocative inefficiency where allocative inefficiency is defined as the deviations from the optimal input allocation. Both Kumbhakar and Wang [68] and Kutlu [72] point

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out negative consequences of lumping the allocative inefficiency with technical efficiency when estimating a cost frontier. Kutlu [72] argues that system estimators perform worse than single equation estimators even when the complex functional form for allocative inefficiency is approximated by a first-order Taylor series. In order to address this issue, Kumbhakar and Tsionas [66] use similar approximations in a Bayesian setting, and the solutions based on the cost function approach seem not easy. Kumbhakar and Wang [67] overcome this issue by using a primal system consisting of a translog production function and first-order conditions of cost minimization. In defense of standard stochastic cost frontier models, Kumbhakar and Wang [67] and Kutlu [72] are typical examples for those studies that find negative results based on changing where and how an error term enters a model. While these negative results put some unrest about cost function estimations, they depend on how the data generating process is determined. Nevertheless, unlike the conventional cost function estimations where researchers generally estimate a cost-input share system, the number of such studies is almost non-existent in the stochastic frontier literature.

Endogeneity in Cost Function Models Using the production function approach is appropriate if the inputs are exogenous. However, researchers often encounter endogenous input choices in the production process. In particular, the factor inputs under a firm’s control may be reallocated to achieve the firm’s objectives. In the case of a stochastic production function by a firm maximizing expected profits [122], all variable inputs can be considered weakly exogenous. However, if the expected profit maximization assumption of Zellner et al. [122] is not accurate, then one potential solution is to use an instrumental variable or control variable approach to address the issue. In many scenarios, the price taking assumption is more reasonable compared to the exogenous factor inputs assumption and good instruments may be hard to find. Hence, a widely used solution is to estimate a cost function rather than a production function. This is one of the reasons why a dual cost function specification may be preferred over a primal production function specification. Exogenous input prices are more likely when the market is competitive, and thus researchers would prefer the cost function approach given that the level of output is dictated by market forces exogenous to the firm. However, cost functions may suffer from endogeneity problems as well if the output fails to be exogenous. Thus, both production and cost functions may suffer from endogeneity. Besides endogenous outputs, other scenarios may lead to endogeneity in the cost function approach. One potential problem occurs when a cost function includes a quality variable where the quality is jointly determined by the costs. Mutter et al. [92] argue that inclusion of the quality variable leads to inconsistent parameter estimates. Some researchers drop the quality variable to avoid such problem, but this does not solve the issue in the stochastic frontier framework. If the quality is cost enhancing and a stochastic frontier model is estimated, the efficiency estimates would be

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inconsistent irrespective of whether the quality variable is included in the frontier. Duncombe and Yinger [27] and Gronberg et al. [44] exemplify studies that point out the endogeneity of output quality in their cost equation. Another potentially endogenous variable used in cost function estimations is the Herfhindahl-Hirschman Index (HHI). This variable is popular in stochastic frontier models due to close connection between market power and efficiency. In particular, it is common to model inefficiency by using HHI as one of the environmental variables. Karakaplan and Kutlu [55, 56] find evidence of endogeneity from HHI. Similarly, Kutlu, Tran, and Tsionas [76] find evidence of endogeneity from another related variable that measures profitability, i.e., return on revenue. The endogeneity problem is more likely to occur in a stochastic frontier setting due to presence of the additional inefficiency term. In particular,  as stated earlier, the  standard models in this literature assume that u∗ , v, and x1 , x2 are all independent from each other. Guan et al. [45] and Kutlu [71] are the earliest studies that aim to solve endogeneity problems in the stochastic frontier setting. These papers relax the independence assumption of x1 and v. Guan et al. [45] achieve this via a twostage method where in the first stage they get the consistent frontier parameter estimates using the GMM and in the second state they estimate efficiency using a standard stochastic frontier model. Kutlu [71] uses a limited information maximum likelihood estimation method (single-stage control function estimation) to solve the endogeneity problem. Tran and Tsionas [114] propose the GMM counterpart of Kutlu [71]. Karakaplan and Kutlu [54, 55] present cross-sectional and panel data variations of Kutlu [71] and extend his method to allow environmental variables  to be endogenous, i.e., allowing v and x1 , x2 to be correlated. In a Bayesian   framework, Griffiths and Hajargasht [42] propose models that allow v and x1 , x2 to be correlated. Using a copula approach, Amsler et al. [7, 8] provide crosssectional models that allow more general correlations, including the correlation  between u∗ and x1 , x2 . The approach requires using a proper copula and may be computationally intensive. Kutlu, Tran, and Tsionas  [76] provide an individual effects panel data model that allows v and x1 , x2 to be correlated, which is a generalization of time-varying heterogeneity as in Wang and Ho [118]. In an appendix, they also provide a copula variation of their model that allows more general correlation structures. However, they argue and illustrate by Monte Carlo simulations that when the heterogeneity term is included, the consequences of violating general correlation assumptions are not serious if the heterogeneity is controlled. Finally, the standard modeling of a cost function does not incorporate agencyrelated aspects into the optimization problem. Kutlu, Mamatzakis, and Tsionas [77] present a model where the manager is a utility maximizer in a quantity-setting oligopoly market. The utility of the manager is a function of profit and her effort level. They assume that higher effort reduces the costs. This introduces an additional structural inefficiency term, which is a specific function of frontier variables. Hence, given that the standard models ignore this structural inefficiency term, the parameter and efficiency estimates from the standard stochastic frontier cost function models would be inconsistent if the assumptions of this model hold. Basically, the solution

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to this problem would be including the structural inefficiency term as a control function to correct the bias. Gagnepain and Ivaldi [33, 34] propose related models where additional terms appear in the cost function due to agency-related problems.

Marginal Cost Estimation Sometimes a researcher is interested in the marginal cost rather than the cost itself. A common application is estimating the cost function and then calculating the marginal cost (e.g., Weiher et al. [119] and [74]). However, in many occasions data on total cost is either not available at all or not available at the desired market level. For example, Weiher et al. [119], Kutlu and Sickles [74], and Kutlu and Wang [75] have airline specific total cost data for the US airlines although these studies are interested in route-airline-specific marginal cost estimates. The new empirical industrial organization literature allows estimation of marginal cost without using total cost data. The marginal cost estimates (along with market power estimates) are obtained by estimating the so-called conduct parameter (conjectural variations) model where a general form of demand-supply system is estimated. Bresnahan [18] and Perloff et al. [94] provide excellent surveys on this topic. Recently, Kutlu and Wang [75] present a methodology that combines the conduct parameter and stochastic frontier methods that enables estimation of market power, marginal cost, and marginal cost efficiency estimates from a demand-supply system. The advantage of studying marginal cost efficiency over cost efficiency is that marginal cost efficiency is directly related to deadweight loss. While both measures are valuable, marginal cost efficiency measure may be more relevant from the antitrust point of view.

Revenue Functions This section presents some important features of a revenue function and issues related to its estimation. As mentioned before, since many of the concepts introduced apply to the revenue function estimation, this section will be brief.

Revenue Function Properties The revenue function gives the maximal amount of revenue a firm can achieve at a x certain level of inputs x ∈ RM + , given technological possibilities and fixed output M

prices p ∈ R+ y . First, the properties of a revenue function are summarized below as these properties play a central role when estimating a revenue function [110]: 1. R(x, p) ≥ 0(non-negativity) 2. R(x, p) is continuous in (x, p) (continuity)8 8A

weaker continuity condition is that R(x, p) is continuous in p and upper semi-continuous in x.

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3. R(x, kp) = kR(x, p), ∀ k > 0(linear homogeneity in p) 4. R (x, p) ˜ ≥ R (x, p) , ∀p˜ ≥ p(monotonicity in p) 5. R(x, p) is convex in p (convexity in p) M

y x where x ∈ RM + and p ∈ R++ are vectors of inputs and output prices, respectively. In practice, Conditions 1 and 2 are automatically satisfied by a proper functional choice for the revenue function. As in the cost function case, Condition 1 may be violated for some functional form choices but, generally, it is satisfied at sample data points. Imposition of Condition 3 is not problematic as well. As in the cost function case, the monotonicity conditions are not problematic in practice. However, again, curvature conditions pose some difficulties when estimating a flexible functional form.

Functional Forms for Revenue Function Estimation Typically, the functional forms used in revenue function estimation are similar to those used in (multiple-output) cost function estimation. Hence, this section is brief. The most widely used revenue function is translog revenue function [23], which is given by:    ln R (x, p) = β0 + j βxj ln xj + j βj ln pj + 12 j,k βxxj k ln xj ln xk   + j,k βxj k ln pj ln xk + 12 j,k βj k ln pj ln pk

(48)

   where β jk = β kj , β xxjk = β xxkj (symmetry), j β j = 1, j β xjk = 0, and k β jk = 0 (linear homogeneity). The output share equations are given by: y

sj (x, p) = βj +

 k

βxj k ln xk +

 k

βj k ln pk .

(49)

Diewert [23] provides the details about Generalized Leontief revenue function. A functional form, which haven’t been mentioned earlier, that is used in the revenue framework is the mean of order of two revenue functions [24]. Diewert considers only one input case though the functional form can be extended to a multi-input scenario in a straightforward way. Using solutions to a set of functional equations, Chambers et al. [20] show that the translog revenue function can be obtained from the Shephard distance function for generalized quadratic functions in the dual price space.

Stochastic Frontier Models for Revenue Functions Unlike the cost function, the relevant stochastic revenue frontier model needs to be slightly modified and is given by:

16 Cost, Revenue, and Profit Function Estimates

ln R = α + x1 β − u + v

661

(50)

where R is the revenue of a productive unit; α is the constant term; x1 is a vector of input variables; u ≥ 0 is the one-sided term that captures cost inefficiency; v is the usual two-sided error term. As in the cost function case, the smaller values of u indicate that the productive unit is more cost efficient, and when u = 0 the productive unit becomes fully efficient. The standard stochastic frontier   assumptions about independence of variables remain the same so that u∗ , v, and x1 , x2 are all independent from each other. In the case of endogenous input variables, estimates from the revenue function would be inconsistent. The endogeneity solutions mentioned for the stochastic cost frontier models can also be applied to the stochastic revenue frontier function estimation. Applications of the revenue function are not as prevalent as the cost and production function, but the revenue function is still applicable in various research questions. Kumbhakar and Lai [69] apply the revenue function to a non-radial and output-specific measure of technical efficiency they propose in a revenuemaximizing framework. They use the maximum likelihood estimation method to estimate a translog revenue-share system. The empirical work by Oliveira and his colleagues [93] use a revenue function to analyze efficiency of hotel companies in Portugal based on the stochastic frontier approach. Mairesse and Jaumandreu [83] study the discrepancies between the cross-sectional and time-series estimates of scales and capital elasticities by estimating the production function as well as the revenue function with two panel datasets. They find that the estimates of the functions have little difference and conclude that the bias from other sources, rather than the lack of firm data on output prices, are more likely to be problematic. Rogers [99] estimate revenue efficiency along with cost and profit efficiency to show the importance of including nontraditional output in bank studies. They find that the standard model understates bank efficiency if nontraditional output is excluded.

Profit Functions This section presents some important features of a profit function and issues related to its estimation. It also talks about a less well-known form of profit function, which has many desirable properties, so-called alternative profit function.

Profit Function Properties The profit function gives the maximal amount of profit for given input and output prices with given technological possibilities. First, this section summarizes properties of a profit function below as these properties play a central role when one estimates a profit function [110]:

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1. 2. 3. 4. 5. 6. 7.

π (w, p) ≥ 0(non-negativity) π (w, p) is continuous in (w, p) (continuity) π (kw, kp) = kπ (w, p), ∀ k > 0(linear homogeneity in (w, p)) π (w, p) ˜ ≥ π (w, p) , ∀p˜ ≥ p(monotonicity in p) π ( w , p) ≥ π (w, p) , ∀ w ≤ w(monotonicity in w) π (w, p) is convex in w (convexity in w) π (w, p) is convex in p (convexity in p) M

y x where w ∈ RM ++ and p ∈ R++ are vectors of input and output prices, respectively. While Conditions 2–5 are relatively easily satisfied, the curvature conditions (Conditions 6 and 7) and Condition 1 need some extra care. In the banking industry, for example, data points with negative profits are not uncommon. However, profit cannot be negative given a concave production function. To use this result, the profit   has to be defined as π (w, p) = p y − w xand used in the model, instead of reported profit. Observed negative profits violate the property and are problematic.

Functional Forms for Profit Function Estimation As discussed earlier in the cost function setting, apparent proper candidates for a profit function are twice differentiable functional forms that are based on a quadratic form. Diewert [23] notes that having a second-order approximation which is homogenous of degree one is a preferred method. However, in this case, the second-order approximation reduces to a first-order approximation. Due to this reason, he considers alternatives such as generalized quadratic in square roots profit function and its special case, the generalized Leontief profit function. The extended profit function of Behrman et al. [14] exemplifies another study that is motivated by the same problem. Now, this section briefly discusses the extended profit function of Behrman et al. [14]. This model is presented using their notation. Let x be the vector of variable inputs and H be the quasi-fixed input used for producing multiple output represented by y with prices p. Further the output and input prices and quantities are combined       as q = (p , w ) and u = (−y , x ) . Then, the generalized Leontief variable profit function can be written as follows: π (q, H ) =

 j,k

1/2 1/2

γj k qj qk

+

 j

γj H qj H 1/2

(51)

where γ jk = γ kj . Therefore, the constant elasticity transformation-constant elasticity of substitution- generalized Leontief variable profit function (CET-CES-GL) can be expressed as: π (q, H ) =

 j

γjj qjε

1/ε

+

 j,k=j

1/2 1/2

γj k qj qk

+

 j

γj H qj H 1/2

(52)

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where γ jk = γ kj . The corresponding variable profit maximizing output supply and input demand equations are given by: uj (q, H ) = γjj qjε−1

 k

γkk qkε

(1−ε)/ε

+

 k=j

−1/2 1/2 qk

γj k qj

+ qj H H 1/2 . (53)

Profit Function with Allocative and Technical Distortions Lovell and Sickles [81] incorporate technical and allocative inefficiency into a profit function in the Generalized Leontief form by assuming wrong price ratios and by allowing the actual output and input to differ from the optimal levels. The output prices p = (p1 , . . . , pm ) > 0 and input prices w = (w1 , . . . , wn ) > 0 are given as exogenous; the profit maximization problem becomes maxy,x py − wx s.t. (y, −x) ∈ T .

(54)

The profit function is useful from the fact that a profit function π and a production possibilities set T both represent the profit-maximizing technology due to a duality relationship. In addition, profit maximizing output and input allocations can be derived using Hotelling’s Lemma: ∇p π (p, w) = y (p, w) , ∇w π (p, w) = −x (p, w) .

(55)

The profit of a firm producing two outputs using two inputs, as an example, is assumed to be the Generalized Leontief form. Then, the profit maximizing output and input equations can be derived from Hotelling’s Lemma and can be modified to include inefficiency as follows  1  1  1 p1 − 2 p1 − 2 p1 − 2 y1 (p, w, φ, θ) = (β11 −φ1 ) +β12 θ12 +β13 θ13 +β14 θ14 , p2 w1 w2 (56)  1  1  1 p1 2 p2 − 2 p2 − 2 +β23 θ23 +β24 θ24 , y2 (p, w, φ, θ) = (β22 −φ2 ) +β12 θ12 p2 w1 w2 (57)  1  1  1 p1 2 p2 2 w1 − 2 +β23 θ23 +β34 θ34 , −x1 (p, w, φ, θ) = (β33 −φ3 ) +β13 θ13 w1 w1 w2 (58)

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 1  1  1 p1 2 p2 2 w1 2 −x2 (p, w, φ, θ) = (β44 −φ4 ) +β14 θ14 +β24 θ24 +β34 θ34 . w2 w2 w2 (59) The parameters φ i ≥ 0 measure the under-production of outputs and excessive usage of inputs due to technical inefficiency. The parameters θ ij > 0, j > i are interpreted as allocative inefficiency. If both technical and allocative inefficiency exist, the observed profit can be expressed 1 1 3 4  − 1 4 1 βij θij 2 +θij2 qi2 qj2 , π (q, φ, θ ) = (βii −φi ) qi + i

i

j

(60)

where q ≡ (p1 , p2 , w1 , w2 ). The change in profit due to technical inefficiency is obtained by π(q) − π (q, φ) =

4 i

φi qi ,

(61)

and the change in profit due to allocative inefficiency is obtained by π(q) − π (q, θ ) =

3 4 i

j

  1 1 1 1 − qi2 qj2 . βij 2 − θij 2 + θij2

(62)

Allocative inefficiency can be further decomposed into output mix inefficiency, input mix and scale inefficiency depending on θ ij . The perceived price

inefficiency, qi ratios θij qj are consistent allocative inefficiency if they satisfy   qj qi qi θij θj k = θik , i < j < k, qj qk qk

(63)

θik = θij θj k , i < j < k.

(64)

which requires

Based on the work of Lovell and Sickles [81], Sickles, Good, and Johnson [111] apply the Generalized Leontief profit function with allocative distortions to the US airline industry by assuming wrong price ratios. The generalized Leontief profit function including output characteristics is expressed as  1 1 1 1 −2 2 π (q, c, t; θ ) = i βii qi + i,j βij θij + θij qi2 qj2 (65) 1 2

1 2

+ i βit qi t + i,j,k δij k qi cj ck , δij k = δikj , ∀i, j = k,

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where q is the vector of input and output prices, c is the vector of output characteristics, and t is a time index. The output and input allocation equations can be derived as  1 1 1 qi 2 di (q, c, t; θ ) = βii + j =i βij θij + βit t + j,k δij k cj2 ck2 , qj

(66)

where d = (y, −x). The output characteristics are approximated by 1 1/2 − 2

ci (q, t) = j k>j γij k qj qk

1

−1

+ j k>j γij kt qj2 qk 2 t + γit t + γi .

(67)

Kumbhakar [63] models technical and allocative inefficiencies in profit maximizing frameworks emphasizing on multi-outputs and multi-inputs. He derives the exact relations between the inefficiencies and profit when translog functions are used to represent technology. In the presence of output technical inefficiency, the firm’s profit maximization problem is max π = p y − w  x y,x

  s.t.F yeu , x = 0,

(68)

where y is the actual output and u ≥ 0 is the technical inefficiency so that yeu is the ∗ = k p , where θ and maximum possible output level. Assume wj∗ = θj wj and pm m m j km are input inefficiency and output inefficiency, respectively. Optimal inputs and outputs are determined by the shadow profit adjusted for efficiency. The efficiency adjusted normalized shadow profit is πˆ ∗ = y1 eu +



p˜ ∗ ym eu m m

−

 j

  wˆ j∗ xj = πˆ ∗ wˆ ∗ , p∗

(69)

eu w ∗

∗ u

∗ = p ∗ /p and p ∗ = p .The normalized j∗ = p1 j , p˜ m where πˆ ∗ = πp∗e , wˆ j∗ = eu w 1 m 1 1 1 actual profit adjusted for efficiency and the shadow profit adjusted for efficiency are related as follows

    1  1 ∗ ∗ e π˜ = πˆ 1 + − 1 Rm + − 1 Qj , m km j θj u A

∗

∗ = where the shadow revenue and cost shares are Rm

−

w j∗ xj π˜ ∗

∂ ln π˜ ∗ ∗ ∗ ∗ , Qj ∂ ln p˜ m j

=

(70) ∂ ln πˆ ∗ ∂ ln wˆ j∗

=

. This transforms into ln π˜ A = ln πˆ ∗ + ln H − u,

(71)

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where H incorporates the shadow revenue and cost shares. The equations that relate the actual revenue and cost shares to the shadow revenue and cost shares are given by A ∗ = Rm Rm

1 1 H km

(72)

1 1 . H θj

(73)

∗ QA j = −Qj

  Using a translog form for πˆ ∗ wˆ ∗ , p∗ given the expressions for the shadow revenue and cost shares, one can obtain the expression for H. The profit function specification is then complete. In the presence of input technical inefficiency, the firm maximizes the profit as follows: maxπ = p y − w  x y,x

  s.t.F y, xe−τ = 0,

(74)

where τ ≥ 0 is interpreted as technical inefficiency and e−τ ≤ 1 as input technical efficiency. Similar to the output technical inefficiency setup, the normalized shadow profit function is     π ∗ (.) ∗ π˜ ∗ w ∗ eτ , p∗ = = y1 + p˜ m ym − wˆ ∗ x e , m j j j p1

(75)

w ∗ eτ

where wj∗ = pj 1 and xje = xj e−τ . Since π˜ ∗ (w ∗ eτ , p∗ ) is not observed, it can be related to the normalized actual profit by    1  1 ˜ ˜j , π˜ = π˜ 1 + − 1 Rm + −1 Q m km j θj A

∗

(76)

w which implies ln π˜ A = ln π˜ ∗ + ln H˜ where p˜ m = ppm1 , w j = p1j , wˆ j = wj eτ . Same procedure follows as in the output technical inefficiency case in which the derived shadow revenue and cost shares can be related to the actual shares. Assuming a translog form for π˜ ∗ gives expressions for the shadow revenue and cost shares.

Stochastic Frontier Models for Profit Functions The stochastic frontier models for profit functions differ from the models for cost and revenue functions in the presence of technical inefficiency. Kumbhakar [64] derives the expressions for the profit function corresponding to different

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assumptions on the underlying production function. In the presence of technical inefficiency, the profit function can be written as π (p, w, u)= π (w, pe−u ), where p is the output price, w is the input price, and e−u ≤ 1 is a measure of technical inefficiency. To illustrate, assume a translog form on actual profit and the estimable profit function is as follows:

ln

    wj wj 1 wk π + ln −u+v = α + αj ln α ln jk p pe−u 2 pe−u pe−u (77)

or in terms of the profit frontier:  π ln = ln π (p, w) + ln h (p, w, u) + v, where p 



wj ln h (p, w, u) = −u 1 − αj − αjk ln p



u − αj k 2

(78)  (79)

is profit technical inefficiency, which is not a constant multiple of u unless αjk = 0 ∀ k, i.e., the underlying production technology is homogenous. The standard stochastic profit frontier models assume that u, v, and the profit frontier variables are all independent from each other. These assumptions can be relaxed as stated in the stochastic cost frontier section. In empirical applications, negative accounting profit is a commonly observed phenomenon. However, the dependent variable for a stochastic profit frontier model is the logarithm of the profit, which is not defined for observations with negative profit. Some studies drop the observations with negative profits and estimate the model with the remaining observations. As Bos and Koetter [17] mention, this method has at least two shortcomings. First, one cannot obtain efficiency estimates for the observations that are dropped. Second, these observations are likely to belong to the least efficient productive units. Hence, dropping these observations may potentially distort efficiency estimates. An alternative method is rescaling π for all firms so that the rescaled π becomes positive. For example, a commonly used recalling is done by adding θ = min (|π − |) + 1 to π where π − = min (π , 0) is the negative part of π . Hence, the stochastic frontier profit model is given as follows: ln (π + θ ) = α + x1 β − u + v.

(80)

Berger and Mester [15], Vander Vennet [116], Maudos et al. [88], and Kasman and Yildirim [57] exemplify some studies that use this rescaling approach. Critics to this approach would ask “Where did this money come from?” Hence, Berger and Mester [15] modify the prediction of profit efficiency as follows:

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E f fπ =

 fˆπ − u − θ fˆπ − θ

(81)



where fˆπ − u −θ is the predicted actual profit and fˆπ −θ is the predicted maximum of profit that could be earned if the productive unit is fully efficient. In order to reflect the actual amounts, the profits are adjusted by θ and thus the standard formula for efficiency calculations does not work.9 Finally, an issue in the estimation of a stochastic frontier profit function is that the risk needs to be included in the model when the production involves risks. Since the risk-taking behavior of a productive unit represents its objective, one would incorrectly consider the risk-averse productive units as relatively inefficient when the risk is not included in the estimation. The studies on financial sectors (e.g., banking) are generally careful about controlling for risk when estimating a profit or alternative profit function.

Alternative Profit Function Alternative profit function, introduced by Humphrey and Pulley [52], is another representation of profits that can be used when the underlying assumptions of standard profit function do not hold. In contrast to the profit function, which takes input and output prices as given, the alternative profit function takes the input prices and output as given, i.e., π (w, y). Hence, the independent variables for an alternative profit function are the same as that of a cost function. The underlying assumption in derivation of the alternative profit function is that the productive units maximize profits by choosing input quantities and output prices. Berger and Mester [15] list four conditions where estimating alternative profit function may provide useful information: 1. There are substantial unmeasured differences in quality of outputs. 2. Outputs are not completely variable so that the productive unit cannot achieve every scale and output mix. 3. Output markets are not perfectly competitive. 4. Output prices are not accurately measured.

9 Bos and Koetter [17] propose an alternative approach to overcome this issue. For observations where the profit is positive, they keep the left-hand-side variable as lnπ , and for those observations where the profit is negative, they replace the left-hand-side variable with 0. They also add an indicator variable to the right-hand side. This indicator variable equals 0 when the profit is positive and equals ln|π − | when the profit is negative. This method has the advantage that it uses all sample points for the estimations. However, when measuring inefficiency, the logarithmic scale breaks down for negative profits. Hence, the interpretation of inefficiency estimates for the observations with negative profits deviates from the standard interpretation. Koetter et al. [59] exemplify a study that uses this approach.

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A model of alternative profit function is very similar to that of a cost function except the dependent variable and the linear homogeneity in input prices assumption. However, in the stochastic frontier setting, an alternative stochastic profit model does not penalize high-quality banks in terms of efficiency, which may not be the case for a stochastic frontier cost model. It is important to note that, unlike the profit function, the alternative profit function is not linearly homogenous in input prices [98]. Hence, linear homogeneity of an alternative profit function is an empirical question and not a theoretical restriction. Restrepo-Tobón and Kumbhakar [98] illustrate that incorrect imposition of linear homogeneity in prices may lead to misleading results.

Multi-output Functional Forms In productivity analysis, data on input and output levels are needed to estimate the production function. The difficulty in obtaining input data and the fact that more companies have integrated production across different segments make it even harder to access to division-level input information. Data on total output and input do not show how the company allocates resources in a certain segment, and thus one cannot estimate the production function for one specific segment. Same problem persists in study of a country’s productivity. In this case, most models of productivity assume one common production function for the whole economy. This does not correctly reflect how a country invests its resources since different industries/ sectors use technology differently. Gong and Sickles [37] develop modeling and estimation methods for multidivisional/multiproduct firms and improve standard assumptions in the productivity and efficiency literature. They develop a model to find input allocations among different divisions given total inputs, outputs from each division, and input prices averaged over the segment. The stochastic frontier model for a company i at time t is yit = f (xit ; β0 ) ezτ evit e−uit

(82)

  where yit is the total output (aggregated); xit = xit1 , xit2 , . . . , xitM is the vector of inputs of M types; f (xit ; β 0 )ezτ is the average production frontier, β 0 = (β 01 , β 02 , . . . , β 0M ) is a vector of M types of parameters, z is a vector including time dummy variables, and τ is a vector of corresponding coefficients; evit is the random shocks to the production; and uit is a one-sided stochastic term related to technical efficiency. To allow different frontiers for different segments, Gong and Sickles [37] introduce segment-specific production frontier (SSPF). In an economy that produces N outputs/segments using M inputs during T periods, the production technology in each segment is characterized in a system of N equations as follows:

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⎧ z1 τ1 vi1t −ui1t ⎪ ⎨ yi1t = f1 (xi1t ; β1 ) e e e .. . ⎪ ⎩ yiN t = fN (xiN t ; βN ) ezN τN eviNt e−uiNt

(83)

where yijt is the observed output and xijt are vectors of inputs (unobserved) of firm i in segment j at time t. The production frontier for segment j is represented by fj (xijt ; β j ). Note that the parameters in the production β j are segment specific. Similar to the single frontier case, zj = (zj2 , zj3 , . . . , zjT ) is a vector of year dummy variables, and τ j = (τ j2 , τ j3 , . . . , τ jT ) is a vector of the corresponding coefficients. The technical efficiency uijt = − η(t − T)u ij is time variant. The random shock

2 . One can use the SSPF framework ν ijt is assumed to be drawn from N 0, σvj to predict division-level efficiency T ˆE ij t = e−uij t . In the case of single-frontier, the SSPF approach predicts firm-level efficiency. It is straightforward to see that the firm-level efficiency for the multidivisional firm T ˆE it is the average of divisionlevel efficiency weighted by the ratio of division-level revenue to firm-level revenue:

T ˆE it =

  Rij t j

Rit

 ˆ T E ij t .

(84)

The SSPF approach incorporates the heterogeneity in production frontiers and has advantage in deriving division-level efficiency compared with a traditional SPF.

Non-parametric Estimation (and Shape Restrictions) No better example exists of a disconnect between the conditions under which a dual relationship is estimated and interpreted than in the case of non-parametric estimation of cost functions. Such relationships have been estimated in the literature for a variety of important industries, most notably in banking services where substantial data in the form of panels of cross sections are publicly available. This issue has been well studied over the last several decades. A number of important papers have contributed to the development of shape restrictions in nonparametric estimation. A short list includes Matzkin [86, 87], Ruud [101], Fox [32], Mammen and Thomas-Agnan [85], Hall and Huang [48], Ait-Sahalia and Duarte [6], Lewbel [80], Shively et al. [108], Du et al. [26], and Wu and Sickles [121]. This section begins with some examples to show how to restrict a function by transforming the function. First, it looks at how to restrict a function’s range. If a function needs to be nonnegative, for example, a common approach is to specify the function as f (x) = (r(x))2 or (x) = er(x) such that f (x) ≥ 0. To further restrict the values of the function to be (0, 1), one can specify the function as (x) = 1+e1r(x) such that 0 < f (x) < 1. In general, a range restriction on a function to take values b−a in (a,b) is f (x) = a + 1+e r(x) such that a < f (x) < b. This restriction transforms a constrained problem (specifying f ) into an unconstrained problem (specifying r)

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and still maintain global compliance with constraints. This contrasts with the kernelbased methods which keep observation-specific compliance as seen in Mukerjee [91] and Mammen [84]. Later studies on penalized kernel-smoothers include Hall and Huang [48], Henderson et al. [50], Blundell et al. [16], Ma and Racine [82], and Du et al. [26]. To impose monotonicity constraints, integration procedures are utilized. Suppose there is a monotone function f (x) and x ∈ [0, 1]. Ramsay [96] represents f (x) as:  f (x) =

x

er(s) ds.

(85)

0

Monotonicity and concavity can be imposed on f (x) by introducing an unconstrained function r(x), such that f  (x) = er(x) > 0 and f  (x) = f  (x)r  (x) so that if r  (x) < 0, then f  (x) < 0. One way to model r(x) is to use an integration transformation as follows: ⎞

⎛  f (x) =

⎟ ⎜  s x ⎟ ⎜ exp ⎜− g (t) dt⎟ ds. ⎠ ⎝ 0  0

(86)

r(s)

  $x It is clear that f  (x) = exp − 0 g (t) dt > 0 and f (x) = −f (x) g (x) .Therefore, if g(x) > 0, monotonicity and concavity naturally follow. Such a positivity constraint can be imposed via functions such as g = x2 or g = exp(x). Wu and Sickles [121] utilize a spline basis for the non-parametric expression for the function g(x) = g(h(x)), where, e.g., g(x) = (h(x))2 . A dth order splines can be written as ⎛

⎞T

⎜ ⎟ (x) = ⎝1, x, . . . , x d , (x − j1 )d+ , · · · , (x − jM )d+ ⎠ ,     power series

(87)

piecewise power series

where (x)+ = max (x, 0) and j1 < · · · < jM are spline knots and thus h(x) = cT Γ (x), where c are the spline coefficients. This leads to a non-parametric production model with monotonicity and curvatures constraints given by:  yi = β0 + β1

xi

  s

T exp − g c (t) dt ds + εi

0

0

= f (xi ; β, c) + εi . Then, one can derive the penalized nonlinear least squares estimator as:

(88)

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min β,c

1 n λR(f ) , (yi − f (xi ; β, c))2 + i=1   n   roughness penalty

(89)

goodness of fit

where R(f ) > 0 measures roughness in f and λ controls the balance between the goodness-of-fit and smoothness such that the model is not over fit. The spline coefficients become closer to 0 as λR(f ) decreases. One can use the common $ 1 2 integrated squared derivatives R(f ) = 0 f (q) (x) dx, q = 1, 2, . . . . to model the penalty. Simar, Van Keilegom, and Zelenyuk [112] propose a non-parametric leastsquares method, which utilize the advantage of the local MLE method with less strict assumptions (and less computational complexity) to analyze efficiency in the context of stochastic frontier. Given a set of i.i.d. random variables (Yi , Xi , Zi ), i = 1, . . . , n where Yi is the output, Xi are the inputs, and Zi can be considered as environmental conditions that affect the production. Setting Xi = x, and Zi = z, one can characterize the output produced as: Y = m (x, z) − u + v,

(90)

where m (x, z) is the production frontier unknown to researchers, u∼D+ (μu (x, z), varu (x, z)), and v∼D(0, varv (x, z)) are independent random variables conditional on (X, Z). The method first estimates an average production function along with some moments of the complex error term. Then the local inefficiency can be computed after identifying the local asymmetry of the error. To estimate the average production function in the first step, rewrite the output equation as: Y = r1 (x, z) + e,

(91)

where r1 (x, z) = m(x, z) − μu (x, z) represents the average production function, e = v − u + μu (x, z). One can estimate r1 (x, z) = E(Y|x, z) using standard nonparametric methods and obtain the residuals eˆi = Yi − rˆ1 (Xi , Zi ) , i = 1, . . . , n. Then, we can consistently estimate r2 (x, z) and r3 (x, z) where rj (x, z) = E(ej |x, z) by: rˆj (x, z) =



j

Wi,h (x, z) eˆi ,

(92)

where Wi, h (x, z) represents the estimation method that depends on the vector of bandwidth h. Next, assume semi-parametric forms of independent u and v as:

u | x, z ∼ N + 0, σu2 (x, z) ,

(93)



v | x, z ∼ N 0, σv2 (x, z) .

(94)

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One can derive expressions in terms of r2 and r3 for σu3 (x, z) and σv2 (x, z), and thus obtain the variance estimates σˆ u2 (x, z) and σˆ v2 (x, z) by plugging rˆ2 and rˆ3 estimated in the first step. Then, individual efficiency scores can be estimated using the method of Jondrow et al. [53] for the general case. One We can also consistently estimate the conditional mean of inefficiency term μˆ u (x, z) after the variance estimates are obtained. Then, the stochastic frontier can be estimated by: m ˆ (x, z) = rˆ1 (x, z) + μˆ u (x, z).

(95)

Non-parametric methods have become essential to analyze productivity. Developments in computational software and statistical methods have contributed greatly to empirical studies on productivity. The challenge remains, however, that estimating non-parametric models is more difficult than imposing restrictions needed for interpreting results from different functional forms. As pointed out in the introductory remarks, this chapter focuses on “how cost, revenue, and profit functions are used to identify and characterize an underlying technology” and “concentrates on the more widely used cost functions to motivate various issues.” Thus, this chapter focuses on parametric functions. There are of course many alternative nonparametric methods to specify both the mean function and the error terms in a stochastic frontier function that can be utilized to estimate the dual cost, revenue, and profit functions or the primal production or distance function. Relatively recent work has also focused on methods to ensure that the regularity conditions in such nonparametric approaches are imposed on the functions estimated. This literature includes the work by Fan, Li, and Weersink [29], Adams, Berger, and Sickles [2, 3], Adams and Sickles [1], Kuosmanen and Kortelainen [70], and Simar, Van Keilegom, and Zelenyuk [112].

Concluding Remarks What this chapter has covered is just a small part of the economic theory and practice enriched by the development of the duality theory. It highlights some of the benefits from the duality theory of the production function and uses the cost function to summarize the benefits. Revenue and profit functions have similar properties and the benefits can also be applied. First, the cost function enables us to derive an easier representation of technology. In the case of multiple outputs, for example, the production function becomes infeasible and only yields an implicit function. However, the cost function can deal with multiple outputs and has become a common practice. Second, the dual approach using the cost function incorporates optimal input allocation from optimizing firms’ behavior, while the primal approach which uses the distance or production function does not contain such information. Third, one can check if a firm is a cost-minimizer based on the precise conditions that the cost function needs to satisfy in order to characterize the technology of a cost-minimizing firm. In addition, one can specify a functional form of the cost

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function suitable for the estimations as long as it satisfies all the precise conditions. Finally, the data on outputs and input prices required to estimate the cost function are easier to obtain than the data on actual output and input levels required in the estimation using the primal approach. This chapter has also illustrated how to specify functional forms of technology in various optimization problems which are consistent with both primal and dual relationships in empirical productivity studies. One important lesson is that allowing flexible functional forms may sacrifice parsimony properties. Researchers always need to consider the benefits and losses from flexibility when choosing functional forms to represent the production technology. Last but not least, one must be consistent when interpreting results from any productivity research with the standards established 40 years ago that are still of great importance today.

Cross-References  Data Envelopment Analysis: A Nonparametric Method of Production Analysis  Distance Functions in Production Economics  Duality in Production  Multiproduct Technologies  Neoclassical Production Economics: An Introduction  Shadow Pricing in Production Economics  Stochastic Frontier Analysis: Foundations and Advances I  Stochastic Frontier Analysis: Foundations and Advances II

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Scale Elasticity and Returns to Scale

17

Victor V. Podinovski and Finn R. Førsund

Contents Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Scale Elasticity and Returns to Scale for Smooth Production Frontiers . . . . . . . . . . . . . . . . . . The Case of a Single Output . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The General Case with Multiple Outputs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Returns to Scale . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Scale Elasticity and Returns to Scale in the VRS Technology . . . . . . . . . . . . . . . . . . . . . . . . . Evaluation of Scale Elasticity and Returns to Scale in the VRS Technology . . . . . . . . . . . . . Technically Optimal Scale . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Economies of Scale and Cost Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Partial Scale Characteristics for Smooth Production Frontiers . . . . . . . . . . . . . . . . . . . . . . . . . Partial Elasticity of Response for Arbitrary Polyhedral Technologies . . . . . . . . . . . . . . . . . . . Global Returns to Scale . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Cross-References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

682 684 685 686 688 690 693 698 700 703 706 711 715 716 716

Abstract

This chapter presents conventional and recent developments of the notions of scale elasticity and returns to scale, in both the neoclassical economics framework and the nonparametric methodology of data envelopment analysis. In addition to the standard development of these notions, this chapter provides a rigorous exposition of their extensions to the case of nonsmooth production V. V. Podinovski () School of Business and Economics, Loughborough University, Loughborough, UK e-mail: [email protected] F. R. Førsund Department of Economics, University of Oslo, Oslo, Norway e-mail: [email protected] © Springer Nature Singapore Pte Ltd. 2022 S. C. Ray et al. (eds.), Handbook of Production Economics, https://doi.org/10.1007/978-981-10-3455-8_23

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frontiers, partial scale characteristics, general polyhedral technologies, and global returns to scale. We show that this broad range of extensions naturally arises from the introduction of an output response function that, in a general setting, describes a proportional response of any subset of inputs and outputs to marginal proportional changes of another subset, as observed on the production frontier. This function is closely related to the directional distance function and provides a natural language for the definition and computation of different scale characteristics in both the neoclassical and nonparametric frameworks. Keywords

Returns to scale · Scale elasticity · Production frontiers · Data envelopment analysis

Introduction The notions of scale elasticity and returns to scale are important characteristics of a production function which has its roots in works of classical and neoclassical economists starting in the middle of the nineteenth century.1 Frisch [31], based on a series of papers starting in the mid-1920s, gives a comprehensive presentation of the neoclassical production theory and the most common mathematical assumptions to be imposed. The two key concepts are scale properties and substitution properties. This chapter is concerned with scale properties only.2 We distinguish between the case of a single output as an explicit function of multiple inputs and the case of an implicit transformation function which allows both multiple outputs and inputs. The single output production function can be stated as y = f (x1 , . . . , xm ) ,

(1)

where y is the single output and (x1 , . . . , xm ) is the vector of inputs. The production function (1) represents the maximum amount of output that can be produced from any given vector of inputs as for example stated in Samuelson [77] and therefore defines what is now called the production frontier. The scale property of production frontiers was expressed in Johnson [39] by the elasticity of output with respect to a uniform change in the scale of all factor inputs named the “elasticity of production.” Frisch [31] coined the term the “passus coefficient.” The term “elasticity of scale” may be the most common one used now in the literature. Frisch characterized general scale properties by introducing

1 See,

for example, the introduction to the work of J. H. von Thünen published in 1863 by Samuelson [78]. 2 See Lloyd [47] for a review of the origins of the concept of substitution.

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a class of production functions (with a single output) obeying the Regular Ultra Passum Law. The latter defines a particular behavior of the scale elasticity along any nondecreasing curve in the input space: the scale elasticity starts out as greater than one, then decreases monotonically and passes the value of one, and continues to decrease after this. Frisch illustrated scale properties by drawing contour lines for constant values of the passus coefficient. The contour line for the passus coefficient equal to one is of special importance because this curve shows the technically optimal scale. The case of multiple outputs and inputs is based on the use of a transformation function written as the implicit relation F (X, Y ) = 0,

(2)

 where X = (x1 , . . . , xm ) ∈ Rm + is the vector of inputs and Y = (y1 , . . . , ys ) s 3 ∈ R+ is the vector of outputs. The transformation function F(X, Y) is usually assumed to be sufficiently smooth and satisfy additional conditions of the implicit function theorem. The definition of the elasticity of scale for this case was not developed until the 1970s [37, 55, 81]. The theory of scale elasticity and returns to scale was later extended and operationalized in the nonparametric methodology of data envelopment analysis (DEA). In contrast with the neoclassical setting in which the transformation function F(X, Y) is assumed to be known and sufficiently smooth, in DEA this function is generally unknown and not differentiable everywhere. Banker et al. [5] and Banker and Thrall [4] demonstrated that the scale properties of the production frontiers in DEA can be evaluated using the appropriate supporting hyperplanes to the production possibility set which in turn are obtained by analyzing the set of shadow prices at a given unit on the production frontier. The latter task is achieved by resorting to linear programming techniques. It is worth noting that while in economics it is standard to characterize scale properties of production frontiers in terms of scale elasticity, in DEA the focus has traditionally been on the less informative qualitative characterization of returns to scale as increasing, constant, or decreasing returns [27]. An explanation of this difference may lie in the fact that in the neoclassical approach, the transformation function F(X, Y) is assumed to be known and the evaluation of scale elasticity is a straightforward task, both conceptually and computationally. In contrast, in DEA models where the efficient frontier is not known explicitly and is not smooth, the notion of scale elasticity is generally undefined on the edges (including the vertex points) of the production frontier and its one-sided analogues are less intuitive. Methods of assessment of returns to scale have been developed in the early pioneering works on DEA. However, the theoretical links between returns to

3 Joint

production was also discussed by the classical and neoclassical economists – see Kurz [46] for a review of the origins.

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scale and the underlying scale elasticity have often been overlooked and have only recently been rigorously investigated. There is a large body of literature devoted to scale characteristics of production frontiers. Diewert et al. [20] and Balk et al. [2] provide reviews of contributions in the neoclassical setting. Banker et al. [8] and Sahoo and Tone [74] summarize contemporary work on the scale elasticity and returns to scale in the field of DEA. The goal of this chapter is to present a unifying exposition of the notions of scale elasticity and returns to scale that reflects recent advances in both the neoclassical and nonparametric DEA settings. In writing this chapter, we pursue several goals. First, we aim to provide an updated overview of the field, with particular emphasis on the new developments and extensions. Well-known results are presented in a reduced form and are given only cursory discussion, with references to the literature. Second, in contrast with some existing work that discusses returns to scale without introducing the notion of scale elasticity, we develop the two notions simultaneously and link one to the other. Third, we show that there is a direct correspondence and analogy between the neoclassical and DEA approaches to the treatment of standard scale characteristics and their various extensions. In simple words, we show that there is parity between the neoclassical framework and DEA with respect to conceptual and computational possibilities to categorize scale on production frontiers. Fourth, we provide a rigorous development of a number of theoretical issues that either have traditionally been overlooked or approached heuristically in previous work. This includes a rigorous treatment of the notion of one-sided scale elasticity in different production technologies and the relation between returns to scale and technically optimal scale (most productive scale size) in arbitrary, including nonconvex, technologies. Fifth, we include and discuss recent extensions to the traditional notions of scale elasticity and returns to scale. This includes different partial scale characterizations, their generalizations to arbitrary nonparametric polyhedral technologies, and the notion of global returns to scale.

Scale Elasticity and Returns to Scale for Smooth Production Frontiers The scale elasticity and the types of returns to scale are characteristics of efficient frontiers of production technologies. Conceptually, a technology T is defined as the set of all feasible input-output combinations (X, Y) that can be used by a production unit:   s T = (X, Y ) ∈ Rm + × R+ |X can produce Y . If the transformation function (2) is known, technology T is defined as   s T = (X, Y ) ∈ Rm + × R+ |F (X, Y ) ≤ 0 .

(3)

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For a freely disposable technology, it is assumed that the function F(X, Y) is strictly increasing in all outputs and strictly decreasing in all inputs, for which, further assuming differentiability, a sufficient condition used by Hanoch [37] is ∂F (X, Y ) ∂F (X, Y ) < 0, i = 1, . . . , m; > 0, r = 1, . . . , s. ∂xi ∂yr

(4)

The inequalities (4) imply that the technology T defined by Eq. (3) is freely disposable in all inputs and outputs and that equality (2) describes efficient production.

The Case of a Single Output Consider the case of a single output as a function of multiple inputs defined by production function (1). Let the unit (xo1 , . . . , xom , yo ) be located on the production frontier, that is, satisfy Eq. (1). Consider an arbitrary proportional change of the input vector (xo1 , . . . , xom ) by the variable multiplier α ≥ 0 and define the corresponding relative change  β (α) of the output yo assuming that the resulting unit  αxo1 , . . . , αxom , β (α) yo remains located on the production frontier. The function β (α) represents the largest amount of the output yo that can be produced given the input vector (αxo1 , . . . , αxom ): β (α) yo = f (αxo1 , . . . , αxom ).

(5)

Note that β(1) = 1. Assuming differentiability of the production function (1), the scale elasticity ε, or the Passus coefficient of Frisch [31], evaluated at the unit (xo1 , . . . , xom , yo ) is the derivative of the function β (α) evaluated at α = 1, that is, 

ε = β (1). Differentiating (5) by α, we obtain the Passus equation of Frisch: εyo =

m  ∂f (xo1 , . . . , xom ) xoi , ∂xi i=1

or, rearranging and assuming yo = 0,

  m  (xo1 , . . . , xom ) , ∇f xo1 , . . . , xom  ∂f (xo1 , . . . , xom ) yo = xoi , ε= ∂xi yo i=1

(6)

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where ∇f (xo1 , . . . , xom ) is the gradient of the function f (x1 , . . . , xm ) evaluated at the point (xo1 , . . . , xom ). The numerator on the right-hand side of Eq. (6) is the scalar product of this gradient and the input vector (xo1 , . . . , xom ). The meaning of the scale elasticity ε follows from equality (5). Namely, ε is the multiplier by which a marginal proportional change α of the input vector (xo1 , . . . , xom ) translates to the relative change β (α) of the output yo observed on the production frontier.  For example, let ε = β (1) = 2. Consider increasing the input vector in some small proportion, for example, by 1%, which corresponds to the increase of α from  α = 1 to α˜ = 1.01. Then, by definition of the derivative β (1) and to the first degree of approximation, β (α) ˜ − β(1)  ≈ β (1) = 2. α˜ − 1 Rearranging the above equality in which β(1) = 1 and α˜ = 1.01, we have β (α) ˜ ≈ 1.02. Therefore, in response to the proportional increase of the input vector by 1%, the maximum amount of the output that can be produced increases by 2%. Similarly, a proportional reduction of the input vector by 1% (corresponding to αˆ = 0.99) results in the  reduction of the maximum amount of the output by 2% (corresponding to β αˆ ≈ 0.98).

The General Case with Multiple Outputs Consider the general case characterized by multiple inputs and multiple outputs in which the efficient boundary (production frontier) of technology T is defined by the transformation function F(X, Y) as in equality (2).4 Let (Xo , Yo ) be an arbitrary unit located on the production frontier such that Xo = 0 and Yo = 0. Then we have F(Xo , Yo ) = 0. Consider an arbitrary proportional change of the input vector Xo by the variable β (α) of the multiplier α ≥ 0 and define the corresponding proportional change   output vector Yo , assuming that the resulting unit αXo , β (α) Yo remains located on the production frontier. Substituting this into Eq. (2), we have   F αXo , β (α) Yo = 0.

(7)

Because the transformation function in Eq. (2) represents efficient production, β (α) is unique for each α ≥ 0 and is equal to the largest proportion of the output vector Yo that can be produced given the input vector αXo . Below we refer to β (α) as the (proportional) output response function.

4 The

case of a single output described by production function (1) is a special case of the more general statement (2). To see this, we can restate (1) as F(x1 , . . . , xm , y) = y − f (x1 , . . . , xm ) = 0.

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Generalizing the single-output case, define the scale elasticity ε(Xo , Yo ) as the derivative of the function β (α) evaluated at α = 1: 

ε (Xo , Yo ) = β (1).

(8)

According to the given definition, the scale elasticity ε(Xo , Yo ) represents the largest proportional change of the vector of outputs Yo possible in the technology (as observed on the production frontier) as a response to a marginal proportional change of the input vector Xo by a factor α. Subject to known regularity conditions assumed by the implicit function theorem and differentiating (7) with respect to α, we obtain the standard formula [37, 55, 81]: 

ε (Xo , Yo ) = β (1) = −

Xo , ∇X F (Xo , Yo ) ,

Yo , ∇Y F (Xo , Yo )

(9)

where ∇ X F(Xo , Yo ) and ∇ Y F(Xo , Yo ) are the partial gradients of the function F(X, Y) evaluated at the point (Xo , Yo ) with respect to the inputs and outputs, respectively. Note that in the case of a single output defined by the production function (1), formula (9) is equivalent to Eq. (6). Remark 1 It is useful to give an equivalent definition of the proportional output response function β (α) based only on the notion of technology T, without a reference to the transformation function, similar to its definition in Starrett [81] and Førsund [27]: β (α) = max {β| (αXo , βYo ) ∈ T , β ∈ R} .

(10)

Defining the function β (α) as in Eq. (10) is particularly useful for the development of the notion of scale elasticity in nonparametric models of technology in which the transformation function is typically unknown.5 In most of such models, equality (10) becomes a linear program. Applying known results of sensitivity analysis to this program at the point (Xo , Yo ) leads to a linear programming method for the evaluation of scale elasticity, even without knowing the explicit transformation function. This is discussed in detail in the next sections. It is worth noting that the function β (α) is closely related to the directional distance function of Chambers et al. [13, 14] and is reciprocal to the gauge function of Rockafellar [73] evaluated at the unit (αXo , Yo ) in the direction of vector gY = Yo (see, e.g., Podinovski et al. [68]).

5 To

avoid dealing with excessive technicalities, we assume that the maximum in formula (10) is attained. This assumption is true for most technologies of practical interest, in particular, for the nonparametric variable returns-to-scale technology used in DEA and discussed in subsequent sections.

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Remark 2 It is clear that there exist an infinite number of different transformation functions F(X, Y) that characterize the same production technology T by formula (3), or its production frontier by equality (2). One important example of this is the characterization of technology T by the directional distance function of Chambers et al. [13, 14] defined as follows:  D (X, Y, gX , gY ) =

sup {δ| (X − δgX , Y + δgY ) ∈ T } , if (X, Y ) ∈ T , − ∞, if (X, Y ) ∈ / T,

(11)

s where (gX , gY ) ∈ Rm + × R+ \ {(0, 0)}. Chambers et al. [14] show that if technology T is freely disposable with respect to all inputs and outputs, then the function D(X, Y, gX , gY ) is a complete function representation of T, that is, we have

  s T = (X, Y ) ∈ Rm + × R+ |D (X, Y, gX , gY ) ≥ 0 .

(12)

In this case the frontier of technology T is given by the equality D(X, Y, gX , gY ) = 0. Taking into account that the vectors gX and gY are fixed, note that the directional distance function D(X, Y, gX , gY ) defines the transformation function in Eq. (2) as follows: F (X, Y ) = −D (X, Y, gX , gY ) , and (12) becomes a special case of Eq. (3).6 If the function D(X, Y, gX , gY ) is sufficiently smooth (with respect to the variable vectors X and Y, for the fixed parameters gX and gY ) and satisfies all conditions of the implicit function theorem at the unit (Xo , Yo ), the scale elasticity ε(Xo , Yo ) is calculated by formula (9), in which F(X, Y) is substituted by D(X, Y, gX , gY ) [2, 33, 83].

Returns to Scale The evaluation of scale elasticity by formula (9) leads to the conventional returns-toscale characterization of the unit (Xo , Yo ) into the following three types: increasing, decreasing, and constant returns to scale. Definition 1 Let F(Xo , Yo ) = 0. Then the unit (Xo , Yo ) exhibits (i) Increasing returns to scale if ε(Xo , Yo ) > 1 (ii) Decreasing returns to scale if ε(Xo , Yo ) < 1

6 Properties of the directional distance function (11) are explored by Chambers et al. [14, 15] and Chambers and Quiggin [12].

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Output

Q

M

P C

B

L

A K Input

0 Fig. 1 Returns to scale for a smooth production frontier

(iii) Constant returns to scale if ε(Xo , Yo ) = 1 Example 1 Consider the technology with a single input x and single output y depicted in Fig. 1. Suppose that its efficient frontier is defined by the production function y = f (x). In this case, the formula for scale elasticity (6) can be restated as follows: ε (xo , yo ) =

  yo xo f  (xo ) = f  (xo ) / . yo xo

(13)

The above formula explains the following conventional interpretation of the scale elasticity in the case of a single input and a single output. Namely, the scale elasticity evaluated at an output radial efficient unit (xo , yo ) is equal to the ratio of its marginal productivity f  (xo ) to its average productivity yo /xo . Consider, for example, unit A in Fig. 1. The marginal productivity evaluated at this unit is equal to the slope of the line KL tangent to the frontier at the point A. The corresponding average productivity is equal to the slope of the ray OA. Because the former is greater than the latter, their ratio is greater than 1, and unit A exhibits increasing returns to scale. Similarly, both the marginal and average productivities evaluated at the unit B are equal to the slope of the ray OM. Therefore, B exhibits constant returns to scale. Finally, the marginal productivity evaluated at the unit C (equal to the slope of the line PQ) is lower than the average productivity (equal to the slope of the ray OC).

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By formula (13), the scale elasticity evaluated at C is less than 1 and therefore unit C exhibits decreasing returns to scale.

Scale Elasticity and Returns to Scale in the VRS Technology In this section we discuss the extension of the notions of scale elasticity and returns to scale to the variable returns-to-scale (VRS) technology of Banker et al. [5]. Further extensions to other nonparametric technologies and their generalizations are presented in subsequent sections.7 VRS technology generated by a finite set of observed units  Let T be the s Xj , Yj ∈ Rm + × R+ , j ∈ J = {1, . . . , n}. We assume that Xj and Yj are nonzero vectors for any j ∈ J, that is, that each observed unit has at least one positive input and one positive output. Denote X and Y the m × n and s × n matrices whose columns are, respectively, the input and output vectors Xj and Yj of the observed units j ∈ J. Following Banker et al. [5], the VRS technology T can be stated as follows:   m+s |Xλ ≤ X, Y λ ≥ Y, 1 λ = 1, λ ∈ Rn+ . T = (X, Y ) ∈ R+

(14)

(We use the superscript  to denote transposed vectors. The vector inequalities mean that the specified inequalities are satisfied for each component.) In contrast with the neoclassical economics approach in which the transformation function is assumed to be known and sufficiently smooth, the task of defining and evaluating the scale elasticity in the VRS technology leads to two obstacles. First, we do not have an explicit formula for the transformation function (2) describing the frontier of the VRS technology and cannot use formula (9) for the calculation of scale elasticity. Below we show that this difficulty can be overcome by using known results of sensitivity analysis applied to the definition of the output response function β (α) by formula (10) which, in the case of the VRS technology, becomes a linear program. Second, the efficient frontier of the VRS technology is generally not smooth and the standard definition (8) of scale elasticity generally does not apply to this frontier. This conceptual difficulty is overcome by defining a one-sided (left-hand and righthand) scale elasticity and giving a corresponding one-sided returns-to-scale characterization of the frontier points. In the context of VRS technology, this approach was pioneered by Banker and Thrall [4]. It was explored further by Fukuyama [32], Førsund and Hjalmarsson [28], Hadjicostas and Soteriou [35], Podinovski et al.

7 The

axiomatic approach used by Banker et al. [5] stipulates that the VRS technology is a convex set. This assumption precludes an essential feature of the neoclassical frontiers, i.e., the Regular Ultra Passum Law of Frisch [31] – for a discussion, see Førsund and Hjalmarsson [29], Olesen and Petersen [51] and Olesen and Ruggiero [52].

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[67], and Podinovski and Førsund [66]. It was extended to more general settings by Chambers and Färe [11], Zelenyuk [83], and Podinovski et al. [68]. It is important to note that in the literature it is often assumed that the notions of scale elasticity and returns to scale apply only to strongly efficient units.8 As shown by the development of these notions in Podinovski et al. [67] and Podinovski et al. [68], it is sufficient that the unit (Xo , Yo ) be only output radial efficient. Using the proportional output response function β (α) defined in Eq.(10), this condition is stated as follows: Assumption 1 The unit (Xo , Yo ) is output radial efficient, that is, β(1) = 1. If the transformation function F(X, Y) is known and the unit (Xo , Yo ) satisfies equality (2) that represents efficient production, Assumption 1 is automatically satisfied and does not need to be mentioned or verified. However, in the VRS technology, typically only some units satisfy Assumption 1 which therefore needs to be verified. Consider any unit (Xo , Yo ) ∈ T that satisfies Assumption 1. (This unit may be observed or not observed.) Denote  the domain of the output response function β (α). Because the VRS technology T is freely disposable in all inputs, it is straightforward to show that  is an unbounded interval that can be stated in the form  = [α ∗ , +∞), where α ∗ > 0. As follows from known results of sensitivity analysis in linear programming, β (α) is a continuous, concave, and piecewise linear function on  [68]. This implies that if α = 1 is not the left extreme point of  (i.e., if α ∗ = 1), then the function β (α) has both the right-hand and left-hand derivatives   β + (1) and β − (1) (because β (α) is a linear function in some neighborhood on each side of α = 1). If α ∗ = 1, then only the right-hand derivative is defined in the  conventional sense. In this case, the left-hand derivative β − (1) is undefined and is often formally taken equal to +∞. The above observations lead to the following definition of the one-sided (righthand and left-hand) scale elasticities ε+ and ε− evaluated at the unit (Xo , Yo ). This definition generalizes the notion of scale elasticity ε defined by Eq. (8) and is effectively used by Banker and Thrall [4]: 

ε+ (Xo , Yo ) = β + (1),  ε− (Xo , Yo ) = β − (1).

(15)

Because the function β (α) is concave on , according to a known result of convex analysis ([73], Theorem 24.1) we have: ε− (Xo , Yo ) ≥ ε+ (Xo , Yo ) .





(16)





unit (Xo , Yo ) ∈ T is strongly efficient if there exists no (X , Y ) ∈ Tsuch that X ≤ Xo , Y ≥ Yo ,   and (Xo , Yo ) = (X , Y ).

8A

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The inequality (16) is satisfied as equality if and only if the function β (α) is differentiable at α = 1, in which case both one-sided scale elasticities are equal to the scale elasticity ε(Xo , Yo ) defined by Eq.(8). This shows that the notion of onesided scale elasticities (15) is a generalization of the conventional notion of scale elasticity to the points on the frontier at which the latter is undefined but the former is defined correctly. The meaning of the one-sided scale elasticities (15) is similar to the meaning of the conventional scale elasticity ε(Xo , Yo ). For example, suppose that ε+ (Xo , Yo ) = 0.5 and ε− (Xo , Yo ) = 2. If we increase the input vector Xo in a small proportion, for example, by 1%, the proportion by which the production of the output vector Yo in the VRS technology can be increased is equal to 0.5% as indicated by the value of the right-hand scale elasticity ε+ (Xo , Yo ). Similarly, if we reduce the input vector Xo proportionally by 1%, the maximum production of the output vector Yo is reduced by 2% as indicated by the left-hand scale elasticity ε− (Xo , Yo ). The notion of one-sided scale elasticity (15) leads to the following characterization of returns to scale of output-efficient units and is due to Banker and Thrall [4]. Definition 2 Let the unit (Xo , Yo ) ∈ T satisfy Assumption 1. Then (Xo , Yo ) exhibits (i) Increasing returns to scale if 1 < ε+ (Xo , Yo ) ≤ ε− (Xo , Yo ) (ii) Decreasing returns to scale if ε+ (Xo , Yo ) ≤ ε− (Xo , Yo ) < 1 (iii) Constant returns to scale if ε+ (Xo , Yo ) ≤ 1 ≤ ε− (Xo , Yo ) Example 2 Consider the VRS technology with a single input and a single output shown in Fig. 2. This technology is generated by the five observed units A, B, C, D, and E, all of which are output radial efficient. Consider unit C. The right-hand marginal productivity at this unit is equal to 2/3, which is the slope of the segment CD. The average productivity of unit C is equal to 6/5. Therefore, the right-hand scale elasticity at C is equal to the ratio (2/3)/(6/5) ≈ 0.56. Similarly, the left-hand scale elasticity is the ratio (2/1)/(6/5) ≈ 1.67. According to Definition 2, C exhibits constant returns to scale.9 Similar analysis shows that the units A and B exhibit increasing returns to scale, and the units D and E exhibit decreasing returns to scale.

9 Note

that unit C actually exhibits decreasing returns to scale if the input is increased (because the right-hand scale elasticity evaluated at this unit is less than 1) and increasing returns to scale if the input is reduced (because the left-hand scale elasticity is greater than 1). This contrasts with the standard definition of constant returns to scale requiring that the scale elasticity be equal to 1 [34].

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Output 11 10

E

9

D

8 7

C

6 5 4

B

3 2

A

1 0

1

2

3

4

5

6

7

8

9

10 11

Input

Fig. 2 Returns to scale in the VRS technology

Evaluation of Scale Elasticity and Returns to Scale in the VRS Technology In DEA, with the exception of very simple cases like in Example 2, we do not know a functional representation of the production frontier and do not know the proportional output response function β (α). Consequently, we cannot use formulae (15) for the calculation of the one-sided scale elasticities. Below we consider the linear programming approach to their evaluation based on the dual characterization of the VRS technology T. The described approach leads to the known returns-toscale characterization of production units based on the minimum and maximum optimal values of the sign-free variable dual to the normalizing equality 1 λ = 1 in the statement (14) of the VRS technology. Let the unit (Xo , Yo ) ∈ T satisfy Assumption 1. Consider assessing its output radial efficiency by solving the following output-oriented multiplier program in which u ∈ Rs+ and v ∈ Rm + are the vectors of output and input weights, respectively, and v0 is a sign-free variable10 :

10 We state program (17) and related programs (18) and (19) in a form consistent with Banker and Thrall [4] and Førsund and Hjalmarsson [28]. Because in these three programs the variable v0 is free in sign, an alternative statement is equally valid in which the “plus” sign before this variable in the objective function and constraints is changed to the “minus” sign as, for example, in Cooper et al. [18]. In the latter case the formulae for the calculation of the one sided-scale elasticities given in this section need to be changed accordingly.

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θ ∗ = min

v  Xo + v0 ,

(17)

subject to u Yo = 1 v  Xj − u Yj + v0 ≥ 0, j = 1, . . . , n, u, v ≥ 0, v0 sign free. Because by Assumption 1, unit (Xo , Yo ) is output radial efficient, we have θ ∗ = 1. Denote the set of all optimal solutions u, v, v0  of program (17). Let v0min and v0max be, respectively, the minimal and maximal values of the scalar v0 taken over the set . For example, v0max is the optimal value of the following linear program: v0max = max

v0 ,

(18)

subject to u Yo = 1, v  Xo − u Yo + v0 = 0, v  Xj − u Yj + v0 ≥ 0, j = 1, . . . , n, u, v ≥ 0, v0 sign free. Note that the feasible region of program (18) is the set of optimal solutions to the multiplier program (17). Indeed, program (18) has the same constraints as program (17) and the additional constraint v Xo − u Yo + v0 = 0. Because u Yo = 1, this additional constraint is equivalent to v Xo + v0 = 1, which is satisfied only by the optimal solutions of program (17). Similarly, to compute v0min , we change the maximization of the objective function in Eq. (18) to its minimization: v0min = min

v0 ,

(19)

subject to u Yo = 1, v  Xo − u Yo + v0 = 0, v  Xj − u Yj + v0 ≥ 0, j = 1, . . . , n, u, v ≥ 0, v0 sign free. Both programs (18) and (19) are feasible and their feasible sets coincide with the set of optimal solutions to program (17). It is clear that in program (18) we

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always have v0max ≤ 1.11 In contrast, the objective function of program (19) may be unbounded below, in which case program (19) does not have a finite optimal solution. The following theorem takes into account that for any unit (Xo , Yo ) ∈ T, the domain of the function β (α) is a closed half-interval  = [α ∗ , +∞). Its proof follows from a more general result established by Podinovski and Førsund [66]. Theorem 1 Let the unit (Xo , Yo ) ∈ T satisfy Assumption 1. Then (i) Program (18) has a finite optimal value v0max and 

ε+ (Xo , Yo ) = β + (1) = 1 − v0max .

(20)

(ii) Program (19) has a finite optimal value if and only if α = 1 is not the left extreme point of , that is, α ∗ = 1. In this case we have 

ε− (Xo , Yo ) = β − (1) = 1 − v0min .

(21)

(iii) Program (19) has an unbounded optimal value if and only if α = 1 is the left extreme point of , that is, if α ∗ = 1. (In this case we can formally define  v0min = −∞ and let ε− (Xo , Yo ) = β − (1) = 1 − v0min = +∞.) Theorem 1 suggests a straightforward computational procedure for the evaluation of one-sided scale elasticities ε– and ε+ at any given unit (Xo , Yo ) ∈ T. This procedure automatically establishes if the unit (Xo , Yo ) satisfies Assumption 1, that is, if it is output radial efficient. This fact allows us to solve programs (18) and (19) on the entire set of observed units J = {1, . . . , n} without preliminary evaluation of their output radial efficiency and selecting its efficient subset.12 To be specific, consider the evaluation of the left-hand scale elasticity ε− (Xo , Yo ) by solving program (19) and using formula (21). The following three outcomes are logically possible.   Case 1. Let program (19) have a finite optimal solution u, ˆ v, ˆ v0min . As any feasible solution to Eq. (19), this optimal solution is also optimal in program (17), and the optimal value of the latter is equal to 1. Therefore, the unit (Xo , Yo ) satisfies Assumption 1, that is, it is output radial efficient. By formula (21), we have  ε− (Xo , Yo ) = β − (1) = 1 − v0min . Case 2. Let program (19) be feasible and its optimal value be unbounded. In this case any feasible solution to this program is also optimal in program (17). Therefore, follows from the second constraint of program (18), taking into account that v Xo ≥ 0 and u Yo = 1. 12 The fact that programs (18) and (19) can be solved on the full set of observed units, without first identifying the subset of output radial efficient units, was suggested in a similar setting by Atici and Podinovski [1]. 11 This

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the unit (Xo , Yo ) satisfies Assumption 1. By statement (iii) of Theorem 1, α = 1 is the left extreme point of the domain . This means that reducing the input vector Xo in any proportion α < 1 leads outside the VRS technology. (Such situation occurs at the unit A in Fig. 2, where there are no units in the technology with the input smaller than that of the unit A.) In this case the left-hand scale elasticity ε− is undefined. For convenience, in this case it is often formally assumed that v0min = −∞ and  ε− (Xo , Yo ) = β − (1) = 1 − v0min = +∞. Case 3. Let program (19) be infeasible. This means that there exists no solution that satisfies all constraints of program (17) at which its objective function attains the value of 1. (Equivalently, the equality v Xo − u Yo + v0 = 0 cannot be satisfied). In this case the unit (Xo , Yo ) does not satisfy Assumption 1, and the notion of onesided scale elasticity evaluated at this unit is undefined. The evaluation of the right-hand scale elasticity ε+ (Xo , Yo ) requires solving  program (18) and using formula (20) to calculate ε+ (Xo , Yo ) = β + (1). As noted, if the unit (Xo , Yo ) satisfies Assumption 1, this program always has a finite optimal solution v0max ≤ 1. Therefore, we can have only the analogues of Cases 1 and 3, as the Case 2 is impossible. It is also worth noting that both programs (18) and (19) are either both feasible or both infeasible. Therefore, if solving either of these programs shows their infeasibility (Case 3), then there is no need to solve the remaining program. Remark 3 The one-sided scale elasticities and the returns-to-scale characterization of units in the VRS technology can also be evaluated by solving the input-oriented multiplier VRS models. Let the unit (Xo , Yo ) ∈ T be both output radial efficient (as required by Assumption 1 for a correct definition of the one-sided scale elasticities) and also input radial efficient. Then the optimal value θˆ of the following inputoriented multiplier VRS model is equal to 1: θˆ = max

u Yo + u0 ,

(22)

subject to v  Xo = 1, v  Xj − u Yj − u0 ≥ 0, j = 1, . . . , n, u, v ≥ 0, u0 sign free. ˆ the set of optimal solutions u, v, u0  of program (22). Let umin and Denote

0 ˆ These two be the minimal and maximal values of u0 taken over the set . umax 0 values can be evaluated by solving programs analogous to programs (18) and (19). Then the one-sided scale elasticities at the unit (Xo , Yo ) are expressed as follows [4, 28, 67]: ε+ (Xo , Yo ) =

1 1 , ε− (Xo , Yo ) = . min 1 − umax 1 − u0 0

(23)

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Remark 4 In the DEA literature, the notion of returns to scale is sometimes extended to the units that are output radial inefficient and, therefore, do not satisfy Assumption 1. A common approach in such cases requires that the unit (Xo , Yo ) be first projected on the boundary of the VRS technology, either in the output or input orientation. If the projection satisfies Assumption 1 (which is always true for output projections but is not necessarily true for the input projections), then its returns-to-scale type is defined correctly and may be assigned to the unit (Xo , Yo ). It is well known that the input and output projections of the same inefficient unit may belong to different types of returns to scale, and therefore, the returns-to-scale characterization of the inefficient units generally depends on the selected projection [7, 27]. Remark 5 Let us mention two known methods that evaluate the returns-to-scale type of the unit (Xo , Yo ) directly, without assessing the one-sided scale elasticities. These methods may be useful when the strength of the characterization measured by the one-sided scale elasticities is unimportant, and only the classification of units into the three types of returns to scale is of interest. The resulting characterizations obtained by these methods are equivalent to those based on Definition 2 that employs the notion of one-sided scale elasticity [6]. The first method was introduced by Banker and Thrall [4] and further discussed by Seiford and Zhu [80] and Cooper et al. [18, p. 138]. In this method, the returnsto-scale characterization of the unit (Xo , Yo ) is obtained by evaluating its input radial efficiency in the constant returns-to-scale (CRS) technology of Charnes et al. [16]. The statement of the CRS technology is obtained from the statement (14) of the VRS technology, by omitting the normalizing equality 1 λ = 1. The unit (Xo , Yo ) exhibits increasing returns to scale if and only if 1 λ < 1 in all optimal solutions to the CRS model measuring its input radial efficiency. Similarly, the unit (Xo , Yo ) exhibits decreasing returns to scale if and only if 1 λ > 1 in all such optimal solutions. The remaining case (i.e., when there exists an optimal solution such that 1 λ = 1) corresponds to the unit (Xo , Yo ) exhibiting constant returns to scale. Applying this method requires solving two linear programs: one maximizing and the other minimizing the term 1 λ. An alternative method was developed by Färe et al. [23, 24] and was further discussed by Färe and Grosskopf [22]. This approach requires the evaluation of the input radial efficiency of the unit (Xo , Yo ) in the three technologies: the actual VRS technology T and its nonincreasing and constant returns-to-scale (NIRS and CRS) extensions. Let ET (Xo , Yo ), E NIRS (Xo , Yo ) and E CRS (Xo , Yo ) be the corresponding input radial efficiencies of the unit (Xo , Yo ) in the three technologies. Then (Xo , Yo ) exhibits constant returns to scale if and only if ET (Xo , Yo ) = ECRS (Xo , Yo ). Otherwise we have ET (Xo , Yo ) > ECRS (Xo , Yo ) and further investigation using the NIRS technology is required. Namely, if ENIRS (Xo , Yo ) = E CRS (Xo , Yo ), then the unit (Xo , Yo ) exhibits increasing returns to scale. Otherwise, (Xo , Yo ) exhibits decreasing returns to scale. The method of Färe et al. [23] was further modified by Kerstens and Vanden Eeckaut [41] and Briec et al. [9]. Podinovski [56, 57] proved that the above

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approaches correctly characterize returns to scale in any convex technology T, including the standard VRS technology.13 If the technology T is not convex, this group of methods does not evaluate the local returns to scale but rather, with a small modification, indicates a direction to the optimal scale of operations. We discuss this in detail is section “Global Returns to Scale.” Remark 6 An alternative to the linear programming approach is a group of direct computational methods [30, 43, 44]. Such methods aim at determining (reconstructing) the section of the production frontier that include all units stated in the form (αXo , βYo ). Identifying this section of frontier is equivalent to identifying the function β (α), and the use of formula (15) becomes straightforward. Direct methods generally offer advantages in computational speed compared to the traditional methods based on solving multiplier DEA models, although their implementation requires good programming skills and fine-tuning of the algorithms. This makes direct methods particularly suitable for the implementation in software packages, especially those that aim at the visualization of production frontiers, which is a computationally demanding task [43].

Technically Optimal Scale In the neoclassical setting, the notion of technically optimal scale was introduced by Frisch [31]. Assuming that a smooth production function (1) satisfies the Regular Ultra Passum Law, the surface of technically optimal scale is the locus of all units characterized by the scale elasticity equal to 1. Under the Regular Ultra Passum Law, any proportional increase of all inputs of a unit located on this surface by a factor α > 1 would lead to an increase of the output by a factor less than α. Similarly, a proportional reduction of all inputs by α < 1 would result in a reduction of the output by a factor greater than α. In both cases the resulting unit would exhibit reduced productivity compared to the original unit. This implies that any unit of technically optimal scale with the vector of inputs (x1 , . . . , xm ) exhibits the highest productivity among all units whose vectors of inputs belong to the ray (αx1 , . . . , αxm ), where α > 0. A similar idea is utilized by Banker [3] who introduces the notion of the most productive scale size (MPSS) in the nonparametric setting of DEA. Although Banker defines MPSS for the VRS technology, his definition is equally applicable to any production technology. As discussed by Førsund [26, 27], the notion of MPSS is identical to the notion of technically optimal scale for smooth production functions

13 The application of the method of Färe et al. [23, 24] to arbitrary convex technologies assumes that we have an operational statement of their cone and NIRS extensions. Obtaining such statements is generally not a simple task. Podinovski and Bouzdine-Chameeva [65] develop a unifying operational statement of such extensions for any polyhedral technology.

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(1) that satisfy the Regular Ultra Passum Law. However, Banker’s definition of MPSS also extends to arbitrary technologies with multiple outputs, including various DEA technologies whose efficient frontiers are not smooth and for which the conventional scale elasticity is undefined. It also applies to nonconvex technologies in which the units characterized by the scale elasticity equal to 1 are not necessarily at MPSS. (We discuss this in detail in section “Global Returns to Scale.”) Let T be any production technology. Following the definition of Banker [3], a unit (Xo , Yo ) ∈ T is at MPSS if, for all units in the form (αXo , βYo ) ∈ T, where α > 0, we have β/α ≤ 1. This is equivalent to the condition β (α) /α ≤ 1, for all α > 0, where the β (α) is the output response function defined by formula (10). Verifying whether the unit (Xo , Yo ) ∈ T is at MPSS requires solving the following program14 : max

β/α,

(24)

subject to (αXo , βYo ) ∈ T , α, β > 0. The unit (Xo , Yo ) is at MPSS if the optimal value of program (24) is equal to 1. In this case it is attained at α = β = 1. As shown by Banker [3], if T is a VRS technology, the supremum in Eq. (24) is attained and is equal to the output radial efficiency of the unit (Xo , Yo ) in the corresponding CRS technology of Charnes et al. [16], which is the cone extension of the VRS technology. It turns out that a similar result is also true for an arbitrary technology T. Define the reference technology T ∗ as the cone extension of technology T:     s ˜ ˜ ˜ ˜ . T ∗ = (X, Y ) ∈ Rm + × R+ |∃ X, Y ∈ T , δ ≥ 0 : X, Y )= δ( X, Y As proved by Podinovski [63], the optimal value of program (24) is equal to the optimal value of the following program utilizing the reference cone technology T ∗ : max

β,

subject to

14 For an arbitrary technology T, the maximum in program (24) may not be attained and needs to be replaced by supremum. To avoid excessive technicalities, we assume that the maximum in program (24) is attained. This assumption is true for most technologies used in DEA, including the VRS technology.

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(Xo , βYo ) ∈ T ∗ , β ≥ 0. This result shows that the optimal value of program (24) stated for technology T, is equal to the inverse of the output radial efficiency of the unit (Xo , Yo ) evaluated in the reference cone technology T ∗ . Summarizing the above results, the unit (Xo , Yo ) ∈ T is at MPSS if and only if it is output radial efficient in the reference cone technology T ∗ generated by T. If T is the conventional VRS technology, Banker [3] and Banker and Thrall [4] show that the unit (Xo , Yo ) is at MPSS if and only if it exhibits constant returns to scale according to Definition 2. If (Xo , Yo ) exhibits increasing or decreasing returns to scale, it needs to increase or, respectively, reduce its size in order to achieve its MPSS. Podinovski [63] proves that the same result is true not only in the VRS technology but in any convex technology T. However, as shown in section “Global Returns to Scale” below, the same relationship between MPSS and the three types of returns to scale is generally invalid if technology T is not convex.

Economies of Scale and Cost Functions The notions of optimal scale, or MPSS in the framework of DEA, and returns-toscale considered in the previous sections are technical characteristics of production frontiers. As shown, all these notions are evaluated as specific characteristics of the output response function β (α) defined by formula (10). In this section, we consider an alternative, and generally different, approach to defining the optimal scale and economies of scale based on the cost minimization framework. Earlier research in this framework was started by Frisch [31], and more recent exposition can be found in Färe et al. [25] and Ray [70, 71]. s Let T ∈ Rm + × R+ be a production technology with m inputs and s outputs. Also, m let w ∈ R+ be the vector of input prices which is assumed to be known. For any output vector Y ∈ Rs+ , define the cost function C(w, Y) as follows:   C (w, Y ) = min w  X| (X, Y ) ∈ T .

(25)

The cost function C(w, Y) is correctly defined for the output vectors Y ∈ Rs+ for which there exists an input vector X ∈ Rm + such that the unit (X, Y) ∈ T. In other words, the function C(w, Y) is defined for all output vectors Y for which   the input requirement set V (Y ) = X ∈ Rm | Y ∈ T is not empty. Below the (X, ) + assumption that V(Y) = ∅ is implicitly assumed without further mention. Let T be the VRS technology given by its statement (14). For a fixed output vector Y, C(w, Y) is found as the optimal value of the following linear program:   C (w, Y ) = min w  X|Xλ ≤ X, Y λ ≥ Y, 1 λ = 1, λ ∈ Rn+ .

(26)

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The minimization of the objective function w X in program (26) is performed with respect to variable vectors X and λ, while the output vector Y and the vector of input prices w are fixed. The cost function C(w, Y) represents the lowest cost of producing the output vector Y possible in the VRS technology T, given the input prices w. Based on the notion of cost function (25), we can now proceed to giving an economic definition of scale efficiency. In the case of a single (scalar) output Y > 0, the average cost of producing this output is defined as AC (w, Y ) =

C (w, Y ) . Y

(27)

Economies of scale are present at the production level Y = Yo if the average cost AC(w, Y) is a decreasing function of Y in some neighborhood of Yo . This implies that AC(w, Y) falls to the right of Yo and increases if we move to the left of Yo . Similarly, diseconomies of scale are present at Y = Yo if AC(w, Y) is an increasing function in some neighborhood of Yo . Let Y ∗ > 0 be an output level at which, for the given input prices, the average cost function AC(w, Y) defined by Eq. (27) attains its minimum. Any such output level Y ∗ is referred to as an efficient scale of production. The economic scale efficiency of any actual output level Y = Yo is defined as the ratio of the minimum average cost AC(w, Y ∗ ) evaluated at any efficient scale Y ∗ to the actual average cost AC(w, Yo ). This definition is generalized to the case of multiple outputs Y ∈ Rs+ . In this case, consider the ray of output vectors {tY| t > 0} and define the ray average cost as RAC (w, t, Y ) =

C (w, tY ) . t

(28)

Scale economies (or diseconomies) at Y = Yo are present if the ray average cost RAC(w, t, Yo ) is a decreasing (respectively, increasing) function of t in some neighborhood of t = 1. For an output vector Yo , the efficient scale of production is defined as Y ∗ = T ∗ Yo , where T ∗ minimizes RAC(w, t, Yo ) with respect to t. The economic scale efficiency of output vector Yo is defined as the ratio of the minimum ray average cost RAC(w, t, Yo ) attained at t = T ∗ to the ray average cost at t = 1. The latter is equal to the cost function C(w, Yo ) evaluated by Eq. (25). In both the single and multiple-output cases, the economic scale efficiency is interpretable as the utmost factor by which the ray average costs associated with the production of the output (vector) Y can be reduced by rescaling production to the efficient scale Y ∗ . Following Ray [71], it is worth highlighting the difference between the notions of (technically) optimal scale or MPSS and efficient scale of production arising in the cost minimization framework. For a production unit (Xo , Yo ) ∈ T, the definition of MPSS by program (24) seeks the highest ratio β/α achievable among the units (αXo , βYo ) ∈ T that have the same input and output structures Xo and Yo (although taken in different quantities α and β) as the actual unit (Xo , Yo ). This definition

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is therefore independent of the vector of input prices w used by the definition of efficient scale of production. Moreover, as pointed out in section “Technically Optimal Scale,” in any convex technology T, the notion of MPSS is equivalent to the notion of local CRS, that is, a DMU (Xo , Yo ) ∈ T is at MPSS if and only if it exhibits CRS in the sense of Definition 2. In contrast, the definition of efficient scale of production depends on the vector of input prices w that is assumed known. Similar to MPSS, the idea is to minimize the use of inputs per unit of quantity of the output vector Yo . (In program (28), this quantity is represented by variable t.) However, in contrast with the notion of MPSS, this minimization is understood in the sense of identifying the minimal total cost of the inputs (which depends on their prices w) required for the production of outputs Yo . For a unit (Xo , Yo ), the efficient scale of production of its output vector Yo is defined by comparing this unit with all units in the form (X, tYo ), t > 0. In particular, the input vector X is not assumed to be in the form αXo as in the definition of MPSS. Assume that the unit (Xo , Yo ) minimizes the (ray) average cost. This involves two assumptions. First, the vector Xo minimizes the cost of producing Yo , that is, we have C(w, Yo ) = w Xo . Second, the output vector Yo represents efficient scale of production. In the case of a single output this means that Yo minimizes the average cost (27). In the case of multiple outputs, this means that T ∗ = 1 is optimal in program (28). Ray [70] proves that any such unit (Xo , Yo ) is at MPSS. The opposite is not true, that is, it is possible that a unit (Xo , Yo ) is at MPSS but does not minimize the (ray) average cost. Let T be a CRS technology. In this case, the cost function C(w, Y) is evaluated by solving program (26) from which we remove the normalizing equality 1 λ = 1. As shown by Ray [70], the cost function C(w, Y) for the CRS technology is homogeneous of degree 1 in vector Y, that is, for any scalar t ≥ 0, we have C (w, tY ) = tC (w, Y ) . Based on this property, Ray [70] proves that in the case of a single output, the average cost function AC(w, Y) defined by Eq. (27) is constant with respect to Y > 0. Similarly, in the case of vector output Y, the ray average cost function RAC(w, t, Y) defined by Eq. (28) is constant with respect to t > 0. In contrast, the (ray) average cost functions (27) and (28) for the VRS technology are not constant and can generally be thought of as having a U-shape [31, 70]. It further follows [70] that the minimum of this function is equal to the (constant) value of the ray average cost function of the corresponding CRS technology. In graphical terms, the straight horizontal line representing constant ray average costs (28) in the CRS technology as a function of variable t > 0 is tangent to the U-shaped curve representing ray average costs in the VRS technology, at the minimum point T ∗ of the latter curve. Remark 7 The economic scale efficiency of the output level Y is defined as the ratio of the minimum (ray) average costs (attained at the efficient scale) to the ray average costs at the actual level Y. Ray [70] provides an alternative method for the calculation of this ratio based on the cost functions C(w, Y) evaluated in the VRS and

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∗ ∗ CRS technologies. Namely, let CVRS and CCRS be the minimum costs of producing ∗ and output (vector) Y in the VRS and CRS technologies, respectively, that is, CVRS ∗ CCRS are the optimal values of linear program (26) and its CRS analogue. Then the ∗ /C ∗ . economic scale efficiency of the output level Y is equal to the ratio CCRS VRS The advantage of this approach is that it avoids solving a nonlinear program aimed at minimizing the ray average cost (28) as required by the definition of efficient scale.

Remark 8 We now turn to the question of practical identification of the efficient scale of production in the VRS technology represented by an optimal solution Y ∗ of program (27) or vector Y ∗ = T ∗ Yo , where T ∗ is optimal in program (28). Both programs are nonlinear and finding their exact solution may be problematic. To overcome this problem, Ray [71] shows that we can first solve the cost minimization linear program (25) with Y = Yo in the CRS technology, that is, program (26) from which the normalizing condition 1 λ = 1 is removed. Let vector λ∗ be optimal in the resulting linear program (together with some input vector X∗ ). Denote ∗ = 1 λ∗ . Then Y ∗ = Yo /∗ represents an efficient scale of production for the output vector Yo . In the general case, the optimal scale of production is not unique, and the described approach will produce only one such scale Y ∗ . The full range of optimal scales can be obtained as follows. Following Ray [71], define ∗max and ∗min as the maximum and minimum of the term 1 λ found over the set of optimal solutions of the CRS analogue of program (26). Identifying ∗max and ∗min requires solving two linear programs. Then the smallest and largest efficient scales of production are ∗ = Y /∗ ∗ ∗ found as Ymin o max and Ymax = Yo /min , respectively.

Partial Scale Characteristics for Smooth Production Frontiers The conventional notion of scale elasticity given by formula (9) and the notion of returns to scale assume that the whole vector of outputs Yo responds to proportional marginal changes of the whole vector of inputs Xo . From a practical perspective, we may also be interested in the elasticity of response of a partial vector of outputs to marginal changes of a partial vector of inputs. For example, in the short run, we may be able to change only some inputs such as labor and materials, but be unable to change the other inputs such as capital or infrastructure.15 This raises a question of short-run scale elasticity that characterizes the response of a subvector of outputs to proportional marginal changes of a subvector of inputs.

15 For

example, Johansen [38] considers a short-run industry production function that requires the evaluation of the scale elasticity with respect to a subset of two inputs (labour and energy) and treating the third input (capital) as fixed. Nelson [50] and Salvanes and Tjøtta [76] explore a similar idea by excluding quasi-fixed inputs in applications to the power generation and distribution sectors.

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In a more general setting, we may consider the elasticity of response of an arbitrary subset B of inputs and/or outputs to marginal proportional changes of an arbitrary subset A of inputs and/or outputs, while keeping the remaining subset C of inputs and outputs constant. Special cases of this general framework are situations in which the subsets A and B include a single input or output. We require that the subsets A, B, and C be mutually exclusive, the subsets A and B be nonempty, while C may be an empty set. To model the described scenario, we represent any unit (Xo , Yo ) in the following form:  (29) (Xo , Yo ) = XoA , XoB , XoC , YoA , YoB , YoC . Depending on the composition of the subsets A, B, and C, some subvectors in the above general formula may be omitted. For example, if the subset A includes only inputs, the subvector YoA is removed. If the set C is empty, then both subvectors XoC and YoC are removed. Assume that the unit (Xo , Yo ) stated in the form (29) is located on the production frontier (this requirement is stated more precisely in Assumption 2 below). We are interested inthe elasticity of proportional response of the joint subvector of inputs  and outputs XoB , YoB observed on the production frontier and caused by a marginal   proportional change of the joint subvector of inputs and outputs XoA , YoA , provided  C C the joint subvector of inputs and outputs Xo , Yo is kept constant. Similar to our development of the notion of scale elasticity, and following the approach of Podinovski and Førsund [66] and Podinovski et al. [68], define the partial proportional response function    β˜ (α) = max β| αXoA , βXoB , XoC , αYoA , βYoB , YoC ∈ T , β ∈ R .

(30)

The above function is a generalization of the output response function (10) and becomes the latter if XoA = Xo and YoB = Yo . We also need to adjust Assumption 1, by Brequiring  that the unit (Xo , Yo ) be efficient in the direction of the joint subvector Xo , YoB . Assumption 2 ˜ β(1) = 1. Note that if the set B contains only outputs and does not contain inputs, then Assumption 2 means that the unit (Xo , Yo ) is weak efficient in the production of subvector YoB of outputs. Assume that the function β˜ (α) is defined in some neighborhood of α = 1 and is at this point. Define the elasticity of response of the  differentiable   joint subvector  XoB , YoB with respect to marginal changes of the joint subvector XoA , YoA : εA,B (Xo , Yo ) = β˜  (1).

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Let the production frontier be defined by the transformation function (2). Talking into account (30), we obtain the following implicit statement of the function β˜ (α):  F αXoA , β˜ (α) XoB , XoC , αYoA , β˜ (α) YoB , YoC = 0. Assuming that the conditions of the implicit function theorem are satisfied and differentiating β˜ (α) by α, we have    A F (X , Y ) + Y A , ∇ A F (X , Y ) XoA , ∇X o o o o o Y   , εA,B (Xo , Yo ) = β (1) = −  B B Xo , ∇X F (Xo , Yo ) + YoB , ∇YB F (Xo , Yo ) 

˜

(31)

A F (X , Y ), ∇ A F (X , Y ), ∇ B F (X , Y ) and ∇ B F (X , Y ) are the where ∇X o o o o o o o o Y X Y partial gradients of the function F(X, Y) evaluated at the point (Xo , Yo ) with respect to the inputs and outputs included in the sets A and B, respectively. Depending on technology T (or transformation function F(X, Y)), the choice of sets A and B and the unit (Xo , Yo ), the elasticity of response εA, B (Xo , Yo ) may be positive, negative, or zero. For example, if each of the sets A and B consists of a single output and the unit (Xo , Yo ) is strongly efficient, then εA, B (Xo , Yo ) < 0, that is, a marginal increase of one output on the production frontier would result in a decrease of the other output.

Remark 9 Consider a special case in which the set A consists only of inputs (but not necessarily of all of them) and the set B consists only of outputs (again, not necessarily of all of them). In this case the elasticity of response εA, B (Xo , Yo ) has a meaning of partial scale elasticity and formula (31) is restated as follows:  A F (X , Y ) XoA , ∇X o o . εA,B (Xo , Yo ) = β (1) = −  B B Yo , ∇Y F (Xo , Yo ) 

˜

(32)

For a freely disposable technology for which we have inequalities (4), formula (32) implies that εA, B (Xo , Yo ) ≥ 0. This allows us to give the following definition of partial returns to scale, which generalizes the standard Definition 2. Definition 3 Let the unit (Xo , Yo ) ∈ T satisfy Assumption 2. Then (Xo , Yo ) exhibits (i) Partial increasing returns to scale if εA, B (Xo , Yo ) > 1 (ii) Partial decreasing returns to scale if εA, B (Xo , Yo ) < 1 (iii) Partial constant returns to scale if εA, B (Xo , Yo ) = 1

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Partial Elasticity of Response for Arbitrary Polyhedral Technologies Following Podinovski et al. [68], below we show how the notion of elasticity of response εA, B (Xo , Yo ) described in the previous section can be defined and computed in a very large and practically important class of polyhedral technologies. The notions of one-sided scale elasticity and returns-to-scale introduced for the standard VRS technology become special cases in this more general development. Technology T is called polyhedral if T is a polyhedral set in the input-output space Rm × Rs , that is, T is the intersection of a finite number of closed halfspaces in Rm × Rs [73]. To simplify notation and formulations, we assume that all inputs and outputs of all units in T are nonnegative and that technology T is strongly disposable with respect to its inputs and outputs. These assumptions are not essential and are not required in the general case considered by Podinovski et al. [68]. We state the freely disposable polyhedral technology T as the set of all units s q ˆ (X, Y ) ∈ Rm + × R+ for which there exist a vector λ ∈ R and vectors of input and m S output slacks SX ∈ R and SY ∈ R such that the following conditions are true: Xˆ λˆ + SX = X, Yˆ λˆ − SY = Y, Uˆ λˆ = Uo , ˆ SX , SY ≥ 0. λ,

(33)

In the statement (33), Xˆ and Yˆ are, respectively, the m × q and s × q input and output data matrices which allow a range of different interpretations. For example, the columns of matrices Xˆ and Yˆ may be the input and output vectors of the observed units, as in the standard VRS and CRS models. These columns may also be used, for example, to incorporate additional information in the model such as the information about the scalability of selected inputs and outputs or production trade-offs.16 The third equality in the statement (33) is used to specify additional conditions on the vector λˆ and is optional. In this equality, the vector Uo is of some dimension p and the matrix Uˆ is of dimension p × q. An example of this equality is the normalizing condition 1 λ = 1 used in the statement (14) of the VRS technology. Another example is the two-stage VRS technology described in Sahoo et al. [75] which includes two normalizing equalities, one for each stage. Often, as in the cone extension of the hybrid returns-to-scale (HRS) technology of Podinovski [61], Uo is a zero vector. In the CRS technology of Charnes et al. [16], the condition Uˆ λˆ = Uo is not used and is omitted from the statement of technology.

16 An

example of the former is the hybrid returns-to-scale (HRS) technology of Podinovski [58] and its multiple HRS analogue [69]. Polyhedral technologies with production trade-offs are dual to multiplier models with weight restrictions [62].

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The class of polyhedral technologies includes most of the known convex nonparametric technologies. This includes the standard CRS and VRS technologies of Charnes et al. [16] and Banker et al. [5] and their extensions by production trade-offs [60, 62], nonincreasing and nondecreasing returns-to-scale (NIRS and NDRS) technologies [21, 79], hybrid returns-to-scale technologies that assume scalability only of a subset of inputs and outputs [58, 61], technologies with multiple component processes [17, 69], many of the known network technologies stated in terms of the inputs and final outputs [40, 75], and convex technologies assuming joint weak disposability of outputs or inputs [45, 48, 49]. Podinovski et al. [68] show that the notion of elasticity of response εA, B (Xo , Yo ) introduced in the previous section for smooth production frontiers is generalizable to the one-sided elasticity of response in any polyhedral technology. Moreover, such one-sided elasticities can be calculated by solving essentially the same linear programs which need appropriate specification depending on the statement (33) of technology T and the choice of the sets A and B. Below we describe this approach in detail.17 Let T be any polyhedral technology stated in the form (33), and let β˜ (α) be the partial proportional response function defined as in Eq. (30). Consider any unit ˜ (Xo , Yo ) ∈ T. Denote ˜ the domain of the function β˜ (α). Clearly, α = 1 ∈ . ˜ ˜ Furthermore,  is a closed interval and β (α) is a continuous, concave and piecewise linear function on ˜ [68]. Define the partial  right-hand and left-hand elasticities of response  of the subvector XoB , YoB with respect to marginal changes of the subvector XoA , YoA as follows: +  (1), εA,B (Xo , Yo ) = β˜+ −  (1). ˜ εA,B (Xo , Yo ) = β−

(34)

˜ we have the following inequality: Because the function β˜ (α) is concave on , − + εA,B (Xo , Yo ) ≥ εA,B (Xo , Yo ) .

(35)

Let us consider a linear programming approach to the evaluation of the partial one-sided elasticities (34). The following theorem is proved in Podinovski et al. [68]. In its statement, the vectors v = (vA , vB , vC ) and u = (uA , uB , uC ) represent the weights corresponding to the inputs and outputs in the sets A, B, and C, respectively. The vector ω is of dimension p, the same as the dimension of vector Uo . Theorem 2 Let the unit (Xo , Yo ) ∈ T satisfy Assumption 2. Then the following four statements are true:

17 The

described approach is also applicable to various polyhedral technologies with undesirable outputs, e.g., to the weakly disposable technology of Kuosmanen [45]. This is considered in detail by Podinovski [64].

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˜ then the right-hand elasticity of (i) If α = 1 is not the right extreme point of , + response εA,B (Xo , Yo ) exists, is finite and equal to the optimal value of the linear program: +  (1) = min εA,B (Xo , Yo ) = β˜+

 A A vA Xo − u A Yo ,

(36)

subject to B −vB XoB + u B Yo = 1,   v Xo − u Yo + ω Uo = 0, v  Xˆ − u Yˆ + ω Uˆ ≥ 0, u, v ≥ 0, ω sign free vector.

˜ then program (36) has an unbounded (ii) If α = 1 is the right extreme point of ,  + optimal value. (In this case we can formally define εA,B (Xo , Yo ) = β + (1) = −∞.) ˜ then the left-hand elasticity of (iii) If α = 1 is not the left extreme point of , − response εA,B (Xo , Yo ) exists, is finite and equal to the optimal value of the linear program: −  (1) = max εA,B (Xo , Yo ) = β˜−

 A A vA Xo − u A Yo ,

(37)

subject to B −vB XoB + u B Yo = 1,   v Xo − u Yo + ω Uo = 0, v  Xˆ − u Yˆ + ω Uˆ ≥ 0, u, v ≥ 0, ω sign free vector.

˜ then program (37) has an unbounded (iv) If α = 1 is the left extreme point of ,  − optimal value. (In this case we can formally define εA,B (Xo , Yo ) = β − (1) = +∞.) Theorem 2 is a generalization of Theorem 1 to the case of an arbitrary polyhedral technology T and arbitrary subsets A and B of inputs and outputs. In this general + case the right-hand elasticity of response εA,B (Xo , Yo ) may be undefined (unlike in the standard case of scale elasticity) because a proportional increase of the joint   subvector XoA , YoA may lead outside the polyhedral technology T. This case is described by part (ii) of Theorem 2. This case is impossible in the VRS technology and is not included in Theorem 1.18 18 If technology T

is the standard VRS technology and the sets A and B are the sets of all inputs and outputs, respectively, then by a simple rearrangement program (36) becomes formula (20) from Theorem 1, and program (37) becomes formula (21). We discuss this in greater detail in Remark 11.

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Theorem 2 leads to a straightforward computational procedure for the evaluation + − of the one-sided elasticities of response εA,B (Xo , Yo ) and εA,B (Xo , Yo ), which is similar to the evaluation of the one-sided scale elasticities discussed in section “Evaluation of Scale Elasticity and Returns to Scale in the VRS Technology.” To be specific, consider the assessment of the left-hand elasticity of response − εA,B (Xo , Yo ). If program (37) has a finite optimal solution, then the left-hand elasticity of − response εA,B (Xo , Yo ) exists and is equal to the optimal value of program (37). − If program (37) has an unbounded optimal value, we formally let εA,B (Xo , Yo ) = +∞. This means that the technology T contains no units whose structure is   A αXo , βXoB , XoC , αYoA , βYoB , YoC , where α < 1, that is, it is impossible to reduce   the joint subvector of inputs and outputs XoA , YoA in technology T while keeping the subvectors XoC and YoC constant. If program (37) is infeasible, then the unit (Xo , Yo ) does not satisfy Assumption 2.19 Remark 10 Similar to Remark 9, let us consider a special case in which the set A consists only of inputs and the set B consists only of outputs.20 In this case, the  XA , objective functions of programs (36) and (37) are replaced by the single term vA o which is always nonnegative. Furthermore, if technology T is freely disposable with respect to all inputs then, similar to the case of the VRS technology, the domain of the function β˜ (α) is a closed unbounded interval ˜ = α, ˜ +∞). This implies that the case (ii) of Theorem 2 is impossible. Taking into account inequality (35) and  − formally letting εA,B (Xo , Yo ) = β − (1) = +∞ if α = 1 is the left extreme point of ˜ we have , + − 0 ≤ εA,B (Xo , Yo ) ≤ εA,B (Xo , Yo ) ≤ +∞. + In the described special case, the one-sided elasticities of response εA,B (Xo , Yo ) − and εA,B (Xo , Yo ) represent the marginal proportional change of the partial vector of outputs YoB with respect to a marginal proportional change of the partial vector of inputs XoA . They are therefore interpretable as the one-sided partial scale elasticities. + Similar to Definition 3, the evaluation of partial scale elasticities εA,B (Xo , Yo ) − and εA,B (Xo , Yo ) leads to the following characterization of partial returns to scale.

Definition 4 Let the unit (Xo , Yo ) ∈ T satisfy Assumption 2. Then (Xo , Yo ) exhibits

19 This

last statement formally follows from Theorem 1 and Proposition 5 proved in Podinovski et al. [68]. 20 Hadjicostas and Soteriou [36] consider the same case for the VRS technology only. Their linear programs for the evaluation of one-sided scale elasticities are special cases of the programs presented by Podinovski and Førsund [66] which in turn are generalized further by programs (36) and (37).

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+ − (i) Partial increasing returns to scale if 1 < εA,B (Xo , Yo ) ≤ εA,B (Xo , Yo ) + − (ii) Partial decreasing returns to scale if εA,B (Xo , Yo ) ≤ εA,B (Xo , Yo ) < 1 + − (iii) Partial constant returns to scale if εA,B (Xo , Yo ) ≤ 1 ≤ εA,B (Xo , Yo )

Remark 11 Let the set A include all inputs and let the set B include all outputs. + − Then the one-sided elasticities of response εA,B (Xo , Yo ) and εA,B (Xo , Yo ) are the + − standard one-sided scale elasticities ε (Xo , Yo ) and ε (Xo , Yo ) evaluated at the unit (Xo , Yo ) on the frontier of the polyhedral technology T stated in the form (33). Let be the set of all optimal solutions u, v, ω of the multiplier model assessing the output radial efficiency of the unit (Xo , Yo ) in technology T. As shown in Podinovski et al. [68], the linear programs (36) and (37) are then equivalently restated as follows:   ω  Uo , (38) ε+ (Xo , Yo ) = 1 − max

u,v,ω∈

ε− (Xo , Yo ) = 1 −

min

u,v,ω∈

  ω  Uo .

(39)

It is now clear that Theorem 1 is a special case of Theorem 2. Indeed, if T is the standard VRS technology, then the vector Uo has a single component equal to 1, and the equalities (38) and (39) are the same as (20) and (21), respectively. The one-sided scale elasticities ε+ (Xo , Yo ) and ε− (Xo , Yo ) can also be expressed ˜ of optimal solutions u, v, ω to the multiplier model measuring the using the set input radial efficiency of the unit (Xo , Yo ), in an arbitrary polyhedral technology T. Podinovski et al. [68] show that these are calculated as follows:  +

ε (Xo , Yo ) = 1/ 1 −

min

    ω Uo ,

(40)

max

    ω Uo .

(41)

˜

u,v,ω∈

 −

ε (Xo , Yo ) = 1/ 1 −

˜

u,v,ω∈

If T is the standard VRS technology, the formulae (40) and (41) become formulae (23). Remark 12 Although most of the DEA literature devoted to the subject of returns to scale and scale elasticity has traditionally focused on the VRS technology, a number of studies have developed specialized techniques for evaluating returns to scale in other polyhedral technologies, often without exploring a link to the underlying scale elasticity. For example, Tone [82] and Korhonen et al. [42] develop methods for the evaluation of returns to scale in technologies expanded by weight restrictions. Sahoo et al. [75] consider the notion of returns to scale in two-stage network models. Atici and Podinovski [1] develop methods for the assessment of a partial elasticity of response in the standard CRS technology.

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The general approach developed by Podinovski et al. [68] and presented above as Theorem 2 allows the evaluation of scale elasticity and its partial analogues, and hence the characterization of returns to scale, in any polyhedral technology, by solving essentially the same appropriately specified linear programs. This effectively removes the need to develop bespoke methodologies for assessing returns to scale in each individual technology. Podinovski et al. [68] illustrate the application of this general approach in the VRS technology expanded by the specification of weight restrictions. Podinovski [63] applies the same approach in a two-stage network technology.

Global Returns to Scale The conventional returns-to-scale characterization simultaneously plays two roles: local and global. To be specific, let us illustrate this using the nonparametric VRS technology of Banker et al. [5]. Assume that the unit (Xo , Yo ) exhibits increasing returns to scale. From the local perspective, a small proportional increase of the input vector of this unit (e.g., by 1%) would result in a larger proportional increase (more than 1%) of the vector of outputs, assuming that the resulting unit remains on the efficient frontier. In other words, a marginal increase of the size of the unit leads to its improved productivity. From the global perspective, if the unit (Xo , Yo ) exhibits increasing returns to scale, then it is not at MPSS, and in order to achieve the latter, the unit (Xo , Yo ) needs to increase the scale of its operations. Therefore, the type of returns to scale exhibited by the unit (Xo , Yo ) in the VRS technology is also indicative of the direction of resizing in which the productivity would increase, both in the immediate local sense and in the global sense as a direction to MPSS. Podinovski [63] proves that the same dual role of the local returns-to-scale characterization is true in any convex production technology. However, if technology T is not convex, the conventional local characterization of returns to scale is generally no longer indicative of the direction to MPSS.21 This observation motivates the development of the concept of global returns to scale by Podinovski [56, 57]. The notion of global returns to scale applies to almost any production technology. The only assumption that is required is that technology T does not allow free and

21 Nonconvex

technologies arise naturally if, for example, some inputs or outputs are represented by ratio measures such as percentages, which is common in managerial applications of DEA. The incorporation of ratio inputs or outputs in the model of technology generally invalidates the assumption of convexity. Olesen et al. [53, 54] develop nonconvex variants of the VRS and CRS technologies in which both volume and ratio types of inputs and outputs are native types of data.

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unlimited production of outputs.22 In particular, technology T is not assumed to be convex or exhibit any particular type of input or output disposability. The types of global returns to scale are indicative of the direction in which a given unit should resize the scale of its operations in order to achieve its MPSS. To introduce this characterization formally, let the unit (Xo , Yo ) be output radial efficient, that is, satisfy Assumption 1. program (24) used for verifying if the unit (Xo , Yo )is at MPSS. Let  Consider ˆ ˆ α, ˆ β be any optimal solution to this program. Then the unit αX ˆ o , βYo is at MPSS. This unit is referred to as a scale reference unit (SRU) of (Xo , Yo ). Clearly, the unit (Xo , Yo ) is at MPSS if and only if αˆ = βˆ = 1 is an optimal solution to program (24). Suppose that (Xo , Yo ) is not at MPSS.  Then two ˆ o possibilities arise: either αˆ > 1 or αˆ < 1. In the former case the SRU αX ˆ o , βY is larger than the unit (Xo , Yo ) and in the latter it is smaller.23 Note that program (24) may have multiple optimal solutions, each defining a different SRU of the unit (Xo , Yo ).24 It is possible that all such SRUs are larger than (Xo , Yo ) or all are smaller than the latter. It is also theoretically possible that some SRUs are larger and some are smaller than (Xo , Yo ). The following definition of the four types of global returns to scale is given by Podinovski [56, 57] and indicates the direction of resizing for the unit (Xo , Yo ) as it changes the scale of its operations towards its MPSS (represented by its SRUs). Definition 5 Let the unit (Xo , Yo ) satisfy Assumption 1. Then (Xo , Yo ) exhibits (i) Global constant returns to scale (G-CRS) if (Xo , Yo ) is at MPSS (ii) Global increasing returns to scale (G-IRS) if all its SRUs are larger than (Xo , Yo ) (iii) Global decreasing returns to scale (G-DRS) if all its SRUs are smaller than (Xo , Yo ) (iv) Global subconstant returns to scale (G-SCRS) if some of its SRUs are smaller and some are larger than (Xo , Yo ), but the unit (Xo , Yo ) itself is not at MPSS In a more general case (which is primarily of academic interest), it is possible that the optimal value of program (24) is not attained and, therefore, the unit (Xo , Yo ) does not have an SRU. In this case, Podinovski [56] defines the notion of an approximate SRU and restates Definition 5 in the latter terms.

22 Technology

 ˆ Yˆ ∈ T such that Xˆ = 0 and T allows free production if there exists a unit X,

Yˆ ≥ 0, Yˆ = 0. Technology T allows if there exists an input vector Xˆ and  unlimited production  ˆ ˆ ˆ output vector Y = 0 such that sup β| X, β Y ∈ T = +∞. [56] proves that αˆ > 1 implies βˆ > 1 and, assuming strong efficiency of the unit (Xo , Yo ), αˆ < 1 implies βˆ < 1. 24 In practice, it may be of interest to identify the smallest and largest SRUs of the unit (X , Y ). o o Provided we have already evaluated the scale efficiency of the unit (Xo , Yo ), identifying its smallest and largest SRUs requires solving two further simple linear programs ([63], Footnote 16). 23 Podinovski

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Output

F E

D

B

C

A

0

Input

Fig. 3 Global returns to scale in a nonconvex technology

The following example illustrates the four types of global returns to scale. Example 3 Consider the nonconvex technology shown as the shaded area in Fig. 3. Consider the local characterization of returns to scale first. Similar to Example 1, we establish that the units A, C, and E exhibit constant returns to scale, D exhibits increasing returns to scale and B and F exhibit decreasing returns to scale. Note that only the unit E is at MPSS. Therefore, this unit is the single SRU of the other five units. By Definition 5, unit E exhibits G-CRS, units A, B, C, and D exhibit G-IRS (because all four are smaller than their SRU E) and F exhibits G-DRS (because it is larger than its SRU E). Note that these global types are inconsistent with the types of local returns to scale. In this example, no unit exhibits the fourth type of global subconstant returns to scale. This type would be observed if the frontier point A were located higher, on the ray OE. In this case both units A and E would then exhibit G-CRS, and units B, C, and D would be classed as exhibiting G-SCRS. Example 4 Figure 4 shows an example of free disposal hull technology of Deprins et al. [19]. It is clear that the unit C is at MPSS and therefore exhibits G-CRS. The units A and B are smaller than C and therefore exhibit G-IRS. The units D and E are larger than C and exhibit G-DRS.

714

V. V. Podinovski and F. R. Førsund Output 11 10

E

9

D

8

C

7 6 5 4

B

3 2

A

1 0

1

2

3

4

5

6

7

8

9

10 11

Input

Fig. 4 Global returns to scale in the free disposal hull technology

It is worth noting that the standard local characterization of returns to scale in the free disposal hull technology is uninteresting. Indeed, the right-hand scale elasticity at each of the observed units in Fig. 4 is equal to zero, and the left-hand scale elasticity is undefined (or can formally be taken equal to +∞). Therefore, by the standard Definition 2 of returns to scale, all observed units exhibit constant returns to scale.25 The above examples show that in a nonconvex technology, the local and global characterizations are generally different. The three types of local returns to scale are based on the notion of scale elasticity and are therefore local characteristics of the production frontier. The four types of global returns to scale are global characteristics. They are based on the direction towards MPSS and are not directly related to the scale elasticity. It is also worth noting that the global type of subconstant returns to scale is primarily of academic interest and should not normally be observed in practical applications. This fact has recently been confirmed by computational experiments by Cesaroni et al. [10]. However, without this type the global characterization of production frontiers would be logically incomplete.26

25 We

can also say that each observed unit exhibits decreasing returns to scale on the right (corresponding to the right-hand scale elasticity equal to zero) and increasing returns to scale on the left (corresponding to the left-hand scale elasticity equal to +∞). 26 Podinovski [63] notes that the G-SCRS type may occur naturally in the nonconvex free replication hull technology developed by Ray and Hu [72].

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Podinovski [63] proves that if technology T is convex, the local and global characterizations coincide. More precisely, the unit (Xo , Yo ) exhibits global constant, increasing, or decreasing returns to scale if and only if it exhibits local constant, increasing, or decreasing returns to scale, respectively. This further implies that in a convex technology, the global type of subconstant returns to scale is impossible. Therefore, any difference between the local and global characterizations may be observed only in a nonconvex technology, as was illustrated by Examples 3 and 4. Podinovski [56] shows that the four types of global returns to scale can be identified by a modification of the reference technology method developed by Färe et al. [23, 24] and further modified by Kerstens and Vanden Eeckaut [41] and Briec et al. [9]. To describe this method, consider any technology T that disallows free and unlimited production. Let ET (Xo , Yo ), E NIRS (Xo , Yo ) and E NDRS (Xo , Yo ) denote the output radial efficiencies of the unit (Xo , Yo ) in technology T and the NIRS and NDRS technologies generated by T, respectively.27 The following theorem is proved by Podinovski [56, 57].28 Theorem 3 Let the unit (Xo , Yo ) satisfy Assumption 1. Then it exhibits (i) (ii) (iii) (iv)

G-CRS if and only if E NDRS (Xo , Yo ) = E NIRS (Xo , Yo ) = ET (Xo , Yo ) = 1 G-DRS if and only if E NDRS (Xo , Yo ) < E NIRS (Xo , Yo ) ≤ ET (Xo , Yo ) = 1 G-IRS if and only if E NIRS (Xo , Yo ) < E NDRS (Xo , Yo ) ≤ ET (Xo , Yo ) = 1 G-SCRS if and only if E NIRS (Xo , Yo ) = E NDRS (Xo , Yo ) < ET (Xo , Yo ) = 1

Conclusion In this chapter we presented an overview of the notions of scale elasticity and returns to scale in different production technologies. The literature on this topic is vast and has been developed over many decades. A particular difficulty in presenting a consistent unifying development of this topic is the fact that it has historically been explored in two different theoretical frameworks. The first and the older one is the neoclassical economics framework ascending from the pioneering works of Frisch [31]. According to this approach, efficient production is represented by a known transformation function which is subject to various economic and technical assumptions that facilitate its analysis. The second approach has started in the seminal works on DEA by Banker et al. [5] and Banker [3] in which the

27 If

T is a free disposal hull technology, the output radial efficiencies E NIRS (Xo , Yo ) and ENDRS (Xo , Yo ) and their input analogues can be evaluated by solving mixed integer linear programs [59]. 28 Podinovski [56, 57] states and proves an analogue of Theorem 3 in the input orientation, that is, by using the input radial efficiencies of the unit (Xo , Yo ) in the three technologies. Its proof is based on the assumption of strong efficiency of the unit (Xo , Yo ). For a similar proof of the output-oriented variant of this theorem presented in this chapter, the weaker Assumption 1 is sufficient.

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whole production technology is defined axiomatically, without making any explicit assumptions about its efficient frontier. We note that despite the differences between the two theoretical frameworks and required assumptions, the fundamental and unifying concept in both approaches is the proportional output response function β (α). For any unit (Xo , Yo ), this function represents the maximum proportion of the output vector Yo that can be produced in the technology as a result of the input vector Xo changing proportionally by a factor α ≥ 0. It is important that the definition of the function β (α) is universal – it does not depend on the theoretical approach used to model the technology and does not rely on any additional assumptions about it.  In the neoclassical approach, the derivative of this function β (α) calculated at α = 1 is the scale elasticity evaluated at the unit (Xo , Yo ). In DEA models, the one-sided derivatives of the function β (α) are the corresponding one-sided scale elasticities. The maximum ratio β (α) /α over the ray α > 0 corresponds to the optimal technical scale in the neoclassical approach and MPSS in nonparametric DEA models. Furthermore, the function β (α) can be redefined to represent the proportional response of a subset of inputs and outputs with respect to proportional changes of another subset of inputs and outputs, which leads to a spectrum of different definitions of partial scale elasticity. Finally, in nonconvex technologies, the ratio β (α) /α may generally attain its maximum over the ray α > 0 at multiple values of the variable α. An investigation of this leads to the global returns-to-scale characterization of production frontiers.

Cross-References  Data Envelopment Analysis: A Nonparametric Method of Production Analysis  Distance Functions in Production Economics  Elasticities of Substitution  Nonconvexity in Production and Cost Functions: An Exploratory and Selective

Review  Reminiscences of “Returns to Scale in Electricity Supply”

References 1. Atici KB, Podinovski VV (2012) Mixed partial elasticities in constant returns-to-scale production technologies. Eur J Oper Res 220(1):262–269 2. Balk BM, Färe R, Karagiannis G (2015) On directional scale elasticities. J Prod Anal 43(1):99– 104 3. Banker RD (1984) Estimating most productive scale size using data envelopment analysis. Eur J Oper Res 17(1):35–44 4. Banker RD, Thrall RM (1992) Estimation of returns to scale using data envelopment analysis. Eur J Oper Res 62(1):74–84 5. Banker RD, Charnes A, Cooper WW (1984) Some models for estimating technical and scale inefficiencies in data envelopment analysis. Manag Sci 30(9):1078–1092

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Nonconvexity in Production and Cost Functions: An Exploratory and Selective Review∗

18

Walter Briec, Kristiaan Kerstens, and Ignace Van de Woestyne

Contents Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Technologies and Distance Functions: Basic Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Axiom of Convexity: Arguments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Convexity and Duality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Convexity and Time Divisibility . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Convexity and Managerial Practice: Some Skepticism Around . . . . . . . . . . . . . . . . . . . . . . Nonparametric Nonconvex Technologies and Value Functions: Free Disposal Assumption and Minimum Extrapolation Principle . . . . . . . . . . . . . . . . . . . . . . . . . . Technologies: FDH and Its Extensions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Economic Value Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Efficiency Decompositions and the Testing of Convexity: A Priori Relations . . . . . . . . . . Empirical Evidence on FDH and Its Extensions: The Impact of Convexity . . . . . . . . . . . . FDH and Its Extensions: Further Methodological Refinements . . . . . . . . . . . . . . . . . . . . . . Mitigating Convexity: A Selection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Partial Convexity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Regular Ultra Passum Law . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . From Generalized Convexity to Nonconvexity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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acknowledge the most helpful comments of R. Chambers and G. Cesaroni on an earlier version. The usual disclaimer applies. W. Briec University of Perpignan, LAMPS, Perpignan, France e-mail: [email protected] K. Kerstens () IESEG School of Management, CNRS, Université de Lille, UMR 9221-LEM, Lille, France e-mail: [email protected] I. Van de Woestyne Research Unit MEES, KU Leuven, Brussel, Belgium e-mail: [email protected] © Springer Nature Singapore Pte Ltd. 2022 S. C. Ray et al. (eds.), Handbook of Production Economics, https://doi.org/10.1007/978-981-10-3455-8_15

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Semilattice Structures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Preliminary Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Abstract

The purpose of this contribution is to provide an overview of developments in nonconvex production technologies and economic value functions, with special attention to the cost function. Apart from a somewhat selective review of theoretical issues, the emphasis is on whether the assumption of convexity makes a difference in practice. Anticipating our conclusion, we argue that traditional convex empirical results differ on average rather markedly from alternative nonconvex ones. This should make the discipline reconsider its traditional relationship with convexity in both theoretical and applied production analysis. Keywords

Nonparametric frontier · Convexity · Production · Cost function · Scale · Productivit

Introduction This contribution focuses on deterministic nonparametric frontier technologies that somehow relax the traditional hypothesis of convexity. Apart from developments in general equilibrium theory with nonconvexities, we are unaware of any developments in empirical production theory that allow to empirically document the eventual impact of the traditional convexity axiom. This explains the narrow and selective focus of this chapter. The seminal article of Farrell [61] introduced a single output multiple inputs deterministic nonparametric frontier technology, but did not establish a link with linear programming. Boles [20] and Charnes et al. [39] are the first economics and operations research articles, respectively, that have given the impetus that made the nonparametric approach to production one of the great success stories in terms of both methodological developments and empirical applications. While the axiom of convexity is traditionally maintained in these nonparametric production models (see Afriat [4], Banker et al. [13], Charnes et al. [39], Diewert and Parkan [50]) as well as in the mainstream empirical economic literature on production analysis, Afriat [4] was probably the first to mention a basic single output nonconvex technology imposing the assumptions of strong input and output disposability. A multiple output version has probably been proposed for the first time in Deprins et al. [49] and these authors suggested the moniker Free Disposal Hull (FDH). The work of Scarf [108–111] may well be considered as an important predecessor of FDH, since he studied activity analysis models based on integer data. For instance, Figure 1 displayed in Scarf [108, p. 3638] resembles the FDH as we know it. Without the pretension to recount the history of the FDH technology in detail, it

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suffices to mention Lovell and Vanden Eeckaut [88, footnote 2] lists another three potential historical sources of the FDH concept. This traditional stress on convex applied production analysis is to some extent surprising, since it is theoretically well-known that important features of technology fundamentally violate the convexity of the production possibility set (see Farrell [62]). First, indivisibility implies that inputs and outputs are not necessary perfectly divisible. Furthermore, scaling down or up the entire production process in infinitesimal fractions may not be feasible. Examples include the start-up and shutdown costs in industries (see, e.g., O’Neill et al. [93] for electricity generation). Scarf [112,113] stresses the importance of indivisibility in selecting among technological options. Second, economies of scale (e.g., modern information technology) and economies of specialization (e.g., Romer [106] on nonrival inputs in the new growth theory) violate the convexity of technology. Third, the existence of positive or negative production externalities also leads to nonconvexities. Thus, the structure of production in society is potentially full of nonconvexities. It should be realized that the natural environment is full of nonconvexities as well (see Dasgupta and Mähler [46] for an overview). Ecologists identify pathways by which ecosystem constituents interact with one another and with the external environment. A large body of empirical work reveals that those pathways often involve transformation possibilities among environmental goods and services that constitute nonconvex sets (e.g., see Boscolo and Vincent [21] on forestry economics). In the words of Dasgupta and Mähler [46]: “The word “convexity” is ubiquitous in economics, but absent from ecology.” This book chapter is structured as follows. Section “Technologies and Distance Functions: Basic Definitions” provides some basic definitions of the traditional axioms underlying technologies and their representation via distance functions. Section “Axiom of Convexity: Arguments” discusses in detail the existing justifications for the axiom of convexity. Section “Nonparametric Nonconvex Technologies and Value Functions: Free Disposal Assumption and Minimum Extrapolation Principle” first focuses on nonconvex FDH with its extensions and the corresponding traditional convex technologies, then followed by a discussion of nonconvex economic value functions as well as efficiency decompositions and tests of convexity that have been conceived in the literature. Next, we offer an empirical perspective on the use of FDH and its extensions on a variety of topics. Finally, we discuss some further methodological refinements. Section “Mitigating Convexity: A Selection” offers a very selective review of several attempts to mitigate the impact of the convexity axiom while avoiding FDH and its extensions. Section “Conclusions” concludes and outlines some future research issues.

Technologies and Distance Functions: Basic Definitions A production technology describes all available possibilities to transform input n vectors x = (x1 , . . . , xm ) ∈ Rm + into output vectors y = (y1 , . . . , yn ) ∈ R+ . The production possibility set or technology T summarizes the set of all feasible n input and output vectors: T = {(x, y) ∈ Rm + × R+ : x can produce y}. Note

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that it may be surprising that the main contributions in this literature continue considering that the technology is a subset of Rm × Rn . In section “Nonparametric Nonconvex Technologies and Value Functions: Free Disposal Assumption and Minimum Extrapolation Principle” we open a perspective on considering the domain Nm × Nn instead. Given our focus on input-oriented efficiency measurement later on, this technolm ogy can be represented by the input correspondence L : Rn+ → 2R+ where L(y) is the set of all input vectors that yield at least the output vector y: L(y) = {x : (x, y) ∈ T } .

(1)

n The radial input efficiency measure is a map E : Rm + × R+ −→ R+ ∪ {∞} that can be defined as:

E (x, y) = min {λ : λ ≥ 0, λx ∈ L(y)} .

(2)

This radial efficiency measure, which is the inverse of the input distance function, indicates the minimum contraction of an input vector by a scalar λ while still remaining in the input correspondence. Obviously, the resulting input combination is located at the boundary of this input correspondence. For our purpose, the radial input efficiency has two key properties (see, e.g., Hackman [68]). First, it is smaller or equal to unity (0 ≤ E (x, y) ≤ 1), whereby efficient production on the isoquant of L(y) is represented by unity and 1−E (x, y) indicates the amount of inefficiency. Second, it has a cost interpretation. Note that more general efficiency measures are around in the literature: one example is the directional distance function introduced by Chambers et al. [38] that is sometimes mentioned in this contribution. n Consider a set of K observations A = {(x1 , y1 ) , . . . , (xK , yK )} ∈ Rm + × R+ . In the following, let us denote K = {1, . . . , K}. Nonparametric specifications of technology can then be estimated by enveloping these K observations in the set A while maintaining some basic production axioms (see Hackman [68] or Ray [104]). We are interested in defining minimum extrapolation technologies satisfying the following assumptions: T 1: (0, y) ∈ T ⇒ y = 0; (0, 0) ∈ T . T 2: T is closed. n T 3: For all (x, y) ∈ T and all (u, v) ∈ Rm + × R+ if (x, −y) ≤ (u, −v), then (u, v) ∈ T . T 4: T exhibits (ı) constant returns to scale (CRS), δT ⊆ T , ∀δ > 0; (ıı) nonincreasing returns to scale (NIRS), δT ⊆ T , ∀δ ∈ (0, 1);(ııı) nondecreasing returns to scale (NDRS), δT ⊆ T , ∀δ > 1; (ıv) variable returns to scale (VRS), when (ı), (ıı), and (ııı) do not hold. T 5: T is convex. We briefly expand on the interpretation of these basic axioms. Axiom (T1) states that there is no free lunch and that inaction is feasible. Axiom (T2) indicates that

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the technology is closed. Axiom (T3) represents strong or free disposability in the inputs and the outputs: inputs can be wasted without opportunity costs, and outputs can be reduced at will. Axiom (T4) defines all four traditional returns to scale hypotheses (i.e., constant, nonincreasing, nondecreasing, and variable (flexible) returns to scale). Finally, the convexity assumption (T5) is traditional, but it is not indispensable.

Axiom of Convexity: Arguments While the axiom of convexity (T5) is traditionally maintained in economics, we develop three types of arguments to put it under scrutiny. Two arguments are related to economic theory. One argument is more pragmatic: in empirical applications, it turns out that managers often object to convexity. Sometimes the motivation to maintain the convexity axiom is just analytical convenience (see, e.g., Hackman [68, p. 2]). We think this is an argument that is valid only if one can show that convex results provide a reasonably good approximation to a potentially nonconvex economic reality.

Convexity and Duality Often duality is invoked as a reason to maintain convexity. Since the main duality relations in economics linking, e.g., production and cost approaches presume some form of convexity, in applied empirical production analysis, researchers feel compelled to maintain the same axioms. It is an open question whether this desire for theoretical consistency is cogent. We explore this viewpoint a little bit. The traditional duality results often fit in a general equilibrium framework that maintains convexity in its simplest forms. But, applied researchers tend to forget that general equilibrium theory has become less attractive as a general normative framework since the Sonnenschein-Mantel-Debreu results appeared in the early 1970s. Almost entirely negative conclusions appeared about the uniqueness and stability of general equilibrium. While uniqueness only occurs under restrictions void of economic realism, instability is the rule rather than the exception since almost any continuous pattern of price movements may occur in general equilibrium (see Ackerman [2]). Furthermore, general equilibrium theory has been developed under more general conditions of nonconvexity on technology and preferences (see Chavas and Briec [41]). Realistically, this involves some process of nonlinear pricing. At the firm level, one may therefore look for proper nonconvex specifications that do justice to the nonconvexities in technology. This may imply recourse to more complex duality relations, but this is simply the price to pay for the gain in realism. The FDH and its extensions can be seen as one example that may fit into such a strategy (see, e.g., Agrell and Tind [5]).

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Convexity and Time Divisibility Several economic theorists interpret convexity of technology solely in terms of time divisibility of technologies and see no other justification for its use. Hackman [68, p. 39] puts things clearly when discussing the axiom of convexity in his textbook: It does have the following “time-divisibility” justification. Suppose input vectors x1 and x2 each achieve output level u > 0. Pick a λ ∈ [0, 1], and imagine operating 100λ% of the time using x1 and 100(1 − λ)% of the time using x2 . At an aggregate level of detail, it is not unreasonable to assume that the weighted average input vector λx1 + (1 − λ)x2 can also achieve output level u.

Jacobsen [70, p. 759] remarks when discussing the quasi-concavity property of the production function: (A.5) implies a time divisibility in the production process.

Shephard [116, p. 15] states about the property of convexity of the input set: Property P.8 is valid for time divisibly-operable technologies. For example, if x ∈ L(u), y ∈ L(u) and θ ∈ [0, 1], the input vector [(1 − θ)x + θy] may be interpreted as an operation of the technology a fraction (1 − θ) of some unit time interval with the input vector x and a fraction θ with y, assuring at least the output rate u.

The added footnote at the end of the last cited phrase reads: “Indeed the input vector [(1 − θ )x + θy] may have no meaning unless so interpreted.” This time divisibility argument basically ignores setup and lead times which make a switch between the underlying activities costly in terms of time. This implies that convexity becomes questionable when time indivisibilities compound all other reasons for spatial nonconvexities (e.g., indivisibilities, increasing returns to scale, economies of specialization, externalities, etc.).

Convexity and Managerial Practice: Some Skepticism Around Decision-makers do not necessarily believe in convexity. This is evidenced in remarks, scattered in the literature, on the problems encountered in communicating the results of traditional efficiency measurement assuming convexity to decisionmakers. We provide some examples of quotes reflecting this doubt of managers to the axiom of convexity. In a study applying convex nonparametric frontier methods to measure bank branch efficiency, Parkan [96, p. 242] notes: The comparison of a branch which was declared relatively efficient, to a hypothetical composite branch, did not allow for convincing practical arguments as to where the inefficiencies lay.

Epstein and Henderson [53, p. 105] report similar experiences in that managers simply question the feasibility of the hypothetical projection points resulting from

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convex nonparametric frontiers when discussing an application to a large publicsector organization: The algorithm for construction of the frontier was also discussed. The frontier segment connecting A and B was considered unattainable. It was suggested that either (1) these two DMUs should be viewed as abnormal and dropped from the model, (2) certain key variables have been excluded, or (3) the assumption of linearity was inappropriate in this organization. It appears that each of these factors was present to some degree.

In a very similar vein, Bouhnik et al. [22, p. 243] state: Equally as important, it is our experience that managers often question the meaning of convex combinations that involve what they perceive to be irrelevant DMUs.

All quotes seem to point to the fact that convexity may well in practice combine observations that are too far apart in terms of input mix, output mix, and/or scale of operations. While one hopes for a rather uniformly dense rather well-spaced cloud of points that avoids the combination of extreme points of production, such extreme combinations apparently occur and are puzzling for managers. In a value efficiency analysis application (a way of incorporating preference information into efficiency analysis), Halme et al. [69, p. 11] also opt for its use with FDH because this matched the preferences of management: The management was also more comfortable providing preference information over existing units than virtual units, and found the results valuable.

Also some researchers concede that nonconvex analysis of production facilitates the practical use of efficiency analysis. For instance, Bogetoft et al. [19, p. 859] declare in this context: In general, allowing the possibility set to be nonconvex facilitates the practical use of productivity analysis in benchmarking. In particular, fictitious production possibilities, generated as convex combinations of those actually observed, are usually less convincing as benchmarks, or reference units, than actually observed production possibilities.

This experience is confirmed by Halme et al. [69, p. 10]: During our long experience of DEA applications we repeatedly encountered the phenomenon that DMs (Decision Maker) are reluctant to evaluate other than existing units.

Obviously, we understand that this is just casual evidence that transpires from the empirical literature. But, it is useful to consider in addition to the other arguments above. Turning to a mathematical argument, notice that there exists some general condition under which a distance function (related to the efficiency measure (2)) can characterize a nonconvex technology. This general condition is independent of the strong disposability assumption (T3) (though we use it in the remainder for computational reasons). One can provide a simple condition considering the radian subset of R ∈ Rd . A subset R of Rd is a radian set if for all λ ∈ [0, 1] and all x ∈ R, λx ∈ R. Equivalently, such a subset is called a starshaped set (see Aliprantis and Border [6] for related concepts). A subset S is co-radian if for all λ ≥ 1, λx ∈ S. In the field of functional analysis in mathematics, a distance function is called a gauge

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function (analogous to the Minkowski functional for symmetrical sets). This is a function that recovers a notion of distance on a linear space. For all subset D of Rd , the gauge function ψD is the map ψD : Rd −→ [0, ∞] defined by: ψD (x) = sup{δ : δx ∈ D},

(3)

with the convention that ψD (x) = 0 if there is no λ ≥ 0 such that λx ∈ A. Paralleling this definition, for all co-radian set, one can define a co-gauge as: ηD (x) = inf{δ : δx ∈ D}.

(4)

This definition implies that for all, respectively, closed radian and co-radian sets R and S: R = {x ∈ Rd : ψR (x) ≥ 1}

and

S = {x ∈ Rd : ηS (x) ≤ 1}

(5)

It follows that a production technology can be characterized from the efficiency measure (2) if and only if the input set L(y) is co-radian for all y ∈ Rm + . Considering an output-oriented efficiency measure, such a characterization applies if and only if the output set is a radian (starshaped) set.

Nonparametric Nonconvex Technologies and Value Functions: Free Disposal Assumption and Minimum Extrapolation Principle Technologies: FDH and Its Extensions While Deprins et al. [49] are commonly acknowledged as the developers of the basic FDH model, Kerstens and Vanden Eeckaut [73] extended this basic model by introducing the possibilities of constant, nonincreasing, and nondecreasing returns to scale. This leads to the definition of three new technologies complementary to the assumption of flexible or variable returns to scale embodied in the basic FDH model. Individual production possibility sets are based upon one production unit (xk , yk ), the strong disposability assumption, and different maintained hypotheses of returns to scale:   n N (xk , yk ) = (x, y) ∈ Rm + × R+ : x ≥ δxk , y ≤ δyk , δ ∈  , where  ∈ {CRS , N DRS , N I RS , V RS }, with: (i) CRS = {δ : δ ≥ 0} ; (ii) N DRS = {δ : δ ≥ 1} ; (iii) N I RS = {δ : 0 ≤ δ ≤ 1} ; (iv) V RS = {δ : δ = 1} .

(6)

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Unions and convex unions of these individual production possibility sets yield the nonconvex technologies on the one hand and the traditional convex models on the other hand: TN C, =



N (xk , yk ) and TC, = Co



k∈K

 N (xk , yk ) ,

(7)

k∈K

where Co stands for the convex hull operator. In addition to this approach based on sets and their operations, an alternative and useful formulation can be proposed making some analogy to the traditional convex model. Let us introduce the following notation: C =



 zk = 1, zk ≥ 0 and N C = zk = 1, zk ∈ {0, 1} .

k∈K

k∈K

A unified algebraic representation of convex and nonconvex technologies under different returns to scale assumptions for a sample of K observations is found in Briec et al. [30]:   n T, = (x, y) ∈ Rm × R : (x, −y) ≥ δz (x , −y ), z ∈ , δ ∈  , k k k k + + k∈K

(8) where  ∈ {N C , C }. First, there is the activity vector (z) operating subject to a convexity (C) or nonconvexity (NC) constraint. Second, there is a scaling parameter (δ) allowing for a particular scaling of all K observations spanning the technology. This scaling parameter is smaller than or equal to 1 or larger than or equal to 1 under nonincreasing returns to scale (NIRS) and nondecreasing returns to scale (NDRS), respectively, fixed at unity under variable returns to scale (VRS), and free under constant returns to scale (CRS). Briec et al. [30, Proposition 1] prove the following result: Proposition 1 ( [30, p. 166]). The nonconvex technologies TNC , are the minimal n extrapolation technologies containing the data A = {(xk , yk ) : k ∈ K} ⊂ Rm + × R+ and satisfying the axioms T 1 to T 4. The same statement for basic FDH solely has earlier been developed in Färe and Li [55]: FDH can be seen as the closest inner approximation of the true, strongly disposable but possibly nonconvex technology. The advantages of this formulation (8) are twofold. First, it offers a coherent formulation of all basic technologies under the four basic returns to scale assumptions (T4) and under both convexity (T5) and nonconvexity. For example, under VRS (i.e., setting δ = 1) and no convexity (i.e., constraint (N C )), one obtains the classical FDH technology:

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  n , TNC ,V RS = (x, y) ∈ Rm × R : (x, −y) ≥ z (x , −y ), z ∈  k k k N C + +

(9)

k∈K

as formulated by Deprins et al. [49]. As another example, under VRS and convexity (i.e., constraint (C )), one retrieves the basic technology defined by Banker et al. [13] and Färe et al. [56]:1   n . TC ,V RS = (x, y) ∈ Rm × R : (x, −y) ≥ z (x , −y ), z ∈  k k k C + +

(10)

k∈K

Second, its pedagogical advantage is that it neatly separates the role of the various assumptions in the formulation of technology. For instance, the restrictions on the scaling parameter (δ) relate directly to the basic definitions of the axioms on returns to scale (T4). Furthermore, the sum constraint on the activity vector z (i.e., constraint (C )) relates to the convexity axiom (T5). In this way, one can avoid confusing statements as found in the literature. For instance, the sum constraint on the activity vector z (i.e., constraint (C )) in the envelopment or primal formulation (10) is often called a “convexity constraint” under the VRS assumption, while the CRS technology has no such constraint in the formulation of Charnes et al. [39] though it also maintains the convexity axiom (see, e.g., Cook and Seiford [44, p. 2–3]). To compute the radial input efficiency measure (2) relative to convex technologies in (8) requires solving a nonlinear programming problem (NLP) for each evaluated observation. As shown in Briec and Kerstens [28, Lemma 2.1], this NLP can be transformed into the familiar linear programming (LP) problems that are known from the literature by substituting wk = δzk . For the nonconvex technologies in (8), the radial input efficiency measure (2) requires computing a nonlinear binary mixed integer program (NLBMIP): see Briec et al. [30, p. 166]. In fact, to reduce the computational complexity of this NLBMIP problem, three distinctive alternative solution methods have been proposed in the literature. First, Podinovksi [99] reformulates all these nonconvex technologies as binary mixed integer programs (BMIP) using a big M technique. Second, starting from an existing LP model for the basic FDH model (9) (see Agrell and Tind [5]), Leleu [85] formulates for all these nonconvex technologies equivalent LP problems. Third, Briec et al. [30] develop for all nonconvex technologies an implicit enumeration strategy to obtain closed form solutions for the radial input efficiency measure (2):2 Proposition 2. Let EN C, denote the radial input efficiency measure defined with respect to technologies TNC , . For all (x, y) ∈ TNC , and k = 1, · · · , K, let us 1 Note

that the convex VRS and NDRS technologies do not satisfy inaction. that the use of enumeration for the basic nonconvex FDH production model (9) has been around in the literature for quite a while: examples include [49, 63, 122], among others.

2 Note

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denote: xki i∈I (x) xi

αk (x) = max

and

yj , j ∈J (yk ) ykj

βk (y) = max

n where for all (x, y) ∈ Rm + × R+ , I (x) = {i ∈ {1, . . . , m} : xi > 0} and J (y) = {j ∈ {1, . . . , n} : yj > 0}. We have, for all (x, y) ∈ TNC , :

EN C, (x, y) =

⎧ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎩

 = V RS ;

min

αk (x)

if

min

βk (y) · αk (x)

if  ∈ {CRS , N I RS };

min

{max {βk (y), 1} · αk (x)} if

(xk ,yk )∈B (x,y) (xk ,yk )∈B (x,y) (xk ,yk )∈B (x,y)

 = N DRS .

with B (x, y) = {(xk , yk ) : δxk ≤ x, δyk ≥ y, δ ∈ }. Briec and Kerstens [28, p. 148–149] refine this analysis and also offer closed form solutions for the output-oriented and graph-oriented efficiency measures. Furthermore, these authors indicate that the computational complexity of enumeration is advantageous compared to the BMIP or LP approaches. Indeed, the maximum (minimum) of a vector with n components can be calculated in the worst case in O(n) arithmetic operations. Thus, to enumerate on the data set with the number of firms K, the number of arithmetic operations is about O(LK(m + n)), where m and n represent the number input and output dimensions and L is a measure of data storage for a given precision. A standard linear program has a O(LK 3 ) polynomial time complexity linked to the number of observed firms K. Since K > m + n in general, the time complexity of enumeration is thus better than LP. In fact, Kerstens and Van de Woestyne [75] empirically document that implicit enumeration is by far the fastest solution strategy followed by BMIP and finally LP.3 Kerstens and Van de Woestyne [76] provide closed form solutions for the directional distance functions under alternative returns to scale assumptions. One can mention that in this nonconvex framework, one can also treat the discrete case by considering that the technology is a subset of Nm × Nn (instead of Rm × Rn ). However, the radial measure (2) involves an assumption of divisibility and is therefore unsuitable. In line with Andriamasy et al. [9], one can overcome this problem by using the directional distance function (see Chambers et al. [38]) and selecting a direction that is the unit vector of Nm × Nn . In principle, the appropriateness of the convexity axiom can be tested for any comparison between convex and nonconvex technologies imposing a similar returns to scale hypothesis. We can define tests for the convexity of technology as a simple ratio between the convex and nonconvex input efficiency measures. Thus, the ratio:

3 This

poor performance is related to the huge size of the LP formulation in Leleu [85].

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CT (x, y) = EC, (x, y)/EN C, (x, y)

(11)

determines a nonparametric local goodness-of-fit test for the convexity of technologies conditional on the scaling law  (see Briec et al. [30, p. 178]).

Economic Value Functions The nonconvex production models have been complemented by nonconvex cost functions with corresponding specific returns to scale assumptions in Briec et al. [30]. Turning to a dual representation of technology, recall that the cost function C : Rn+ × Rm + −→ R+ ∪ {∞} defines the minimum costs to produce an output vector y given a vector of semi-positive input prices (w ∈ Rm + ): C(y, w) = inf {w · x : x ∈ L(y)} .

(12)

Briec et al. [30, p. 175–176] establish a local duality result between the nonconvex cost functions and the nonconvex FDH and its extensions. The computation of the cost function (12) relative to convex nonparametric technologies TC, again requires an NLP to be solved for each evaluated observation. As above, this NLP can be transformed into the familiar LP problem that is known from the literature (e.g., Hackman [68]). The cost function (12) relative to the nonconvex technology TN C, involves computing a NLBMIP as mentioned above. Again, to reduce the computational complexity of this NLBMIP problem, three distinctive solution methods can be pursued. First, following the Podinovksi [99] approach, one can transform these nonconvex cost functions to BMIPs. Second, Leleu [85] formulates for all these nonconvex cost functions equivalent LP problems. Third, Briec et al. [30] develop for all nonconvex cost functions an implicit enumeration strategy yielding closed form solutions. For all y ∈ Rn+ , let us denote:   V (y, xk , yk ) = x ∈ Rm + ; (x, y) ∈ N (xk , yk )

(13)

By construction, we have: 



CN C, (y, w) = min w · x : x ∈

 V (y, xk , yk ) .

(14)

k∈K (k)

By defining CN C, (y, w) = min{w · x : x ∈ V (y, xk , yk )}, we obtain: (k)

CN C, (y, w) = min CN C, (y, w). k∈K

(15)

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Interestingly the above properties can be derived from the standard background of convex analysis (see Clarke [43] and Rockafeller and Wets [105] for references).4 Given a closed subset D of Rd , let δD : Rd −→ R ∪{−∞} be the indicator function defined as:  0 if x ∈ D δD (z) = (16) −∞ if x ∈ /D One can then show that:  (w), inf{w.z : z ∈ D} = inf{w.z − δD (z) : z ∈ Rd } = δD

(17)

 (w) stands for the conjugate of δ . Suppose moreover that for all k ∈ K, where δD D  Dk is a closed subset of Rd and that D = k∈K Dk .

(w) = inf{w.z − δk∈K Dk (z) : z ∈ Rd } (18)    = inf{w.z − max δDk (z) : z ∈ Rd } = inf min w.z − δDk (z) : z ∈ Rd

  δD (w) = δ

k∈K Dk

k∈K

k∈K

(19)  (w). = min inf{w.z − δDk (z) : z ∈ Rd } = min δD k k∈K

(20)

k∈K

Along this line we obtain for all k ∈ K: (k)

CN C, (y, w) = δV  (y,xk ,yk ) (w)

and

CN C, (y, w) = min δV  (y,xk ,yk ) (w). k∈K

(21) Notice that a similar method applies for efficiency analysis. The next result is then derived. Proposition 3. Let CNC, (y, w) denote the cost function with respect to technologies TNC , . For all (y, w) ∈ Rn+ × Rm + , we have:

CNC, (y, w) =

⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨

min {w · xk : yk ≥ y}

if  = V RS ;

min {βk (y)w · xk }

if  = CRS ;

k∈K

k∈K

min {βk (y)w · xk } if  = N I RS ; ⎪ ⎪ ⎪ {k:βk (y)≤1} ⎪   ⎪ ⎪ ⎩min max {βk (y), 1} w · xk if  = N DRS ; k∈K

4 This

point was suggested to the authors by R. Chambers.

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where J (y) = {j : yj > 0} and βk (y) = maxj ∈J (yk ) Proposition 2.

yj ykj

are defined as in

Remark that Ray [104, Section 10.2] shows that the basic FDH cost function yields the same result as the Weak Axiom of Cost Minimization (WACM) as defined by Varian [123]. This is intuitively obvious since WACM only imposes convexity of the input set, and thus this partial convexity yields the same cost function as the one not imposing convexity at all. Now, there is a property of the cost function in the outputs worthwhile spelling out. Some seminal contributors to axiomatic production theory state that the cost function is nondecreasing and convex (nonconvex) in the outputs when convexity of technology is assumed (rejected) (e.g., Färe [54, p. 87], Jacobsen [70, p. 765], Shephard [116, p. 227], or Shephard [117, p. 15]). A central result established in Briec et al. [30] is that cost functions based on convex technologies are always smaller or equal to cost functions based on nonconvex technologies. Proposition 4 ( [30, p. 171]). The convex and nonconvex cost functions CC, and CN C, , respectively, satisfy the following properties: (a) For all (y, w) ∈ Rn+ × Rm + , CC, (y, w) ≤ CN C, (y, w). (b) In the single output case, if  = CRS , then CC, (y, w) = CN C, (y, w). Both cost functions are only equal in the case of CRS and a single output. Proposition 4 can be conceived as a more detailed result spelling out the precise impact of convexity on the above property of cost functions in the outputs. Obviously, these results can also be transposed to other economic value functions. Revenue functions based upon convex technologies are higher than or equal to revenue functions based upon nonconvex technologies. Only in the single input and CRS case, both these revenue functions coincide. For the long-run profit function, by contrast, the use of convex technologies or nonconvex technologies is logically indistinguishable. However, for any other restricted profit function, one obtains the result that profit is higher or equal when tangent to a convex instead of a nonconvex technology. Also the appropriateness of the convexity axiom can be tested by comparing convex and nonconvex value functions imposing a similar returns to scale hypothesis. A simple test of the convexity of, e.g., the cost function can be defined as a simple ratio between the convex and nonconvex cost functions. Thus, the ratio: CC (y, w) = CC, (y, w)/CN C, (y, w)

(22)

determines a nonparametric local goodness-of-fit test for the convexity of cost functions conditional on the scaling law  (see Briec et al. [30, p. 178]). Obviously,

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this convexity test in Definition 22 is similar in structure to the test earlier developed in Definition 11.

Efficiency Decompositions and the Testing of Convexity: A Priori Relations While Farrell [61] provided the first measurement scheme for the evaluation of Technical and Allocative Efficiency in a frontier context, Färe et al. [57] and Seitz [115] both offer alternative extended efficiency taxonomies. Because it is in our opinion the most widespreadly used, we stick in this contribution to the conceptual framework developed in Färe et al. [57, pp. 3–5]. The radial efficiency measure (2) used relative to different technologies entails the different concepts in this efficiency taxonomy of Färe et al. [57]. By conditioning the notation of the radial efficiency measure (2) on, e.g., a particular returns to scale hypothesis, it is straightforward to provide a formal characterization of all efficiency notions in the following definition (see, e.g., Briec et al. [30, p. 179]). The following input-oriented efficiency notions are identified: (a) (b) (c) (d) (e)

Technical Efficiency T E (x, y) = E,VRS (x, y). Overall Technical Efficiency OT E (x, y) = E,CRS (x, y). Scale Efficiency SCE (x, y) = E,CRS (x, y)/E,V RS (x, y). Overall Efficiency OE (x, y, w) = C,CRS (y, w)/(w · x). Allocative Efficiency AE (x, y, w) = OE (x, y, w)/OT E (x, y).

While Technical Efficiency (T E (x, y)) requires production on the boundary of the VRS technology, Overall Technical Efficiency (OT E (x, y)) necessitates that production is situated on the boundary of the CRS technology. Scale Efficiency (SCE (x, y)) reflects a social goal and is measured by the ratio between the actual (VRS) and ideal (CRS) technological configurations. Overall Efficiency (OE (x, y, w)) requires computing a cost function relative to a CRS technology (C,CRS (y, w)) and taking the ratio between minimal and observed costs (w · x). Allocative Efficiency (AE (x, y, w)) is a residual term computed by the ratio of OT E (x, y) and OT E (x, y).5 Since E,CRS (x, y) ≤ E,V RS (x, y), evidently 0 < SCE (x, y) ≤ 1. The embeddedness of technologies in terms of returns to scale assumptions determines the relations between these efficiency measures. These static efficiency concepts are mutually exclusive, and their radial measurement yields a multiplicative decomposition: 5 This

decomposition ignores structural efficiency or congestion. Recently, an attempt was made to develop new methods to measure strong forms of hypercongestion for convex and nonconvex technologies alike in Briec et al. [31]. This new methodology is empirically illustrated in Briec et al. [32]. Abad and Briec [1] transpose this methodology toward the modeling of bad outputs using a by-production framework.

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OE (x, y, w) = AE (x, y, w) · OT E (x, y)

(23)

where OT E (x, y) = T E (x, y) · SCE (x, y). To develop tests for convexity, we clarify the relationship between convex and nonconvex decompositions: n Proposition 5 ( [30, p. 180]). For all (x, y) ∈ Rm + × R+ , the relations between convex and nonconvex decomposition components are: (a) OT EC (x, y) ≤ OT EN C (x, y); (b) T EC (x, y)≤T EN C (x, y); (c) OEC (x, y, w)≤OEN C (x, y, w).

Thus, while three out of the five above efficiency notions can be ordered with respect to the impact of convexity, there is no a priori ordering possible for the nonconvex and convex scale (SCE (x, y)) and Allocative (AE (x, y, w)) Efficiency components. Though the underlying efficiency measures can be ordered, it is not possible to order the ratios between these efficiency measures. Nonparametric goodness-of-fit tests for the convexity of the efficiency components based upon constant returns to scale technologies and cost functions, respectively, are provided by the following ratios (see Briec et al. [30, p. 181]): CRT E(x, y) = OT EC (x, y)/OT EN C (x, y)

(24)

CRCE( x, y, w) = OEC (x, y, w)/OEN C (x, y, w).

(25)

and

Several methods have been proposed in the literature to obtain qualitative information regarding global returns to scale (e.g., see Seiford and Zhu [114]). Since these methods are not suitable for nonconvex technologies, Kerstens and Vanden Eeckaut [73, Proposition 2] generalize an existing goodness-of-fit method to suit all technologies. Including a fourth returns to scale case only relevant for nonconvex technologies (see Podinovksi [98]), the following proposition summarizes this method. Proposition 6 ([35, p. 579]). Conditional on the optimal efficient point, technology T,V RS is globally characterized by: (a) (b) (c) (d)

CRS : E,N I RS (x, y) = E, N DRS (x, y) = E,V RS (x, y); IRS : E,N I RS (x, y) < E,N DRS (x, y) ≤ E,V RS (x, y); DRS : E,N DRS (x, y) < E,N I RS (x, y) ≤ E,V RS (x, y); SCRS : E,N I RS (x, y) = E,N DRS (x, y) < E,V RS (x, y);

where IRS, DRS, and SCRS stand for increasing, decreasing, and sub-constant returns to scale, respectively.

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Table 1 Nonconvex and convex cost estimates: a selection Article Balaguer-coll et al. [11] Briec et al. [30] Cummins and Zi [45] De Borger & Kerstens [47] Grifell-Tatjé & Kerstens [67] Viton [124]

Ratio CC (y, w) (in %) 58.87 97.76 50.55 77.59 90.85 79.82 87.64 92.77

Remarks CRS

Actual Ideal 1 Output 4 Outputs

Essentially, these CRS, NIRS, and NDRS technologies are auxiliary to determine the position of an observation relative to the true flexible (i.e., VRS) returns to scale technology. Recently, Mostafaee and Soleimani-Damaneh [92] propose a more elaborated taxonomy of global returns to scale characterizations for nonconvex technologies based on results of Mostafaee and Soleimani-Damaneh [91].

Empirical Evidence on FDH and Its Extensions: The Impact of Convexity This subsection focuses on the key question: does nonconvexity matter in empirical applications when compared to traditional convex analysis? We provide some evidence for a selection of four economic topics: (i) cost functions, (ii) efficiency decompositions, (iii) productivity growth, and (iv) capacity utilization.

Cost Function Results In Table 1 we list a small selection of studies that report the results of convex and nonconvex frontier cost estimates. The first column lists the authors of the article, the second column reports the ratio CC (y, w) as defined in Definition 22, and the third column eventually provides a remark.6 The Balaguer-Coll et al. [11] study on Spanish municipalities reveals that convex costs are only 58.87% of nonconvex costs at the sample average. Analyzing the US life insurance industry, Cummins and Zi [45] even report 50.55% on average for CC (y, w): this means that convex cost is about half of the nonconvex costs. The De Borger and Kerstens [47] analysis of Belgian municipalities shows that convex costs are only 77.59% of convex costs. In a study of Spanish electricity distribution,

6 In

case the study does not report cost estimates but rather overall efficiency ratios, one can obtain CC (y, w) = CC, (y, w)/CN C, (y, w) by taking the ratio of the corresponding overall efficiency ratios OEC (x, y, w)/OEN C (x, y, w). The observed cost in each of the denominators of OE (x, y, w) cancels out.

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Grifell-Tatjé and Kerstens [67] report a ratio of 90.85% when using data from the actual network and of 79.82% when using data from an ideal engineering network. The Briec et al. [30] study lists a ratio of 97.76%, but this study imposes CRS and therefore meets one of the two conditions for equality (see Proposition 4). The Viton [124] article is a bit a special case in that the author compares WACM and traditional convex cost estimates: since WACM coincides with a nonconvex estimate, this amounts to an implicit test of convexity. He reports a ratio of 87.64% under a single output specification (meeting again one of the two conditions for equality, Proposition 4) and a ratio of 92.77% under a multiple output specification. In conclusion, it is undeniable that convexity has an important to huge impact on cost estimates and hence on Overall Efficiency.

Efficiency Decomposition From the efficiency decomposition discussed in section “Efficiency Decompositions and the Testing of Convexity: A Priori Relations,” the overall efficiency component has already been discussed in section “Cost Function Results.” Therefore, we focus on technical efficiency components in this part. As established in Proposition 5, T EC (x, y) ≤ T EN C (x, y). There is an abundance of studies reporting efficiency measures computed relative to basic convex (10) and nonconvex (9) technologies. We focus on just a few examples. For instance, Stroobants and Bouckaert [120] compare libraries in the Flemish region and report substantial differences between convex and nonconvex results for three specifications (though no statistical tests are reported). As another example, Mayston [90] evaluates UK economics departments and finds substantial differences at the sample level (though again no statistical tests are reported). Cesaroni et al. [35, p. 582–583] report on the decomposition OT E (x, y) = T E (x, y) · SCE (x, y) for five secondary data sets. These authors find that convex and nonconvex OT E (x, y) is only significantly different for two data sets, while convex and nonconvex SCE (x, y) happens to be significantly different for all data sets and convex and nonconvex T E (x, y) for most data sets. The same authors also focus on conflicting cases in returns to scale determination using Proposition 6: e.g., switches from increasing returns to scale (IRS) to decreasing returns to scale (DRS), from CRS to IRS, and from CRS to DRS. While one data set has no conflicting cases, four data sets find conflicting cases ranging between 6.98% and 39.02% of observations. Finally, these authors explore the markedly different patterns of ray average productivity curves under convex and nonconvex technologies. Chavas and Kim [42, p. 69–70] report on convex and nonconvex T E (x, y) and SCE (x, y): while no statistical tests are reported, the descriptive statistics seem to be markedly different. Cesaroni and Giovannola [34, p. 128–129] establish results for alternative convex and nonconvex cost-based efficiency components similar to the above: though no statistical tests are mentioned, the descriptive statistics are clearly different beyond doubt.

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Productivity Growth Kerstens and Van de Woestyne [74] report empirical results for the immensely popular Malmquist productivity index (e.g., Färe et al. [60]) as well as for the Hicks-Moorsteen Total Factor Productivity (TFP) index (defined by Bjurek [17]) under various specifications of technology. For both indices, it turns out that convex and nonconvex results for both CRS and VRS yield different descriptive statistics, though no formal tests are provided regarding the statistical significance of these differences. Kerstens and Managi [72] focus on the Luenberger productivity indicator which is defined in terms of the differences between directional distance functions (see [37]) using basic convex (10) and nonconvex (9) technologies. Analyzing a huge data set of petroleum wells, their findings can be summarized as follows. First, productivity change is on average smaller under nonconvexity, and the resulting distributions are significantly different. Second, substantially more observations tend to push the frontier outward under nonconvexity and are thus involved in creating technological change. Third, both β-convergence and σ -convergence are being tested for and happen to occur only under nonconvexity, not under the traditional convexity axiom. In a follow-up study of Chinese banks, Barros et al. [15] also find that the Luenberger productivity change is on average smaller under nonconvexity. Testing differences in productivity with respect to scale and ownership does not yield different patterns according to convexity. Finally, Ang and Kerstens [10] study productivity of US agriculture at the state level using the Luenberger-Hicks-Moorsteen TFP indicator (introduced by Briec and Kerstens [27]) again using basic convex (10) and nonconvex (9) technologies. These authors report a higher TFP change under nonconvexity, and the resulting distributions are significantly different.

Capacity Utilization Johansen [71] introduces the notion of plant capacity as the maximum output vector that can be produced with existing equipment with unrestricted variable inputs per unit of time. Färe et al. [59] transpose this notion into a multi-output frontier framework by using a combination of two output-oriented efficiency measures: one relative to a technology including the variable inputs and another one excluding the variable inputs. Walden and Tomberlin [125] report average output-oriented plant capacity estimates that vary between 52% and 84% in the cases of a basic convex (10) and a basic nonconvex (9) technology, respectively. Kerstens et al. [79] argue that the output-oriented plant capacity utilization is unrealistic when the amounts of variable inputs needed to reach the maximum capacity outputs are not available. This is related to the attainability issue already noted by Johansen [71]. These authors illustrate empirically that the scaling of variable inputs is less implausible for nonconvex compared to traditional convex technologies. Cesaroni et al. [36] define an alternative input-oriented plant capacity notion by using a combination of two sub-vector input-oriented efficiency measures only

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aimed at reducing the variable inputs: one relative to a standard technology and one relative to a technology with the minimum output level per dimension among all observed units. While these authors report average output-oriented plant capacity estimates that are 92% and 89% for the convex (10) and nonconvex (9) technologies, respectively, these apparent small differences nevertheless represent distributions that turn out to be statistically significantly different. For the average input-oriented plant capacity estimates, they report numbers of 120% and 121% for the convex (10) and nonconvex (9) technologies, respectively: again these apparent small differences reflect distributions that are statistically significantly different. It goes without saying that such differences may well have potentially huge implications in the design of policies to combat overcapacity in fisheries. Kerstens et al. [77] report results from a short-run Johansen sector model allowing for the reallocation of production between firms that is developed in two steps. In the first step, output-oriented plant capacity estimates are computed. In the second step, the industry model minimizes the industry use of fixed inputs in a radial way such that total production is maintained at the current total level by reallocating production among firm capacities. From the 398 vessels in the fleet, the convex plant capacity estimates lead to maintain only 330 vessels, while the nonconvex estimates maintain 357 vessels. Thus, the required decommissioning effort resulting from the short-run Johansen sector model is larger under convexity. Kerstens et al. [78] aim to compare empirically technical and economic capacity notions on both convex and nonconvex technologies. After defining these capacity notions, an empirical comparison is performed using a secondary data set containing data of French fruit producers. Two key empirical conclusions are that all these different capacity notions follow different distributions and also that these distributions almost always differ under convex and nonconvex technologies.

FDH and Its Extensions: Further Methodological Refinements One can mention a whole series of methodological refinements and variations that have been introduced in the literature related to methods initially developed in a convex setting. First, traditional radial efficiency measures in FDH models yield potentially huge amounts of slacks and surpluses since the efficient subset is limited to the corner points; nonradial input-, output-, and graph-oriented efficiency measures have been evaluated and found particularly relevant in the basic FDH model by De Borger et al. [48]. Portela et al. [101] focus on some alternative graph-oriented (or nonoriented) efficiency measures in the same context. Following up on Ebrahimnejad et al. [55] Fukuyama et al. [64] develop least-distance efficiency measures for FDH technologies that satisfy a strong monotonicity property. Second, in the spirit of Bouhnik et al. [22] who proposed lower bound restrictions on the intensity variables to avoid unreasonable optimal activity vectors in a convex setting, Mairesse and Vanden Eeckaut [89] develop for these nonconvex production models lower and upper bound restrictions to the scaling of observations.

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Third, several types of extreme points (including anchor points) can be distinguished in FDH (see Soleimani-damaneh and Mostafaee [119]). Fourth, Soleimanidamaneh [118] develops a dynamic FDH production model that can be recursively solved by means of simple enumeration. Fifth, Tavakoli and Mostafaee [121] are the first to develop a network structure production model that opens up the black box of production via parallel and sequential production processes in a nonconvex world. These authors obtain closed form solutions for the basic efficiency measures under FDH and its extensions. Sixth, there is some work on the construction of three-dimensional sections of the efficient frontier for nonconvex models via enumeration methods as developed supra (see Krivonozhko and Lychev [80–83], Krivonozhko et al. [84]). Finally, Tulkens [122] was the first to propose a Free Replicability Hull (FRH) by allowing for integer replications of all observations, eventually complemented by upper bounds on the integer replication process. It turns out that this FRH is computationally quite challenging (see Ehrgott and Tind [52]). In a similar vein, Green and Cook [66] define a nonconvex technology containing all observations as well as all composite observations obtained by simple aggregation. This Free Coordination Hull (FCH) can eventually also be complemented by an upper bound on the number of observations being aggregated. Thus, most of the analysis that has been developed for convex technologies can somehow be transposed to FDH and its extensions. This simply illustrates that this rich body of analytical results is not necessarily jeopardized when opting for nonconvex technologies.

Mitigating Convexity: A Selection It should be clear by now that if one drops the convexity axiom altogether, then FDH and its extensions are the straightforward technological and economic value function choices to consider. However, some people have sought to mitigate the impact of convexity in a variety of ways. This section offers a selection of approaches defining some alternative to the traditional convexity axiom and somehow avoiding FDH and its extensions.

Partial Convexity Several authors have attempted to relax the convexity axiom somewhat. Petersen [97] initiated a small literature aimed at maintaining convexity in input space and in output space solely, but not in the graph of technology. The implementation of this relaxed set of assumptions is corrected by Bogetoft [18] with restrictions on the dimensionality of the production technology. Bogetoft et al. [19] relax these restrictions on the dimensionality of the input and output spaces, while Post [102] improves upon the latter article by proposing a procedure that avoids computational problems in large-scale applications.

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This relaxed assumption is justified by appeal to, for instance, the law of diminishing marginal rates of substitution in the input space or to the idea of diminishing marginal rates of transformation in the output space. However, it is not clear how time divisibility can be applied in the context of this partial convexity notion. Furthermore, one may question whether there really is, for instance, a law of diminishing marginal rates of substitution in the input space. For example, Brokken [33] summarizes three studies revealing that there are increasing marginal rates of substitution of grain for roughage in beef production. Therefore, the law of diminishing marginal rates of substitution is questionable. Podinovski [100] introduces the idea of partial convexity between certain subsets of inputs and subsets of outputs and derives BMIP for the traditional efficiency measures. Leleu [86] proposes new LP formulations combining aspects of convex and nonconvex production models across dimensions for all returns to scale assumptions and for the directional distance efficiency measure. While Podinovski [100, p. 555–556] justifies his partial convexity approach by appealing to divisibility arguments pertaining to specific inputs and/or outputs, one may wonder whether time divisibility is by definition related to the whole production process and that setup times and indivisibilities destroy convexity altogether rather than only in some subset of dimensions. Finally, Chavas and Kim [42] adopt a different strategy to combine convex and nonconvex models by defining the technology as a union of neighborhoodbased local representation of the technology each of which is convex. Obviously, the union of convex technologies needs not be convex. By choosing very small or very large neighborhoods, the technology as a union of neighborhood-based local representations of the technology converges to the nonconvex technology (9) or the convex technology (10), respectively. An obvious problem of the whole approach is the neighborhood choice and its impact on productivity and efficiency analysis.

Regular Ultra Passum Law Olesen and Petersen [94] intend to make convex models (10) suitable to estimate optimal scale size by augmenting these with two additional maintained hypotheses which imply that the frontier is consistent with smooth curves along rays in input and in output space that obey the Regular Ultra Passum (RUP) law (i.e., monotonically decreasing scale elasticities). This RUP law implies that the production frontier must be S-shaped along any expansion path in input space. Obviously, such technologies are nonconvex in input-output space. Olesen and Petersen [94] focus on the multiple inputs single output case. Olesen and Ruggiero [95] continue from there and focus on production technologies that are input homothetic. This allows to maintain convexity in input and in output space but to allow for nonconvexities in input-output space. This homotheticity assumption mainly serves to simplify the estimation procedure. Also this presentation assumes only one output.

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In a sense, imposing the RUP law in this context again focuses on allowing for nonconvexities in input-output space, just as in section “Partial Convexity.” Therefore, the same reservations prevail. Furthermore, there are long-standing misgivings on the use of homothetic structures in production theory as in Olesen and Ruggiero [95]. Already Samuelson and Swamy [107, p. 592] conclude: “Empirical experience is abundant that the Santa Claus hypothesis of homotheticity in tastes and in technical change is quite unrealistic.”

From Generalized Convexity to Nonconvexity We now focus on a modification of the CES − CET model introduced by Färe et al. [58] that is a generalization of the traditional convex approach (10). This CES − CET model has two parts: the output part is characterized by a Constant Elasticity of Transformation specification, and the input part is characterized by a Constant Elasticity of Substitution specification. Consider a generic map φr : Rd+ → Rd+ defined as φr (z) = (z1r , . . . , zdr ). For all r > 0, this function is an isomorphism 1/r 1/r from Rd+ to itself, and its reciprocal is defined on Rd+ as φr−1 (z) = (z1 , . . . , zd ). Given a subset B = {uk : k ∈ K}k∈K of Rd+ , from Ben-Tal [16], one can define its φr -generalized convex hull as:     zk φr (uk ) : zk = 1, zk ≥ 0 . Coφr (B) = φr −1 k∈K

(26)

k∈K

Notice that this set is not convex in the “usual” case which corresponds to the case where r = 1. The CES − CET model can then be defined as the set:    n −1 TC,γ ,δ = (x, y) ∈ Rm zk φγ (xk ) , + × R + : x ≥ φγ k∈K

y≤

φδ−1



(27)

  zk φδ (yk ) , zk = 1, zk ≥ 0 ,

k∈K

k∈K

where γ and δ > 0. Paralleling Banker et al. [13], this construction is derived from the notion of generalized convex hull defined in (26). For such a class of models, the radial efficiency measure (2) can be computed making some obvious linear transformations. Notice that Ravelojaona [103] has proposed a nonlinear version of the directional distance function (see Chambers et al. [38]) that can also be computed by linear programming methods. Boussemart et al. [23, p. 334] state that a production technology T is said to be homogeneous of degree α if for all λ > 0: (x, y) ∈ T ⇒ (λx, λα y) ∈ T .

(28)

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This technology has also been termed “almost homogeneous technology of degree 1 and α.” This degree of homogeneity of the technology has direct implications for the nature of returns to scale. Proposition 7 ( [23, p. 334]). Assume that the production technology T satisfies T1–T4. Moreover, suppose that T is homogeneous of degree α. (a) If α > 1, then T satisfies strictly increasing returns to scale; (b) if 0 < α < 1, then T satisfies strictly decreasing returns to scale. Thus, these homogeneous technologies exhibit either strictly increasing or strictly decreasing returns to scale according to their degree of homogeneity. Therefore, one can say that if the technology is homogeneous of degree α, then it satisfies α-returns to scale. Obviously, strictly increasing returns to scale imply nonconvexity of technology. Boussemart et al. [23] propose to relax the definition proposed in Färe et al. [58] by considering the following production model:  alpha n TC,γ ,δ = (x, y) ∈ Rm + × R+ :

x ≥ φγ−1

 k∈K

y ≤ φδ−1



 zk φγ (xk ) ,

(29)

 zk φδ (yk ) , zk ≥ 0 .

k∈K

alpha

where γ and δ > 0. TC,γ ,δ satisfies an α-returns to scale assumption with α = γ δ . This technology differs from the one proposed by Färe et al. [58] because it suppresses the constraint k∈K zk = 1. While their model is not compatible with an α-returns to scale assumption, model (29) satisfies axioms (T1)–(T4) and satisfies α-returns to scale under a suitable specification of α. alpha

Proposition 8 ( [23, p. 336]). The production technology TC,γ ,δ defined in (27) satisfies: (a) strictly increasing returns to scale if and only if γ /δ > 1; (b) strictly decreasing returns to scale if and only if γ /δ < 1; (c) constant returns to scale if and only if γ /δ = 1; Furthermore, this notion of α-returns to scale has also been extended to FDH and its extensions (see Boussemart et al. [23, p. 336]). In empirical applications, γ and δ are a priori parameters: optimal parameter values can be determined by applying a goodness-of-fit method. This can be done using a grid search method. For example, Leleu et al. [87] analyze four types of intensive care units and find overwhelming evidence of increasing returns to scale, but at the hospital level most institutions operate under decreasing returns to scale.

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More recently, Boussemart et al. [24] attempt to endogenize γ and δ using global optimization tools. They propose a tractable procedure to find an optimal value of α under a generalized FDH technology. This approach fully endogenizes α and estimate its value by linear programming. For each firm k ∈ K, we consider an individual technology defined by:  n 1/γ Qγ ,δ (xk , yk ) = (x, y) ∈ Rm xk , y ≤ λ1/δ yk , λ ≥ 0 . + × R+ : x ≥ λ

(30)

The global technology is then the union of individual technologies as follows: TN C,γ ,δ =



Qγ ,δ (xk , yk ).

(31)

k∈K

For all k, j ∈ K, let us denote: (k)

Eγ ,δ (xj , yj ) = min{θ : (θ xj , yj ) ∈ Qγ ,δ (xk , yk )}.

(32)

By definition, one has Eγ(k) ,δ (xk , yk ) = 1. From Boussemart et al. [24], one can show that:  δ/γ   (k) Eγ ,δ (xj , yj ) = βk (yj ) . αk (xj )

(33)

where for all k, αk (xj ) and βk (xj ) as in Proposition 2. Notice that this result generalizes the one defined in the VRS case. It follows that: EN C,γ ,δ (xj , yj ) = min{θ : (θ xj , yj ) ∈ TN C,γ ,δ }  δ/γ   = min βk (yj ) . αk (xj ) . k∈K

(34) (35)

By defining α = γ /δ, using the fact that any efficiency score is obtained in closed form, one can then find α  which maximizes the quantity M defining an index of goodness of fit as: M(A; α) =

 k∈K

EN C,γ ,δ (xj , yj ) =

 k∈K

min k∈K

 1/α   βk (yj ) . αk (xj )

(36)

subject to the constraint that (xj , yj ) ∈ TN C,γ ,δ for all j ∈ K. Taking the logarithm it is then easy to convert this optimization problem to a linear program. An empirical application is proposed in Boussemart et al. [24]. In the same vein, based on Charnes et al. [40], we now consider the piecewise Cobb-Douglas (CD) model. Let us define the map φ0 : Rd++ −→ Rd++ defined as φ0 (u) = (ln(u1 ), . . . , ln(ud )) . This function is a bijective function from Rd++ to

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itself, and its reciprocal is defined on Rd++ by φ0−1 (u) = (exp(u1 ), . . . , exp(ud )) . This piecewise Cobb-Douglas model can be written as:   λ  λ  m+n :x≥ xk k , y ≤ yk  , λk = 1, λ ≥ 0 . TCD = (x, y) ∈ R++ k∈K

k∈K

k∈K

This model is a generalized convex model derived from the notion of generalized convexity analyzed by Ben-Tal [16]. A general taxonomy is provided in the next subsection.

Semilattice Structures In mathematics, a partially ordered set S for which every two elements have a supremum contained in S is called an upper-semilattice. Hence for some dimension d ∈ N, the partial order defined by u ≤ w if ui ≤ wi for all i ∈ {1, . . . , d}, with u, w ∈ Rd+ , realizes upper-semilattice structures in Rd+ . The supremum of u and w is determined by u ∨ w = (max(u1 , w1 ), . . . , max(ud , wd )). Note that the operator ∨ can be seen as taking the component-wise maximum. Following Briec and Horvath [25], a subset L ⊂ Rd+ is said to be a Bconvex set, if ∀u, w ∈ L, ∀t ∈ [0, 1] : u ∨ tw ∈ L. Obviously, B-convex subsets determine a special class of upper-semilattice structures in Rd+ of which the mathematical properties are analyzed in detail in Briec and Horvath [25]. Briec and Horvath [26] impose B-convexity on technologies in production economics as a substitute for convexity (and nonconvexity in the sense of FDH) and study general properties of these technologies and related cost functions. Starting from the set of n K observations A = {(x1 , y1 ), . . . , (xK , yK )} ⊂ Rm + × R+ , the following B-convex nonparametric technology is defined:  Tmax = (x, y) ∈

Rm +

× Rn+

:x≥



zk xk , y ≤

k∈K



zk yk ,

k∈K



 zk = 1, zk ≥ 0 ,

k∈K

(37) with the notation  k∈K





uk = max(uk1 ), . . . , max(ukd ) ∈ Rd+ , k∈K

k∈K

for uk = (uk1 , . . . , ukd ) ∈ Rd+ , (k ∈ K), expanding the operator ∨ to multiple vectors. Notice the structural similarity with (10) by replacing summation with component-wise maximum. Dual to the notion of an upper-semilattice, a lower-semilattice is defined as a partially ordered set S for which every two elements have an infimum contained

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in S. Applied to Rd+ , this infimum of u, w ∈ Rd+ is determined by u ∧ w = (min(u1 , w1 ), . . . , min(ud , wd )). Obviously, the operator ∧ takes the componentwise minimum of both vectors. Using this dual notion, Adilov and Yesilce [3] define a subset L ⊂ Rd+ ∪ {+∞}d to be inverse B-convex if ∀u, w ∈ L, ∀t ∈ [1, +∞] : u ∧ tw ∈ L, and study its properties. By analogy with the B-convex case, Briec and Liang [29] define the following inverse B-convex nonparametric technology:  Tmin = (x, y) ∈

Rm +

× Rn+

:x≥



zk xk , y ≤

k∈K



zk yk ,

k∈K



 zk = 1, zk ≥ 0 ,

k∈K

(38) with the notation  k∈K

  uk = min(uk1 ), . . . , min(ukd ) ∈ Rd+ , k∈K

k∈K

for uk = (uk1 , . . . , ukd ) ∈ Rd+ , (k ∈ K). Compared with (10), summation is now replaced with component-wise minimum. This type of production technologies allows to take into account the situation where the inputs exhibit complementarity. In such a case, the structure of the input set is similar to that of the Leontief production function. Radial efficiency measurements can be computed with respect to both technologies Tmin and Tmax by using enumeration algorithms developed in Briec and Horvath [26] and Briec and Liang [29]. These new production models have recently been applied in, e.g., energy (Andriamasy et al. [7]), transportation (Barros et al. [14]), and the tourism industry (Goncalves et al. [65]). Coming back to the model proposed by Färe et al. [58] Andriamasy et al. [8] show that these production technologies are the Painlevé-Kuratowski lower [upper] limit of the sequence of production technologies TC,r,r that are derived from technology CES − CET (27) by setting γ = δ = r 7 : Limr−→∞ TC,r,r = Tmax .

(39)

m In addition id A ⊂ Rm ++ × R++

Limr−→−∞ TC,r,r = Tmin ,

7 The

(40)

Painlevé-Kuratowski lower [upper] limit (sometimes also called Peano limit) of the sequence of sets {En }n∈N is denoted Lin→∞ En [Lsn→∞ En ]. For a set of points p for which there exists a sequence {pn } of points such that pn ∈ En for all n and p = limn→∞ pn , a sequence {En }n∈N of subsets of Rm is said to converge, in the Painlevé-Kuratowski sense, to a set E if Lsn→∞ En = E = Lin→∞ En , in which case we write E = Limn→∞ En .

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and finally Limr−→0 TC,r,r = TCD .

(41)

Andriamasy et al. [9] consider a class of closely related nonparametric production models Max-Plus algebra. Let the semi-ring  built  on the so-called   us consider  Rmax = R ∪ −∞ , ⊕, ⊗ composed of the set R ∪ −∞ which is defined by the maximization operation as addition s ⊕ t := max (s, t) and the usual addition operation as multiplication s ⊗ t := s + t. −∞ and 0 are, respectively, the neutral element of the “addition” ⊕ and the “multiplication” ⊗. One can derive from this algebraic structure the following production model:   n T⊕ := (x, y) ∈ Rm (zk ⊗ x k ), + × R+ : x ≥ k∈K

y≤



k∈K

(42)

(zk ⊗ y k ), max zk = 0, z ∈ RK . k∈K

This model is called a Max-Plus nonparametric estimation of the production technology. The efficiency of firms can be meaningfully evaluated using the directional distance function introduced by Chambers et al. [38] for which some closed form has been provided in Andriamasy et al. [9]. Paralleling the standard technology TC,CRS , it is quite natural to define a graph translation homothetic Max-Plus nonparametric model of the technology. This is done by dropping the last constraint in equation (42). The following technology is Max-Plus convex and satisfies a graph translation homothetic (denoted th) assumption:    n T⊕th := (x, y) ∈ Rm (zk ⊗ x k ), y ≤ (zk ⊗ y k ), z ∈ RK . + × R+ : x ≥ k∈K

k∈K

(43) Notice that these types of algebraic structures have more recently been considered by Baldwin and Klemperer [12] to analyze discrete demand types and to prove the existence of an equilibrium with indivisibilities.

Preliminary Conclusions This selection is by definition incomplete and somewhat subjective. For instance, we ignore Hackman [68, p. 135] who introduces the notion of projective convexity. As another example, Kleine [80] offers a series of production models with general or individual bounds on activity levels potentially leading to nonconvexities. Our limited overview just offers a perspective on a non-negligible literature seeking alternatives to the convexity axiom.

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Conclusions Section “Technologies and Distance Functions: Basic Definitions” laid the foundations by providing basic definitions of the traditional axioms underlying technologies and their representation via distance functions. Section “Axiom of Convexity: Arguments” has focused on existing justifications for the axiom of convexity. Apart from duality reasons that often seem to be misunderstood, we have stressed the time divisibility argument and its weakness when indivisibilities also affect the time dimension (e.g., setup times). Furthermore, we have cited some evidence that decision-makers often have a hard time understanding the results from convex analysis and sometimes almost explicitly object to its use. Section “Nonparametric Nonconvex Technologies and Value Functions: Free Disposal Assumption and Minimum Extrapolation Principle” started by a discussion of the nonconvex FDH and its extensions and also their corresponding convex technologies. The focus was on computational problems related to the need to solve nonlinear binary mixed integer programs. Three solution strategies were discussed: (i) BMIP, (ii) LP, and (iii) an implicit enumeration strategy, whereby the latter turns out to be most efficient from a computational point of view. The ensuing discussion of nonconvex economic value functions also touched upon these computational problems and the same three solution strategies. Thereafter, the focus moved to some popular efficiency decomposition and the formulation of basic tests of convexity on the technology and on the cost function. After this methodological analysis, we switched to an empirical perspective on the use of FDH and its extensions grouped under four headings: (i) cost functions, (ii) efficiency decompositions, (iii) productivity growth, and (iv) capacity utilization. A final subsection discussed a series of methodological refinements of FDH and its extensions revealing that almost all refined analysis developed for convex technologies can somehow be transposed to FDH and its extensions. Section “Mitigating Convexity: A Selection” has offered a selective review of attempts to mitigate the impact of the convexity axiom while avoiding FDH and its extensions. We focused extensively on partial convexity, the imposition of Regular Ultra Passum laws, α-returns to scale, and semilattice structures. This review is nowhere complete and reflects our own interests and biases. An attempt to summarize the current state of affairs may be that the alternatives for traditional convex technologies have now been around for a decade or so. Empirical results reveal that convexity matters not only for the technology but also for economic value functions. The latter may surprise some, but it reveals that the issue of imposing convexity or not cannot be taken lightly. We consider attempts to mitigate convexity while steering away from FDH and its extensions not very successful at the moment. Therefore, unless we manage to renew the axiomatic foundations of production theory in a fundamental way, it may be hard to ignore using FDH and its extensions as well as its value functions and even harder to ignore its empirical results. An open question is to what extent existing empirical methodologies need to be re-examined to be able to cope with nonconvexities: given the local nature of some of the results, new standards may need to be established.

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This lack of standards to report nonconvex results as well the need to go beyond traditional convex optimization that is often considered a cornerstone for economic analysis may well contribute to its negligence.

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Contents Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Notation and Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Technology, Output, and Input Distance Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Regularity Conditions R.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Productivity Measurement: The Case of Single Output and Single Input . . . . . . . . . . . . . . . . Absolute Versus Relative Measures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Decomposition of Productivity Change: Single Input and Single Output Case . . . . . . . . . . Multiple Outputs and Inputs: The Index Number Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . What Are Index Numbers? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Measuring Quantity Aggregates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Measures of Output and Input Quantity Change as an Aggregate of Commodity-Specific Changes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Decomposition of Changes in Revenues and Costs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Index Numbers Based on Quantity Aggregates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Specification of Functional Form for the Output Aggregates . . . . . . . . . . . . . . . . . . . . . . . . Specification of Functional Forms for Input Aggregates . . . . . . . . . . . . . . . . . . . . . . . . . . . Index Number Approach to Measuring Quantity Change . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Direct Approach to Quantity Index Numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Indirect Measures of Quantity Change . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Direct Versus Indirect Measures of Quantity Change: Which One to Use? . . . . . . . . . . . . . Axiomatic Approach to Index Numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Notation for the Axiomatic Approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Axioms and Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Economic Theoretic Approach to Output and Input Quantity Index Numbers . . . . . . . . . . . . Notation and Basic Framework . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Economic-Theoretic Approaches to Measurement of Output Quantity Change . . . . . . . . . Direct Measures of Quantity Change in the Presence of Price Data . . . . . . . . . . . . . . . . . . Indirect Output Quantity Index Numbers Using Output Price Index Numbers . . . . . . . . . .

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Indirect Output Quantity Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Direct Quantity Index Based on Malmquist Distance Function . . . . . . . . . . . . . . . . . . . . . . Input Quantity Index Numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Special Topics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Use of Quantity Aggregates to Measure Quantity Change . . . . . . . . . . . . . . . . . . . . . . . . . . Transitivity and Quantity Index Numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

795 796 800 802 802 803 808 815 817

Abstract

The chapter provides an overview of the index number methods and approaches for measuring changes in output and input quantities as well as measuring changes in output and input prices. The problem of generalization to the case of multiple outputs and inputs is discussed. Two major alternative approaches to this problem form the core of this chapter. The first approach uses a framework where aggregates of output and input quantities are first computed and the resulting aggregates are used to measure changes in output and input quantities. Issues relating to the choice of functional form for the aggregates are discussed. The second approach is the standard index number approach where index number formulae are used to directly measure changes in output and input quantities without having to first measure quantity aggregates. Index number formulae used in direct and indirect measurement of quantity change are discussed. The axiomatic and economic theoretic approaches to index numbers along with their implications for the choice of an appropriate index number formula are described. The problem of multilateral comparisons, the transitivity requirement, implications of transitivity for the choice of appropriate formulae along with a few recommendations to the practitioner are presented in the last section. Keywords

Output and input aggregates · Index numbers · Axiomatic approach · Economic theoretic approach · Laspeyeres · Paasche · Fisher and T‚ornquist · Lowe and Young indices · Transitivity

Introduction Index number theory and practice are important tools used by researchers, analysts and policy makers at the national and international level. Most national statistical offices around the world compile measures of changes in consumer prices which are published in the form of consumer price index (CPI) on a monthly, quarterly and annual basis. Central banks in different countries use measures of core inflation for the purpose of setting interest rates in their macroeconomic policy making. Economists rely on regular updates on macroeconomic aggregates such as gross

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domestic product (GDP) at current and constant prices. Users are also familiar with indices like the Dow Jones Index measuring changes in stock market prices for 30 large companies listed on stock exchanges in the United States; and the human development index (HDI) published regularly in the United Nations’ Human Development Report. The HDI tracks performance of nations in three different dimensions: income, health as measured by life expectancy, and education. The objective of this chapter is to provide an overview of index number theory and practice as it relates to productivity measurement. Index numbers have a long history and tradition. William Fleetwood was one of the earliest (1707) to use the idea of index numbers and measures of price change and to come to the conclusion that five pounds in fifteenth century would cost around 30 pounds at the beginning of the eighteenth century. His conclusion was based on prices of essentials like bread, drink, meat, cloth, and books. Since then contributions in this area have been steady and research on index numbers for measuring temporal and spatial price changes is continuing. Some of the earliest known index numbers are due to Dutot, Carli, Young, Lowe and Jevons; Laspeyres [39], Paasche [49], Fisher [28], and Törnqvist [62]. Most of these index numbers are still in vogue and play a central role in the construction of price and quantity index numbers. For a comprehensive account of the history of index numbers, the reader is referred to Diewert [17] and to a more recent account in Balk [6]. Index numbers can be used to measure price and quantity changes from the perspective of a consumer or a producer. On the consumer side, emphasis is on the construction of consumer price index or the cost of living index numbers. The economic theoretic approach employed in this context necessarily depends on the theory of consumer behavior. The Konus [37] and the Allen [2] indices are two theoretical index numbers used for making price and real expenditure comparisons. The axiomatic approach to index numbers, in contrast, focuses on the properties expected of price and quantity index numbers. On the production side, the scope of index numbers is somewhat wider as there are data on output prices and quantities as well as data on input quantities used in the production process and their prices. This means that index number methods are needed for compiling output as well as input quantity and price index numbers. The economic theoretic approach in this case relies on the production technology; revenue maximization behavior; input cost minimization or profit maximizing behavior of the producer. The axiomatic approach for these index numbers, however, is similar to that used in the consumer price indices and comparisons of real expenditure. An added dimension on the production side is the need to compile measures of productivity change and then to identify various components of productivity change. This chapter will focus purely on index numbers for output and input level comparisons across firms or movements over time. Measurement of productivity change and identification of sources of productivity change are discussed in other contributions in this Handbook. Index numbers are also used in empirical studies as a tool to reduce dimensionality in the process of identifying production frontiers. Techniques like the data envelopment analysis (DEA) and stochastic frontier analysis (SFA) make use of

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micro- or firm-level data and often suffer from the curse of dimensionality. In order to be able to apply these methods, index numbers are used in the first stage to reduce the dimension of the output or input vectors. For example, in the case of agriculture, agricultural output includes thousands of agricultural commodities. It is difficult to identify production frontiers with such high-dimensional data using DEA or SFA. Index numbers are used to reduce the dimension to a manageable number of composite agricultural commodities like cereals, pulses, vegetables, milk, livestock, and animal products. Index number techniques used for this purpose are the same as index numbers used to measure quantity change or to make comparisons of levels of output at the firm level. As the current chapter presents only essential elements of index number construction, readers and practitioners are advised to refer to major reference works like the ECE-ILO Manual on the Consumer Price Index (2014) [21]1 and the Producer Price Index (PPI) Manual [35]2 for further details. In addition, there are a large number of publications, working papers, and economic measurement course materials available on the website of Erwin Diewert whose pioneering work has helped shape the modern approach to index numbers.3 As the main focus of this Handbook is on productivity measurement, a useful reference on various measurement and index number related issues is OECD [48], Measuring Productivity – OECD Manual Measurement of Measurement of Aggregate and Industry-level Productivity Growth. This chapter is organized as follows. Section “Notation and Preliminaries” establishes the notation and preliminaries for the chapter. Section “Productivity Measurement: The Case of Single Output and Single Input” examines, using the simple one output and one input case, the link between productivity measurement, price and quantity movements, profitability ratio, and changes in the terms of trade. The section also discusses various components of productivity change. Section “Multiple Outputs and Inputs: The Index Number Problem” focuses on the multi-output and multi-input case and introduces the notion of fundamental index number decomposition. Section “Index Numbers Based on Quantity Aggregates” focuses on measuring quantity change using quantity aggregates. Section “Index Number Approach to Measuring Quantity Change” presents the standard index 1 The

latest version of the CPI Manual can be found at: www.ilo.org/global/statistics-anddatabases/WCMS_331153/lang%2D%2Den/index.htm. The Manual is an expanded revision of Consumer price indices: An ILO manual, published in 1989. The current version is largely due to the efforts of the Inter-secretariat Working Group on Price Statistics (IWGPS) which included the participation of: the International Labour Office (ILO); the International Monetary Fund (IMF); the Organisation for Economic Co-operation and Development (OECD); the Statistical Office of the European Communities (Eurostat); the United Nations Economic Commission for Europe (UNECE); and the World Bank. 2 The PPI Manual can be found on: www.imf.org/en/Publications/Manuals-Guides/Issues/2016/ 12/30/Producer-Price-Index-Manual-Theory-and-Practice-16966. This manual is produced by the Inter-secretariat Working Group on Price Statistics with lead role taken by the International Monetary Fund. 3 Diewert’s website can be accessed through: https://economics.ubc.ca/faculty-and-staff/w-erwindiewert/

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number theory and practice which attempts to measure quantity change directly. Section “Axiomatic Approach to Index Numbers” is devoted to the axiomatic approach to index numbers. Section “Economic Theoretic Approach to Output and Input Quantity Index Numbers” reviews the economic theoretic approach to index numbers. Relationship between the theoretical constructs and operational index number formulae is discussed. The concept of exact and superlative index numbers is introduced. Section “Special Topics” deals with topics that are of special interest. In particular the section discusses the transitivity requirement and its implication for the selection of suitable index number formulae for measuring output and input quantity change. Section “Conclusion” offers guidance to practitioners regarding the choice of appropriate index number methodology for productivity measurement.

Notation and Preliminaries Consider the case where there are N outputs and K inputs indexed by i = 1, 2, . . . , N and k = 1, 2, . . . , K. Let {piτ , qiτ ; i = 1, 2, . . . , N and τ = s, t} represent, respectively, output prices and output quantities in periods s and t.4 The input price and quantities are, respectively, denoted by {wkτ , xkτ ; k = 1, 2, . . . , K and τ = s, t}. Let p, w, q and x, respectively, represent vectors of output and input prices and vectors of output and input quantities. All prices and quantities are assumed to be strictly positive. Inner-products of vectors, say p and q and w and x, denoted by p · q and w · x represent the sum of products of elements of the two respective vectors: p·q ≡

N 

pi · qi

w·x ≡

i=1

K 

wk · xk

k=1

The output revenue shares and input cost shares denoted by, riτ and vkτ , are defined by: piτ · qiτ piτ · qiτ = riτ ≡ N pτ · qτ i=1 piτ · qiτ

i = 1, 2, . . . , N; τ = s, t

and wkτ · xkτ wkτ · xkτ = vkτ ≡ K wτ · xτ k=1 wkτ · xkτ

k = 1, 2, . . . , K; τ = s, t

The revenue and cost shares are all strictly positive, by assumption, and add up to unity over respective domains.

4 When

we consider comparisons across M firms we use indices j and k (= 1, 2, . . . , M).

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Technology, Output, and Input Distance Functions This chapter covers a number of basic concepts from production economics. The state of technology, represented by S, is defined as all pairs of input and output vectors (x, q) where x can produce q. Throughout this chapter, where relevant, the production technology is assumed to satisfy the following set of regularity conditions. A formal treatment of these conditions can be found in Färe and Primont [26].

Regularity Conditions R.1 • With N outputs and K inputs, production technology S is a nonempty closed N +K subset of the nonnegative orthant R+ . 5 • If x  0, then there exists q  0 such that (x, q) ∈ S. This means positive inputs lead to positive amounts of outputs. • Free disposability of inputs and outputs: If (x1 , q1 ) ∈ S; x2 ≥ x1 ; q2 ≤ q1 then (x2 , q2 ) ∈ S . • The technology set is convex. That is, if (x1 , q1 ) ∈ S and (x2 , q2 ) ∈ S then for any 0 ≤ λ ≤ 1, ( λx1 + (1 − λ)x2 , λq1 + (1 − λ)q2 ) ∈ S. • For any given input vector, x, the output set, P(x) = {q; (x, q) ∈ S} is closed, bounded and convex. This means that finite amounts of inputs cannot produce unlimited outputs. • For any given output vector, q, the input set consisting of all input vectors that can produce q, L(q) = {x : (x, q) ∈ S}, is a closed set. The output and input distance functions are used at various places in this chapter. Shephard (1970) introduced the notion of distance functions which have subsequently assumed prominence in efficiency and productivity measurement. The output distance function is the radial distance [27] between the observed output vector from a specific production possibility frontier determined by the state of technology and a prespecified input vector. Then the output distance function is defined as6 : Do (x, q : S) = minλ>0 { λ| (x, q/λ) ∈ S} = minλ>0 { λ| (q/λ) ∈ P (x)}

(1)

x  y means that each element of the vector x, xi , is strictly greater than the corresponding element yi in vector y. In contrast x > y means that xi ≥ yi for all i with strict inequality holding for at least one i. Further, x ≥ y simply means that xi ≥ yi for all i. 6 This is not as rigorous a definition as one would find in standard expositions on this subject (see [26]). But this definition suffices for the purposes of this chapter. 5 Here

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The input distance function is similarly defined: Di (x, q : S) = maxρ>0 { ρ| (x/ρ, q) ∈ S} = maxρ>0 { ρ| (x/ρ) ∈ L(q)} (2) The output and input distance functions play a critical role in the construction of output and input quantity index numbers. See Coelli et al. [12] and Sickles and Zelenyuk [55] for more details and extensive discussion of these distance functions.

Productivity Measurement: The Case of Single Output and Single Input Consider the simplest case where a firm produces only one commodity using a single input. Let q and x represent, respectively, the quantities of output produced and input used in the production process. In addition, suppose the output and input prices are known and represented by p and w, respectively. In this case the following measures of performance of the firm can be used. The first measure of productivity, denoted by PROD, is simply the ratio of output to input: P ROD =

output q = input x

(3)

This measure shows the number of units of output produced per one unit of input. An alternative measure of performance can be defined using profitability ratio, denoted by PFR, which shows the amount of revenue generated per one unit cost. It is given by PFR =

p·q total revenue = total cost w·x

(4)

Both of these measures can be computed directly using data on output, input, and the respective output and input prices. These measures are intuitive and are of practical importance.

Absolute Versus Relative Measures Measures in Eqs. (3) and (4) are both simple real numbers and are not informative unless they are compared with the performance of another firm or compared over time for the same firm. For purposes of illustration, consider productivity and performance of a given firm in two periods, t and t + 1. Then P ROD t =

qt xt

P ROD t+1 =

qt+1 . xt+1

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Productivity change over time, denoted by PRODt, t + 1 , is the ratio of productivity levels in these two periods P ROD t,t+1 =

P RODt+1 qt+1 /xt+1 qt+1 /qt output growth = = = P RODt qt /xt xt+1 /xt input growth

(5)

In this simple case, productivity growth over the period is equivalent to output growth discounted by input growth.7 Consider firm’s performance using change in profitability ratio over the two periods. P F Rt = P F R t,t+1 =

pt qt wt xt

P F R t+1 =

pt+1 qt+1 wt+1 xt+1

P F Rt+1 pt+1 qt+1 /wt+1 xt+1 qt+1 /qt pt+1 /pt = = × P F Rt pt qt /wt xt xt+1 /xt wt+1 /wt

(6)

= Productivity Growth × Change in Terms of Trade Equation (6) is important from a producer view point as it implies that changes in profitability are driven by changes in productivity (ability to produce more output with a given input) as well as changes in terms of trade measured by movements in output prices relative to input prices. This, in turn, implies that profitability can change even if productivity remains the same or, alternatively, profitability could fall even when there are improvements in productivity.

Decomposition of Productivity Change: Single Input and Single Output Case Even though this chapter is about index numbers and measurement of output and input quantity change and productivity change, it is useful to examine the components or drivers of productivity change in this simplest case where a single input is used in producing a single commodity. The discussion below draws material from Section “Specification of Functional Forms for Input Aggregates” of Coelli et al. [11], pp. 100–103. Following Eq. (5), productivity change from period s to t is given by P ROD st =

qt /qs output change = xt /xs input change

Suppose production is governed by production technologies in the periods s and t which are represented by production functions fs (x) and ft (x), respectively. Under 7 Similar

interpretation holds when two firms A and B are compared instead of a single firm compared over two periods. In that case, these ratios represent relative levels rather than growth.

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the assumption that the observed output quantities are the maximum possible under technologies existing in these periods and hence the firm is technically efficient in both periods, the productivity index in (5) can be written as: P ROD st =

qt /xt ft (xt ) /xt = qs /xs fs (xs ) /xs

(7)

However, if the firm is technically inefficient in both periods then the observed production is only a fraction of the maximum feasible indicated by the production function. In this case qτ = δτ × fτ (xτ ) for τ = s, t where 0 ≤ δτ ≤ 1

(8)

Here δ s and δ t presents levels of technical efficiency. Substituting (8) into (7) we have P ROD st =

δt ft (xt ) /xt × δs fs (xs ) /xs

(9)

The first part of the right hand side of Eq. (9) represents efficiency change component of productivity change. Suppose, further, that quantity of input used in the two periods is the same with xt = xs = x∗ then Eq. (9) simplifies to: P ROD st =

δt ft (x∗ ) × δs fs (x∗ )

(10)

Equation (10) simply states that in the absence of input change, productivity growth is driven by two factors: technical efficiency change and technological change. In practice, input quantities do change and hence xt = xs . In the case of a single input, the input in period t can be written as a scalar multiple of input in period s. Therefore, we have xt = κxs where κ > 0. In this case, Eq. (9) can be written as: P ROD st =

1 δt ft (κxs ) /xt δt ft (κxs ) /κxs δt ft (κxs ) × × = × = × δs fs (xs ) /xs δs fs (xs ) /xs δs fs (xs ) κ (11)

This means that the scale of input usage has changed by a factor of κ. Suppose the production function in period t is homogeneous of degree ε(t), then Eq. (11) can be written as: P ROD st =

ft (xs ) δt × κ ε(t)−1 δs fs (xs )

(12)

Equation (12) identifies three drivers of productivity change: (i) the first component on right hand side of Eq. (12) shows technical efficiency change; (ii) the second

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D. S. Prasada Rao

component shows the effect of change in the scale of operations and thus represents the scale effect; and (iii) the last component shows the contribution of technical change. A few comments are notable in this context. First, if the production function exhibits constant returns to scale in period t then productivity change is driven solely by technical change and technical efficiency change. Second, measures of technical change and the scale effect in Eq. (12) are conditional on expressing period t input as a multiple of period s input. A different answer results if period s input is expressed as a multiple of period t input. It is likely that measures of both the scale effect and technical change are different. This difference vanishes if technology in both periods exhibits constant returns to scale. In the more general and realistic case where a vector of outputs is produced using a vector of inputs – the case of multiple inputs and outputs – it is necessary to consider the contribution of changes in the output mix and the input mix to the observed productivity change. There is vast literature on the decomposition of productivity change and identifying the contribution of various drivers of productivity change. This is subject matter for other chapters in this Handbook.

Multiple Outputs and Inputs: The Index Number Problem Consider the more general and realistic case where a firm produces N outputs using K inputs.8 In addition there are input and output data for two periods s and t (or for two firms).9 In this case, let (qs , qt ); (xs , xt ); (ps , pt ); (ws , wt ) represent vectors of output and input quantities and prices, respectively, in periods s and t. The problem is how to generalize the measures of productivity and profitability defined in Eqs. (3) and (4) and to measure changes in productivity and profitability. Index numbers can be used in deriving suitable measures in this multi-output and multi-input case.

What Are Index Numbers? The standard notion of index number used in the discipline of economic statistics is that an index number is a real number which measures changes in a set of related variables. This is the concept that underpins the construction of consumer and producer price index numbers as well as standard price and quantity index numbers. 8 Inputs

may be classified as intermediate inputs and primary inputs (labor and capital) or as KLEMS representing capital, labor, energy, materials and service inputs. This kind of distinction is important in the case of measuring productivity within different sectors of the economy, e.g., agriculture, manufacturing etc. However, for the purpose of this chapter this distinction is not critical. 9 For purposes of exposition the case of two periods is considered but all the results and considerations equally apply to the case of two firms.

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An alternative concept of index number could be that it is a real number that measures the size of a given vector of variables. This concept leads to measures such as the aggregate or the average size of a vector of variables. These two alternative notions of index numbers are implicit in the following description of the index number problem than can be found in Diewert [20]’s introductory chapter on Index Numbers and Measurement Economics. Diewert states: “The question that this book addresses is: how exactly should the microeconomic information involving possibly millions of prices and quantities be aggregated into a smaller number of price and quantity variables? This is the basic index number problem.”10 In the literature on productivity measurement, the traditional approach has been to use index numbers as measures of change in a set of related variables and therefore index numbers are used as measures of changes in prices and quantities over time or as a measure of difference in levels in prices and quantities associated with different firms. This treatment can be found in the seminal works of Diewert [13], Caves et al. [10], Diewert [16], Balk [6], the CPI and PPI Manuals, and in a standard textbook treatment provided in Coelli et al. [11, 12]. Based on this discussion, three strands of index number problems relevant for productivity measurement and performance assessment can be identified.

Measuring Quantity Aggregates In its simplest form, construction of index numbers based on economic aggregates involves the computation of output and input aggregates. Let Q and X, respectively, be real valued functions representing output and input aggregates. These are expressed as functions of the quantities of outputs produced and inputs used in the process. Thus, Q = Q (q1 , q2 , . . . , qN )

X = X (x1 , x2 , . . . , xK )

(13)

Once the output and input aggregates are constructed, productivity in the presence of multiple outputs and inputs can simply be measured, similar to the single output and single input case, as the ratio P ROD =

Q Q (q1 , q2 , . . . , qN ) aggregate output = = aggregate input X X (x1 , x2 , . . . , xK )

(14)

The output and input quantity indices measuring changes in outputs produced and inputs used are simply the ratios of the corresponding aggregates. The indices measuring changes in outputs and inputs are defined as:

10 Reader

[11, 12].

may find alternative working definitions of index numbers from Balk [6] and Coelli et al.

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D. S. Prasada Rao

Qst ≡

Q (q1t , q2t , . . . , qN t ) Q (qt ) = and Q (qs ) Q (q1s , q2s , . . . , qN s )

X (x1t , x2t , . . . , xKt ) X (xt ) = Xst ≡ X (xs ) X (x1s , x2s , . . . , xKs )

(15)

where Qst and Xst , respectively, represent output and input quantity index numbers measuring the changes in output and input levels from period s to period t.11 Implementation of this approach hinges on how the quantity aggregates in Eq. (14) are measured. This approach is discussed further in Section “Multiple Outputs and Inputs: The Index Number Problem.”

Measures of Output and Input Quantity Change as an Aggregate of Commodity-Specific Changes This approach advocates measuring overall change in outputs and inputs directly using index number methods. Here the required quantity index is measured using both price and quantity data pertaining to the two periods. The output and input quantity index numbers are then defined as: Qst ≡ Qst (qt , pt ; qs , ps ) and Xst ≡ Xst (xt , wt ; xs , ws )

(16)

where Qst and Xst are real valued functions satisfying a number of properties like positivity, continuity, monotonicity, and others.12 We discuss a number of index number formulae in the Section “Index Numbers Based on Quantity Aggregates,” but it is useful to consider the intuition behind the index number approach. First focus on the output quantity index number measuring change in the output vector over time. For commodity i (=1, 2, . . . , N), growth or change in the output, from base year s to comparison period t, is given by the ratio qit /qis . Given this observed growth in each of these outputs, an overall index may be defined as a weighted average of the growth in individual items. If weights reflect the importance of each commodity as measured by the revenue share of the commodity in period s defined in Section “Notation and Preliminaries,”13 the output quantity index is given by

11 Index numbers always identify the reference or base period and the comparison or current period.

In Equation (15), periods s and t are the base and current periods for which the index is defined. properties are usually discussed under the axiomatic approach to index numbers which is considered further in Section “Index Number Approach to Measuring Quantity Change” of this chapter. 13 On the other hand, one may choose to use expenditure shares in the current or comparison period, t. 12 These

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N N N   qit · pis qit qit pis qis qt · ps Qst = · ris = · N = i=1 = N qis qis qs · ps i=1 pis qis i=1 qit · pis i=1 i=1

(17)

This quantity index is generally referred to as the Laspeyres index [39] index. Several features of this formula are worth noting. First, this index does not require a quantity aggregate to be defined for each period, instead it starts with growth in output of each commodity as building blocks and the index is then defined as a weighted average of the change in different commodities. Second, the last ratio in this equation suggests that this index can be interpreted as the ratio of value of output in periods t and s both evaluated at the output prices in period s, the base period. Third, weights used in Eq. (17) which are revenue shares in period s can be replaced by revenue shares in period t or by an average of the shares in both periods. Finally, it is not necessary that the quantity index makes use of an arithmetic average. One may use geometric or harmonic means of commodity specific output quantity changes. A range of index number formulae based on these alternative specifications are discussed in Section “Index Numbers Based on Quantity Aggregates.”

Decomposition of Changes in Revenues and Costs From the perspective of the firm, performance is reflected in the revenue, costs, and profits generated out of its operations. In periods t and s, total revenue and costs are given by: Rt =

N 

K 

qit · pit = qt · pt Ct =

xkt · wkt = xt · wt

i=1

k=1

N 

K 

(18) Rs =

qis · pis = qs · ps Cs =

i=1

xks · wks = xs · ws

k=1

The revenue and cost changes over the two periods are then given by Rst =

Rt qt .pt = Rs qs · ps

and Cst =

Ct xt .wt = Cs xs .ws

(19)

In Eq. (19), total revenues and costs in periods t and s are both observed. The index number problem is often stated as one of identifying the price change and quantity change that drive the revenue and cost changes. For example, suppose the total revenue of a firm has increased from $1500 to $2000 which gives a value of 1.33 for Rst indicating a 33% increase in the revenue. The firm manager might wish to know how much of this increase is due to increase in output prices and what part is due to increase in outputs produced. The index number problem is often described as finding Qst and Pst which respectively represent quantity change and price change over the period s to t such that the value or revenue change is equal to the product

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D. S. Prasada Rao

of price and quantity change Rst =

Rt qt .pt = = Qst · Pst . Rs qs · ps

(20)

The change in costs from period s to t can be similarly decomposed into input quantity change and input price change Cst =

Ct xt .wt = = Xst · Wst Cs xs .ws

(21)

This decomposition is often described as a test or an axiom in the index number literature. The axiomatic or test approach is discussed in Section “Axiomatic Approach to Index Numbers.” An important corollary of this decomposition of value change is that, for example, if revenue change is observed to be 1.33, then estimates of quantity and price change have to be consistent with this observation. It also implies that, in Eq. (20), the knowledge of two out of the three entities, Rst , Qst and Pst will determine the third. For example, if revenue change is observed and if it is easy to compute the price change, Pst , the quantity change can be simply obtained as: Qst =

Rst Pst

(22)

Similarly, changes in input costs can be decomposed into input price change and input quantity change. Based on this decomposition, it is possible to decompose change in profitability ratio. This is given by P F R st =

P F Rt Rt /Ct Rt /Rs = = P F Rs Rs /Cs Ct /Cs

=

Qst · Pst Revenue change = Cost change Xst · Wst

=

Qst Pst · = Productivity change × Terms of trade change Xst Wst

(23)

Equation (23) is a multi-output and multi-input generalization of the decomposition of profit change described in Eq. (4). This equation underscores the important link between price and quantity index numbers and the value change. Ignoring this link would create a disconnection between measures of quantity change, price change, and the measure of value change.

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Index Numbers Based on Quantity Aggregates This review of index numbers begins, within the context of measuring productivity change, with the approach that builds on the notion of output and input quantity aggregates. Use of quantity aggregates and price aggregates in the context of index numbers is not new (see [20]). However, this approach has reemerged in the literature as it has recently been advocated in a series of papers on productivity measurement by O’Donnell [44–47]. The approach in these papers underpins the DPIN software developed by O’Donnell [43] which is available through the website of the Centre for Efficiency and Productivity Analysis (CEPA) (URL: www.uq.edu. au/economics/cepa). In simple terms, this approach advocates the construction of input and output aggregates in periods s and t denoted respectively by Q(qs ), Q(qt ), X(xs ), and X(xt ). The required indices are then given by: Output index = Qst ≡

Q (qt ) X (xt ) ; Input index = Xst ≡ ; Q (qs ) X (xs )

Qst Productivity change = P ROD st ≡ Xst

(24)

Implementation of this approach requires details as to how these aggregates can be computed. This in turn requires a description of the basic properties expected of these aggregates. O’Donnell [43, 46] lists the following properties for the output aggregates14 : Property 1 Q(.) is a real-valued function which is nonnegative. For any given output vector which is nonnegative, the aggregate is nonnegative. Property 2 Output aggregate is nondecreasing in its arguments. If the output of one of the commodities increases, keeping other outputs fixed at given levels, the output aggregate must increase or at least remain the same. Property 3 The output aggregate function is linearly-homogeneous. This means that if all the outputs are multiplied by a positive constant then the output aggregate is also multiplied by the same constant. That is, for any λ > 0, Q (λq1 , λq2 , . . . , λqn ) = λQ (q1 , q2 , . . . , qN ). These properties may be considered as a minimal set of properties to be satisfied by output and input aggregates.

Specification of Functional Form for the Output Aggregates In order to implement this approach, it is necessary to specify the functional form for Q (.) and X (.). O’Donnell ([43], p. 3) provides a long list of functional

14 Focus

here is on the output aggregate and similar properties are expected of the input aggregate.

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D. S. Prasada Rao

forms that satisfy the three properties listed above. In defining some specific types of aggregates, observed price data are used to combine quantities of different commodities. Thus, the observed price vectors in periods s and t, ps and pt , are used below. Here s represents the base period (or reference firm) and t represents the current or comparison period (or firm). Laspeyres-type aggregate: Base period prices are used to compute the output aggregate. It is given by: QL (qτ ) ≡ ps · qτ =

N i=1

pis · qiτ τ = s, t

(25)

In this case the output quantity aggregate is simply the total value of the output in each period s and t evaluated at prices observed in the base period, s. Paasche-type aggregate: This aggregate uses current or comparison period prices to compute the output aggregate and it is given by: QP (qτ ) ≡ pt · qτ =

N i=1

pit · qiτ τ = s, t

(26)

This aggregate represents the total value of output in periods s and t evaluated at prices observed in the current period, t. Fisher-type aggregate: The Fisher aggregate is defined as the geometric mean of the Laspeyres and Paasche- type of aggregates defined above. QF (qτ ) ≡ [(ps · qτ ) . (pt · qτ )]0.5   0.5  N N = pis · qiτ · pit · qiτ τ = s, t i=1

(27)

i=1

It is important to note that the Fisher-type aggregate is not derived by evaluating output quantities at geometric averages of prices in the base and current periods. The Fisher-type aggregate in (27) may be considered as a symmetric average of the Laspeyres and Paasche quantity aggregates based on the geometric average. Thus, it is possible to define alternatives to Fisher-type aggregates by taking the arithmetic or harmonic averages leading to (QL (qτ ) + QP (qτ )/2 or 2QL (qτ ) · QP (qτ )/[QL (qτ ) + QP (qτ )], τ = s, t. The use of Laspeyres, Paasche, and Fisher type aggregates (in Eqs. 25, 26 and 27) leads to the standard Laspeyres, Paasche, and Fisher quantity indices widely used for measuring quantity change. Lowe aggregate: The Lowe aggregate is a general specification that includes Laspeyres and Paasche specifications as special cases. This aggregate makes use of a fixed reference price vector, pR to evaluate quantities in both periods. It is given by: QLowe (qτ ) ≡ pR · qτ =

n i=1

piR qiτ τ = s, t

(28)

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Lowe index requires nomination of the reference price vector. If the reference price vector selected is same as the base period price vector, we have the Laspeyres aggregate. Similarly, use of current period price vector results in the Paasche aggregate. The Lowe aggregate also includes the Edgeworth-Marshal and Walsh indices (see [6] for a description of these indices) as special cases where the reference price vector is an arithmetic or geometric average of the prices in the two periods. When working with multiperiod or multifirm data the Lowe aggregate may be defined using an average of prices in all the periods. No, consider a series of aggregates which make use of the Malmquist output distance function defined in Section “Notation and Preliminaries” of this chapter. Alternative quantity aggregators based on Malmquist distance are listed below. These aggregates form the basis for the Hicks-Moorsteen index [8]. Laspeyres-type Malmquist quantity aggregate: This is defined using base period production technology and the input vector represented, respectively, by Ss and xs . The aggregate is given by  QM−L (qτ ) ≡ Do xs , qτ : S s τ = s, t

(29)

Paasche-type Malmquist aggregate: This aggregate is defined using the current period technology and input vector represented, respectively, by St and xt . The aggregate is defined as:  QM−P (qτ ) ≡ Do xt , qτ : S t τ = s, t

(30)

Hicks-Moorsteen-type Malmquist aggregate: This aggregate is defined along the lines of the Fisher index. It is defined as the geometric average of the Laspeyres and Paasche-type Malmquist aggregates.  0.5

 τ = s, t QM−H M (qτ ) = Do xs , qτ : S s · Do xt , qτ : S t

(31)

Färe-Primont Malmquist Aggregate: This aggregate function is similar in structure to the Lowe index defined above. It depends upon an arbitrarily selected reference input vector, xR , and reference technology, SR . It is defined as:  QM−F P (qτ ) ≡ Do xR , qτ : S R τ = s, t

(32)

Aggregates defined above satisfy all the three Properties 5.1 to 5.3. These quantity aggregates have been employed in the past in measuring productivity levels and productivity change over time. A distinguishing feature of the Malmquistbased aggregators is that these aggregates do not require any price information and therefore can be used even when price data are not available. On the down side, it

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D. S. Prasada Rao

is necessary to know the production technology in the form of the output distance function – both the functional form and values of the parameters in the function. In implementing the quantity aggregates approach as illustrated through Eqs. (25), (26), (27), (28), (29), (30), (31), and (32), the practitioner has to make some conscious decisions regarding the type of aggregator, price vector in the case of Lowe index, and the choice of reference technology and reference input vector in the case of Fare-Primont Malmquist distance based aggregate.

Specification of Functional Forms for Input Aggregates Specification of functional forms for input aggregates is similar to output aggregates discussed in the previous section. In defining input quantity aggregates use is often made of input prices. As input quantities cannot be summed directly, input prices are used in the process of aggregating quantities. The input quantity and price vectors in the two periods are represented by (xτ , wτ : τ = s, t). Measures similar to those in Eqs. (25), (26), (26), (27), and (28) used in defining output quantity aggregates will lead to the following input quantity aggregates. K

XL (xτ ) ≡ ws · xτ = XP (xτ ) ≡ wt · xτ =

k=1

K

k=1

wks · xkτ τ = s, t

wkt · xkτ τ = s, t

XF (xτ ) ≡ [(ws · xτ ) . (wt · xτ )]0.5   0.5  K K = wks · xkτ · wkt · xkτ τ = s, t k=1

XLowe (xτ ) ≡ wR · xτ =

k=1

K k=1

wkR xkτ τ = s, t

where X(.) represents the quantity aggregate and wR denotes a reference input price vector used in the Lowe aggregate function. The Malmquist input distance function based aggregates makes use of the input distance function Di (x, q : S). Based on this distance function, we can define four input aggregates similar to those in Eqs. (29), (30), (31), and (32). These are:  XM−L (xτ ) ≡ Di xτ , qs : S s τ = s, t  XM−P (xτ ) ≡ Di xτ , qt : S t τ = s, t  0.5

 τ = s, t XM−H M (xτ ) = Di xτ , qs : S s · Di xτ , qt : S t  XM−F P (xτ ) ≡ Do xτ , qR : S R τ = s, t

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where qR represents a preselected reference quantity vector and SR represents reference production technology. The Färe-Primont type aggregate will differ depending on the choice of the reference technology and reference output vector. The output and input aggregates discussed in this section can be used in measuring changes in output and input vectors over time or the level differences across two firms. Measures of input and output change are simply defined, see Eq. (24), as the ratio of the respective quantity aggregates over time and for measures of productivity levels across firms use the ratio of the respective aggregates over two firms at the same point in time.

Index Number Approach to Measuring Quantity Change This section describes the traditional index number theoretic approach to measuring quantity changes. As before, the exposition focuses on methods for measuring output quantity change but these are equally applicable to measurement of input quantity change. These methods are designed to make comparisons between two sets of observations, over time or across two firms. Without loss of generality, discussion below focuses on comparisons over time. As index numbers measure changes in a set of related variables, it is necessary to identify one of the periods or firms as the base or reference period and the other as the current or comparison period. This means that the index number measures changes in output quantities from the base period to the current period. For comparisons between firms, use the terms reference firm and the comparison firm. In the literature there are two alternative approaches to the construction of quantity index numbers. The first approach is the direct approach whereby the index number formulae are designed to measure output quantity change. The second approach, known as the indirect approach, where the output price change is measured first and then quantity change is measured as the ratio of the observed value (revenue in this case) change and the price change. The revenue change is measured by the ratio of revenues in periods s and t, shown in equation below. Revenue change = Rst ≡

Rt qt .pt = Rs qs · ps

The indirect approach is based on the fundamental index number decomposition which states that measures of value change must equal the product of measures of quantity change and price change. In the case of outputs and revenues, revenue change is decomposed into output quantity change and output price change, as shown below. Rst = Qst · Pst = Output quantity change × Output price change. The output quantity change can be measured indirectly as:

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D. S. Prasada Rao

Rst = Qst · Pst ⇒ Qst =

Rst Pst

The indirect input quantity index numbers can similarly be defined using Cst , Wst , and Xst which, respectively, represent measures of cost change, input price change, and input quantity change. See Section “Indirect Measures of Quantity Change” for further discussion on the decomposition of value (revenue or cost) change.

Direct Approach to Quantity Index Numbers Under this approach quantity index numbers are computed directly using observed price and quantity data. In presenting various formulae a further distinction is made between two different approaches. The first approach makes use of a fixed price vector to compare quantity levels. The second approach computes an average of changes in quantities of different products over the two periods. This approach is essentially one of obtaining a measure of central tendency. The formulae presented here refer to output quantities but are equally applicable for measuring changes in input quantities when output quantities and prices are substituted by input quantities and input prices.

Fixed Price Approach As the title suggests, application of this approach requires specification of a reference set of prices at which output quantities are evaluated. The most generic index under this class is the Lowe index. indices like the Laspeyres, Paasche, and Walsh indices are similar to the Lowe index in formulation but with subtle differences. Lowe Index: The Lowe index [40] makes use of an arbitrarily chosen reference vector of prices, pR . The Lowe index is defined as: N QLowe st

i=1 ≡ N

piR qit

i=1 piR qis

(33)

The Lowe index is the ratio of revenues associated with output quantities in periods s and t evaluated at fixed reference prices. Any change in the revenue ratio in Eq. (33) reflects changes in quantities since reference prices are fixed. The selection of the reference price vector is a critical decision in the implementation of the Lowe index. In practice use is made of the average of output prices in both periods s and t or all the time periods in the analysis if quantity changes are measured over several periods. If quantity comparisons are made across several firms from a given industry, then the reference price vector may be an average of the prices faced by different firms. Laspeyres Index: The Laspeyres quantity index [39] is in a way a special case of the Lowe index but it differs from the Lowe index from both operational and

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economic perspectives. Laspeyres index has its origins in the construction of fixed basket consumer price index numbers. Faced with the choice of the reference price vector for the implementation of the Lowe index, the Laspeyres index suggests the use of base period, s. The choice is driven by the fact that the reference price vector is operationally relevant for the firm in at least one of the periods. From an economic point, firm’s decisions on output quantities in the base period would have been governed by output prices prevailing in that period. Thus, the Laspeyres quantity index is defined as: N QL st

i=1 ≡ N

pis qit

i=1 pis qis

=

ps · qt ps · qs

(34)

Economic intuition of the Laspeyres index is simply that one output quantity vector is deemed to be bigger than another if it generates more revenue at the prices prevailed in the base period. For example, at the base period prices if the revenues generated at base and current period quantities are $10 million and $12 million, respectively, then the Laspeyres quantity index implies a 20% increase in output quantities. Paasche Index: The Paasche index [49] is defined as the change in revenue generated from outputs in the two periods when current price vector, pt , is used for evaluating output. This index is also a special case of the Lowe index where the reference price vector is the current period price vector, pt . The Paasche quantity index is defined as: N QPst

i=1 ≡ N

pit qit

i=1 pit qis

=

pt · qt pt · qs

(35)

The Paasche index compares the revenue generated by current period output quantities at current period prices with the revenue that would have been generated in the base period at current period prices. As the same prices are used in evaluating revenues generated by quantity vectors in periods s and t, any change in the revenue is attributed to quantity change. If pit = λ · pis ; i = 1, 2, . . . , N where λ is strictly positive, then the Laspeyres and Paasdche indices coincide. N

QPst

=

i=1 pit qit N i=1 pit qis

N

=

i=1 λ · pis qit N i=1 λ · pis qis

N i=1 = N

pis qit

i=1 pis qis

= QL st

However, in practice price changes are not proportional across all commodities. In that case these indices differ, sometimes quite significantly, if the price structures exhibit major shifts. When comparisons are being made over time, one would expect the price structures to be similar when the periods are not too distant. However, firms in different spatial locations are likely to face different prices and relative

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D. S. Prasada Rao

price structures. Differential price structures are likely to be encountered if the firms are located in different countries or in different regions of a country. For example, relative prices of agricultural commodities are likely to differ significantly across different states within the USA. Fixed Price Indices based on Symmetric Averages of Base and Current Period Prices: The Laspeyres and Paasche indices are usually considered to be two extremes since these indices utilize prices prevailing in the base and current periods. In cases where price structures may be changing over time, it may be prudent to make use of an average of the prices in the two periods. The resulting price index would be: N ∗ i=1 pi qit Q∗st = N ∗ i=1 pi qis pi∗

where

(pis + pit ) √ ; or pis · pit ; or  = 2

(36)

2 1 pis

+

1 pit

; i = 1, 2, . . . , N

In Eq. (36), the arithmetic, geometric, and harmonic means of prices in the two periods are used in evaluating quantity vectors in the two periods. The quantity index resulting from the use of the arithmetic average of prices is known as the Edgeworth-Marshall index [22, 42]. Similarly, the Walsh index [62] makes use of geometric average of prices in the current and base periods. Fisher Index and Symmetric averages of the Laspeyres and Paasche Indices: Since the Laspeyres and Paasche indices are seen as two extremes relying on the base or the current period prices, Irving Fisher [28] suggested the use of a geometric average of the Laspeyres and Paasche indices. The Fisher index is defined as:

1/2

P = QFst ≡ QL st · Qst



N i=1 pis qit N i=1 pis qis

N

i=1 pit qit

· N

i=1 pit qis

0.5 (37)

The Fisher index possess several important axiomatic and economic theoretic properties which are discussed in this and the following section. It is possible to use arithmetic or harmonic averages of the Laspeyres and Paasche quantity indices but none of them have properties similar to that of the Fisher index. The Lowe, Laspeyres, Paasche, and Fisher indices discussed in this section play an important role in measuring changes in output quantities, changes in input quantities and measurement of productivity change.

Direct Indices Based on Statistical Averages In this section the problem of measuring changes in the output quantity vector is considered as a problem of obtaining a suitable average or a measure of central tendency of observed growth in outputs of different commodities. Given the output quantities in two periods, {q1s , q2s , . . . , qNs } and {q1t , q2t , . . . , qNt } , there are N measures of quantity change, one for each commodity, {qit /qis : i = 1, 2, . . . , N}.

19 Index Numbers and Productivity Measurement

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Then the index number problem is one of obtaining a suitable measure of central tendency of these N ratios. Unweighted Measures If all commodities are treated as equally important, then the following two index numbers due to Carli [9] and Jevons [36] are of importance.

QCarli st

N 1  qit ≡ N qis

QJevons st

≡

i=1

 N   qit 1/N i=1

(38)

qis

These indices are essentially arithmetic and geometric averages of quantity change observed for each commodity. These indices were proposed and in use well before the Lowe, Laspeyres, Paasche, and Fisher index numbers. Quantity Indices Using Revenue Share Weighted Averages Use of unweighted averages as is the case in Eq. (38) means that quantity change in each commodity is considered equally important. This is an untenable position from the perspective of an economist as not all commodities produced are considered to be of equal importance. Usually a weighted average with weights reflecting the economic importance of different commodities is recommended. A measure of economic importance of output of a particular commodity is its contribution to the total revenue or its share in the revenue (defined in Section “Notation and Preliminaries”). Revenue shares may refer to the base period, s, or to the current period, t, or an average of shares in both periods. It is also possible to use some hypothetical or reference set of weights which have no relationship with revenue shares observed in the base and current periods. Weighted Arithmetic Averages Base-period Revenue Share Weights: Laspeyres Quantity Index Using the base period revenue shares, ris , as weights in an arithmetic average formula lead to the Laspeyres index. It can be seen by N

N qit ris · = i=1 i=1 qis



pis qis

N

i=1 pis qis

qit · qis



N i=1 = N

pis qit

i=1 pis qis

= QL st

Current Period Revenue Share Weights: Paasche Quantity Index

If a harmonic mean of quantity ratios is used in conjunction with current period revenue shares, rit , it results in the quantity index. N

1

i=1 rit

·

qis qit

= N

1

pit qit i=1 N p q i=1

it it

·

qis qit

N i=1 pit qit = N = QPst p q i=1 it is

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D. S. Prasada Rao

Weighted Geometric Averages A number of commonly used formulae can be derived as geometric averages of quantity ratios with revenue share weights. Out of these the most commonly used are the Törnqvist index [62] and the geometric-Young (G-Y) index [62]. These indices can be built from the Laspeyeres and Paasche-type indices shown below. Geometric-Laspeyres: This index is defined using base period revenue shares. QG−L ≡ st

 N   qit ris i=1

(39a)

qis

Geometric- Paasche: This index is defined using current period revenue shares. QG−P ≡ st

 N   qit rit i=1

(39b)

qis

T‚ornqvist Index: The T‚ornqvist (TT) index is then defined, similar to the Fisher index, as the geometric mean of the geometric-Laspeyres and geometric-Paasche indices. It is given by: N   rit 1/2   ris +rit N  N  ris 

1/2 

2 q q qit it it TT G−L G−P Qst ≡ Qst · Qst = · = qis qis qis i=1

i=1

i=1

(40) The TT index is similar to the Fisher index in its formulation – defined as a symmetric average of the geometric-Laspeyres and geometric-Paasche indices. While Fisher index cannot be shown to be a weighted arithmetic average of quantity change observed for each commodity, the TT index is indeed a weighted geometric average of quantity ratios where weights are the arithmetic average of shares in the base and current periods. The TT index is an important formula for constructing consumer and producer price and quantity index numbers. It also plays an important role in productivity measurement. Though the TT index does not satisfy all the axiomatic properties that the Fisher index does, numerical values of TT index in most practical situations are quite close to the Fisher index. So, in practical situations it really does not matter whether one chooses the Fisher or TT index for measuring quantity change. Geometric-Young Index The G-Y index is similar in concept to the Lowe index but defined in terms of quantity change rather than quantity aggregates. Thus G-Y index is more flexible operationally and it carries intuitive statistical interpretation. This index uses item specific weights that need not necessarily correspond to revenue shares in the base or current periods. If ri denotes the weight for i-th commodity ratio, then the G-Y index is defined as:

19 Index Numbers and Productivity Measurement

QG−Y st

≡

779

 N   qit ri i=1

qis

(41)

Empirical implementation of the G-Y index requires specification of the weights. The index is particularly useful when comparisons are being made over several periods or across a set of firms. In that case the G-Y index can be anchored on the average of revenue shares, averaged over time or across different firms. After we discuss the axiomatic framework for index numbers, we will note that the G-Y index has a unique property that makes it a particularly useful tool.

Indirect Measures of Quantity Change Measures of output quantity change are often derived indirectly whereby price changes are measured first and then quantity changes are obtained indirectly through the standard decomposition of revenue change15 formula: N i=1 Rst = N

pit qit

i=1 pis qis

= Pst · Qst

This decomposition is also known as the product test. Given this decomposition, Qst can be derived indirectly if Pst is known. The indirect quantity index is then given by N N Rst 1 i=1 pit qit i=1 pit qit /Pst Qst = = N · =  N Pst Pst i=1 pis qis i=1 pis qis

(42)

Revenue in period t adjusted for price change from period s to t = Revenue in pertiod s The numerator is revenue in period t after the effect of price change over the period s to t is removed. It is usually referred to as constant price revenue aggregate.16 Equation (42) means that the indirect quantity change measure is a ratio of the base and current period revenues after the effect of price change is eliminated using an appropriate price index. In order to implement (42), we need to select a price index number formula. Price index formulae are similar to quantity indices and most standard price index

15 In

the case of input quantity changes, changes in total cost are decomposed into input price change and input quantity change. Here input price change is measured first and input quantity change is measured indirectly. 16 This terminology is more common in the case of national accounts aggregates such as GDP. National statistical offices regularly publish GDP in current prices and GDP in constant (base period) prices.

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D. S. Prasada Rao

number formulae can be derived by simply interchanging price and quantities in the formulae discussed in Section “Direct Approach to Quantity Index Numbers.” As this is a mechanical exercise, we do not list them.

Quantity Index Based on the Laspeyres Price Index This index is defined as the ratio of the revenue change index and Laspeyres price index. L

I QPst =

N N N Rst i=1 pit qit / i=1 pis qis i=1 pit qit = = = QPst N N N PstL i=1 pit qis / i=1 pis qis i=1 pit qis

(43)

L

where I QPst represents the indirect quantity index derived using the Laspeyres Price index. From Eq. (43), it can be seen that this indirect index is the same as the Paasche-Quantity index. An implication of Eq. (43) is that if the Laspeyres price index is a suitable measure of price change then Paasche Quantity index must be used in order to satisfy the index number decomposition condition. In this sense the Paasche quantity index is the dual of the Laspeyres price index.

Quantity Index Based on the Paasche Price Index This index is defined as the ratio of the revenue change index and the Paasche price index. P I QPst

N N N pit qis Rst i=1 pit qit / i=1 pis qis = P = N = i=1 = QL N st N Pst i=1 pit qit / i=1 pis qit i=1 pit qis

(44)

P

where I QPst represents the indirect quantity index derived using the Paasche price index. From Eq. (44) it can be seen that the Laspeyres quantity index is dual to the Paasche price index.

Quantity Index Based on the Fisher Price Index The Fisher price index is the geometric mean of the Laspeyres and Paasche price index numbers. Thus, the Fisher-based indirect quantity index is given by:     Rst Rst 0.5 Rst 0.5 P 0.5 L 0.5 F = · = Qst · Qst = QFst I QPst =  1/2 L P L P P P Pst · Pst st st (45) The second last expression on the right hand side follows from Eqs. (43) and (44). This is an important result which shows that the Fisher quantity index is dual to the Fisher price index and therefore Fisher price and quantity indices are self-dual to each other.

19 Index Numbers and Productivity Measurement

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Indirect Quantity Indices with Geometric Price Indices Since the revenue shares are additive and the geometric indices are multiplicative there are no interesting dual properties attached to indirect geometric quantity indices. We simply note here that indirect quantity indices are computed using Eq. (42) along with one of the geometric price indices defined in equations (39a, b) to (41).

Direct Versus Indirect Measures of Quantity Change: Which One to Use? This is an important question. When a practitioner is working with a data set that contains both price and quantities of commodities, this question assumes significance. Fortunately this is an issue that was adequately addressed by Allen and Diewert [3]. Their recommendation is to use the index that is most reliable out of the two alternatives. An important conclusion from their paper is that if price and quantity ratios – commodity specific changes in prices and quantities over time – do not show much variability it really does not matter whether direct or indirect measures are used and, in this case, most index number formulae will result in numerical values close to each other. Their main recommendation refers to the case where price and quantity relatives exhibit significant variability across commodities. In this case, Allen and Diewert [3] suggest a procedure which involves the following steps: (i) run regressions where ln (pit /pis ) and ln (qit /qis ) are regressed separately on a constant; (ii) compute residual sum of squares of these regressions; and (iii) choose the index that is associated with the smallest residual sum of squares. Properties of the residual sum of squares as measures of reliability are established in Allen and Diewert [3]. The only comment to add is that this measure of reliability can be improved upon by running weighted least squares with revenue shares as weights reflecting the relative importance of difference commodities included in index computation.

Axiomatic Approach to Index Numbers The axiomatic approach to index numbers offers a number of criteria which can be used in evaluating the usefulness of various index number formulae. These criteria are usually referred to as axioms or tests. This approach serves two important functions. First and foremost it helps the user or practitioner in choosing an appropriate index number formula which can be a real issue when one is confronted with a myriad of formulae serving the same purpose. The second role of the axiomatic approach is that it helps us to narrow the choice set of formulae by restricting to those which satisfy a given set of axioms. The earliest attempts to identify a number of desirable criteria date back to Fisher [28]. Section “Absolute Versus Relative Measures” of Balk [6] sketches the historical development of axiomatic approach. A formal treatment of the axioms and proofs of a number

782

D. S. Prasada Rao

of useful results can be found in Eichorn and Voeller [24], Eichorn [23], Diewert [16], and Balk [4, 6]. Balk [6] makes a distinction between axioms and tests. Time reversal, factor reversal, and some others are classified under tests whereas properties like identity and linear homogeneity are included in the list of axioms. Diewert [16] and  Chap. 19, “Index Numbers and Productivity Measurement”, of the Producer Price Index Manual [35] refer to all the properties as tests. This chapter considers axioms and tests as properties of index numbers and discuss them under the single heading, axioms. The axiomatic theory treats prices and quantities of commodities as independent nonstochastic variables. This treatment of prices and quantities differs from the economic theoretic approach where prices and quantities are assumed to be functionally related.

Notation for the Axiomatic Approach A slightly more elaborate notation is used to discuss axioms for index numbers. For simplicity the focus is on axioms for index numbers measuring changes in prices and quantities from the base period, s, to current or comparison period, t.17 In the axiomatic approach, price and quantity index numbers are considered as real valued functions of prices and quantities in these two periods. The following notation: Pst = P (pt , qt , ps , qs )

Qst = Q (pt , qt , ps , qs )

where Pst and Qst represent, respectively, price and quantity index numbers is used. Prices and quantities are assumed to be strictly positive. The price and quantity index functions are assumed to be strictly positive, continuous, and differentiable.18 Price and quantities here may refer to outputs or inputs.

Axioms and Discussion The following is a set of widely canvassed axioms in index number literature. All the axioms are stated for quantity indices but equally applicable for price index numbers. Axiom 1 – Strong Identity Axiom: If all the quantities in the current period are equal to quantities in the base period, then the quantity index must equal 1. If qt = qs = q then Qst (pt , q, ps , q) = 1

(46)

17 In most standard textbook expositions on index numbers base and current periods are denoted by

“0” and “1” respectively. and differentiability are mathematical requirements that ensure smoothness of the index numbers derived.

18 Continuity

19 Index Numbers and Productivity Measurement

783

This axiom implies that changes in prices over the period s to t do not influence the output quantities. This has the implication that output prices and quantities are independently determined, which may not be consistent with microeconomic theory of producer behavior. Axiom 2 – Weak Identity Axiom: If prices and quantities in period t are the same as those observed in period s, that is, pt = ps ; qt = qs then the quantity index must equal 1. Thus Qst (ps , qs , ps , qs ) = 1

(47)

This axiom links price and quantity movements and this identity axiom applies only when the price-quantity pair remains the same and, therefore, weaker than the strong identity axiom. Axiom 3 – Linear homogeneity in current period quantities: If the output vector in current period is multiplied by a positive constant, λ, then the resulting index is a multiple of the quantity index by the same constant λ. This means Qst (pt , λqt , ps , qs ) = λ · Qst (pt , qt , ps , qs )

for λ > 0.

(48)

Axiom 4 – Monotonicity: The quantity index must be increasing in current period quantities and decreasing in base period quantities. Qst (pt , q, ps , qs ) > Qst (pt , qt , ps , qs ) for all q > qt ; Qst (pt , qt , ps , q) < Qst (pt , qt , ps , qs ) for all q > qs

(49)

Axiom 5 – Homogeneity of degree zero in quantities: If quantity vectors in both periods are multiplied by the same factor then the quantity index remains unchanged. Qst (pt , λqt , ps , λqs ) = Qst (pt , qt , ps , qs ) for λ > 0

(50)

Axiom 6 – Commensurability or dimensional invariance: The quantity index must be independent of the units in which output quantities are measured. If all the quantities are scaled using a diagonal matrix, N × N with positive elements, then the quantity index remains unchanged. Thus, we have  Qst −1 pt , qt , −1 ps , qs = Qst (pt , qt , ps , qs ) where N xN > 0 (51) Axiom 6 – Proportionality: If quantities in period t are a constant multiple of quantities in period s or equivalently qit /qis = λ (λ > 0) for all i, then the quantity index must equal λ. Qst (pt , λqs , ps , qs ) = λ where λ > 0

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D. S. Prasada Rao

It can be seen that Axiom 1, the Strict Identity Axiom, is a special case of Axiom 6 where λ is set to 1. Axiom 7 – Mean value axiom: A quantity index number must lie within the smallest and largest changes in quantities from period s to t, or observed quantity ratios. This means     q1t q2t q1t q2t qN t qN t min , , ., ., ., , , ., ., ., ≤ Qst (pt , qt , ps , qs ) ≤ max q1s q2s qN s q1s q2s qN s (52) Axiom 8 – Time Reversal Test: The quantity index measuring changes from period s to t must be the reciprocal of the quantity index measuring changes from period t to s. Thus, the test requires Qst (pt , qt , ps , qs ) = 1/Qts (ps , qs , pt , qt )

or

Qst (pt , qt , ps , qs ) · Qts (ps , qs , pt , qt ) = 1

(53)

This axiom implies consistency between comparisons from s to t and comparisons in reverse from t to s. Axiom 9 – Circularity test: Suppose we are interested in comparisons involving several periods, say 1 to T then the circularity test requires: Q12 (p2 , q2 , p1 , q1 ) ×Q23 (p3 , q3 , p2 , q2 ) × · × QT −1T (pT , qT , pT −1 qT −1 ) · QT 1 (p1 , q1 , pT , qT ) = 1 (54) The circularity test requires consistency in quantity comparisons over several periods and suggests that chained comparisons from periods 1 to T must equal the reciprocal of comparison between T and 1. Axiom 10 – Transitivity Axiom: The circularity test is often stated in the form of  a transitivity axiom which states that for any three periods s, t, and t the following condition on pairwise output quantity comparisons holds: Qst (pt , qt , ps , qs ) = Qtt  (pt , qt , pt  , qt  ) · Qt  s (pt  , qt  , ps , qs )

(55)

The transitivity axiom can be interpreted as follows. Suppose a comparison between time periods s and t is desired. There are two alternative approaches. The first is a direct comparison between the two periods s and t. The second is to make an indirect chained comparison using the product, Qtt  (pt , qt , pt  , qt  ) · Qt  s (pt  , qt  , ps , qs ) through a third time period, t’. Transitivity axiom stipulates that the direct and the chained comparison must yield the same value for the index. The circularity test and transitivity axiom are similar in concept but not the same. This can be established from Eqs. (54) and (55). Transitivity implies identity test and the circularity test. The circularity test together with the identity axiom implies

19 Index Numbers and Productivity Measurement

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transitivity. However, most index number formulae satisfy the time reversal test but very few index number formulae satisfy the circularity and transitivity axioms. The following two axioms relate price and quantity indices to value change. Axiom 11: Factor test or product test: This test simply requires that the product of a quantity index and a price index used must be equal to their value index. N i=1 pit qit Qst (pt , qt , ps , qs ) · Pst (pt , qt , ps , qs ) = N = Vst i=1 pis qis

(56)

Equation (56) places no restriction on the formula chosen to compute Qst and Pst . For example if Laspeyres index to measure price changes and Paasche index to measure quantity changes are chosen, these two index numbers together satisfy the Factor test. A stronger requirement would be to insist that the same formulae be used to compute both price and quantity changes. Axiom 12 – Factor Reversal Test: According to this test, the product of price and quantity comparisons derived using the same formula must equal the value change. If Qst represents a quantity index formula and the price index uses the same formulae but with prices and quantity vectors interchanged then their product must equal value change. Thus N i=1 pit qit Qst (pt , qt , ps , qs ) · Qst (qt , pt , qs , ps ) = N = Vst i=1 pis qis

(57)

where Qst (qt , pt , qs , ps ) denotes a price index obtained using the same formula that is used for the quantity index and by interchanging prices and quantities. The factor reversal test represents a more stringent requirement than the factor test and consequently it is satisfied by only a few index number formulae. We introduce two more axioms, though not standard, that play an important role in characterizing the Fisher index. Axiom 13 – Price Reversal Test: This test states that the quantity index remains unchanged if the price vectors in periods s and t are reversed. Thus Qst (pt , qt , ps , qs ) = Qst (ps , qt , pt , qs )

(58)

This axiom reflects the general philosophy of the axiomatic approach that prices and quantities are independently determined. Axiom 14: Value dependence Axiom. This axiom states that the quantity index can be expressed as a function of value aggregates formed out of prices and quantities in the two periods expressed as: Qst (pt , qt , ps , qs ) = f

 N i=1

pit qit ;

N i=1

pis qis ;

N i=1

pit qis ;

N i=1

 pis qit (59)

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D. S. Prasada Rao

This axiom restricts the form of the quantity index number. The Laspeyres, Paasche and Fisher indices, among others, satisfy this axiom. However, none of the geometric indices satisfy this axiom. The tests listed above is a selected list designed to give a feel for the axiomatic approach to index numbers. There are many variants of these tests but it is sufficient to focus on the tests listed here. For an in-depth coverage of the tests and their interrelationships, the reader may refer to Diewert [16], of the Producer Price Index Manual [35] and Balk [6]. An important logical question that arises is whether there exists an index number formula that satisfies all these axioms. If such an index exists, that index can be used for measuring output and input quantity changes. Applying these tests to the formulae discussed so far, it is easy to check that all the indices satisfy positivity, continuity, commensurability, and the bounds tests. Laspeyres and Paasche indices do not satisfy time reversal test or the factor reversal test. However, Lapeyres price index paired with Paasche quantity index and Paasche price index paired with Laspeyres quantity index satisfy the product test. The Fisher index satisfies the time reversal test as well as the factor reversal test. None of the geometric indices listed, geometric-Laspeyres, geometric-Paasche, Törnqvist, and geometric Young indices satisfy the factor reversal test.19 With the exception of the geometric-Young index, none of the others satisfy the circularity test. The axioms are also used to characterize different index numbers. In what follows, a few important results are stated (without proofs) to conclude this section. Proofs of these results are available from Eichorn and Voeller [24], Diewert [16], IMF [35], and Balk [6] just to mention a few. Result 1: The only quantity index number Qst (pt , qt , ps , qs ) that satisfies the axioms of positivity, time reversal test, price reversal test and the factor reversal test is the Fisher price index. In this result the price reversal test in Eq. (58) is critical. A similar result characterizing the Fisher index using the value dependence axiom is also available. Result 2: A quantity index Qst (pt , qt , ps , qs ) satisfies the linear homogeneity axiom, factor reversal test and the value dependence test if and only if the index is the Fisher index. Now we state an impossibility theorem which establishes that we cannot find an index number formula that satisfies a small set of axioms. Result 3: There does not exist an index number formula that simultaneously satisfies the identity axiom, the circularity test and the factor reversal test. There are several other impossibility theorems for price and quantity index numbers which can be found in the book by Eichorn and Voeller [24].

19 The

only geometric or log-change index that satisfies the factor reversal test is the Sato-Vartia index [53, 62]. The Sato-Vartia index is similar to the Törnqvist index but the weights used are based on logarithmic averages of expenditure shares in the current and base periods.

19 Index Numbers and Productivity Measurement

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If one is seeking a formula that satisfies the circularity test or transitivity 20 the following result due to Funke et al. [31] is important. Result 4: A quantity index Qst (pt , qt , ps , qs ) satisfies monotonicity, linear homogeneity, identity, commensurability and circularity tests if and only if the index is of the geometric-Young form: Qst (pt , qt , ps , qs ) =

 N   qit wn n=1

qis

with wn > 0, n = 1, 2, . . . , N and

N 

wn = 1

i=1

(60) This index is sometimes referred to as a Cobb-Douglas type index. This is indeed an important result if one is selecting index number formula that is required to satisfy transitivity as well as a few other relatively basic properties listed in Result 4. A point to note here is that weights in the geometric-Young index in Eq. (60) must be independent of the time periods under consideration. This rules out indices such as the geometric-Laspeyres, geometric-Paasche and the Törnqvist index. In concluding this section it is useful to remind the readers that the focus has been mainly on output quantity index numbers. Similar axioms can be stated for output price index numbers as well as input price and input quantity index numbers. Input quantity index numbers can be obtained by simply replacing {pt , qt , ps , qs } with input price and quantity vectors for the two periods, {wt , xt , ws , xs } and all the axioms and results are then applicable for these vectors.

Economic Theoretic Approach to Output and Input Quantity Index Numbers Economic theoretic or economic approach to index numbers has a long history. In his seminal work, Konus [37] articulated an economic theoretic approach to measure changes in the cost of living using the theory of consumer behavior. The Konus index serves as a starting point for any discourse on consumer price index numbers. Ragnar Frisch [30] in his survey article on index numbers referred to this approach as the functional approach. This approach explicitly recognizes the existence of a functional relationship between price and quantity data arising out of decision making by consumers. On the production side, the work of Fisher and Shell [29] offered a producer behavior perspective to producer price index numbers – their contributions form the basis for the exposition in this section. However, it is the path-breaking work of Diewert [13] followed by Caves et al. [10] that has provided

20 Transitivity is a property which is considered essential in the context of multilateral comparisons

of prices and quantities. See Section “Economic-Theoretic Approaches to Measurement of Output Quantity Change” for further discussion on multilateral comparisons.

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D. S. Prasada Rao

a framework to establish a systematic link between the Laspeyres, Paasche, Fisher, and Törnqvist index numbers and producer behavior operating within the confines of a production technology. Diewert [15, 16] and many of his works21 have been instrumental in establishing economic theoretic properties of these index numbers which were until then considered to be either statistical or heuristic or axiomatic. In particular Diewert has shown that the Fisher and Törnqvist index numbers are exact measures of output and input quantity and price change under certain specific forms of the production technology and under the assumption that the observed input and output quantities are both technically and allocatively efficient. Exposition in this section draws heavily from Diewert [15].22 It is impossible to provide a comprehensive review of all the material available on this topic. Further, some of the work in this area relies on sophisticated economic and mathematical analysis. The principal objective of this section is to offer the reader a flavor of what is available and provide an overview of developments in this area. Throughout this section the focus is on bilateral comparisons involving a firm over two time points. In order to limit the size of this section, the focus is on output related measurement issues. Input related measures can be derived by following similar concepts and measures with appropriate changes. Under a given technology, a firm’s objective, for example, could be to maximize revenue by making decisions on outputs produced given a vector of inputs and a vector of output prices. Just as easily, the firm’s objective could be to minimize the cost of producing a given output vector with a given vector of input prices.

Notation and Basic Framework The notation and framework for this section is the same as that discussed in Section “Notation and Preliminaries.” Reference will be made to the regularity conditions R.1 listed in that section. The notion of revenue function plays an important role in the discussion on output price and output quantity index numbers. Given the production technology, the revenue function,23 r (p, x), is defined as the maximum revenue feasible with a given vector of input endowments and output prices. This function is defined as: r (p, x) ≡ maxq {p.q : (x, q) ∈ S}

21 Research

(61)

work and contributions of Diewert in the area of index numbers can be found and downloaded from his website: https://economics.ubc.ca/faculty-and-staff/w-erwin-diewert/ 22 Of all the papers on this subject, Diewert [15] is easily one of the best expositions where the treatment is focused solely on the economic theoretic approach and the matter is developed quite systematically. The earlier work of Fisher and Shell [29] is another systematic exposition on this subject but Diewert [15] is comprehensive in its coverage of this subject matter. 23 Note that this terminology differs slightly from that used by Diewert [15] but this terminology is more intuitive.

19 Index Numbers and Productivity Measurement

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Properties of the revenue function can be derived under the regularity conditions R.1. Important properties of the revenue function include nonnegativity, linear homogeneity, continuity, convexity in output price vector p; and it is nondecreasing in p. The revenue function is nondeceasing and concave in input vector x.

Economic-Theoretic Approaches to Measurement of Output Quantity Change The economic theoretic approach to measurement of output quantity changes depends on whether output price data are available. In most practical situations, price data are available and it is reasonable to assume revenue maximizing behavior by the firms. When output price data are available, there are two options. The first option is to measure output changes directly using the Samuelson and Swamy [52] approach. Alternatively, one may first obtain a measure of price change discussed in Fisher and Shell [29] and use it to measure output change applying the deflation method. In the case where price data are not available, then a direct approach based on the Malmquist output distance function is used in measuring output quantity change. Price data are usually not available for the output of nonmarketed goods and services such as education, health, and police services. Material in this section is divided into the following four sections each dealing with a different economictheoretic approach to the construction of output quantity index numbers. Direct approach in the presence of output price data: Under this approach, the output quantity index is measured directly but it is based on the notion of a revenue function introduced in Eq. (61). Fisher and Shell approach to output price index numbers: In the presence of output price data, economic-theoretic measurement of price change is considered using the Fisher and Shell approach. Indirect or deflation approach to output quantity index numbers: This approach makes use of the fundamental index number decomposition applied to the production side whereby change in the revenue from period s to t is expressed as a product of price change and output quantity change. In this case, the indirect measure is defined as the ratio of value change and a measure of price change. Essentially this approach compares value output in the base period with value of output in the current period after adjusting for movements in output prices and thus referred to as the deflation approach . Direct approach based on Malmquist output distance function: This approach provides a measure of quantity change without the need for any price data and it is anchored on the Malmquist output distance function leading to what is usually referred to as the Malmquist output quantity index . As these approaches are all anchored on the underlying production technology, empirical implementation and numerical measures of quantity indices depend in principle on the knowledge of the production technology. However, in practice, technology used by the firms is not known. In many practical situations, the only observations available are the actual quantities of inputs used and outputs produced

790

D. S. Prasada Rao

along with data on input and output prices faced by the firms. These can be a random sample from firms operating in the industry. In many practical situations where index number methods are used only two data points – either observations for a firm at two different points of time or for two firms at a given point of time – are available. On the face of it, this looks like a mission impossible, but under certain assumptions it can be shown that these measures of output change can be reasonably well approximated by standard index number formulae such as the Törnqvist index and the Fisher index.

Direct Measures of Quantity Change in the Presence of Price Data The approach described here is due to Samuelson and Swamy [52] and further discussion of the approach can also be found in Sato [54]. It makes use of the existence of production technology S which satisfies the regularity conditions R.1 stated in Section “Notation and Preliminaries” and the availability of price and quantity data for both outputs and inputs. Here the output index is defined as the ratio of the maximum revenue generated in periods s and t using production technologies available in these periods, Ss and St ; input quantity vectors xs and xt at a common reference price vector pR . Using the revenue function defined in Eq. (61), the Samuelson-Swamy (SS) output index is defined as: QSS st ≡

r t (pR , xt ) r s (pR , xs )

(62)

This index depends on the choice of the reference price vector. If the base period output price vector is the reference vector, then the resulting Laspeyres-type SS output index is given by: = QSS−L st

r t (ps , xt ) r s (ps , xs )

(63)

If the firm is technically and allocatively efficient in both periods, revenue maximizing behavior implies that the observed revenue in the base period equals the denominator in (63). Then, QSS−L = st

r t (ps , xt ) r t (ps , xt ) =  N r s (ps , xs ) i=1 pis qis

(64)

Similarly, Paasche-type SS index can be defined using the current period price vector as the reference vector. Then, QSS−P st

N pit qit r t (pt , xt ) = si=1 = s r (pt , xs ) r (pt , xs )

(65)

19 Index Numbers and Productivity Measurement

791

If there is sufficient data, for example data on a large number of firms in both years, then it would be possible to compute indices in (64) and (65) by first identifying production technologies in periods s and t. The SS index in Eq. (62) can be expressed as: QSS st =

r s (pR , xt ) r t (pR , xt ) × s s r (pR , xt ) r (pR , xs )

(66)

Equation (66) shows that the SS output index is the product of two components. The first component provides a measure of technical change as it measures the change in revenue, for a given reference price vector pR and input quantity vector, xt , that is purely attributable to change in technology from period s to t. The second component is a measure of change in revenue purely attributable to change in input vector xs to xt .

Indirect Output Quantity Index Numbers Using Output Price Index Numbers The indirect approach to the measurement of output change was first described in Fisher and Shell [29]. Under this approach, a suitable output price index is first measured and the quantity index is then derived indirectly using the fundamental index decomposition of revenue (value) change into output price change and output quantity change. Then n pit qit Revenue change = ni=1 i=1 pis qis Price index = Pst (pt , qt , ps , qs ) n n Indirect Quantity index ≡ pit qit / pis qis /Pst (pt , qt , ps , qs ) i=1

i=1

(67) In order to derive the quantity index defined in Eq. (67), it is necessary to obtain an economic theoretic measure of output price change. Here one may employ the Fisher-Shell price index24 which uses the revenue function to define the output price index.

The Fisher-Shell Output Price Index The Fisher and Shell (FS) (1972) output price index, PstF −S , is defined as the ratio of maximum revenues generated under prices prevailing in periods s and t using a

24 This

prices.

is somewhat similar in concept to the Konus index used in measuring changes in consumer

792

D. S. Prasada Rao

reference technology, SR , and a fixed reference input vector xR . The FS price index is given by  R R (p , x ) t R PstF −S = Pst pt , ps , x R , S R ≡ R R (ps , xR )

(68)

In the case of singe output, this price index reduces to the ratio of output prices in the two periods. Apart from the assumptions on the production technology, under the assumption that the observed output quantities in periods t and s are optimal in that they are solutions to the problem of maximizing revenue given the technology prevailing in these periods and for the input vectors available. This means that these observations represent output vectors that are technically and allocatively efficient. Thus, R t (pt , xt ) =

N 

pit qit = pt · qt and R s (pt , xt ) =

i=1

N 

pis qis = ps · qs

(69)

i=1

Equation (69) provides information helpful to link the FS output price index concept in (68) to the price index numbers discussed in Section “Index Number Approach to Measuring Quantity Change.” The following results are stated without proof. Result 5: If the price vectors are strictly positive, the input vector is nonnegative with at least one positive input, and if the reference technology satisfies the standard regularity conditions of production technology and that a nontrivial output vector is feasible with the given input vector, then the FS price index in Eq. (68) is within the minimum and maximum observed price ratios.  mini

   pit pit FS , i = 1, 2, . . . , N ≤ Pst ≤ maxi , i = 1, 2, . . . , N pis pis

(70)

For proof, see Diewert [15], p. 1058. This means that the FS price index satisfies the standard bounds-axiom.

FS- Laspeyres and FS-Paasche Output Price Index Numbers In empirical implementation of the Fisher-Shell index, the most obvious choices for the reference technology and the input vectors would be those prevailing in periods t and s. When the base period input vector and production technology are used, it leads to the FS-Laspeyres index indicating that the index relies on base period information. Similarly FS-Paasche index uses period t technology and the current/comparison period input vector. These are defined as: PstF S−L ≡

R s (pt , xs ) R t (pt , xt ) F S−P and P ≡ st R s (ps , xs ) R t (ps , xt )

(71)

19 Index Numbers and Productivity Measurement

793

Result 6: Under the assumption that the regularity conditions R.1 hold in periods t and s and that the observed outputs are optimal in periods t and s (technically and allocatively efficient), then the FS-Laspeyres and FS-Paasche indices satisfy the following inequalities. PstF S−L =

R s (pt , xs ) R s (pt , xs ) pt · xs = ≥ = PstL s R (ps , xs ) ps · xs ps · xs

(72)

PstF S−P =

pt · xt pt · xt R s (pt , xt ) = s ≤ = PstP s R (ps , xt ) R (ps , xt ) ps · xt

(73)

and

This means that the standard Laspeyres and Paasche output price indices discussed in Section “Index Number Approach to Measuring Quantity Change” provide bounds for the economic-theoretic FS output price index numbers. This is the first link between the economic theoretic output price index and the index number formulae based on statistical and other considerations. Proofs of Result 6 and the results that follow can be found in Diewert [17]. Result 7: Combining results 5 and 6, the following bounds hold for the FSLaspeyres and FS-Paasche indices:   pit pt · xs F S−L = ≤ Pst ≤ maxi , i = 1, 2, . . . , N ps · xs pis   pit pt · xt mini , i = 1, 2, . . . , N ≤ PstF S−P ≤ = PstP pis ps · xt PstL

(74) (75)

Equations (74) and (75) provide useful upper and lower bounds for the unknown FS-Laspeyres and FS-Paasche index numbers which can be computed using observed data on output prices and quantities even in the absence of any knowledge about the actual production technology. The following result shows that under certain conditions there exists a theoretical FS index for which the Laspeyres and Paasche indices simultaneously provide lower and upper bounds respectively. Result 8: Suppose the price and input vectors in periods s and t are strictly positive and that technologies in period s and t are well-behaved, then there exists a constant λ, 0 ≤ λ ≤ 1, such that the following bounds hold for the FS index defined for production technology25 S ∗ = {(1 − λ)S s + λS t } and the input vector x ∗ = {(1 − λ)x s + λx t }:

25 The

convex combination of technologies Ss and St is the convex hull of the two technologies. For example, this implies that for a given input vector x, the production possibility set under the convex combination of technologies would be the smallest convex set that contains the production possibilities sets in periods s and t, Ss and St .

794

D. S. Prasada Rao

PstLP =

 pt · xt pt · xs ≤ PstF S pt , ps , x ∗ , S ∗ ≤ = PstP ps · xs ps · xt

(76)

Bounds given in Eq. (76) are important since they bound the unknown FisherShell output price index defined for a weighted average of the input vector and the corresponding convex combination of technologies. This inequality means that if an average of the standard Laspeyres and Paasche indices is taken, then it is likely to get close to the unknown theoretical output price index. These bounds offer support to the use of the Fisher index. Second, if the Laspeyres and Paasche indices are numerically close then the unknown theoretical index lies within a small numerical interval. The next result is similar to the result proved by Diewert [13, 14] where he established the notion of exact and superlative consumer price index numbers. This version focuses on the production side counterpart and offers a result for the FS output price index. Result 9: Suppose the regularity conditions R.1 hold and that the observed output vectors in periods s and t are revenue maximizing under respective technologies and output prices (i.e., observed outputs are technically and allocatively efficient). Further, let the output revenue function at time τ (τ = s,t) in the two observed periods has translog functional form: ln R τ (p, x) = a0τ +

K 

bkτ ln xk +

k=1

+

N 

+

k=1 l=1

aiτ ln pi +

i=1 K N  

K K 1  τ bkl ln xk ln xl 2

N N 1  τ aij ln pi ln pj 2

(77)

i=1 j =1

τ cik ln pi ln xk

τ = s, t

i=1 k=1 τ = bτ ; a τ = a τ for all i, j = 1, 2, . . . , N and k, l = 1, 2, . . . , K τ = where bkl lk ij ji s, t. Assume further that the second order coefficients associated with prices in Eq. (77) are the same in both periods, i.e., aijs = ajt i for all i and j, then the geometric average of Laspeyres-FS and Paasche-FS output price index numbers is exactly equal to the Törnqvist index in Eq. (40):

 wi1 +wi2 N 

1/2  R s (p , x ) R t (p , x ) 1/2  2 pit t s t t F S−L F S−P Pst · t · Pst = s = = PstT T R (ps , xs ) R (ps , xt ) pis i=1

(78) This result is quite important as it forges a link between index number theory and practice. The left-hand-side of Eq. (78) is an economic theoretic concept based on the Fisher-Shell (1972) approach and the right-hand-side of the equation is a

19 Index Numbers and Productivity Measurement

795

practitioners’ measure of output price change which is a weighted geometric average of commodity specific price change. Result 10: Suppose all the conditions stated in Result 9 hold and that the revenue function is given by a quadratic function. If the second order coefficients corresponding to output prices are identical in both periods s and t then the geometric average of FS Laspeyres and FS Paasche indices is exactly equal to the Fisher output price index. Thus

1/2  R s (p , x ) R t (p , x ) 1/2 t s t t PstF S−L · PstF S−P · = R s (ps , xs ) R t (ps , xt )  1/2 N N i=1 pit qis i=1 pit qit = N · N = PstF i=1 pis qis i=1 pit qit

(79)

Results 9 and 10 are important in establishing links between economic theoretic measures of output price change and the commonly used Fisher and Törnqvist index numbers which can be computed using observed price and quantity data without the need for any additional information or knowledge about the production technology.

Exact and Superlative Index Numbers At this stage it is useful to introduce the reader to the notion of exact and superlative index numbers which were introduced in a path-breaking contribution by Diewert [13]. An index number is said to be exact if it is equal to the economic theoretic index when the revenue function takes a particular form. Result 9 shows that the Törnqvist index is exact for the translog revenue function, and we have the result that Fisher index is exact for the quadratic revenues function, respectively. Diewert [13] designates an index to be superlative if it is exact for a given revenue function which provides a second order approximation to the unknown revenue function. Since translog function provides a second order approximation to the unknown revenue function, Törnqvist index is a superlative index. Similarly, the Fisher index is also superlative since it is exact for a quadratic revenue function which also provides a second order approximation to the unknown revenue function. These results hold only when observed output vectors are technically and allocatively efficient for the prevailing technologies and output prices. An advantage with the TT and Fisher indices is that they are numerically very close to each other, almost to four significant digits [14]. Therefore, empirically it does not matter as to which index is chosen.

Indirect Output Quantity Index Returning to Eq. (67) and the indirect output quantity index based on Fisher-Shell output price index, QSS st is given by:

796

D. S. Prasada Rao

QFstS

=

 N i=1

 N =

 N pit qit / pis qis /PstF S (pt , qt , ps , qs ) i=1

N

i=1 pit qit /

 N



i=1 pis qis

PstF S (pt , ps , x)

=

N

i=1 pit qit /



i=1 pis qis

(80)

  R R pt , x R /R R ps , x R

where the technology and input vector in reference period R are used in defining the Fisher-Shell Index. The following result can be easily established. The indirect quantity index in Eq. (80) satisfies properties of positivity, continuity, commensurability, and time reversal test. However, it fails the homogeneity property which states that if the output vector in period t is a scalar (positive) multiple, λ, of output vector in period s then the output quantity index must be equal to the scalar ([15], p. 1071). Note here that implementation of the Fisher-Shell output index requires the selection of a reference price vector and a reference technology. Results 9 and 10 suggest that the true Fisher-Shell price index in (80) may be replaced by exact or superlative indices such as the Fisher and the Törnqvist price indices.

Direct Quantity Index Based on Malmquist Distance Function The direct quantity index based on the Malmquist distance function is usually referred to as the Malmquist output index. The Malmquist quantity index, denoted R by QM st , is defined using a reference technology, S and a reference input vector, xR . It is defined as: QM st ≡

DoR (xR , qt ) DoR (xR , qs )

(81)

There are several advantages with the Malmquist output index. First, the Malmqust output index in (81) does not require any output price data. This is an advantage when it comes to measuring output growth in nonmarketed goods and services like education and health. Second, the Malmquist index satisfies the homogeneity property which is not satisfied by the indirect index. The Malmquist output index in (81) depends on the reference technology and reference input vector used unless the technology satisfies homothetic separability condition ([15], p. 1071) which states that the output distance function must have the following form: Dot (x, q) =

1 g(q)h (t, x)

(82)

where g(.) and h(., .) are appropriately defined functions (see [15], pages 1069 and 1071). In general, the homothetic separability condition does not hold empirically

19 Index Numbers and Productivity Measurement

797

and therefore it is necessary to select the reference technology and the reference quantity vector before the Malmquist quantity index in Eq. (81) can be computed. The most natural choices available are the base and current period input vectors and technologies. This leads to two alternative forms of the Malmquist index.

Malmquist-Laspeyres Index As the title suggests, here the technology and input vector associated with the base period are selected. QM−L ≡ st

Dos (xs , qt ) Dos (xs , qs )

(83)

If the production vector qs is on the frontier of the output set associated with input vector xs and technology in period s, we have Dos (xs , qs ) = 1. In this case, the Malmquist-Lasdpeyres index simplifies to QM−L = Dos (xs , qt ). st

Malmquist-Paasche Index Use of information from the comparison or current period leads to the MalmquistPaasche index: QM−P ≡ st

Dot (xt , qt , ) Dot (xt , qs )

(84)

If the production vector qt is on the frontier of the output set with input vector xt and technology in period t, then the Malmquist-Paasche index simplifies to QM−P = 1/Dot (xt , qs ). st

Malmquist-Fisher Index In the spirit of the Fisher index, the Malmquist-Fisher index is defined as the geometric mean of the –Malmquist-Laspeyres and Paasche indices in Eqs. (83) and (84).  ≡ QM−F st

Dos (xs , qt ) Dot (xt , qt ) · Dos (xs , qs ) Dot (xt , qs )

1/2 (85)

The Laspeyres, Paasche, and Fisher versions of the Malmqust output index require the knowledge of the production technology or equivalently the output distance function so that all the distances in Eqs. (83), (84), and (85) can be evaluated. In the absence of this key information, the following results, stated without proof, from Diewert [15] are quite useful. Result 11: Suppose the regularity conditions R.1 hold and that the observed output vectors in periods s and t are revenue maximizing under respective technologies and output prices (i.e., observed outputs are technically and allocatively efficient). The Malmquist-Laspeyres output index in (83) is bounded from below by the Laspeyres output quantity index.

798

D. S. Prasada Rao

QM−L ≡ st

Dos (xs , qt ) ps · qt ≥ QL st = Dos (xs , qs ) ps · qs

Result 12: Suppose the regularity conditions R.1 hold and that the observed output vectors in periods s and t are revenue maximizing under respective technologies and output prices (i.e., observed outputs are technically and allocatively efficient). The Malmquist-Paasche output index is bounded from above by the Paasche quantity index.  QM−F ≡ st

Dos (xs , qt ) Dot (xt , qt ) · Dos (xs , qs ) Dot (xt , qs )

1/2 ≤ QPst =

pt · qt pt · qs

Result 13: Suppose the regularity conditions R.1 hold and that the observed output vectors in periods s and t are revenue maximizing under respective technologies and output prices (i.e., observed outputs are technically and allocatively efficient). Suppose further that technologies in periods s and t are sufficiently well behaved so that the distance functions are well defined for convex production technologies. There exists a positive scalar λ such that the Malmquist index defined using a convex combination of the production technologies in periods s and t, λSs + (1 − λ)St and weighted average of input vectors in periods s and t, λxs + (1 − λ)xt is bounded by the Laspeyres and Paasche Quantity index numbers. Stated formally, this result gives the following bounds: ps · qt D ∗ (x∗ , qt ) ≤ QPst ≤ o∗ ps · qs Do (x∗ , qs ) pt · qt = ; where S∗ = λSs + (1 − λ) S t and x∗ = λxs + (1 − λ) xt pt · qs

QL st =

This result implies that by using the Fisher index it may be possible to get closer to the unknown Malmquist quantity index. This theorem is particularly useful when the Laspeyres and Paasche indices computed using observed price and quantity data are numerically close to each other. The following result due to Caves et al. [10] provides a direct link between the Törnqvist index, Fisher Index and the Malmquist-Fisher theoretical index in Eq. (85). This result is similar in content to Result 9. Result 14: If the following conditions hold: (i) the production technology satisfies the standard regularity conditions stated in R.1 hold; (ii) the output vector, output price vector and input vector are all strictly positive26 ; (iii) the output vectors in periods s and t are technically and allocatively efficient, respectively, for the input vectors and technologies in these two periods; and (iv) the distance function assumes the following translog functional form:

26 All

elements of these vectors are strictly positive.

19 Index Numbers and Productivity Measurement

ln Doτ (x, q) = a0τ +

K 

bkτ ln xk +

k=1

+

N 

aiτ lnqi +

K N  

K K 1  τ bkl ln xk ln xl 2 k=1 l=1

i=1

+

799

1 2

N N  

aijτ lnqi lnqj

i=1 j =1

τ cik lnqi ln xk

τ = s, t

i=1 k=1

where τ = bτ ; a τ = a τ for all i, j = 1, 2, . . . , N and k, l = 1, 2, . . . , K τ = s, t; bkl lk ij ji and (v) aijs = ajt i for all i, j = 1, 2, . . . , N, then the economic theoretic Malmquist output index based on the Malmquist-Fisher type index in Eq. (85) is exactly equal to the Törnqvist output index. That is: 

Dos (xs , qt ) Dot (xt , qt ) · Dos (xs , qs ) Dot (xt , qs )

1/2 =

 N   qit i=1

wi1 +wi2 2

qis

= QTstT

(86)

This result is usually restated to indicate that the Törnqvist quantity index is exact for a translog output distance function. This index is also considered superlative as the translog distance function provides a second order approximation to any unknown output distance function. A result similar to Result 14 can be established to show that if all the conditions in Result 14 hold and further if the distance function is quadratic with identical second order coefficients in both periods s and t, then the theoretical Malmquist-Fisher type index is exactly equal to the Fisher index. In this case: 

Dos (xs , qt ) Dot (xt , qt ) · Dos (xs , qs ) Dot (xt , qs )



1/2 =

N i=1 pis qit N i=1 pis qis

N

i=1 pit qit

· N

i=1 pit qis

1/2 = QFst (87)

Equation (87) shows that the theoretical Malmquist index is equal to the Fisher index introduced in Section “Index Number Approach to Measuring Quantity Change” provided the distance function is quadratic. These two results are significant since they show that under plausible assumptions and conditions on the distance function and data collected, it is possible to compute theoretical Malmquist output index using only price and quantity data from these two periods. The main assumption here is that the distance function is quadratic and has the same second order coefficients associated with the outputs. The Fisher index is thus known as an exact and superlative index number. These indices can be considered as a good approximation to the unknown theoretical indices. So far three alternative theoretical measures of output quantity change have been considered – the direct index due to Samuelson and Swamy [52]; the indirect

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index based on deflated value change with the Fisher-Shell output deflator; and the Malmquist output index based on the notion of output distance function. This section is concluded with two key points. The first point is that of all the three indices, the Malmquist output index is the only index that satisfies the homogeneity condition. The second point is that all the three indices coincide when the production technology satisfies the homothetic separability condition (see Theorem 5, [15], p. 1067).

Input Quantity Index Numbers This chapter so far has entirely focused on economic theoretic approach to output quantity index numbers. For productivity measurement, input quantity index numbers are equally important. The general apparatus used in the construction of input quantity index numbers is much the same as that used for output index numbers. The main difference is that the input quantity indices make use of input cost functions and Malmquist input distance functions. Given the production technology, the cost function is defined as the minimum cost of producing a given output vector q when the input price vector is w and the technology is represented by S. This function is defined as: c (w, q) ≡ minx {w.x : (x, q) ∈ S} Using the cost function, the indirect input quantity index is defined as the change in input costs after adjusting for movements in price change. If our focus is on comparisons over two periods, s and t, the indirect input quantity index27 is defined using these measures: K k=1 wkt xkt Cost change = K k=1 wks xks Input Price index = I P st (wt , xt , ws , xs )   K K wkt xkt / wks xks /IP st (wt , xt , ws , xs ) Indirect Quantity index I Qst ≡ k=1

k=1

In order to derive the indirect input quantity index, we need a measure of input price change. The input price index, following Fisher and Shell [29], is defined as:  C R (w , q ) t R I P Fst −S = I P st wt , ws , qR , S R ≡ R C (ws , qR )

27 Denote

input quantity and price index numbers as IQ and IP- this notation helps to distinguish this from the output quantity and price index numbers.

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As in the case of output price index numbers, under assumptions similar that in Section “Notation and Basic Framework” and under the assumption the observed input quantities are optimal for the input prices and output quantities prevailing in the two periods, that is, C t (wt , qt ) =

K 

wkt xkt = wt · xt and C s (ws , qs ) =

k=1

K 

wks xks = ws · xs ,

k=1

then the equivalent of Results 5 to 14 can be proved. The Laspeyres and Paasche input price index numbers, I P FstS−L and I P FstS−P usually serve as upper and lower bounds for the true input price index. The main message is that under the assumptions stated in Results 9 to 14, and under the assumption that the input cost function is of the translog functional form, the input price index can be computed using the Törnqvist input price index number

1/2  C s (w , q ) C t (w , q ) 1/2 t s t t I P FstS−L · I P FstS−L · = C s (ws , qs ) C t (ws , qt )  cis +cit K   2 wit = = I P TstT wis k=1

where ckτ =

w x K kτ kτ k=1 wkτ xkτ

τ = s, t represents the cost share of input k in period τ

(=s,t). Further, if the input cost function is quadratic then we have

1/2  C s (w , q ) t s I P FstS−L · I P FstS−L · = C s (ws , qs )  K k=1 wkt xks = K k=1 wks xks

 C t (wt , qt ) 1/2 C t (ws , qt ) 1/2 K k=1 wkt xkt · K = I P Fst k=1 wks xkt

Similar results also hold in the case of Malmquist input quantity index numbers. The theoretical input quantity index is defined using the Malmquist input distance function: I QM st ≡

DiR (xt , qR ) DiR (xs , qR )

Under various regularity conditions on the technology set and assumptions on the form of the input distance function and under the assumption that observed input quantity vectors are technically and allocately efficient, the Fisher and Törnqvist quantity index numbers are exact and superlative for the Malmquist input quantity

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index when the distance functions are, respectively, quadratic or translog functional form.

Summary The economic theoretic approach to output and input price and quantity indicators is well developed. This section has just touched upon the basic elements of this approach. The main objective of this section is to give an overview rather than to provide a detailed description of all the methods and to offer proofs of various results. Interested reader will benefit from reading the original works of Fisher and Shell [29], Samuelson and Swamy [52], Sato [54], Malmquist [41], Diewert [13, 15, 16], and Balk [5]. Regarding the choice between compiling direct and indirect quantity indices, the general advice is to use direct quantity indices based on Malmquist distance function as they satisfy the homogeneity condition. In the case where the revenue and cost functions, input and output distance functions satisfy homothetic separability assumption then the direct and indirect measures coincide and there is no need to choose between the alternative approaches. The basic take-home message is that empirical implementation of the economic theoretic indices requires complete knowledge of the functional form for the revenue, cost, Malmquist input and output distance functions. However, in the absence of such information, economic theoretic approach can provide empirically feasible measures of output price, output quantity, input price, and input quantity index numbers based on observed price and quantity data when certain assumptions and regularity conditions are satisfied. This approach is particularly well suited when limited data are available for a firm over two periods or only for two firms at a given point of time. In such cases, under some reasonable assumptions, the Fisher and Törnqvist indices provide useful tools for measuring output and input quantity change which can then be used in measuring productivity change.

Special Topics This chapter, thus far, has provided an extensive survey of various index number approaches for measuring output and input quantity change that can in turn be used to measure productivity change. This section is devoted to two special topics that deserve special attention and additional discussion. The first topic addresses a question which is somewhat fundamental in that it concerns the choice between the two alternative approaches which are both designed to measure output and input quantity change. The first approach, discussed in Section “Index Numbers Based on Quantity Aggregates” of this chapter, advocates construction of quantity (output or input) aggregates before measuring quantity change. Under this approach, aggregate quantity levels are computed first and change is measured using these levels. The second approach, discussed in Sections

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“Index Number Approach to Measuring Quantity Change,” “Axiomatic Approach to Index Numbers,” and “Economic Theoretic Approach to Output and Input Quantity Index Numbers,” in contrast constructs measures of quantity change directly using standard index number methods. The practitioner is then faced with a choice between these two approaches. Section “Use of Quantity Aggregates to Measure Quantity Change” below is devoted to this problem of choosing between these two approaches. The discussion so far has typically focused on comparisons over two periods s and t or across two firms. However, in practical situations one may be required to make multilateral comparisons across a number of firms or measure change over several periods of time. Section “Transitivity and Quantity Index Numbers” focuses on the problem of multilateral comparisons of quantity change or levels and discuss the role of transitivity requirement. The reader may recall that transitivity and the circularity axioms were discussed in the context of the axiomatic approach to index numbers but in this section the main concern is on the implications of transitivity for the choice of index number methods for measuring quantity change.

Use of Quantity Aggregates to Measure Quantity Change The approach described in Section “Index Numbers Based on Quantity Aggregates” where output quantity change is measured by the ratio of quantity aggregates: Qst =

Q (q1t , q2t . . . , qN t ) Q (qt ) = Q (qs ) Q (q1s , q2s . . . , qN s )

where the aggregator function Q(qτ ); τ = s, t is required to satisfy a minimal set of conditions: nonnegativity, nondecreasing in arguments, and linear homogeneity. In the implementation of this approach based on quantity aggregates advocated by O’Donnell [44, 46, 47], a few aspects need special consideration. (1) Are there other axioms, in addition to the three conditions, that need to be considered in selecting quantity aggregator functions? If so, what are the implications of the additional conditions? This aspect is considered in Section “An Additional Axiom for Quantity Aggregates.” (2) One of the indices canvassed in O’Donnell [44, 46] is the Lowe index which makes use of value aggregates evaluated at a fixed price vector. This index is attractive since it is consistent with the notion of additivity of quantity aggregates (see Section “An Additional Axiom for Quantity Aggregates” below) and it leads to transitive comparisons. Despite these useful properties, the Lowe index can lead to strange ordering of output quantity vectors. This issue is considered in Section “The Lowe Index.” (3) If the quantity aggregate approach for measuring quantity change is adopted, then it is natural and symmetric to use a similar approach for measuring price change based on price aggregates. Will the measures of quantity change and price change based on respective aggregates be consistent with observed changes in the revenue or costs? Will the product of quantity change and price change, each based on respective aggregates in periods

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s and t, be equal to the observed revenue change or change in input costs? This issue is discussed in Section “Value Decomposition and the Use of Quantity and Price Aggregates.” (4) The use of quantity aggregates to measure quantity change is implicitly based on the assumption that movements in prices and quantities are independent. Economic theory suggests that these are not independent. How does one evaluate validity of this assumption using observed data? This question is answered in Section “Are Price and Quantity Data Independent?”

An Additional Axiom for Quantity Aggregates The current treatment of quantity aggregates uses a minimal set of properties for the quantity aggregate. The properties of nonnegativity, nondecreasing in its arguments and linear homogeneity listed in O’Donnell [44, 46] is indeed a minimal set. While these properties are desirable, these may not capture all the characteristics expected of an aggregate function. Now consider an additional property which is an intuitive requirement one would expect from a quantity aggregate. This is the additivity property which requires the quantity aggregate of the sum of two quantity vectors to be equal to the sum of quantity aggregates of the two quantity vectors. Stated formally, the quantity aggregate Q(q) is said to satisfy additivity if for two output vectors, q1 and q2 , we have: Q (q1 + q2 ) = Q (q1 ) + Q (q2 ) for all q1 , q2 ≥ 0

(88)

This type of additivity ensures that if a firm produces q1 in the first half of the year and q2 in the second half of the year, then the quantity aggregate for the year would be the same as the sum of quantity aggregates in the first and second half of the year. The additivity axiom is also important if one were aggregating output of firms over a geographical region. If we add additivity to the three properties already listed, then the class of functions which satisfy these four properties together would be more restricted. The question is how restricted is such a class? The answer to this can be found from the solution to the first fundamental Cauchy equation: f (x + y) = f (x) + f (y). This is similar to (88) which is in vector form. The following result provides the answer. Result 15: If the function Q (q) is a real-valued function that maps vectors in the N to R , and if the function Q (.) nonnegative orthant of the N-dimensional space, R+ + satisfies, in addition to properties (1) to (3), the additivity property stated in (Eq. 88) then there exists a vector of positive constants c = {c1 , c2 , . . . , cN } such that Q (q) =

N i=1

ci qi f or all q = (q1 , q2 , . . . , qN )

(89)

Proof of this for the scalar case is in Diewert [19] and in earlier works of Aczèl [1] and Eichorn [23]. The multivariate case with multiple outputs has been considered in Kuczma and Gilányi [38].

19 Index Numbers and Productivity Measurement

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This result implies that once we additivity is imposed in addition to axiomatic properties (1) to (3), then the class of functions narrows to a simple linear function  shown in Eq. (89). The constants in this equation, ci s, may be interpreted as prices at which outputs are evaluated. Such an aggregate is consistent with the notion of a quantity aggregate evaluated at a given set of prices. This function is indeed the Lowe aggregate and it results in the Lowe quantity index discussed in Section “Index Numbers Based on Quantity Aggregates.” Discussion presented here is just to alert the practitioner that it is important to be clear about what properties one wishes to see in the aggregate and that it may in turn restrict the choice of an appropriate aggregator function.

The Lowe Index Let pR represent a vector of output prices which serves as the reference price vector in defining the Lowe aggregate. Then the value of the quantity aggregate in periods s and t, is given by: QLowe ≡ τ

N i=1

piR · qiτ τ = s, t

(90)

where {piR : i = 1, 2, . . . , N} are fixed constants. Lowe quantity index is then given by: QLowe = st

N i=1 piR qit = N QLowe (q ) s t i=1 piR qis QLowe (qt ) t

(91)

The use of Lowe index for making price and quantity comparisons is not new. Since the time Lowe [40] proposed this index, it has been the subject of considerable research. Hill [34] provides an overview of the use of the Lowe index for temporal and spatial price and quantity comparisons. Balk and Diewert [7] examine the nature and extent of substitution bias induced by the Lowe index in the context of intertemporal price comparisons. Here attention is drawn to the issues that arise when the Lowe quantity index in Eq. (91) is used for productivity comparisons. The Lowe index may lead to counter-intuitive conclusions if one uses this index somewhat mechanically. Suppose the quantity vectors in the two periods, qt and qs have the same value aggregate when evaluated at the reference prices of the Lowe index. Then at the Lowe reference prices N i=1

piR qit =

N i=1

piR qis = δ > 0

(92)

Further suppose that these two output vectors are produced using the same input vector, x, in both periods. Then we have the Lowe output quantity index equal to 1 and the input quantity index is also equal to 1. Then productivity index comparing

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D. S. Prasada Rao

Fig. 1 Illustration of Lowe output aggregate

periods s and t also equals 1, with the conclusion that there has been no productivity change. In the diagram below, the two-commodity case is depicted. Here points A and B represent output vectors in periods t and s. The straight line associated with reference price vector pR represents the iso-revenue line representing all the quantity vectors which result in the same amount of revenue which is δ in Eq. (92). Since both output vectors are produced using the same input vector, x, the use of Lowe index leads to the conclusion that there has been no productivity change when we move from output qs at point B to qt at point A and the Lowe productivity index equals 1 (Fig. 1). However, it can be seen from the diagram that there has been a shift in the production possibility frontier associated with technology Ss in period s to technology St in period t representing a significant technological change. In fact the production possibility frontier in period t completely dominates the frontier in s – with the implication that the firm is more productive in period t compared to period s. Such a shift in the frontier should imply productivity change driven by technical change. However, the Lowe index shows no productivity change! This illustration serves to remind the users of the Lowe index of some of the consequences of using the Lowe index to measure output and input quantities and their subsequent use in measures of productivity levels and change.

Value Decomposition and the Use of Quantity and Price Aggregates From an economist’s perspective and that of a firm, the link between changes in revenues, costs and profitability and movements in outputs produced, inputs used, and changes in terms trade are quite important. These links are discussed in detail in Sections “Productivity Measurement: The Case of Single Output and Single Input”

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and “Multiple Outputs and Inputs: The Index Number Problem.” If the approach of constructing output and input quantity aggregates is implemented, then the link hinges on the fundamental decomposition of value aggregates into quantity and price aggregates below. If Q (q) and P (p) represent the output and price aggregates then this decomposition is satisfied if Q (q) · P (p) =

N i=1

qi pi = V

(93)

If there exist functions Q (q) and P (p) such that decomposition in Eq. (93) holds then the following decomposition of value change into quantity and price change holds: Vst ≡

Q (qt ) P (pt ) Vt Q (qt ) · P (pt ) = · = Qst · Pst = Vs Q (qs ) · P (ps ) Q (qs ) P (ps )

(94)

From a firm’s perspective this decomposition is useful in their decision making and therefore the usefulness of the aggregates approach hinges on the existence of Q (q) and P (p) satisfying Eq. (94). The following theorem due to Eichorn [23], p. 144 discussed in Diewert [19] sheds some light on the existence of these functions. Result 16: Consider the case where the number of commodities, N, is greater than 1. Then then there do not exist any functions Q (q) and P (p) that satisfy the following two conditions: (i) Q (q) > 0 if q  0 and P (p) > 0 if p  0, and (ii) Q (q).P (p) = N i=1 pi qi = V Proof of this result is quite simple and available in Diewert [19]. The implication of Result 16 is significant if this approach is expected to serve as a tool in the analysis of revenue and cost changes and its decomposition into input and output changes and changes in the terms of trade. This result shows that the use of the aggregates approach is of limited use when the scope of the analysis extends beyond just providing measures of productivity change.

Are Price and Quantity Data Independent? The construction and use of quantity aggregates in measuring quantity change is predicated on the premise that price and quantity data are somehow independent. Some of the axioms discussed Section “Axiomatic Approach to Index Numbers” also assume independence of price and quantity vectors. For example, strong identity axiom assumes that quantity vectors would remain the same even when the prices are different in the two periods. Microeconomic theory of the firm and optimizing behavior of the firms – revenue maximization, cost minimization, or profit maximization – implies that decisions of the firms depend on observed input and output prices. Therefore, one would expect significant correlation between prices and quantities. The question then is how does one know if such correlation

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D. S. Prasada Rao

exists between prices and quantities (outputs or inputs)? The answer to this question can be found in the following decomposition of the spread between the Laspeyres and Paasche indices. This decomposition was provided by von Bortkiewicz [62]. Bortkiewicz has shown that the percentage difference between the Laspeyres and Paasche price or quantity index can be written as (see [6], p. 64):    N  pit /pis PstP PstP − PstL QPst − QL qit /qis st −1= = = ris −1 −1 PstL PstL QL PstL QL st st i=1 (95)

The right hand side of the Laspeyres-Paasche spread in Eq. (95) is a weighted measure of covariance between relative price changes and the relative quantity changes. If the covariance is positive, implying that output quantities increase when output prices increase, then Eq. (95) shows that the Paasche index would be greater than the Laspeyres index – a result stated in Section “Economic Theoretic Approach to Output and Input Quantity Index Numbers.” If the prices and quantities refer to inputs then one would expect the covariance to be negative in which case the Paasche index would be lower than the Laspeyres index. An important implication of Eq. (95) is that if the relative price movements and relative quantity movements are uncorrelated, one would expect the Laspeyres and Paasche indices (price as well as quantity) to be close to each other. For a practitioner the recommendation would then be to compute the Laspeyres and Paasche indices and if there is a significant difference between these two, then the assumption of independence between price and quantity data is not justified. This in turn gives an indication as to the type of axioms one may wish to impose on the index numbers compiled.

Transitivity and Quantity Index Numbers Transitivity is one of the axioms within the axiomatic framework for index numbers discussed in Section “Axiomatic Approach to Index Numbers.” It is useful to discuss the context in which this axiom assumes significance and examine some of the practical issues when choosing an appropriate index number formula that satisfies transitivity. The need for transitivity of quantity and price index numbers arises when multilateral comparisons are being undertaken. Multilateral comparisons refer to the case when comparisons between every pair of firms, regions, countries, time periods, or a combination of all of these are undertaken. Let us consider the simplest case where output index numbers are compiled for all pairs of firms within a crosssection of firms.

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Multilateral Comparisons Suppose there are M firms producing N outputs using K inputs. In this case data for all these firms are represented by: {pj , qj , wj , xj ; j = 1, 2, . . . , M} which, respectively, represent output price and quantity vectors of dimension N and input price and quantity vectors of dimension K. For the purpose of exposition, consider output quantity index numbers. In the case of multilateral comparisons, the aim is to fill the following matrix of quantity index numbers, Q: ⎡

QM×M

Q11 ⎢ Q21 =⎢ ⎣ Qj 1 QM1

Q12 . . . . Q22 . . . . Qj 2 . . . . QM2 . . . .

Q1k . . . Q2k . . . Qj k . . . QMk . . .

⎤ Q1M Q2M ⎥ ⎥ Qj M ⎦ QMM

(96)

where Qjk (j, k = 1, 2, . . . , M) represents output quantity index for firm k with firm j as the base. If this index, for example, is 1.20 then output of firm k is 20% higher than that of the firm j. By definition, all the diagonal elements are equal to 1. In principle, elements of this matrix can be filled by applying any of the formulae discussed in Section “Index Number Approach to Measuring Quantity Change.” Discussion in Sections “Axiomatic Approach to Index Numbers” and “Economic Theoretic Approach to Output and Input Quantity Index Numbers” suggests that either the Fisher or the Törnqvist index could be used here. These index numbers are intuitive and have useful axiomatic properties and from an economic theoretic perspective these are exact and superlative. An important issue arises in this context. For example, when Fisher (or TT) index is used, direct comparison between two firms, say 1 and 2, will be different from an indirect comparison through another firm, say firm 3. The direct and indirect comparisons in this case are, respectively, given by QF12 and QF13 · QF32 .  QF12

≡

N i=1 pi1 qi2 N i=1 pi1 qi1

 =

N i=1 pi1 qi3 N i=1 pi1 qi1

N ·

i=1 pi2 qi2 N i=1 pi2 qi1

N

i=1 pi3 qi3

· N

i=1 pi3 qi1

1/2 = QF13 · QF32 

1/2 ×

N i=1 pi3 qi2 N i=1 pi3 qi3

N

pi2 qi2 · i=1 N i=1 pi2 qi3

1/2

It is easy to check this fact using any numerical data. This means that use of the Fisher index formula (or for that matter most other formulae) will result in a matrix of comparisons that are not internally consistent and hence not transitive. A formal definition of transitivity is given below. Definition: A matrix of quantity index numbers Q is said to satisfy transitivity if for all j, k and l (=1, 2, . . . , M) Qj k = Qj l · Qlk

(97)

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D. S. Prasada Rao

Transitivity is really an accounting or consistency requirement which will ensure internally consistent assessments of output levels and productivity levels. This is a requirement that is currently imposed on international comparisons of prices and real expenditures conducted by the World Bank as a part of the International Comparison Program. Details of the methods used by the World Bank can be found in [18, 50, 51]. In the context of international comparisons, an additional requirement of base invariance or country symmetry is also imposed which requires symmetric treatment of all the countries involved in price and real expenditure comparisons. An implication of adherence to transitivity requirement is that inclusion of an additional firm into comparisons will affect all the quantity comparisons. Is Transitivity a Natural Requirement? Before turning attention to implications of transitivity for the choice of a suitable formula, it is useful to recognize that transitivity is an accounting requirement and it is usually not naturally satisfied when comparisons are being made. For example, transitivity is violated even in the simplest case of measuring distance between two N-dimensional vectors, x and y. Distance between these vectors can be measured using the Euclidean distance measure, d(x, y), which is given by:

d (x, y) =

N 

1/2 (xi − yi )

2

i=1

This distance measure is symmetric but it does not generally satisfy transitivity. In fact, the distance measure satisfies triangular inequality: d(x, y) ≤ d(x, z) + d(z, y) with equality holding if and only if these three vectors lie on the same hyperplane. In two-dimensions, this means that the Euclidean distance satisfies transitivity only in the cases where all the three vectors x, y and z line on a straight line. Another example where transitivity is automatically satisfied is in the case of exchange rates of currencies. In the absence of any arbitrage and differential transaction costs, exchange rates between pairs of currencies satisfy transitivity. But in practice, even these do not satisfy transitivity strictly. These two examples serve as a cautionary note but these are not presented as an argument to discard the notion of transitivity or to diminish the importance of transitivity in the context of multilateral output and productivity comparisons.

Alternative Approaches to Making Transitive Comparisons There are basically two approaches to index numbers that satisfy transitivity. The first approach is to restrict to the use of only those index number methods that satisfy transitivity. The second approach is to construct transitive multilateral comparisons using binary index numbers as building blocks. The second approach was first proposed by Gini [32, 33] but was independently proposed and popularized by Éltet˝o and K‚oves [25] and Szulc [56], and hence the method is referred to as the GEKS method.

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Index Number Formulae that Are Transitive Several index numbers described in Sections “Index Numbers Based on Quantity Aggregates” and “Index Number Approach to Measuring Quantity Change” automatically satisfy transitivity. For example, the Lowe index, the geometric Young index, or the Malmquist quantity index applied to multilateral comparisons with a specific choice of reference technology and reference input quantity vector satisfy transitivity. The Lowe Index

The Lowe index in the context of multilateral quantity index numbers takes the following form. For any pair of firms j and k, the index is defined as: N QLowe jk

= i=1 N

piR qik

i=1 piR qij

for all j, k = 1, 2, . . . , M

(98)

where piR represent the “reference” or fixed price vector used to evaluate quantity vectors of all the firms. It is easy to see that the Lowe index satisfies the identity, positivity, circularity, and transitivity axioms. It does not satisfy the factor reversal test. Implementation of the Lowe index requires the choice of the reference prices used in Eq. (98). The obvious choices are: (i) Choose price vector associated with one of the firms as the reference price vector. In this case indices are transitive but one firm is given a special treatment. Note that results can differ significantly when one firm’s price vector is replaced by another firm’s price vector. (ii) In order to avoid the problem of choosing between different firms to specify the reference price vector, one may choose the average of all the price vectors as the reference vector. In this case the arithmetic, geometric, or harmonic average of price vectors of all the M firms can be employed. For commodity i pAM = i

M   1/M 1 M pij pij and pGM = i = 1, 2, . . . , N i j =1 M

(99)

j =1

A problem with this approach is that all the firms are treated as equally important in defining the average price. If firms differ significantly in size, one may consider a weighted average. (iii) A slightly more complicated but a realistic choice is to use a quantity weighted average of the prices faced by the M firms in the comparison. For each item i, use the average: M i=1 pi =  M

pij qij

j =1 qij

=

M  j =1

qij pij · M

j =1 qij

i = 1, 2, . . . , N

(100)

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D. S. Prasada Rao

It is clear that one can think of many other reference price vectors for use in conjunction with the Lowe index formula in Eq. (98). Logically and from an economic view point the quantity-weighted average price vector in Eq. (100) is to be preferred. This choice is less important if all the price vectors are close to each other. It is useful to remind the reader that the use of Lowe index in Eq. (98) may lead to quantity indices that are counter-intuitive. In view of the discussion and Fig. 1, the general recommendation is not to use the Lowe index for purposes of measuring output, input quantity and productivity changes. If one were to use Lowe index for this purpose, it is important to examine and report sensitivity of the results to alternative choices of the reference price vector. The Geometric Young Index

Recall that the geometric Young index (GY) for quantity comparisons (see Section “Index Number Approach to Measuring Quantity Change”) is given by:

Young Qj k

=

 N   qik wi i=1

qij

0 ≤ wi ≤ 1 and

N i=1

wi = 1; j, k = 1, 2, . . . , M (101)

The GY index is similar in structure to the Cobb-Douglas function and it is also similar to the Törnqvist quantity index. The main point to note here is that the weights wi are fixed for all pairwise comparisons. Numerical values of the index depend upon the choice of the weights wi . It is meaningful to use weights based on output revenue shares. The most commonly used set of weights are: wi =

pij qij 1 M rij where rij = N j =1 M i=1 pij qij

for all i = 1, 2, . . . , N

(102)

By construction the GY index is transitive and satisfies a number of properties. It is useful to recall the following important theorem, stated as Result 4 in Section “Axioms and Discussion,” which has strong implications for the choice of quantity index numbers which are transitive. This result is stated again here. Result: A quantity index Qst (pt , qt , ps , qs ) satisfies monotonicity, linear homogeneity, identity, commensurability and circularity tests if and only if the index is of the Geometric-Young form and

Qst (pt , qt , ps , qs ) =

 N   qit wn i=1

qis

with wi > 0, i = 1, 2, . . . , N and

N  i=1

wi = 1

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This result and its proof is originally from Funke et al. [31] but Balk ([6], pp. 97–98) provides a simpler proof.28 This result is indeed quite powerful since it shows that if monotonicity, linear homogeneity, identity, commensurability, and circularity tests are to be satisfied then the choice is limited to the use of the Geometric Young index in (102). This result has important implications for the recently developed approach by O’Donnell [44, 46]. As in the case of Lowe index, output quantity comparisons from the GY index are influenced by the particular choice of the set of weights used. In this regard, it is useful to conduct an analysis of sensitivity of quantity comparisons to the choice of weights and then make a judicious choice regarding the weights. The Malmquist Output Quantity Index

The Malmquist output quantity index was introduced in Section “Direct Quantity Index Based on Malmquist Distance Function.” The Malmquist index is given by: QM jk =

DoR (xR , qk )  DoR xR , qj

for all j, k = 1, 2, . . . , M

(103)

The Malmquist index requires the specification of a reference technology and a reference input vector. In the case of binary comparisons, it is usually defined as the geometric average of the Malmquist index computed using the technology and input vectors of the base and reference firms leading to the following index:  QM jk

=

1/2 j Do xj , qk Dok (xk , qk ) · k j Do xk , qj Do xj , qj

(104)

Note that the index in Eq. (104) is not transitive in general. However, the index would be transitive if all the binary comparisons make use of the same reference technology and reference input vector. Two remarks are in order. 1. The numerical values of the quantity index critically depend on the choice of the technology and the input vector except in the case where the technology is such that the output distance function is separable and homothetic, that is, j Do (q, x) = 1/g(q)h (j, x). In this case the value of the Malmquist index is independent of the choice of reference technology and the reference input vector xR . 2. However, the assumption of homothetic separability does not hold in general. When this condition does not hold, the choice of reference technology and input vector is indeed critical. If it is cross-sectional data, all the firms may be using

28 The

original theorem and the proof in Balk [6] refer to GY index for price comparisons. But the same proof holds for quantity index numbers.

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the same technology and in this case choice of the reference input vector xR is critical. Again the recommendation is to examine the sensitivity of productivity comparisons resulting from different choices and select a plausible reference input vector.

‚ves-Szulc (GEKS) Approach ˝ o Gini-Élteto-K The GEKS approach proposed in Gini [32, 33], Éltet˝o and K‚oves [25], and Szulc [56] is a simple approach that works on the following logic. For making binary comparisons the Fisher, Törnqvist or some other suitable formula may be selected. This choice may depend upon the fact that these indices satisfy a number of axiomatic and economic theoretic properties. However, using these formulae results in a matrix of multilateral quantity index numbers which are not transitive. For example, use of the Fisher index leads to: ⎡

QFM×M

QF ⎢ Q11 ⎢ F = ⎢ F21 ⎣ Qj 1 QFM1

QF12 . . . . QF22 . . . . QFj2 . . . . QFM2 . . . .

QF1k . . . QF2k . . . QFjk . . . QFMk . . .

⎤ QF1M QF2M ⎥ ⎥ ⎥ QFjM ⎦ QFMM

(105)

This matrix of comparisons is not transitive. The GEKS approach is then to construct a matrix of transitive comparisons, denoted by QGEKS M×M , where elements of the matrix exhibit the property that the GEKS indices deviate the least from the Fisher binary indices. If Fisher index is the primary choice then the GEKS approach provides a way to make minimal changes to the Fisher indices resulting in transitive comparisons. The GEKS method solves the following minimization problem:

minQGEKS jk

s.t.QGEKS jk

2 M  M

 ln QGEKS − ln QFjk jk

j =1 k=1 = QGEKS jl

× QGEKS lk

(106)

for all j, k and l = 1, 2, . . . M

The solution to the optimization problem in Eq. (106) has the following closed form: QGEKS jk

M

1/M  QFjl · QFlk = j, k = 1, 2, . . . , M

(107)

l=1

The GEKS quantity comparisons have an intuitive interpretation. For a comparison between j and k, it is possible to make indirect comparisons through each of the firms in the comparison. Since the Fisher index is not transitive, different numerical values result from these indirectly linked comparisons. The GEKS index in Eq. (107) can be seen as a simple geometric average of all these M chained comparisons.

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The GEKS index is a formula for multilateral comparisons. This is the main aggregation method used in international comparisons of prices and real expenditure at the World Bank over the last two decades. This has been used as the aggregation procedure by the Eurostat for comparisons across the EU member countries since 1990. The GEKS index is also the method proposed by Caves et al. [10] for multilateral output and input quantity index numbers based on the Malmquist distance functions and for multilateral productivity comparisons based on the Malmquist productivity index.29 The GEKS index satisfies many of the axioms discussed in Section “Axiomatic Approach to Index Numbers.” However, it fails the strong form of identity test but satisfies the weak form of identity test. It also fails to satisfy the proportionality axiom and monotonicity axiom in extreme cases. Despite the deficiencies associated with GEKS index, it is a commonly used method for multilateral comparisons. The GEKS method is preferred over the Lowe and GY indices as these methods are anchored on a single selected reference price vector in the case of the Lowe Index and a reference set of weights in the case of GY index. In addition, the Malmquist index requires a reference technology and reference input vector. In conclusion to this section, transitivity is an important operational requirement when multilateral output and input quantity index numbers are considered. If one wishes to strictly adhere to the axiomatic approach, the result by Funke et al. [31] is quite important and in this case the Geometric Young index is the only option. The GEKS approach provides an alternative but it fails some important tests. However, it has the advantage that it does not rely on any choice with respect to the weights or reference technology or input vectors.

Conclusion This chapter is about the use of index numbers to measure output and input growth generally and on their use in productivity measurement. This chapter is not about efficiency and productivity measures and the related task of productivity decomposition. These aspects are dealt with in other chapters in this Handbook, of particular relevance are  Chap. 20, “Conceptualization and Measurement of Productivity Growth and Technical Change: A Nonparametric Approach”;  Chap. 8, “Stochastic Frontier Analysis: Foundations and Advances I”, and  Chap. 9, “Stochastic Frontier Analysis: Foundations and Advances II”, and  Chap. 10, “Data Envelopment Analysis: A Nonparametric Method of Production Analysis”. The main objective of the chapter has been one of providing the practitioner with an overview of the index number methods and approaches available for measuring changes in output and input quantities. These methods are equally applicable for

29 The

GEKS index is also being increasingly used in inter-temporal comparisons of prices based on scanner data.

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measuring changes in output and input prices. In this chapter two alternative approaches to this problem are discussed in detail. The first approach, discussed in Section “Index Numbers Based on Quantity Aggregates,” utilizes the notion of aggregates of output and input quantities and uses the resulting aggregates to measure changes in output and input quantities. The second approach, discussed in Sections “Index Number Approach to Measuring Quantity Change,” reviews the standard index number approach where index number formulae are used to directly measure changes in output and input quantities without having to first measure quantity aggregates. The axiomatic and economic theoretic approaches are discussed in Sections “Axiomatic Approach to Index Numbers”and “Economic Theoretic Approach to Output and Input Quantity Index Numbers.” Section “Special Topics” discusses a few additional topics that bring the whole chapter together. The problem of multilateral comparisons and the need to compile transitive measures of output and quantity change are discussed in Section “Transitivity and Quantity Index Numbers.” The main take home message for the practitioner is that the index number approach to quantity and productivity comparisons does not always lead to a single index number formula. When bilateral comparisons involving only two periods or two firms are required, the general recommendation is to use the Fisher or Törnqvist index numbers formulae. These indices possess useful axiomatic and economic theoretic properties and are known to be numerically close to each other. However, the problem becomes somewhat complex when multilateral comparisons (Section “Transitivity and Quantity Index Numbers”) are required as the recommended Fisher and Törnqvist indices fail to satisfy transitivity property. The often recommended Gini-Éltet˝o-K‚oves-Szulc method of measuring quantity changes in a multilateral context satisfies transitivity but fails to satisfy the strong identity test (although it satisfies the weak identity test) and the proportionality test and may fail the monotonicity test in extreme cases (Section “Transitivity and Quantity Index Numbers”). It is in this context, the approach based on quantity aggregates, discussed in Section “Index Numbers Based on Quantity Aggregates,” assumes relevance. The practitioner interested in using the quantity aggregates to measure quantity changes must be aware of the main issues associated with the use and implementation of the quantity aggregates to measure quantity change (discussed in Section “Use of Quantity Aggregates to Measure Quantity Change”). The empirical implementation of this approach requires choices to be made at several stages. To start with, one has to articulate the set of properties to be satisfied by the quantity aggregate (see Section “Use of Quantity Aggregates to Measure Quantity Change”). The subsequent choice of an appropriate aggregator function such as the Lowe, Geometric Young, or a Malmquist aggregate function (Section “Specification of Functional form for the Output Aggregates”) followed by the choice of selection of weights, reference vectors, and reference technology (in the Malmquist index case) are critical to this approach. The resulting measures of output and input quantity changes are likely to be highly sensitive to the choices made in this process. Currently no concrete recommendations regarding these choices are available. Keeping this in perspective, it is important to report sensitivity of the

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results arising out of various choices made in measuring quantity changes using quantity aggregates along with results from the use of direct index number approach and the GEKS method. Where the results show significant differences in results from alternative choices, it is necessary to outline the reasons and to defend any particular choice made for the final analysis.

References 1. Aczèl J (1966) Lectures on functional equations and their applications. Academic, New York/London 2. Allen RGD (1949) The economic theory of index numbers. Economica NS 16(99):197–203 3. Allen RC, Diewert WE (1981) Direct versus implicit superlative index number formulae. Rev Econ Stat 63:430–435 4. Balk BM (1995) Axiomatic price index theory: a survey. Int Stat Rev 63:69–93 5. Balk BM (1998) Industrial Price, quantity, and productivity indices: the micro-economic theory and an application. Springer, New York 6. Balk B (2008) Price and quantity index numbers – models for measuring aggregate change and difference. Cambridge University Press, New York 7. Balk BM, Diewert WE (2009) The Lowe consumer price index and its substitution bias. Chapter 8. In: Diewert WE, Balk BM, Fixler D, Fox KJ, Nakamura AO (eds) Price and productivity measurement – volume 6 index number theory. Trafford On Demand Pub 8. Bjurek H (1996) The Malmquist total factor productivity index. Scand J Econ 98(2):303–313 9. Carli GR (1764) Del Valore e della Proporzione de’ Metslli Monerysyi. In Scrittori Classici Italiani di Economia Politica< aprte Moderna, XIII, 297–366 10. Caves DW, Christensen LR, Diewert WE (1982) Multilateral comparisons of output, input, and productivity using superlative index numbers. Econ J 92:73–86 11. Coelli TJ, Rao DSP, Battese GE (1998) An introduction to efficiency and productivity measurement. Kluwer Academic, New York 12. Coelli TJ, Rao DSP, O’Donnell CJ, Battese GE (2005) An introduction to efficiency and productivity measurement. Springer, New York 13. Diewert WE (1976) Exact and superlative index numbers. J Econ 4:115–145 14. Diewert WE (1978) Superlative index numbers and consistency in aggregation. Econometrica 46:883–900 15. Diewert (1983) The theory of the output Price index and the measurement of real output change. In: Diewert WE, Momtmarquette C (eds) Price level measurement: proceedings from a conference sponsored by statistics Canada, Ottawa, pp 1049–1108 16. Diewert WE (1992) Fisher ideal output, input and productivity indices revisited. J Prod Anal 3:211–248 17. Diewert WE (1993) The early history of Price index research. In: Diewert WE, Nakamura A (eds) Essays in index number theory. Amsterdam, North-Holland 18. Diewert WE (2013) Methods of aggregation above the basic heading level Chapter 5. In: World Bank (ed) Measuring the real size of the world economy. World Bank, Washington, DC, pp 93–119 19. Diewert WE (2015a) Functional equations, Chapter 2. In: Index number theory and practice. Lecture Notes, Department of Economics, University of British Columbia, Vancouver 20. Diewert WE (2015b) Early approaches to index number theory, Chapter 1. In: Index number theory and practice. Lecture Notes, Department of Economics, University of British Columbia, Vancouver 21. ECE-ILO (2014) Manual of the consumer price index. ILO, Geneva 22. Edgeworth FY (1925) The plurality of index numbers. Econ J 35:379–388 23. Eichorn W (1978) Functional equations in economics. Addison-Wesley, Reading

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24. Eichorn W, Voeller J (1976) Theory of the price index, Lecture notes in economics and mathematical systems 140. Springer, Berlin 25. Éltet˝o O, K‚oves P (1964) On an index computation problem in international comparisons (in Hungarian). Statiztikai Szemle 42:507–518 26. Färe R, Primont D (1995) Multi-output production and duality: theory and applications. Kluwer Academic, Boston 27. Farrell M (1957) The measurement of productive efficiency. J Roy Stat Soc A (General) 120(3):253–290 28. Fisher I (1922) The making of index numbers: a study of their varieties, tests and reliability. Houghton Mifflin Company, Boston/New York 29. Fisher, F.M. and K. Shell (1972), “The pure theory of the National Output Deflator”, pp. 49-113 in The economic theory of price indices by F.M. Fisher and K. Shell, Academic, New York 30. Frisch R (1936) Annual survey of general economic theory: the problem of index numbers. Econometrica 4:1–38 31. Funke H, Hacker G, Voeller J (1979) Fisher’s circular test revisited. Schweizerische Zeitschrift f˝ur Volkswirtschaft und Statistik 115:677–687 32. Gini C (1924) Quelques considerations au Sujet de la Construction des Nombres Indices des Prix et des Questions Analogues. Metro 4:3–162 33. Gini C (1931) On the circular test of index numbers. Int Rev Stat 9(2):3–25 34. Hill TP (2009) Lowe indices. Chapter 9. In: Diewert WE, Balk BM, Fixler D, Fox KJ, Nakamura AO (eds) Price and productivity measurement – volume 6 index number theory. Trafford On Demand Pub 35. IMF (2004) Produce Price Index (PPI). Manual, Washington, DC 36. Jevons WS (1865) On the variation of prices and the value of the currency since 1782. J Stat Soc Lond 28:2940–2320 37. Konus AA (1924) The problem of the true index of the cost of living, translated in Econometrica, 7 (1939), pp 10–29 38. Kuczma M, Gilányi A (2009) An introduction to the theory of functional equations and inequalities: cauchy’s equation and Jensen’s inequality. Birkhäuser, Basel 39. Laspeyres E (1871) Die Berchnung einer Mittlerrrn Waarenpreisstrigertung. Jahrbücher für National‚okonomie und Statistik 16:296–314 40. Lowe J (1823) The present state in regard to agriculture, trade and finance, 2nd edn. Longman, Hurst, Rees, Orne & Brown, London 41. Malmquist S (1953) Index numbers and Indsifference surfaces. Trabajos de Estadistica 4: 209–242 42. Marshall A (1887) Remedies for fluctuations of general prices. Contemp Theatr Rev 51:355– 375. Reprinted in Memorials of Alfred Marshall, edited by A.C. Pigou, Macmillan, London, 1925) 43. O’Donnell CJ (2011) DPIN 3.0: a program for decomposing productivity index numbers. Available on Center for Efficiency and Productivity Analysis website: www.uq.edu.au/ economics/cepa/dpin.php; School of Economics, University of Queensland, Brisbane 44. O’Donnell CJ (2012a) Alternative indices for multiple comparisons of quantities and prices. Centre for Efficiency and Productivity Analysis Working Papers WP05/2012. School of Economics, University of Queensland, Brisbane 45. O’Donnell CJ (2012b) Nonparametric estimates of the components of productivity and profitability change in U.S. agriculture. Am J Agric Econ 94(4):873–890 46. O’Donnell CJ (2016) Using information about technologies, markets and firm behaviours to decompose a proper productivity index. J Econ 190(2):328–340 47. O’Donnell CJ (2018) Productivity and efficiency analysis: an economic approach to measuring and explaining managerial performance. Springer, Singapore 48. OECD (2001) Measurement of aggregate and industry-level productivity growth – productivity Manual. OECD, Paris

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49. Paasche H (1874) Ueber die Preisentwicklung derr Letzten Jable nack dern Hamburger B‚orsennotierungen. Jahrbücher für National‚okonomie und Statistik 23:168–178 50. Rao DSP (2013a) The framework for the international comparison program (ICP). Chapter 1. In: World Bank (ed) Measuring the real size of the world economy. World Bank, Washington, DC, pp 13–45 51. Rao DSP (2013b) Computation of basic heading purchasing power parities (PPPs) for comparisons within and between regions. Chapter 4. In: World Bank (ed) Measuring the real size of the world economy. World Bank, Washington, DC, pp 93–119 52. Samuelson S, Swamy S (1974) Invariant economic index numbers and canonical duality: survey and synthesis. Am Econ Rev 64:566–593. 58, pp 434–442 53. Sato K (1976a) The ideal log-change index number. Rev Econ Stat 58:223–228 54. Sato K (1976b) The meaning and measurement of the real value added index. Rev Econ Stat 58:434 55. Sickles R, Zelenyuk V (2019) Measurement of productivity and efficiency: theory and practice. Cambridge University Press, Cambridge, UK 56. Szulc B (1964) Index numbers of multilateral regional comparisons (in Polish). Przeglad Statysticzny 3:239–254 57. Shephard RW (1970) Theory of Cost and Production Functions. Princeton University Press, Princeton 58. Törnqvist L (1936) The Bank of Finland’s consumption Price index. Bank Finl Mon Bull 16(10):27–34 59. Vartia YO (1976) Ideal log-change index numbers. Scand J Econ 3:121–126 60. von Bortkiewicz L (1923) Zwck und Struktur einer Preisindexzahl. Nordisk Statistisk Tidsskrift 2:369–408 61. Walsh CM (1901) The measurement of general exchange-value. Macmillan, New York 62. Young A (1812) An inquiry into the progressive value of money in England as marketed by the Price of agricultural products. Hatchard, Piccadilly

Conceptualization and Measurement of Productivity Growth and Technical Change: A Nonparametric Approach

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Subhash C. Ray

Contents Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The Theoretical Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The Production Possibility Set . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Distance Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Technical Efficiency and Distance Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Input and Output Sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Technical Change . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Change in the Technology Versus Change in the Technique: A Clarification . . . . . . . . . . . Productivity Change in Continuous Time . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Productivity Change in Discrete Time . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A Hicks-Moorsteen Productivity Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The Tornqvist Productivity Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The Fisher Productivity Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Profitability, Terms of Trade, and Productivity Indexes . . . . . . . . . . . . . . . . . . . . . . . . . . . . Malmquist Productivity Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Malmquist Productivity Index with Multiple Outputs and Inputs . . . . . . . . . . . . . . . . . . . . Allowing Technological Change . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Allowing Returns to Scale Effect . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Biennial Malmquist Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Directional Distance Function and Luenberger Productivity Indicator . . . . . . . . . . . . . . . . Relation Between Tornqvist and Malmquist Productivity Indexes . . . . . . . . . . . . . . . . . . . . Relation Between Fisher and Malmquist Productivity Indexes . . . . . . . . . . . . . . . . . . . . . . Nonparametric Decomposition of the Fisher Productivity Index . . . . . . . . . . . . . . . . . . . . . Relation Between Malmquist Productivity Index and Luenberger Productivity Indicator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Data Envelopment Analysis and a Nonparametric Measurement of Productivity Change . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . DEA Models for Measuring the Malmquist Productivity Index . . . . . . . . . . . . . . . . . . . . . .

822 823 823 825 827 827 828 828 829 838 838 839 841 842 844 847 848 849 851 852 855 858 860 863 864 864

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Biennial Malmquist Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . DEA Model for the Directional Distance Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Cross-References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

865 868 868 869 869

Abstract

The famous Solow residual measures productivity change as the difference between the growth rates of output and input. Under constant returns to scale and in the absence of any technical inefficiency, it serves as a measure of technical change. In neoclassical production economics, productivity change can be measured alternatively from the production, cost, profit, or distance functions. In continuous time analysis, one measures the rates of productivity and technical change. In discrete time, one measures indexes of productivity and technical change over time. This chapter describes the Tornqvist, Fisher, and Malmquist productivity indexes along with the Luenberger productivity indicator based on the directional distance function and how they relate to one another. Also discussed is the relation between productivity and profitability of a firm. The relevant nonparametric DEA models for measuring the Malmquist index and the Luenberger productivity indicator are formulated for nonparametric analysis of productivity, technical change, and change in efficiency.

Keywords

Solow residual · Shephard distance function · Directional distance function · Data envelopment analysis

JEL Classification Numbers

D24, C61

Introduction An appropriate starting point for any discussion of productivity growth and technical change in the neoclassical production economic framework is the seminal paper by Solow on technical progress and productivity change [34]. The famous Solow residual measuring the difference between the rates of growth in output and inputs is interpreted as the rate of technical progress. Solow assumed constant returns to scale, which is quite appropriate in the context of his macroeconomic model. When applied to an individual producer, one needs to allow variable returns to

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scale. Further, changes in technical efficiency may account in part for a higher or lower rate of growth in output. It is now generally accepted that in addition to technical progress, returns to scale effects of a change in inputs along with changes in technical efficiency may also contribute to the Solow residual. The principal objective of this chapter is to explain how to isolate technical progress, scale effects, and efficiency change as three distinct components of productivity change measured empirically using the nonparametric method of data envelopment analysis (DEA).1 The paper unfolds as follows. Section “The Theoretical Background” presents a brief overview and basic assumptions of the neoclassical production theory and considers alternative ways to measure technical change and productivity growth in continuous time using production, cost, profit, and distance functions as the analytical framework. Section “Productivity Change in Discrete Time” revisits the issues in discrete time using index numbers. In particular, Tornqvist, Fisher, and Malmquist productivity indexes and their interrelations are considered. Also described is the more recent Luenberger productivity indicator measured by the differences between directional functions. Section “Data Envelopment Analysis and a Nonparametric Measurement of Productivity Change” presents the relevant nonparametric DEA models for measuring the Malmquist index and the Luenberger productivity indicator. Section “Conclusion” is the conclusion.

The Theoretical Background The Production Possibility Set n be a vector Consider an industry producing m outputs from n inputs. Let x ∈ R+ m of inputs and y ∈ R+ an output vector. Then the input-output pair is a feasible production plan if and only if y can be produced from x. The set of all feasible production plans constitute the production possibility set:

  m n can be produced from x ∈ R+ T = (x, y) : y ∈ R+

(1)

It is conventional to make the following assumptions about the production possibility or the technology set: (a) T is closed and bounded. (b) T is convex. (c) Inputs are freely disposable. This implies that from any y if (x0 , y) ∈ T and x1 ≥ x0 , then (x1 , y) ∈ T. 1A

comparable analysis of productivity change using the parametric method of stochastic frontier analysis (SFA) may be found in  Chap. 21, “Modeling Technical Change: Theory and Practice” by Kumbhakar in this volume.

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(d) Outputs are freely disposable. This implies that from any x if (x, y0 ) ∈ T and y1 ≤ y0 , then (x, y1 ) ∈ T. If, in addition to (a)–(d) one also assumes constant returns to scale (CRS), we get (e) (x, y) ∈ T implies that (kx, ky) ∈ T for all k ≥ 0. It is often convenient to define the production possibility set indirectly by means of a parametrically specified function. For single-output technologies, one can define the production function y∗ = f (x) as the maximum output producible from the input (vector) x. Equivalently, the production possibility set is defined as T = {(x, y) : y ≤ f (x)}

(2)

In practice, the output produced by a firm from the input bundle x may be strictly lower than f (x) due to technical inefficiency. One can incorporate efficiency in the production function as y = f (x)τ ; 0 ≤ τ ≤ 1.

(3)

Alternatively, τ=

y . f (x)

(4)

When multiple outputs are produced from multiple inputs, a simple representation of the technology through a production function is no longer possible. One may however specify a transformation function: n m F (x, y) = a; x ∈ R+ , y ∈ R+ , a ∈ R1.

(5)

The production possibility set can then be defined as     n m T = (x, y) : F x, y ≤ 0; x ∈ R+ . , y ∈ R+

(6)

For any explicit functional specification of the transformation function, free disposability of inputs and outputs would imply ∂F ≤ 0, i = 1, 2, . . . , n; ∂xi

(7a)

∂F ≥ 0, j = 1, 2, . . . , m; ∂yj

(7b)

and

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825

Further, convexity of T would imply that F(x, y) is a concave function.

Distance Function It is apparent from (6) above that the transformation function provides a criterion for separating feasible input-output bundles from those which are infeasible. However, an input-output bundle is not efficient unless F(x, y) = 0. When F(x0 , y0 ) < 0 for any input-output bundle (x0 , y0 ), (7a) and (7b) would imply that it would be possible to increase one or more output(s) or to reduce one or more (inputs) without violating feasibility. Shephard [32, 33] defined the (output) distance function as  1 D y (x, y) = min λ : x, y ∈ T . λ

(8)

It is clear that if F(x, y) = 0, Dy (x, y) = 1. On the other hand, for F(x, y) < 0, < 1. Thus, an alternative, and equivalent, characterization of the production possibility set is Dy (x, y)

    n m . T = (x, y) : D y x, y ≤ 1; x ∈ R+ , y ∈ R+

(9)

At this point, it will be useful to distinguish between weak (or radial) efficiency and strong (or non-radial) efficiency. An input-output bundle (x0 , y0 ) is weakly output efficient if     / T. x 0 , y 0 ∈ T and a > 1 ⇒ x 0 , ay 0 ∈

(10a)

Similarly, (x0 , y0 ) is weakly input efficient if     / T. x 0 , y 0 ∈ T and b < 1 ⇒ bx 0 , y 0 ∈

(10b)

By contrast, (x0 , y0 ) is strongly (non-radially) output efficient if     / T. x 0 , y 0 ∈ T and y 1 ≥ y 0 ⇒ x 0 , y 1 ∈

(11a)

Similarly, (x0 , y0 ) is strongly (non-radially) input efficient if     / T. x 0 , y 0 ∈ T and x 1 ≤ x 0 ⇒ x 1 , y 0 ∈

(11b)

Weak output efficiency rules out any proportional increase in all outputs, but increase in some outputs may be possible. The vector inequality in (11a) rules out

826

S. C. Ray

any increase in any output without increasing inputs. It is clear that Dy (x, y) = 1 implies that the bundle (x, y) must be weakly (or radially) efficient even though it may or may not be strongly efficient. Listed below are some properties of the output-oriented distance function: O1. Dy (x, y) is non-decreasing in y. That is, for any input bundle x, y1 ≥ y0 ⇒ Dy (x, y0 ) ≤ Dy (x, y1 ). O2. Dy (x, y) is non-increasing in x. That is, for any output bundle y, x1 ≥ x0 ⇒ Dy (x0 , y) ≥ Dy (x1 , y). O3. Dy (x, y) is homogeneous of degree one in y. That is, Dy (x, αy) = αDy (x, y). O4. Dy (x, y) is convex in y. Comparable to the output distance function, one can define the input distance function:  D x (x, y) = max δ :

1 x, y δ

∈T

(12)

Again, for any feasible input-output bundle (x, y), Dx (x, y) ≥ 1. Hence, another way to define the production possibility set is     n m . , y ∈ R+ T = (x, y) : D x x, y ≥ 1; x ∈ R+

(13)

As argued above, if Dx (x0 , y0 ) = 1, (x0 , y0 ) is weakly (but not necessarily strongly) input efficient. The analogous properties of the input-oriented distance function are: I1. Dx (x, y) is non-decreasing in x. I2. Dx (x, y) is non-increasing in y. I3. Dx (x, y) is homogeneous of degree one in x. I4. Dx (x, y) is concave in x. In general, there is no specific relationship between the output- and the inputoriented distance functions for the same input-output bundle (x, y). However, under CRS, the distance functions are inverses of one another. This can be shown as follows. We know from the of the output-oriented distance function in (8) above  definition  1 ∗ that λ is min λ : x, λ y ∈ T . However, under the CRS assumption, that would mean λ∗ = min λ : (λx, y) ∈ T = max θ :



1 x, y θ



Hence, using the subscript C to refer to a CRS technology,

∈ T.

(14)

20 Conceptualization and Measurement of Productivity Growth and Technical . . . y

DC (x, y) =

1 . DCx (x, y)

827

(14a)

Technical Efficiency and Distance Function Farrell [21] defined the output-oriented technical efficiency of an input-output bundle (x, y) as τy (x, y) =

1 ; ϕ ∗ = max ϕ : (x, ϕy) ∈ T . ϕ∗

(15)

Comparison of (15) with (8) shows that τy (x, y) = D y (x, y) .

(15a)

On the other hand, input-oriented Farrell efficiency is τx (x, y) = min θ : (θ x, y) ∈ T .

(16)

Hence, D x (x, y) =

1 . τx (x, y)

(16a)

Input and Output Sets Yet another equivalent characterization of the technology is through the families of input sets and output sets. For any given output vector y0 , the input set is       n : x, y 0 ∈ T V y 0 = x ∈ R+

(17)

Two things may be noted. First, for any production possibility set T, there is a m+n n. family of input sets, one for each output vector. Second, while T ⊂R+ , V (y)⊂R+ 0 In an analogous way, for any input vector x , the output set is       m : x0, y ∈ T P x 0 = y ∈ R+ Again, T yields a family of output sets, one for each input vector.

(18)

828

S. C. Ray

Technical Change In our discussion, so far, we have assumed that the production possibility set, T, remains unchanged over time. In reality, however, due to advancement in technical knowhow, the frontier of the production possibility may shift outward making it possible to produce a bigger output bundle than before from the same input bundle. This is interpreted as technical progress. One can accommodate technical change by dating the production possibility set over time as   m n T t = (x, y) : y ∈ R+ can be produced from x ∈ R+ in period t

(19)

Thus, in the single-output case,     n T t = (x, y) : y ≤ f x, t ; x ∈ R+ in period t

(20)

Similarly, for the multiple output case,     n m T t = (x, y) : F x, y; t ≤ 0; x ∈ R+ ; y ∈ R+     n m ; y ∈ R+ ⇐⇒ (x, y) : D y x, y; t ≤ 1; x ∈ R+

(21)

Change in the Technology Versus Change in the Technique: A Clarification It is important to clear up a common misconception about technical progress. Often an increase in the capital labor ratio is erroneously regarded as an improvement in the technology. This amounts to confusing a movement along an isoquant with a shift in the isoquant. Consider a firm that uses 2 machines (K) and 50 workers (L), and suppose that the production function is √ Q = 2 KL.

(22)

Further assume that there is no technical inefficiency. Then, its output (Q) will be 20. Now suppose that in the next period, it uses 20 machines and 5 workers. So long as the same production function applies in both periods, again its output will be 20. The two input bundles (K = 2, L = 50) and (K = 20, L = 5) are two points in the same isoquant for Q=20. Even though the second input bundle shows a much higher use of capital per worker, it represents input substitution and does not imply any technical progress. Only if the production function itself changes between the two periods, we can talk about technical progress. This will be associated with a shift of the isoquant toward the origin because less of labor or capital (or a combination of both) will be required to produce the same level of output as before. Solow is quite

20 Conceptualization and Measurement of Productivity Growth and Technical . . .

829

clear on this point and used the phrase “technical change” as a shorthand expression for any kind of shift in the production function ([34], p. 312). A point on an isoquant represents a technique. Thus, movement from one point to another on the isoquant amounts to a change in the technique. By contrast, the technology is the assortment of all available techniques to produce any given level of output, and a change in the technology implies a shift in the production function resulting in a shift in the isoquant.

Productivity Change in Continuous Time Deriving the Solow Measure of Technical Progress Solow considered an aggregate production function for the entire economy with output, Q, as a function of capital, K, and labor, L, and technical change causing a neutral shift in the production function over time without altering the marginal rate of substitution between capital and labor: Q = A(t)f (K, L)

(23)

Thus dQ d(f (K, L) dA(t) = A(t). + f (K, L) dt dt dt  ∂f dL dA(t) ∂f dK + + f (K, L) = A(t). ∂K dt ∂L dt dt Defining get

dQ dt

˙ dK = K, ˙ dL = L, ˙ dA(t) = A, ˙ ∂f = fK , and = Q, dt dt dt ∂K  ˙ Q fK K K˙ fL L L˙ A˙ = + + Q f K f L A

∂f ∂L

= fL , we

(24)

Under perfect competition and treating the output price as the numeraire, the ∂Q input prices of capital and labor will be equal to their marginal productivities ∂K = ∂Q A(t)fK and ∂L = A(t)fL , respectively. In that case, the partial elasticities of output ∂Q

∂Q

.K

.L

with respect to the inputs, ∂KQ = fKfK = sK and ∂KQ = fLf L = sL , become the corresponding shares of K and L in the total output. Under the assumption of CRS, f (K, L) = fK K + fL L so that the factor shares add up to unity. At this point, one may define the rate of change in the aggregate output as X˙ K˙ L˙ = sK + sL X K L

(25)

830

S. C. Ray

Then the (multi-factor) productivity change can be seen to be equal to the rate of technical change: ˙ M˙ Q X˙ A˙ = − = . M Q X A

(26)

As already noted above, in Solow’s analysis CRS was the default returns to scale. Moreover, he assumed full technical efficiency. We now relax the CRS assumption and also accommodate inefficiency.

Productivity Change at the Firm Level with Variable Returns to Scale and Inefficiency2 Consider a one-output multiple-input production function at the firm level allowing the returns to scale to vary across different input bundles and the presence of technical inefficiency in the observed output. n be a bundle of n inputs. The production Let y be a scalar output and x ∈ R+ function can then be written as y = f (x, t) τ ; 0 ≤ τ ≤ 1

(27)

where t is an index of time to capture technical change and τ is a measure of efficiency, which coincides with the Shephard output distance function. Then dy df (x, t) dτ ≡ y˙ = τ. + f (x, t) dt dt dt

 ∂f dx i ∂f dτ + =τ + f (x, t) ∂xi dt ∂t dt

(28)

i

Define Thus

∂f ∂xi

≡ fi (i = 1, 2, . . . , n) and

∂f ∂t

≡ ft .

 fi xi x˙i τ˙ y˙ ft = + . + y f xi f τ

(29)

i

It can be easily seen that εi = ∂∂ lnlnxyi = ffi xi is the partial elasticity of output with respect to the ith input. We may now define a measure of the rate of growth in the total input as

2 Sections

“Productivity Change at the Firm level with Variable Returns to Scale and Inefficiency”, through “Productivity Growth from Growth Accounting”, and “Measuring Productivity Growth from the Cost Function” builds upon the model in Denney, Fuss, and Waverman [14].

20 Conceptualization and Measurement of Productivity Growth and Technical . . .

 ˙  fi xi x˙i  x˙i X = = εi X 1 f xi xi i

831

(30)

i

Then we get the measure of total (or multi) factor productivity growth as  ˙  ˙ M y˙ ft X τ˙ = − = + . M 1 y X 1 f τ Here

ft f

(31)

is the rate of autonomous shift in the production function due to technical

change, and ττ˙ is a measure of the change in technical efficiency over time. While the measure of total factor productivity growth in (31) is theoretically correct, the measure of the rate of change total input in (30) is not quite satisfactory. The reason is that unless CRS holds, the partial elasticities (εi ) do not add up to unity. This creates the paradox that if εi < 1 (as is the case when diminishing i

returns to scale hold at the observed input bundle)even  when all inputs increase at X˙ will be less than 5%! One the same rate (say by 5%), the total input growth X 1 way out of this paradox is to redefine the aggregate input growth rate as  ˙  εi x˙i  x˙i  X εi = = ηi ; ηi = . ε0 = εi X 2 ε0 xi xi ε0 i

i

(32)

i

Note that ε0 is the scale elasticity that equals, falls short of, or exceeds unity under constant, diminishing, and increasing returns to scale, respectively. With this alternative measure of the growth rate in the aggregate input, the paradox mentioned above does not arise. Note also that  ˙  X 1 X˙ = . X 2 ε0 X 1

(33)

Thus the total factor productivity growth rate based on this alternative measure of growth rate of total input is  ˙  ˙  ˙  ˙  ˙ M X X y˙ y˙ X X = − = − + − M 2 y X 2 y X 1 X 1 X 2  ˙  ˙  M X 1 = + 1− M 1 ε0 X 1  ˙  X τ˙ 1 ft + + 1− = . f τ ε0 X 1

(34)

In this decomposition of productivity growth, the last term on the right captures the returns to scale effect. Under CRS, ε0 equals unity and the scale component

832

S. C. Ray

disappears. Under DRS, ε0 is less than unity and the scale effect is negative. The opposite is true under IRS when ε0 exceeds unity.

Productivity Growth from Growth Accounting By far, the most common approach to measuring growth of total input is to create an average of growth rates of the individual inputs weighted by their respective cost shares. This yields the measures  ˙   wi xi  x˙i  X = ;C = wi xi X 3 C xi

(35)

 ˙  ˙ M X y˙ = − . M 3 y X 3

(36)

i

i

and

As an aside, consider, at this point, the cost minimization problem of the firm: min C = w  x : A(t)f (x, t) .τ = y.

(37)

n is a strictly positive vector of input prices. We know from the Here w ∈ R++ first-order conditions for a minimum that for each input, wi = λA(t)fi τ and also that the Lagrange multiplier λ = ∂C ∂y . Thus

 y w x λA(t) (fi xi ) .τ y ∂C fi xi wi xi i i = = = C C y A(t)f (x, t) τ C ∂y f

(38)

 ˙    wi xi  x˙i X y ∂C  fi xi x˙i . = = X 3 C xi C ∂y f xi

(39)

Thus,

i

Further, Thus,

y ∂C C ∂y

=

∂ ln C ∂ ln y

i

≡ εcy is the flexibility of cost with respect to output.3

 ˙  ˙  X X y ∂C  fi xi x˙i . = εcy = . X 3 C ∂y f xi X 1 i

Hence,

3 The

inverse of εcy is a metric of overall scale economies.

(40)

20 Conceptualization and Measurement of Productivity Growth and Technical . . .

833

 ˙  ˙  ˙  ˙  ˙ X X M y˙ y˙ X X = − = − + − M 3 y X 3 y X 1 X 1 X 3  ˙  ˙  X  M + 1 − εcy = M 1 X 1  ˙  X ft τ˙  = . + + 1 − εcy f τ X 1

(41)

Again, the last term on the right in (41) measures the scale effect. Under CRS, average cost remains constant so that εcy = 1 and there is no scale effect. Under diminishing returns, the average cost is increasing, and εcy > 1 so that the scale effect is negative. In the case of increasing returns, the scale effect is positive.

Measuring Productivity Growth from the Cost Function We now show how the measure of productivity growth can be derived from a firm’s dual cost function. Denoting the vector of its input prices by w, the dual cost function can be written as C = C (w, y, t) .

(42)

The right-hand side of (42) equals the right-hand side of (37) above by definition. Differentiating (42) with respect to t, we get  ∂C dwi ∂C dy ∂C dC = + + . dt ∂wi dt ∂yi dt ∂t

(43)

i

That is C˙ =

 ∂C ∂C w˙ i + y˙ + Ct . ∂wi ∂yi

(44)

i

Also recall that the dual cost equals the cost of the least cost output bundle and C=



(45)

wi xi .

i

From (45) we get C˙ =

 i

wi x˙i +



xi w˙ i .

(46)

i

Comparing (44) and (46) and recognizing that by Shephard’s lemma for each input i, we get

∂C ∂wi

= xi

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S. C. Ray

 ∂C y˙ + Ct = wi x˙i . ∂yi

(47)

 wi xi x˙i Ct ∂C y y˙ + = . ∂yi C y C C xi

(48)

i

Hence,

i

That is εcy

 wi xi x˙i Ct y˙ + = = y C C xi i

 ˙  ˙ X X = εcy . X 3 X 1

(49)

Hence,  ˙  ˙ M y˙ 1 Ct X = − =− . M 1 y X 1 εcy C

(50)

Of course, under constant, diminishing, or increasing returns to scale, εcy is equal to, greater than, and less than unity, respectively. In this representation, technical progress is captured by − CCt , the rate of autonomous downward shift of the cost function.

Measuring Productivity Growth from the Profit Function4 To wrap up the discussion of the one-output case, we show how productivity growth can be measured from a firm’s dual profit function: π = π (p, w, t) = max py − w x : (x, y) ∈ T t ⇐⇒ π (p, w, t) = max py − w  x : y ≤ f (x, t)

(51)

In (51) above, p is the price of output, w the input price vector, and Tt the production possibility set at time t. As before differentiating (51) with respect to t, we get π˙ =

∂π dp  ∂π dw i ∂π dπ = + + . dt ∂p dt ∂wi dt ∂t

(52)

i

This time, by Hotelling’s lemma,

4 This

∂π ∂p

= y, and

∂π ∂wi

= −xi so that (52) leads to

section extends Ray and Segerson [29] by explicitly including inefficiency.

20 Conceptualization and Measurement of Productivity Growth and Technical . . .

π˙ =

 dπ = y.p˙ − xi w˙ i + πt . dt

835

(53)

i

Also, π = py −



(54)

wi xi

i

where (x, y) is the profit maximizing input-output bundle. Again, from (54) π˙ =

  dπ = y.p˙ + py˙ − xi w˙ i − wi x˙i . dt i

(55)

i

Thus a comparison of (53) and (55) yields πt = py˙ −



wi x˙i .

(56)

i

Dividing both sides of (56) by py, we get y˙ 1  πt = − wi x˙i py y py i

(57)

C  wi xi x˙i y˙ = − y py C xi i

For profit maximization, p =

∂C ∂y

so that

C py

=

C ∂C ∂y .y

=

1 εcy

. Thus, (57) becomes

y˙ 1  wi xi x˙i πt = − py y εcy C xi i

 1 X˙ y˙ = − . y εcy X 3

(58)

Hence, by virtue of (49) above πt y˙ = − py y

 ˙  ˙ X M = . X 1 M 1

(59)

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S. C. Ray

Productivity Measurement with Multiple Outputs5 m from the input We now consider a firm producing a vector of m outputs y ∈ R+ n vector x ∈ R+ . Let its production technology be represented by the Shephard (output) function: D y (x, y, t) =

1 ;ϕ ≥ 1 ϕ

(60)

Alternatively, D y (x, y, t) .ϕ = 1

(61)

Differentiating (61) with respect to t, dϕ dD y (x, y, t) ϕ dD y (x, y, t) =ϕ + D y (x, y, t) dt dt dt ⎛ ⎞ y  ∂D y dx i  ∂D y dy j ∂D ⎠ =ϕ⎝ + + ∂xi dt ∂yl dt ∂t i

(62)

j

+ D y (x, y, t)

dϕ =0 dt

Dividing (62) through by Dy (x, y, t)ϕ, we get ⎛

⎞ y   ∂D y 1 dx i   ∂D y 1 dy j ∂D 1 ⎠ dϕ 1 ⎝ + + = 0 (63) + ∂xi D y dt ∂yj D y dt ∂t D y dt ϕ i

j

Recall that the distance function is increasing in outputs and decreasing in ln D inputs, and define its partial output and input elasticities as δj = ∂∂ ln yj = ∂D y yj ∂yj D y

y

y

(j = 1, 2, , , m) ; μi = ∂∂lnlnDxi = − ∂D ∂xi Then the expression in (63) becomes ⎛ ⎝

 j

xi Dy

(i = 1, 2, , , n).

⎞ y y˙j  x˙i D ϕ˙ δj − μi ⎠ + ty + yj xi D ϕ

(64)

Define the growth rates of aggregate output and input derived from the function as

5 This

section is based on Lovell [23].

20 Conceptualization and Measurement of Productivity Growth and Technical . . .

837

 ˙ D  y˙j Y = δj Y yj

(65)

 ˙ D  X x˙i = μi X 1 xi

(66)

j

i

Note that the output distance function

isy increasing (non-decreasing) and homogeneous of degree 1 in outputs. Thus Dj yj = D y (x, y, t), and therefore j

 Djy yj Dy

j

However, unless CRS holds,

=



δj = 1.

(67)

j

μi = 1. As a result the growth rate of aggregate

j

input is not homogeneous of degree 1 in the growth rates of individual inputs. To solve this problem, we define the new weights ωi = μμ0i , μ0 = μi , and a different i

measure of the growth rate of aggregate input  D  ˙ D  x˙i 1  x˙i 1 X˙ X = ωi = μi = . X 2 xi μ0 xi μ0 X 1 i

(68)

i

Using (65 and 66) in (64), we get  ˙ D  D  D y M y˙ x˙ D ϕ˙ ≡ − = − ty − M 1 y x 1 D ϕ

(69)

The alternative measure of total factor productivity growth is  ˙ D  ˙ D  ˙ D M Y X ≡ − M 2 Y X 2  ˙ D  ˙ D  ˙ D  ˙ D Y X X X = − + − Y X 1 X 1 X 2  ˙ D  ˙ D M M = + (μ0 − 1) M 1 M 2  ˙ D y M D ϕ˙ = (μ0 − 1) − ty − M 2 D ϕ

(70)

838

S. C. Ray

In (70), the first term represents the returns to scale effect and disappears under constant returns when μ0 equals unit, the second term captures productivity growth measured by the rate of autonomous shift in the distance function over time, and the last term is the rate of change in technical efficiency over time.

Productivity Change in Discrete Time A Hicks-Moorsteen Productivity Index Consider a firm producing a single output y from a single input x. Let (xt , yt ) (t = 0, 1) represent its input-output pairs in two successive time periods. Then its productivity index in period 1 with period 0 treated as the base year is

π1,0

AP1 = = AP0

y1 x1 y0 x0

.

(71)

π1,0

AP1 = = AP0

y1 y0 x1 x0

.

(72)

This can also be written as

Define the output and input quantity ratios Qy = π1,0 =

Qy . Qx

y1 y0

and Qx =

x1 x0 .

Then

(73)

If the proportionate increase in output is greater than the increase in input, π 1, 0 > 1, and productivity is higher in period 1. In almost all realistic cases, however, firms use multiple inputs and often produce multiple outputs. Even in a simple one-output two-input case, there will be two partial average productivities corresponding to  the two inputs. Suppose that the input vectors in the two periods are x t = x1t , x2t , t = 0, 1 Now the partial average productivities are AP t1 = xytt and AP t2 = xytt in the two different periods 1 2 t = 0 and 1. Consequently, measuring a productivity index as the ratio of average productivities becomes problematic. In the case of multiple outputs and multiple inputs, the problem is even more complicated. One way to solve this problem is to create aggregate measures of inputs and outputs and measure total factor productivity (rather than partial productivity for individual inputs).  that the output vectors produced by the firm in the two  Suppose periods are yt = y1t , y2t , t = 0, 1. We use the aggregator functions Yt = a y1t , y2t and Xt = b x1t , x2t to measure the total factor productivity index:

20 Conceptualization and Measurement of Productivity Growth and Technical . . .

π1,0 = T F P 1,0 =

Y1 X1 Y0 X0

=

  a y11 ,y21  0 0 a y1 ,y2   b x11 ,x21  0 0 b x1 ,x2

.

839

(74)

Define the output and input quantity indexes as    Qy =

a y11 ,y21   a y10 ,y20

and Qy =

b x11 ,x21 .  b x10 ,x20

Then

T F P 1,0 =

Qy . Qx

(75)

The only restrictions on the aggregator functions are that they must be nonnegative, increasing (non-decreasing) in each argument, and homogeneous of degree 1. That is, if the quantity of any input (output) increases, the aggregate input (output) also increases (does not decrease), and if every input (output) increases by the same proportion, so does the aggregate input (output).6 The total factor productivity index defined in (75) is the ratio of the output and input quantity indexes. 7

The Tornqvist Productivity Index By far the most popular quantity index number is the Tornqvist index measured by a weighted geometric mean of the relative quantities from the two periods. Consider the output quantity index first. Suppose that m outputs are   involved. The 0 output vectors produced in periods 0 and 1 are, respectively, y 0 = y10 , y20 , . . . , ym   1 . The corresponding output price vectors are p 0 = and y 1 = y11 , y21 , . . . , ym    0 0  0 1 1 , respectively. Then, the Tornqvist p1 , p2 , . . . , pm and p = p11 , p21 , . . . , pm output quantity index in period 1 with period 0 as the base is  T Qty

Here, vjt =

=

pjt yjt

m

1

pkt ykt

y1 1 y1 0

v1t 

y2 1 y2 0

v2t



ym 1 .... ym 0

vmt  m ; vjt = 1

(76)

1

(t = 0, 1) . is the share of output i in the total value of the

output bundle in period t. Of course, the value shares of the individual outputs are, in general, different in the two periods. In practical applications, for vjt , one uses the arithmetic mean of vj 0 and vj 1 . This leads to 6 It is important to emphasize that although such arbitrary quantity aggregators yield a theoretically

valid productivity index, they may not provide any insight into the extent of technical change between the two periods. 7 For a detailed discussion of index numbers, see  Chap. 19, “Index Numbers and Productivity Measurement” by Prasada Rao in this volume.

840

S. C. Ray

 T Qy =

0

y1 1 y1 0

0

1

1  v1 +v 2

y2 1 y2 0

1

2 v2 +v 2



m vm +v 2 0

ym 1 .... ym 0

1

=



T Q0y .T Q1y .

(77)

It may be noted that in the single-output case, the Tornqvist output quantity index trivially reduces to the ratio of output quantities in the numerator of (73). This is also true when the quantity ratio remains unchanged across all outputs.   Similarly, let the input vectors in the two periods be x 0 = x10 , x20 , . . . , xn0   and x 1 = x11 , x21 , . . . , xn1 . The corresponding input price vectors are w 0 =    0 0  w1 , w2 , . . . , wn0 and w 1 = w11 , w21 , . . . , wn1 . Then, the Tornqvist input quantity index is  T Qtx

=

x1 1 x1 0

s1t 

x2 1 x2 0

s2t



xn 1 .... xn 0

snt  n ; sit = 1; t = 0, 1.

(78)

1

Here, wit xit sit = ; t = 0, 1. is the share of input j in the total cost of the input bundle. n 1

wkt xkt

Again, in practice, one uses the average of the cost share of any input in the two periods. Thus, the Tornqvist input quantity index is T Qx =

 T Q0x .T Q1x .

(79)

The Tornqvist productivity index is the ratio of the Tornqvist output and input quantity indexes. Thus, πT Q =

T Qy . T Qx

(80)

When TQy > TQx , output in period 1 has grown faster (or declined slower) than input as a result of which productivity has increased in period 1 compared to what it was in period 0. It may be noted that the Tornqvist productivity index can be measured without any knowledge of the underlying technology so long as data are available for the input and output quantities as well as the shares of the individual inputs and outputs in the total cost and total revenue, respectively.8

8 However, unless one assumes that the cost or revenue shares correspond to cost-minimizing/profit-

maximizing behavior by the producer, one cannot extract technical change from the Tornqvist productivity index.

20 Conceptualization and Measurement of Productivity Growth and Technical . . .

841

The Fisher Productivity Index An alternative to the Tornqvist index of productivity is the Fisher index where one uses Fisher indexes of output and input quantity in the multi-factor productivity index measure. It may be noted that the Fisher quantity (or price) index is itself the geometric mean of the relevant Laspeyres and Paasche indexes. The Laspeyres output quantity index is the value ratio of the two output vectors at base period prices and is measured as m

LQy =

pj 0 yj 1

1

m

.

(81)

pj 0 yj 0

1

It is easy to see that LQy =

m

λj 0



1

where λj 0 =

pj 0 yj 0 m

pk 0 yk 0 1

yj 1 yj 0



is the same as vj 0 defined above.

Thus, while the Tornqvist quantity index is a weighted geometric mean of the quantity relatives, the corresponding Laspeyres index is a similarly weighted arithmetic mean. The Paasche output quantity index, for which we evaluate the current and base period output bundles at current period prices, is measured as m

P Qy =

1 m

pj 1 yj 1 .

(82)

pj 1 yj 0

1

Thus, P Qy =

m

1

μj 1



yj 1 yj 0



where μj 1 =

pj 1 yj 0 . m

pk 1 yk 0 1

The Fisher output quantity index is the geometric mean of the Laspeyres and Paasche output quantity indexes.9 Hence, F Qy =

 LQy .P Qy .

(83)

In an analogous manner, the Laspeyres, Paasche, and Fisher input quantity indexes are obtained as

9 The

geometric mean is a merged relative score that satisfies a number of important postulates. The interested reader should refer to Aczel [1].

842

S. C. Ray n

LQx =

1 n

wj 0 xj 1 ,

(84)

,

(85)

wj 0 xj 0

1 n

P Qx =

1 n

wj 1 xj 1 wj 1 xj 0

1

and F Qx =



(86)

LQx .P Qx ,

respectively. The resulting Fisher productivity index is πF =

F Qy . F Qx

(87)

It may be noted that the Tornqvist and Fisher indexes are derived from the geometric and arithmetic means of ratios of the output and input quantities. In practical applications their numerical values are generally quite close.

Profitability, Terms of Trade, and Productivity Indexes10 Consider again the input and output quantity and price vectors from the two periods (xt , yt ) and (wt , pt ) for t = 0,1. Then, in any period, one can define profitability measures: Proft =

pt y t (t = 0, 1) . w t x t

(88)

Following O’Donnell [26], one may construct a profitability index in period 1 with period 0 as the base as

Prof1,0

Prof1 = = Prof0

p1 y 1 w1 x 1 p0 y 0 w0 x 0

.

The right-hand side of (89) can be alternatively expressed as

10 This

section is based in part on O’Donnell [26].

(89)

20 Conceptualization and Measurement of Productivity Growth and Technical . . .

Prof1,0

Prof1 = = Prof0

p1 y 1 p0 y 0 w1 x 1 w0 x 0

Prof1,0

Prof1 = = Prof0

p1 y 1 p0 y 0 w1 x 1 w0 x 0

843

=

p1 y 0 p0 y 0 w1 x 0 w0 x 0

p1 y 1 p1 y 0 w1 x 1 w1 x 0

(90a)

=

p1 y 1 p1 y 0 w1 x 1 w1 x 0

p1 y 0 p0 y 0 w1 x 0 w0 x 0

(90b)

or

A geometric mean of (90a) and (90b) yields  Prof1,0 = 

p1 y 0 p1 y 1 p0 y 0 p0 y 1 w1 x 0 w1 x 1 w0 x 0 w1 x 0

 

p1 y 1 p0 y 1 p1 y 0 p0 y 0 w1 x 1 w0 x 1 w1 x 0 w0 x 0

    P p 1 , p 0 F Qy .  = = T T 1,0 πF 1 0 F Q W w ,w x

(91)

In (91) above    1 0 1 1 P p1 , p0 = pp0 yy 0 pp0 yy 1 is a Fisher index of output prices   w1 x 0 w1 x 1  and W w 1 , w 0 = w 0 x 0 w 1 x 0 is a Fisher index of output prices. Thus, the two components of the profitability index are the Fisher productivity   P p1 ,p0

FQ

index πF = F Qyx and a terms of trade factor T T 1,0 = W w1 ,w0 . When output ( ) prices increase faster (slower) than input prices, the terms of trade factor is greater (less) than 1, and profitability index is greater (less) than the productivity index. In this chapter the principal focus is on explaining the rate of growth in productivity measured by the rate of growth in the output beyond the rate of growth in inputs in terms of technical change, change in technical efficiency, and returns to scale effects of input change. This is in the tradition of Schultz [31] and Jorgenson and Griliches [22]. O’Donnell [26] argues, however, that changes in the so-called terms of trade factor (TT1, 0 ) affect productivity under competitive profitmaximizing behavior of a firm. In a single-output single-input case, the profit maximization problem is max py − wx s.t.y ≤ f (x).

(92)

From the first-order conditions for a maximum, one gets (i) f  (x) = w p and (ii)     w y = f (x). This leads to the input demand x ∗ = f  −1 w p = x p output supply      w function y ∗ = f x w = y w p p . When p goes up (i.e., the TT goes down),     w x w p goes down, and hence the output supply y p also goes down. Finally, if the production function is concave, output declines less than proportionately with

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S. C. Ray

the input. As a result, average productivity goes up. Hence, there would be an inverse relationship between changes in the terms of trade and productivity change.

Malmquist Productivity Index Both the Tornqvist and Fisher productivity indexes are essentially descriptive measures based on the observed input and output data. By contrast, the Malmquist productivity index introduced by Caves, Christensen, and Diewert (CCD) [6] is a normative index and is based on a reference technology. We first consider the one-output one-input case and assume that the input-output quantities in the two periods are (x0 , y0 ) and (x1 , y1 ). Further, assume that the production function in both periods is y∗ = f (x).We can then write the productivity index as π1,0 =

y1 x1 y0 x0

=

y1 f (x1 ) f (x1 ) . x1 y0 f (x0 ) f (x0 ) . x0

(93)

.

Alternatively, we could invert the production function to get the input requirement function x∗ = f−1 (y) ≡ g(y) and write the productivity index as π1,0 =

y1 x1 y0 x0

=

y1 g(y1 ) g(y1 ) . x1 y0 g(y0 ) g(y0 ) . x0

=

∗ y1 x1 g(y1 ) . x1 ∗ y0 x0 g(y0 ) . x0

=

y1 g(y1 ) .τx y0 g(y0 ) .τx

(x1 , y1 ) (x0 , y0 )

.

(94)

CCD proposed two different measures of the productivity index defined by ratios of distance functions – one input oriented and the other output oriented. Their inputoriented Malmquist productivity index for (x1 , y1 ) relative to (x0 , y0 ) is MP I x (x1 , y1 ; x0 , y0 ) =

D x (x0 , y0 ) . D x (x1 , y1 )

(95)

On the other hand, their output-oriented Malmquist productivity index for the same input-output pairs is MP I y (x1 , y1 ; x0 , y0 ) =

D y (x1 , y1 ) . D y (x0 , y0 )

(96)

CCD assume Dx (x0 , y0 ) equals unity in (95). As a result, MPIx (x1 , y1 ; x0 , y0 ) = They explicitly recognize, however, that the input- and outputoriented Malmquist productivity indexes “will differ from each other by a factor that reflects the returns to scale of the production structure” ([6], p. 1402). Figure 1 shows, for the one-output one-input case, how the Malmquist productivity index in (95) differs from the intuitive measure of productivity change. (Dx (x1 , y1 ))−1 .

20 Conceptualization and Measurement of Productivity Growth and Technical . . .

845

1

0

Fig. 1 CCD input-oriented Malmquist productivity index (VRS)

Points A and B in this diagram denote the input-output bundles (x0 , y0 ) and (x1 , y1 ), respectively. Because Dx (x0 , y0 ) equals unity, A is located on the production frontier shown efficient projection of point B is the point  bythe curve f(x).The input-oriented By1 Ox1 C x1∗ , y1 and D x (x1 , y1 ) = Ox ∗ = Cy . Further, dividing both the numerator and 1 1 the denominator of Dx (x1 , y1 ) by Oy1 , we obtain D (x1 , y1 ) = x

By1 Oy1 Cy1 Oy1

=

Oy1 Cy1 Oy1 By1

=

AP (C) . AP (B)

(97)

Hence, MP I x (x1 , y1 ; x0 , y0 ) =

AP (B) D x (x0 , y0 ) . = x D (x1 , y1 ) AP (C)

(98)

Thus, AP (B) MP I x (x1 , y1 ; x0 , y0 ) = = AP (C)



AP (B) AP (A)



AP (A) . AP (C)

(99)

In the CCD input-oriented Malmquist productivity index shown in (99), the first factor is the productivity index, while the second factor is the returns to scale effect.

846

S. C. Ray

0

Fig. 2 CCD input-oriented Malmquist productivity index (CRS)

We can contrast this with the case where the production function exhibits CRS. g(y1 ) g(y0 ) 1) 0) = f (x We would then get f (x x1 x0 and also y1 = y0 , and (94) above would reduce to π1,0 =

y1 x1 y0 x0

=

y1 g(y1 ) g(y1 ) . x1 y0 g(y0 ) g(y0 ) . x0

=

x1∗ x1 x0∗ x0

=

D x (x0 , y0 ) τx (x1 , y1 ) = Cx . τx (x0 , y0 ) DC (x1 , y1 )

(100)

This is shown in Fig. 2, where the production function f(x) is a ray through the origin. This time, AP(A) and AP(C) are equal due to CRS. Hence the ratio of the distance functions correctly measures the productivity index. π1,0 =

y1 x1 y0 x0

=

τyC (x1 , y1 ) τyC (x0 , y0 )

=

τxC (x1 , y1 ) . τxC (x0 , y0 )

(101)

That is, productivity change between period 0 and period 1 is simply the change in technical efficiency relative to a CRS production function. Unless, CRS holds, however, the ratio of technical efficiencies (input and output oriented) will differ from the productivity index11 .

11 For

√ a simple example, consider the production function f (x) = 2 x and its inverse f −1 (y) =

g(y) =

y2 4.

This time, τy (xk , yk ) =

yk √ 2 xk

(k = 0, 1) . Clearly,

τy (x1 ,y1 ) τy (x0 ,y0 )

=

y √1 2 x1 y √0 2 x0

=

y1 x1 y0 x0

.

20 Conceptualization and Measurement of Productivity Growth and Technical . . .

847

Fig. 3 Geometry of the output-oriented Malmquist productivity index

Malmquist Productivity Index with Multiple Outputs and Inputs Although the one-input one-output example of the Malmquist productivity index easily generalizes to the m-output n-input technology, the input- and output-oriented Malmquist productivity indexes in (95) and (96) correctly measure productivity change only when the distance functions are measured relative to a CRS frontier. Thus,

π1,0 = MP I

C



x ,y ;x ,y 1

1

0

0



   y  DC x 1 , y 1 DCx x 0 , y 0 = y  0 0 = x  1 1 DC x , y DC x , y

(102)

In reality, of course, CRS may not hold in many situations. It is important to recognize, however, that the Malmquist productivity index should be measured by the ratio of the CRS distance functions even when the technology does not exhibit CRS globally. Below, in the discussion of a multiplicative decomposition of the Malmquist productivity index, we identify the specific contribution of the returns to scale factor to overall productivity change.

848

S. C. Ray

Allowing Technological Change We now consider technological change over time. In that case, there will be two different production possibility sets in the two different time periods, and we will get two different measures of the distance function (or technical efficiency) for y the same input-output bundle. Let Dt (x, y) be the VRS output distance function evaluated at the input-output bundle (x, y) relative to the technology from period y t(=0, 1). Further, DCt (x, y) is the output function when CRS is assumed. We will now have two alternative measures of the Malmquist index depending on which of the two technologies (t = 0 and t = 1) is used as the benchmark. Fare, ¨ Grosskopf, Lindgren, and Roos (FGLR), [17] provided a decomposition of the Malmquist index into two distinct components representing change in technical efficiency and technical change measured by the shift in the frontier. Consider first the MPI relative to the technology from period 0: MP I C 0

 y   y    D y x 1 , y 1  DC1 x 1 , y 1 DC0 x 1 , y 1 C0 1 1 0 0   . (103) x , y ; x , y = y  0 0 = y  0 0 . DC0 x , y DC0 x , y DC1 x 1 , y 1

Similarly, with period 1 technology as the benchmark, MP I C 1

 y   y    D y x 1 , y 1  DC1 x 1 , y 1 DC0 x 0 , y 0 C1 1 1 0 0   . (104) x , y ; x , y = y  0 0 = y  0 0 . DC1 x , y DC0 x , y DC1 x 0 , y 0

Taking the geometric mean of (103) and (104), one gets    D y x 1 , y 1  D y x 0 , y 0  D y x 1 , y 1  C1 C0 C0  . MP I C x 1 , y 1 ; x 0 , y 0 = y  0 0  . y  0 0 . 1 1 D DC0 x , y DC1 x , y C1 x , y

(105)

The first factor on the right-hand side  y  DC1 x 1 , y 1  = T EC y  DC0 x 0 , y 0

(106)

represents technical efficiency change (TEC) between the two periods. A value of TEC greater (less) than 1 indicated increase (decline) in technical efficiency in period 1 compared to period 0. Each ratio inside the second factor represents the shift in the production frontier measured at the input-output bundle in the two different periods. The geometric mean of the two is the second factor 

 y   y  DC0 x 0 , y 0 DC0 x 1 , y 1   = TC   . y DC1 x 0 , y 0 DC1 x 1 , y 1

(107)

20 Conceptualization and Measurement of Productivity Growth and Technical . . .

849

representing technical change (TC). A value of TC greater than 1 implies an outward shift of the technology or technical progress.

Allowing Returns to Scale Effect In a subsequent paper, Fare, ¨ Grosskopf, Norris, and Zhang (FGNZ) [18] relaxed the CRS assumption and offered the following decomposition: 



 DC1 x 1 ,y 1   D y x 1 , y 1  D y x 0 , y 0  D y x 1 , y 1  D y (x 1 ,y 1 ) C0  . y1  . C0  MP I C x 1 , y 1 ; x 0 , y 0 = 1y  0 0  . y  D0 x , y DC1 x 0 , y 0 DC1 x 1 , y 1 DC0(x 0 ,y 0) y

D0 x 0 ,y 0

(108) As in FGLR [17], the first two terms are interpreted as technical efficiency change (TEC) and technical change (TC). The last term in the right-hand side of (108)   DC1 x 1 ,y 1 y 1 1 D1 (x ,y ) y

DC0 (x 0 ,y 0 ) y D0 (x 0 ,y 0 )

= SEC

(109)

is the ratio of scale efficiencies in period 1 and period 0 and is described as scale efficiency change (SEC). Ray and Desli (RD) [30] argued that assuming VRS to measure technical efficiency change and CRS to measure technical change within the same decomposition is not internally consistent and proposed the following decomposition:   MP I x 1 , y 1 ; x 0 , y 0   D y (x 1 ,y 1 ) D y (x 1 ,y 1 )         C0 C1 (110) y y y y y D0 x 0 , y 0 D0 x 1 , y 1  D1 x 1 , y 1 D0 (x 1 ,y 1 ) D1 (x 1 ,y 1 )      . . = y  0 0 . . .  D y (x 0 ,y 0 ) D y (x 0 ,y 0 ) y D0 x , y D1 x 0 , y 0 D1 x 1 , y 1 C0 C0 y y D0 (x 0 ,y 0 ) D0 (x 0 ,y 0 ) The first factor in (110) measures technical efficiency change (TEC) exactly the same way as in (108), but the second factor measures technical change (TC) by the shift in the VRS frontier and is consistent with the first factor where technical efficiency change is also measured with reference to VRS frontiers. RD call the last factor on the right-hand side of (110)

850

S. C. Ray

  D y (x 1 ,y 1 ) D y (x 1 ,y 1 )  C0 C1  D y (x 1 ,y 1 ) D y (x 1 ,y 1 ) 1  0  D y (x 0 ,y 0 ) . D y (x 0 ,y 0 ) = SCF C0 C0 y y D0 (x 0 ,y 0 ) D0 (x 0 ,y 0 )

(111)

a scale change factor (SCF). Unlike SEC in (109), it has no clear intuitive interpretation and is more like a residual incorporating both scale efficiency change and what is described by some authors as “scale bias of technical change.” The Malmquist productivity index and its three-factor decomposition are shown in Fig. 3. The points A (x0 , y0 ) and B (x1 , y1 ) show the input-output bundles for the same firm in period 0 and period 1.The curves y∗ = f0 (x) and y∗ = f1 (x) show the production functions in periods 0 and 1. The productivity index is π1,0 ==

y1 x1 y0 x0

=

AP (B) . AP (A)

(112)

This can be expressed as π1,0 = =

AP (C) AP (D) AP (E) AP (B) AP (B) = . . . AP (A) AP (A) AP (C) AP (D) AP (E) AP (B) AP (E) AP (A) AP (C)

In (113) the first factor

(113)

AP (D) AP (E) . . . AP (C) AP (D)

AP (B) AP (E) AP (A) AP (C)

on the right is

y1 f 1 (x1 ) y0 f 0 (x0 )

=

τy1 (x1 ,y1 ) τy0 (x0 ,y0 )

= T EC, which 1

f (xo ) (D) measures technical efficiency change, the second factor AP AP (C) = f 0 (x0 ) = T C (x0 ) is a measure of the autonomous shift in the production function due to technical SE 1 (x1 ) AP (E) change measured at x0 , and the last factor AP (D) = SE 1 (x0 ) measures the relative scale efficiency of inputs x1 and x0 relative to the production function from period 1. In a completely analogous way,

π1,0 =

AP (B) AP (A)

=

AP (B) AP (E) AP (F ) AP (C) AP (E) . AP (F ) . AP (C) . AP (A) AP (E) AP (F ) . AP (F ) . AP (C) .

=

AP (B) AP (E) AP (A) AP (C)

(114)

In (114) π1,0 =

τy1 (x1 , y1 ) f 1 (x1 ) SE 0 (x1 ) AP (B) = 0 . . . AP (A) τy (x0 , y0 ) f 0 (x1 ) SE 0 (x0 )

The RD decomposition is the geometric mean of (114) and (115).

(115)

20 Conceptualization and Measurement of Productivity Growth and Technical . . .

851

Biennial Malmquist Index The biennial Malmquist index introduced by Pastor, Asmild, and Lovell [27] provides the same decomposition as in Ray and Desli [30] but avoids a linear programming infeasibility problem associated with the Ray-Desli decomposition of the Malmquist index that arises when an input-output bundle from one period is infeasible relative to the frontier in the other period. Instead of using a periodspecific production possibility frontier, they estimate the technical efficiency of a production unit with reference to a biennial production possibility frontier which is empirically constructed from the pooled input-output data from two consecutive periods. The reference technology set TB is empirically constructed from the pooled data from both periods t and t + 1 (a simple graphical illustration of the biennial production possibility frontier, for single-output single-input case, is given in section “Biennial Malmquist Index” and in Fig. 6). Using the output-oriented technical efficiency scores with reference to a CRS biennial frontier, the biennial Malmquist productivity index of the firm s producing a single output from multiple inputs is measured as12 Mc

B

  T Ec B x t+1 , y t+1  s s t t t+1 t+1   . xs , ys ; xs , ys = T Ec B xst , yst

(116)

The decomposition of this biennial Malmquist productivity index is   Mc B xs t , ys t ; xs t+1 , ys t+1 = T EC × T C × SEC

(117)

where

TC =

  T Ev t+1 xs t+1 , ys t+1 T EC = , T Ev t (xs t , ys t )     T Ev B xs t+1 , ys t+1 /T Ev t+1 xs t+1 , ys t+1 T Ev B (xs t , ys t ) /T Ev t (xs t , ys t )

(118)

(119)

and SEC =

12 Since

    T Ec B xs t+1 , ys t+1 /T Ev B xs t+1 , ys t+1 T Ec B (xs t , ys t ) /T Ev B (xs t , ys t )

.

(120)

the biennial Malmquist index of productivity uses the biennial CRS production possibility set, which includes the period t and t + 1 sets, one need not calculate a “geometric mean” of two productivity indexes while measuring it.

852

S. C. Ray

Directional Distance Function and Luenberger Productivity Indicator Building upon Luenberger’s [24] benefit function, Chambers, Chung, and Fare ¨ [10] introduced the directional distance function (DDF) to measure the distance of an observed input-output bundle from the frontier of the PPS  in a direction chosen by n and g y = g y , g y , . . . , g y ∈ R m be the analyst. Let g x = g1x , g2x , . . . , gnx ∈ R+ m + 1 2 two direction sub-vectors. Then the DDF can be defined as   − →  0 0 x y D x , y ; g , g = max β : x 0 − βg x , y 0 + βg y ∈ T . (121) It is clear that one can recover the radial output-oriented model by setting gx = 0 and gy = y0 . In that case, β in (121) would equal (ϕ − 1) in (15) above. An interesting choice of the direction for projection would be (gx , gy ) = (x0 , y0 ). That leads to   − →  0 0 x y D x , y ; g , g = max β : (1 − β) x 0 , (1 + β) y 0 ∈ T . (122)

Output ( )

In that case β is the maximum percentage by which all outputs can be expanded and all inputs can be contracted simultaneously. In Fig. 4, A is the observed bundle (x0 , y0 ). The point B (gx = − x0 , gy = y0 ) defines the direction of movement. The point C on the production frontier shows the maximum feasible movement within the production possibility set in the direction AC parallel to OB. In this case, the directional distance function is β = OB = OD OB . While the Malmquist productivity index measured by the geometric mean of ratios of Shephard distance functions corresponding to the technologies of two different periods continues to be the most commonly used approach for measurement of productivity change, in recent years an alternative measure of productivity change

∗

= ( )

C 0

B

A

D 0

Fig. 4 Directional distance function

Input (X)

20 Conceptualization and Measurement of Productivity Growth and Technical . . .

853

known as the Luenberger productivity indicator introduced by Chambers, Färe, and Grosskopf (CFG) [11] has gained wide popularity.13 Consider, as before, two CRS technologies – TC0 for period 0 and TC1 period 1. Further, let (xt , yt ) be the input-output of a firm period t (=0, 1). Next consider a direction g = (−gx , gy ).Then two possible measures of productivity change between period 0 and period 1 are L0

     −  −  →0  →0  x1, y1 , x0, y0 ; g = D x0, y0; g − D x1, y1; g

(123)

with reference to TC0 and L1

     −  −  →1  →1  x1, y1 , x0, y0 ; g = D x0, y0; g − D x1, y1; g

(124)

with reference to TC1 .        − →t  − →t  If D x 0 , y 0 ; g > D x 1 , y 1 ; g ⇐⇒ Lt x 1 , y 1 , x 0 , y 0 ; g > 0, (x0 , y0 ) is farther away from the period t frontier than (x1 , y1 ) in the direction g. This implies higher productivity in period 1 than in period 0. CFG [11] define the Luenberger productivity indicator as L

     1       L0 x 1 , y 1 , x 0 , y 0 ; g x1, y1 , x0, y0 ; g = 2      +L1 x 1 , y 1 , x 0 , y 0 ; g  1 − →0  0 0  − →0  1 1  D x ,y ;g − D x ,y ;g = 2   →0  1 1  − →0  0 0  − + D x ,y ;g − D x ,y ;g

(125)

This can also be expressed as  L

    −  −  →0  →1  x1, y1 , x0, y0 ; g = D x0, y0; g − D x1, y1; g +

1 2 

  − →1  0 0  − →0  D x , y ; g − D x0, y0; g (126)

 −  →0  − →1  + D x1, y1; g − D x1, y1; g

13 The



Luenberger productivity indicator was first formulated in a working paper by Chambers [7]. Subsequently, it appeared in a number of papers including Chambers, Chung, and Fare ¨ ([10], [12]); Chambers and Pope [9]; Chambers, Fare, ¨ and Grosskopf [11]; and Chambers [8].

854

S. C. Ray

Fig. 5 Luenberger productivity indicator

The first term inside the square brackets represents change in (directional) technical (in)efficiency. A positive value of this difference means that the level of (in)efficiency is lower in period 1 compared to period 0. This implies a positive technical efficiency change (TEC). The other term represents technical change (TC). The two differences inside the second term show shifts in the frontier between period 0 and period 1 measured from (x0 , y0 ) and (x1 , y1 ) in the direction g. A positive value of the difference implies an outward shift of the frontier. In Fig. 5, the lines OR0 and OR1 show the CRS production frontiers in period 0 and period 1. The points a and b show the input-output combinations of a firm in period 0 and period 1, respectively. The projection of a in the direction g is the point c on OR0 and the point e on OR1 . The corresponding projections of b are the points d on OR0 and the point f on OR1 . One way to measure the productivity − →0 differences between b and a is to compare the distances ac = D C (x0 , y0 ; g) and − →0 bd = D C (x1 , y1 ; g) from the OR0 line. A positive value of this difference implies that productivity is higher at b than at a. Similarly, one could compare the distances − →1 − →1 ae = D C (x0 , y0 ; g) and bf = D C (x1 , y1 ; g) from the OR1 line. The Luenberger productivity indicator shown in (126) is 12 [(ac − bd) + (ae − bf )] . Based on the directional distance functions, the Luenberger productivity indicator depends critically on the choice of the direction of projection. As mentioned

20 Conceptualization and Measurement of Productivity Growth and Technical . . .

855

above, a popular choice for the direction is (gx = xt , gy = yt ). This, of course, would mean that the directions of projection will be different in the two periods 0 and 114 .

Relation Between Tornqvist and Malmquist Productivity Indexes CCD define the Malmquist output quantity index as   D y y 1 , x  1 0 Qy y , y ; x = y  0  D y ,x

(127)

Because the output function is homogeneous of degree 1 and is (weakly) ∂D y monotonic in outputs, for any output vector y, D y (y, x) = ∂yj yj is a valid j

output aggregator. Corresponding to the two different input bundles, there are two measures of the Malmquist output quantity index, Q0y

  D y y 1 , x 0    D y y 1 , x 1  1 0 0 1 1 0 1 y , y ; x = y  0 0  and Qy y , y ; x = y  0 1  . D y ,x D y ,x

(128)

Similarly, the Malmquist input quantity indexes are Q0x

  D x y 0 , x 1    D x y 1 , x 1  1 0 0 1 1 0 1 x , x ; y = x  0 0  and Qx x , x ; y = x  1 0  . D y ,x D y ,x

(129)

At this point, consider the revenue maximization problem R (y1 , y2 ; x1 , x2 ) = max p1 y1 + p2 y2 s.t.D y (x1 , x2 ; y1 , y2 ) = 1

(130)

where (p1 , p2 ) are the output prices and (x1 , x2 ) are the given input quantities. The Lagrangian for the constrained maximization problem is L = p1 y1 + p2 y2 + λ (1 − D y (x1 , x2 ; y1 , y2 ))

(131)

and the corresponding first-order conditions for a maximum are ∂L ∂D y = pj − λ = 0 (j = 1, 2) ∂yj ∂yj 14 Afrashian

(132)

and Ahn [2] have extended the CFG [11] decomposition of the Luenberger productivity indicator to identify separately a change in direction component.

856

S. C. Ray

∂L = 1 − D y (x1 , x2 ; y1 , y2 ) = 0 ∂λ

(133)

Thus, (131), (132) together imply that pj yj ∂ ln D y

= vj = (j = 1, 2) . pk yk ∂ ln yj

(134)

k

Here vj is the revenue share of output j defined earlier in the context of the Tornqvist index. Similarly, for the cost minimization problem for a given pair of outputs, C (x1 , x2 ; y1 , y2 ) = min w1 x1 + w2 x2 s.t.D x (x1 , x2 ; y1 , y2 ) = 1

(135)

Arguing as above, we can get wi xi ∂ ln D x

= si = (j = 1, 2) . wk xk ∂ ln xi

(136)

k

As before, si is the cost share of input i. CCD have shown that when both input-output bundles (xk , yk ) (k = 0, 1) are elements of the same production possibility set (i.e., there no technological difference between them) and have translog distance functions with identical linear, quadratic, and interaction parameters, the Malmquist and Tornqvist output and input quantity indexes will be identical. We show that with a two-output two-input example, for simplicity. Consider now the output distance function 1 1 ln D y (x1 , x2 ; y1 , y2 ) = α0 + α1 ln y1 + α2 ln y2 + α11 (ln y1 )2 + α22 (ln y2 )2 2 2 1 + α12 (ln y1 ) . (ln y2 ) + β1 ln x1 + β2 ln x2 + β11 (ln x1 )2 2 1 + β22 (ln x2 )2 + β12 (ln x1 ) . (ln x2 ) + γ11 (ln y1 ) (ln x1 ) 2 + γ12 (ln y1 ) (ln x2 ) +γ21 (ln y2 ) (ln x1 ) +γ22 (ln y2 ) (ln x2 ) (137) For the distance function in (137),

20 Conceptualization and Measurement of Productivity Growth and Technical . . .

∂ ln D y = α1 + α11 (ln y1 ) + α12 (ln y2 ) + γ11 (ln x1 ) + γ12 (ln x2 ) ; ∂ ln y1 ∂ ln D y = α2 + α21 (ln y1 ) + α22 (ln y2 ) + γ21 (ln x1 ) + γ22 (ln x2 ) ; ∂ ln y2 Because the output function is homogeneous of degree 1 in (y1 , y2 ), ∂ ln D y ∂lny2

= 1. This in its turn implies

857

(138)

∂ ln D y ∂lny1

+

lα1 + α2 = 1; α11 + α12 = α21 + α22 = 0;

(139)

γ11 + γ12 = γ21 + γ22 = 0; Also, by Young’s theorem, α 12 = α 21 . In light of (139), CCD use Diewert’s quadratic identity [15] formulated in CCD [6] as a translog identity to show that 1   1    1 1 2 v1 + v10 ln y11 − ln y10 + v21 + v20 ln y21 − ln y20 = ln Q0y .Q1y 2 2 (140) Hence, the Malmquist output quantity index

Qy =



 Q1y .Q0y =

v1 +v1  v2 +v2 2 2 y1 . 20 = T Qy 0

y11 y10

0

1

y2

1

(141)

In a comparable manner, one can use a translog input distance function to show that    Q1y .Q0y Qy 0 1 0 1 MP I x x , x ; y , y =  = (142) 1 0 Qx Qx .Qx Recall that CCD propose two alternative measures of the Malmquist productivity index, one output oriented and the other input oriented, and in the absence of CRS, they will provide differing measures of productivity change. Further, it should be noted that if the two units face different technologies, the Malmquist productivity indexes (whether input or output oriented) will be technology-specific. In that case, one should use a geometric mean of the technology-specific indexes. CCD ([6], theorems 3 and 4) have shown that the input- and output-oriented Malmquist productivity indexes will differ from the Tornqvist productivity index by a scale factor. The difference will disappear when CRS holds.

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S. C. Ray

Bjurek [4] proposed a different measure of the Malmquist productivity index as    Q1y .Q0y T Qy = . MP I x 0 , x 1 ; y 0 , y 1 =  T Qx Q1x .Q0x

(143)

Relation Between Fisher and Malmquist Productivity Indexes Fare ¨ and Grosskopf [16] used the duality between the distance function and the dual cost function to establish equivalence between the Fisher and Malmquist productivity indexes under certain conditions. We have assumed that the production possibility set T and, hence, the input requirement set V(y) = (x : (x, y) ∈ T) is convex. By definition of a distance t function, x ∗t = D x xx t ,y t lies on the boundary of V(yt ). Therefore, by the supporting ( ) hyperplane theorem, there exists an input price vector w∗ 0 such that w ∗0 x ∗0 =

  w ∗0t x 0   = C w ∗0 , y 0 0 0 x ,y

Dx

(144)

In other words,   w ∗0 x 0  = Dx x 0, y 0 .  C w ∗0 , y 0

(145)

Similarly, there will be an input price vector wˆ ∗1 for which   w ∗1 x 1  = Dx x 1, y 1 .  C w ∗1 , y 1

(146)

At this point, they assume that there is no allocative inefficiency and, consequently, w∗ t = wt , (t = 0, 1). This amounts to assuming that   C w0 , y 0 =

  w 0t x 0 w 1t x 1   ; C w1 , y 1 =   ; Dx x 0, y 0 Dx x 1, y 1

  C w0 , y 0 =

  w 0t x 1 w 1t x 0   ; C w1 , y 0 =  ; x 0 1 x D x ,y D x0, y0

(147)

As argued by Balk [3], while the first two assumptions in (147) are relatively innocuous, the other two are quite problematic15 . While the input mix of the x1

15 Balk

points out a typo in of FG [16]. See the footnote 3 on page 681 of Balk [3].

20 Conceptualization and Measurement of Productivity Growth and Technical . . .

859

bundle may be allocatively efficient for the isoquant for the output bundle y1 , there is no reason why its efficient radial projection will be cost minimizing for the isoquant for the y0 input bundle as well. Following Fare ¨ and Grosskopf, one can express the input-oriented Malmquist productivity index of CCD in (95) as 

MP I x x , x ; y , y 0

1

0

1



  Dx x 0, y 0 = x  1 1 D x ,y .  =

w 0 x 0 w 1 x 0 . w 0 x 1 w 1 x 1

12   1 1   0 1  C w ,y C w ,y .    . C w1 , y 0 C w0 , y 0

1 2

. (148)

Finally, they assume profit maximization under CRS which implies (a) that for each output j, the price pj equals its marginal cost and also that the cost function is homogeneous of degree 1 in outputs. That is  ∂C  ∂C = pj and C (w, y) = yj = pj yj . ∂yj ∂yj j

j

In consequence, (148) would reduce to 

MP I x x , x ; y , y 0

1

0

1



 =

p1 y 1 p0 y 1 . p1 y 0 p0 y 0

1 2

F Qy = = πF . 1  F Qx w1 x 1 2

(149)

w0 x 1 . w0 x 0 w1 x 0

Balk [3] argues that although the conditions under which FG show equivalence between the Malmquist and Fisher productivity indexes are unlikely to hold except in very special cases, an approximate equivalence can be established for the general case.     0 1 1 0 Because C w 0 , y 1 ≤ D xwx 1x,y 1 and C w 1 , y 0 ≤ D xwx 0x,y 0 , combined with ( ) ( ) the first two equalities in (147), one can get16 

w0 x 0 C (w0 ,y 0 ) 

w0 x 1 C (w0 ,y 1 )

  Dx x 0, y 0 ≤ x  1 1 ≤ D x ,y

w1 x 0 C (w1 ,y 0 ) w1 x 1 C (w1 ,y 1 )

(150)

As argued by Balk [3] (see page 681),

16 By

    ∈ V y 1 . Hence, D x x11 ,y 1 x 1 ∈ V y 1 . But, ( )   0 1 x : x ∈ V(y1 ). Hence, C w 0 , y 1 ≤ D xwx 1x,y 1 . Similarly, for the other ( )

  definition, D x x 1 , y 1 = max θ :

C(w0 , y1 ) = min w0 inequality.



1 1 θx

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S. C. Ray

⎤1 ⎡  2  0 0 w0 x 0 w1 x 0 x ,y ⎢ C (w0 ,y 0 ) C (w1 ,y 0 ) ⎥    ⎣ 0 1 ⎦ w1 x 1 w x Dx x 1, y 1 1 1 0 1 C w ,y ) C (w ,y ) ( 1        w 0 x 0 w 1 x 0 2 C w 0 , y 1 C w 1 , y 1 .    = . .  w 0 x 1 w 1 x 1 C w0 , y 0 C w1 , y 0 Dx

 =

1 2

(151)

   12   C w0 ,y 1 C w1 ,y 1 . C (w0 ,y 0 ) C (w1 ,y 0 )

F Qx

Again, assuming profit maximization under CRS,  0 0 x ,y   x D x1, y1 Dx



p0 y 1 p1 y 1 . p0 y 0 p1 y 0

1 2

=

F Qx

F Qy = πF . F Qx

(152)

Nonparametric Decomposition of the Fisher Productivity Index As was recognized before, the Fisher productivity index is a descriptive rather than a normative measure. It is, nonetheless, possible to use the dual representation of an empirically constructed best practice technology to decompose the Fisher productivity index into a number of economically meaningful factors. We now consider an analogous decomposition of the Fisher productivity index introduced by Ray and Mukherjee [28]. As explained before, the Fisher productivity index is the geometric mean of a Laspeyres and a Paasche productivity index. Consider the Laspeyres index first. For simplicity, assume that the firm produces a single output from multiple inputs. Suppose that we are measuring the productivity index for firm k. The output quantities produced by the firm are yk0 in period 0 (the base period) and yk1 (in period 1) the current period. The observed input bundles are xk 0 and xk 1 in the two periods. The corresponding input price vectors are wk0 and wk1 . Then the Laspeyres productivity index becomes L=

yk1 yk0

’

.

wk0 xk1

’

wk0 xk0

At this point, recall the dual cost function for period t C t (w, y) = min w ’ x : (x, y) ∈ T t

(153)

20 Conceptualization and Measurement of Productivity Growth and Technical . . .

861

where Tt is the production possibility set in period t. Then the Laspeyres productivity index can be expressed as

L=

 yk1  C 1 wk0 ,yk1  yk0  C 0 wk0 ,yk0

  C 1 wk0 ,yk1

’

wk0 xk1   C 0 wk0 ,yk0

(154)

.

’

wk0 xk0

But, following the Farrell decomposition of the cost efficiency, we can write       C 1 wk0 , yk1 = T E 1 xk1 , yk1 .AE 1 xk1 , yk1 ; wk0 ’ wk0 xk1

(155)

    where T E 1 xk1 , yk1 is the technical efficiency of the input-output pair xk1 , yk1 in   period 1 and AE 1 xk1 , yk1 , wk0 is the allocative efficiency of the input mix of the bundle xk 1 at input price wk 0 in period 1. In an analogous manner,       C 0 wk0 , yk0 = T E 0 xk0 , yk0 .AE 0 xk0 , yk0 ; wk0 . ’ wk0 xk0

(156)

    C 0 wk0 ,yk0  T E 1 xk1 , yk1 .AE 1 xk1 , yk1 ; wk0 . yk0 L=     C 1 wk0 ,yk1  . T E 0 xk0 , yk0 .AE 0 xk0 , yk0 ; wk0 . yk1

(157)

Thus,

This can be further manipulated to get  L=

   1    1   0 0  ⎡ C 1 wk0 ,yk0 ⎤ 0 1 x , yk1 AE xk , yk1 ; wk C wk , yk0 y  ⎦.  k      ⎣ 1  k0 C wk0 ,yk1 T E 0 xk0 , yk0 AE 0 xk0 , yk0 ; wk0 C 1 wk0 , yk0

T E1

yk1

(158) Similar manipulations of the Paasche productivity index

P =

yk1 yk0

’

wk1 xk1

’

wk1 xk0

(159)

862

S. C. Ray

lead to the decomposition  P =

        0 1  ⎡ C 0 wk1 ,yk0 ⎤ T E 1 xk1 , yk1 AE 1 xk1 , yk1 ; wk1 C wk , yk1 y  ⎦.   ⎣ 0  k0     C wk1 ,yk1 C 1 wk1 , yk1 T E 0 xk0 , yk0 AE 0 xk0 , yk0 ; wk1 yk1

(160) Now define   T E 1 xk1 , yk1  ; T EI = T E 0 xk0 , yk0

       AE 1 x 1 , yk1 ; w 0 AE 1 x 1 , yk1 ; w 1 k k k k   .  ; AEI = AE 0 xk0 , yk0 ; wk0 AE 0 xk0 , yk0 ; wk1       0  C 0 w , yk0 C 0 w 1 , yk1 k k ; .  T CI =   0 C 1 wk , yk0 C 1 wk1 , yk1

(161)

(162)

(163)

and    C 1 w0 ,yk0  C 0 w1 ,yk0   k k  y yk0  .  ACI =  1  k0 . 0 1 0 C wk ,yk1 yk1

C wk ,yk1 yk1

(164)

Then, πF =

√

L.P = (T EI ).(AEI ).(T CI ).(ACI ).

(165)

In this factorization, the four terms on the right-hand side relate to (a) technical efficiency change, (b) allocative efficiency change, (c) technical change, and (c) change in scale economies, respectively. The first, TEI, obviously shows the increase (decrease) in technical efficiency in period 1 relative to what it was in period 0. The factor AEI is itself the geometric mean of two ratios, each of which shows the relative allocative efficiency of the input bundle from period 1 compared to the bundle from period 0. The allocative efficiencies are measured using the same technology and input prices for both bundles. TCI is a dual measure of technical change. It shows the autonomous shift of the cost function between the two periods evaluated alternatively at the input price and output quantity levels from the two periods. Finally, the factor ACI shows the relative (dual) scale efficiencies of the output levels from the two periods. When any one of the two ratios under the square root sign in this factor is greater than unity, it implies that along the dual cost function for the technology and input prices specified, the average cost is lower at the output level in the current period than at the output level from the base period.

20 Conceptualization and Measurement of Productivity Growth and Technical . . .

863

That is, the current period output is relatively more scale efficient. This contributes positively to productivity growth. A note of caution is in order here. As with all nonparametric models based on cross-period DEA, some components of this decomposition of the Fisher productivity index may be unavailable. This will be the case when the output level from one period is larger than the maximum output observed in the other period. In that case, the input requirement set relevant for the cross-period cost minimization problem would be empty.

Relation Between Malmquist Productivity Index and Luenberger Productivity Indicator We first consider the relationship between the Shephard distance function and the directional distance function under CRS.   For any input-output pair (x, y), D (x, y) = min λ : x, λ1 y ∈ T ⇒   − → F x, λ1 y = 0, while D (x, y; g x , g y ) = max β : F (x + βg x , y + βg y ) = 0. Hence, setting (gx , gy ) = (−x, y), we get F((1− β)x, (1 + β)y) = 0. Now, under 1−β 1 we get F x, 1+β CRS, F(tx, ty) = 0. Setting t = 1−β 1−β y = 0. Hence, λ = 1+β . In other words, D y (x, y) =

− → 1 − D (x, y; −x, y) . − → 1 + D (x, y; −x, y)

(166)

That is,     − → − → ln D y (x, y) = ln 1 − D (x, y; −x, y) − ln 1 + D (x, y; −x, y)

(166a)

From (96) above,        ln MP I y x 0 , x 1 ; y 0 , y 1 = ln D y x 1 , y 1 − ln D y x 0 , y 0      − →y  − →y  = ln 1 − D x 1 , y 1 ; −x 1 , y 1 − ln 1 − D x 0 , y 0 ; −x 0 , y 0      − →y  − →y  − ln 1 + D x 1 , y 1 ; −x 1 , y 1 − ln 1 + D x 0 , y 0 ; −x 0 , y 0 (167) Using a first-order Taylor’s series approximation, Boussemart et al. [5] show17 that

17 See

their proposition 2 in pp 399–400.

864

S. C. Ray

        ln MP I y x 0 , x 1 ; y 0 , y 1  −L1 x 1 , y 1 , x 0 , y 0 ; g

(168)

Data Envelopment Analysis and a Nonparametric Measurement of Productivity Change Unlike in a parametric approach like stochastic frontier analysis (SFA), in data envelopment analysis, one makes a minimal number of fairly general assumptions about the underlying production technology to construct an approximation to the production possibility set using observed values of inputs and outputs with any explicit specification of a functional form of the production, cost, or function.  One starts with the data set of input-output bundles D = x j , y j , j = 1, 2, . . . , N that  and assumes   all observed input-output bundles are feasible. That is x j , y j ∈ D ⇒ x j , y j ∈ T . Then under the assumptions of free disposability of input and outputs and convexity of the production possibility set, an empirical approximation of T based on the observed data points is

S=

⎧ ⎨ ⎩

(x, y) : x ≥

N 

λj x j ; y ≥

j =1

N 

λj y j ;

j =1

N 

λj = 1; λj ≥ 0; j = 1, 2, . . . , N

j =1

⎫ ⎬ ⎭

(169)

DEA Models for Measuring the Malmquist Productivity Index In order to quantify the different components of the Malmquist index, we need to compute various contemporaneous and cross-period efficiency measures. The contemporaneous technical efficiency/distance function measures are obtained solving for each period (t = 0, 1) the standard output-oriented DEA LP problems   1 τyt xt0 , yt0 = ∗ ϕ where ϕ ∗ = max ϕ  j λj yt ≥ ϕyt0 ; s.t. 

j j

λj xt ≤ xt0 ;

j

 j

λj = 1; λj ≥ 0; (j = 1, 2, . . . , N )

(170)

20 Conceptualization and Measurement of Productivity Growth and Technical . . .

Of course, for CRS the constraint

865

λj = 1 is dropped. For measurement of

j

cross-period efficiency   of the input-output bundle from period t against the period s technology τys xt0 , yt0 , one needs to solve the problem: max ϕ  j s.t. λj ys ≥ ϕyt0 ; 

j j

λj xs ≤ xt0 ;

(171)

j



λj = 1; λj ≥ 0; (j = 1, 2, . . . , N )

j

In solving the cross-period optimization problem in (171), one encounters infeasibility whenever any input from the bundle xt0 is smaller than the smallest j value of the corresponding input across all input bundles xs (j = 1, 2, .., N) . This is an inherent LP problem with the VRS assumption and is not a limitation of the RD decomposition in particular. By using the CRS frontier to evaluate crossperiod distance functions for measuring technical change, the FGNZ decomposition avoids the infeasibility problem. However, at the conceptual level, it lacks internal consistency. Another problem that applies to the Malmquist productivity index in general is that it fails the circularity test in the sense that a direct measure of productivity change between periods 0 and 2 needs not be the same as the product of the change between periods 0 and 1 and between 1 and 2. To avoid the infeasibility problem, one may use a “sequential technology” as the base. As mentioned above in section “Relation Between Fisher and Malmquist Productivity Indexes,” Pastor, Asmild, and Lovell [27] proposed a “biennial” Malmquist productivity index.

Biennial Malmquist Index18 The appropriate  t t  DEA model to estimate period t output-oriented technical efficiency T EB c xs , ys of firm s, with reference to a CRS biennial production possibility set, is

18 This

section is based on Deb and Ray [13].

866

S. C. Ray

ϕs∗ = Max ϕ nk  

subject to

λkj yjk ≥ ϕyst ;

k=t,t+1 j =1 nk  

λkj xjk

≤

(172)

xst ;

k=t,t+1 j =1

λkj ≥ 0;  t t  1  where nk is the number of observed firms in period k and T E B c xs , ys = ϕs∗ .  t t B Period t output-oriented technical efficiency T E v xs , ys of firm s, with reference to a biennial VRS frontier, can be estimated by the following DEA model: φs∗ = Max ϕ nk  

subject to

λkj yjk ≥ φyst ;

k=t,t+1 j =1 nk  

λkj xjk ≤ xst ;

(173)

k=t,t+1 j =1 nk  

λkj = 1;

k=t,t+1 j =1

λkj ≥ 0;  t t  1  where nk is the number of observed firms in period k and T E B v xs , ys = φs∗ . Figure 6 provides an illustration of the biennial production possibility frontier and measure of output-oriented technical efficiency with reference to it for a firm, producing a single output from a single input, observed in two time periods t and t+1 (point A and B, respectively). The VRS frontiers for period t and t + 1 are indicated by K0 L0 M0 -extension and K1 L1 M1 -extension, respectively. The rays through origin OP0 and OP1 represent the CRS frontiers for period t and period t + 1, respectively. The biennial VRS frontier is indicated by the broken line K1 L1 DFM0 -extension, and the biennial CRS frontier in this case coincides with that of period t + 1. Output-oriented technical efficiency of the firm with reference to CRS biennial frontier in period t is TEc B (xt , yt ) = (AXt /QXt ), and that for period t + 1 is TEc B (xt + 1 , yt + 1 ) = (BXt + 1 /RXt + 1 ). Similarly with reference to the VRS biennial frontier, TEv B (xt , yt ) = (AXt /DXt ) and TEv B (xt + 1 , yt + 1 ) = (BXt + 1 /FXt + 1 ) show the levels of technical efficiency for the firm in period t and t + 1, respectively. Therefore, the biennial Malmquist productivity index is

20 Conceptualization and Measurement of Productivity Growth and Technical . . .

867

Fig. 6 Biennial Malmquist index and its decomposition

Mc

B

  BX t+1 /RX t+1  t t t+1 t+1  . x ,y ;x ,y =  AX t /QXt

(174)

The decomposition of this Malmquist productivity index is Mc

B

  BX t+1 /EX t+1  t t t+1 t+1  x ,y ;x ,y =  AX t /CX t     BXt+1 /F X t+1 / BXt+1 /EX t+1     × AX t /DX t / AX t /CX t     BXt+1 /RX t+1 / BX t+1 /F Xt+1     × AXt /QX t / AX t /DXt

(175)

Here   BX t+1 /EX t+1   = T EC (technical efficiency change) AXt /CX t     BX t+1 /F Xt+1 / BXt+1 /EX t+1     = T C (technical change) AXt /DX t / AX t /CX t

(176a)

(176b)

868

S. C. Ray

and 

   BXt+1 /RX t+1 / BX t+1 /F Xt+1     = SEC (scale efficiency change) (176c) AXt /QX t / AXt /DX t

DEA Model for the Directional Distance Function For an arbitrary choice of (gx , gy ), the relevant VRS DEA problem will be max β s.t.

N

j =1

N

j =1 N

j =1

j

y

λj yr − βgr ≥ yr0 ; (r = 1, 2, . . . , m) ; j

λj xi + βgix ≤ xi0 ; (i = 1, 2, . . . , n)

(177)

λj = 1;

λj ≥ 0; (j = 1, 2, . . . , N) ; β unrestricted The flexibility of the DDF is apparent from the fact that it can be radial (setting gx = 0 or gy = 0), bi-radial (setting gx = x0 and gy = y0 ), or completely non-radial for arbitrary choice of (gx , gy ).

Conclusion This chapter offers a broad overview of how a measure of technical change can be extracted from alternative measures of total factor productivity change obtainable from production, cost, profit, or distance functions. In the nonparametric approach, the focus is on index numbers. Appropriate DEA models are formulated for measuring Malmquist or biennial Malmquist productivity indexes and the Luenberger productivity indicator. Some topics not considered in this chapter are measurement of technical change without the convexity assumption (through Free Disposal Hull analysis [35] or output and input biases of technical change). For a detailed discussion of index numbers and their properties, one should look into the relevant chapter in this volume. An excellent discussion of the Malmquist index can be found in Färe, Grosskopf, and Roos [19].

20 Conceptualization and Measurement of Productivity Growth and Technical . . .

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Cross-References  Index Numbers and Productivity Measurement  Modeling Technical Change: Theory and Practice

References 1. Aczel J (1990) Determining merged relative scores. J Math Anal Appl 150:20–40 2. Afsharian M, Ahn H (2014) The Luenberger indicator and directions of measurement: a bottoms-up approach with an empirical illustration to German savings banks. Int J Prod Res 52(20):6216–6233 3. Balk B (1993) Malmquist productivity indexes and fisher productivity indexes comment. Econ J 103:680–682 4. Bjurek H (1996) The Malmquist total factor productivity index. Scand J Econ 98:303–313 5. Boussemart J, Briec W, Kerstens K, Poutnieau J (2003) Luenberger and Malmquist productivity indices: theoretical comparisons and empirical illustration. Bull Econ Res 55(4):391–405 6. Caves DW, Christensen LR, Diewert WE (1982) The economic theory of index numbers and the measurement of input, output, and productivity. Econometrica 50(6):1399–1414 7. Chambers RG (1996) A new look at exact input, output, productivity, and technical change measurement. Working papers 197840, University of Maryland, Department of Agricultural and Resource Economics 8. Chambers RG (2002) Exact nonradial input, output, and productivity measurement. Economic Theory 20:751–765 9. Chambers RG, Pope RD (1996) Aggregate productivity measures. Am J Agric Econ 78(5):1360–1365 10. Chambers RG, Chung Y, Färe R (1996) Benefit and distance functions. J Econ Theory 70:407– 419 11. Chambers RG, Färe R, Grosskopf S (1996) Productivity growth in APEC countries. Pac Econ Rev 1:181–190 12. Chambers RG, Chung Y, Färe R (1998) Profit, directional functions, and Nerlovian efficiency. J Optim Theory Appl 98:351–364 13. Deb AK, Ray SC (2014) Total factor productivity growth in Indian manufacturing: a biennial Malmquist analysis. Indian Econ Rev 49(1):1–25 14. Denney M, Fuss M, Waverman L (1981) The measurement and interpretation of total factor productivity in regulated industries with an application to Canadian telecommunications. In: Cowing TG, Stevenson RE (eds) Productivity measurement in regulated industries. Academic, New York, pp 179–218 15. Diewert WE (1976) Exact and superlative index numbers. J Econ 4:115 16. Fare ¨ R, Grosskopf S (1992) Malmquist productivity indexes and fisher ideal indexes: comment. Econ J 102:158–160 17. Fare ¨ R, Grosskopf S, Lindgren B, Roos P (1994) Productivity developments in Swedish hospitals: a Malmquist output index approach. In: Charnes A, Cooper WW, Lewin AY, Seiford LM (eds) Data envelopment analysis: theory, methodology and applications. Kluwer Academic, Boston. (Originally presented at a Conference on new uses of DEA in management and public policy, University of Texas, Austin, September 27–29, 1989) 18. Fare ¨ R, Grosskopf S, Norris M, Zhang Z (1994) Productivity growth, technical progress, and efficiency change in industrialized countries. Am Econ Rev 84(1):66–83 19. Fare ¨ R, Grosskopf S, Roos P (1994) Malmquist productivity indexes: a survey of theory and practice. In: Fare ¨ R, Grosskopf S, Russell R (eds) Index numbers: essays in Honor of Sten Malmquist. Kluwer Academic, Boston, 1998, pp 127–190

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20. Fare R, Grosskopf S, and Roos P (1998) Malmquist productivity indexes: a survey Of theory and practice; in Fare R, Grosskopf S, and Russell R (eds) Index Numbers: Essays in Honor of Sten Malmquist. Boston: Kluwer Academic Press, 127–190 21. Farrell M.J (1957) The Measurement of Technical Efficiency. Journal of the Royal Statistical Society Series A, General, 120, Part 3, 253–81 22. Jorgenson DW, Griliches Z (1967) Explanation of productivity change. Rev Econ Stud 34(3):249–283 23. Lovell CAK (2003) The decomposition of productivity indexes. J Prod Anal 20:437–458 24. Luenberger DG (1992) Benefit functions and duality. J Math Econ 21:461–481 25. Moorsteen RH (1961) On measuring productive potential and relative efficiency. Q J Econ 75(3):451–467 26. O’Donnell C (2012) Nonparametric estimates of components of productivity and profitability change in U.S. agriculture. Am J Agric Econ 94(4):873–890 27. Pastor JT, Asmild M, Lovell C (2011) The biennial Malmquist productivity change index. Socio Econ Plan Sci 45:10–15 28. Ray SC, Mukherjee K (1996) Decomposition of the fisher ideal index of productivity: a nonparametric dual analysis of U. S. Airlines data. Econ J 106:1659–1678 29. Ray SC, Segerson K (1991) A profit function approach to measuring the rate of technical progress: an application to U.S. manufacturing. J Prod Anal 2:39–52 30. Ray SC, and E. Desli (1997) Productivity Growth, Technical Progress, and Efficiency Change in Industrialized Countries: Comment“, American Economic Review, Vol 87, No. 5, Dec. 1997 pp 1033–1039. 31. Schultz T (1956) Reflections on agricultural production, output, and supply. J Farm Econ 38(3):748–762 32. Shephard RW (1953) Cost and Production functions princeton: princeton university press 33. Shephard RW (1970) Theory of Cost and Production Functions (Princeton) 34. Solow RM (1957) Technical change and the aggregate production function. Rev Econ Stat 39(3):312–320 35. Tulkens H, Eeckaut PV (1995) Non-parametric efficiency, progress and regress measures for panel data: methodological aspects. Eur J Oper Res 80(3):474–499

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Contents Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Modeling TC: The Single Output Case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Production Function Approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The Cost Function Approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Profit Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Price Function Approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Specification and Estimation Issues . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Production Function Approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Estimation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Production Function Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Cost Function Approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Profit Function Approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Multiple Outputs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The Primal Approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The Dual Approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . TC Measures Induced by Management/Exogenous Factors and Time . . . . . . . . . . . . . . . . . . . Management Variables as Technology Shifter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Specification of the IDF with Multiple Outputs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . TC in Cross-Sectional Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Technical Change from Other Indirect Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Revenue Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Indirect Production Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Technical Change and TFP Change . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . TC and Profit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . TC as a Component of TFP Change . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Formulating and Estimating TC Without Estimating Profit/Cost Function . . . . . . . . . . . . .

872 874 874 877 878 879 879 879 883 883 886 894 894 894 897 898 899 901 903 904 904 905 906 906 908 909

S. C. Kumbhakar () Department of Economics, State University of New York at Binghamton, Binghamton, NY, USA Inland Norway University of Applied Sciences, Lillehammer, Norway e-mail: [email protected] © Springer Nature Singapore Pte Ltd. 2022 S. C. Ray et al. (eds.), Handbook of Production Economics, https://doi.org/10.1007/978-981-10-3455-8_27

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Models with Technical Inefficiency . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . TC and Technical Inefficiency . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Estimation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . TC in Production Models with Good and Bad Outputs . . . . . . . . . . . . . . . . . . . . . . . . . . . . Productivity and Profitability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . TC and Factor Productivity with One Variable Input . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Concluding Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

911 911 913 915 918 919 920 920

Abstract

This chapter focuses on models dealing with technical change (TC). We discuss TC under both single output and multiple outputs cases. For the single output case, we specify the technology in terms of a (i) production (input or output distance) function, (ii) cost function, and (iii) profit function. Each of these cases is discussed under disembodied approaches, which are not associated with a specific input and embodied (factor-augmenting (FA)) approaches, which are specific to input(s). In the disembodied approach, we mostly focus on time driven TC which includes models with the time trend and general index approaches. First, we discuss each of the cases in a standard neoclassical setup without technical inefficiency followed by a brief discussion of some of the models with technical inefficiency. We also discuss TC and total factor productivity (TFP) in models with good and bad outputs. Finally, we link TFP change with profitability change. Keywords

Technical change · Total factor productivity · Panel data · Production function · Cost function · Profit function · Distance function · Technical inefficiency

JEL Classification Numbers

D24

Introduction “Productivity isn’t everything, but in the long run it is almost everything. A country’s ability to improve its standard of living over time depends almost entirely on its ability to raise its output per worker.” (Paul Krugman, The Age of Diminishing Expectations [1]). Similarly, Robert Solow argued in his famous paper, A Contribution to the Theory of Economic Growth [2], that an economy cannot grow in the long run without technological progress. More specifically, growth in per capita real income for an economy in the long run will be determined by technological progress.

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Both Krugman and Solow referred to macro aspects of economic growth. Solow, in particular, focused on the role of technical progress in steady state long run growth. However, in this chapter, our focus will be on micro perspective and therefore we will not be discussing economic well beings or aggregate economic growth. Instead, our focus will be on individual firms (producers), and therefore, we will concentrate on the impact of technical change (TC) on a firm’s cost, profit, revenue, returns to outlay, etc. While pursuing this goal, we will be discussing various micro models that are used to estimate TC. Similar to the macro models, we define TC as a shift in firm-level technology, which can occur due to many factors. However, the main idea is that more output is produced without increasing inputs if there is technical progress (positive TC), that is, in a production function formulation, the production function shifts upward. Alternatively, less input is required (i.e., cost is lower given input prices) to produce a given level of output. It can also be examined from many other forms, where the technology can be represented by profit, revenue, and distance functions. This shift in firm-level technology can take place in different forms and due to many different factors. As a result, TC can be examined in many ways, although the production function formulation is used the most. TC can also be viewed as endogenous or exogenous. In endogenous TC, the factors causing the shift are decision variables to the firm. For example, a firm can invest in state-of-the-art computer (IT capital) to speed up TC. If firms do not decide on such variables, TC can be viewed as exogenous. Similarly, TC can be embodied in some specific inputs or disembodied. If TC is associated with capital of recent vintage, it is said to be embodied in capital. On the other hand, TC may not be associated with any particular input in which case it is disembodied. Finally, TC can be neutral (not directed towards any input in particular) or non-neutral. If TC is non-neutral, it is biased. Skill biased TC is an example of non-neutral TC. In the Solow model, unitary returns to scale are assumed. In the micro models that we will be focusing on, returns to scale (RTS) will be unconstrained – either constant (not unitary) or observation specific, depending on the functional form used. Nonconstant RTS will be assumed when we focus on flexible functional forms. Since the estimate of TC is likely to depend on whether one assumes RTS to be unitary or not, which is a testable hypothesis, our models will be unconstrained in relation to RTS. Thus, although our main focus is TC, we will be frequently discussing RTS. Another reason for emphasizing RTS is that TC is only one component of total factor productivity (TFP) change. Although our discussion of TFP change will be limited, because it is discussed in detail in another chapter of this volume (Ray [68]), we cannot ignore RTS entirely. In this chapter, we focus on time driven TC (a shift taking place over time), as well as a shift due to some exogenous (to the firm) factors in parametric models. That is, we review models in which TC is exogenous and non-neutral. Perhaps we will be abusing the terminology “embodied TC” by labeling the factor-augmenting (FA) TC models as embodied TC models, especially when TC is embodied in the inputs of recent vintage proxied by time in panel models. For the cross-sectional models, embodiments are associated with exogenous factors such as management.

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By the same token, we can use the term disembodied TC to mean a shift that is not attributed to a particular input. These notions can also be linked to neutral and non-neutral TC. As a result, different classifications are not completely separate. Although one can discuss TC using time series data for a firm, region, state, country, etc., our focus will mostly be on the use of panel (time-series of cross section) data so that we can concentrate on time as the shifter in the absence of exogenous factors. We will also be discussing TC in cross-sectional models when TC is driven by exogenous factors. Our discussion of TC in this chapter will cover single as well as multiple outputs. For the single output case, we will specify the technology mostly but not entirely in terms of a (I) production (input or output distance) function, (II) cost function, and (III) profit function. The definition of TC and its interpretation will differ depending on whether the technology is specified in terms of a production, cost, profit, or revenue function. Finally, each of these cases will be discussed under disembodied and embodied (FA) approaches. In the disembodied approach, TC is not associated with a specific input, which is the case in the FA approach. We will mostly focus on time driven TC in the disembodied approach, including the time trend and general index approaches. First, we discuss each of the cases without technical inefficiency followed by a brief discussion of some models with technical inefficiency. We also discuss TC and TFP in models with good and bad outputs. Finally, although our focus in this chapter is not on TFP, we link TFP change with profitability change for an easy interpretation of TFP change. We use an input or output distance function to represent multiple outputs in a primal framework. Since the dual cost/profit functions with multiple outputs are straightforward generalizations of the single output cost/profit functions, our discussion on the multiple output cost/profit functions with and without inefficiency is not extensive.

Modeling TC: The Single Output Case In this section, we discuss modeling (disembodied) exogenous TC via (i) the production function Approach, (ii) the cost function approach, and (iii) the profit function approach.

Production Function Approach First, we consider models where TC is time driven.

The Time Trend (Continuous Time) Model In the time trend (TT) formulation of TC, the production function (see, for example, Chambers [3] for the properties) is specified as y = f (x, t)

(1)

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where y is output, x is a vector of inputs and t is the time-trend variable. Mathematically, t can be viewed as a factor of production like x. Although x is mostly treated as the input vector, it can also include environmental variables (facilitating inputs). In measuring TC, we consider the shift in the production function over time, ceteris paribus. TC, in this formulation, is defined as TCp (x, t) = ∂ln y/∂t = ∂ln f (x, t) /∂t.

(2)

The subscript p on TC in (2) is used to indicate that TC is based on a production function formulation. A positive (negative) value of TC indicates technical progress (regress). It shows the rate at which output changes while holding input levels unchanged. In other words, output increases over time without increasing the input quantities when technical progress takes place. This is shown in the figure below where output increases from yA to yB moving forward in time from t0 to t1 (t1 > t0 ), holding x unchanged at xA . If x is labor, the figure shows that output per worker (average product of labor) increases over time, ceteris paribus. Because of this, the average product of different inputs is likely to change depending on the functional form of f (.).

y = f ( x, t1 )

y

y = f ( x, t0 )

yB yA

0

xA

x

TC is said to be Hicks neutral if the production function y = f (x, t) can be written as y = μ(t)f (x) where μ(t) is the shift function. Note that here TC (the shift function) is not embodied in any particular input, and this is why it is labeled as neutral. Thus, productivity of all the inputs changes by the same proportion μ(t) under Hicks neutral TC. Consequently, the rate of technical substitution between any two inputs (the ratio of marginal products), is constant under Hicks neutral TC.

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The General Index (Discrete Time) Formulation In the general index (GI) approach, instead of treating t as a continuous variable, an index A(t) is defined using the time dummies. The production function with this index is written as y = f (x, A(t)) where A(t) =



(3)

λt DT t , DTt being the time dummies, and λt are the associated

t

coefficients (parameters). TC is then defined as   TCp (x, A(t)) =  ln f x, A(t) = ln f (x, A(t)) − ln f (x, A (t − 1))

(4)

Here, the partial derivative is replaced by finite changes, holding x unchanged. Conceptually, it is no different from the TT model. The only difference is in the algebraic formulation and estimation, which will come later.

The Factor-Augmenting (Embodied in Time) Approach As noted before, we view the factor-augmenting approach as a form of embodied TC for which the production function is written as   y = f x∗

(5)

where x∗ = x or xj∗ = λj (t) xj , j = 1, . . . , J. The efficiency (factor augmenting) factors λj > 0 are functions of time. Thus, TC is embodied in all the inputs, although one can set λj = 1 for some j so that TC is augmenting in some specific inputs. For example, TC might be embodied only in new capital. In a way, the FA approach is more intuitive because it shows that the shift in the technology is taking place because efficiency of one or more factor inputs is changing over time. For example, if labor becomes more productive because of learning with experience, it is natural to think that technical progress is due to the fact that labor is becoming more productive over time. TC is Harrod neutral if the production function y = f (K, L, t) = f (K, λL (t)L), λL > 1. That is, under a Harrod neutral TC, only the efficiency of labor increases. Such an interpretation is not always possible with the TT representation. Meaning that the TT and FA formulations are not identical for all types of production function. TC in the FA model is obtained from        TCp x ∗ = ∂ln y/∂t = ∂ln y/∂ln xj∗ ∂ln xj∗ /∂t =

 j

j

T Cj =

 j

Ej λ˙ j

where Ej = ∂ln y/∂xj∗ and λ˙ j = ∂ln λj /∂t.

(6)

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Thus, the overall TC can be decomposed into input-specific components. However, the input-specific components cannot be identified for all types of production functions (Cobb-Douglas (CD), for example).

The Cost Function Approach Duality in production theory shows that all the features of the production function can be captured by a dual cost function under cost minimization behavior. So instead of using a production function to represent the technology, one can also use a cost function which is a function of input prices and output. To accommodate facilitating inputs, we can add them separately and treat them like output variables which are exogenous. Similar to the production function, to model TC, we append the shifter as an argument in the cost function. In the TT approach, the shifter is t and the neoclassical cost function (see, for example, Chambers [3], Fuss and McFadden [4] for its properties) is C = C (w, y, t)

(7)

where w is the input price vector. If the production function shifts upward (technical progress), the cost function will shift downwards, ceteris paribus. Thus, it is natural to define TCc as TCc (w, y, t) = −∂ln C/∂t ≡ −C˙ t

(8)

where the subscript c indicates that TC is based on a cost function formulation. A positive (negative) value of TC indicates technical progress (regress). TC in a cost function approach is the rate of cost diminution and it is generally not the same as TC from the production function unless the cost elasticity of output is unitary (i.e., ∂ln C/∂ln y = 1). This can be shown as follows: From the cost minimization problem:    = wx + θ y − f (x, t) where θ is the Lagrange multiplier, which has an optimum value that is θ = ∂C/∂y. Since at the optimum, the value of the Lagrangian  is the same as the optimum cost, we can get ∂C(w, y, t)/∂t = −θf(x, t)/∂t = −∂C/∂y × ∂f(x, t)/∂t ⇒ ∂ln f(x, t)/∂t = TC(x, t) = −T C c (w, y, t)/ECY , where ECY = ∂ln C/∂ln y is the cost elasticity of output.

(9)

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In the GI approach, the cost function is expressed as   C = C w ∗ , y, A (t)

(10)

In the FA approach, the neoclassical cost function is   C = C w ∗ , y, t

(11)

where w∗ = Bw and Bj = 1/λj .       ∂ln C/∂ln wj∗ ∂ln wj∗ /∂t TCc w ∗ , y, t = −∂ln C/∂t = − =

j

 j

Sj∗ λ˙ j , Sj∗ = ∂ln C/∂ln wj∗ .

(12)

If there is technical progress, λ˙ j < 0 and therefore, TC will be positive. Intuitively, technical progress in a FA model means an increase in input productivity by  λj , which is equivalent to a decrease in input prices by λj and a decrease in cost by j Sj∗ λ˙ j .

Profit Function Following neoclassical duality theory, the technology for a profit maximizing firm can be expressed as π = π (p, w, t)

(13)

where p is the output price and w is the vector of input prices. See Chambers [3], Fuss and McFadden [4] for properties of profit functions. Here TC is defined as TCπ (p, w, t) = ∂ln π/∂t ≡ π˙ t

(14)

Again, the subscript π on TC indicates that it is based on a profit function. Note that to use the profit function approach for estimating TC, we need profit to be positive, especially when one uses the popularly used functions such as the CD or translog. Similar to the production function approach, technical progress (regress) will mean a positive (negative) value of π˙ t . The numerical value of TC in a profit function is different from those in a production and cost function. The relationship can easily be established. The Lagrangian for the profit maximization problem is  = py − wx + μ(y − f (x, t) where μ is the Lagrange multiplier. Using the Envelope theorem, which gives μ = − p, and at the optimum π (p, w, t) = py − wx, we get ∂π (w, p, t)/∂t = − μ ∂f (x, t)/∂t ⇒ π ∂ln π (w, p, t)/∂t = py ∂ln f (x, t)/∂t. Thus, TCp (p, w, t) = (R/π ) TC(x, t) = Eπp TC(x, t)

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where Eπ p = ∂ln π (w, p, t)/∂ln p = py/π using Hotelling’s lemma. In the GI model, the profit function can be expressed as π = π (p, w, A(t)) where A(t) is an index of time dummies. The TC can then be defined as   TCπ p, w, A(t) = ln π (w, p, A(t)) = ln π (w, p, A(t)) − ln π (w, p, A (t−1)) . (15) For this, profit has to be positive. Since in reality profit can be negative for many firms, the profit function is not very popular. Furthermore, it requires information on both output and input prices which are often difficult to get. Further, without enough variations in prices, the profit function cannot be reliably estimated. ∗ = In the FA approach, we write the profit function as π = π (p∗ , w) where pm pm Bm and TC is       ∂ln π/∂ln wj∗ ∂ln wj∗ /∂t ∂ln π/∂t = TC p, w ∗ = j

= j P S j λ˙ j , P S j = wj xj /π

(16)

Price Function Approach Jorgenson [5] suggested the use of a Price Function Approach to model TC. The price function is based on the long-run equilibrium condition of zero profit (revenue equals cost under constant returns to scale). Jorgenson [6] shows that all the features of the underlying technology can be obtained from the price function. Thus, this approach is useful for estimating aggregate TC along with other features of the aggregate technology. In this approach, output price is specified as a function of input prices and the time trend. TC is then defined as negative of output price change between two time periods, holding everything else unchanged. See Jin and Jorgenson [7] for details. In many micro applications, the use of constant returns to scale assumption is not a norm, so this approach is not very useful for modeling TC for individual producers who may be operating under increasing or decreasing RTS.

Specification and Estimation Issues Production Function Approach Time Driven Disembodied TC Here the production function is y = f (x, t). We start our discussion with a CobbDouglas (CD) formulation. Although this is much simpler and more restricted than a flexible functional form, such as the translog, the parameters in the CD formulation have economic meaning and are therefore easy to interpret. Thus, it might be helpful

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in understanding the issues. Because of this, the CD formulation is still used in many applications. We skip the firm and time subscripts to avoid notational clutter and use them when it is absolutely necessary. In specifying the technology, assumptions are often  made on RTS. For the production function in (1), it is defined as RTS = j ∂lny/∂lnxj which is the percentage change in output for a change in all the inputs by a constant percentage. Thus, the question is, if the scale of operation is increased by ρ% by what percentage will output increase? If all the inputs change by ρ > 1% but output changes by ρ h , then RTS is said to be increasing (decreasing) if h > 1 ( 1. Similarly, it is decreasing (DRS) if RTS < 1. Finally, it is constant RTS (CRTS), when RTS = 1. RTS for a firm may not be increasing, decreasing, or constant for all levels of outputs, unless the underlying is restrictive. For a more flexible function, RTS is allowed to change with output level which is more intuitive.

The Cobb-Douglas Case: Time Trend Formulation ln y = α0 +

 j

αj ln xj + αt t

TC = A˙ t = αt and RT S =

 j

(17) αj

Both are constants for all t and for all firms. Furthermore, TC is neutral because it is not related to any input. Similarly, RTS is also constant but not unitary. TC can be biased towards some inputs – non-neutral technical change affects productivity (efficiency) of different inputs differently. More specifically, it is said to be biased towards input j if its share in total cost increases over time. Alternatively, TC is said to be input j using (saving) if Bj (t) = ∂Sj /∂t > 0 ( 0 then efficiency of input j increases over time at the rate bj . TC = ∂lny/∂t =

  j

αj +

 r

αj r (ln xr + ln λr )

      Sj bj ∂ln λj /∂t = j

(32) Biasj = ∂Sj /∂t =

 r

αj r (∂ln λr /∂t) =

 r

αj r br

(33)

which is a constant but differs across inputs.        ∂ln y/∂ln xj∗ ∂ln xj∗ /∂ln xj ∂ln y/∂ln xj = j j     αj + = αj r (ln xr + ln λr ) = Sj

RTS =

j

r

(34)

j

which is observation specific. Thus, RTS changes with input levels and is nonconstant unless restrictions are imposed on the parameters.

Estimation We now introduce the firm and time subscript, i and t.

Production Function Models The TT Model The TT model (CD) is ln yit = α0 +

 j

αj ln xj it + αt t + vit

(35)

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S. C. Kumbhakar

where v is the error term. Assuming the input variables are exogenous, which is usually assumed in the literature, the OLS can be used to estimate the parameters. If the input variables are not exogenous/predetermined, we need to use instrumental variables to take care of the endogeneity. Once the parameters are estimated, we can get an estimate of TC from the coefficient of the time trend variable (t), that is, TC = αt RTS =

 j

(36) αj

(37)

Both TC and RTS are constant.

The GI Model Here the model is ln yit = α0 +

 j

αj ln xj it + A(t) + vit

(38)

which can be estimated using OLS with time-dummies. TC is then estimated from TCt = A(t) − A (t − 1) RTS =

 j

αj

(39) (40)

Here, TC is time varying but exactly the same for all firms. On the other hand, RTS is the same for all firms in every year.

Extensions of TT and GI Models Here the model is  ln yit = α0 + αj ln xj it + θ (i) A(t) + vit

(41)

j

Since both θ (i) and A(t) are firm and time-specific parameters, the model can be estimated using nonlinear least squares. However, we need to set θ (i) = 1 for one i for identification. Once the parameters are estimated, TC can be obtained from TCit = θ (i) [A(t) − A (t − 1)]

(42)

 which is both firm and time-specific. However, RTS = j α j is the same for all observations. The main problem with the CD models in all the above cases is that RTS does not vary across firms and over time. This is very restrictive. One can easily avoid this problem by using a flexible functional form such as the translog.

21 Modeling Technical Change: Theory and Practice

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The TT Model (Translog) ln yit = α0 + +

 j

 j

αj ln xj it + αt t +

1   αj k ln xj it ln xkit + αtt t 2 j k 2 (43)

αj t ln xj it t + vit

The TT model, like the CD model, can be estimated using OLS, assuming that the inputs are not correlated with the noise term. Once the parameters are estimated, TC, RTS, Bias, etc., can be computed using the formulas discussed earlier (Eqs. (24)–(26)).

The GI Model The translog GI model adds a noise (error) term. ln yit = α0 + +



 j

αj ln xj it + A(t) +

j

1   αj k ln xj it ln xkit j k 2

(43a)

αj t ln xj it A(t) + vit

One of the parameters in A(t) = at DTt is to be normalized to 1 for identification. Since A(t) appears interactively with lnxj , the model is nonlinear and can be estimated using nonlinear least squares (NLS). Once the parameters are estimated, TC, RTS, Bias, etc., can be computed using the formulas discussed earlier (Eqs. (28)–(30)).

Factor Augmenting TT Model ln yit = α0 + +

 j

  αj ln xj it + ln λj

  1   αj k ln xj it + ln λj t (ln xkit + ln λkt ) + vit j k 2

(44)

If we assume that ln λjt = bj t, that is, factor augmentation is input specific and time varying, the above model becomes nonlinear and can be estimated using NLS. Once the parameters are estimated, TC, RTS, Bias, etc., can be computed using the formulas discussed earlier (Eqs. (32)–(34)).

Factor Augmenting GI Model The model is similar to the TT model above, except that the factor augmentation is modeled differently, that is,

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S. C. Kumbhakar

ln yit = α0 +

 j

  αj ln xj it + ln λj t

  1   + αj k ln xj it + ln λj t (ln xkit + ln λkt ) + vit j k 2

(45)

where ln λjt = bj (t) = bj (DTt ). Since DTt are time dummies and bj (t) are both input specific and time varying parameters, the above model is nonlinear and NLS can be used to estimate the parameters. Once the parameters are estimated, TC, RTS, Bias, etc., can be computed from TCit =

 j

+ +

  αj bj (t) − bj (t − 1)

  j

k

j

k

 

  αj k ln xj it + ln λj t {bk (t) − bk (t − 1)}

(46)

  αj k {ln xkit + ln λkt } bj (t) − bj (t − 1)

Sj t = αj +

 k

αj k {ln xkit + ln λkt }

(47)

αj k [bk (t) − bk (t − 1)]

(48)

and the bias is Bj t = Sj t =

 k

which is input specific and varies over time. Finally, RTS is RTS =

    αj + αj k {ln xkit + ln λkt } j

(49)

k

Cost Function Approach Time Trend Formulation in Generalized Leontief Cost Function One can use many different parametric forms of the cost function, although the translog is the most popular. The generalized Leontief (GL) function, which is also flexible, is often used. The generalized Leontief (GL) cost function was first introduced by Diewert [9]. RTS in this chapter was assumed to be unitary. The cost function was specified as shown below (the time trend is added to allow technical change). ⎡



C (w, y, t) = y ⎣

k

k

bj k wj 1/2 wk 1/2 +



⎤ bj t wj t ⎦

(50a)

j

 with bjk = bkj , j, k = 1, . . . , J. TC in this formulation is TCc = y j bjt wj which is quite restrictive compared to the more flexible specification introduced later. The

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GL cost function is further generalized in Diewert and Wales [10] in which the generalized GL cost function (double generalized) is C (w, y, t) =

 k

bj k wj 1/2 wk 1/2 y +



k

bj wj +

j



bj t wj ty

j

⎛ ⎞ ⎛ ⎞ ⎛ ⎞    + bt ⎝ αj wj ⎠ t + byy ⎝ βj wj ⎠ y 2 + btt ⎝ γj wj ⎠ t 2 y j

j

j

(50b) with bjk = bkj , j, k = 1, . . . , J. Note that if all bjk for j = k are constrained to be non-negative, only then is the cost function i globally concave. See Diewert and Wales [10] for identifying restrictions on the parameters. Since this restriction is too restrictive, Diewert and Wales [10] did not recommend its use in empirical applications. There are other forms of the GL cost function which are not fully flexible (constant RTS imposed) that are used in the literature. C (w, y, t) = y

 j

bj k wj 1/2 wk 1/2 +



k

bj t wj t +



j

γj t wj t 2

(51)

j

Another form suggested by Diewert and Wales [10] is the Normalized quadratic cost function

C (w, y, t) = y

⎧ ⎨ ⎩

βj wj + g(w) +



j

βj t wj t +

j

 j

γj t wj t 2

⎫ ⎬ ⎭

(52)

where g(w) = w Bw/2θ  w, B = [β jk ] is a J × J symmetric matrix and θ = [θ 1 , . . . , θ J ] > 0 is a vector of non-negative predetermined constants, not all zero. In order to identify all the parameters in B, it is assumed that Bw∗ = 0 for some chosen w∗ . TC for the above cost functions are [∂C(w, y, t)/∂t]/C(w, y, t).

Time Trend and General Index Models (Translog) The translog cost function for the TT model can be written as ln C = α0 +

 j

αj ln wj + αy ln y + αt t

1   αj k ln wj ln wk + αyy ln y ln y + αtt t 2 j k 2   + αjy ln wj ln y + αj t ln wj t + αyt ln y t + v.

+

j

j

(53)

888

S. C. Kumbhakar

The cost function above is assumed  to satisfylinear homogeneity  (in w) and symmetry restrictions. These are: α = 1, α = 0 ∀ k, j j j jk j α jy = 0,  j α jt = 0 and α jk = α kj . One way of generalizing this TT model is to replace the trend variable t by A(t) where A(t) (t = 1, . . . , T) has one parameter for each time period. This model is labeled as the general index (GI) model by Baltagi and Griffin [11]. Thus, assuming a translog form, the GI model can be written as ln C = α0 +

 j

αj ln wj + αy ln y + A(t)

1   αj k ln wj ln wk + αyy ln y ln y j k 2   + αjy ln wj ln y + αj t ln wj A(t) + αyt ln y A(t) + v +

j

(54)

j

The above cost function is assumed to satisfy linear homogeneity (in w) and symmetry restrictions. These restrictions are the same as before. Technical change in the TT (TC_TT0) and GI (TC_GI0) models can be expressed as   TC_TT0 = − αt + αtt t + αj t ln wj + αyt ln y , (55) j

TC_GI0 = − {A(t) − A (t − 1)}

 j

αj t ln wj + αyt ln y

.

(56)

Both measures of TC are firm-specific and time-varying. It is to be noted that there are restrictions built-in on the nature of TC in the TT model. First, the pure (neutral) component, −{α t + α tt t}, either increases or decreases linearly. Second, with unbalanced panel data, it is not clear whether the trend variable for a firm entering in period τ (1 < τ < T) should start from τ or be rescaled to start from 1. Both of these problems are avoided in the GI model by estimating one parameter for each time period in A(t). In translog models, TC can be biased towards a particular input(s). For input j (j = 1, . . . , J) bias in TC can be measured from IBj = ∂Sj /∂t, where Sj = ∂ln C/∂ln wj = wj xj /C is the cost share of input j [12]. A positive (negative) value of IBj indicates that TC is relatively jth input using (saving). A zero value of IBj implies that TC is neutral (not biased towards any particular input). Thus, TC is skilled bias if the share of skilled workers increases over time. In the TT model, IBj = α jt which is a constant for all time periods, and its sign is determined by the sign of α jt . Thus, although the cost function representing the TT model is flexible, measures of technological biases (inputs and scale) derived from it are neither firm-specific nor time-varying. Stevenson [12] avoided this problem in a TT model by appending some third order terms in the translog cost function. These additional terms allow technological biases to change among firms and over time. However, in the GI model, IBj = α jt [A(t) − A(t − 1)] varies over time. The sign of

21 Modeling Technical Change: Theory and Practice

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IBj is the same (opposite) as that of A(t) − A(t − 1), if α jt is positive (negative). RTS can be obtained from  αjy ln wj + αyt t (57) RTS_TT0−1 = αy + αyy ln y + j

RTS_GI0−1 = αy + αyy ln y +

 j

αjy ln wj + αyt A (t)

(58)

in the TT and GI models, respectively.

Generalizations of Time Trend and General Index Models In both the TT and GI models, TC and RTS are firm-specific, because the w and y variables vary among firms. However, input and scale biases in TC are the same for all firms in both models. This undesirable feature can be eliminated by extending both the TT and GI models in which the α yt , αj t parameters are firm-specific, that i , α i , i = 1, . . . , N. See Kumbhakar, Nakamura, and is, αyt , αj t are replaced by αyt jt Heshmati [13] for details on this model. In doing so, no restrictions are imposed on technological bias measures. The cost of this generalization is that JN additional parameters are to be estimated. Thus, TC in the TT1 and GI1 models can be expressed as   TT_TT1 = − αt + αtt t + αji t ln wj + αy t i ln y ,

(59)

  i αji t ln wj + αyt ln y . TT_GI1 = − {A(t) − A (t − 1)} 1 +

(60)

j

j

It can be seen from (59) and (60) that firms producing the same output level and i , facing the same input prices will experience different rates of TC as long as αyt αji t parameters are not the same for all firms. For the same reason, input and scale biases in TC are firm-specific although the formulas for calculating these biases are the same as those in the TT0 and GI0 models. Another advantage of the above generalization is that the TT0 and GI0 models become special cases of the TT1 and GI1 models – specification of which can be tested using likelihood ratio (LR) tests. Formulas for calculating RTS, for the TT1 and GI1 models, are the same as those i , αi . in (55)–(58), except that αyt , αj t are now firm-specific and are replaced by αyt jt Thus, estimates of RTS, for firms producing the same output level and facing the same input prices, will be different. Consequently, the scale components will also be different for these firms. The same is true for input and scale biases, which are firm-specific if the αyt , αj t parameters are firm-specific. In addition to these two extensions, several other extensions of the TT and GI models are considered in the Kumbhakar et al. [13] paper. These extensions are parsimonious in comparison with the TT1 and GI1 models, and yet flexible enough to generate firm-specific biases in TC. In the TT2 (GI2) model, they specify i = θ (i)α and α i = θ (i) α , where θ (i) are firm-specific parameters. Thus, αyt yt jt jt the input and scale biases are not allowed to be completely free as in the TT1 (GI1)

890

S. C. Kumbhakar

model, because the θ (i) parameters are constrained to be the same for all inputs and output. Consequently, the TT2 (GI2) model has only (N–1) additional parameters compared to the TT0 (GI0) model. Note that one of the θ parameters is normalized to unitary for identification. The TT0 (GI0) model can be obtained as a special case of the TT2 (GI2) model by restricting θ (i) = 1 for all i. Such restrictions are testable econometrically. One can justify the above extensions of the TT and GI models because of their flexibility. Empirically, we often deal with firms that are heterogeneous in size. Such heterogeneity may not always be captured by firm-effects (fixed or random) in the error term (which essentially makes the intercept firm-specific). It might have some second-order effects as well. Since we are mostly interested in analyzing TC and RTS properties of the production technology in an industry that is characterized by firms of different sizes and product mixes, one often focuses on models in which the scale and TC components can vary across firms, irrespective of variation in input prices and output quantities. In other words, one can let the data determine whether such second order effects are present or not. This means that before imposing such constraints a priori, one can test whether scale and TC components are affected by firm size or not. Since the TT0 (GI0) model is nested in TT1 and TT2 (GI1 and GI2), the LR test can be used to determine the appropriateness of these extensions.

Factor Augmenting Approach The cost function, which is dual to the FA production function in (5), can be written as C = C (w, ˜ y) ,

(61)

where w˜ j = Bj (t)wj and Bj (t) = 1/λj (t) ∀ j, λj (t) ≥ 1. Thus, an increase in efficiency of an input (λj (t) ≥ 1) is equivalent to a decrease (Bj (t) ≤ 1) in its effective price. Assuming a translog form for the cost function in (61), we get ln C = α0 +



αj ln w˜ j + αy ln y

j

⎫ ⎧ (62) ⎬  1 ⎨  + αj k ln w˜ j ln w˜ k + αyy ln y ln y + αjy ln w˜ j ln y ⎭ 2⎩ j

k

j

where w˜ j = Bj (t)wj and Bj (t) are parameters (j = 1, . . . , J and t = 1, . . . , T). The above cost function is assumed to satisfy the symmetry and linear homogeneity (in w) ˜ restrictions. It is to be noted that efficiency of each input in the above cost function model is completely flexible (subject to the normalizing constraints Bj (1) = 1 for all j). The rate of efficiency change in input j can be obtained from ln λj (t) − ln λj (t − 1) = − (ln Bj (t) − ln Bj (t − 1)) for t = 2, · · · , T, and j = 1, · · · , J.

21 Modeling Technical Change: Theory and Practice

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The different neutrality hypothesis postulated above can be tested via the cost function in (62) with appropriate parametric restrictions. For example, technical change is Hicks neutral if the RTS is unitary (i.e., α y = 1, α yy = 0, α jy = 0 ∀ j), and Bj (t) = B(t) ∀ j. That is, the cost function under the Hicks neutral TC can be written as C = B(t)·C(w, y) where the cost function C(w, y) is homogeneous of degree 1 in y (RTS = 1). Similarly, technical change is Harrod neutral if RTS = 1 and Bj (t) = 1 for all j= labor. Furthermore, technical change is Solow neutral if RTS = 1 and Bj (t) = 1 for all j= capital. The measure of overall technical change (TC_FA) in the above model can be obtained from T C_F A = [ln C(t)− ln C(t−1)]|w,y unchanged =



  Sj ln Bj (t) − ln Bj (t − 1)

j

+ ≡

  1  αj k ln Bj (t) − ln Bj (t − 1) {ln Bk (t) − ln Bk (t − 1)} 2



j

k

STCj

j

(63) where Sj =

 wj xj w˜ j x˜j ∂ln C = = αj + = αj k ln w˜ k + αjy ln y ∂ln w˜ j C C

(64)

k

and   1 STCj = ln Bj (t) − ln Bj (t − 1) Sj + αj k {ln Bk (t) − ln Bk (t − 1)} 2

!

k

(65) which is the share of input j in the overall technical change TC_FA. The above formula has several important features. First, the rate of overall technical change is defined by the change in efficiency of each input, as well as the cost share of each input (Sj ). The cost shares as shown in (64) are also affected by factor productivity of each input via ln w˜ k = ln wk + ln Bk . Second, the share of input j in the overall technical change is given by STCj which is defined in (65). Note that STC j is affected by a change in efficiency of other inputs (k = j) directly through the αj k {ln Bk (t) − ln Bk (t − 1)} component and indirectly via Sj . Third, k

by rewriting (63) as

892

S. C. Kumbhakar

T C_F A =

 j

+

ln Bj (t)− ln Bj (t−1)





! 1 αj k {ln Bk (t)− ln Bk (t − 1)} αj + 2 k



αjy ln y ln Bj (t) − ln Bj (t − 1)



j

+

 j

  αj k ln wk ln Bj (t) − ln Bj (t − 1)

k

(66) one can decompose TC into the following components: pure 

ln Bj (t) − ln Bj (t − 1)

j



! 1 αj k {ln Bk (t) − ln Bk (t − 1)} , αj + 2 k

(67) scale related 

  αjy ln y ln Bj (t) − ln Bj (t − 1) ,

(68)

j

and input price related  j

  αj k ln wk ln Bj (t) − ln Bj (t − 1) .

(69)

k

Bias in technical change associated with input j (j = 1, . . . , J) in the present model can be measured from I bj =



αj k {ln Bk (t) − ln Bk (t − 1)}

(70)

k

signs of which will depend on efficiency change in each input as well as the α jk parameters. Like the GI model, these measures are time-varying, but are also related to efficiency of all inputs. Consequently, input bias in technical change associated with an input j depends on efficiency change in every input. If efficiency factors are the same for all inputs, that is, Bj (t) = B(t) ∀ j, and the underlying production function is homogeneous, then Ibj = 0 ∀ j. Another possibility is that α jk = 0 ∀ j, k, which implies that the production function is Cobb-Douglas. The neutrality hypothesis arising from either way is testable. One can also test the neutrality hypothesis with constant returns to scale (Hicks neutral technical change).

21 Modeling Technical Change: Theory and Practice

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Other forms of neutrality hypotheses representing Harrod and Solow neutral technical change can also be tested. Scale bias in technical change in the factor augmenting model can be obtained from Scb =



αjy {ln Bk (t) − ln Bk (t − 1)} .

(71)

j

Thus, unlike the other models, scale bias in the factor augmenting  model has input-specific components which can be estimated. If Bj (t) = B(t) ∀ j and j αjy = 0, the scale bias in technical change is zero.

Symmetric Generalized McFadden Cost Function Cost functions, such as the translog, tend to violate economic theoretical restrictions (see Diewert and Wales [10]). Imposition of global concavity in input prices often destroys flexibility of the translog cost function. To avoid this problem, Diewert and Wales [10] suggested a functional form (Symmetric Generalized McFadden (SGM)) that is globally concave. The SGM cost function is C (w, t, y) = g(w)y +



ηj wj +

j



⎛ ηjj wj y + b ⎝

j



⎛ + att ⎝



γj wj ⎠ ty

j

⎛ ⎞ ⎛ ⎞   φj w j ⎠ t + ηY Y ⎝ βj wj ⎠ y 2 + at ⎝ j

⎞

(72)

j

⎞ λj wj ⎠ t 2 y,

j = 1, . . . , J

j

where the function g(w) is defined as g(w) = 

1 w  Sw 2 θ w

(73)

where θ ≡ [θ 1 , . . . , θ j ] > 0J is a vector of non-negative constants not all zero, and  S is an J × J symmetric negative semidefinite (NSD) matrix such that S w = 0. w can be considered as the point of approximation, that is, the point at which the data is normalized to unitary. Following theorem 10 of Diewert and Wales [10], it can be shown that C(·) is globally concave in w if the matrix S is negative semidefinite. If S is not NSD then negative semi-definiteness can be imposed following the technique developed by Wiley, Schmidt, and Bramble [14]. One can reparametrize S   as S = −   where  is an upper triangular matrix. Such reparameterization does not affect the flexibility of C(·) and its global behavior. The θ parameters in g(w)

894

S. C. Kumbhakar

are assumed to be known. To minimize the number of parameters, while keeping the function flexible, the γ j , φ j , β j , and λj are also assumed to be known. See Kumbhakar [15] for an application of the SGM model. The translog cost function described above can be generalized to accommodate multiple outputs. In general, the cost function is C = C(w, y, t) where w and y are vectors of input prices and output. Kumbhakar [16] generalizes the SGM cost function to the multioutput case that makes it relatively easy to estimate different aspects of a production technology. The model is estimated using the system of demand functions (normalized by y) which are derived from ∂C(.)/∂Wj = xj (.). Once the parameters are estimated, TC can be obtained from TC = (1/C(.)) (∂C(.)/∂t) where ∂C(.)/∂t ⎛ ⎞ ⎛ ⎞    = b y + at ⎝ φj wj ⎠ + 2att ⎝ λj wj ⎠ ty j

(74)

j

and C(.) is replaced by its predicted value. Other applications include Kutlu, Liu, and Sickles [17] who discussed properties of the SGM as well as some other cost functions.

Profit Function Approach Under the behavioral assumption of profit maximization with competitive input and output markets, the profit function can be expressed as π = π (w, p, t) [3, 4, 18], where w (p) is a vector of input (output) prices. The profit function, similar to the cost function, can handle multiple outputs. The profit function is homogeneous of degree 1 in p and w. One can estimate the profit function after imposing the linear homogeneity restrictions, using any parametric functions. For the CD and the translog functions, profit has to be positive for all observations. TC can then be computed using the formula defined earlier in the estimated profit function. Note that to estimate the profit function, one needs data on input and output prices, which should have enough variations (both cross-sectionally and over time) in them to get reliable estimate of the parameters.

Multiple Outputs The Primal Approach Output Distance Function Although most of the production processes consist of multiple outputs, multiple output production functions are rarely used in practice [19]. The problem lies in using the function econometrically. Even if inputs are assumed to be exogenous,

21 Modeling Technical Change: Theory and Practice

895

it is not straightforward to accommodate multiple outputs in a single equation production function when outputs are endogenous. That is, using one output as the left-hand side variable (which will be treated as endogenous) and others as the right-hand side variables (which are assumed to be exogenous) will give rise to the endogeneity problem, which cannot be avoided no matter which output appears on the left-hand side of the equation. To avoid this, one can use distance functions, for example, output distance functions (ODF). An easy way of describing it is to start with a transformation function T(y, x, t) = A. For identification of the parameters in it using a parametric form such as the CD or translog, we assume T(.) to be homogeneous of degree 1 in outputs which gives the ODF [20]. With the homogeneity assumption, the ODF can be expressed as   ln y1 = mO ln yˆm , ln x, t + ln A

(75)

where yˆm = ym /y1 , m = 2, . . . , M. One can use CD or translog form for the mO (.) function. The translog ODF with TT specification is ln y1it = β0 +



βj ln xj it +

j =1

 1  βj k ln xj it ln xkit + αm ln yˆmit 2 j =1 k=1

m=2

1  + αmn ln yˆmit ln yˆnit 2 m=2 n=2

+



δmj ln xj it ln yˆmit

m=2 j =1

+



 1 + αt t + αtt t 2 + αj t ln xj it t 2

(76)

j

βmt ln yˆmit t + vit ,

m=2

If inputs are exogenous (which is rarely true) for a revenue maximizing firm, output ratios will be exogenous (competitive output markets) especially when the output prices are exogenous. In such a case, the ODF does not suffer from the endogeneity (of outputs) problem. The ODF is invariant to the choice of the numeraire output which is the output that is used to impose the homogeneity restriction. Similar to the production function approach, one can obtain RTS from  j ∂ln y1 /∂ln xj and TC from ∂ln y1 /∂t. Note that the interpretation of TC here is similar to the production function model. Holding the inputs unchanged, TC denotes a proportional change in all the outputs. This means the output ratios yˆm appearing in the right-hand side of the  ODFare unchanged. Because of this, TC from the ODF can be obtained from TC y, ˆ x, t = ∂ln y1 /∂t. It is a radial measure and shows the rate at which all the outputs would be increased holding inputs unchanged. Since an increase in all the outputs by k% increases revenue by k% as well, TC in ODF can have revenue interpretation (the percentage at which the revenue is changed due to a change in all the outputs by 1%). A positive (negative) value  of TC indicates technical progress (regress). Similarly, RTS is defined as RTS = j ∂ ln y1 /∂ ln xj . It

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is a radial measure, meaning that RTS shows the rate at which all the outputs change when the inputs are changed by a constant percentage. The GI formulation in the ODF can be obtained from the above TT formulation by replacing the time trend variable t above by A(t) and dropping the t2 term. TC can be computed by taking finite differences, that is, lny1it − ln y1i, t − 1 holding the inputs xjit and yˆmit unchanged.   ˆ A1 (t)x1 , . . . , AJ (t)xJ For FA modeling, we write the ODF as yˆ1 = mI F y, and then use a parametric functional form (such as the translog) for mIF (.). The rest of the model is similar to our earlier discussion of TC in a FA model.

Input Distance Function (IDF) If outputs are assumed to be exogenous (which is the case for service industries such as electricity, transportation, hospital, telephone, post offices, etc.), one can specify the technology in terms of an IDF which is obtained from the transformation function by making it homogeneous of degree 1 in inputs. Thus, the IDF can be expressed as   ln x1 = mI ln y, ln x, ˆ t

(77)

where ln xˆ = ln (x2 /x1 ) , ln (x3 /x1 ) , . . . , ln (xJ /x1 ) . One can use a CD or a translog form for the mI (.) function. The translog IDF can be written as (Kumbhakar and Lien [21], Kumbhakar [20], and many others) ln x1it = β0 +



βj ln xˆj it +

j =2

 1  βj k ln xˆj it ln xˆkit + αm ln ymit 2 m j =2 k=2

1  αmn ln ymit ln ynit 2 m n  + δmj ln xˆj it ln ymit + αt t +

(78)

m j =2

  1 + αtt t 2 + αj t ln xˆj it t + βmt ln ymit t, 2 m j =2

For a cost minimizing firm with exogenous outputs, input ratios will be exogenous if input prices are exogenous (competitive input market) and therefore the IDF does not suffer from the endogeneity (of outputs) problem. See for example, Sipilainen et al. [22]. The ODF is invariant to the choice of the numeraire output (the output that is used to impose the homogeneity  restriction). Similar to the cost function approach, one can obtain RTS from m ∂ln x1 /∂ln ym and TC from −∂ln x1 /∂t. Note that here TC is a measure of cost diminution holding outputs and input proportions unchanged. That is, TC is measured radially by examining the rate at which all the inputs can be reduced. Thus, for example, if all the inputs are

21 Modeling Technical Change: Theory and Practice

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reduced by k%, cost is also reduced by k% and hence we have the cost interpretation of TC in IDF. Instead of using the TT formulation, one can use the GI as well as FA approaches in specifying the IDF (ODF). The expressions for TC and RTS can be derived similarly. Because of this, we do not repeat them here.

The Dual Approach In a primal approach (using the production function, ODF, and IDF), no economic behavioral assumption is made explicitly so far as the use of inputs and/or production of output are concerned. In a dual approach, economic behavior is explicitly used. In the dual cost function approach, the economic behavior is cost minimization subject to a given output level.

Cost Function The cost function with multiple outputs can be written as C = C(w, y, t) where w and y are input price and output vectors, respectively. Since the cost function assumes output(s) to be exogenous, a vast majority of the applications are from service sectors such as banking, provision of health services (hospitals, nursing homes, etc.), water, gas, and electricity distribution, transportation services (airlines, railroad, trucking, etc.), post office, and many others. The only difference between a single and multiple outputs is that y is scalar in a single output case and it is a vector for multiple outputs. The translog cost function with multiple outputs and time trend TC can be written as ln Cit = β0 +



βj ln wj it +

j

+



j

αm ln ymit +

m

+

 m

+



1  βj k ln wj it ln wkit 2 k

1  2

m

δmj ln wj it ln ymit

j

αmn ln ymit ln ynit

n

 1 + αt t + αtt t 2 + αj t ln wj it t 2

(79)

j

βmt ln ymit t,

m

which is homogeneous of degree 1 in input prices. Other restrictions are symmetry, that is, β jk = β kj , α mn = α nm . TC from (79) is TC (w, y, t) = αt + αtt t +

 j =1

αj t ln wj it +



βmt ln ymit

m

which is non-neutral. Input bias of TC is ∂Sjit /∂t = α jt where

(80)

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S. C. Kumbhakar

Sj it = ∂ln Cit /∂ln wj it = βj +



βj k ln wkit +



δmj ln ymit + αj t t.

(81)

m

k

Thus, bias is input specific but constant for all firms. Finally, RTS is defined as the inverse of. RTSit =



∂ln Cit /∂ln ymit ⎤ ⎡    ⎣αm + = αmn ln ynit + δmj ln wj it + βmt t ⎦ m

m

n

(82)

j

which is observation specific. There are many special cases of the above cost function. Some of these are: (i)  separability  (δ mj = 0), (ii) homogeneity (which implies constant RTS,   (δmj = 0, αmn = 0, βmt = 0), and (iii) unitary RTS (δ mj = 0, αmn =0, βmt =0, m m m m αm = 1).

Profit Function The profit function with multiple outputs is similar to that of the single output. The only difference is that output price (p) is replaced by a vector, that is, π = π (p, w, t). The profit function is homogeneous of degree 1 in input and output prices, which can be easily imposed by writing π (.) as π/p1 = π (p, ˜ w, ˜ t), where p˜ m = pm /p1 , m = 2, . . . , M and w˜ j = wj /p1 , j = 1, . . . , J . Since TC is defined in the same way and has the same interpretation as in the single output case, we are not discussing it again to avoid repetition.

TC Measures Induced by Management/Exogenous Factors and Time So far, in the models we considered, TC is not only exogenous but purely time driven (manna from heaven). Sometimes TC can be explained by some exogenous (predetermined) variables. In this case, TC can be decomposed into a time driven component and a component that is induced by predetermined variables (z). It could also be driven by endogenous variables. However, we are not addressing endogenous TC here. Thus, if something is endogenous, for modeling purposes we will be treating it as predetermined (decided sometimes in the past) to avoid the endogeneity problem. First, we discuss the primal approach for a single output (production function). Then we address TC in the context of multiple outputs (distance function).

21 Modeling Technical Change: Theory and Practice

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Management Variables as Technology Shifter Here the z variables (dummy) represent different management types, say, from 1 to 5 as shown in Triebs and Kumbhakar [69]. One can also think of different types of innovation, R&D, etc. Finally, it is possible to combine management induced and time driven TC in a production function. In the following, we consider both management induced (z) and time driven (t) TC in a translog production function.

Both z and t Are Continuous The translog production function with the time-invariant z variable can be written as (Triebs and Kumbhakar [69]) ln yit = β0 +



βj ln xj it +

j

+

 j

1  1 βj k ln xj it ln xkit + βt t + βtt t 2 2 2 j

k

 1 βj t ln xj it t + βz zi + βzz zi2 + γj z ln xj it zi + δzi t 2

(83)

j

from which the time driven and management induced TC, respectively, are T C 1it = ∂ln yit /∂t = βt + βtt t +

 j

T CM1it = ∂ln yit /∂zi = βz + βzz zi +

βj t ln xj it + δzi



(84) γj z ln xj it + δt

The z variable(s) can be both firm-specific and time-varying. If z is treated as a discrete (binary) variable, one needs to drop the z2 term in (83) and therefore TCM1it will  be given by TCM1it = ln yit |{lnxjit , zi = 1} − ln yit |{lnxjit , zi = 0} = β z + γ jz ln xjit + δt.

Multiple Discrete Management Variables zm and Time Trend (Continuous) If there are multiple management variables that are discrete in nature, one can M  θm zmi and use it the translog production define a management index M (zi ) = m=1

function which is (Triebs and Kumbhakar [69]) ln yit = β0 + +



 j

j

βj ln xj it +

1  1 βj k ln xj it ln xkit + βt t + βtt t 2 2 2 j

βj t ln xj it t + M (zi ) +

k



(85) γj ln xj it M (zi ) + δM (zi ) t,

j

Like before, TC in the above formulation can have two components: one driven by time and the other induced by management variables. Since there are zm

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S. C. Kumbhakar

management variables which may not be all 1 for a firm i, one can define TC induced by management in many different ways. For simplicity, we consider the case where zmi = 1 ∀ m vs. zmi = 0 ∀ m = 1 in defining management induced TC. The time driven and management induced TC are: TC2it = ∂ln yit /∂t = βt + βtt t +



βj t ln xj it + δM (zi )

j

    T C M2it = ln yit | ln xj it , M (zmi = 1∀m) − ln yit | ln xj it , M (zmi = 0∀m) ⎞ ⎛  = M (zi ) ⎝1 + γj ln xj it + δt ⎠ j

(86) where M(zi ) = M(zmi = 1 ∀ m). Note that although the zm variables (and therefore M(zi )) are time-invariant, TCM2it is not time-invariant.

TC with Continuous Management and General Time Index Assuming that there is only one continuous management variable, z, we write the translog production function as (Triebs and Kumbhakar [69]) ln yit = β0 +



βj ln xj it +

j

 1  βj k ln xj it ln xkit +A(t) + βj t ln xj it A(t) 2 j

1 + βz zi + βzz zi2 + 2



k

j

γj z ln xj it zi + δzi A(t)

j m

(87) where A(t) =

T 

λt Dt .

t=1

TC measures associated with (87) are, respectively, ⎛ TC3it = (A(t) − A (t − 1)) ⎝1 +



⎞ βj t ln xj it + δzi ⎠ ,

j

T C M3it = βz + βzz zi +



(88)

βj m ln xj it + δmt A(t)

j

TC with Management Index and Time Index To accommodate both management and time indexes, we write the translog production function as (Triebs and Kumbhakar [69])

21 Modeling Technical Change: Theory and Practice

ln yit = β0 +



βj ln xj it +

j

+ M (zi ) +



901

 1  βj k ln xj it ln xkit +A(t) + βj t ln xj it A(t) 2 j

k

j

γj ln xj it M (zi ) + δM (zi ) A(t)

j

(89) where A(t) =

T 

λt Dt and M (zi ) =

t=1

M 

θm zmi .

m=1

The time driven and management induced TC measures for this model are ⎛ ⎞  βj t ln xj it + δM (zi )⎠ , TC4it = (A(t) − A (t − 1)) ⎝1 + j

⎛ T C M4it = M (zi ) ⎝1 +



⎞

(90)

γj ln xj it + δA(t)⎠

j

Specification of the IDF with Multiple Outputs The Translog Specification with Time Trend The translog IDF with the time trend formulation is written as ln x 1it = α0 +

M 

βm ln ymit +

m=1

J M  

δmj ln ymit ln x˜j it +

m=1 j =2

J 

αj ln x˜j it + αt t

j =2

⎞ ⎛ J M M J   1 ⎝  + βmn ln ymit ln ynit + αj k ln x˜j it ln x˜kit + αtt t 2 ⎠ 2 j =2 k=2

m=1 n=1

+

M  m=1

αmt ln ymit t +

J j =2

βj t ln x˜j it t + vit (91)

Sometimes researchers are interested in estimating the impact of innovation, innovation systems, R&D, management, etc., on productivity. One can view such effects as TC because these exogenous and/or predetermined variables, shift the technology, ceteris paribus. Thus, one can measure the effect of innovation, R&D, and management type variables from the shift in technology, and therefore, it can be labeled as TC. In particular, for the innovation/management induced TC, one can (i) define an innovation (management) index or several indices, or (ii) use Likert scale variables to define dummy variable for innovation. These innovation dummies can then be

902

S. C. Kumbhakar

used to define a general innovation index similar to general time index of Baltagi and Griffin [11]. This index (indices) can be constructed in a cross-sectional framework as well. Thus, one can define TC in a cross-sectional model also. See Heshmati and Kumbhakar [23] for this type of formulation.

IDF with Many Innovation Indices as Shifters Following Heshmati and Kumbhakar [24], we specify an IDF with many innovation indices   q  lnx1it = lnDI x˜ , y, t, Tq zit

(92)

 q q where Tq zit , q = 1, . . . , Q are Q innovation indices based on a vector of zit  q external factors (labeled as technology shifters). Further, we specify Tq zit as ⎛ ⎞ Pq Pq    q q q q Tq zit = ⎝ ϕp zpit ⎠ , ϕp = 1∀q p=1

(93)

p=1

where Pq is the number of technology shifters in the innovation index Tq (·). For each of the indices, one can restrict the sum of the weights to be unity, so that the weights can be interpreted as “importance” of each shifter on the innovation indices (technology shifters).   q  To complete the model, a translog function for ln DI x, ˜ y, t, Tq zit is assumed, that is,

ln x1it = α0 +

M 

βm ln ymit +

m=1

+

Q M  

Q 

+

 q γmq ln ymit Tq Zit +

Q  J 

J   q γqj Tq zit ln x˜j it + αj ln x˜j it

q=1 j =2

j =2

 1 ⎝  βmn ln ymit lnyYnit + αj k ln x˜j it ln x˜kit 2 M

M

J

Q  R 

J

j =2 k=2

m=1 n=1

+

m=1 j =2

q=1

m=1 q=1

⎛

J M   q  λq Tq zit + δmj ln ymit ln x˜j it



q

λqr Tq zit

⎞ M   r Tr zit + αtt t 2 ⎠ + αt t + αmt ln ymit t

q=1 r=1

+

J  j =2

βj t ln x˜j it t +

m=1 Q 

 q λqt Tq zit t + vit

q=1

(94)

21 Modeling Technical Change: Theory and Practice

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In this specification, technology shifts (given input quantities) are allowed in terms of both time trend and innovation indices. The time driven technical change (TCT ) for this model is TCT = −

∂ln x1it ∂t

Q M J      q  = − αt + αtt t + αmt ln ymit + βj t ln xj it + λqt Tq zit " #$ % m=1 j =2 q=1 " " #$ % #$ % P ure T C N on−neutral T C

innovation index comp

(95) Innovation induced technical change associated with the qth index (TCZq ) is ∂ln x1it  q ∂Tq zit ⎡ ⎤ Q M J     q = − ⎣ λq + γmq ln ymit + γqj ln x˜j it + λqr Tq zit + λqt t ⎦

TCZq = −

j =2

m=1

q=1

(96) Thus, the overall innovation induced TC is TCz =

Q q

TCZq

(97)

Note that TC, in this setup, can be defined for each innovation index, which can be interpreted as the percentage cost diminution for a one-point change (or one standard deviation change) in the innovation index. This, as mentioned before, follows from the duality between the IDF and the cost function. As noted in the outset, TC is a shift in the technology without changing the inputs. So, in principle, if there are variables other than inputs that shift the technology, the shift can be viewed as TC so long as the shift variables are exogenous or predetermined. If we take this broad view of TC, then we can model it in a crosssectional setup. Below we illustrate this in the context of innovation indices using an IDF.

TC in Cross-Sectional Data Since TC is defined as a shift in the technology, it is possible to define TC in crosssectional models also. The IDF with the innovation index using cross-section data can be expressed as

904

S. C. Kumbhakar

  q  ˜ y, Tq zi ln x1i = ln DI x,

(98)

 q q where Tq zi are Q (q = 1, . . . , Q) innovation indices and zi are external factors  q that shift the technology shifters. As before, we specify Tq zi as '  &  q Pq Pq q q q Tq zi = ϕp zpi , ϕp = 1∀q p=1

p=1

(99)

where Pq is the number of technology shifters in the technology index Tq (·). For each of the indices, we restrict the sum of the weights to be unity, so that we can interpret the weights as “importance” of each shifter on the technology components. The overall TC can then be calculated from Q TCz = −

∂ln DI  q Zit

q=1 ∂Tq

(100)

where M J Q  q ∂ln DI  q  = λq + γmq ln ymi + γqj ln x˜j i + λqr Tq zi m=1 j =2 q=1 ∂Tq Zi (101) In this setup, TC can be defined for each innovation index, which can be interpreted as the percentage cost diminution for a one-point change in the innovation index. Note that all the variables used to define the innovation index are assumed to be exogenous (predetermined).

Technical Change from Other Indirect Functions Revenue Function We considered cost minimization and profit maximization as behavioral objectives of firms. These are widely used in the literature. However, a firm may be maximizing revenue, given the inputs. If inputs are exogenously given, then profit maximization is the same as revenue maximization. This is because if inputs and input prices are exogenously given, so is cost. Therefore, maximizing profit is the same as maximizing revenue . In such a case, the technology can be represented by the dual revenue function defined as R = R (p, x, t)

(102)

where R = py is revenue and p is the output price vector. TC, for the revenue function, can then be defined as ∂lnR(.)/∂t, holding output prices and input quantities

21 Modeling Technical Change: Theory and Practice

905

fixed. In this case, TC represents a shift in the revenue function, meaning the percentage change in revenue over time, ceteris paribus. That is, TCR (p, x, t) = ∂ln R(.)/∂t

(103)

TCR can be related to TC in the ODF for the multiple output case, and to the production function for a single output. For a single output R = py = pf (x, t) ⇒ ∂ln R/∂t = ∂lnf (x, t)∂t ≡ TCp (x, t). Thus, TC from the revenue and production function are theoretically the same. Intuitively, if output is increased by k% due to a shift in the production function, revenue will also increase by k% holding output price unchanged. However, empirical results are likely to be different because data used for estimation of the revenue function are different from those used in a production function estimation. For the multiple output case, the technology can be represented with various forms. If one uses an output distance function (ODF) which without inefficiency is nothing, but a transformation function T(y, x, t) = A with homogeneous of degree 1 in outputs  imposed for identification [20], the ODF can be expressed as ln y1 = ln ψ y, ˆ x, t where yˆ = y2 /y1 , . . . , yM /y1 . Assuming that TC changes all outputs radially, TCodf = ∂ln y1 /∂t = ∂ln ψ y, ˆ x, t /∂t. We can give it a revenue interpretation by writing R (p, x, t) =



pm ym = p1 y1 1 +

m



 pˆ m yˆm

⇒ ∂ln R/∂t = ∂ln y1 /∂t,

m=2

(104) holding output ratios unchanged. Note that, pˆ m = pm /p1 , m = 2, . . . , M. Thus,   TCR (p, x, t) = TCodf y, ˆ x, t

(105)

that is, TC from the revenue function is the is the same as TC from the IDF. The interpretation is the same as the single output case. TCodf is the percentage change in revenue, holding output prices and inputs unchanged.

Indirect Production Function In formulating the profit maximization problem, we assume that producers do not face any budget constraint. In reality, the producers may face a budget constraint. In such a case, profit maximization, given output price, is the same as output maximization if input cost is exactly the same as the exogenously given budget. Thus, Max π = pf (x, t) − wx subject to C 0 = wx

906

S. C. Kumbhakar

is the same as Max y = f (x, t) subject to C 0 = wx where C0 is the exogenously given budget. That is, the input allocation under profit maximization that satisfies the exogenously given budget is the same as output maximization. The above problem gives solutions of input demand functions which can be plugged into the production function. This gives the indirect production function (IPF) which can be written as y = ϕ(w, C0 , t) which is homogeneous of degree zero in w and C0 . TC from the IPF can be derived from     TCipf w, C 0 , t = ∂ln ϕ w, C 0 , t ∂t

(106)

which shows the percentage change in output over time, keeping input prices and the budget constraint unchanged. TCp (x, t) shows the rate at which output changes over time, holding input quantities constant. On the other hand, TCipf (w, C0 , t) shows the rate at which output changes, holding input prices and budget unchanged. If input quantities and input prices are unchanged, then the budget constraint is also unchanged. Thus, the two problems are theoretically the same. For example, if TCp (x, t) increases output by k% holding x unchanged, TCipf (w, C0 , t) will be the same if input prices do not change. While TCp (x, t) does not assume any economic behavior and it does not depend on input prices, TCipf (w, C0 , t) is based on an explicit economic behavior and it depends on input prices. Thus, although, in both TCp (x, t) and ϕ(w, C0 , t), we measure the rate of change in output over time, these functions depend on different sets of variables and therefore, empirically, the estimates of f (x, t) and ϕ(w, C0 , t) are likely to be different.

Technical Change and TFP Change TC and Profit Since our discussion is focused on technical change, from an individual producer’s point of view, a question that arises naturally is how TC affects productivity and profitability for an individual producer. The underlying assumption is that profit is the ultimate goal of a producer in making production decisions. To show the link, we first relate profit with productivity and then link it to TC. For this, we define productivity as the ratio of output produced to an input index, instead of using the ratio of output to labor (which is labor productivity). In a one input case, productivity is nothing but the average product (AP) of the input. So, the question is whether high productivity is always good to a producer. Alternatively, is low productivity necessarily bad to a producer? The answer depends on input and output prices. If real wage (w/p) increases faster than AP, productivity in value terms (py/wx) will be lower. Since py/wx = R/C

21 Modeling Technical Change: Theory and Practice

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is the return to the outlay, it can be lower (higher) even if AP is higher (lower). Thus, low productivity (AP) in physical terms is not always bad judging from return to the outlay. Can productivity for a producer be increased by using more labor? More generally, can it be input driven? To answer this, assume that there is no inefficiency. Go back to the production function, which shows that productivity declines as x is increased (follows from concavity of the production function). Labor productivity can be increased in the short run by using more capital. But capital is not free. Thus, productivity cannot increase simply by using more inputs. Input driven growth is not sustainable in the sense that it cannot increase productivity [2, 25]. The production function has to shift up (technical progress) to increase productivity and increase returns to the outlay. This is the ideal situation as can be seen in the figure below.

Is high productivity always good for a producer from the point of view of profit? That is, how does productivity affect profit? For this, we link profit (π ) to productivity starting from the definition of profit π = py − wx ⇒ y = π /p + (w/p)x. From the figure below, it can be seen that the optimal x, x∗ (associated with maximum profit) is given by the point of tangency between the straight line y = π /p + (w/p)x and the production function, y = f (x). If x < x∗ a firm could increase its profit by increasing the use of x, although it will reduce its productivity. That is, high productivity is associated with low profitability. Increase in input usage might increase profit but lower productivity.

908

S. C. Kumbhakar

(p /p)*

TC as a Component of TFP Change Production Function Approach From the discussion above, it is clear that input productivity (partial) is affected by TC defined as a shift in the production function. In a multiple inputs case, productivity can be measured by changes in (y/x) where x is an index of aggregate input (input aggregator). In such a case, productivity change is called total factor productivity (TFP) change as opposed to partial productivity change, which is change in y/xj , j= 1, . . . , J. Starting fromthe production function and the definition of TFP change T F˙ P as y˙ − x˙ ≡ y˙ − j Sj x˙j , it is straightforward to show that [13], [63] T F˙ P = TCp + (RTS − 1)

 j

λj x˙j = TCp + Scale,

(107)

where the “dot” over a variable means its rate of change and Sj = wj xj /C is the cost share of input j. In deriving the result in (107), we assumed that input allocations are optimal (no misallocation/allocative inefficiency). RTS and λj in the scale component are defined as RTS =

    ∂ln y/∂ln xj ≡ εj , λj = εj /RTS j

j

(108)

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It is clear from (107) that TC plays an important part in total factor productivity change. In fact, if one assumes a technology that exhibits unitary returns to scale (as in Solow [2]), total factor productivity change will be solely determined by TC. That is, productivity can only be increased by shifting the production function upward (technical progress). This can be done via some policy instruments such as investment in R&D, changes in management, etc.

Cost Function Approach Starting from the accounting definition of cost (C = wx) and the neoclassical cost function, C = C(w, y, t), it is shown by Denny, Fuss, and Waverman [26] that    Sj x˙j = 1 − RT S −1 y˙ − C˙ t = Scale + TCc (109) T F˙ P = y˙ − j

where RTS = [∂lnC/∂ln y]−1 . As argued before, TC in (107) and (109) are not exactly the same. They have different interpretations and are derived from two different representations of the technology.

Profit Function Approach Starting from the accounting definition of profit (π = py − wx) and the neoclassical profit function π = π (w, p, t), it is straightforward to show that [27] T F˙ P = (π/py) (∂ln π/∂t) + (RTS − 1)

 j

Sj x˙j = TCπ + Scale

(110)

Note that TCπ is the change in profit as a ratio of total revenue and therefore it not the same as (∂ ln π /∂t). Thus, the interpretation of TC and its magnitude differs depending on whether one uses a production, cost or profit function.

Formulating and Estimating TC Without Estimating Profit/Cost Function TC from a Production Function Formulation In modeling TC, we used either a primal or a dual approach to specify the technology and derive measures of TC therefrom. In this subsection, we consider formulations where TC is formulated differently, although its definition is exactly the same.  For this we start with the production function y = f (x, t) which gives y˙ = j λj x˙ j + TC (x, t), where λj = ∂lny/∂lnxj are input elasticities. If firms maximize profit in choosing the inputs and output and the markets are perfectly competitive, λj = ∂lny/∂lnxj = wj xj /py = Rj which can be  obtained from the observed data on inputs, output, and their prices. Thus, y˙ − j Rj x˙j = TC (x, t) which can be computed from data without doing any econometric estimation. In this framework, TC will be observation specific. Note that although the profit maximizing assumption is used here, TC is a function of input quantities x and t. However, one can argue that since the input demand functions in a profit maximizing

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case depend on w and p, TC can be expressed in terms of w, p, and t. This framework is pointed out in Chambers [3]. If firms minimize cost and output is exogenously given (as is the case in the service industries), then using the first order conditions of cost minimization, we can express input elasticities as λj = ∂lny/∂lnxj = Ecy Sj , Ecy = ∂lnC/∂lny. Thus, T (x, t) = y˙ − Ecy (y, w, t) j Sj x˙j which can be written in a regression format ˙ y˙ = T (x, t) + Ecy (y, w, t) x,

(111)

 where x˙ = j Sj x˙ j and Ecy (.) is a function of w, y, and t because the cost function depends on them. Furthermore, since the input demand functions (in a cost minimizing model) depend on w, y, and t, we can write y˙ = TC (w, y, t) + Ecy (y, w, t) x˙ ≡ β0 (w, y, t) + β1 (w, y, t) x˙ + v,

(112)

which is a semiparametric smooth coefficient (SPSC) regression model after adding the noise term v. Since the intercept and slope coefficients are nonparametric functions of w, y, and t, the TC = β 0 (w, y, t) and RTS (which is the reciprocal of Ecy = β 1 (w, y, t)) are much more flexible than a parametric function such as the flexible translog production function, for example. The regressors in this SPSC model are y˙ and x˙ which are computed from data. To estimate the model (the functional coefficients), we need data on input prices in addition to output quantities. If prices are constant (the same for all firms), the model still works although β 0 , β 1 will be functions of only y and t. Note that here we are not using a cost function per se, although the cost minimization assumptions are invoked.

TC from a Cost Function Formulation Here we start from the cost function C(w, y, t) in which cost minimization conditions are explicitly where C = wx gives  used. The log derivative of C = C(w, y, t) ˙ C˙ = j Sj w˙ j +Ecy y+ ˙ C˙ t which is rewritten as C− ˙ j = Ecy (w, y, t) y− ˙ j Sj w  TCc (w, y, t). Defining q˙ = C˙ − j Sj w˙ j (which is data), the above expression becomes q˙ = β1 (w, y, t) y˙ + β0 (w, y, t)

(113)

After adding an error term to it, we get a SPSC model in which the intercept function represents TC and the slope is the reciprocal of RTS. Being nonparametric, both TC and RTS are flexible which does not depend on functional form of the cost function. Note that TC from (112) is not likely to be the same as the one from (113).

TC from TFP Change  The Divisia index defined as y˙ − j Sj x˙j is used as a measure of nonparametric (growth accounting) TFP change. Starting from the production function y = f (x, t) yields

21 Modeling Technical Change: Theory and Practice

T F˙ P = TCp + (RTS − 1)

 j

λj x˙j = TCp + SCALE

911

(114)

in which all the notations are defined before. If firms minimize cost, and there are  no mistakes in input allocations, λj = Sj . Defining x˙ = j Sj x˙j as the change in the aggregate x as before, (114) gives (after adding error terms) a regression function of the form T F˙ P = (RTS(x, t) − 1) x˙ + TC(x, t) ≡ β0 (x, t) + β1 (x, t) x˙ + v

(115)

which is a SPSC regression function. The intercept function gives an estimate of TC(x, t) = β 0 (x, t) and the slope function estimates RTS(x, t) − 1 = β 1 (x, t) (scale economies). Kumbhakar and Sun [28] used an IDF and their estimating equation used the rate of change in the input, input ratios, and output variables. The coefficients are nonparametric functions of outputs, input ratios, and time. They linked the estimated coefficients to TFP change, RTS, and TC. The main advantage of their approach is that they did not use any functional form assumption on the IDF.

Models with Technical Inefficiency So far, we assumed that the production process is fully efficient. However, a firm can be technically inefficient. That is, firms may either not be able to produce the maximum possible output (output-oriented inefficiency) or they may be using more inputs than necessary (input-oriented inefficiency). Instead of repeating our discussion of inefficiency in all the cases discussed so far, we focus on the discussion in terms of TFP change. Details can be found in Kumbhakar and Lovell [29], Kumbhakar, Wang and Horncastle [30]. See also the chapters by Kumbhakar et al. [66, 67] in this volume.

TC and Technical Inefficiency Note that although technical inefficiency can be defined in a cross-sectional setup, here we focus on panel models because our objective is to model TC.

The Production Function Approach There is a plethora of panel models that include technical inefficiency. We start with the production function and write it as (Kumbhakar [60]) yit = f (xit , t) exp (−uit ) , where u ≥ 0 is output-oriented technical inefficiency. All other variables are defined before. Output-oriented technical inefficiency, u, measures the proportion

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by which actual output (yit ) falls short of maximum possible output (labeled as the frontier output f (xit , t)). Output technical efficiency (or simply technical efficiency) is then defined by yit /f (xit , t) = exp(−uit ) ≤ 1. For small values of uit , we use the relationship that technical efficiency (exp(−uit )) equals 1 minus technical inefficiency, uit using exp (−uit ) 1 − uit . 1 Given the input vector xit , output can change because of a shift in the technology (which is labeled as TC) as well as a change in technical efficiency (TEC). Differentiating the production function totally and using the definition of TFP change (as before), we get T F˙ P = (RTS − 1)



λj x˙j + TC + TEC +

j

  λj − Sj x˙j ,

(116)

j

where every term is defined as before except TEC = −∂u/∂t. In (116) we allow the possibility of input misallocation. Subscripts i and t are omitted  to avoid notational clutter. Defining the scale component as SCALE = (RTS − 1) j λj x˙j    and input misallocation (or price effects) as MISAL = j λj − Sj x˙j , the above relationship in (116) decomposes TFP change into SCALE, TC, TEC, and MISAL. Note that MISAL captures either deviations of input prices from the value of their marginal products, that is, wj = pfj , or departure of marginal rate of technical substitution from the ratio of input prices (fj /fk = wj /wk ) which is often labeled as allocative inefficiency. It is clear that exclusion of inefficiency (both technical and allocative) is likely to bias estimates of TFP change. If technical efficiency is time-invariant (i.e., TEC = −∂u/∂t = 0), the above decomposition of TFP change shows that TEC does not affect TFP change. Under unitary RTS assumption, the TFP change formula is identical to the one derived in Nishimizu and Page [31], viz., T F˙ P = TC − ∂u/∂t +



 λj − Sj x˙j = TC + TEC + MISAL.

(117)

j

Note that all the components of TFP change are firm specific and vary over p time. With inefficiency, one can define TFP in a potential sense, that is, TFPit = p yit exp (uit ) /xit where xit is an input aggregator. This gives T F˙ Pit = y˙it −  x˙it + u˙ it = T F˙ Pit + u˙ it and x˙it = Sj it x˙j it . Thus, potential TFP change j

adjusted for inefficiency will depend on temporal pattern of inefficiency change (u˙ it ). Consequently, TFP change ignoring inefficiency is likely to be biased (unless it is time invariant). If inefficiency declines over time  (u˙ itp< 0), the discrepancy  between the potential TFP change and TFP change T F˙ Pit − T F˙ Pit = u˙ it will decline.

1 Kumbhakar

and Hjalmarsson [61] were first to seperate TC from technical efficiency.

21 Modeling Technical Change: Theory and Practice

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Estimation The components of TFP change defined in the preceding section can be estimated from parametric production, cost, and profit functions. In this subsection, we discuss in detail the production function approach in which inputs are assumed to be exogenous (not decision variables).2 We assume that the production function is defined by a translog function, viz., ln yit = a0 + +





j

aj ln xj it + at t +

1  1 aj k ln xj it ln xkit + att t 2 2 2 j

k

(118)

aj t ln xj it t − uit + vit ,

j

  where v it is the i.i.d.N 0, σv2 random noise term, i = 1, . . . , N indexes producers, and t = 1, . . . , T indexes time. This is not the state-of-the-art stochastic frontier model. Greene [32, 33] defined models that include either random or fixed firm-effects along with uit and vit . Although the presence of the firm-effects will not change the decomposition formula discussed below, it will change the estimation method and the parameter estimates. Further developments by Colombi et al. [34] and Kumbhakar et al. [35] decompose inefficiency into time-invariant and time-varying components. Again, these models will not change the decomposition formula because the time-invariant effects do not count in TFP change. Again, the estimation method will be different. See Kumbhakar, Wang, and Horncastle [30] for discussion of TFP change decomposition for some of these new models. Estimation of the above models requires some assumption about the behavior of technical inefficiency over time. If it is assumed to be time-invariant, the TFP change is not affected by the presence of technical inefficiency. Thus, it might not be an interesting specification to study TFP change. There are several specifications that make the technical inefficiency term time-varying. In Kumbhakar [36] and Battese and Coelli [37], uit = ui g(t) where g(t) is a parametric function of time, and ui is a nonnegative random variable with half-normal (or exponential) distribution. In Lee and Schmidt [38], uit = ui λt where λt are time-effects (represented by time dummies) and ui can be either fixed or random producer-specific effects with no distributional assumption. Cornwell, Sickles, and Schmidt [39] specified it as uit = a1i + a2i t + a3i t2 where a1i , a2i , a3i are producer-specific parameters. Since time appears in a linear fashion as a regressor in the production function, as well as in uit , the parameters associated with the time variable in the production function and in uit cannot, in general, be identified. For example, it is not possible to separate the effects of technical change and technical efficiency change in the Cornwell, Sickles, and Schmidt [39] model. In the specification uit = ui g(t) the time effects, captured by g(t), are identified by positing a nonlinear specification by Kumbhakar 2 We

are not surveying the literature that deals with input endogeneity.

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[36] and Battese and Coelli [37]. In the specification uit = ui λt , no assumption is made on the temporal pattern of inefficiency. However, a restriction is imposed in both specifications. The temporal pattern of technical inefficiency is assumed to be the same for all producers. With these caveats in mind, let us consider the   model in which v it is i.i.d.N 0, σv2 , uit = ui g(t), ui ∼ i.i.d.N + μ, σu2 , and v it independent of ui for all i and t. Based on these assumptions, one can derive the probability density function of the composite error term v it − uit , and hence the log-likelihood function for the model in (118). Once the ML estimators of the parameters of the production function in (118), the parameters in g(t) as well as σu2 , σv2 , and μ are obtained, the Jondrow, Lovell, Materov, and Schmidt [40] (JLMS) formula can be used to obtain estimates of uit (see Kumbhakar and Lovell [29] for details). Since our interest here is estimation of TFP change and its components that are given in (116), we need to get estimates of RTS, SCALE, TC, and TEC, all of which can be computed once the parameters (including u) in (118) are estimated. These are RTSit =



εj it =



j

SCALEit = (RTSit − 1) TCit = at + att t +

( aj +

j

j

j

) aj k ln xkit + aj t t ,

k







λj it x˙j it

aj t ln xj it , and TEC = −

(119) ∂g(t) ∂uit = −ui ∂t ∂t

Once these components are estimated, the TFP change can be computed for each producer at every point in time. If price information is available, one can calculate Sj directly from the data. Although dated, there are two main advantages of the specification in uit = ui λt over uit = ui g(t). First, the temporal pattern of technical inefficiency is completely flexible (other than the normalizing restrictions for identification). Second, no distributional assumption is made on ui which can be estimated from the coefficient of firm dummies (in the fixed effects model). The production function in (118) can be estimated using the nonlinear least squares method without any distributional assumption on v. Since the mean of v it is zero, one can use the time mean of the composed error term (residual after estimation) to estimate ui and then uit = ui g(t). Since the decomposition and estimation using a cost or profit function are similar, we decided not to discuss them here. Details can be found in Kumbhakar [13] for single as well as multiple outputs. The decomposition results are similar to those of a single output. One can also add facilitating inputs (z) in the production function. The corresponding TFP change formula will have an extra component associated with the rate of change in the z variables. In a cost function, one can add allocative inefficiency. However, estimation of both technical and allocative inefficiency requires a system approach, either the production function and the first-order conditions of cost minimization (as

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in Schmidt and Lovell [41], Kumbhakar [42, 43], Kumbhakar and Wang [44], Kumbhakar and Lai [45], Tsionas and Kumbhakar [64]) or the cost function and cost share equations (Kumbhakar and Tsionas [46], Greene [47, 48], Kumbhakar and Lovell [29], Kumbhakar et al. [30], among others). Modeling technical and allocative inefficiency in a profit function is more complicated unless the underlying production function is restrictive (Cobb-Douglas) (see Kumbhakar [42], Kumbhakar et al. [30]). For a profit function with flexible production function, see Kumbhakar [49].

TC in Production Models with Good and Bad Outputs In many cases, production of desirable (good) outputs generates some unintended (bad) outputs. The special issue of Empirical Economics (2018) edited by Kumbhakar and Malikov [65] presents some papers dealing theoretical and econometric models to address the issues arising from bad outputs. Interested readers are advised to check the papers in the special issue. In this subsection, we consider a modeling approach in the spirit of Fernandez et al. [50, 51], Forsund [52], Murty et al. [53], and Malikov et al. [54] in which production of both good and bad outputs are considered. Specifically, we model the firm’s production process as a system of simultaneous production technologies for desirable and undesirable outputs. In this setup, desirable outputs are produced by transforming inputs via the conventional transformation function, satisfying all standard assumptions. The by-production of undesirable outputs is treated as the socalled “residual generation technology” which models production of bad outputs. The production function for bad outputs depends on production of good output(s), as well as inputs that might be specifically used to mitigate production of bad outputs. By separating the generation of undesirable outputs from that of desirable outputs, we ensure that the former are not modeled as inputs in the same production function. An advantage of the by-production system approach is that it can distinguish between technical efficiency and environmental efficiencies that are undesirable output-specific. It can also differentiate between traditional technical productivity and environmental (“green”) productivity. This framework allows for technical inefficiency in the production of, say, one desirable output and environmental inefficiency in the by-production of multiple undesirable outputs. For example, in electricity generation, the good output is electricity which is produced using (good inputs) labor and capital. Production of electricity also produces unintended bad outputs such as SO2 and NOx . The technologies for the production of these bad outputs are separated so that the arguments are good output and possibly bad inputs. An alternative to the by-production model is to consider a single technology which can be represented by F(y, xg , xb , z) = 1 where y, xg , xb , z are vectors of M good outputs, J good inputs, K pollution generating (bad) inputs and Q bad outputs. The monotonicity assumptions on these variables consist of Fy ≥ 0, Fxg ≤ 0, Fxb ≤ 0 and Fz ≤ 0 where Fy , Fxg , Fxb , Fz are partial derivatives of F (.).

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Since Fxg ≤ 0, Fxb ≤ 0 and Fz ≤ 0, there is no difference between z, xg and   xb in F y, xg , xb , z = 1 from a pure mathematical point of view. That is, bad outputs can be treated as inputs (both xg and xb ), and since inputs are assumed to be freely disposable, so are bad outputs. This violates axioms of production theory and is criticized in the environmental production literature (e.g., see Färe and Grosskopf [56] and Färe et al. [55]). Further, if, for example, in estimating the input distance function all bad outputs are treated as inputs, both bad outputs and inputs can be scaled back by the same proportion, holding good outputs constant. However, this approach violates a basic engineering requirement that a reduction in bad outputs requires the usage of more good inputs, holding good outputs and inefficiency constant. See Kumbhakar and Tsionas [46] for other problems. Given the problems in modeling bad outputs in terms of a single equation distance function, Fernandez et al. [50], Førsund [52], and Murty et al. [53] proposed a bi-production approach which uses two separate technologies to model good and bad outputs. The former describes the textbook type production process, where inputs (good and bad) are transformed into desirable outputs, and the process does not depend on bad outputs. Furthermore, it satisfies all the standard properties, most importantly the free-disposability property. The latter can be viewed as a residual generation technology which models the production of bad outputs as a function of good outputs and xb . Inefficiency is allowed in each technology, thereby distinguishing technical inefficiency from environmental inefficiency. We write the bi-production system in terms of the stochastic transformation function where P separates the environmental residual generation functions for each undesirable output. This is similar to Kumbhakar and Tsionas [57], that is,   f x, θ −1 y, t = exp {v0 }     Hp y, λp Bp , t = exp vp

(120) ∀

p = 1, 2, . . . , P

where θ ≤ 1 and λp ≤ 1 are technical and environmental efficiencies, respectively; and (v0 , vp ) are the noise terms. We rewrite the above system in the log form as ln yt = ln f (xt , t) − u0,t + v0,t ln Bp,t = ln hp (yt , t) + up,t + vp,t

(121)

where, for convenience, we define f (·) = [F(·, 1)]−1 and hp (·) = [Hp (·, 1)]−1 ; and u0, t = − ln θ t ≥ 0 and up, t = − ln λp, t ≥ 0 (p = 1, 2, . . . , P) as technical and environmental inefficiencies, respectively. Total differentiation of the above equations with respect to t yields  ∂ln f (xt , t) ∂ln xj,t ∂ln f (xt , t) ∂u0,t d ln yt = + − dt ∂ln xj,t ∂t ∂t ∂t J

j =1

21 Modeling Technical Change: Theory and Practice

917

∂ln hp (yt , t) ∂ln yt ∂ln hp (yt , t) ∂up,t d ln Bp,t = + + dt ∂ln yt ∂t ∂t ∂t

∀p

where we have made use of ∂v0, t /∂t = ∂vp, t /∂t = 0 since (v0 , vp ) are the i.i.d. noise terms. After some rearranging, we get the following Solow [25] type (Divisia) technical productivity index (TFPG): ∂ln f (xt , t) ∂u0,t d ln yt  ∂ln f (xt , t) ∂ln xj,t − = − , dt ∂ln xj,t ∂t ∂t ∂t % #$ % #$ " " j =1 J

TFPG =

TTC

(122)

TEC

along with the similarly defined environmental productivity index (EPG) for each p from & EPGp = − =− "

∂ln hp (yt , t) ∂ln yt d ln Bp,t − dt ∂ln yt ∂t

∂ln hp (yt , t) ∂up,t − ∂t ∂t % #$ %" #$ ETCp

∀

'

p = 1, 2, .., P

(123)

EECp

Furthermore, (122) and (123) provide a meaningful way to decompose productivity indices into technical change and efficiency change components. The conventional technical productivity index TFPG equals the sum of the technical change TTC = ∂lnf (xt , t)/∂t, which measures the temporal shift in the production frontier, and technical efficiency change TEC = − ∂u0, t /∂t which measures the movement toward (away from) the frontier. Similarly, the Bp -oriented environmental productivity index EPGp is decomposed into environmental technological change ETCp = − ∂ln hp (yt , t)/∂t and environmental efficiency change EECp = − ∂up, t /∂t. Note the conceptual difference between the definition of a “technological progress” for desirable outputs and that for undesirable outputs. For a desirable output y, the technological progress corresponds to the case of TTC > 0, that is, an outward shift in the production frontier over time, whereas for an undesirable output Bp , the technological progress corresponds to ETCp < 0, that is, an inward shift in the residual generating frontier over time. Thus, the residual generating frontier Hp (·) (p = 1, . . . , P) is defined as the minimum quantity of undesirable output generated when producing a given quantity of desirable outputs subject to the material balance condition. To implement it empirically, one needs to specify functional forms on ln f (.) and ln hp (.) and estimate the system jointly. Malikov et al. [54] used translog functional forms for ln f (.) and ln hp (.) (p = 1, . . . , P). Random firm effects, in ln f (.) and ln hp (.), were also introduced so that the estimates are not contaminated by firm effects in the technologies of both good and bad output production. Malikov et al. [54] used a Bayesian approach to estimate the system.

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Productivity and Profitability The concept of productivity change in a micro model is not quite intuitive because even if productivity change is positive it is not clear whether profitability of a firm will be increasing over time or not. To make the concept more intuitive and practical, we examine change in profit as a percentage of total cost as opposed to profit. This is useful because in practice, actual profit is often negative. If profit is negative but change in profit is positive (things are improving), then the percentage change in profit will be negative and there is no way to separate this from the case in which change in profit is negative (things are getting worse) but actual profit is positive. This problem can be avoided by expressing profit change as a percentage of cost or revenue. Differentiating profit π (= py − wx totally, we get )   py 1 ∂π ˙ − wj xj w˙ j + Sj x˙j . Using the expression for TFP C ∂t = C {p˙ + y} j

j

change, we can express the above as  . R R 1 ∂π = p˙ + y˙ − Sj w˙ j + T F P C ∂t C C

(124)

j

We call the left-hand side of the above equation profitability change (change in profit per unit of cost). It is positive when a change in profit is positive, irrespective of profit being positive or negative. This will not be the case if we use rate of change in profit as a profitability change measure, because negative profit with a negative profit change will be identical to a positive profit change with positive profit. The above decomposition gives three additional components for profitability change and these are related to output and output price changes as well as input price changes. This result holds irrespective of the number of outputs. Note that TFP change is a physical measure and it cannot describe performance of a firm in terms of profit. Since profit depends on input and output prices, the decomposition in (124) is more appealing. Even if TFP change is positive and large, profitability change can be low or even negative if change in output price is negative and input price change is positive and large. To show this explicitly, we write the profitability change as  . 1 ∂π R sj w˙ j + T F P = R˙ − C ∂t C

(125)

j

so that profitability change depends on change in TFP which is a physical measure, plus the financial measures which include rates of change in revenue (R) and input prices. The profitability measure is more important to an individual producer instead of TFP change. For a country, the TFP change is more meaningful because a positive TFP change means higher average output, which in turn means higher wellbeing no

21 Modeling Technical Change: Theory and Practice

919

matter whether higher average output goes to the workers or not. It is possible to derive the profitability change formula with facilitating inputs (z). In such a case, there will be an extra term associated with the profitability change formula.

TC and Factor Productivity with One Variable Input If we assume that labor is the only variable input in the production process, that is, y = f (L, z, t) where z is a vector of quasi-fixed inputs, then labor productivity is the same as TFP. This might be the mindset of people when they do not distinguish TFP from labor productivity. In such a case, change in labor productivity,    y˙ ˙ j j z˙ j + TC, where L = ∂ln f (.)/∂ln L < 1. Thus, labor L = (L − 1) L + productivity may not increase with an increase in labor use, even  TC > 0 unless  with  y˙ ˙ TC > (1 − L ) L + j j z˙ j . With multiple variable inputs, L = (L − 1) L˙ +  j x˙j + TC. Thus, labor productivity will depend on whether nonlabor inputs are j =L

increasing over time or not, along with the other two terms (TC and L ). Therefore, one cannot ignore the other factors while talking about labor productivity. This is a common mistake in many official statistics. If there is one variable input but many outputs, one can specify the technology as L = ψ(z, y, t) which is the labor requirement function introduced in Diewert [58], Kumbhakar and Hjalmarsson [59]. Like a cost function, the labor requirement function gives the minimum amount of labor required to produce a vector of outputs, ceteris paribus. Assuming a TT specification with translog functional form as Kumbhakar and Hjalmarsson [59, 62] do, the labor requirement function (without z) is ln Lit = β0 +



βm ln ymit

m

+

 1  1 βkl ln ykit ln ylit + αt t + αtt t 2 + λkt ln ykit t + vit 2 2 k

l

k

(126) where ym are M different services which are exogenously given. If a GI formulation is used, the model becomes ln Lit = β0 + +





m

βm ln ymit +

1  βkl ln ykit ln ylit + A(t) 2 k

l

λkt ln ykit A(t) + vit

k

where A(t) =

 t

δt DT t , DTt being the time dummies.

(127)

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S. C. Kumbhakar

TC in the above models (126) and (127) are TC_TTit = αt + αtt t +



λkt ln ykit * +  TC_GIit = [A(t) − A (t − 1)] 1 + λkt ln ykit k

(128)

k

A negative sign on TC means labor requirement is reduced (labor saving) over time, ceteris paribus. This is equivalent to technical progress in the cost function model. If there is more than one variable input and outputs are exogenous, the best strategy is to use the IDF which is invariant to the choice of numeraire input. This strategy works not only for estimating TC but also inefficiency and change in technical efficiency. For more details, there are many papers that use IDF with inefficiency and decompose output growth and TFP change (Kumbhakar [20], Sipilainen et al. [22], among others).

Concluding Remarks The literature on TC and productivity is extensive. Our discussion of it in this chapter does not cover every aspect of it. Nor does our discussion completely cover all the topics written in papers regarding TC and productivity (which would be a monograph). We concentrated on the models that are mostly used in the microeconomics literature3 . In particular, we focused on parametric models based on different forms of production, cost, and profit functions to model TC in a single output case. For multiple outputs, we used cost, profit, and distance function formulations. Most of our models and discussions excluded detailed treatment of technical inefficiency, especially the recent models. This is done to avoid repetition and to keep a reasonable length. Although most of our models use panel data, we added a section on modeling TC in a cross-sectional setup when shift in the technology is explained by exogenous factors. Finally, we added a section linking TC to TFP and also used a model to include both good and bad outputs. TFP is also linked to a measure of profitability change.

References 1. Krugman P (1997) The age of diminishing expectations. MIT Press, Cambridge 2. Solow RM (1956) A contribution to the theory of economic growth. Q J Econ 70:65–94 3. Chambers RG (1988) Applied production analysis. Cambridge University Press, Cambridge 4. Fuss M, McFadden D (1978) Production economics: a dual approach to theory and applications volume I: the theory of production. North-Holland, Amsterdam

3 See

Ray [68] in this volume for other formulations.

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5. Jorgenson DW (1986) Econometric methods for modeling producer behavior. In: Griliches Z, Intriligator MD (eds) Handbook of econometrics, vol 3. North-Holland, Amsterdam, pp 1841– 1915 6. Jorgenson DW (2000) Econometric modeling of producer behavior. The MIT Press, Cambridge 7. Jin H, Jorgenson DW (2009) Econometric modeling of technical change. J Econ 157:205–219 8. Kumbhakar SC, Heshmati A, Hjalmarsson L (1999) Parametric approaches to productivity measurement: a comparison among alternative models. Scand J Econ 101:405–424 9. Diewert WE (1971) An application of the Shephard duality theorem: a generalized Leontief production function. J Polit Econ 79:481–507 10. Diewert WE, Wales TJ (1987) Flexible functional forms and global curvature conditions. Econometrica 55:43–68 11. Baltagi BH, Griffin JM (1988) A general index of technical change. J Polit Econ 96:20–41 12. Stevenson R (1980) Measuring technological bias. Am Econ Rev 70:162–173 13. Kumbhakar SC, Nakamura S, Heshmati A (2000) Estimation of firm-specific technological bias, technical change and total factor productivity: a dual approach. Econ Rev 19(4):493–515 14. Wiley DE, Schmidt WH, Bramble WJ (1973) Studies of a class of covariance structure models. J Am Stat Assoc 68:317–323 15. Kumbhakar SC (1992) Allocative distortions, technical progress, and input demand in U.S. airlines: 1970–1984. Int Econ Rev 33(3):723–737 16. Kumbhakar SC (1994) A multiproduct symmetric generalized McFadden cost function. J Prod Anal 5:349–357 17. Kutlu L, Liu S, Sickles R (2020) Cost, revenue, and profit function estimates. In: Ray SC, Chambers R, Kumbhakar SC (eds) Handbook of production economics, vol 1. Springer Nature, Singapore 18. Lau LJ (1978) Applications of profit functions. In: Fuss M, McFadden DL (eds) Production economics: a dual approach to theory and applications, Volume I: The theory of production. Elsevier, Amsterdam 19. Hasenkemp G (1976) Specification and estimation of multiple-output production functions, Lecture notes in economics and mathematical systems, vol 120. Springer-Verlag, Berlin/Heidelberg 20. Kumbhakar SC (2013) Specification and estimation of multiple output technologies: a primal approach. Eur J Oper Res 231:465–473 21. Kumbhakar SC, Lien G (2009) Productivity and profitability decomposition: a parametric distance function approach. Food Econ Acta Agric Scand C 6:143–155 22. Sipilainen T, Kumbhakar S, Lien G (2014) Performance of dairy farms in Finland and Norway from 1991–2008. Eur Rev Agric Econ 41:63–86 23. Heshmati A, Kumbhakar SC (2011) Technical change and total productivity growth: the case of Chinese provinces. Technol Forecast Soc Chang 78:575–590 24. Heshmati A, Kumbhakar SC (2013) A general model of technical change with an application to the OECD countries. Econ Innov New Technol 23:25–48 25. Solow RM (1957) Technical change and the aggregate production function. Rev Econ Stat 39:312–320 26. Denny M, Fuss M, Everson C, Waverman L (1981) The measurement and interpretation of total factor productivity in regulated industries, with an application to Canadian telecommunications. In: Cowing TG, Stevenson RE (eds) Productivity measurement in regulated industries. Academic Press, New York, pp 179–218 27. Kumbhakar SC (2002) Productivity measurement: a profit function approach. Appl Econ Lett 9:331–334 28. Kumbhakar SC, Sun K (2012) Estimation of TFP growth: a semiparametric smooth coefficient approach. Empir Econ 43:1–24 29. Kumbhakar SC, Lovell CAK (2000) Stochastic frontier analysis. Cambridge University Press, Cambridge 30. Kumbhakar SC, Wang H-J, Horncastle AP (2015) A practitioner’s guide to stochastic frontier analysis using Stata. Cambridge University Press, New York

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31. Nishimizu M, Page JM (1982) Total factor productivity growth, technological Progress and technical efficiency change: dimensions of productivity change in Yugoslavia, 1965–78. Econ J 92:920–936 32. Greene W (2005) Fixed and random effects in stochastic frontier models. J Prod Anal 23: 7–32 33. Greene W (2005) Reconsidering heterogeneity in panel data estimators of the stochastic frontier model. J Econ 126:269–303 34. Colombi R, Kumbhakar SC, Martini G, Vittadini G (2014) Closed-skew normality in stochastic frontiers with individual effects and long/short-run efficiency. J Prod Anal 42:123–136 35. Kumbhakar SC, Lien G, Hardaker JB (2014) Technical efficiency in competing panel data models: a study of Norwegian grain farming. J Prod Anal 41:321–337 36. Kumbhakar SC (1990) Production frontiers, panel data, and time-varying technical inefficiency. J Econ 46:201–211 37. Battese GE, Coelli TJ (1992) Frontier production functions, technical efficiency and panel data: with application to paddy farmers in India. J Prod Anal 3:153–169 38. Lee Y, Schmidt P (1993) A production frontier model with flexible temporal variation in technical efficiency. In: Fried H, Lovell CAK, Schmidt S (eds) The measurement of productive efficiency. Oxford University Press, Oxford 39. Cornwell C, Schmidt P, Sickles RC (1990) Production frontiers with cross-sectional and timeseries variation in efficiency levels. J Econ 46:185–200 40. Jondrow J, Lovell CAK, Materov IS, Schmidt P (1982) On the estimation of technical inefficiency in the stochastic frontier production function model. J Econ 19:233–238 41. Schmidt P, Lovell CAK (1979) Estimating technical and allocative inefficiency relative to stochastic production and cost frontiers. J Econ 9:343–366 42. Kumbhakar SC (1987) The specification of technical and allocative inefficiency in stochastic production and profit frontiers. J Econ 34:335–348 43. Kumbhakar SC (1997) Modeling allocative inefficiency in a translog cost function and cost share equations: an exact relationship. J Econ 76:351–356 44. Kumbhakar SC, Wang H-J (2006) Estimation of technical and allocative inefficiency: a primal system approach. J Econ 134:419–440 45. Kumbhakar SC, Lai H-p (2019) Technical and allocative efficiency in a panel stochastic production frontier system model. Eur J Oper Res 278:255–265 46. Kumbhakar SC, Tsionas EG (2005) The joint measurement of technical and allocative inefficiencies: an application of Bayesian inference in nonlinear random-effects models. J Am Stat Assoc 100:736–747 47. Greene WH (1980) On the estimation of a flexible frontier production model. J Econ 13: 101–115 48. Greene WH (1993) The econometric approach to efficiency analysis. In: Fried HO, Lovell CAK, Schmidt SS (eds) The measurement of productive efficiency: techniques and applications. Oxford University Press, Oxford, pp 68–119 49. Kumbhakar SC (2001) Estimation of a profit function when profit is not maximum. Am J Agric Econ 83:1–19 50. Fernandez C, Koop G, Steel MFJ (2002) Multiple-output production with undesirable outputs: an application to nitrogen surplus in agriculture. J Am Stat Assoc 97:432–442 51. Fernandez C, Koop G, Steel MFJ (2005) Alternative efficiency measures for multiple-output production. J Econ 126:411–444 52. Forsund F (2009) Good modelling of bad outputs: pollution and multiple-output production. Int Rev Environ Resour Econ 31:1–38 53. Murty S, Russell RR, Levkoff SB (2012) On modeling pollution-generating technologies. J Environ Econ Manag 64:117–135 54. Malikov E, Kumbhakar SC, Tsionas EG (2015) Bayesian approach to disentangling technical and environmental productivity. Econometrics 3:443–465 55. Färe R, Grosskopf S, Noh D-W, Weber WL (2005) Characteristics of a polluting technology: theory and practice. J Econ 126:469–492

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56. Färe R, Grosskopf S (2003) Nonparametric productivity analysis with undesirable outputs. Am J Agric Econ 85(4):1070–1074 57. Kumbhakar SC, Tsionas EG (2016) The good, the bad and the technology: endogeneity in environmental production models. J Econ 190:315–327 58. Diewert WE (1974) Functional forms for revenue and factor requirements functions. Int Econ Rev 15:119–130 59. Kumbhakar SC, Hjalmarsson L (1995) Decomposing technical change with panel data: an application to the public sector. Scand J Econ 97:309–323 60. Kumbhakar SC (2000) Estimation and decomposition of productivity change when production is not efficient: a panel data approach. Econ Rev 19:425–460 61. Kumbhakar SC, Hjalmarsson L (1993) Technical efficiency and technical progress in Swedish dairy farms. In: Fried H, Schmidt S, Lovell CAK (eds) The measurement of productive efficiency: techniques and applications. Oxford University Press, New York 62. Kumbhakar SC, Hjalmarsson L (1998) Relative performance of public and private ownership under yardstick competition: electricity retail distribution. Eur Econ Rev 42:97–122 63. Kumbhakar SC, Heshmati A (1996) Technical change and total factor productivity growth in Swedish manufacturing industries. Econ Rev 15(3):275–298 64. Kumbhakar SC, Tsionas EG (2020) On the estimation of technical and allocative efficiency in a panel stochastic production frontier system model: some new formulations and generalizations. Eur J Oper Res 287(2):762–775 65. Kumbhakar SC, Malikov E (2018) Good modeling of bad outputs, special issue (ed. Kumbhakar and Malikov). Empir Econ 54:1–308 66. Kumbhakar SC, Parmeter CF, Zelenyuk V (2020) Stochastic frontier analysis: foundations and advances I. In: Ray SC, Chambers R, Kumbhakar SC (eds) Handbook of production economics, vol 1. Springer Nature, Singapore 67. Kumbhakar SC, Parmeter CF, Zelenyuk V (2020) Stochastic frontier analysis: foundations and advances II. In: Ray SC, Chambers R, Kumbhakar SC (eds) Handbook of production economics, vol 1. Springer Nature, Singapore 68. Ray S (2020) Conceptualization and measurement of productivity growth and technical change: A nonparametric approch. In: Ray SC, Chambers R, Kumbhakar SC (eds) Handbook of production economics, vol 1. Springer Nature, Singapore 69. Triebs T, Kumbhakar SC (2018) Management in production: from unobserved to observed. J Prod Anal 49:111–121

Economics of Externalities: An Overview

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Jean-Paul Chavas

Contents Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Motivations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A General Equilibrium Analysis of Efficiency under Externalities . . . . . . . . . . . . . . . . . . . . . Efficient Pricing under Externalities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Efficient Policies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Externalities arise when the decisions of an agent have direct effects on the welfare of others. This chapter presents an overview on the economics of externalities. Relying on Pareto efficiency, the analysis is presented in a general equilibrium framework and evaluates the efficient management of externalities. The investigation also focuses on the role of non-convexity and transaction costs. It covers alternative institutional setups, including markets, government interventions, and contracts. It reexamines how efficient management of externalities remains consistent with aggregate profit maximization under transaction costs and non-convexity. It indicates how pricing can support an efficient allocation under externalities, but this may require nonlinear pricing under non-convexity.

The author would like to thank two anonymous reviewers for useful comments on an earlier draft of the chapter. J.-P. Chavas () University of Wisconsin, Madison, WI, USA e-mail: [email protected] © Springer Nature Singapore Pte Ltd. 2022 S. C. Ray et al. (eds.), Handbook of Production Economics, https://doi.org/10.1007/978-981-10-3455-8_13

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And it explores how the minimization of transaction costs is an integral part of the efficient management of externalities. Keywords

Externalities · Pareto efficiency · Non-convexity · Transaction costs

Introduction Externalities are common in economics: they arise whenever an agent makes decisions that affect the welfare of other agents [1, 2]. The agents can be producers or consumers. And the externalities can be positive (when they improve the welfare of others) or negative (when they have adverse effects on others). Examples abound. Pollution is a negative production externality when it causes damages to crops or public health. Second-hand smoke is a negative consumer externality when a smoker has adverse effects on the health of nearby nonsmokers. Bees generate positive production externality when, besides producing honey, they also provide pollination services to surrounding crops and orchards. An individual getting vaccinated against a communicable disease involves a positive consumer externality when it reduces the odd of infection for both himself and other people nearby. Thus, externalities are prevalent. But how they should be managed is less clear. One argument is that externalities make unregulated markets inefficient (when market prices do not reflect the full social cost or benefit of goods and services). In this case, recommendations are often made for government interventions implementing policies that can “internalize” the externalities (e.g., [1, 3–5]). Others have argued that externalities can be managed by contracts among the affected agents, in which case an efficient outcome can be attained without government intervention [6]. This debate raises two important questions. First, how to characterize an efficient allocation in the presence of externalities? Second, how can we evaluate the role of markets versus non-market institutions in the efficient management of externalities? These are the key questions addressed in this chapter. It is well known that, in the absence of externalities and under convexity conditions, competitive markets can implement an efficient allocation [7]. This has stimulated much interest in relying on market institutions and decentralized decision-making to support efficient allocations. But externalities create significant challenges to market allocations [8]. First, by affecting the ability to decentralize decision-making in an efficient manner, externalities undermine the efficiency of unregulated markets (e.g., [5]). Second, as argued by Starrett [9] and Baumol and Bradford [10], production externalities can imply non-convex technologies. In addition, environmental goods often generate externalities while being produced in non-convex ecosystems [11]. It is well known that, under non-convexity and uniform pricing, markets can fail to be efficient (e.g., [12–17]). Chavas and Briec [13] showed that nonlinear pricing can be required under non-convexity. Following Chavas [18], it means that nonlinear pricing may be needed to manage externalities

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efficiently. Third, as emphasized by Coase [6] and others, various institutional choices (including contracts and the legal system) can also be used to manage externalities. Relying on the classical Pareto efficiency criterion, this chapter provides a broad overview on the economics of externalities. By definition, externalities imply the effects of some agents on others. Managing externalities means developing some coordination scheme among these agents, which typically involves the use of resources. We call the costs of these resources “transaction costs.” They include information cost, search cost, and enforcement cost. The economics of transaction costs has been analyzed by Coase [19], Foley [20], Hahn [21], Williamson [22], Williamson and Winter [23], Shleifer [24], and others. Transaction costs vary across economic institutions and are relevant in the evaluation of externality management. This applies to market as well as nonmarket institutions (including government and contracts). The efficient management of externalities suggests a need to integrate Coase’s [6] analysis of contracts with the role of transaction costs and non-convexity. This chapter argues that the Coase theorem (stating that the efficient management of externalities is consistent with aggregate profit maximization) holds under transaction costs and non-convexity. This result applies to production externalities as well as coordination/exchange externalities. In addition, the efficient management of consumption externalities is consistent with the minimization of aggregate consumer expenditure. In this context, we examine how transaction costs play a role in efficiency analysis and how the minimization of transaction costs is an integral part of efficient allocations. Such results apply under non-convexity provided that we allow for nonlinear pricing. In the presence of non-convexity, the efficient management of externalities becomes more complex. First, the classical dichotomy often made in economics between nonmarket institutions and competitive markets (exhibiting uniform pricing for all market participants) is not very helpful. Indeed, when externalities generate non-convexity, markets exhibiting nonlinear pricing may be needed to implement efficient allocations. Second, within market institutions, the implementation of nonlinear pricing (where prices vary across agents) requires the identification and implementation of price discrimination schemes. Such choices would be made by firm managers and/or policy makers (a “visible hand”) and would typically involve transaction costs. This indicates that transaction costs play a role in the management of externalities under markets as well as nonmarket institutions. The chapter is organized as follows. Section 2 motivates the analysis of efficiency under externalities using graphical arguments. A general equilibrium characterization of Pareto efficiency under externalities, non-convexity, and transaction costs is presented in Sect. 3. The analysis goes beyond Baumol and Oates [1] in several ways. First, it relies on a benefit function as an intuitive measure of aggregate welfare (the maximization of aggregate benefit providing a convenient representation of efficiency). Second, the linkages between the maximization of aggregate benefit and pricing give useful insights into the role of markets in supporting efficient allocations under externalities. For example, the approach identifies

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conditions under which the implementation of simple Pigouvian taxes/subsidies is inappropriate (when uniform pricing is inefficient). Third, the analysis stresses the role of transaction costs in choosing institutions and policies. Policy implications for the efficient management of externalities are discussed in Sect. 4. Finally, Sect. 5 concludes.

Motivations To motivate the analysis, we examine a production process involving two outputs (y1 , y2 ) produced under a given set of resources. We consider alternative situations, all illustrated in Fig. 1. We start the analysis with a situation where there are no externalities. Without externalities, the outputs satisfy (y1 , y2 ) ∈ Y0 ⊂ R2 , where Y0 denotes the feasible set without externalities. We assume that the feasible set Y0 is convex, corresponding to a technology exhibiting diminishing marginal productivity. We now introduce externalities in the analysis. P1: An externality arises when the decisions of one agent have direct effects on the welfare of other agents. Coase [6] discusses an example with two agents: a rancher managing livestock and a farmer producing crops, the externality coming from straying livestock that destroy crops growing on the neighboring farm. In this context, letting y1 be livestock production and y2 be crop production, there is a negative externality of y1 on y2 . From P1, at least two agents are required. Indeed, in the Coase example, if the farmer and the rancher were to merge into a single firm, then the management of crop losses

1

A Y0

B E1 Y1

E2 Y2

C

Fig. 1 Feasible set under negative externalities

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due to livestock damages would reduce to an issue of internal management (in which case the externality would be “internalized”). We examine two externality scenarios. Under scenario 1, the outputs satisfy (y1 , y2 ) ∈ Y1 ⊂ R2 with Y1 ⊂ Y0 and the feasible set Y1 is convex. Under scenario 2, (y1 , y2 ) ∈ Y2 ⊂ R2 with Y2 ⊂ Y0 but the feasible set Y2 is non-convex. These two scenarios are illustrated in Fig. 1. Both scenarios represent negative externalities that have adverse effects on productivity and reduce the feasible set. As showed in Fig. 1, introducing externalities implies an inward shift in the boundary of the feasible set from isoquant (A B C) to isoquant (A E1 C) under scenario 1 and to isoquant (A E2 C) under scenario 2. Note that all isoquants go through the points A and C: there is no externality when only one product is produced (i.e., when either y1 = 0 or y2 = 0). But when products are produced (i.e., when y1 > 0 and y2 > 0), then negative externalities between y1 and y2 implies a reduction in productivity (due to crop losses generated by straying livestock in the Coase example). As discussed below, the distinction between scenario 1 (where Y1 is convex) and scenario 2 (where Y2 is non-convex) will prove important. While Fig. 1 reflects negative externalities reducing the feasible set, it also shows that, when external effects are large enough (under scenario 2), this reduction is associated with a shift from a convex set Y0 to a non-convex set Y2. P2: Externalities can lead to non-convexity in the production set. Property P2 has been noted in the literature by Starrett [9], Baumol and Bradford [10], and Dasgupta and Maler [11]. We discuss below how non-convexity creates significant challenges to the analysis of economic efficiency under externalities. This section focuses on a graphical analysis involving two firms (a rancher and a farmer in the Coase example) and one consumer. Firm 1 (the rancher) produces output y1 (livestock production) while firm 2 (the farmer) produces y2 (crop output), with y1 having negative effects on the production of y2 . This example will help motivate the more general analysis presented in the rest of the paper. First, consider scenario 1 where negative externalities imply that Y1 ⊂ Y0, the feasible set Y1 being convex. This scenario is illustrated in Fig. 2. Figure 2 shows that the efficient allocation is given by point E1. Indeed, point E1 generates the highest possible utility level represented by the indifference curve (D E1 D’). At the efficient point E1, the slope of the indifference curve (D E1 D’) is tangent to the isoquant line (A E1 C) (the upper bound of the feasible set Y1). And in the neighborhood of point E1, the slope of both lines is equal to the slope of the line (F E1 F ) in Fig. 2. In turn, this slope is equal to (−p2 /p1 ), where (p1 , p2 ) are the (social) shadow prices of (y1 , y2 ). When the feasible set is convex, taking prices (p1 , p2 ) as given, the line (F E1 F ) is the budget line supporting the efficient point E1 for the two producers and the consumer. But what if producers failed to take into consideration the external effects of their decisions on others? In general, a failure to internalize production externalities would affect production incentives. In the Coase example, if the external cost of the externality imposed on the farmer is neglected, the rancher would have incentives to produce y1 beyond what is socially optimal, leading to a market equilibrium where

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D

F A

∗ 1

E1 E1’

D’

Y1 0

C

F’

Fig. 2 Efficiency under externalities and a convex feasible set Y1 (scenario 1)

  prices p1 , p2  differ from (p1 , p2 ). This is illustrated in Fig. 3 for the market good y1 . Figure 3 distinguishes between a social supply curve (where production decisions reflect the presence of externalities) and a private supply curve (where externalities are ignored in an unregulated market). In Fig. 3, the efficient point  E1 corresponds to p1 , y1∗ and is situated at the intersection of the social supply curve and demand market equilibrium is given by  curve.  But the unregulated   point E1’ where p1 , y1 differs from p1 , y1∗ because production decisions ignore external effects. Such a market equilibrium would necessarily be inefficient. In this context, government intervention can help. It can be done in at least two ways. First, government could set regulations/standards or issue permits/quotas stipulating the efficient quantities to be produced (i.e., y1∗ and y2∗ in Figs. 2 and 3). Second, government could implement Pigouvian taxes/subsidies that reflect the social cost of the externality [5]. When the feasible set is convex, such Pigouvian taxes/subsidies would restore efficiency [1, 3, 5].1 P3: In unregulated markets facing externalities, government policy can help restore efficiency. This can be done through regulations using standards/permits/quotas; or through price policies imposing Pigouvian taxes/ subsidies when the feasible set is convex.

1 In

case where Pigouvian taxes/subsidies are not fiscally neutral, attaining efficiency requires redistribution of any fiscal surplus/deficit to consumers through lump sum payments [1, 3, 5]. Otherwise, Pigouvian taxes would not achieve “first-best” efficiency.

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Fig. 3 Market equilibrium under externalities

But what if the feasible set is not convex? We now turn our attention to scenario 2 where negative externalities are associated with the feasible set Y2 being nonconvex. This scenario is illustrated in Fig. 4. The efficient allocation corresponds to point E2 in Fig. 4. Indeed, point E2 generates the highest possible utility level represented by the indifference curve (D E2 D’). Again, at the efficient point E2, the slope of the indifference curve (D E2 D’ is tangent to the isoquant line (A E2 C) (the upper bound of the feasible set Y2). And in the neighborhood of E2, the slope of both lines is equal to the slope of the line (F E2 F ) in Fig. 4. What is new in Fig. 4 is that the linkages between the slope of the line (F E2 F ) and efficiency breaks down. Indeed, if the two firms faced prices given by the slope of (F E2 F ), they would have incentive to produce at point C, where only output y2 is produced. This is showed in Fig. 4 where the lines (F E2 F ) and (C C) have the same slopes (reflecting the same relative prices), but aggregate revenue is higher at point C than at point E2. Yet, point C is inefficient as it generates lower utility, the indifference curve (C C ) being lower than the indifference curve (D D’). In other words, under the non-convexity of the set Y2, the slope of line of (F E2 E’) no longer defines global prices that can support an efficient allocation. Note that this slope still provides a local measure of the shadow prices of (y1 , y2 ) in the neighborhood of the efficient point E2. But this local measure does not provide a global representation of efficient pricing. This raises the question: Does there exist a pricing scheme that can support an efficient allocation? As showed in Fig. 4, an efficient pricing scheme is given by the budget line (G E2 G’), the slope of this line reflecting the shadow relative prices of (y1 , y2 ). Like the line (F E2 F ), the line (G E2 F ) is tangent to both the indifference curve (D E2 D’) and the isoquant (D E2 D’) in the neighborhood of

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Fig. 4 Efficiency under externalities and a non-convex feasible set Y2 (scenario 2)

the efficient point E2. But unlike (F E2 F ), the budget line (G E2 G’) provides incentives for the firms to produce at the efficient point E2. And in contrast with (F E2 F ), the budget line (G E2 G’) is nonlinear. Interpreting the slope of budget lines as measures of relative prices, we deduce that a pricing scheme supporting an efficient allocation must involve nonlinear pricing. Noting that this statement holds only under the non-convexity of the feasible set (i.e., under scenario 2), this gives the following result. P4: In a market economy with externalities, nonlinear pricing may be required to support an efficient allocation when the feasible set is non-convex. Knowing from property P2 that externalities can lead to non-convexity of the feasible set, property P4 indicates that linear pricing may not support an efficient allocation. In such a situation, uniform Pigouvian taxes/subsidies mentioned in property P3 would not be efficient.2 Such complexities raise a more fundamental question: What is the most efficient way to manage externalities? Trying to identify the appropriate institutional and policy response to externality issues has generated a debate among economists and policy makers. At least three different lines of arguments have been explored. A first line is that externality issues can be managed through government intervention. This includes regulation and/or Pigouvian taxes/subsidies chosen to make market

2 As

discussed below, government intervention can still be helpful to achieve efficiency through quantity regulations and/or through nonlinear pricing policies.

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prices reflect the social cost of the externalities [3, 5]. But as noted above, standard Pigouvian taxes may not always lead to efficient allocations. A second line is to note that production externality issues can be resolved when the affected firms merge. In this case, the externalities can be “internalized” under efficient management by the merged firm. A third line is that externalities can be managed through contracts between the affected parties [6]. In the Coase example mentioned above, this would involve a contract between the rancher and the farmer that would stipulate how livestock is managed to reduce or eliminate crop damages. For example, a contract could stipulate building a fence to prevent livestock from straying and destroying the crop grown on the neighboring farm. This could be efficient if the cost of building a fence is less than the crop damages. As stressed by Coase [6], under this contract option, there is no need for government intervention. So, how can externalities be managed efficiently? There are many institutional options, including government intervention, mergers, and contracts.3 In general, it is difficult to tell which option is better. If each option could be implemented under perfect information and at zero cost, then all options may be seen as equally efficient, in which case efficiency alone would not provide much guidance.4 But most situations involve imperfect information and the management of externalities is typically costly. This suggests introducing transaction costs explicitly in the analysis. Defining transaction cost as the cost of the resources used in the process of coordination/exchange, this will provide additional insights on how externalities can be managed efficiently. This is illustrated in Fig. 5 under two institutional options, A and B, where institution B exhibits larger transaction costs (i.e., uses more resources) than institution A in managing externalities, ceteris paribus. Resources used in externality management are no longer available for consumption. As a result, the feasible set for consumer goods is larger under institution A than under institution B. As showed in Fig. 5, it follows that institution A is deemed efficient compared to institution B. Indeed, point EA attained under institution A generates a higher level of utility (as given by the indifference curve (D EA D’)) than point EB attained under institution B. This generates the following result. P5: Transaction costs are relevant in the management of externalities: ceteris paribus, institutions exhibiting lower transaction costs are deemed more efficient. Property P5 states that transaction costs play an important role in the efficient management of externalities. It indicates that each institutional option (government intervention, merger, or contract) may be desirable on efficiency ground when associated with low transaction costs. Alternatively, any option involving large

3 Other options include individual transferable permits/quotas [25], and Varian [26]’s scheme involving a two-step mechanism that can implement efficient allocations as subgame-perfect equilibria under externalities. 4 Note that equity considerations (not addressed in this chapter) can also play a role in evaluating alternative externality management strategies.

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G

D

A

EB

EA

D’ G’ C

2

Fig. 5 Efficiency under externalities and transaction costs

transaction costs may be inefficient. To the extent that information costs are included among transaction costs, property P5 would apply to information: efficient management of externalities should be handled by institutions that have good access to information. These arguments suggest that government intervention may be seen as desirable when government has good information about the externalities and government action can be implemented at relatively low cost (e.g., [5]).5 Similarly, contracts provide a good option to manage externalities when contractual costs are low and the affected agents are well informed [6, 24]. Finally, the merger option may be desirable when the management of the merged firm is well informed and effective in “internalizing” the externalities [19, 27].

A General Equilibrium Analysis of Efficiency under Externalities Based on the motivating example presented in Sect. 2, we now present a general model of externalities. Following Chavas [18], consider an economy consisting of m commodities and n economic agents. We distinguish between two groups of agents: consumers and producers. Let Nc be the set of nc consumers, and Ns the set of ns

5 In this context, when comparing government pricing policies versus government standards/quotas,

economists often follow Pigou [5] and argue in favor of pricing policies on the ground that they are easier to implement and require less information (especially in the presence of heterogeneous agents). These issues are further discussed in Sect. 4.

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producers. The set of all agents is N = Nc ∪ Ns = {1, 2, . . . , n}, where n = nc + ns . The i-th consumer chooses a consumption bundle xi = (xi1 , . . . , xim ) ∈ Xi ⊂ Rm , i ∈ Nc , where the feasible set Xi is assumed to be convex and to have a lower bound, i ∈ Nc . Let x = (x1 , . . . , xnc ), where x ∈ X = X1 × · · · × Xnc . Consumer preferences for the i-th consumer are represented by a continuous utility function ui (x), i ∈ Nc . The utility functions are general and allow for non-zero income effects. And given x = (x1 , . . . , xnc ), they allow for externalities among consumers (as the consumption decision of a consumer can affect the utility of other consumers). The allocation of m commodities among the n agents also involves production and exchange activities. The production activities of the j-th producer are denoted by yj = (yj1 , . . . , yjm ) ⊂ Rm , j ∈ Ns . When the k-th commodity is a consumer good, yjk is the nonnegative quantity produced by the j-th producer (yjk ≥ 0), j ∈ Ns , and xik is quantity consumed by the i-th consumer (xik ≥ 0), i ∈ Nc . Let y =   the nonnegative y1 , . . . , yn s ∈ Y ⊂ Rm ns , where Y is the feasible set for production activities. In general, the feasible set Y represents a joint production process, allowing for externalities among producers. Exchange can take place among agents. Let the vector tji = {tjik :k = 1, . . . , m}∈Rm denote the exchanged quantities of commodities provided by the j-th agent to the i-th agent, j, i ∈ N. When the k-th commodity is a consumer good, tjik ≥ 0 is the quantity traded from the j-th agent to the i-th consumer. We consider the case where coordination and exchange among agents can be costly and involve the use of resources. Let z = (z1 , z2 , . . . , zn ), where zi = (zi1 , . . . , zim ) ∈ Rm is the vector of commodities used by the i-th agent in coordination and exchange activities, i ∈ N. The costs of z are the transaction costs from coordination and exchange activities among the agents. Such costs include transportation cost, information cost, search cost, contractual cost, and enforcement cost. The feasible set for (z, t) is denoted by Z, with (z, t) ∈ Z. Again, this allows for externalities in coordination and exchange among agents. Assume that the sets Y, Z, and X are closed, that (0, 0) ∈ Z, and that the set     ∩ X y : , . . . , y ∈ Y has a non-empty interior. Importantly, (y ) 1 m i∈Nc i j ∈Ns j we allow the sets Y and Z to be non-convex. In this context, a feasible allocation is defined as a vector (x, y, z, t) satisfying  i∈N

xi ≤

tj i ≤ yj − zj , j ∈ Ns ,

 j ∈N

tj i − zi , i ∈ Nc ,

(1)

(2)

where x ∈ X, y ∈ Y, (z, t) ∈ Z. Equations (1) and (2) are commodity balance constraints. When the k-th good is an output and a consumer good, yjk ≥ 0 is the quantity produced by the j-th producer, tjik ≥ 0 is the quantity traded from the j-th agent to the i-th agent, and (1) implies that the j-th producer cannot sell more than its production yjk net of resources used in coordination and exchange zjk , j ∈ Ns . And when the k-th commodity is an input, then yjk ≤ 0 where yjk  is the input quantity

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used by the j-th producer, tjik ≤ 0 where tjik  is the quantity traded from the j-th agent to the i-th agent, and (1) implies that yjk  ≤ i ∈ N |tjik | − zjk , j ∈ Ns , meaning that the use of the  k-th input by the j-th producer yjk  cannot exceed its availability from exchange ( i ∈ N | tjik | ) net of zjk.. Similarly, when the k-th commodity is a consumer good, we have xik ≥ 0, and (2)  implies that the i-th consumer cannot consume more than what it can acquire ( j ∈ N tjik ) net of zi , i ∈ Nc . And when the k-th commodity is a “consumer bad” (e.g., pollution), let xik ≤ 0 where xik  is the quantity of the k-th bad facing the ith consumer, and let tijk ≤ 0 where tijk  is the quantity of the k-th bad exchanged from the i-th agent to the j-th agent, i ∈ Nc . This shows that a feasible allocation in (1, 2) applies under general conditions, including situations of externalities between producers and consumers where production/trade activities have adverse effects on consumer welfare. Finally, note that the analysis can include dynamics and uncertainty. Following Debreu [7] and using a state-contingent approach, each decision can be defined to be specific to a given time period and a given state of nature representing uncertainty (e.g., weather conditions). The feasible sets X, Y, and Z would then reflect the information available to each of the n agents. In this case, externalities may arise when information and learning involve social networks [28]. Our analysis focuses on efficient allocations, relying on the classical Pareto efficiency criterion: a feasible allocation (x∗ , y∗ , z∗ , t∗ ) is Pareto efficient if there is no other feasible allocation (x, y, z, t) that can make one individual better off without making anyone else worse off. Our analysis of Pareto efficiency relies on a benefit function. Consider the commodity bundle g = (0, . . . , 0, 1) ∈ Rm + where the m-th commodity is “money” treated as a private good that that can be exchanged costlessly among the n individuals. We assume that consumer preferences are non-satiated in the m-th good (money). Following Luenberger [29], using g as a reference bundle, define the aggregate benefit function as B (x, U ) = max β

 i∈Nc

≥ Ui , i ∈ Nc

  βi : (xi − βi g) ∈ Xi , ui x1 − β1 g, . . . , xnc − βn c g

 (3)

  if there is a feasible β = β1 , . . . , βn c , = − ∞ otherwise, where U = (U1 , . . . , Unc ). The benefit function B(x, U) in (3) gives the largest amount of the bundle g that consumers facing utilities U are willing to give up to reach consumption x. The function B(x, U) in (3) provides a general

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measure of aggregate consumer benefits under consumption externalities.6 When g = (0, . . . , 0, 1) ∈ Rm + with the m-th commodity being money, we set the unit price of the bundle g to be 1, meaning that our welfare measurements involve monetary evaluations. Yet, the analysis allows for production and consumption externalities with respect to the first (m − 1) goods. And it allows for transaction costs to arise when exchange in the first (m − 1) commodities takes place. Next, define a maximal allocation as an allocation (x, y, z, t) solving the following maximization problem V (U ) = max {B (x, U ) : equ.(1a) − (1b), x ∈ X, y ∈ Y, (z, t) ∈ Z} . x,y,z,t

(4)

And (x, y, z, t) is said to be zero maximal if, in addition, U is chosen in (3) such that V(U) = 0. The following result was obtained by Luenberger [29, 30] and Chavas [18]. Proposition 1: A Pareto efficient allocation is equivalent to a zero-maximal allocation given in (3) with U = (U1 , . . . , Unc ) being chosen such that V(U) = 0. Proposition 1 holds in the presence of externalities, non-convexity, and transaction costs. It means that the investigation of Pareto efficiency under externalities can be based on the analysis of zero-maximal allocations. The function V(U) in (3) has an intuitive interpretation: it is the distributable surplus that maximizes aggregate benefit [29, 32]. From (3), a maximal allocation makes aggregate benefit as large as possible. And zero-maximality means that this surplus must be entirely redistributed to consumers. In this context, Proposition 1 states that Pareto efficiency involves the maximization of aggregate benefit and then its complete redistribution. In addition, the zero-maximality condition V(U) = 0 is an implicit equation for U = (U1 , . . . , Unc ) that characterizes the Pareto utility frontier. Noting that V(U) is non-increasing in U, the set of U that satisfies V(U) ≥ 0 defines the space of reachable utility levels, and {U : V(U) = 0} identifies the upper bound of this space as the Pareto utility frontier. In this case, moving along the Pareto utility frontier corresponds to efficient allocations associated with different welfare distributions among consumers. These results establish the characterization of efficiency under externalities, transaction costs, and non-convexity under general conditions. By identifying the efficient quantities produced, consumed, and traded, Proposition 1 can be used in the evaluation of contracts and government regulations in the presence of externalities. When externalities are evenly distributed, Proposition 1 6 Equation

(3) includes as a special case the situation where there is no consumption externality, in which case individual benefit can be evaluated one consumer at a time and aggregate benefit is just the sum of individual benefits across all consumers [30, 31].

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is consistent with government regulations imposing uniform standards over space. But efficient contracts/standards would change depending on the nature and extent of the externalities. For example, efficient standards would vary across space when the externalities are spatially distributed. In this situation, efficient regulation would require information on the spatial nature of the externalities. In addition, nonconvexity can provide incentives to concentrate external effects in a small area, in which case spatially uniform standards would be inefficient [33].

Efficient Pricing under Externalities We now examine the role of markets and pricing in controlling externalities. Studying the linkages between efficiency and market allocations is not new (e.g., [34, 35]). Here, we follow Chavas [18] to explore the economics of efficiency and pricing in the presence of externalities under general conditions (including nonconvexity (from P2) and transaction costs (from P5)). Let F be the set of continuous and non-decreasing functions f from Rm to R that satisfy the translation property: f (y + α g) = α + f (y) for any y ∈ Rm and any α ∈ R. Consider the generalized Lagrangian functional L:    

  fj yj − fj L (x, y, z, t, f, h, U ) = B (x, U ) + t j i + zj j ∈Ns i∈N  

   + hi tj i − zi − hi (xi ) , i∈Nc

j ∈N

(5)   where f = f1 , . . . , fn s ∈ Fs = F × · · · × F and h = (h1 , . . . , hnc ) ∈ Fc = F × · · · × F are “penalty functions” associated with constraints (1) and (2), respectively. Given g = (0, . . . 0, 1), the m-th commodity is used as a numeraire good. When the unit price of g is normalized to be equal to 1, then fi and hi can be interpreted as monetary values of goods associated with the i-th agent, i ∈ N. In this context, we interpret the functions f and h as reflecting pricing schemes. Note the Lagrangian in (4) is “generalized” in the sense that it allows the functions (f, h) to be nonlinear. This is an extension of the standard Lagrangian approach applied under convexity where the penalty functions (f, h) are taken to be linear, their slopes being Lagrange multipliers measuring the shadow prices of constraints (e.g., [36, 37]). As illustrated in Fig. 4, the presence of non-convexity requires us to consider nonlinear penalty functions (and nonlinear pricing as discussed below). For a given U, consider the case where the generalized Lagrangian in (4) has a saddle-point (x∗ , y∗ , z∗ , t∗ , f∗ , h∗ ) ∈ [X × Y × Z × Fs × Fc ] that satisfies     L (x, y, z, t, f, h, U ) ≤ L x ∗ , y ∗ , z∗ , t ∗ , f ∗ , h∗ , U ≤ L x ∗ , y ∗ , z∗ , t ∗ , f, h, U (6)

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for any x ∈ X, y ∈ Y, (z, t) ∈ Z, f ∈ Fs , h ∈ Fc .7 The first inequality in (6) implies the maximization of the Lagrangian with respect to (x,y, z, t) and the second inequality implies its minimization with respect to (f, h). In addition, we have L(x∗ , y∗ , z∗ , t∗ , f∗ , h∗ , U) = V(U), implying that the Lagrangian approach in (6) provides a dual formulation to the maximization of aggregate benefit given in (4). The Lagrangian approach also gives the efficient pricing scheme (f∗ , h∗ ). Chavas [18] obtained the following result. Proposition 2: A Pareto efficient allocation satisfies E (h, U ) = Minx

 i∈Nc

πs (f ) = Maxy πT (f, h) = Maxz,t  –

 hi (xi ) : ui (x) ≥ Ui , i ∈ Nc ; x ∈ X ,

 j ∈Ns



j ∈Ns

i∈Nc

fj

hi

   fj yj : y ∈ Y ,

(8)





i∈N

tj i –zi j ∈N

 tj i + zj : (z, t) ∈ Z ,

(7)

V (U ) = I nff,h {πs (f ) + πT (f, h) –E (h, U ) : f ∈ Fs , h ∈ Fc } ,

(9)

(10)

with U = (U1 , . . . , Unc ) being chosen such that V(U) = 0. Proposition 2 gives a dual representation of Pareto efficiency under externalities, allowing for externalities, non-convexity, and transaction costs. It provides useful information on the role of markets in the efficient management of externalities. We interpret the function fj as measuring the value of goods associated with the j-th producer, j ∈ Ns , and the function hi as measuring the value of goods associated with the i-th consumer, i ∈ Nc . This interpretation applies to market institutions as well as nonmarket institutions (in which case the functions f ’s and h’s represent valuations associated with implicit markets; see Rosen [39] and Ekeland et al. [40]). Equations (7, 8, and 9) follow from the first inequality in (6), while Eq. (10) follows from the second inequality in (6). Equation (7) states that, conditional on U and the pricing scheme h, consumption x is chosen to minimize aggregate consumer expenditure, i∈Nc hi (xi ). Equations (8, 9) are profit maximizing conditions. In    Eq. (8), j ∈Ns fj yj is the aggregate value from all production activities and πs (f ) is the largest possible aggregate profit given the pricing scheme f. In Eq. (9),

      – is the aggregate value from h t –z i∈Nc i j ∈N j i i j ∈Ns fj i∈N tj i + zj

7 The

conditions needed for the existence of a saddle-point in (6) are mild and are expected to hold under fairly general conditions. See Gould [38], Bertsekas [36], and Chavas and Briec [13]. In this chapter, we assume that these conditions hold.

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coordination and exchange activities, and πT (f, h) is the largest possible profit from trade, conditional on f and h. Equation (10) establishes linkages between aggregate profit, πs (f ) + πT (f, h), net of aggregate expenditure, E(h, U). Note that the function V(U) in (10) is the same as the distributable surplus V(U) defined in (3), making it clear that V(U) is a welfare measure with a monetary interpretation. Then, Eq. (10) states that the distributable surplus V(U) is obtained by choosing the pricing functions f and h that minimize aggregate profit minus aggregate expenditure. Finally, choosing U such that V(U) = 0 means that, under Pareto efficiency, the distributable surplus V(U) is entirely redistributed. Denote the solution to Eq. (10) by (f∗ , h∗ ). From proposition 2, given the aggregate expenditure function E(h, U) in (7) and the aggregate profit functions πs (f ) in (8) and πT (f, h) in (9), Eq. (10) defines the pricing scheme (f∗ , h∗ ) that yields the distributable surplus V(U). Then, choosing U such that V(U) = 0 in (10) implies that πs (f∗ ) + πT (f∗ , h∗ ) − E(h∗ , U) = 0. This corresponds to an aggregate budget constraint where aggregate consumer expenditure E(h∗ , U) equals aggregate profit πs (f∗ ) + πT (f∗ , h∗ ). This provides a basis supporting the measures commonly used in national accounts: the total value of goods and services can be measured equivalently from the production side πs (f∗ ) + πT (f∗ , h∗ ), or from the consumption side E(h∗ , U). And with {U : V(U) = 0} defining the Pareto utility frontier, moving along this utility frontier corresponds to efficient allocations associated with different distributions of income among consumers. Recall that our analysis covers general consumer preferences and allows for income effects. In the presence of income effects, it means that efficient allocations would change along the Pareto utility frontier (as production, consumption, trade, and monetary values would typically change under different income redistributions). Proposition 2 generalizes the Coase theorem [6]. Indeed, Proposition 2 and expression (8) imply that aggregate profit maximization is consistent with Pareto efficiency. This is the essence of the Coase theorem [6], making it clear that aggregate profit maximization is at the heart of Pareto efficiency, with or without externalities. It indicates that a failure to maximize aggregate profit would be inconsistent with efficiency. But the analysis in Coase [6] was presented assuming no transaction costs. As argued in Chavas [18], aggregate profit maximization remains a valid characterization of the efficient management of externalities under two important generalizations: (1) under transactions costs; and (2) in the presence of non-convexity in production and exchange activities. These are important generalizations of the Coase theorem. They stress the generality of the arguments presented by Coase [6] and their importance leading him to win the Nobel Prize in 1991. Another important result is given in Eq. (9): aggregate profit maximization for exchange activities is also consistent with Pareto efficiency. This is an insight that was explored in Coase [19] but not in Coase [6]. Thus, Proposition 2 provides a nice integration of Coase’s two seminal papers. First, the consistency between aggregate profit maximization and Pareto efficiency applies to production

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as well as coordination/exchange activities. Importantly, this result holds in the presence of externalities in consumption and coordination/exchange activities and under non-convex technology. Second, as stressed in [19], assessing the relative efficiency of markets versus nonmarket institutions requires evaluating the role of transaction costs. As argued in Chavas [18], this is a “missing piece of the puzzle” in Coase [6]. Identifying which institutions can manage externalities efficiently depends on their associated transaction costs. The efficient management of externalities is associated with the institutions that maximize aggregate profit, including both aggregate profit from production activities (as stated in (8)) and aggregate profit from coordination/exchange activities (as stated in (9)). The profit maximization condition in (9) implies the minimization of transaction costs. In other words, efficient institutions are the ones that maximize aggregate profit (as discussed in Coase [6]) as well as minimize transaction costs (as stressed in Coase [19]). It is sometimes argued that the Coase analysis implies that the efficient management of externalities is independent of the assignment of property rights [41]. In general, this argument is false. Indeed, reassigning property rights typically affects the distribution of income. In the presence of income effects, this would affect consumption decisions and thus resource allocation [42]. Thus, given the empirical prevalence of income effects, it is incorrect to assert that the efficient management of externalities is independent of property rights. Note that this argument holds under very general conditions: it applies with or without transaction cost; and it applies in the presence of non-convexity. In other words, it is inappropriate to argue that the assignment of property rights is irrelevant in the efficiency evaluation of externalities. While Eqs. (7, 8) state that aggregate profit maximization is consistent with Pareto efficiency, what does it say about decentralized decision making? In general, in the presence of externalities among firms, production and exchange activities cannot be fully decentralized. Indeed, production externalities would make firmlevel profit maximization inefficient. As illustrated in Fig. 3, efficiency requires that external effects among producers be explicitly taken into consideration. Similarly, under coordination/exchange externalities (e.g., due to social learning), firm-level profit maximization applied to trading firms would be inefficient (by failing to consider external effects among traders). In such cases, some coordination schemes are needed among the agents facing externalities. How does Proposition 2 relate to the standard welfare theorems establishing close linkages between Pareto efficiency and markets (e.g. [43])? As illustrated in Fig. 2, externalities do not always lead to non-convexity. Under convexity, the analysis of efficiency is simpler. In this case, the separating hyperplane theorem applies, meaning that there exists a hyperplane separating the feasible set from the efficient consumption set [37], the slope of the hyperplane measuring prices supporting an efficient allocation. Then, the functions f and h in (7)–(10) can be taken to be linear:  fj (yj ) = psi yj , j ∈ Ns , and hi (xi ) = pci xi , i ∈ Nc , where the p s are prices reflecting the

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social value of the commodities.8 In Fig. 2, the separating hyperplane is given by the line (F E1 F ). The slope of the hyperplane is equal to (−p2 /p1 ), where (p1 , p2 ) being the social prices supporting an efficient allocation. But as discussed in Sect. 2, under decentralized decisions, neglecting the externalities would affect supply/demand  decisions, leading to inefficiency as observed prices p1 , p2  differ from efficient prices (p1 , p2 ) (as illustrated in Fig. 3). Under this scenario, inefficiency can be restored by government intervention imposing Pigouvian taxes/subsidies reflecting the social cost of the externalities. As argued by Starrett [9] and Baumol and Bradford [10], externalities can lead to non-convexity. This is the scenario illustrated in Fig. 4, where negative externalities are large enough to make the feasible set non-convex. The investigation of nonconvexity is not new (e.g., [13–17]). It is well known that introducing non-convexity for Y or Z can invalidate the standard welfare theorems. The reason is that, under non-convexity, the separating hyperplane theorem no longer holds and cannot be used to identify efficient prices. This is illustrated in Fig. 4 where the linear pricing line (F E2 F ) fails to support the efficient allocation E2 (as it would provide incentives to produce at the inefficient point C). But as discussed in Sect. 2, the nonlinear pricing line (G E2 G’) would support the efficient allocation E2. This pricing line corresponds to a hypersurface separating the feasible set from the efficient consumption set. Again, the slope of this separating hypersurface provides information about prices. The separating hypersurface being nonlinear implies nonlinear pricing.9 This is a scenario where the pricings schemes (f, h) in (7)–(10) must be nonlinear. Interestingly, Eq. (7) still associates efficiency with aggregate expenditure minimization. And Eqs. (8, 9) associate efficiency with aggregate profit maximization. Thus, the problem created by non-convexity does not come from aggregate profit maximization (which continues to hold under efficiency). The problem comes from uniform pricing. Under non-convexity, efficiency can be attained by moving away from uniform pricing and implementing a nonlinear pricing scheme [13]. In Fig. 4, this involves moving from the uniform price line (F E2 F ) to the nonlinear pricing line (G E2 G’). This argument applies to externalitydriven non-convexity of the production set Y. But it also applies to possible non-convexity in the feasible set Z. For example, externalities among traders could also generate non-convexity in Z. Again, such non-convexity invalidates the separating hyperplane theorem, implying that uniform prices may fail to support an efficient allocation and require nonlinear pricing schemes f∗ and h∗ as identified in (10).

8 Note

that, under convexity and in the absence of externalities, the analysis would then reduce to the standard welfare theorems establishing close relationships between Pareto efficiency, decentralized decisions, and competitive markets (e.g., [7, 43]). 9 The line (G E2 G’) in Fig. 4 implies that the relative price (p /p ) declines with y , indicating 2 1 2 that the price p2 decreases with y2 . This is a situation of “volume discount” commonly observed in nonlinear pricing (e.g., [44]).

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This stresses the importance of nonlinear pricing in efficiency analysis under non-convexity. When externalities generate non-convexity, this implies that economists and policy makers should not insist on uniform pricing in the efficiency evaluation of externalities. Indeed, nonlinear pricing may be required to implement an efficient allocation in the presence of externalities. As noted by Wilson [44], nonlinear pricing schemes are commonly observed in many markets. The challenges of implementing nonlinear pricing in support of efficiency are discussed below. Finally, while the efficient pricing schemes (f∗ , h∗ ) were identified in (10), one issue remains: how to discover and implement such pricing schemes? Under nonmarket institutions (e.g., contracts), these pricing schemes need not be explicit: they would be shadow prices in implicit markets supporting an efficient allocation [39, 40]. Alternatively, under market institutions, the pricing schemes (f∗ , h∗ ) are an explicit part of a market economy. Under linear pricing (where fj (yj ) = psi yj , ∗ and p ∗ are easy to identify and j ∈ Ns , and hi (xi ) = pci xi , i ∈ Nc ), the prices psi ci implement: they are the market-clearing prices that satisfy the commodity balance Eqs. (1 and 2), respectively. As illustrated in Fig. 2, linear pricing can always support an efficient allocation under convexity, and identifying efficient schemes reduces to finding the “right prices”. In competitive market prices in the absence of externalities, the market clearing prices correspond to Adam Smith’s “invisible hand.” In the presence of externalities, this can involve Pigouvian taxes that make market prices equal to social prices [1, 3, 5]. But non-convexity and nonlinear pricing make economic evaluations more complex. In this case, the nonlinear pricing schemes (f∗ , h∗ ) have two roles to play: they clear the markets; and they provide the proper incentives to implement an efficient allocation. This second role arises as a separate function only under nonconvexity. This raises the question: In situations where the role of pricing goes beyond just “clearing the markets,” who choose the nonlinear pricing schemes (f∗ , h∗ )? In this case, the pricing strategy is chosen by the managers in charge of marketing and/or the policy makers in charge of pricing policy. These are scenarios where implementing an efficient allocation requires a “visible hand” as pricing decisions are made by managers and/or policy makers. Such pricing decisions require the use of information and managerial skills that are typically not costless. This implies that, under non-convexity and nonlinear pricing, transaction costs would be relevant in the evaluation of pricing schemes supporting efficient allocations. In such situations, transaction costs would play a role in the efficient management of externalities under markets as well as nonmarket institutions. These issues are further discussed in the next section.

Efficient Policies The analysis presented in the previous section provides two approaches to the characterization of Pareto efficiency in the presence of externalities: Proposition 1 identifies efficient production, consumption, and trade; Proposition 2 relies on a dual generalized Lagrangian approach that also evaluates the role of pricing.

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From a policy viewpoint, can we use these two Propositions to recommend whether externalities should be managed through market-based mechanisms (e.g., as opposed to regulations)? The answer to this question is: No. Indeed, Propositions 1 and 2 provide alternative representations of the same efficient allocations. This means that Proposition 1 does not imply that regulation is a preferred solution to an externality problem. Similarly, Proposition 2 does not imply that market-based mechanisms provide superior means of controlling externalities. How can economists assist policy makers in choosing policies supporting efficient allocations in the presence of externalities? Typically, tradeoffs exist between regulatory approaches versus market-based approaches to externality control. If Propositions 1 and 2 do not assist in evaluating these tradeoffs, economists may make recommendations on ideological grounds (e.g., market-loving economists argue in favor of market-based solutions, while government-loving economists argue in favor of regulations). This is not desirable: ideological arguments do not help identify the tradeoffs between alternative policy options. Yet, economists can make constructive contributions to the policy-making process by focusing attention on the role of transaction costs. As first argued by Coase [19], transaction costs play a fundamental role as they affect the limits of organizations and firms and the functioning of markets (e.g., [20–24]). Such arguments also apply to institutions affecting the management and control of externalities [24, 45]. Two issues have the subject of special attention (e.g., [4]). First, when externalities can be reduced through technological innovations, any policy response to externalities needs to provide proper incentives to support innovations. Second, information is typically costly and is an important part of transaction costs. Access to information is crucial in the design and implementation of efficient allocations (e.g., [46, 47]). Without good information, any management or policy would fail to assess the nature and extent of externalities. This is a scenario where poorly informed decision-makers would fail to provide efficient management of externalities. This argument is often presented against government regulations: poorly informed regulators would fail to develop and implement efficient policies dealing with externalities. This is a likely scenario when externalities vary across space and obtaining information about local externalities is difficult (e.g., [48]). But poor information is not specific to regulators. For example, when externalities are global and there are economies of scale in obtaining information about externalities, private agents may be less informed than policy makers about the exact nature of externalities. In general, obtaining information about externalities or new technologies is costly. The cost and utilization of this information typically vary across policy options. Higher information cost would contribute to a downward shift in the Pareto utility frontier, indicating that economic efficiency would improve under institutions that have lower information costs and a better access to information. This argument applies across institutions, including markets, contracts, and regulation. Shleifer [24] has argued that the prevalence of regulations in market economies can be explained in part by the failure of courts and their (in)effectiveness in settling

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contract disputes. This raises the question of identifying the institution(s) that can obtain good information and use it to implement an efficient control of externalities. Our analysis has examined the role of pricing. Figure 2 indicates that externalities do not always generate non-convexity, in which case linear pricing and Pigouvian taxes/subsidies are appropriate. But Fig. 4 shows that externalities can generate nonconvexity that requires nonlinear pricing. More generally, externalities are not the only possible source of non-convexity. Another source of non-convexity involves the presence of fixed cost. It is well known that fixed cost can make competitive markets inefficient (e.g., [12, 14, 28]). For example, under fixed cost and increasing returns to scale, marginal cost pricing is inefficient: marginal cost would be lower than average cost and competitive firms would exit (as marginal cost pricing would generate negative firm profit, revenue being insufficient to cover the fixed cost). Yet, nonlinear pricing can support an efficient allocation (when many consumers buy at marginal cost, but some consumers pay a higher price that allows firms to cover their fixed cost). For this reason, some industries have moved in the direction of using nonlinear pricing (e.g., [44]). An interesting example is the case of block pricing for electricity (e.g., [49–52]). The electricity industry has three important characteristics: (1) electricity-generating power plants have large fixed cost (thus exhibiting non-convexity); (2) the demand for electricity varies over time (e.g., peak demand occurring during heat wave); and (3) fuel-based power plants pollute the air (thus generating negative externalities). One may think about implementing a Pigouvian tax to cover the social cost of the externality. But in this case, a uniform Pigouvian tax is not efficient: a “high uniform price” would be needed to cover the fixed cost of building an additional power plant to supply electricity during peak periods. The inefficiency comes from two sources: the “high” electricity price would have negative impacts on the welfare of most consumers; and when active, the additional power plant would increase pollution. With fixed cost creating nonconvexity, the efficient nonlinear pricing would involve many consumers paying the (lower) marginal social cost of electricity (including the cost of pollution), while some consumers pay higher prices inducing a reduction in their electricity consumption during peak periods. In this case, the higher price paid by some consumers would play two roles: (1) covering the fixed cost of power plant, thus making the pricing scheme sustainable; and (2) reducing the demand for electricity during peak periods. This last effect creates a double dividend: it saves on the cost of building an additional power plant; and it reduces pollution during peak periods. In this case, uniform Pigou taxes would be inefficient. But while nonlinear pricing can be efficient, it also has distributional effects [49]. And to achieve efficiency, nonlinear pricing faces the difficulties of identifying the price-responsive consumers who are going to pay higher prices as well as the nature and timing of the price discrimination scheme [50]. Addressing these difficulties requires more information (compared to uniform pricing) and creates significant challenges for designing and implementing market-based policies that can achieve efficiency. Previous environmental policies can shed useful lights on the relative effectiveness of alternative policy options to externality control. Experiences have varied

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greatly depending on the nature and extent of the externality [53]. For illustration purpose, we consider three examples. First, the depletion of the stratospheric ozone layer and its link with pollution were identified in the 1970s. This generated a strong regulatory response leading to the 1987 Montreal Protocol and the international banning of ozone-depleting chemicals [54, 55]. Evidence indicates that this strong policy response was effective: healing of the Antarctic ozone layer is in progress [56]. Second, acid rain was another environmental issue identified as an effect of sulfur dioxide (SO2 ) emission from US coal-fired power plants. This is a situation where externalities vary across space (as some power plants pollute more than others). The US policy response was a “cap and trade” acid rain program enacted under the 1990 Clean Air Act [57, 58]. The program consisted in the government setting maximum emission allowances for US electric power plants while permitting a market exchange for allowances among plants. The program has been credited as a great success: it drastically reduced SO2 emissions from US coal-fired power plants and it did so at a lower cost than a comparable command-and-control regulation [59]. The US acid rain program illustrates how market-based policies can provide a flexible and cost-effective way of reducing pollution. Our last example involves climate change. Nordhaus [60] identifies climate change and its linkages with greenhouse gases (GHG) emission as the “ultimate challenge” for economics. (Nordhaus received the Nobel Prize in economics in 2018 for his research on climate change.) The production of GHG from human activities (mostly carbon dioxide CO2 generated from burning fossil fuels) contributes to increasing atmospheric temperature on earth with significant long-term effects on global climate [61]. Attempts to reduce GHG emission has led to the 1997 Kyoto Protocol and the 2015 Paris Agreement. The debate has focused on evaluating the social cost of carbon and on considering the imposition of Pigouvian taxes on GHG emission. At this point, there are some disagreements. On the policy side, the United States withdrew from the Paris agreement in 2017. On the economic side, uncertainties related to climate dynamics and economic valuations have made it difficult for economists and policy makers to agree on precise estimates of the social price of carbon [60], thus lessening the political support in favor of a Pigouvian tax on GHG. Finally, there is some concern that a carbon tax may not be an effective way of dealing with GHG externalities (e.g., [62]). Patt and Lilliestam [62] argue that the most effective response to climate change issues will be technological, implying that current policies should be supporting technological innovations that reduce our reliance on fossil fuels. Patt and Lilliestam [62] contend that a carbon tax alone would not be enough to move the world economy away from fossil fuels toward low-carbon technologies. These examples illustrate the complexities of designing and implementing efficient policies dealing with externalities. In general, the nature and magnitude of externalities matter. So does the information available on their effects. In the context of pollution, the presence (or absence) of close substitutes to the pollutants or to the polluting technology plays a role. Finally, policy making always depends on the bargaining ability of different interest groups to deal with each other. This argument

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applies at the local level, national level, and international level (e.g., [53]). This stresses the importance of the political economy of externalities.

Conclusion This chapter has investigated the efficiency of resource allocation in the presence of externalities. The analysis applies to markets as well as nonmarket institutions (including contracts and government). We argue that externalities can lead to nonconvexity and the need for nonlinear pricing. In this context, while simple Pigouvian taxes/subsidies can help, they are inappropriate when uniform pricing is inefficient. We also examine the effects of transaction costs on efficient allocations and coordination/exchange activities. Reducing transaction costs is an integral part of the efficiency of resource allocation. This argument applies to contracts, regulations as well as market-based policies and their relative abilities to control externalities. The analysis provides useful insights into the efficiency of alternative governance structures dealing with externalities.

References 1. Baumol WJ, Oates WE (1988) The theory of environmental policy, 2nd edn. Cambridge University Press, Cambridge 2. Laffont JJ (2008) Externalities. In: Durlauf S, Blume LE (eds) The new Palgrave dictionary of economics, 2nd edn. Palgrave Macmillan, London 3. Baumol WJ (1972) On taxation and the control of externalities. Am Econ Rev 62:307–322 4. Cornes R, Sandler T (1996) The theory of externalities, public goods and club goods. Cambridge University Press, Cambridge 5. Pigou AC (1920) The economics of welfare. Macmillan, London 6. Coase RH (1960) The problem of social cost. J Law Econ 3:1–44 7. Debreu G (1959) Theory of value. Wiley, New York 8. Arrow K (1969) The organization of economic activity: issues pertinent to the choice of market versus non-market allocation. In: The analysis and evaluation of public expenditures: the PPB system. Government Printing Office, Washington, DC 9. Starrett D (1972) Fundamental non-convexities in the theory of externalities. J Econ Theory 4:180–199 10. Baumol WJ, Bradford DF (1972) Detrimental externalities and non-convexity of the production set. Economica 39:160–176 11. Dasgupta P, Maler KG (2003) The economics of non-convex ecosystems: introduction. Environ Resour Econ 26:499–525 12. Brown DJ (1991) Equilibrium analysis with non-convex technologies. In: Hildenbrand W, Sonnenschein H (eds) Handbook of mathematical economics, vol 4. Elsevier, Amsterdam 13. Chavas JP, Briec W (2012) On efficiency under non-convexity. Econom Theor 50:671–701 14. Guesnerie R (1975) Pareto optimality in non-convex economies. Econometrica 43:1–29 15. Khan MA (1999) The Mordukhovich normal cone and the foundations of welfare economic. J Public Econom Theor 1:309–338 16. Khan MA, Vohra R (1987) An extension of the second welfare theorem to economies with nonconvexities and public goods. Q J Econ 102:223–242 17. Mordukhovich BS (2005) Nonlinear prices in nonconvex economies with classical pareto and strong Pareto optimal allocations. Positivity 9:541–568

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52. Joskow PL, Wolfram CD (2012) Dynamic pricing of electricity. Am Econ Rev 102(3):381–385 53. Sandler T (2017) Environmental cooperation: contrasting international environmental agreements. Oxford Econ Pap 69(2):345–364 54. Haas PM (1991) Policy responses to stratospheric ozone depletion. Glob Atmos Change 1(3):224–234 55. Morrisette PM (1989) The evolution of policy responses to stratospheric ozone depletion. Nat Resour J 29(3):794–820 56. Solomon S, Ivy DJ, Kinnison D, Mills MJ, Neely RR III, Schmidt A (2016) Emergence of healing in the Antarctic ozone layer. Science 353(6296):269–274 57. Joskow PL, Schmalensee R (1998) The political economy of market-based environmental policy: the U.S. acid rain program. J Law Econ 41(1):37–84 58. Stavins RN (1998) What can we learn from the grand policy experiment? Lessons from SO2 allowance trading. J Econom Perspect 12(3):69–88 59. Chan HR, Chupp BA, Cropper ML, Muller NZ (2018) The impact of trading on the costs and benefits of the acid rain program. J Environ Econ Manag 88:180–209 60. Nordhaus W (2019) Climate change: the ultimate challenge for economics. Am Econ Rev 109(6):1991–2014 61. Velders GJM, Andersen SO, Daniel JS, Fahey DW, McFarland M (2007) The importance of the Montreal protocol in protecting climate. Proc Natl Acad Sci 104(12):4814–4819 62. Patt A, Lilliestam J (2018) The case against carbon prices. Joule 2:2494–2498

Shadow Pricing in Production Economics

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Rolf Färe, Shawna Grosskopf, and Dimitris Margaritis

Contents Introduction: What Is a Shadow Price? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Primal Representation of Technology: Distance Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . Calculus and Dual Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Pricing Inputs and Outputs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . One Input Price Is Known . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Total Cost Is Known . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Pricing Inputs with a Single-Output Technology and CRS . . . . . . . . . . . . . . . . . . . . . . . . . . Pricing Outputs and Their Characteristics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Pricing Outputs When One Output Price Is Known . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Total Revenue Is Known . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Cost and Revenue Indirect Pricing Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Cost Indirect Pricing Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Revenue Indirect Pricing Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Sub-cost and Sub-revenue Indirect Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Pricing Inputs and Outputs: A Profit Maximization Approach . . . . . . . . . . . . . . . . . . . . . . . . .

953 954 958 960 960 963 966 967 967 970 972 972 977 981 983

R. Färe () Department of Economics and Department of Agricultural and Resource Economics, Oregon State University, Corvallis, OR, USA Department of Economics and Department of Applied Economics, School of Public Policy, Oregon State University, Corvallis, OR, USA Department of Agricultural Economics, University of Maryland, College Park, MD, USA e-mail: [email protected] S. Grosskopf Department of Economics, School of Public Policy, Oregon State University, Corvallis, OR, USA e-mail: [email protected] D. Margaritis Department of Accounting and Finance, University of Auckland Business School, Auckland, New Zealand e-mail: [email protected] © Springer Nature Singapore Pte Ltd. 2022 S. C. Ray et al. (eds.), Handbook of Production Economics, https://doi.org/10.1007/978-981-10-3455-8_16

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Appendix A: Catalog of Shadow Pricing Rules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Input Pricing Rules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Output Pricing Rules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Pricing Inputs, Indirect Approaches . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Pricing Outputs, Indirect Approaches . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Price and Quantity Mixed, Indirect Approaches . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Pricing Under CRS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Pricing Inputs and Outputs When Total Profit  and (x, y) Are Known . . . . . . . . . . . . . . Appendix B: Functional Forms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . More Formal Exposition of Calculus and Primal and Dual Spaces . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Abstract

This chapter is devoted to outlining production theoretical approaches to identifying shadow prices. Shadow prices have a long history in economics; they are perhaps most familiar from welfare economics and benefit-cost analysis. The focus here is narrower – shadow prices derived in a production theoretical framework. We seek to identify prices that are consistent with resource use or opportunity cost that would induce the decision-maker to choose the observed quantity vector. Market prices – if they exist – may be consistent with this condition, but there are many cases where they are not, often referred to as cases of market failure. We begin with the function representations of technology used to identify shadow prices, i.e., distance functions and their dual associated value functions. We emphasize the relationship between primal and dual spaces through their connection with calculus. Perhaps the most familiar example of our approach is Shephard’s lemma, which uses calculus and duality to find the optimal input quantities associated with the cost function, i.e., going from price space to quantity space. Here we adopt a dual Shephard’s lemma approach: begin with quantity space representations of technology, and use duality and calculus to find the associated dual support prices. These include shadow pricing of inputs or their characteristics, shadow prices of outputs, as well as pricing of inputs and outputs in a profit function and directional distance function framework. We include an appendix devoted to the choice of appropriate functional forms which accommodate the underlying structure of technology and the calculus. A more detailed theoretical development of the role of calculus in our approach to shadow pricing suggested to us by Robert Chambers is also included in an appendix. Keywords

Shadow pricing · Production economics · Shephard distance functions · Directional distance functions · Cost pricing · Revenue pricing

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Introduction: What Is a Shadow Price? “Let x be a given vector of quantities. Then p is a shadow price for vector x if x is an optimal choice given p” (Daniel Primont, private communication, August 29, 2016). Building on [1], this chapter is devoted to outlining production theoretical approaches to identifying shadow prices. Shadow prices have a long history in economics; they are perhaps most familiar from welfare economics and benefit-cost analysis (see, e.g., [2]). Although the focus here is narrower – shadow prices derived in a production theoretical framework (see [3] for a consumer-based approach) – the definition is motivated by basic welfare economics notions. In simple undergraduate textbook terms, we seek to identify prices that are consistent with resource use or opportunity cost that would induce the decision-maker to choose the observed quantity vector. Market prices – if they exist – may be consistent with this condition, but there are many cases where they are not, often referred to as cases of market failure. Among other examples are the cases of public goods, positive and negative externalities, and price distortions due to regulation. For example, production decisions may be made with respect to shadow prices rather than observed market prices. In this sense the wedge between these two prices reflects the effects of exogenous constraints faced by the producer; see [4–8], and [9] among many others. We begin with some preliminaries to focus on the production setting. The starting point includes the basic function representations of technology which we use to identify shadow prices, i.e., various distance functions and their dual associated value functions. We emphasize the relationship between primal and dual spaces through their connection with calculus. Perhaps the most familiar example which exploits the relationship between duality and calculus is Shephard’s lemma. After the preliminaries we turn to various cases of shadow pricing beginning with pricing inputs or input characteristics. Shadow prices of outputs are next followed by pricing of inputs and outputs in a profit function and directional distance function framework. Our basic empirical focus here is on parametric specifications of technology to facilitate the use of calculus to identify the shadow prices. We include an appendix devoted to the choice of appropriate functional forms which accommodate the underlying structure of technology.1 Also included in an appendix is a more detailed theoretical development of the role of calculus in our approach to shadow pricing suggested to us by Robert Chambers.We summarize all of our pricing models in yet another appendix.

1 Although

not included in this chapter, nonparametric estimation is possible. For how to apply calculus to estimation with, for example, data envelopment analysis, see [10].

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Primal Representation of Technology: Distance Functions In this chapter we derive shadow pricing rules using nine different distance functions as our representations of technology.2 They are function representations of technology that accommodate multiple input quantities, as do production functions. In contrast to production functions, they accommodate multiple output quantities as well. The distance functions we employ may be classified into two groups: those that are defined as radial scalings (like Shephard-type distance functions) and the second group dubbed directional distance functions. In order to provide some intuition and perhaps clarification, we begin with the underlying technology sets and the properties required to ensure existence of our distance functions and the resulting properties inherited by the distance functions. We show how these functions are defined and construct a chart summarizing these distance functions and relationships. The basic technology sets underlying our distance functions include the input set consisting of all input vectors x = (x1 , . . . , xN ) ∈ N + L(y) = {x ∈ N : x can produce y ∈ M },

(1)

and the output set of all output vectors y = (y1 , . . . , yM ) ∈ M + P (x) = {y ∈ M : x ∈ L(y)}.

(2)

We assume free disposability of inputs and outputs (to ensure the existence of our distance functions) and that the technology set may be defined as T = {(x, y) : x ∈ L(y)}

(3)

and is closed and strictly convex. This ensures that our functions are differentiable and globally optimal. Note that these three sets represent the same technology, hence (x, y) ∈ T ⇔ x ∈ L(y) ⇔ y ∈ P (x). Before formally introducing the various distance functions and their relationships, we first summarize key properties of the two types of distance functions that they inherit from the technology and their definitions. We illustrate for the input side; the output side is similarly derived. The standard radial input distance function is defined as Di (y, x) = sup{λ > 0 : x/λ ∈ L(y)} and the directional input distance function is

2 The

sub-cost and sub-revenue distance functions are not included in this count.

(4)

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 i (x, y; gx ) = sup{β ∈  : x − βgx ∈ L(y)}, D

(5)

where gx is a direction vector. Given the conditions above, both of these are function representations of the technology so that >  Di (y, x) > = 1 ⇔ x ∈ L(y) ⇔ Di (x, y; gx ) = 0.

(6)

In terms of our shadow prices, the most important properties of these two types of distance functions are (i) positive homogeneity of Di (y, x) in x which ensures that Euler’s theorem holds ∇x Di (y, x)x = Di (y, x)

(7)

 i (x, y; gx ) and (ii) the translation property for D  i (x + αgx , y; gx ) = D  i (x, y; gx ) + α. D

(8)

Parallel to (7), the inner product of the gradient of the directional distance function and the direction vector is one3 which implies that  i (x + αgx , y; gx ) = ∇x D  i (x, y; gx ). ∇x D

(9)

We now can define our distance functions – providing a function representation of the underlying sets defined above, which we can ultimately estimate. See the appendix on functional forms for details on parameterization of these functions. Beginning with input-based radial and directional distance functions, we have Di (y, x) = sup{θ : x/θ ∈ L(y)}  i (x, y; gx ) = sup{β : (x − βgx ) ∈ L(y)} D I Di (p/r, x) = sup{λ : x/λ ∈ L(y), py > = r}  i (p/r, x; gx ) = sup{β : (x − βgx ) ∈ L(y), py > r}, ID =

(10)

 i (p/r, x; gx ) denote revenue indirect input distance where I Di (p/r, x) and ID functions, both Shephard-type and directional distance functions, respectively. These functions are appropriate for entities operating with revenue targets py. On the output side we have Do (x, y) = inf{θ : y/θ ∈ P (x)}  o (x, y; gy ) = sup{β : (y + βgy ) ∈ P (x)} D

3 This

is readily shown by differentiating (8) with respect to α and then setting α equal to zero.

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I Do (w/c, y) = inf{θ : y/θ ∈ P (x), wx < = c}  o (w/c, y; gy ) = sup{β : (y + βgy ) ∈ P (x), wx < c}, ID =

(11)

 o (w/c, y; gy ) denote cost indirect output distance where I Do (w/c, y) and ID functions which are appropriate for entities operating under a budget constraint, typical of the public sector. We also have the directional distance function defined on the technology set:  T (x, y; gx , gy ) = sup{β : (x − βgx , y + βgy ) ∈ T }. D

(12)

The relationships among these distance functions are depicted in Fig. 1.4 The numbers in parentheses refer to the more detailed explanations of the relationships below the schematic. Input Distance Functions (Left-Hand Side of Fig. 1)  i (x, y; gx ), then (see [11]) (1) If we set gx = x in D  i (x, y; gx ) = 1 − 1/Di (y, x). D (2) I Di (p/r, x) and Di (y, x) are dual to each other (see [12]): Di (y, x) = inf {I Di (p/r, x) : py > = r} p/r

IDi (p/r, x)

(2)

(4)  i (p/r, x; gx ) ID

Di (y, x)

(9)

 o (x, y; gy ) D

 i (x, y; gx ) D

(10)

(11)  T (x, y; gx , gy ) D

Fig. 1 Distance functions

4 Many

(6)

(5)

(1) (3)

Do (x, y)

thanks to Maryam Hasannasab for the figure.

IDo (w/c, y) (8)

(7)

 o (w/c, y; gy ) ID

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I D i (p/r, x) = sup{Di (y, x) : py > = r} y

 i (x, y; gx ) are dual to each other (see [13]):  i (p/r, x; gx ) and D (3) ID  i (x, y; gx ) = sup{ID  i (p/r, x; gx ) : py > r} D = p/r

 i (x, y; gx ) : py > r}  i (p/r, x; gx ) = sup{D ID = y

(4) To our knowledge this relationship is not available in the literature, but we derive it here. Let gx = x; then  i (p/r, x; gx ) = sup{β : (x − βx) ∈ L(y), py > r} ID = > = sup{β : Di (y, x(1 − β)) = 1, py > = r} > sup{β : Di (y, x)(1 − β)) > = 1, py = r} > = 1 + sup{1 − β) : (1 − β) > = 1/Di (y, x), py = r} 1 =1+ I Di (p/r, x) Output Distance Functions (Right-Hand Side of Fig. 1) (5) As in (1), these two distance functions are dual to each other. Set gy = y; then (see [11])  o (x, y; gy ) = 1 − 1/Do (x, y). D (6) These functions are dual to each other (see [12]): Do (x, y) = inf {I D o (w/c, y) : wx < = c} w/c I D o (w/c, y) = sup{Do (x, y) : wx < = c}. y

(7) These functions are dual to each other (see [13]):  o (x, y; gy ) = sup{D  o (x, y; gy ) : wx < c} D = w/c

 o (x, y; gy ) : wx < c}  o (w/c, y; gy ) = sup{D ID = y

(8) Set gy = y (the proof is left to the reader); then  o (w/c, y; gy ) = 1 − 1/I Do (y, x) ID

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(9) Under CRS we have Di (y, x) = 1/Do (x, y). (10) Set gy = 0; then  T (x, y; gx , 0).  i (x, y; gx ) = D D (11) Set gx = 0; then  T (x, y; 0, gy ).  o (x, y; gy ) = D D

Calculus and Dual Spaces The purpose of this section is to provide some basic intuition with respect to the role of calculus in deriving shadow prices. In particular we highlight the role of the derivative of a function – which takes us from the primal space to the dual space on which the function is defined.5 An important example from economics is Shephard’s lemma [18], which we use to illustrate this concept. We provide a sketch of this lemma and begin by recalling the definition of the input requirement set L(y) as L(y) = {x : x can produce y}

(13)

M where x ∈ N + is an input vector and y ∈ + an output vector. The cost function is defined as

C(y, w) = min wx s.t. x ∈ L(y), x

(14)

where w ∈ N + denotes input prices. Shephard’s lemma states that the partial derivative of the cost function with respect to any input price wn yields the associated cost minimizing input quantity xn , i.e., ∂C(y, w)/∂wn = xn , n = 1, . . . , N,

5 For

(15)

a more formal treatment which requires more advanced calculus concepts than those we use here, we refer the reader to the appendix, which was included with collaboration of Robert Chambers.

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or even more compactly this can be written in terms of the gradient vector of the cost function as w C(y, w) = x.

(16)

The gradient vector can be written in more detail as  w C(y, w1 , . . . , wN ) =

∂C ∂C ,..., ∂w1 , ∂wN

 .

(17)

Thus the derivative taken with respect to a price wn yields a quantity xn , where the two variables are dual to each other. We state this idea in more general terms. Let l : 2 →  be a function with domain 2 and range . This function is linear if q, q o , q 1 ∈ 2 l(q o + q 1 ) = l(q o ) + l(q 1 )

(18)

and since this is Cauchy’s first equation, under continuity it implies l(aq) = al(q), a ∈ .

(19)

One can prove that a function is linear if and only if (see [14] and [15]) there exist real numbers b1 , b2 such that l(q) = b1 q1 + b2 q2 .

(20)

The dual space to 2 , (2 )∗ consists of all continuous functionals (f1 , f2 , f3 ) given by (f1 + f2 )q = f1 (q) + f2 (q)

(21)

(af )q = a(f (q)).

(22)

2 = (2 )∗ ,

(23)

and

The following holds

i.e., if 2 is the quantity space and (2 )∗ is the price space, they coincide.6 We note that in economics, where we think of quantities being in primal space and prices in

6 This

is true here, but in general it is not. See, e.g., [16].

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dual space, we consider them as different. We typically can “consume” quantities, but not prices. To relate the dual space to calculus, let g :  → .

(24)

This function is differentiable if there exists an l, an element of  = ∗ that is a linear function () such that lim

h→0

g(q + h) − g(q) − l(h) = 0. h

(25)

By homogeneity of l, we have lim

h→0

g(q + h) − g(q) − l(1) = 0. h

(26)

and hence dg(q) = l(1) ∈ ()∗ . dq

(27)

dg(q)/dq ∈ ()∗ ,

(28)

Therefore

belongs to the dual space of q. If g : N → , then the gradient vector is  q g(q1 , . . . , qN ) =

∂g ∂g ,..., ∂q1 ∂qN



∈ (N )∗ .

(29)

In this chapter we use these concepts to derive shadow price models. Specifically, we model technologies in terms of our previously defined distance functions, which are typically functions we define in quantity space and apply calculus to derive the corresponding dual space shadow prices.

Pricing Inputs and Outputs One Input Price Is Known In this section we assume that we observe one input price, as well as input quantities M x ∈ N + and output quantities y ∈ + . Let the one known input price be w1 ; our goal is to show that we can retrieve shadow prices for the remaining (unknown)

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input prices (w2 , . . . , wN ) using our known data w1 , x, and y. Recall that Shephard input distance function is defined as Di (y, x) = sup{θ : (x/θ ) ∈ L(y)}

(30)

L(y) = {x : x can produce y}, y ∈ M +

(31)

where

is the input requirement set. For our purpose, two properties of the distance function introduced earlier in (6) and (7) are key, namely, (i) x ∈ L(y) if and only if Di (y, x) > = 1, and (ii) Di (y, θ x) = θ Di (y, x), θ > 0. We refer to (i) as the representation property and (ii) as the homogeneity of degree +1 in inputs. The homogeneity property follows from the definition of the distance function, while the representation property holds if inputs are weakly disposable, i.e., x ∈ L(y), θ > = 1 ⇔ θ x ∈ L(y).

(32)

If we denote input prices by w = (w1 , . . . , wN ) ∈ N + , then the cost function is given by C(y, w) = min wx s.t. x ∈ L(y) x

(33)

which due to the representation property can be written as C(y, w) = min{wx : Di (y, x) > = 1}, x

(34)

or if written as a Lagrangian problem C(y, w) = min wx − μ(Di (y, x) − 1). x

(35)

The first-order conditions associated with this problem are w − μx Di (y, x) = 0.

(36)

Using our first-order conditions, we can start to identify the unknown or shadow price w2 from

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w2 ∂Di (y, x)/∂x2 = w1 ∂Di (y, x)/∂x1

(37)

which we can solve for w2 as w2 = w1

∂Di (y, x)/∂x2 , ∂Di (y, x)/∂x1

(38)

where the right-hand side consists of known data w1 , x, y. Recall that our preferred parameterization of Shephard input distance functions is the translog function, which implies that no zeros are allowed in the data. To allow for zeros, one may specify a quadratic function, which is our preferred parameterization for the directional distance function. The directional input distance function is defined as  i (x, y; gx ) = sup{β : x − βgx ∈ L(y)} D

(39)

where gx ∈ N + , gx = 0 is the directional vector which determines in which direction the input vector x is contracted toward the boundary of L(y). This distance function satisfies the representation property  i (x, y; gx ) > 0 if and only if x ∈ L(y) D =

(40)

as well as the translation property  i (x − αgx , y; gx ) = D  i (x, y; gx ) − α, α ∈ . D

(41)

The translation property (which is an additive analog of the multiplicative homogeneity property associated with Shephard distance functions) follows from the definition of this distance function. The representation property holds when the technology satisfies g-disposability (note technology is g−disposable if x ∈ L(y); then x ∈ L(y) where x = x + λgx , λ > = 0).  Since Di (x, y; gx ) satisfies the representation property, we can rewrite our cost minimization definition as  i (x, y; gx ) > 0}, C(y, w) = min{wx : D = x

(42)

or as a Lagrangian problem  i (x, y; gx ). C(y, w) = min wx − μD x

(43)

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The associated first-order conditions7  i (x, y; gx ) = 0 w − μx D

(44)

may be used as the basis for our next pricing rule for unknown input price w2 w2 = w1

 i (x, y; gx )/∂x2 ∂D ,  i (x, y; gx )/∂x1 ∂D

(45)

where again we are assuming that w1 , x, y, gx , are known. It is important to note that the resulting value of w2 depends on the choice of the direction vector gx , i.e., it may change as gx changes.

Total Cost Is Known In this section we assume that we observe total cost c = wx =

N 

wn xn ;

(46)

n=1

however, we do not observe individual input prices wn . In this case we can derive the following input pricing rule w=c

x Di (y, x) . Di (y, x)

(47)

To verify this rule, recall our first-order conditions for cost minimization with the Shephard input distance function w = μx Di (y, x).

(48)

We start by deriving a useful interpretation of the multiplier μ. Multiply both sides of the first-order condition by x, wx = μx Di (y, x)x,

(49)

and by the homogeneity property of the distance function in the input vector (see (7)), we have

 i (x, y; gx ) and gx is equal to one by the that the inner product of the gradient vector of D translation property.

7 Note

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c = μDi (y, x)

(50)

μ = c/Di (y, x) μ = C(y, p). Inserting this into our first-order conditions results in w = C(y, w)x Di (y, x).

(51)

Multiplying both sides by x and applying Euler’s theorem to the distance function; then c = wx = C(y, w)Di (y, x)

(52)

C(y, w) = c/Di (y, x).

(53)

which implies

We can now use the right-hand side to substitute for the unobserved C(y, w) to yield our pricing rule: w = C(y, w)x Di (y, x) =c

x Di (y, x) . Di (y, x)

(54)

Thus given observed data on c, y, x, and Shephard input distance function, the input price vector w can be estimated. To relate the above pricing rule to earlier literature, assume that inputs are technically efficient in the sense of Farrell, i.e., Di (y, x) = 1.

(55)

Also assume that we seek to find cost-deflated input prices: wˆ = w/c.

(56)

In this case the pricing rule becomes wˆ = x Di (y, x),

(57)

and this expression for a single output can be found in [17] as well as in [18]. Again, if there are zeros in the data, the translog function associated with the Shephard input distance functions may not be suitable. The directional distance

23 Shadow Pricing in Production Economics

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 i (x, y; gx ) is an alternative. Rewriting the first-order condition for cost function D minimization  i (x, y; gx ) = 0 w − μx D

(58)

we can derive the input pricing rule as w=c

 i (x, y; gx ) x D .  i (x, y; gx )x x D

(59)

To verify this rule, we again need to interpret the Lagrangian multiplier under the directional distance function constraint. We set up a perturbation of the cost minimization problem as8  i (x, y; gx ) − α)} ˆ C(y, w, α) = min{wx − μ(D x

(60)

 i (x, = wαgx + min{xw ˆ xˆ − μD ˆ y; gx )} x

by the translation property, where xˆ = x − αgx . Thus ˆ C(y, w, α) = αwgx + C(y, w).

(61)

Differentiating with respect to α ˆ ∂ C(y, w, α)/∂α = μ = wgx .

(62)

So in the directional distance function case, the multiplier may be interpreted as the value of the direction vector gx . Inserting this expression into the first-order conditions yields  i (x, y; gx ). w = wgx x D

(63)

Multiplying both sides by x and noting that c = wx, we have  i (x, y; gx )x c = wgx x D

(64)

= wgx multiplying both sides of (58) by gx noting that the inner product of the gradient of the directional distance function and the direction vector is equal to one, and then use this result to obtain (63) directly.

8 Alternatively, we could obtain μ

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or wgx =

c  i (x, y; gx )x x D

(65)

 i (x, y; gx ) from above to substitute for wgx , From this and using w = wgx x D we derive the pricing rule

w=c

 i (x, y; gx ) x D .  i (x, y; gx )x x D

(66)

This pricing rule is in terms of the directional input distance function and therefore will depend on the choice of the directional vector gx . Note that in contrast to the pricing rule for the radial input distance function given by (54), here we have in the denominator of (66) the inner product of the gradient of the distance function times the input vector rather than simply the input distance function since we cannot apply Euler’s theorem.

Pricing Inputs with a Single-Output Technology and CRS Here we relate our Shephard input distance function model to those of [19] and [20] by assuming that only one output y ∈ + is produced (e.g., house value). In addition we assume that technology exhibits CRS (constant returns to scale), i.e., L(λy) = λL(y), λ > 0 or equivalently Di (λy, x) = (1/λ)Di (y, x), λ > 0.

(67)

In this case the distance function is also independent of output, i.e., w=c

x Di (1, x) . Di (1, x)

(68)

Since CRS and single output yield a distance function Di (y, x) = 1/yDi (1, x), y > 0,

(69)

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when the distance function takes a value of one, Di (y, x) = 1, which signals technical efficiency, we have y = Di (1, x)

(70)

y = F (x)

(71)

or

where F (x) is a production function. In their work on pricing, [19] takes F (x) to be a CES function, while [20] takes F (x) to be a Cobb-Douglas function.

Pricing Outputs and Their Characteristics Pricing Outputs When One Output Price Is Known In this section we derive shadow prices for outputs when only one output price is known: we seek to derive shadow prices for the outputs which do not have observable output prices that reflect resource use or opportunity cost. This follows the outline of the previous treatment of shadow pricing inputs and their characteristics. We began the treatment of input shadow prices with the representation of technology using Shephard input distance function. Here we turn to Shephard’s output distance function as representation of technology of a multiple output, multiple input technology. Recall that this function is defined on the output set P (x) = {y : x can produce y}

(72)

Do (x, y) = inf{θ : y/θ ∈ P (x)}.

(73)

as

We exploit two of the properties of the output distance function to identify our shadow prices, namely, (i) y ∈ P (x) if and only if Do (x, y) < =1 and (ii) Do (x, λy) = λDo (x, y), λ > 0.

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The first condition is our representation property, which establishes the distance function as a complete characterization of the underlying output set, with the provision that outputs are weakly disposable, i.e., < y ∈ P (x), 0 < = θ = 1 ⇔ θy ∈ P (x).

(74)

The second condition – homogeneity in outputs – follows from the definition of Do (x, y) and is important for determining its parameterization. The value function associated with the output distance function is the revenue N function which depends on the vector of output prices p ∈ M + and inputs x ∈ + and is the result of maximizing revenues py given technology R(x, p) = max{py : y ∈ P (x)}

(75)

R(x, p) = max{py : Do (x, y) < = 1}

(76)

y

or using (i) above y

The Lagrangian formulation of this problem is R(x, p) = max py − μ(Do (x, y) − 1), y

(77)

where μ is the Lagrangian multiplier. The first-order conditions are p − μy Do (x, y) = 0.

(78)

Suppose the price of y1 is known to equal its observed market price p1 ; then we can solve for the shadow price of say output y2 using our first-order conditions p2 ∂Do (x, y)/∂y2 = p1 ∂Do (x, y)/∂y1

(79)

and p2 = p1

∂Do (x, y)/∂y2 ∂Do (x, y)/∂y1

(80)

where the right-hand side consists of observed data x, y, p1 . This model applies to any pm , m = 2, . . . , M and of course an estimate of the distance function. Translog is our preferred parameterization of the output distance function. However, if there are zeros in the data, our preferred alternative is a quadratic form which allows for zeros and is consistent with the properties of the directional output distance function, to which we turn next.

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The directional output distance function is defined as  o (x, y; gy ) = sup{β : (y + βgy ) ∈ P (x)}, D

(81)

where gy ∈ M + , gy = 0 is the directional output vector which specifies the direction in which the output vector is projected onto the boundary of the output set P (x). It  o (x, y; gy ) is a function of gy , since the choice of this is important to note that D vector affects the resulting output price. This distance function, like the directional input distance function, satisfies representation  o (x, y; gy ) > 0 if and only if y ∈ P (x) D =

(82)

and the translation property  o (x, y + αgy y; gy ) = D  o (x, y; gy ) − α. D

(83)

The translation property follows from the definition of the distance function, and representation holds if the technology P (x) is gy disposable. This property  o (x, y; gy ), as is discussed in a later section. is important for parameterizing D The revenue function R(x, p) is also the value function associated with the directional output distance function. Given output prices p ∈ M + , the revenue maximization problem is R(x, p) = max{py : y ∈ P (x)}

(84)

 o (x, y; gy ) > 0.} R(x, p) = max{py : D =

(85)

y

or

y

In Lagrangian form  o (x, y; gy ), R(x, p) = max py − μD y

(86)

where, as before, μ is the Lagrangian multiplier. The first-order conditions for this problem are  o (x, y; gy ) = 0. p − μy D

(87)

Suppose again that one output price is known, say p1 . Then we may derive the shadow prices of outputs m = 2, . . . , M from the first-order conditions as

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p m = p1

 o (x, y; gy )/∂ym ∂D ,  o (x, y; gy )/∂y1 ∂D

(88)

where the right-hand side consists of the data we have assumed are observable – (p1 , y, x) and the chosen direction vector gy . If one price is known, using our pricing rules, we can also derive the remaining prices above as well as total revenue, since M  ∂Do (x, y)/∂ym r = p1 y1 + ym ∂Do (x, y)/∂y1

(89)

m=2

or in terms of the directional distance function r = p1 y1 +

M   o (x, y; gy )/∂ym ∂D y  o (x, y; gy )/∂y1 m ∂D

(90)

m=2

In the next section, we assume that revenue r but no pm is known.

Total Revenue Is Known We assume that total revenue r = py =

M 

pm ym

(91)

m=1

is known, but none of the output prices pm , m = 1, . . . , M are known. This leads to two output pricing rules: one in terms of the Shephard output distance function and another in terms of the directional output distance function, i.e., y Do (x, y) Do (x, y)

(92)

 o (x, y; gy ) y D ,  o (x, y; gy )y y D

(93)

p=r and p=r

respectively. Recall that the first-order conditions associated with the revenue maximization problem with Shephard’s output distance function are p − μy Do (x, y) = 0

(94)

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or p = μy Do (x, y).

(95)

In order to derive a pricing rule using these conditions, we need to interpret the multiplier μ. We will prove that μ = R(x, p),

(96)

i.e., the multiplier equals the value function, maximal revenue R(x, p). As we did for the input distance function case, we begin by multiplying both sides of the firstorder condition above by y. Summing and accounting for Euler’s theorem since the output distance function is homogeneous of degree 1 in y, we have py − μDo (x, y).

(97)

Rearranging yields μ=

py Do (x, y)

= R(x, p)

(98)

as desired. Inserting this into the first-order condition p = R(x, p)y Do (x, y).

(99)

Multiply both sides by y and apply Euler’s theorem to obtain r = py = R(x, p)Do (x, y)

(100)

R(x, p) = r/Do (x, y).

(101)

p = R(x, p)y Do (x, y)

(102)

or

Inserting this into

yields our pricing rule p=r

y Do (x, y) , Do (x, y)

(103)

where as before the right-hand side consists of observed data r, x, y, as well as estimates of the directional distance function.

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Cost and Revenue Indirect Pricing Models In the earlier sections of this chapter, we used Shephard and directional distance functions to provide us with a function representation of technology based on input and output quantities. Here we generalize that approach to the case in which the decision-maker faces a budget constraint and revenue target, which were termed indirect models by Shephard; see [21]. He recognized that these models were especially useful for the case of service and public sectors. Following Shephard’s terminology, the representation of technology in this section is indirect distance functions.

Cost Indirect Pricing Models The pricing models discussed earlier used the classic Shephard and directional distance functions as their representation of technology, defined in terms of input and output quantity vectors. Here we define indirect distance functions which are defined in terms of normalized input prices and output quantities. Thus this pricing model makes use of prices, much like the classic competitive model that sets the prices of outputs equal to marginal cost. We begin with the traditional output set P (x) = {y : x can produce y}, x ∈ N +.

(104)

Input prices are w ∈ N + and let observed cost be denoted as c. The associated budget constraint is c = wx =

N 

wn xn .

(105)

n=1

We modify the traditional output set by introducing the budget constraint and defining the cost indirect output set as the union of all output sets which satisfy the budget constraint, i.e., all those output sets with input vectors which cost no more than the given c I P (w/c) = {y : y ∈ P (x), wx < = c} = {y : y ∈ P (x), w/c < = 1}.

(106)

We can now define a radial (Shephard-type) distance function on the cost indirect output set I P (w/c) as I Do (w/c, y) = inf{θ : y/θ ∈ I P (w/c)}.

(107)

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As in the direct distance function case, this function is homogeneous of degree +1 in the scaled output vector I Do (w/c, λy) = λI Do (w/c, y), λ > 0.

(108)

It also satisfies the representation property I Do (w/c, y) < = 1, if and only if y ∈ I P (w/c)

(109)

I P (w/c) = {y : I Do (w/c, y) < = 1}.

(110)

or

If outputs are strongly disposable, i.e.,

y ∈ I P (w/c), y < = y ⇔ y ∈ I P (w/c).

(111)

We are now ready to introduce the cost indirect pricing models, which are derived using revenue maximization. We define the cost indirect revenue function as I R(w/c, p) = max{py : y ∈ I P (w/c)} y

= max{py : I Do (w/c, y) < = 1}, y

(112)

where the second equality follows from the representation property. As usual, p ∈ M + is a non-negative output price vector, and one can prove that the output set I P (w/c) is compact; therefore the maximum exists. The Lagrangian formulation of the maximization problem is I R(w/c, p) = max py − μ(I Do (w/c, y) − 1), y

(113)

where μ is the Lagrangian multiplier. The first-order conditions are p − μy I Do (w/c, y) = 0.

(114)

Suppose that we know p1 and seek the price of output m, pm , m = 2, . . . , M; then using the first-order conditions, we have ∂I Do (w/c, y)/∂ym pm = , p1 ∂I Do (w/c, y)/∂y1

(115)

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which yields the pricing model p m = p1

∂I Do (w/c, y)/∂ym , m = 2, . . . , M. ∂I Do (w/c, y)/∂y1

(116)

We have shown that if input prices w, total cost c, outputs y and output price p1 are known, we may compute the remaining output prices, pm = 2, . . . , M. We note that normalized input prices are used here. In the case in which technology exhibits constant returns to scale, i.e., L(λy) = λL(y)λ > 0,

(117)

then the cost function is homogeneous of degree +1 in outputs C(λy, w) = λC(y, w), λ > 0.

(118)

In turn, we can then prove that C(y, w/c) = I Do (w/c, y),

(119)

in which case we may substitute the cost function for our cost indirect output distance function. Using this in our maximization problem, we have max(py − C(y, w)) y

(120)

which yields the standard competitive result that output prices equal marginal cost p = y C(y, w).

(121)

Again under CRS, we may use the indirect output distance function in the maximization problem max(py − I Do (w/c, y))

(122)

p = y I Do (w/c, y)

(123)

y

which yields the pricing rule

Next assume that we know total revenue r = py =

M  m=1

pm ym ;

(124)

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then we can derive the following pricing rule using the cost indirect output distance function p=r

y I Do (w/c, y) . I Do (w/c, y)

(125)

Recall that revenue maximization under the constraint I Do (w/c, y) < = 1 yields the first-order constraints p − μI Do (w/c, y) = 0.

(126)

Following our earlier sections, one can prove that μ = I R(w/c, p).

(127)

Inserting this into the first-order conditions yields p = I R(w/c, p)y I Do (w/c, y).

(128)

Next multiply both sides by y and apply Euler’s theorem to arrive at r = py = I R(w/c, p)I Do (w/c, y)

(129)

I R(w/c, p) = r/I Do (w/c, y)

(130)

or

yielding the pricing rule p=r

y I Do (w/c, y) . I Do (w/c, y)

(131)

Thus if we know revenue r but not individual prices, we can derive a pricing rule given w, c and y. Next we consider the indirect functions in terms of directional distance functions. Thus define the cost indirect directional output distance function as  o (w/c, y; gy ) = sup{β : (y + βgy ) ∈ P (x), wx < c}, ID =

(132)

where gy ∈ M + , gy = 0 is the directional vector. This function satisfies the two properties:  o (w/c, y; gy ) − α  o (w/c, y + αgy ; gy ) = ID ID

(133)

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and I Do (w/c, y; gy ) > = 0 if and only if y ∈ I P (w/c),

(134)

the translation and representation properties, respectively. The cost indirect revenue maximization problem is I R(w/c, p) = max{py : y ∈ I P (w/c)} y

(135)

 o (w/c, y; gy ) > 0} = max{py : ID = y

where the last equality holds due to the representation property. The Lagrangian formulation of the problem is  o (w/c, y; gy ), I R(w/c, p) = max py − μID y

(136)

where μ is the multiplier. The first-order conditions associated with this problem are  o (w/c, y; gy ) = 0. p − μy ID

(137)

Now if the price of one output is known, say p1 , the prices of the other outputs follow from p m = p1

 o (w/c, y; gy )/∂ym ∂ ID .  o (w/c, y; gy )/∂y1 ∂ ID

(138)

Given w, c, y and gy together with p1 , the prices pm = 2, . . . , M may be calculated from the expression above. In addition, we can then find total revenue as r = py =

M 

(139)

pm ym

m=1

= p1 y1 +

M  m=2

p1

 o (w/c, y; gy )/∂ym ∂ ID y .  o (w/c, y; gy )∂y1 m ∂ ID

What if we know revenue r but not the individual output prices? Following our earlier models, we can derive a pricing model

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p=r

977

 o (w/c, y; gy ) y ID ,  o (w/c, y; gy )y y ID

(140)

 o (w/c, y; gy ) is a gradient vector. where we note that y ID We leave the derivation to the reader.

Revenue Indirect Pricing Models A parallel set of pricing rules can be developed for our other indirect models – namely, the revenue indirect cases. Here the decision-maker seeks to minimize costs but must also meet a minimal revenue target. Thus the decision-maker must choose input quantities to minimize costs given input prices as well as a revenue target r. More formally the revenue indirect cost minimization problem is I C(p/r, w) = min wx s.t. x ∈ I L(p/r), x

(141)

where the revenue indirect input set is I L(p/r) = {x : x ∈ L(y), py > = r} = {x : x ∈ L(y), py/r > = 1}.

(142)

This set models the condition that feasible input vectors x ∈ L(y) must generate at least total revenue r, given output prices p ∈ M +, r = py =

M 

pm ym .

(143)

m=1

The input distance function defined on I L(p/r) is called the revenue indirect input distance function, and it is defined as I Di (p/r, x) = sup{λ : x/λ ∈ I L(p/r)}.

(144)

This function is homogeneous of degree +1 in inputs: I Di (p/r, θ x) = λI Di (p/r, θ x), θ > 0.

(145)

It also satisfies the representation condition I Di (p/r, x) > = 1 if and only if x ∈ I L(p/r),

(146)

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or I L(p/r) = {x : I Di (p/r, x) > = 1}

(147)

We can now define the cost function defined on I L(p/r) I C(p/r, w) = min{wx : x ∈ I L(p/r)}

(148)

which by the representation property can also be written as I C(p/r, w) = min{wx : I Di (p/r, x) > = 1}.

(149)

The Lagrangian formulation of this cot function is I C(p/r, w) = min wx − μ(I Di (p/r, x) − 1),

(150)

x

with first-order conditions w − μx I Di (p/r, x) = 0.

(151)

Here again if we know one input price, say w1 , the other input prices may be estimated from the above as ∂I Di (p/r, x)/∂xn ∂I Di (p/r, x)/∂x1

(152)

∂I Di (p/r, x)/∂xn , n = 2, . . . , N. ∂I Di (p/r, x)/∂x1

(153)

wn /w1 = or equivalently wn = w1

Using this result we can compute the total cost: c = wx =

N  n=1

wn xn = w1 x1 +

N  ∂I Di (p/r, x)/∂xn n=2

∂I Di (p/r, x)/∂x1

xn .

(154)

If we assume that technology satisfies CRS, we can prove that the revenue function R(x, p/r) = max{p/ry : y ∈ P (x)}, r > 0

(155)

equals the revenue indirect input distance function R(x, p/r) = I Di (p/r, x).

(156)

23 Shadow Pricing in Production Economics

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Thus under CRS, we may substitute the revenue function for I Di (p/r, x) in our pricing models. For example, if we define profit maximization as the least costly way to meet the minimum revenue target min R(x, p) − wx,

(157)

x R(x, p) = w,

(158)

x

we find that

i.e., marginal revenue equals input price. Of course under CRS, we could also consider the problem min I Di (p/r, x) − wx

(159)

x I Di (p/r, x) = w.

(160)

x

which gives us

If we observe total cost c = wx =

N 

wn xn

n=1

but do not know individual input prices, we can develop the following pricing rule w=c

x I Di (p/r, x) . I Di (p/r, x)

(161)

From the first-order conditions w − μx I Di (p/r, x) = 0

(162)

μ = I C(p/r, w),

(163)

w = I C(p/r, w)x I Di (p/r, x).

(164)

and the condition that

we have

Multiplying both sides by x and applying Euler’s theorem yields c = wx = I C(p/r, w)I Di (p/r, x)

(165)

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which gives us the pricing rule for input prices w=c

x I Di (p/r, x) . I Di (p/r, x)

(166)

Our next indirect pricing rule uses the directional indirect input distance function, which is defined as  i (p/r, x; gx ) = sup{β : (x − βgx ) ∈ L(y), py > r}, ID =

(167)

where gx ∈ N + , gx = 0 is the directional vector. As a directional distance function, this satisfies the translation and representation properties:  i (p/r, x − αgx ; gx ) = ID  i (p/r, x; gx ) − α ID

(168)

 i (p/r, x; gx ) > 0 if and only if x ∈ L(y), py > r, ID = =

(169)

and

respectively. The revenue indirect cost minimization problem is I C(p/r, w) = min{wx : x ∈ I L(p/r)} x

(170)

where the revenue indirect input set is defined as I L(p/r) = {x : x ∈ L(y), py > = r}.

(171)

Employing the representation property, we may reformulate the minimization problem as  i (p/r, x; gx ), I C(p/r, w) = min wx − μID x

(172)

where μ is the Lagrangian multiplier. The first-order conditions for this problem are  i (p/r, x; gx ) = 0. w − μx ID

(173)

If one input price, say w1 , is known, the remaining input prices may be calculated as wn = w1

 i (p/r, x; gx )/∂x1 ∂ ID .  i (p/r, x; gx )/∂xn ∂ ID

(174)

23 Shadow Pricing in Production Economics

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This can be used to calculate total cost: c = wx =

N 

wn xn

n=1

= w1 x1 +

N  n=2

w1

 i (p/r, x; gx )/∂x1 ∂ ID x .  i (p/r, x; gx )/∂xn n ∂ ID

(175)

We now assume that we know total cost but that none of the individual input prices are known. Our pricing rule in this case is w=c

 i (p/r, x; gx ) x ID .  i (p/r, x; gx )x x ID

(176)

To derive this expression we have used the fact that the Lagrangian multiplier μ equals the indirect cost function μ = I C(p/r, w).

(177)

Using this together with the first-order conditions yields the pricing rule above. We leave the details to the reader.

Sub-cost and Sub-revenue Indirect Models In this section we generalize the pricing models from sections “One Input Price Is Known” and “Total Cost Is Known” by defining sub-cost and sub-revenue indirect distance functions. This accommodates short- and long-run models as well as models with mixed quality and price data. We use inputs and normalized input price in the indirect sub-cost distance function, similarly for the revenue indirect functions. Beginning with the indirect sub-cost case, let inputs x = (xs , x−s ) ∈ N +,

(178)

where s is a subset of {1, . . . , N} and −s is its complement. A sub-budget cost indirect output set can then be defined as SI P (ws /cs , x−s ) = {y : y ∈ P (x), ws xs < = cs , x = (xs , x−s ), }

(179)

where cs is the sub-budget cost for the n = 1, . . . , s inputs. The radial sub-budget indirect output distance function is SI Do (ws /cs , x−s , y) = inf{θ : y/θ ∈ SI P (ws /cs , x−s )}.

(180)

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The corresponding directional distance function is SID o (ws /cs , x−s , y; gy ) = sup{β : (y + βgy ) ∈ SI P (ws /cs , x−s )}.

(181)

Both functions satisfy a representation property; thus we can define the following revenue maximization problems max py s.t. SI Do (ws /cs , x−s , y) < = 1,

(182)

max py s.t. SID o (ws /cs , x−s , y; gy ) > = 0.

(183)

y

and y

The first problem yields the pricing models pm = p1

∂SI Do (ws /cs , x−s , y)/∂ym , m = 2, . . . , M, ∂SI Do (ws /cs , x−s , y)/∂y1

(184)

when one price, p1 , is known, and p=r

y SI Do (ws /cs , x−s , y) y SI Do (ws /cs , x−s , y)y

(185)

when p1 and r are known. Verification is left to the reader. The second maximization yields the following pricing rules p m = p1

∂ SID o (ws /cs , x−s , y; gy )/∂ym ∂ SID o (ws /cs , x−s , y; gy )/∂y1

(186)

y SID o (ws /cs , x−s , y; gy ) y SID o (ws /cs , x−s , y; gy )y

(187)

and p=r

for the case in which p1 and r are known. Details are left to the reader. Finally, we turn to the sub-budget version of the revenue indirect models, whose reference input requirement set is SI L(ps /rs , y−s ) = {x : x ∈ L(y), ps ys > = rs , y = (ys , y−s )},

(188)

where rs is the sub-revenue and y = (ys , y−s ) with s ⊂ {1, . . . , M} and s− is its complement.

23 Shadow Pricing in Production Economics

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The radial and directional distance functions defined on this set are SI Di (ps /rs , y−s , x) = sup{θ : x/θ ∈ SI L(ps /rs , y−s )}

(189)

and SID i (ps /rs , y−s , x; gx ) = sup{β : (x − βgx ) ∈ SI L(ps /rs , y−s )},

(190)

respectively. Both distance functions satisfy a representation property which allows us to define min wx s.t. SI Di (ps /rs , y−s , x) > = 1,

(191)

min wx s.t. SID i (ps /rs , y−s , x; gx ) > = 0.

(192)

x

and x

The four pricing rules associated with these two minimization problems are wn = w 1 w=c

x SI Di (ps /rs , y−s , x) SI Di (ps /rs , y−s , x)

wn = w1 w=c

∂SI Di (ps /rs , y−s , x)/∂xn , n = 2, . . . , N ∂SI Di (ps /rs , y−s , x)/∂x1

∂ SID i (ps /rs , y−s , x; gx )/xn , n = 2, . . . , N ∂ SID i (ps /rs , y−s , x; gx )/x1

x SID i (ps /rs , y−s , x; gx ) , x SID i (ps /rs , y−s , x; gx )x

(193) (194) (195) (196)

respectively.

Pricing Inputs and Outputs: A Profit Maximization Approach In most of our earlier pricing sections, we have focused on input prices and output prices separately. And in each case, we assume that we know or observe at least one input (output) price or total costs (total revenues), respectively. In this section we show how knowledge of one input price can be used to price outputs and how one output price can be used to price inputs. We also consider using total costs to shadow price outputs and total revenues to shadow price inputs. The key idea is to exploit profit maximization as the optimization criterion, rather than cost minimization or revenue maximization. When the goal is to maximize profit, we are seeking optimal quantities of both inputs and outputs, providing the

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“crossover” from inputs to outputs. In fact this allows us to simultaneously find shadow prices for both inputs and outputs. The traditional profit maximization problem takes the form max(py − wx) s.t. (x, y) ∈ T , x,y

(197)

where T is the technology set, i.e., T = {(x, y) : x can produce y}.

(198)

N As usual, x ∈ N + , w ∈ + are our input vectors and their prices, respectively. M Similarly, we have y ∈ + , p ∈ M + on the output side. Next we formulate the Lagrangian representation of the profit maximization problem, but we directly represent technology as the directional technology distance function, noting that this function satisfies representation and translation:

 T (x, y; gx , gy ) = sup{β : (x − βgx , y + βgy ) ∈ T } D

(199)

In addition we will make use of the radial input and output distance functions.9 In order to justify the use of the Shephard-type radial distance functions, we appeal to the following equivalencies: (x, y) ∈ T ⇔ x ∈ L(y) ⇔ y ∈ P (x),

(200)

i.e., the three sets T , P (x), L(y) model the same technology. Both the input and output distance functions satisfy the representation property L(y) = {x : Di (y, x) > = 1} and P (x) = {y : Do (x, y) < = 1}, respectively. It follows that technology T may be expressed in terms of these distance functions T = {(x, y) : x ∈ L(y)} = {(x, y) : Di (y, x) > = 1} and T = {(x, y) : y ∈ P (x)} = {(x, y) : Do (x, y) < = 1}. 9 We

leave development of the corresponding theory for the directional input and output distance functions to the reader.

23 Shadow Pricing in Production Economics

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The last two expressions verify our use of the radial input and output distance functions as constraints in the profit maximization model. We begin by using the output distance function as the technology constraint, max(py − wx) − μ(Do (x, y) − 1), x,y

(201)

where μ is the Lagrangian multiplier. The first-order conditions with respect to outputs are p − μy Do (x, y) = 0

(202)

w − μx Do (x, y) = 0.

(203)

and with respect to inputs

Our first pricing rules are10 pm = w 1

∂Do (x, y)/∂ym , m = 1, . . . , M, ∂Do (x, y)/∂x1

(204)

∂Do (x, y)/∂xn , n = 1, . . . , N. ∂Do (x, y)/∂y1

(205)

and the second rule is w n = p1

Thus we may price outputs ym , m = 1, . . . , M using information on one input price, say w1 , as well as information on input and output quantities (x, y). Similarly, we may also price inputs xn , n = 1, . . . , N given one output price, say p1 , together with data on (x, y). To show how we may use data on total revenue r = py =

M 

pm ym ,

m=1

together with input and output quantities to solve for shadow prices for inputs, consider the first-order constraints for outputs from our profit maximization problem p = μy Do (x, y)

(206)

and multiply both sides by y and apply Euler’s theorem to the right-hand side (recalling that Do (x, y) is homogeneous of degree +1 in y); then

10 We

may also derive rules for the “same” price.

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r = py = μDo (x, y);

(207)

μ = r/Do (x, y).

(208)

thus

Inserting this into the first-order constraints for inputs, we have w=r

x Do (x, y) , Do (x, y)

(209)

which shows that total revenue r together with quantities (x, y) can be used to solve for shadow prices of inputs. This may be compared to the pricing rule from Section 3.2. w=c

x Di (y, x) . Di (y, x)

(210)

The next two pricing rules for outputs make use of Shephard input distance function Di (y, x). Again maximizing profit, now using Di (y, x) as the technology constraint, we have max py − wx − μ(Di (y, x) − 1) x,y

(211)

with the associated first-order conditions with respect to y p − μy Di (y, x) = 0

(212)

and for the conditions with respect to x w − μx Di (y, x) = 0,

(213)

where μ is the Lagrangian multiplier. From these conditions we can develop the output pricing rule pm = w1

∂Di (y, x)/∂ym , m = 1, . . . , M, ∂Di (y, x)/∂x1

(214)

∂Di (y, x)/∂xn , n = 1, . . . , N. ∂Di (y, x)/∂y1

(215)

and for the inputs w n = p1

Thus observing one input or output price, together with quantities (x, y), we can price outputs pm , m = 1, . . . , M or inputs wn , n = 1, . . . , N .

23 Shadow Pricing in Production Economics

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We may also use total cost c = wx =

N 

wn xn

n=1

together with (x, y) to price outputs. To verify this claim, multiply the first-order constraints for inputs by x and use Euler’s theorem to obtain c = wx = μDi (y, x),

(216)

μ = c/Di (y, x).

(217)

yielding

Inserting this expression into the first-order constraints for outputs yields p=c

y Di (y, x) , Di (y, x)

(218)

a pricing rule for outputs based on total costs c and input distance function Di (y, x) and its gradient with respect to outputs y. Contrasting this rule to that for inputs w=c

x Di (y, x) , Di (y, x)

(219)

we see that the pricing rule for inputs or outputs depends on which gradient vector of Di (y, x) we choose. Recall, however, that this latter rule was derived under cost minimization while the former was derived under profit maximization. Next we return to the directional technology distance function to derive what we call “crossover” shadow pricing rules. We begin with the profit maximization problem with directional technology distance function constraint  T (x, y; gx , gy ) > 0, max(py − wx) s.t. D = x,y

(220)

or as a Lagrangian problem  T (x, y; gx , gy ), max(py − wx) − μD x,y

(221)

where μ is the Lagrangian multiplier. The first-order constraints with respect to outputs are  T (x, y; gx , gy ) = 0 p − μy D

(222)

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and with respect to inputs  T (x, y; gx , gy ) = 0. −w − μx D

(223)

From these expressions we can derive two crossover pricing rules:11 pm = w 1

 T (x, y; gx , gy )/∂ym ∂D , m = 1, . . . , M,  T (x, y; gx , gy )/∂x1 ∂D

(224)

 T (x, y; gx , gy )/∂xn ∂D , n = 1, . . . , N.  T (x, y; gx , gy )/∂y1 ∂D

(225)

and for inputs w n = p1

Next we assume that in addition to input and output quantities (x, y), that total profit is known  = py − wx =

M 

pm ym −

m=1

N 

wn xn

n=1

. Given this information, two additional pricing rules can be developed: one for outputs p=

 T (x, y; gx , gy ) y D  T (x, y; gx , gy )y + x D  T (x, y; gx , gy )x y D

(226)

and for inputs w = −

 T (x, y; gx , gy ) x D .  T (x, y; gx , gy )y + x D  T (x, y; gx , gy )x y D

(227)

We go through the derivation of the expression for output shadow prices and leave the input pricing derivation to the reader.  T (x, y; gx , gy ), it follows that the Lagrangian Using the translation property of D μ takes the form μ = wgx + pgy .

(228)

may also price say pm , m = 2, . . . , M for outputs if we know p1 . Similar results hold for input pricing.

11 We

23 Shadow Pricing in Production Economics

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Inserting this into the first-order constraints with respect to outputs and multiplying by the output vector y gives us  T (x, y; gx , gy )y py = (wgx + pgy )y D

(229)

with the corresponding expression for the input case which is  T (x, y; gx , gy )x. wx = −(wgx + pgy )x D

(230)

If we sum this expression and insert μ = (wgx + pgy ) =

   T (x, y; gx , gy )x y DT (x, y; gx , gy )y + x D

(231)

into the first-order constraints for outputs, it yields the desired output shadow pricing rule.

Appendix A: Catalog of Shadow Pricing Rules This appendix summarizes the pricing rules we developed in sections “Introduction: What Is a Shadow Price?,” “Primal Representation of Technology: Distance Functions,” and “Calculus and Dual Spaces.” They are organized by the type of distance function used to represent technology – whether radial or directional – as well as by what type of variable is being priced. We also explicitly state what data are required to estimate the shadow price.

Input Pricing Rules (x, y) is known and one input price is known: ∂Di (y, x)/∂xn , n = 2, . . . , N ∂Di (y, x)/∂x1

(232)

 i (x, y; gx )/∂xn ∂D , n = 2, . . . , N  i (x, y; gx )/∂x1 ∂D

(233)

wn = w1

wn = w1

(x, y) and total cost c are known: w=c

x Di (y, x) Di (y, x)

(234)

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w=c

 i (x, y; gx ) x D .  i (x, y; gx )x x D

(235)

(x, y) and one output price are known: ∂Di (y, x)/∂xn , n = 1, . . . , N ∂Di (y, x)/∂y1

(236)

∂Do (x, y)/∂xn , n = 1, . . . , N. ∂Do (x, y)/∂y1

(237)

w n = p1

w n = p1

Output Pricing Rules (x, y) and one output price are known: ∂Do (x, y)/∂ym , m = 2, . . . , M, ∂Do (x, y)/∂y1

(238)

 o (x, y; gy )/∂ym ∂D , m = 2, . . . , M.  o (x, y; gy )/∂y1 ∂D

(239)

p m = p1

pm = p1

(x, y) and total revenue r are known: y Do (x, y) Do (x, y)

(240)

 o (x, y; gy ) y D  o (x, y; gy )y y D

(241)

x Di (y, x) Di (y, x)

(242)

p=r

p=r (x, y) and total cost c are known:

p=c (x, y) and one input price are known: pm = w 1

∂Di (y, x)/∂ym , m = 1, . . . , M, ∂Di (y, x)/∂x1

(243)

pm = w 1

∂Do (x, y)/∂ym , m = 1, . . . , M. ∂Do (x, y)/∂x1

(244)

23 Shadow Pricing in Production Economics

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Pricing Inputs, Indirect Approaches (p/r, x) and one input price are known: wn = w1

∂I Di (p/r, x)/∂xn , n = 2, . . . , N, ∂I Di (p/r, x)/∂x1

(245)

wn = w1

 i (p/r, x)/∂xn ∂ ID , n = 2, . . . , N.  i (p/r, x)/∂x1 ∂ ID

(246)

(p/r, x) and total cost c are known: w=c

x I Di (p/r, x) , I Di (p/r, x)

w n = w1

 i (p/r, x) x ID .  i (p/r, x)x x ID

(247)

(248)

Pricing Outputs, Indirect Approaches (w/c, y) and one output price are known: p m = p1

∂I Do (w/c, y)/∂ym , m = 2, . . . , M, ∂I Do (w/c, y)/∂y1

(249)

pm = p1

 o (w/c, y)/∂ym ∂ ID , m = 2, . . . , M.  o (w/c, y)/∂y1 ∂ ID

(250)

(w/c, y) and total revenue are known: y I Do (p/r, x) , I Do (p/r, x)

(251)

 o (w/c, y; gy ) y ID ,  o (w/c, y; gy )y y ID

(252)

p=r

p=r

Price and Quantity Mixed, Indirect Approaches Pricing inputs when (ps /rs , y−s , x) and one input price are known:

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∂SI Di (ps /rs , y−s , x)∂xn , n = 2, . . . , N, ∂SI Di (ps /rs , y−s , x)∂x1

(253)

∂ SID i (ps /rs , y−s , x; gx )∂xn , n = 2, . . . , N, ∂ SID i (ps /rs , y−s , x; gx )∂x1

(254)

wn = w1

wn = w1

Pricing inputs when (ps /rs , y−s , x) and total cost c are known: x SI Di (ps /rs , y−s , x) , SI Di (ps /rs , y−s , x)

(255)

x SID i (ps /rs , y−s , x; gx ) , x SID i (ps /rs , y−s , x; gx )x

(256)

w=c

w=c

Pricing outputs when (ws , cs , x−s , y) and one output price are known: ∂SI Do (ws /cs , x−s , y)/∂ym , m = 2, . . . , M ∂SI Do (ws /cs , x−s , y)/∂y1

(257)

∂ SID o (ws /cs , x−s , y; gy )/∂ym , m = 2, . . . , M ∂ SID o (ws /cs , x−s , y; gy )/∂y1

(258)

pm = p1

pm = p1

(ws , cs , x−s , y) and total revenue r are known: y SI Do (ws /cs , x−s , y) SI Do (ws /cs , x−s , y)

(259)

y SID o (ws /cs , x−s , y; gy ) . y SID o (ws /cs , x−s , y; gy )y

(260)

p=r

p=r

Pricing Under CRS

p = y C(y, w)

(261)

w = x R(x, p).

(262)

23 Shadow Pricing in Production Economics

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Pricing Inputs and Outputs When Total Profit  and (x, y) Are Known

p=

 T (x, y; gx , gy ) y D  T (x, y; gx , gy )y + x D  T (x, y; gx , gy )x y D

(263)

w=

 T (x, y; gx , gy ) x D .  T (x, y; gx , gy )y + x D  T (x, y; gx , gy )x y D

(264)

Appendix B: Functional Forms In this chapter we make use of two types of distance functions as the representation of the production technology: radial (Shephard-type distance functions) and directional distance functions. Our pricing rules are derived in terms of derivatives of these functions, which in general would be estimated parametrically, typically requiring specification of a functional form. The purpose of this appendix is to provide guidance in those choices based on the properties of the functions as well as flexibility of the specification. By their definitions these two types of distance functions satisfy distinct properties, which as we shall see imply different functional forms. The Shephard distance functions are based on a radial scaling – or multiplication – of a vector, which implies that the distance function is homogeneous in that scaled vector. In contrast the directional distance function, rather than homogeneity of the outputs or inputs, satisfies the translation property. To formalize these properties, we begin with some notation. Let F : 2 → F (q1 , q2 ) ∈ 

(265)

be a function, which is homogeneous of degree +1 in q if F (λq) = λF (q), λ > 0, q = (q1 , q2 ).

(266)

This function satisfies the translation property if F (q + αg) = F (q) + α, g ∈ 2 , α ∈ .

(267)

In addition to these properties, we seek a functional form that is “flexible,” i.e., it allows for interaction terms and second-order terms. Thus we want our functions to

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belong to the family of generalized quadratic functions.1 More formally, let h :  →  and ρ −1 :  → , then we say that F is generalized quadratic if there exist real constants ai , aij such that F (q1 , q2 ) = ρ −1 (ao +

2 

ai h(qi ) +

2 2  

aij h(qi )h(qj )).

(268)

i=1 j =1

i=1

When ao and aij = 0, then this reduces to a quasi-linear function [25]. When F is generalized quadratic as well as homogeneous, then [24] showed that F can take two forms: the translog F (q1 , q2 ) = ao +

2 

ai ln(qi ) +

2 2  

aij ln(qi ) ln(qj )

(269)

i=1 j =1

i=1

and, alternatively, the mean of order ρ form ρ/2 ρ/2

F (q1 , q2 ) = (a11 q1 + a22 q2 + a12 q1 q2 )1/2 ,

(270)

where ai and aij are appropriately restricted to meet homogeneity. Thus the radial distance function can be parameterized as a translog or mean of order ρ function. Since the mean of order ρ function has only second-order parameters aij terms but no first-order parameters ai , we prefer the translog form. This form is also differentiable, which allows us to use calculus to derive and estimate shadow prices. When F is generalized quadratic and satisfies the translation property (rather than homogeneity), [26] show that the functional form must be either a quadratic function F (q1 , q2 ) = ao +

2 

ai qi +

2 2  

aij qi qj ,

(271)

i=1 j =1

i=1

or an unnamed function  1 (ln aij exp(λqi ) exp(λqj )) 2α 2

F (q1 , q2 ) =

2

(272)

i=1 j =1

1 This

terminology is due to [22]. This form has been dubbed transformed quadratic by [23] and Taylor series approximation by [24].

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with the appropriate restrictions on the parameters ai , aij to satisfy translation. Again, we favor the form that has both first- and second-order parameters (the quadratic functional form) over the unnamed function which has only second-order parameters. Thus we recommend the quadratic functional form – which is also differentiable – for estimation of the directional distance function. We have not considered a number of problems associated with these estimating these distance functions and their value counterparts. For examples of econometric applications that address some of these issues, including endogeneity and choice of direction vectors, see [27] and [28].

More Formal Exposition of Calculus and Primal and Dual Spaces We gratefully acknowledge this appendix which was suggested and sketched out for us by Robert G. Chambers. Suppose we have an arbitrary vector space C; we denote its dual space C ∗ as the space of linear functional defined on C (for a reference, see [16]). For N it is well-known that N ∗ = N . Note that the usual notion of a gradient (using the Gâteaux definition) for a smooth function f : N →  is as the singleton set ∇f (x) = {m ∈ N : mv = lim

λ→0

f (x + λv) − f (x) for all v ∈ N }, λ

or alternatively the gradient is defined as ∇f (x) ∈ N = N ∗ such that ∇f (x)v = lim

λ→0

f (x + λv) − f (x) λ

for all v ∈ N , which in turn means that the directional derivative is linear. Hence, gradients are elements of the dual space. This is the generalization of the definition used in the earlier section on calculus and dual spaces that were employed for real spaces. As earlier, we assume properties (disposability, convexity, and closedness of technology set T ) to ensure existence, differentiability, and optimality of the distance functions which then are function representations used to identify shadow

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prices.2 As before (focusing on the input distance functions for simplicity), these distance functions are of two types: radial (Shephard-type) distance functions Di (y, x) = sup{λ > 0 : x/λ ∈ L(y)} and the directional distance function  i (x, y; gx ) = sup{β ∈  : x − βgx ∈ L(y).} D The properties of these functions that are relevant for shadow prices include positive homogeneity of Di (y, x) in x which ensures that ∇x Di (y, x) is homogeneous  i (x, y; gx ), which implies of degree zero in x (Euler’s theorem) and translation for D that for α ∈   i (x + αgx , y; gx ) = ∇x D  i (x, y; gx ). ∇x D These properties have implications for the elements of the dual space N associated with the these gradient vectors, yielding

∗

∇x Di (y, x) x=1 Di (y, x) and  i (x, y; gx )gx = 1. ∇x D This means that the hyperplanes generated by the respective gradients of Di (y, x)  i (x, y; gx ) value x and gx at 1, respectively, i.e., x and gx are the respective and D x Di (y,x)  i (x, y; gx ) define “dual (valuation)” hyperplanes and ∇x D numeraires and ∇D i (y,x) passing through the numeraires. In economics we are generally interested in the linear functions of, e.g., x associated with prices. As an example, the question is equivalent to asking what is the cheapest bundle of inputs consistent with producing a given output. So define ∗ C : M × N →  as C(y, w) = min{wx : x ∈ L(y)}. For our two types of input distance functions, the associated Lagrangeans are wx − μDi (y, x)

2 Following

[10], it is important to distinguish the linear programming notion of a shadow price from the economic notion of a shadow price. The economic notion presumes efficiency, whereas the linear programming notion does not.

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 i (x, y; gx ). wx − ϕ D Taking account of the homogeneity and translation properties of the distance functions, the FOC simplify to wx = μ since ∇x Dxi (y,x) = Di (y, x) = 1 (by the constraint) and wg = ϕ, respectively. Returning to a restatement of Daniel Primont’s definition (in real form) put into the cost function context, we have the following: for a given vector of quantities ∗ x, w = (w1 , w2 , . . . , wN ) ∈ N is a real shadow price vector for it if x is optimal for w. According to that definition, 

∂Di ((x, y) ∂x1

−1

∇x Di (x, y)

and 

 ∂Di ((x, y)  i (x, y; gx ) ∇x D ∂x1

are real shadow price vectors for x; all suitably normalized gradients may be interpreted as real shadow price vectors. Returning to the cost function and the relationship of our shadow prices to Shephard’s lemma, and maintaining the assumptions and properties from above, note that by the duality between cost and distance functions, we may write C(y, w) = min{ x

wx } Di (y, x)

and C(y,

w w  i (x, y; gx )}. ) = min{ x−D x wgx wgx

Given the curvature conditions we have imposed, we obtain w ∇x Di (x, y) x = ⇔ ∇w C(y, w) = wx Di (y, x) Di (y, x) and w  i (x, y; gx ) ⇔ ∇w/wx C(y, w ) = x. = ∇x D wgx wgx

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This provides the key link between shadow pricing results (on the left) and their dual manifestations (on the right). In a smooth world, the relationship above is what Shephard proved, i.e., Shephard’s lemma. Using these definitions and results and extending them to revenue and profit provides the structure for deriving the remaining “pricing rules” in this chapter.

References 1. Färe R, Grosskopf S, Margaritis D (2019) Pricing non-marketed goods using distance functions. In: Collaboration with Robin Sickles, Chenjun Shang, Maryam Hasannasab and William L. Weber. NOW Publisher, Inc, Hanover 2. Warr PG (1982) Pricing rules for non-traded commodities. Oxf Econ Pap 34:305–325 3. Diewert EW (2003) Hedonic regressions: a consumer theory approach. In: Feenstra RC, Shapiro MD (eds) Scanner data and price indexes. University of Chicago Press, Chicago, pp 317–348 4. Toda Y (1976) Estimation of a cost function when the cost is not minimum: the case of Soviet manufacturing industries. Rev Econ Stat 58:259–268 5. Lau LJ, Yotopoulos PA (1979) Resource use in agricultural applications of the profit function to selected countries: the methodological framework. Stanford Food Res Inst Stud 17:11–22 6. Atkinson S, Halvorsen R (1980) A test of the relative and absolute price efficiency in regulated utilities. Rev Econ Stat 62:81–88 7. Kumbhakar SC, Battacharyya A (1992) Price distortions and resource-use inefficiency in Indian agriculture: a restricted profit function approach. Rev Econ Stat 74:231–239 8. Wang J, Wailes EJ, Cramer GL (1996) A shadow price frontier measurement of profit efficiency in Chinese agriculture. Am J Agric Econ 78:146–156 9. Starrett DA (2000) Shadow pricing in economics. Ecosystems 3(1):16–20 10. Chambers RG, Färe R (2008) A calculus for data envelopment analysis. J Prod Anal 30: 169–175 11. Färe R, Grosskopf S (2004) New directions: efficiency and productivity. Kluwer Academic Publishers, Boston 12. Färe R, Primont D (1995) Multi-output production and duality: theory and applications. Kluwer Academic, Norwell 13. Färe R, Primont D (2006) Directional duality theory. Econ Theory 29:239–247 14. Flemming W (1977) Functions of several variables. Springer, Berlin 15. Aczél J (1987) A short course on functional equations. D. Reidel Publishing Co., Dordrecht 16. Luenberger DG (1969) Optimization by vector space methods. Wiley, New York 17. Shephard RW (1970) Theory of cost and production functions. Princeton University Press, Princeton 18. Shephard RW (1953) Cost and production functions. Princeton University Press, Princeton 19. Thorsnes P (1997) Consistent estimates of the elasticity of substitution between land and nonland inputs in the production of housing. J Urban Econ 42:98–108 20. McMillen DP (2003) The return of centralization to Chicago: using repeat sales to identify changes in house price distance gradients. Reg Sci Urban Econ 33:287–304 21. Shephard RW (1974) Indirect production functions. Mathematical Systems in Economics, No. 10, Verlag Anton Hain, Meisenheim Am Glan 22. Chambers RG (1988) Applied production analysis: a dual approach. Cambridge University Press, Cambridge 23. Diewert EW (2002) The quadratic approximation lemma and decomposition of superlative indexes. J Econ Soc Res 28:63–88 24. Färe R, Sung KJ (1986) On second order Taylor’s series approximations and linear homogeneity. Aequationes Mathematicae 30:180–186

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25. Aczél J (1966) Lectures on functional equations and their applications. Academic, New York 26. Färe R, Lundberg A (2006) Parameterizing the shortage function, mimeo. Department of Economics, Oregon State University 27. Atkinson S, Tsionas MG (2018) Shadow directional distance function with bads: GMM estimation of optimal directions and efficiencies. Empir Econ 54:207–230 28. Atkinson S, Primont D, Tsionas MG (2018) Statistical inference and efficiency of production with bad inputs and outputs using latent price and optimal directions. J Econometr 204(2): 131–146

Capacity and Capacity Utilization in Production Economics

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Contents Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Conceptual Foundations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Technological Approach to Defining Capacity and CU . . . . . . . . . . . . . . . . . . . . . . . . . . . An Economic Optimization Approach to Defining Capacity and CU . . . . . . . . . . . . . . . . Additional Considerations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Other Capacity and Utilization-Related Concepts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Measurement of Capacity and CU . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Macroeconomic Approaches . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Microeconomic Frontier-Based Approaches . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Microeconomic Optimization-Based Approaches . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Concluding Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1002 1004 1005 1008 1014 1017 1021 1021 1023 1028 1029 1030

Keywords

Capacity · Capacity utilization · Dual measures · Primal measures · Quasi-fixed inputs

D. Squires NMFS, Southwest Fisheries Science Center, La Jolla, CA, USA Department of Economics, University of California San Diego, La Jolla, CA, USA e-mail: [email protected] K. Segerson () Department of Economics, University of Connecticut, Storrs, CT, USA e-mail: [email protected] © Springer Nature Singapore Pte Ltd. 2022 S. C. Ray et al. (eds.), Handbook of Production Economics, https://doi.org/10.1007/978-981-10-3455-8_7

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Introduction The concepts of capacity and capacity utilization (CU) help explain many economic phenomena, including investment behavior, productivity measurement, inventory behavior, entry/exit into an industry, market power, pricing, and profitability [35, 107, 124, 132, 136, 166]. These concepts are sometimes employed to indicate the strength of aggregate demand, especially throughout the business cycle and consequent inflationary pressures, and the exploitation pressures placed upon renewable resource stocks. Understanding the role of measured capacity in economic fluctuations also helps understand theories of the business cycle. Central banks, such as the Federal Reserve Board in the United States, other government economists, and academic macroeconomists and industrial organizational economists developed the early notions of capacity to address these issues.1 Similarly, capacity utilization has been studied within the contexts of operations research and business strategy [27, 103, 124]. Capacity expansion models from the operations research literature have been used to explain both the amount and the timing of firm investment or disinvestment to alter capacity and to relate industry capacity expansion to nonstrategic and measurable factors, such as the growth rate and variability of demand, the number of firms and plants in the industry, and the degree of capital intensity and investment economies of scale.2 This approach also considers the role of the firm’s business strategy. For example, excess capacity may be used as a strategy to protect firms’ market share [5, 46, 54] or to allow the firm to seize opportunities that may develop [85]. Capacity strategies can also allow a firm to preempt competitors by taking advantage of market changes before its competitors respond [58]. This chapter provides an overview of the literature on capacity utilization, focusing primarily on the work within production economics but also noting the literature from related fields, especially macroeconomics. We focus on the two broad approaches that have been taken to defining capacity and CU, an engineering or technological approach based on production possibilities and an economic

1 Notable

among the initial economists who contributed are Cassels [28], Chenery [33, 34], Klein [113], Klein and Summers [115], de Leeuw [50], Schultze [160], Hickman [91, 92], Wilson and Eckstein [186], Klein et al. [116], and Klein and Preston [114]. 2 The optimal timing of capital investment projects to expand capacity distinguishes between incremental and more substantive increases in capacity. This discussion also distinguishes between timing within the business cycle, building capacity when the need for it develops at peaks or countercyclically at the lower point of a cycle, and long-term objectives of the firm, such as expanding to maintain the firm’s competitive position and market share. In response to economic shocks, Greenwood et al. [86, 87] allow intensity of capital stock use to vary, and Kydland and Prescott [121] and Bils and Cho [21] allow the workweek of capital to respond to examine variations in CU by the intensity or the period of time for capital utilization. Bresnahan and Ramey [24] and Cooley et al. [42] allow CU to vary by the fraction of the capital stock used in production.

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approach based on optimizing firm behavior [91].3 Under both approaches, capacity utilization (CU) is defined as the ratio of actual output to some reference level of output, but the reference level differs under different CU approaches. Under the technological approach, the reference level is typically some notion of a realistically sustainable maximum level of output. Alternatively, under an economic optimization approach, it is defined as some notion of “optimal” output. We discuss both of these approaches, focusing first on the theoretical/methodological underpinnings of these concepts and then turning to some of the relevant empirical literature aimed at measuring capacity and/or CU. Our intention is not to provide an exhaustive literature review, but rather to introduce and overview this literature and the multitude of issues. The sections of this chapter are organized as follows. “Conceptual Foundations” develops the conceptual foundations for capacity and CU. “Technological Approach to Defining Capacity and CU” discusses the technological approach to defining capacity and CU. Several different approaches or variations are discussed, since the literature developed in different contexts and with different emphases. “An Economic Optimization Approach to Defining Capacity and CU” then discusses the economic optimization approach. Again, we discuss several different approaches that differ by the behavioral objective (cost minimization, profit maximization, or revenue maximization) as well as whether they are based on primal or dual specifications of the firm’s problem. We then turn briefly to some extensions to the basic theory to account for multiple outputs and/or multiple inputs, other stocks (such as stocks of natural capital), regulatory constraints, dynamic decision-making and adjustment, uncertainty and stochastic demand, and imperfect competition. This section then briefly surveys other capacity and utilization-related concepts, which are sometimes conflated with the canonical approach. These include input capacity based on quasi-fixed inputs, capital (as opposed to capacity) utilization, variable input utilization, and the link between capacity utilization and productivity. Following this conceptual overview, “Measurement of Capacity and CU” develops the empirical measurement of capacity and CU. “Macroeconomic Approaches” develops primal technical-engineering measures of capacity and CU. Distinctions are made between different approaches that developed within different contexts and slightly different analytical frameworks and emphases but that are fundamentally the same (each with their own strand of literature), including the US Federal Reserve; peak-to-peak; full employment maximum output; and an explicit production function. “Microeconomic Frontier-based Approaches” discusses measurement based on technological-economic methods that draw from microeconomic foundations and the theory of the firm. These include both primal frontier methods based on production or distance functions and dual methods based on cost, profit, or revenue functions. “Microeconomic Optimization-based Approaches” discusses the plant

3 The

analysis of capacity, investment to increase capacity, and CU can also be approached from the capital budgeting and finance literatures. This chapter does not include a review of this strand of the capacity discussion [118, 158].

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capacity model of CU developed by Johansen [98]. Finally, “Concluding Remarks” provides concluding comments.

Conceptual Foundations As noted above, capacity utilization (CU) is usually defined as the ratio of actual output (Y) to some measure of its “capacity,” which is a reference level of output (YR ) generally thought to be the level that can or should be produced given the existing input base, prices, states of technology and the environment, and firm management. Thus, the primal measure of CU is simply CU Y = YYR . The inverse, 1/CUY = YR /Y, indicates that the amount of output could increase if the existing capacity were fully used (under the technological approach) or optimally used (under the economic optimization approach) [132]. Capacity and CU are typically viewed as short-run concepts. Earlier views allowed for capacity in either the short or the long run. For example, Cassels [28] states: “Careful distinction must always be made between the excess capacity of fixed factors which exists for the short run and the excess capacity of all factors which may be present over the long run . . . .” However, the contemporary view is that capacity and CU arise due to scarcity or fixity or quasi-fixity of one or more inputs that are available to utilize for the production of one or more outputs.4 The stock of labor is sometimes considered as one of these fixed factors, but the most commonly specified fixed or quasi-fixed input, forming the usual capacity base available for production, is the amount of physical capital stock (K) – which we use hereafter unless otherwise noted.5 In addition, in some contexts, firms face other constraints in the short run (and potentially the long run as well) that operate as restrictions on their choices and should thus be treated as “fixed” when defining capacity and CU. For example, in sectors with renewable resources (e.g., fisheries) or non-nonrenewable resources (e.g., mining), the natural capital stock forms a second stock in a stock-flow production technology that is fixed for individual firms. Similarly, a firm might be regulated by a binding quota or predetermined demand that should be reflected in the measure of capacity and CU. We return to the role of these additional constraints below.

4 Fixed

factors are factors that must receive payment whether or not any output is produced. Quasifixed factors must be paid only if the firm decides to produce a positive amount of output. Another definition of quasi-fixed factors is factors of production that can be adjusted in a time period, the short run, but will not be adjusted all the way to the full static equilibrium level because of constraints such as adjustment costs. 5 Issues arise regarding consistently aggregating K from individual capital stocks and different vintages of each capital stock and incorporating multiple stocks in the capacity base [22, 96].

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Technological Approach to Defining Capacity and CU Measures Based on Maximum Sustainable Output Technological notions of capacity and CU measures were discussed early on by Cassels [28] and further developed and given rigor for empirical application by Gold [82], Klein [113], Klein and Summers [115], Klein et al. [116], Klein and Preston [114], Hickman [92], and Johansen [97]. Klein and Summers [115] state that full capacity is “the maximum sustainable level of output which the industry can attain within a very short time if the demand for its product were not a constraining factor, and when the industry is operating its existing stock of capital at its customary level of intensity.” The technological approach thus defines the reference level of output YR as the maximum output (Y0 ) that may be produced given the firm’s short-run capital stock K – or more generally, assuming some factors of production are fixed in the short run that include K but can also include labor and other factors [113, 116, 166]. Capacity is “wasted” when the output produced is lower than what is maximally producible from the quantities of variable and fixed inputs used. Capacity remains “underutilized” when the quantity of variable inputs is less than what is required to produce the maximum output producible given the fixed inputs. Shapiro [166] gives an economic interpretation to this approach: “Capacity is best thought of as the level of output where the marginal cost curve becomes steep. If this region of the cost curve is sometimes relevant, the relationship between capacity and output should be non-linear. If capacity is tight, growth in capacity limits growth in output.” This concept of capacity includes the qualification that capacity represents a realistically sustainable maximum level of output rather than some higher unsustainable short-term maximum [44]. Additional assumptions include that the elasticity of substitution between variable and fixed factors of production (i.e., the elasticity of intensity, [57]) is very low and that movements in production arise from shifts in demand rather than shifts in production possibilities [166]. Under these assumptions, short-run changes in output equal short-run changes in utilization of fixed factors. Thus, the US Federal Reserve, Census Bureau, and Defense Logistics define full production capability as the maximum level of production that an establishment could reasonably expect to attain under reasonable and realistic operating conditions fully utilizing the machinery in place and ready to operate. Under this definition of CU, clearly CU is always less than or equal to 1, i.e., CU Y = YY0 ≤ 1. When CUY < 1, the firm has the potential for greater production, given K, without having to incur major expenditures for new capital or equipment [115]. This implies that some of the capital stock is not fully utilized while full capital utilization and technical efficiency would yield Y0 . However, in practice, due to sharply rising marginal costs as production starts to approach Y0 , full sustainable CU is typically considered, in practice, as attained at a production level that is lower than Y0 . This raises practical questions about defining the full employment or full utilization level of variable inputs and a precise distinction between variable and fixed factors [44, 135]. Klein et al. [116] state that full capacity can be described

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as a full input point on an aggregate production function and that full capacity should be defined as an “attainable level of output that can be reached under normal input conditions” and “fully employing” the variable inputs, “without lengthening accepted working weeks, and allowing for usual vacations and normal maintenance,” given the current technology and keeping fixed factors at their current levels. However, some notion of what constitutes normal conditions must still be determined. For example, is the capacity of a plant and equipment determined by the production of this plant and equipment operating throughout the day or season or year, and should downtime for repair and maintenance, institutional constraints such as holidays, and the like be considered? The answer varies by the type of technology and institutional factors that constitute issues such as “normal” downtime [44]. Short-run output varies with technology type in different ways according to (1) duration and (2) intensity or speed of operations. Alternatively, capacity output can be defined based on applying variable inputs without limit, to the point at which the marginal productivity of all variable inputs falls to zero. As seen below, Färe [65] and Färe et al. [67] rigorously develop this approach within the context of Data Envelopment Analysis. Finally, note that the above definition of capacity output is based on the notion of a best-practice production frontier of Debreu [52], Koopmans [117], and Farrell [[67, 72].6 However, in practice firms might not be operating on their production frontiers because of technical inefficiency. In the presence of inefficiency, the observed output used to calculate the standard measure of primal CU may differ from the capacity output due either to the quasi-fixity of inputs or to technical inefficiency. This would result in a downward bias relative to the frontier measures of capacity and in the CU measure. To address this, Färe et al. [67] proposed an alternative primal CU measure that is the ratio of technically efficient output, YTE , TE to capacity output: CU TY E = YY 0 . This CU measure is consistent in that both the numerator and the denominator are technically efficient. It provides a CU measure that indicates deviations from capacity output due solely to quasi-fixed inputs. Such a ratio is “unbiased” in that it is not directly influenced by measured technical inefficiency.

Johansen’s Plant Capacity The most commonly used approach to defining capacity based on production possibilities employs a primal definition of capacity output – plant capacity – developed by Johansen [97] and refined by Färe [65].7 In contrast to the above 6 Output-oriented

technical efficiency occurs when firms produce the maximum output attainable for a given set of inputs, given the state of technology, environmental conditions, and, in natural resource industries, the resource stock. Output-oriented technical inefficiency occurs when the actual output is less than the technically efficient level of output. Technical efficiency is measured from a best-practice production frontier – the production frontier estimated from the input-output observations for firms with the observed best practice. 7 Again, empirical application uses the notion of the best-practice production frontier of Debreu [52], Koopmans [117], and Farrell [72].

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measure, the plant capacity concept of capacity is defined as “the maximum amount that can be produced per unit of time with existing plant and equipment, provided the availability of variable factors of production is not restricted.” Färe [65] developed a formal proof of the existence for Johansen’s definition of capacity. Färe [65] identified Johansen’s concept of capacity as a strong definition of capacity because it is unbounded. He also developed a weaker notion, which only requires that for fixed inputs output is bounded when there is no restriction on variable inputs. The strong definition implies the weak definition, but the reverse does not hold. When the Johansen plant capacity definition is applied in an industry, the assumption of unbounded variable inputs is problematic [109], again raising the question of how to measure capacity in practice. Complicating factors include how to define customary and normal operating procedures, the number of shifts, vacations, etc. [44, 115, 116]. The maximum potential variable input level, such as three full 8-hour shifts a day every day of a week and every week in a year for similar plants, may not match the observed practice (and hence the data). The maximum must then be defined relative to historically observed maximum variable input usage in a fairly recent period for similar plants (see, e.g., [106, 112]). These norms, rules, and practices can also vary by industry. When the Johansen plant capacity concept is employed in a regulated industry in which the variable input use may be less than the theoretically unrestricted levels, the resulting capacity estimates are likely to be more realistically obtainable than the strict definition indicates [73]. Capacity output may also be limited by budget constraints. Ray et al. [153] develop a restricted version of Johansen’s [97] plant capacity, utilizing a restricted version of Shephard’s [167] indirect production function that takes explicit account of input prices. Capacity output is then the maximum quantity that the firm can produce given a specific quantity of the quasi-fixed factor and an overall budget constraint for its choice of variable inputs. The firm can use any variable input bundle within an overall expenditure constraint. Estimates of plant capacity using individual firm-level data face several limitations [105]. Horizontally summing these firm-level capacity outputs across firms gives a measure of aggregate industry capacity output.8 Comparing this aggregate industry capacity output to current aggregate industry output provides a measure of CU. The plant capacity measure, however, does not allow reallocation of inputs and outputs across firms [98]. In turn, this does not allow assessing the industry’s optimal restructuring and configuration. Nonetheless, the plant capacity approach has been widely applied, notably to fisheries, hospitals, banking, and power plants along with other industries [55, 68, 102, 104–106, 108–112, 129, 138, 139, 142, 151, 153, 155, 177, 179, 181–184]. The Johansen-Färe plant capacity measure of capacity for the individual producer can be extended to the industry model using a multi-product, frontier-based version

8 The

horizontal summation of output to achieve an aggregate industry output implies a private good. A public good would require vertical summation to achieve the aggregate industry provision of the public good.

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of the short-run Johansen industry model [53, 105, 106, 179]. Industry capacity is the sum of firm capacities with possible reallocation of fixed inputs across producers. The short-run Johansen [98] industry model analyzes the industry structure due to the underlying ex post firm-level production structure. Investment decisions imply a putty-clay production structure. Thus, firms may choose ex ante from a catalogue of production options exhibiting smooth substitution possibilities, but most face fixed coefficients ex post investment. Firms then have a capacity that is entirely conditioned by the investment decisions they have made. The short-run industry nonetheless exhibits substitution possibilities when inputs and outputs can be reallocated across the production units that comprise the industry. Over time, substitution and technical change can be traced via shifts in successive short-run industry models.

An Economic Optimization Approach to Defining Capacity and CU In contrast to the technological approach described above, the economic optimization approach to defining capacity and CU defines the reference level of output based on economic optimizing behavior. Part of the stimulus for the development of this approach was an attempt to understand the impact of rising energy prices on capacity and CU following the 1973 oil price shocks triggered by the Iranian Revolution and the emergence of OPEC and the subsequent very high rates of global inflation. Even though rates of economic growth remained relatively high, investment and average labor productivity were lower than expected, and the explanatory power of existing CU measures for inflation and other macroeconomic indicators had fallen off sharply. The development of this approach was also spurred by advances in applied microeconomic theory in the form of duality theory and flexible functional forms, developments in econometrics, advances in computing power and software, and the advent of increasingly larger and more comprehensive data sets. These developments strengthened the link between capacity/CU and economic theory based on explicit behavioral objectives of the firm. Early, and indeed foundational, work in this new, microeconometric approach to capacity and CU included Berndt and Morrison [17], Morrison [132–134], Berndt and Fuss [14], and Hulten [94]. Although this approach was implicit in earlier work, such as that by Cassels [28], Klein [113], Friedman [79], and Hickman [92] among others, developments in duality theory, econometrics, computing power and software, and data sets stimulated an explosion of applied theory and empirical applications. The economic optimization approach is based on defining the reference level of output as the level that is economically optimal (Y*) [28]. Because the stock K is taken as fixed or quasi-fixed in the short run, short-run fluctuations in demand are accommodated by changes in the amount of variable inputs used in production [132, 135]. If the demanded output level differs from the production level that would be optimally supported by K, utilization of K will not be economically optimal. The economically optimal production level given short-run constraints on adjustment,

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i.e., the firm’s “capacity,” is defined as the output level that would be produced in steady-state equilibrium given prices, technology, and the current levels of K. At this production level, i.e., if Y = Y∗ , given input prices and technology, there is no incentive for the firm to change levels of the fixed input stocks. Thus, while capacity output and CU are inherently short-run concepts (since they reflect the existence of one or more fixed factors), the capital stock or the capacity decision is a long-run concept [107]. As is developed below, an economic CU different from one indicates that the firm faces incentives to invest or disinvest in K and therefore is not in longrun equilibrium. Since capacity and CU are inherently short-run concepts, they should be based on the short-run objectives of the firm. The economic behavioral objective of the firm falls into one of three broad behavioral objectives [163, 164, 171]: (1) cost minimization, where output levels are exogenous or predetermined and some inputs are variable while others are fixed or quasi-fixed; (2) revenue maximization, where one or more outputs are endogenous and all remaining outputs and all inputs are fixed or quasi-fixed; or (3) a restricted (variable) profit function, where one or more outputs are endogenous and variable and some inputs are variable while all other outputs and inputs are fixed or quasi-fixed. Each of these can be used to define both a primal measure described above, CUY , and analogous dual measures. Below we provide an overview of these measures under the alternative behavioral objectives.

Primal Measures of CU Based on Cost-Minimizing Behavior Three different economic definitions of capacity output have been proposed for defining a primal measure of CU for a cost-minimizing firm. Each of these approaches addresses the short-run excessive or insufficient utilization of fixed or quasi-fixed inputs. The first defines capacity output as the output at which the shortrun average cost (SRAC) curve reaches its minimum. This definition was originally proposed by Cassels [28] and subsequently adopted by others, including Chenery [33, 34], Hickman [92], Wilson and Eckstein [186], Berndt and Morrison [17], and Berndt and Hesse [16]. This approach emphasizes exploiting short-run economies of scale. However, Klein [113] noted that Cassel’s definition was difficult to apply empirically, since L-shaped long-run average cost curves without a well-defined minimum were more likely to be observed [16]. A second definition of economic capacity based on cost-minimizing behavior defines capacity as the output determined by the minimum of the long-run average cost curve (LRAC) [28, 92] rather than the minimum of the SRAC. However, this approach has not been used much in practice, perhaps because it is so closely intertwined with scale economies [105, 152]. The third definition is based on defining capacity output as the output at which the LRAC and SRAC curves are tangent when the stock K is fixed. This approach was originally suggested by Klein [113] and then further developed by Friedman [79] and Morrison [132], Berndt and Fuss [14], and Hulten [94]. This third definition coincides with the first measure under long-run constant returns to scale, but more generally reflects a capacity output level that is a steady state in the sense that the firm does not have an incentive to change output if input prices, stocks of

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fixed inputs, and technology remain unchanged. Intuitively, this economic notion of capacity means that for a given output level and state of technology and prices, the firm is using the stock of capital that allows that output to be produced at the lowest average cost. If demand falls short of the output level corresponding to the tangency of the SRAC and LRAC curves so that CUY < 1, there is excess capital, and the firm faces incentives to disinvest to lower K. If, on the other hand, demand exceeds the level of output supported in steady state by the existing stock K, then CUY > 1, which implies that there is insufficient K and the firm faces incentives to invest.

Dual Measures of CU Based on Cost-Minimizing Behavior The measures of CU described above are based on an “output gap,” i.e., a comparison of actual vs. capacity output. However, CU can be equivalently measured in terms of the cost gap that exists when Y = Y∗ [132]. This dual CU measure contains information on the difference between the current short-run or temporary equilibrium and the long-run equilibrium in terms of the implicit costs of divergence ∗ from long-run equilibrium. It is defined as CU C = CC , where C is the firm’s actual cost and C* is the firm’s shadow cost. The shadow cost is its cost when capital is valued at its shadow price (rather than its actual price), where the shadow value is defined as the negative of the derivative of the variable cost function with respect to K. More specifically, if G(Y, W, K) represents the firm’s short-run variable cost as a function of output, variable input prices (W), and the fixed capital input K, then  ∗  PK − PK K G (Y, W, K) + PK∗ K C∗ = =1+ (1) CU C = C G (Y, W, K) + PK K C where PK is the actual price of K and PK∗ = −GK is the shadow price of K. Clearly, C∗ > C and hence CUC > 1 when PK∗ = −GK > PK . This corresponds to the case where the valuation of an incremental unit of K is higher than its actual cost and the current level of K is therefore less than the cost-minimizing level, K*. Equivalently, it means that Y > Y* and hence CU Y = YY∗ > 1, when Y* is defined as the output level where long-run and short-run average costs will be equal, i.e., the SRAC and LRAC curves will be tangent, which corresponds to the level of output where GK (Y, W, K) + PK = 0. Thus, while differing in magnitude, both the primal and dual measures imply that the firm has an incentive to invest to increase its capital stock. Conversely, when the observed K is larger than K*, then the shadow price of K will be less than PK and Y < Y*. In this case, both measures signal that the firm has an incentive to reduce its capital stock or disinvest.

Profit and Revenue-Based Measures of CU The above economic measures of capacity output assume cost minimization of exogenous or predetermined output given one or more fixed or quasi-fixed inputs. This approach is particularly apt when a firm produces a standardized product in anticipation of stable market demand and competes on the basis of price and low cost arising from economies of scale. However, firms may have behavioral objectives

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SRMC

$

P = MC SRAC LRAC

A B

C

D E

Y

Fig. 1 Alternative measures of capacity output

other than minimizing the cost of a predetermined level of output, and one or more outputs may be endogenous, i.e., choice variables for the individual firm. Squires [171], Berndt and Fuss [15], Segerson and Squires [163], Fousekis and Stefanous [78], Kim [107], Coelli et al. [41], and Briec et al. [26] extended the concept of capacity and CU to the case where firms choose their output level(s) to maximize short-run profit. Assuming an objective of short-run profit maximization rather than cost minimization has implications for both primal and dual measures of CU. In the context of a single-product firm,9 under cost minimization Y∗ is typically defined based on the point of tangency of the SRAC and LRAC curves, as discussed above. Kim [107] and Briec et al. [26] advocate defining Y* in this way even when the firms choose output endogenously. However, this is not necessarily an equilibrium output level, even in the short run. The short-run profit-maximizing output level is the output level at which the firm maximizes variable profit, i.e., at which price equals shortrun marginal cost or P = SRMC. This output level differs from the output level at which SRAC and LRAC curves are tangent under any reasonable conditions, as illustrated in Fig. 1 (discussed in more detail below). For this reason, Squires [171] and Segerson and Squires [163, 164] implicitly advocate and Coelli et al. [41] explicitly advocate defining Y* as the short-run profit-maximizing output level when defining primal measures of CU. Defining Y* as the short-run profit-maximizing choice of output also allows for a comparable dual measure to be easily defined. In particular, under the assumption of profit maximization, we can define a CU measure based on the profit gap (rather than the cost gap) that stems from the firm being out of long-run equilibrium. This profit-gap measure of CU is given by CU π = ππ∗ , where π and π ∗ are actual and

9 The

definition becomes more complex when there are multiple products as discussed below.

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shadow profit, respectively. More specifically, if H(P, W, K) is the short-run variable (restricted) profit function where P is the output price and W denotes the vector of variable input prices, then π ∗ = H (P , W, K)−PK∗ K, where PK∗ is now the shadow value of capital given by HK . As a result   ∗ PK − PK K π H (P , W, K) − PK K CU π = ∗ = =1+ π H (P , W, K) − PK∗ K π∗

(2)

Segerson and Squires [163]. As with the dual cost-based measure CUC , this profitbased dual measure will signal incentives to invest (CUπ > 1) when PK∗ > P , or, equivalently, K < K∗ and Y > Y∗ . Conversely, CUπ < 1 signals incentives to reduce the capital stock. Revenue maximization is a special case of profit maximization for multi-product firms when all inputs are quasi-fixed or fixed and at least one output is endogenous (some outputs may be quasi-fixed). In this case, the profit-based CU measure can be readily adapted by simply reinterpreting H in (2), giving R(P, K). Alternatively, the cost-based measure in (1) can be applied. Because short-run variable costs are zero if all inputs are quasi-fixed or fixed, in this context the cost-based measure reduces P∗ to CU C = PKK , where now the shadow price of capital is defined as its marginal contribution to revenue [164]. While the magnitudes of these different measures differ, they all consistently imply incentives to increase capital when CU > 1 and to reduce capital when CU < 1. The different measures of capacity output described above, including the primal technological definition and the definitions based on cost-minimizing and profitmaximizing behavior, are depicted graphically in Fig. 1 (taken from Coelli et al. [41]). The first three measures are based solely on the cost curves. Point B defines capacity output as the output at which the short-run average total cost (SRAC) curve reaches its minimum. Point E defines capacity as the output determined by the minimum of the LRAC curve rather than the minimum of the SRAC curve. Point A defines capacity as the output at which the LRAC and SRAC curves are tangent. The fourth measure, based on profit-maximizing behavior and labeled point C, defines capacity as the output at which marginal revenue (here, price P) equals SRMC. Finally, the fifth measure, labeled point D, is the primal plant capacity measure of Johansen [97].

Extensions to Multi-product Firms The above discussion of alternative measures of CU focused on a single-product firm. However, many contexts of interest involve firms that produce multiple products under joint production. A number of authors have considered CU for multi-product firms with joint production. For example, Squires [171], Berndt and Fuss [15], Segerson and Squires [162–164], Fousekis and Stefanous [78], Kim [107], Coelli et al. [41], Briec et al. [26], and De Borger et al. [49] extended the concept of capacity and CU when the firm’s behavioral objective is short-run profit maximization and there are multiple endogenous outputs. Similarly, Segerson

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and Squires [163, 164], Färe et al. [71], and Lindebo et al. [125] developed a revenue-based economic concept of capacity for a multi-product firm. As explained here, it is quite straightforward to extend the dual measures of CU to allow for multiple products, but more challenging for primal measures. The economic duality approach readily extends from the single-product to the multiple-product case with a single quasi-fixed input, because these measures are based on the ratio of scalar measures of costs or profits even when output is not a scalar. Consider first the cost-based dual measure of CU defined in (1). By simply reinterpreting Y as a vector of output levels, this definition can be applied to multiproduct firms as well [162]. Similarly, by interpreting P as a vector of output prices in (2), the profit-based dual measure also extends to multi-product firms. The fact that output is a vector does not limit the use of these measures since cost and profit are still scalars, and hence the cost or profit gap is still easily defined. However, extension of the primal measures is less straightforward, and the restrictiveness of these extensions is one reason to prefer the use of dual measures when studying CU in multi-product firms. Segerson and Squires [162] observe that a consistent scalar measure of output in multi-product firms exists if all outputs are homothetically separable from inputs. In this case, a direct analogue of the single-product primal measure of capacity utilization can be developed for the multiproduct firm or industry. They specifically examined aggregate output for which the long- and short-run cost functions are tangent given the stock of physical capital.10 When the production technology is not homothetically separable in outputs so that a consistent composite output is not possible, Segerson and Squires [162] suggest two alternative ways of defining a primal capacity utilization measure. Both approaches required restrictive assumptions: (1) outputs must move along a ray, giving a ray measure of capacity utilization, or (2) only one output can adjust (giving a partial measure of capacity utilization).11,12 Eilon and Soesan [60] suggested a similar primal ray measure (due to a constant output mix) by constructing a full capacity envelope curve that defines the maximum possible output level for each output mix and then measuring CU as the ratio of observed

10 Gold

[82, 83] suggested using output prices as weights to create a price-weighted sum of actual output levels divided by the price-weighted sum of the maximum possible levels of each output. 11 Segerson and Squires [162] show the equivalence of the ray measure to homothetic output separability due to the linear output expansion path with homothetic output separability (where linear homogeneity is imposed upon the output aggregator function; see [22], Lemma 3.3.a) and relate the (short-run) ray average cost to the work of Baumol, Panzar, and Willig [12] rather than the other major production economics strand of the capacity literature that uses the output distance functions of Shephard [167]. 12 The numerical value of this CU measure will vary across products, and therefore it is not unique for a given firm. However, it can be shown under certain conditions that these measures provide a consistent indication of whether the firm’s capacity is currently under- or overutilized and the same answer is given regardless of which output is used to measure CU.

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output to maximum output, holding the output mix constant.13 Coelli et al. [41] suggested a ray primal measure using a short-run profit function giving an approach analogous to Johansen’s [97] plant capacity but at the corresponding short-run profit-maximizing scale of production when variable input prices are zero. Felthoven and Morrison Paul [74] describe two additional primal approaches using a stochastic multi-product asymmetric transformation function: (1) iteratively or simultaneously solve values of capacity outputs and the capacity level of the variable input to obtain values where each variable corresponds to the simultaneous conditions of marginal product of the variable input equals zero and capacity output ratio(s) equal the observed ratio(s) or (2) iteratively or simultaneously solving for capacity output values that satisfy the profit- (or revenue-)maximizing condition between the marginal rate of product transformation and the negative of the price ratio(s) and a condition on capacity variable input use. In addition to Felthoven and Morrison Paul [74]’s application of a multi-product asymmetric transformation function, Färe et al. [71] measure CU in multi-product firms through the multi-product distance functions of Shephard [167]. This approach is not only readily estimated through either econometrics or Data Envelopment Analysis, but it also allows for undesirable outputs (“bads”) in addition to desirable outputs. This approach has spawned a large number of applications in the area of environmental performance, environmental damages, and fisheries, all too numerous to cite here, and even conservation (e.g., [75]). This approach also allows for the possibility that “too much” of the fixed factor may be employed by allowing for the directional distance function expansions of outputs (giving the output orientation of Johansen) and contracting inputs in the capacity measure.

Additional Considerations Multiple Fixed Inputs As discussed above, the dual economic approach, which is based on cost, profit, or revenue optimization and economic duality, readily accommodates multi-product production under joint production. However, this is predicated on the existence of a single quasi-fixed input that is the focus of incentives to invest or disinvest. When there are multiple quasi-fixed factors, optimization-based measures of capacity and CU are less obvious and more limited measures must be applied to define and measure capacity and capacity utilization [15, 162]. While it might be tempting to assume that movements along the long-run average total cost curve correspond to plant expansions derived from (proportional) increases in all quasi-fixed inputs, Chambers [31] shows that this is clearly not the case. Movements along the average total cost curve indicate plant expansion activities only when there is a single fixed input. Thus, as formally demonstrated by Berndt and Fuss [15], it may not be

13 Such

an aggregation approach (constant output mix) can be justified by Leontief aggregation of outputs or more generally Lewbel’s Generalized Composite Commodity Theorem [123].

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possible to determine the maximum capacity output or rate of capacity utilization when there are multiple quasi-fixed or fixed factors. The indeterminacy problem is more severe in the presence of multiple products. Ray et al. [154] adopt a linear programming approach and build off of Ray [152] to develop a CU measure based on minimum short-run ray average cost under alternative scale assumption when there are multiple outputs and/or multiple fixed inputs. The concept of ray average costs inherently keeps multiple outputs in fixed proportions and thereby satisfies Leontief output aggregation to create a composite output. This being a short-run analysis, the scale of the fixed input vector does not change, but ray expansion of the output vector serves to change the average ray fixed cost.

CU in Natural Resource Industries The concepts of capacity and CU have been of particular interest in natural resource industries where output depends not only on the firm’s input but also on the resource stock. In fact, an explosion of conceptual and especially empirical work on capacity and CU arose in part as a response to emerging and critical practical problems in natural resource economics and concerns about sustainable exploitation of natural resources and pressures on natural resource stocks. The concept of capacity in industries exploiting a natural resource stock, whether renewable or non-renewable, must deal with two stocks of capital, physical capital K and natural capital N.14 The stock of natural capital, however, differs from physical capital, since natural capital is not under the control of the individual firm and hence should not be treated as a discretionary input. In addition, the resource stock imposes an upper limit on the total output that can be produced [108, 109, 147]. That is, regardless of the expansion of K and increased utilization of variable inputs, for given input prices and technology, output cannot exceed some level determined by N. Thus, key conceptual issues that arise in this context are the addition of the stock of natural capital, the observed and sustainable capacity output flows from this stock, and the requirement to compare capacity output not just to observed output for a measure of CU but to the sustainable target output for the industry [63, 64, 108, 109, 137, 138, 142, 147]. Ill-structured, incomplete, or severely attenuated property rights, regulatory structure, regional specificity, multiple stocks for different species, mobility of physical capital, and externalities all further complicate capacity issues in these industries [110]. Capacity and CU Under Regulatory Constraints Up to this point, the discussion has assumed that firms face constraints in the form of fixed or quasi-fixed inputs and possibly predetermined output (in costbased measures). In many contexts, firms can face additional constraints from,

14 Note that the multiple quasi-fixed stock problem of K

and N differs from the problem of multiple quasi-fixed inputs discussed above, since N is not directly controlled by the firms.

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for example, government regulations, which can and should be factored into CU measures as well [51]. For example, Averch and Johnson [9] demonstrated that regulation designed to ensure some rate of return on capital for the firm will induce some “excess” capacity and lower the rate of CU. Similarly, regulations in the form of rations, quotas, or other input or output restrictions can impact production and investment decisions that in turn affect CU. For example, in an effort to promote sustainable fisheries, fisheries managers often impose limits on harvests or inputs that can impact capacity and CU [163, 172, 173, 175, 185].

Dynamics The concepts of capacity and CU discussed above can also be extended to incorporate dynamic decision-making and adjustment. The primal measure is then based on behavior of the firm’s supply along an optimal path to the steady state, while the dual measure is based on the behavior of the dynamic value function in the stock of the quasi-fixed input. For example, Morrison [132] and Fousekis and Stefanous [78] extended the static (single-period) concept of capacity to a dynamic concept in which there is multi-period adjustment of K that reflects the costs of adjustment. Abel [1] allows firms to choose optimal utilization rates of quasi-fixed factors, and capital investment is negatively related to capital utilization along the path to steady state, but capital utilization and investment are positively related. Dynamic models, based on dynamic optimization (e.g., minimizing the present value of costs or maximizing the present value of profits), can help to explain the gradual adjustment that occurs in capital stocks as they move over time toward some target level of quasi-fixed inputs [132]. They can also be modified to incorporate expectations about future output demand and input prices [133] or to account for changes in natural resource stocks over time. Uncertainty CU can also be affected by uncertainty. Stochasticity in demand is one important source of uncertainty that impacts CU [176]. When demand is stochastic, the firm faces potential losses when capacity turns out to be too high or too low. If future demand is higher than the chosen capacity can handle, the firm will lose potential revenue from sales. Conversely, if capacity turns out to be too high relative to demand, the firm must bear the cost of excess capacity. Hence, in an uncertain environment, the choice of capacity (and hence CU) depends on both the distribution of future demand and the magnitude of these two potential losses arising from a mismatch of capacity and demand [176]. Investment that is largely irreversible, so that expenditures are mostly sunk costs that cannot be recovered, creates an additional source of uncertainty [6, 38, 143, 144]. In addition, the investments can be delayed, giving the firm an opportunity to wait for new information to arrive about prices, costs, and other market conditions before it commits resources [7, 143, 144]. Irreversibility makes investment especially sensitive to various forms of risk, such as uncertainty over future

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product and input prices and operating costs that determine cash flows, uncertainty over future interest rates, and uncertainty over the cost and timing of the investment itself. Irreversibility can arise through capital that is firm or industry specific, so that a different firm or industry cannot productively use the capital, and hence the capital becomes a sunk cost. Government regulations or institutional arrangements can also make capital expenditures a sunk cost. Irreversibility and sunk costs in turn impact, PK∗ , the shadow value of capital given by HK in Equation (2), and therefore CU π = ππ∗ . In natural resource industries, uncertainty through fluctuations in natural resource stock size and production quotas, and more generally environmental variability, introduces another source of uncertainty over the optimal level of capacity [88]. In industries where property rights are not fully complete and structured, individual production rights can be imposed and tradable between firms. These production rights can then be considered a distinct type of capital in which investment is required for a firm to operate, and the CU measure can be adjusted to account for this additional source of investment ([185].

Imperfect Competition Imperfect competition can also lead to excess capacity, since it causes each firm to produce below its cost-minimizing full capacity in order to earn higher profits. Studies that have examined the relationship between imperfect competition and capacity include Cassels [28], Gabsewicz and Poddar [81], Fagnart et al. [61, 62], Kim [107], Todorova [180], and Dixon and Savagar [59].

Other Capacity and Utilization-Related Concepts Input Capacity Based on Quasi-Fixed Inputs The measures of capacity and CU discussed above focus on output levels or their dual representation through costs or profits. However, as noted above, in many industries, such as fisheries, regulations (designed, e.g., to promote sustainability or other objectives) may limit the number of firms or plants, output per firm, and/or the capital stock. Focusing only on output-based CU levels begs the question of what should be the optimal input levels or the number of operating units. For example, the literature on fisheries repeatedly stresses the potential for “overcapitalization” or “excess harvesting capacity,” i.e., too many resources chasing too few fish or production that is wasteful or not at minimum cost or maximum net benefit to society. In this case, the need is for information on capacity and CU that is based on output levels, but expressed in terms of inputs, typically the capacity base given by K [108]. This gives rise to the concept of input capacity, in which the output levels are taken as exogenously fixed at the industry level or at the level of the individual firm, and these in turn determine the corresponding appropriate level of K without restrictions on the availability of variable inputs (borrowing from Johansen’s [97]

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output-oriented plant capacity). The potential contraction in the capacity base K that can still support existing output levels can be imputed and compared to the existing capacity or capital level to create input-oriented CU measures. This input-oriented CU measure can be formalized by first defining a measure of potential capital K as the minimum amount of capital that, in a given unit of time, produces the existing or target output under customary and usual working conditions, provided the availability of variable factors of production is not restricted K [13].15,16 Formally, CU K = Potential , where the subscript K indicates that it is K a capital input-oriented measure of capital utilization. Note that, as with technical output-based measures of CU, this measure can never exceed 1. A value of CUK < 1 indicates the potential capacity or capital contraction that could be achieved and still maintain current or target output levels. This definition is closely related to capital utilization (see discussion below) or even input-oriented technical inefficiency, where the key difference is the absence of any restriction on the availability of variable inputs.17 Alternatively, input-based CU can be defined in a manner analogous to the output-based definition of CU based on economic optimization. For example, under the assumption of cost minimization, CU for capital could be defined as CU K = K∗ K , where K* is the long-run cost-minimizing level of capital given the level of output. In this case, CUK can be greater or less than 1. As with the previous output definitions based on optimizing behavior, it provides a signal regarding incentives to invest when CUK > 1 or disinvest when CUK < 1.

Capacity Utilization Versus Capital Utilization Capacity utilization, whether based on outputs or quasi-fixed inputs, is sometimes conflated with capital utilization. However, these are two distinct concepts that coincide only under some very specific conditions. There are several definitions of capital utilization.18 Betancourt [18] refers to capital utilization as the duration of operations of productive processes. Betancourt and Clague [20] define capital utilization as the proportion of time that capital is working productively. Bosworth and Dawkins [23] refer to capital utilization as the timing of input flows and in particular to shift work and overtime. A traditional

15 More recently, Cesaroni et al. [29, 30] define a measure of the input-oriented plant capacity measure that compares variable input levels relative to the amount of variable inputs compatible with a zero output. This measure is discussed below. 16 The minimum stock of capital needed to produce a given level of output can be derived using the factor requirements function or inverse production function (see [57, 167], and [108]). The factor requirements function depicts the production possibilities set and relates the minimal amount of an input required to produce a vector of outputs: Kt = g(Y1t , Y2t , . . . , YMt ),where the outputs are exogenously fixed and K is an endogenous stock. 17 Input-oriented technical inefficiency relates to how much the input use of a firm could contract if used efficiently in order to achieve the same output level. 18 General summaries of capital utilization are given in Betancourt [19], Betancourt and Clague [20], and Winston [187].

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strand of the literature refers to the ratio of actual usage to a maximum value derived from technical-engineering specifications [76]. Schworm [161], Hulten [94, 95], and Lee [122] define capital utilization as the ratio of capital services to the stock of capital. However, the endogenous capital utilization literature allows the rate of capital utilization to vary with input prices and output levels or price and hence is a choice variable to firms. The rate of capital utilization, whether in the form of the length of time that capital is operated or the intensity of its operation per unit of time, is determined by profit maximization. The concept of capital utilization, which relates to underutilizing a given capital stock, is distinct from the economic concept of capacity utilization described above. Capital utilization captures how much of the existing capital stock is being utilized, and CU provides information about short-run vs. long-run equilibrium levels of output and capacity output relative to one or more fixed factors that includes the capital stock and economic incentives for investment and disinvestment. These measures coincide only when there is only one fixed input, all variable inputs are in fixed proportions to the fixed input, and production is characterized by constant returns to scale [4, 13, 15]. One context in which confusion over the distinction between capital utilization and capacity utilization has arisen is fisheries. The fisheries economics literature [36–38], which typically assumes a single composite input (fishing effort), has often conflated a version of capital utilization (i.e., utilization of fishing effort) with capacity utilization.19 Although the specification of the canonical bioeconomic model meets the restrictive conditions for these two concepts to be the same (see, e.g., [36, 47, 89]), this model overlooks that the production relationship between effort and catch is a stock-flow relationship, in which effort is applied to the stock of natural capital (here fish) to produce a flow of output. Hence, variations in the utilization of fishing effort do not provide a direct and linear relationship with variations in catch, even under constant returns to scale in effort, and full utilization of effort can give different catch levels depending upon the size of the stock of natural capital. The effort aggregator function for rivalrous inputs must be linearly homogeneous for consistent aggregation.20 Thus, although capital stock measures are often used to measure potential or capacity output, as well as capital utilization, the equivalence of these measures must be understood to be valid only under very limited conditions.

Variable Input Utilization Capacity utilization is also distinct from variable input utilization, which measures the ratio of optimal use of a variable input to observed use [65]. As with economic measures of capacity utilization, the behavioral objective determines the nature of

19 See

Kirkley and Squires [108] for a review of this literature.

20 When there are externalities beyond those due to the common resource stock, such as congestion

or knowledge spillovers with technical change, then linear homogeneity is no longer enough to ensure consistent aggregation [174].

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the optimal input use, which can be based on cost minimization or maximization of revenue, profit, or output. When the rate exceeds (falls short of) unity, that variable input is overutilized (underutilized) relative to optimal utilization, and the firm or plant has a shortage (surplus) of that variable input. The difference between variable input utilization and capital utilization is that the former refers to a variable input and the latter refers to a fixed or quasi-fixed factor. Moreover, the variable input-based measures involve the returns to variable inputs, given the existing capital base, that result from an expansion in scale of production holding the capital factors fixed [142]. Both this and the output-based measures of CU entail imputing the potential for expansion of output, given fixed inputs. However, the input-based measure reflects returns to a particular input, rather than returns to scale, and the measures will only be numerically the same with constant returns to the variable input.

Capacity Utilization and Productivity Capacity utilization has also played an important role in studies of productivity. Productivity measures within the Denison-Kendrick-Jorgenson-Griliches-Solow framework (see [99, 100]) traditionally assumed that producers are in long-run equilibrium, so that the firm’s output is always at the long-run equilibrium level, i.e., the point of tangency between the short-run unit or average total cost curve and the long-run unit or average cost curve. However, firms may instead be in shortrun or temporary equilibrium [14, 16, 94, 134] due to unexpected demand shocks that in turn lead to under- or overutilization of capacity or sudden changes in factor prices. Temporary equilibrium, such as that associated with the business cycle, can bias measured productivity growth away from its long-term growth path. Periods of low growth and low demand alternate with periods of high growth and growth above long-term trends. Stocks of capital and other quasi-fixed or fixed factors of production are difficult to rapidly adjust, so that periods of low growth and low demand are associated with underutilization of capital, other quasi-fixed factors, and capacity. The total factor productivity (TFP) approach to measuring productivity assumes that the total growth rate of real output is comprised of two components: one that reflects movements along the aggregate production function due to growth rates of the factor inputs and a residual that reflects shifts in the aggregate production function due to changes in the efficiency of production or total factor productivity (TFP) [94, 170]. However, for the reasons described above, the TFP residuals calculated using capital stock data fluctuate procyclically along with the rate of utilization [14, 94, 96], and these fluctuations in the rate of service flows and utilization of the stocks in turn tend to obfuscate the longer-term movements in components of the TFP residual. Thus, when accounting for changes in utilization, changes in TFP should be decomposed into three components: changes in variable inputs, changes in both the stocks and utilization of quasi-fixed or fixed factors, and productivity [14, 94]. The resulting decomposition indicates the amount by which

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the efficiency-driven change is cyclical and how much is driven by the long-run trend [14].21 An alternative approach to the Berndt-Fuss one uses Johansen’s [97] concept of plant capacity to allow for variations in CU in a primal, nonparametric specification of technology [48, 189]. This approach decomposes the Malmquist productivity index into technical efficiency change, variations in plant capacity utilization, and production frontier shifts.

Measurement of Capacity and CU Given the conceptual foundations described above, we turn now to an overview of the empirical measurement or estimation of capacity and CU. The measures can be classified as macroeconomic and microeconomic (firm level), where the latter include both frontier-based and optimization-based approaches.

Macroeconomic Approaches United States (US) Federal Reserve The US Federal Reserve calculates macroeconomic measures of output, capacity, and CU for the United States’ industrial sector, comprised of industries within the manufacturing, mining, and electric and gas utilities subsectors, creating the Federal Reserve’s Industrial Production and Capacity Utilization index [44, 131, 149, 166].22 The Federal Reserve and the Census Bureau have adopted a definition of capacity that assumes the full employment of all variable factors of production and the use of only the equipment in place and ready to operate to give “full production.”23 The CU measure is the ratio of the actual level of output to this definition of sustainable maximum level of output or capacity, defined as the maximum output each plant in a given industry can maintain within the framework of a realistic work schedule, taking account of normal downtime and assuming

21 The

Berndt-Fuss [14] approach to the Solow residual does not, however, remove the procyclical component of the residual that can arise, for example, through entry and exit of firms over the business cycle. This approach to utilization also does not generalize to multiple capital goods. 22 The Federal Reserve’s capacity and CU measures are derived through a complex procedure [44, 131, 148, 166]. Since 1990, the Bureau of the Census’s Survey of Plant Capacity forms the primary source of utilization rates. The Federal Reserve combines the available survey evidence on utilization with their industrial production indices to obtain a consistent system of output, capacity, and CU [44, 166]. 23 The questions that the surveys forming the measures asked pertaining to rates of utilization have changed somewhat over time. Nonetheless, the definitions appear close enough that the time series are treated as a single series without any ad hoc adjustments.

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sufficient availability of inputs to operate the machinery and equipment that is in place. In principle, this is the engineering-technical measure discussed by Klein [113], Klein and Summer [115], Klein et al. [116], etc. However, as Shapiro [166] observes, “The Federal Reserve procedure, moreover, mixes engineering and economic notions of capacity, particularly in its assumption that seasonal peaks in output are unsustainable.”

Peak-to-Peak Measurement of CU The peak-to-peak approach of Klein [113], Klein and Summers [115], and Klein and Preston [114], also known as trend through peaks, is another macroeconomic measure. It is an interpolation of peak values of output per unit input over time, adjusted for technical change [35, 114, 132, 135, 149]. This nonparametric approach intends to reflect the maximum attainable output, given K. Peaks in production per unit of K are used to represent full capacity, and linearly interpolated outputcapital stock ratios between peak years are used, in conjunction with data on K, to estimate the maximum attainable output between peak years (essentially a straight line is drawn between peaks). This approach calculates the ratio of a composite output to K (a ratio of output to an input), identifying the peak values over time, assuming that these peaks depict the full CU given normal operating and economic conditions. This approach is essentially a partial productivity measure, Y/K, and as such suffers from the standard problem of ignoring scarcity of other factors besides K and input substitution possibilities. Ragan [149] and Christiano [35] discuss many of the limitations and extensions to this approach, such as modified trend through peaks of Dhrymes [56] and demand for capital of Hickman [92]. The Wharton School index of CU is an example of a peak-to-peak measure [114, 149]. Cyclical peaks for each of the component indices of the Federal Reserve Board’s Index of Industrial Production are denoted, and then linear segments between successive peaks are fitted. The trend lines through peaks are assumed to represent an index of capacity output, on a base of actual output, which is the same base used in the Federal Reserve Board’s Index of Industrial Production. Harris and Taylor [90] estimate Cobb-Douglas and CES production functions using peak output data to estimate capacity output, which in turn is used to estimate CU. This CU estimate, used as a benchmark, is used to evaluate estimates produced by the Wharton method and the output-capital method. Full Employment Maximum Output Another macroeconomic measure of capacity and CU depends on the notion that the full employment level determines the possible or potential capacity output [80, 135, 140]. This approach is based on the “maximum” level of labor input, although it can include other inputs that are assumed to be important to “fully” employ. Incorporating only a single input excludes the impact that other inputs can have on the maximum output. Incorporating multiple inputs yields a measure that is essentially the same in principle as the Federal Reserve measure that assumes the full employment of all variable factors of production. However, it raises the question of how to define “full” or “maximum” levels of these other inputs.

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Production Function Approach Another macroeconomic approach, which is a variant of the Federal Reserve and full employment maximum approach, was developed based on the production function. Klein [113] and Klein and Preston [114] develop the production function approach to capacity and CU for a firm or industry, and Christiano [35] provides a comprehensive survey. The production function for actual operations may be specified as: Y = f (L, D), where (following Klein) Y denotes output as a flow, L denotes labor measured as actual employment and a flow, and D here denotes the actual flow of capital services. At full capacity, YC = f (Lf , K), where YC denotes full capacity output, Lf denotes the input flow of the fully employed work force, and K denotes the stock of capital that is fully utilized. Capacity output is thus the production flow associated with the input of fully utilized manpower, capital, and other relevant factors of production. Since actual capital and labor cannot be assumed fully utilized, this method requires some adjustment of the inputs [8, 35, 165], which gives rise to the utilization-adjusted production function approach. This approach (1) uses available surveys to estimate the “natural rates” of capacity and labor utilization above which inflation begins to accelerate; (2) estimates a production function with utilizationadjusted capital and labor inputs; and (3) defines potential output as the level of output obtained when both capital and labor are at their estimated natural rates [8, 141]. This basic approach, employed by the International Monetary Fund, requires the existence of an aggregate production function and the natural rate of unemployment and hence is a macroeconomic measure. This approach can also be estimated with adjustments for the mean age of the capital stock to account for vintage effects [8, 35].

Microeconomic Frontier-Based Approaches Production economists have developed empirical methods for estimating measures of capacity and CU based on microeconomic foundations and the theory of the firm. These include both frontier methods (based on production or distance functions) and optimization-based methods (based on cost, profit, or revenue functions). The frontier methods are described here, followed by a description of optimization-based methods in “Microeconomic Optimization-based Approaches.”

Production Frontier Methods The capacity measures generated from frontier models in some cases have been deemed technological-economic, because the analysis restricts the production technology to those observed in the data, which inherently captures underlying economic decisions [70, 91]. Within this framework, the capacity estimate refers to the maximum potential or frontier level of output that could be produced given the fixed factors, states of technology and environment, and unconstrained variable factors. Empirically estimating the production technology by frontier methods involves fitting the production function to the data points representing

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observed output-input combinations. This can be estimated nonparametrically or parametrically. Data Envelopment Analysis: A Nonparametric Frontier Approach Several nonparametric frontier primal approaches are available to estimate capacity output and CU for both output-oriented and input-oriented approaches. The most widely applied approach, proposed by Färe et al. [67],24 is based on Data Envelopment Analysis (DEA) developed by Charnes et al. [32], which implements the Johansen [97]-Färe [65] plant capacity definition. DEA has also been used to develop and measure output-oriented, economic capacity and CU for the behavioral objectives of profit maximization [26, 41], revenue maximization [70, 125], and cost minimization [49, 154]. DEA is a form of activity analysis or linear programming that builds off of Farrell [72, 151]. This approach is typically deterministic, so that the frontier envelops all observations, and no allowance is made for outputs that could potentially lie above the best-practice frontier due to stochastic factors. DEA uses linear programming to construct a piecewise linear representation of the production frontier. This is usually done nonparametrically, i.e., without an explicit functional form, although a parametric approach has been used [2]. These piecewise linear frontiers envelop the observations as tightly as possible subject to certain minimal production axioms. Deviations from the frontier can result from technical inefficiency and variable input use that differs from the firms that define the best-practice frontier. The Färe et al. [67] DEA specification, an output-oriented DEA linear program, computes the maximum proportionate increase in outputs when variable inputs are allowed to vary and be fully utilized but fixed inputs are bounded by their observed value for each observation or firm.25 The chapter entitled  “Stochastic Frontier Analysis: Foundations and Advances I” in this volume develops this DEA model. The radial DEA approach to capacity measurement effectively converts the multiple products into a single composite output through a radial expansion of outputs. Outputs kept in fixed proportions for different input levels give Leontief aggregation and the ray measure of capacity and CU considered by Segerson and Squires [162]. The DEA approach effectively converts the heterogeneous capital stock (multiple fixed factors) into a single measure of the capital stock, i.e., a composite fixed factor, to solve the indeterminancy problem raised by Berndt and Fuss [15]. Stochastic Production Frontier The stochastic production frontier represents the efficient relationship between inputs and outputs [3, 84, 130, 178]. That is, the firms or plants that produce

24 Eilon and Soesan [60] suggested but did not empirically implement the use of linear programming. 25 Other nonparametric frontier models are available that are consistent with the underlying production technology [184], including the free disposal hull model and the “order-m” frontier, which have been used to study efficiency in other industries [25, 45, 169].

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the maximum potential output given a bundle of inputs define the best-practice production frontier and are technically efficient. Firms or plants that produce less output given comparable input bundle, technology, etc. are below the frontier and are technically inefficient. Inputs can be flows or stocks, depending upon the situation and specification. The production frontier could also include control variables such as the state of the environment. The stochastic approach allows for random or stochastic effects upon production, such as weather, bad luck, machine breakdowns, and the like. The stochastic production frontier can be extended to include an equation that explains the technical inefficiency according to exogenous or predetermined variables and that is simultaneously estimated by maximum likelihood [10, 11, 40, 120, 156]. The integrated approach provides consistency in assumptions about the distribution of the inefficiencies. While stochastic frontier techniques have been developed primarily to estimate technical efficiency, they can be readily modified to measure capacity utilization [101, 108–110, 142]. Stochastic Multi-product Distance Function The single-product stochastic production frontier discussed above does not allow for situations where firms produce multiple outputs. To address this concern, Kirkley and Squires [108] and Färe and Grosskopf [66] proposed the primal multioutput stochastic distance function for multi-product technologies, which permits various assumptions about how the composition of multiple outputs may change when operating at full capacity.26 Felthoven and Morrison-Paul [74] empirically estimate such a function for an Alaskan fishery. Fousekis [77] and Felthoven [73] estimated a stochastic ray distance function [126] for multi-product fisheries. Such an approach maintains outputs in fixed proportions and effectively converts the multiple outputs into a single output. (See chapters on  “Index Numbers and Productivity Measurement,” and  “Stochastic Frontier Analysis: Foundations and Advances I” in this volume for more detailed discussion.) As with the stochastic production frontier, the inefficiency function can be explicitly modeled and simultaneously estimated with the stochastic multi-product distance function [93]. The multi-product distance function can be extended to an input orientation [39]. Rather than looking at how the output vector can be proportionally expanded with the input vector held fixed, an input-oriented distance function considers by how much the input vector can be proportionally contracted with the output vector held fixed. The input distance function is defined on the input set, L(Y), as DI (Y, X) = max {ρ : (X/ρL(Y))} [167]. The input set, L(Y), represents the set of all input vectors X that can produce the output vector Y, where the properties of L(Y) are given by Shephard [167]. Under constant returns to scale, the input distance function is equivalent to the inverse of the output distance function (i.e.,

26 Other options for considering multiple outputs in a general stochastic frontier framework include

the use of polar coordinates, canonical regression, and instrumental variables, although all of these options have some limitations.

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DO = 1/DI ) [69]. Econometric estimation of the input-oriented stochastic multiproduct distance function is similar to the output-oriented distance function, where homogeneity of degree one in inputs is imposed on a parametric form of DI (Y, X), such as the translog [39]. Inputs are normalized by one of the inputs, say Xn . The stochastic multi-product distance function can also be extended to the directional distance function [66]. The directional distance function is associated with an explicit direction in which efficiency is gauged. It requires specification of a direction vector, so that outputs are expanded and inputs are contracted in that direction. We do not develop this approach further, but note that it can be adapted to account for capacity and CU. Finally, as with the single-product production functions and frontiers, the parametric multiple-product stochastic distance function approach raises identification issues due to potential endogeneity of the regressors (outputs, inputs), which can lead to biased and inconsistent parameter estimates [157]. However, semiparametric alternatives are available for both the output and input distance functions. The earliest approach is Corrected Ordinary Least Squares, implemented by Lovell et al. [128] for multi-product stochastic distance functions.27 Pitt and Lee [145] and Schmidt and Sickles [159], reviewed by Sickles [168], developed this approach within a panel data context.28 The advantage of this semi-parametric approach is that it avoids the need to specify particular parametric distributions of the inefficiency term. Although the distribution of the inefficiency term is one-sided, the terms are intrinsically latent and unobservable components. The assumption that inefficiency is time-invariant is quite strong, although the model is relatively simple to estimate if efficiency is specified as a fixed parameter instead of as a random variable. (Greene [84] proposed flexible one-sided distributions such as the gamma, which allows the distribution to be shaped by the data.) Sickles [168] reviews other panel frontier estimators that differ from the parametric approach. Stochastic Ray Production Functions Löthgren [126, 127] proposed the stochastic ray production function as an alternative to the stochastic multiproduct distance function. Felthoven [73] and Felthoven

27 The

COLS function is fitted in two steps.  first step involves interpreting the unobservable  The term − ln DOj as a two-sided i.i.d. N 0, σv2 distributed random error, vj , and estimating the (translog) distance function using ordinary least squares (OLS). In the second step, the OLS estimate of the intercept parameter, γ 0 , is adjusted by adding the largest negative OLS residual to it, so that the function no longer passes through the center of the observed data but bounds the data from above (i.e., envelops the data as a frontier). The distance measure for firm j, j = 1, 2, . . . , J, is then calculated as the exponent of the corrected OLS residual, forming the technical inefficiency term.    28 The fixed-effects model can be written as [159]: lnY j t = β0 + lnXj t + vj t − uj = β0 − uj +   lnXj t + vj t = αj + lnXj t + vj t . Once αˆ j are available, uˆi = maxi αˆ j – αˆ j ≥ 0. The approach assumes that the most efficient  unit  in the sample is 100% efficient. Firm-specific efficiency can be obtained from: T ˆE j = exp −uˆ j . Cornwell et al. [43] extend the model to allow for time-varying efficiency, and Rashidghalam et al. [150] give further specifications and discussion.

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and Morrison-Paul [74] empirically applied this approach to a primal estimate of capacity. The ray frontier is conditioned on the polar-coordinate angles representing the output mix and the inputs [127, 77]. All random error terms are specified to affect the outputs radially, given the output mix that is exogenous under the specification. A stochastic version of the ray production function can also be defined. As with the stochastic production frontier, partitioning the input vector into fixed and variable inputs, XK and Xv , and only specifying XK in the stochastic ray production function gives a measure of Johansen’s plant capacity output. Another option estimates with both XK and Xv and evaluates at maximal values of Xv , which would differ from Johansen’s plant capacity output since Xv is bounded. Estimation can take several approaches. As with the stochastic production frontier, the inefficiency function can be explicitly modeled and simultaneously estimated with the stochastic ray production function.  In this case, ui is distributed as a truncation at zero of the distribution N δZ, σu2 [93]. Alternatively, Corrected Ordinary Least Squires (COLS) can be used [128], as discussed above. Estimates of industry capacity output are then obtained by horizontal summation of the individual firm capacity outputs [73]. As with the single-product stochastic production frontier models, CU measures can be constructed as the ratio of technically efficient output to capacity output. Firm-level technical efficiency scores (i.e., estimates of DO (Yi , Xi )) can be computed as TEi = E[exp(ui |ei )], where ei denotes the composite error term. Kumbhakar and Lovell [119] provide further details on the specific formula used in the conditional expectation and the likelihood function for the maximum likelihood estimation. Nonparametric Deterministic Frontier for Minimum Cost- and Profit-Based Capacity The above measures focus on use of frontier methods to estimate technologicaleconomic measures of capacity rather than cost- or profit-based measures. However, nonparametric frontier methods can also be used to estimate measures of capacity and CU that explicitly incorporate profit-maximizing behavior. For example, Coelli et al. [41] developed linear programming models to estimate their short-run profit measure of capacity and CU. Their approach involves the following steps: (1) develop a piecewise linear capacity possibility frontier, essentially applying the Johansen [97] plant capacity framework, to obtain a measure of ray capacity, (2) estimate the maximum short-run profit for each firm, (3) estimate the output-oriented technical efficiency of each firm using the standard DEA linear programming approach, and (4) use a fourth and final linear programming model to measure their ray economic capacity measure for each firm. Similarly, Briec et al. [26] propose the following approach to implement the nonparametric, deterministic frontier-based approach to minimum cost- and profitbased notions of capacity and CU: (1) estimate the short-run minimum average total cost by solving a variable cost function relative to a constant returns to scale technology [146], (2) estimate the long-run minimum average total cost by computing a total cost function relative to a constant returns to scale technology [151], and (3) given fixed inputs but endogenous outputs, such that installed capacity

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is utilized ex post at a tangency level, compute the output level and costs at the tangency between the short-run and long-run average cost functions. Briec et al. [26] also develop comparable economic capacity under variable profit maximization with endogenous multiple outputs. Capacity can also be incorporated into decompositions of static economic efficiency that integrate primal and economic (dual) notions of CU using nonparametric production frontiers [41, 49]. For example, Coelli et al. [41] use ray economic capacity (the largest radial expansion or contraction of the output vector coinciding with the largest possible profit) to decompose the gap between observed and maximum short-run profit into unused capacity, technical efficiency, input-mix allocative inefficiency, and output-mix allocative inefficiency.

Microeconomic Optimization-Based Approaches The methods described in the previous section are based on the estimation of frontier models that explicitly link inputs to outputs. These typically yield output-based, i.e., primal, measures of capacity and CU. Dual measures of CU, on the other hand, are typically estimated through parametric estimation of variable cost functions G(Y, W, K), variable profit functions H(P, W, K), or revenue functions R(P, K). The most important issue in choosing whether to estimate a cost, profit, or revenue function is whether the firm minimizes costs for a given output bundle or maximizes revenue or profits. A related determinant is whether outputs are endogenous. Endogenous outputs can be incorporated by estimating a profit or revenue function (depending on whether some inputs are variable or not), or by estimating a cost function that includes marginal cost functions for outputs and uses instrumental variables to account for endogenous outputs Y. Typically translog or generalized Leontief flexible functional forms are specified, although other functional forms are occasionally used. The minimum variable cost function G(Y, W, K), in which the firm minimizes costs for a fixed (exogenous or predetermined) output bundle, is simultaneously estimated with conditional demand functions for variable inputs derived using Shephard’s Lemma, using either Zellner’s seemingly unrelated regression (SUR) or maximum likelihood (ML) (both typically iterated to convergence) to account for correlation of errors across equations. Estimating a system of equations gives greater efficiency of parameter estimates. The variable profit function H(P, W, K) is simultaneously estimated with variable input demand and variable output supply equations obtained through Hotelling’s Lemma, again using either SUR or ML (iterated to convergence). The revenue function R(P, K) is simultaneously estimated with variable supply functions, using either SUR or ML estimation. When data are time series, either a time trend or time fixed effects are included to account for changes over time, including exogenous technical change. When data are panel, firm fixed or random effects account for unobserved differences among firms due to differences in management skills (e.g., managers) or other characteristics of the firm or the environment in which it operates (along with either time fixed effects

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or a linear time trend). When data are available on managers, a three-way fixedeffect model accounts for manager skill, firm characteristics, and time fixed effects [188]. Standard errors should be heteroscedastic and autocorrelation consistent. Tests for homothetic output separability or Hicks-Leontief aggregation can reduce the number of outputs and give a two-stage production process in which some outputs are combined into an aggregate output under the assumption of revenue maximization and under homothetic output separability. Similar approaches can aggregate variable inputs.

Concluding Remarks The concepts of capacity and capacity utilization were originally and largely developed to address issues and questions in macroeconomics and industrial organization. Developments in microeconomic theory (especially production economics) and econometrics, increased availability of data (especially firm level), and the emergence of other issues, such as energy shocks, renewable resource management (especially overcapacity in fisheries), and appropriate levels of capacity in other industries such as health care (e.g., hospitals) and banking, all stimulated further refinements in the concepts of capacity and capacity utilization and further development in methods of measurement. The emergence of readily available firm-level data led to an orientation away from industry-level production technologies to firmlevel specifications. Two main developments in production economics contributed to the analysis of capacity and capacity utilization. The important development in the production economics of duality led to a reorientation away from the primal to the dual specification of production technology and different firm behavioral objectives of cost minimization, revenue maximization, and profit maximization. The second main development in production economics was the concept of economic efficiency of Debreu [52], Koopmans [[72, 117], and Shephard [167], notably production frontier concepts and the decomposition of economic efficiency into its components of technical, allocative, and scale efficiency. Much of the analysis of capacity and capacity utilization in this vein utilizes Shephard’s [167] distance function. Empirical methods, borrowed from operations research – especially Data Envelopment Analysis [32] – supplementing the development of econometrics, have been applied to the distance function framework. The emergence of Data Envelopment Analysis (coupled with the frequent problem of a paucity of cost data, especially at the firm level) led to a resurgence of primal approaches and applications, especially at the firm level. Data Envelopment Analysis facilitated decompositions of capacity and capacity utilization to account for technical efficiency. These further developments in capacity and capacity utilization arising from duality and economic efficiency (especially the distance function framework), refinement of the concepts and measurement (especially econometrics and Data Envelopment Analysis), and application to new issues occurred almost entirely

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within the field of production economics. The distinct two strands of capacity and capacity utilization within macroeconomics and production economics (now the entire approach of industrial organization applications) continue to reflect their different orientation, purposes, and even sources of data and methods of measurement.

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Aggregation of Efficiency and Productivity: From Firm to Sector and Higher Levels

25

Valentin Zelenyuk

Contents Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The Aggregation Problem: A Brief Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The Essence of the Aggregation Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The Evolution of the Aggregation Literature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Aggregation of Efficiency Scores . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Individual Primal and Dual Efficiency Scores . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Group Primal and Dual Efficiency Scores . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The Fundamental Aggregation Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Understanding the Fundamental Aggregation Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . Aggregation of Aggregates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Price-Independent Weights . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Aggregation of Productivity Indexes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Individual Malmquist Productivity Indexes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Aggregation Problem: Inter-temporal Perspective . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Aggregation of the MPIs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Geometric vs. Harmonic Averaging . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Decomposition and Aggregation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Aggregation for Scale Measures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Aggregation with Possibility of Reallocation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Aggregate Technology and Measures with Reallocation . . . . . . . . . . . . . . . . . . . . . . . . . . Reallocation vs. No Reallocation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Aggregate vs. Individual Reallocative Measures of Efficiency . . . . . . . . . . . . . . . . . . . . .

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The author acknowledges support of the University of Queensland and from the ARC grants (ARC FT170100401). V. Zelenyuk () School of Economics and Centre for Efficiency and Productivity Analysis (CEPA), The University of Queensland, Brisbane, QLD, Australia e-mail: [email protected] © Springer Nature Singapore Pte Ltd. 2022 S. C. Ray et al. (eds.), Handbook of Production Economics, https://doi.org/10.1007/978-981-10-3455-8_19

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Remarks on Estimation of Aggregate Scores . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Concluding Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Cross-References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Abstract

Here we consider various cases where researchers are interested in measuring aggregate efficiency or productivity levels or their changes for a group of decision-making units. These could be an entire industry composed of individual firms, banks, and hospitals or a region composed of sub-regions or countries, or particular sub-groups of these units within a group, e.g., sub-groups of public vs. private or regulated vs. non-regulated firms, banks, or hospitals within the same industry, etc. Such analysis requires solutions to the aggregation problem – some theoretically justified approaches that can connect individual measures to aggregate measures. Various solutions are offered in the literature, and our goal is to try to coherently summarize at least some of them in this chapter. This material should be interesting not only for theorists but also (and perhaps more so) for applied researchers, as it provides exact formulas and intuitive explanations for various measures of group efficiency, group scale elasticity, and group productivity indexes and refers to original papers for more details. Keywords

Efficiency · Productivity · Aggregation · Industry efficiency · Duality JEL Classification Numbers

D24, C43, L25

Introduction An aggregate perspective is very important for theory and perhaps even more so in practice. Even if a researcher estimates the efficiency of individual units, she/he might still (and usually do) want to have just one or a few aggregate numbers that summarize the individual estimates. Such aggregate numbers would be especially useful if the number of individual units is too large to report all of them and especially to comprehend them all for understanding the overall picture. Indeed, hardly anyone would want to read hundreds of individual efficiency scores and would rather demand a summary – some aggregate efficiency or productivity measures that will give a big picture about the efficiency or productivity situation in the industry or sub-groups of interest within it. The key question here is therefore: How to meaningfully aggregate the individual efficiency and productivity scores or indexes? A natural answer would be: “Take an average!” But, which one? Is it arithmetic, geometric, harmonic, or any other?

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And, much more importantly: Should it be a weighted or a non-weighted average? Or, more generally: What are the most meaningful (from economics point of view) ways to aggregate the individual efficiency and productivity scores or indexes, of potentially many individuals, into one number representing aggregate efficiency of productivity of a group? The goal of this chapter is to provide some answers to these fundamentally important questions, by summarizing the recent developments in the literature. In a nutshell, the results summarized here provide applied researchers with the formulas for group efficiency, group scale elasticity, and group productivity indexes. Importantly, in all these formulas, the weights of aggregation have a fairly intuitive economic meaning, yet they are not ad hoc but derived mathematically via economic theoretical reasoning.1

The Aggregation Problem: A Brief Background The problem of finding a measure (a score, an index) representing a group of individual measures is called an aggregation problem – a problem that has been studied in many fields, including economics. In the field of productivity and efficiency analysis, this problem have been raised starting, at least, with the classical works of [17, 19], and later followed up by [30], and most thoroughly theoretically scrutinized by [1] and critically evaluated by [54]. More recently this important analytical problem was addressed and to some extent resolved by [7, 12–14, 36, 48, 48, 56] and most recently [31, 33], to mention a few. Here, we will briefly summarize the essence of the key results from these and other works.2

The Essence of the Aggregation Problem As in the general context, the most important issue here is the choice of weights in the aggregation. To vividly illustrate the point, consider an example of an industry with many firms, most of which are small, while a very few large firms take most of the industry share.3 Now suppose that those small firms are very efficient and suppose for simplicity of computation they are 100% (or nearly that) efficient.

1A

different area of the aggregation questions that focuses on the aggregation of inputs or aggregation of outputs for a firm (e.g., to reduce the dimension of the model) is not considered here and can be found in [6, 11, 51, 53] and the references therein. We also do not consider the question of aggregation of indexes with respect to different references (e.g., time periods) for the same firm, which can be found in [16] and the references therein. 2 This chapter is a substantially revised, extended, and elaborated material that I presented earlier, in Chapter 5 of [47]. 3 While this is a generic example, a reader might have realized that many industries in the real world have a similar composition, often resembling the so-called Pareto principle, more casually

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Meanwhile, suppose those big firms are much less efficient, say 50% efficient.4 If, for such an example, a researcher were to use the simple (equally weighted) arithmetic average, then such aggregate efficiency score would indicate the industry is close to 100% efficient! On the other hand, if another researcher wanted to use a weighted arithmetic average, then a dramatically different conclusion might be reached – depending on the weighting scheme. Indeed, if one takes the market shares as the relative weights and uses them in the weighted arithmetic average, then such an aggregate efficiency score will indicate that the industry is closer to 50% efficient. Thus, one would reach a dramatically different conclusion with opposite policy implications than from the equally weighted average! The essence of the problem here is in the nature of efficiency scores – by construction, they are “standardized” so that they are between 0 and 1, and, while this gives some advantages, the side effect of such standardization is that they lose the information about the relative weights of the firms that obtained these scores. Clearly, one may try to justify some other weights that may imply very different conclusions and thus different policy implications and this, in turn, emphasizes the importance of having justifications for the choice of weights.

The Evolution of the Aggregation Literature The early key ideas that attempted to take into account the economic weights of firms when aggregating their efficiency can be found in the seminal work of [17], where he proposed the concept of Structural Efficiency of an Industry. To be precise, [17] considered a single-output case and proposed taking the weighted arithmetic average of efficiency scores of individual firms in that industry, where the weights were the observed output shares of the firms within the industry. Importantly, note that Farrell had not given any formal theoretic justifications for such an aggregation scheme at that time and, in particular, had not justified why output shares were to be used for aggregating the input-oriented technical efficiency scores that he considered.5 Farrell also did not explain how to apply his idea for a multiple-output case. These limitations were perhaps among the main reasons for why Farrell’s concept of Structural Efficiency of an Industry had not attained a wider use in

known as “the 80/20 rule” postulating that about 80% share (e.g., of wealth, sales, etc.) is taken up by about 20% of members of a group. 4 Lower efficiency of large firms is not unusual and often was reported in the literature. It can arise, for example, due to the greater complexity of being a larger organization involving greater levels of hierarchy and thus implying potentially greater principal-agent problems or requiring more inputs or higher costs than needed for producing the same level and the same quality of output. 5 Indeed, later in this chapter, we will see that output shares are more coherent with output orientation, while for the input orientation it would be more natural to use the cost shares.

25 Aggregation of Efficiency and Productivity: From Firm to Sector and Higher . . .

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practice, where many tended to just use the equally weighted averages to report on the aggregate efficiency of industries or sub-groups within them. About two decades later, Farrell’s ideas were revisited by [19] who proposed to estimate efficiency scores of an “average firm,” constructed as the average of inputoutput allocations. It is easy to construct an example that will show that such a measure can indicate high inefficiency even though all firms are technically efficient, and this was considered (incorrectly) as a drawback and, apparently, was one of the reasons why this measure is also rarely used in practice. The fundamental ideas of [17, 19] were then revisited by a very important (yet for a long time overlooked) work of [30], who attempted to synthesize the two approaches with additional assumptions, focusing on the data envelopment analysis (DEA) context and on the use of the so-called shadow prices in DEA. At the turn of the last century, [1] were the first to scrutinize the problem on pure theoretical grounds and derived several important, yet “negative” results – they proved the impossibility of a solution of the aggregation problem in a general setup. What this implied was that some additional assumptions or structure were needed to arrive at a “positive result.”6 Such additional assumptions and structure were discovered by [12]: In addition to the usual assumptions of production theory, they followed [25]’s work on aggregation in economics, adapting it to the context of efficiency analysis. Differently from [25], however, they assumed an additive structure for the aggregate technology being the set-wise summation of the individual output (rather than technology) sets, for given input allocations. Adding this structure to the standard regularity conditions of production theory and with the so-called “law of one price” assumption (as in Koopmans), [12] then involved the principles of economic optimization to derive a theoretically justified weighting scheme for aggregation of individual efficiencies into a group efficiency. In turn, this theoretical framework provided the grounds of economic theory for the weighting scheme of [17, 30] and circumvented the impossibility theorems of [1]. The approach of [12] was then used to derive many other interesting and useful aggregation results, e.g., for aggregation of directional distance functions in [8], aggregation of scale elasticities in [15], and scale efficiencies in [59]; for aggregation of Malmquist and Hicks-Moorsteen productivity indexes [31, 33, 56]; etc. The goal of this chapter is to summarize these aggregation results and give some insights on future developments.

Aggregation of Efficiency Scores While most of the discussion here will be theoretical, it would be helpful for a reader to keep in mind that a typical empirical context of this methodology is a study

6 This

is not entirely surprising, e.g., recall that very strong assumptions are needed to establish positive aggregation results in consumer theory.

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of efficiency or productivity (or their changes) of an economic system consisting of different decision-making units (hereafter DMUs), e.g., industry consisting of firms or a particular bank or other institutions consisting of many branches, or a comparison of distinct groups within such a system (e.g., regulated vs. nonregulated, foreign vs. local, private vs. public firms, etc.). To get to the group level, we first need to briefly refresh the key concepts and notions for the individual level, which we do in the next subsection.

Individual Primal and Dual Efficiency Scores Without loss of generality, suppose the system is a group (e.g., industry, sector, etc.) consisting of n DMUs, where for each DMU k ∈ {1, 2, . . . , n} we will use vector k ) ∈ N to denote N inputs that the DMU k utilizes to produce a x k = (x1k , . . . , xN + k ) ∈ M . For generality of the aggregation vector of M outputs, y k = (y1k , . . . , yM + results, we will allow for each DMU k to employ technology that is potentially different from those used by other DMUs, and we assume it can be characterized by the technology set  k , defined in general terms as7  k ≡ {(x k , y k )

:

x k can produce y k }.

(1)

An equivalent characterization of technology can also be given via the output sets P k (x k ) ≡ {y k

:

x k can produce y k }, x k ∈ RN +.

(2)

An important advantage of the aggregation results that we summarize here is their generality with respect to characterization of technology. Indeed, we do not assume any particular production or transformation function (e.g., Cobb-Douglass, Leontieff, CES), rather we allow for a very wide class of technologies that satisfy usual regularity axioms of production theory, and in particular: A1: The technology set  k is closed. A2: The output correspondence P k (x k ) is bounded ∀x k ∈ N +. A3: There is no “free lunch,” i.e., nothing cannot produce something, i.e., k ≥ 0 for m = 1, . . . , M, y k = 0 ). (0N , y k ) ∈ /  k , ∀y k ≥ 0M (i.e., ym M . A4: It is possible to produce nothing, i.e., 0M ∈ P k (x k ), ∀x k ∈ N + A5: Outputs and inputs are freely (strongly) disposable, i.e., (x 0 , y 0 ) ∈  k =⇒ (x, y) ∈  k , ∀y  y 0 , ∀x  x 0 . To employ the results from the duality theory in economics, we also need some

7 In

the discussion of economic theoretical foundation here we mainly use framework developed by [45, 46] and further refined in many works and concisely outlined in [10] and [47].

25 Aggregation of Efficiency and Productivity: From Firm to Sector and Higher . . .

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convexity assumptions. At the beginning we only assume that the output sets are convex, i.e., o 1 k k 8 A6: y o , y 1 ∈ P k (x k ), x k ∈ N + ⇒ δy + (1 − δ)y ∈ P (x ), ∀δ ∈ [0, 1]. With these conditions, the output-oriented [46] distance function Dok : N +× → 1+ ∪ {+∞}, defined as

M +

Dok (x k , y k ) ≡ inf{θ > 0 : y k /θ ∈ P k (x k )}, θ

(3)

gives a complete characterization of the technology of a DMU k, in the sense that Dok (x k , y k ) ≤ 1

⇔

y k ∈ P k (x k ).

(4)

A closely related concept is the Farrell output-oriented measure of technical efficiency, defined as OT E k (x k , y k ) ≡ sup{θ > 0 : θ y k ∈ P k (x k )} = 1/Dok (x k , y k ).

(5)

θ

Furthermore, let p = (p1 , . . . , pM ) ∈ RM ++ be the vector of corresponding output 9 prices then the dual characterization of P k (x k ) is obtained from the revenue M 1 function: R k : N + × ++ → + ∪ {+∞} R k (x k , p) ≡ sup{py y

:

y ∈ P k (x k )},

(6)

and the related efficiency measure for a DMU k in the dual framework would then be the revenue efficiency (also referred to as the overall output efficiency), defined formally as RE k (x k , y k , p) ≡ R k (x k , p)/py k .

(7)

From the duality theory for the revenue function [10, 46], we then have R k (x k , p) ≥ py k /Dok (x k , y k ),

(8)

which leads to another notion – a measure of the output-oriented allocative (in)efficiency, defined as a multiplicative residual that turns (8) into equality, i.e.,

8 For

theoretical results we do not require convexity of  k , although when implementing in practice one may impose it when choosing a particular estimator or particular functional form for technology. 9 Note that for the aggregation results, a necessary assumption is the so-called Law of One Price, i.e., here it implies that all firms face the same output prices.

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OAE k (x k , y k , p) ≡ RE k (x k , y k , p)/OT E k (x k , y k ),

(9)

and so, we have a useful decomposition: RE k (x k , y k , p) = OT E k (x k , y k ) × OAE k (x k , y k , p),

(10)

This decomposition (10) is a stepping-stone for deriving the aggregation results, as will be apparent below.

Group Primal and Dual Efficiency Scores Let us consider a sub-group l (l = 1, . . . , L), consisting of nl DMUs within the original group of n DMUs. Such sub-grouping can be based on various exogenous criteria such as geographic regions, ownership structures, regulation regimes, etc. For each group l (l = 1, . . . , L), let the input allocation among DMUs within the sub-group l be Xl = (x l,1 , . . . , x l,nl ), and let the total of output vectors over all  l l firms in the l th group be Y = nk=1 y l,k . A cornerstone in the derivation of the aggregation results is the structure of the aggregate technology. In the context of output orientation, it is natural to assume a linear structure of aggregation of the output sets, as was done in [12]: For each l group l (l = 1, . . . , L), the aggregate output set P (Xl ) is the Minkowski sum of the individual output sets across all DMUs k (k = 1, . . . , nl ) within the group l, i.e.,10 l

P (X ) ≡ l

nl 

P l,k (x l,k ).

(11)

⊕k=1 l

As a result of such a structure, P (Xl ) would inherit the regularity conditions imposed on the individual output sets. In particular, note that the Minkowski sum of convex sets is also a convex set.11 Thus, convexity of the individual output sets l imposed by A6 ensures convexity of P (Xl ). l It is also worth noting that the aggregation structure defined by P (Xl ) presumes no reallocation of inputs across the individuals k ∈ {1, . . . , nl } and so depends not on the total sum of all the inputs but on the particular allocation Xl = (x l,1 , . . . , x l,nl ). This structure also assumes there are no externalities across firms.

use ⊕ to distinguish the summation of sets (also called “Minkowski summation”) from the standard summation; e.g., see [37]. 11 For example, see [26,44], and a more recent work of [37], as well as references therein. For other examples involving Minkowski summation in economics, see Shapley–Folkman-Starr theorem and related results [50]. 10 We

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Now, using the l th sub-group technology (11), one can define the sub-group revenue function as l

l

R (Xl , p) ≡ sup{py : y ∈ P (Xl )},

(12)

y

which, analogous to (7), gives rise to the l th sub-group revenue efficiency measure l

l

l

l

RE (Xl , Y , p) ≡ R (Xl , p)/pY .

(13)

The Fundamental Aggregation Results Having specific formulas for the efficiency measures defined with respect to individual technologies and with respect to aggregate technologies raises questions regarding the relationship between them. Ideally, one may want to establish their equality, so that the latter can be obtained from the former via some feasible computations, at least under some clear and reasonable conditions. Formally, the goal is to find fRE (·) such that l

l

RE (Xl , Y , p) = fRE (RE 1 (·), . . . , RE nl (·)).

(14)

In words, the goal is to find some aggregation function, which we call fRE (·), that can relate the aggregate measure (13) to the individual measures (7), for all firms k ∈ {1, . . . , nl } in a group of interest and do so in some meaningful way in the sense that the group measure should represent the group. Finding a function fRE (·) is not a difficult problem – there is an abundance of well-studied functions offered by mathematicians. It is the “meaningful way” aspect that is the most challenging and, as with many (if not all) notions in economics, depends on the views and assumptions of a researcher. The goal therefore is to make the choice grounded on and derived from some clear assumptions and if one does not like some assumptions then one may try to replace them with others and, possibly, derive new aggregation results. This is the approach we discuss here. In particular, we also consider it as desirable that the decomposition of revenue efficiency into technical efficiency and allocative efficiency that we have at the individual level is also maintained at the aggregate level, so that we have l

l

l

RE (·) = OT E (·) × OAE (·),

(15)

where l

OT E (·) = fT E (OT E 1 (·), . . . , OT E nl (·)),

(16)

and l

OAE (·) = fAE (OAE 1 (·), . . . , OAE nl (·)),

(17)

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where fT E (·), fAE (·) are also some aggregation functions to be found (potentially different from each other and from fRE (·)) so that they ensure the aggregate measures are related to the individual analogues. Such functions can be found using the following fundamental theorem. Theorem 1. For each group l (l = 1, . . . , L), the maximal revenue of the sub-group of DMUs feasible from (Xl , p) is equal to the sum of the maximal revenues of all its member DMUs feasible from their (x l,k , p), k = 1, . . . n, i.e., l

R (Xl , p) =

nl k=1

R l,k (x l,k , p).

(18)

This theorem is from [12], and it can be viewed as the revenue analogue to the [25] theorem of aggregation of the profit functions, while the cost or input-oriented analogue can be found in [7]. More importantly, this theorem provides a key to our aggregation problem, and so it is important to understand the economic intuition behind it: The theorem says that the sum of the revenues of individual revenue-maximizing DMUs in a sub-group is the same as the revenue optimized over the aggregate technology (11) for this subgroup, provided these DMUs face the same (e.g., equilibrium) output prices (and other regularity conditions hold). That is, whether optimized individually or as a group, the same revenue is attained under the “Law of One Price” (e.g., equilibrium price level) for all the outputs. The theorem above assumes full revenue efficiency (and full information) and so, a natural question is: Why do we consider full revenue efficiency when we want to measure output oriented inefficiency? And the answer is: Because we need it to set a benchmark against which the inefficiency will be measured. This is in the same fashion as how we choose the maximal output as the benchmark (although not assuming it to be reached by each firm) so that the actual output can be measured relative to it, in the output oriented context of efficiency measurement.12 From this fundamental theorem (as well as its cost and profit analogues), one can then get many useful results for the aggregation of the efficiency scores, some of which we summarize below, starting with the following corollary that first appeared in [12] and is an immediate consequence of (18). Corollary 1. For each group l (l = 1, . . . , L), we have l

l

RE (Xl , Y , p) =

nl 

RE l,k (x l,k , y l,k , p) × S l,k ,

(19)

k=1

12 In the input-oriented context, such a benchmark will be the cost function, while in the framework

where both input and output vectors can be changed when measuring efficiency (e.g., for efficiency based on the directional distance function or hyperbolic measures), the natural benchmark will be the profit function. We will briefly discuss these cases later in the chapter.

25 Aggregation of Efficiency and Productivity: From Firm to Sector and Higher . . .

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where l

S l,k = py l,k /pY , k = 1, . . . , nl .

(20)

In words, this corollary states that the weighted sum of the revenue efficiencies of individual revenue-maximizing firms in a sub-group is the same as the revenue efficiency with respect to the aggregate technology (11) for this same sub-group, provided these firms face the same output prices and the standard regularity conditions of production theory hold. In turn, this corollary implies another useful result, which gives the weighting schemes for the technical and allocative efficiencies into their group analogues, preserving the decomposition like (10) also at the aggregate level. We summarize this important result in the next corollary (also first appeared in [12]). For each group l (l = 1, . . . , L), the aggregate revenue efficiency can be decomposed multiplicatively into the weighted sum of the technical efficiencies (where the weights are the actual revenue shares), and the weighted sum of the allocative efficiencies (where the weights are the revenue shares corrected for technical inefficiency) of all its member DMUs. We summarize this formally in the next corollary. Corollary 2. For each group l (l = 1, . . . , L), we have l

l

l

nl 

RE (Xl , Y , p)

l

l

= OT E × AE ,

(21)

where OT E ≡

OT E l,k (x l,k , y l,k ) × S l,k ,

(22)

l,k OAE l,k (x l,k , y l,k , p) × Sae ,

(23)

k=1

and l

OAE ≡

nl  k=1

where S l,k ≡

py l,k pY

l

l,k , Sae ≡

p(y l,k OT E l,k (x l,k , y l,k )) , k = 1, . . . , nl . n l p k=1 (y l,k OT E l,k (x l,k , y l,k ))

(24)

In the next subsection, we provide some intuition behind these important results from which many other results can be derived.

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Understanding the Fundamental Aggregation Results Before going further, it is worth making a few intuitive remarks that should help in clarifying the fundamental results on aggregation summarized in the previous subsection. First of all, it should be clear that if L = 1, then the aggregate measures above are the efficiency measures for the entire group. Second, note that the measure (22) can be viewed as a multi-output generalization of Farrell’s measure of Structural Efficiency of an Industry, ( [17], p. 261–262). Third, recall that in the context of aggregation over industries, [3] derived a similar weighting scheme, using different arguments than outlined here and after imposing more restrictive assumptions. Meanwhile, [30] proposed the same weights and decomposition of the aggregate revenue (although defined differently than above) into aggregate technical efficiency and aggregate allocative efficiency measures.13 Fourth, note that while the technical efficiency is constructed to be a price independent measure of efficiency, the aggregation weights for obtaining the (sub)group technical efficiency derived above depend on prices. This might be viewed as undesirable. On the other hand, note that these weights were not chosen arbitrarily or in an ad hoc way, but came out as a result of imposing an economic criterion of optimizing behavior, which researchers also often consider as a benchmark when making their choice of orientation in measuring efficiency. Intuitively, if one wants to account for an economic importance of a DMU that obtained the particular “standardized” efficiency score then, since prices contain important economic information, it shall not be surprising that the weights derived using the economic optimization principle are price-dependent. Another consideration is more practical: price information may be unavailable (or unreliable) in a given study. To circumvent this problem, one may use the shadow prices [30]. Alternatively, one may impose an extra assumption to make the derived weights price-independent, as we outline in section “Price-Independent Weights”. Fifth, a condition often referred to as the ‘Law of One Price’ was assumed to enable feasibility of the derivations of these aggregation results. Importantly, note that this is a necessary assumption for obtaining a positive result in the stated aggregation problem. To be more precise, it is a necessary condition to establish an equivalence between the aggregate notions of efficiency (defined with respect to the aggregate technology and optimized as a group) and the dis-aggregate notions of efficiency (defined with respect to the individual technologies and optimized independently by each individual in the group). In other words, this “Law of One Price” condition can be viewed as the condition of an equilibrium that ensures the system reaches the same outcome whether optimized individually and then

13 To

be precise, [30] used a similar framework, yet without explicit relationship to the maximal revenue defined on the sum of the output sets and without noticing the theoretical link via the analogue of [25] theorem, and focusing on the DEA framework.

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aggregated or optimized over the aggregate technology by a group (e.g., a “central planner” for the group). In this sense, the weights derived from this framework can be viewed as “optimal weights,” in the sense that they are derived from a framework where the system has reached equivalent optimal outcomes from both the aggregate and the dis-aggregate sides. On the other hand, without this assumption, the general impossibility theorems of [1] are in action, which lead to much more disappointing conclusions for practitioners (since they ensure the impossibility of the equivalence) than this quite common condition in economic theory. Indeed, this condition is coherent with many economic models (perfect competition, Cournot-type oligopoly, etc.), where the notion of economic equilibrium indeed implies a common price. As many other theoretical assumptions, it is, of course, simplifying the reality (e.g., see [20, 27] for a discussion). In practice, it is of course possible to use the same formulas for the weights but with different prices and then compute an aggregate by averaging the individual efficiencies using such “nonoptimal” (or ad hoc) weights, and they can be viewed as approximations of the “optimal weights” derived above. The problem is that such an aggregate is not guaranteed to be equivalent to the aggregate obtained with respect to the aggregate technology, yet it may have another useful meaning that might be appealing from another perspective (e.g., it can be regarded as an aggregate efficiency that accounts for the price variation across the observations and thus showing the gap relative to the aggregate based on the “optimal weights”). Finally, it should not be surprising that establishing positive aggregation results in economics requires extra and perhaps relatively strict assumptions. A good example would be the fairly strong conditions imposed to obtain the well-known in economic theory solutions to aggregation of demands, whether over goods or over consumers. Similarly, and as mentioned above, in the context of efficiency analysis, [1] analyzed a more general aggregation problem (without considering optimization behavior) and arrived at several impossibility results, concluding that very strong assumptions on the technology are needed for establishing positive aggregation results. The approach summarized above circumvents such assumptions by resorting to the optimization behavior (as a benchmark against which inefficiency of actual performance is measured) along with the other assumptions described above.

Aggregation of Aggregates We now look at the case when a researcher wants to aggregate further, over already aggregate efficiency scores, i.e., across some sub-groups within a larger group. For example, suppose there is some partitioning of interest of the entire group into L non-intersecting and exhaustive sub-groups l = 1, . . . , L. Let Y ≡ nl L   n k y l,k be the total output across all DMUs in all the sub-groups. k=1 y = l=1 k=1

Also let the input allocation among firms within all the groups be denoted with

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X = (X1 , . . . , XL ). If (11) is true for all l = 1, . . . , L, then we must have P (X) =

n 

L 

P k (x k ) =

⊕k=1

l

P (Xl ) =

⊕l=1

nl L  

P l,k (x l,k ),

(25)

⊕l=1 ⊕k=1

i.e., the aggregate output set of all groups together is the Minkowski sum of the group output sets, over l = 1, . . . , L. Thus, P (X) would inherit its properties from the properties of sub-group technologies, which in turn are inherited from the regularity conditions imposed on the output sets of individual DMUs. Using the group technology (25), one can define the group revenue function as R(X, p) ≡ sup{py : y ∈ P (X)},

(26)

y

which, similar to (7), gives rise to the group revenue efficiency measure RE(X, Y , p) ≡ R(X, p)/pY .

(27)

An immediate consequence of the previous theorem and of (25) is summarized in the next corollary. Corollary 3. The maximal revenue of the entire group of DMUs feasible from (X, p) is equal to the sum of maximal revenues of all its (non-intersecting) subgroups of DMUs feasible from (Xl , p), l = 1, . . . , L, i.e., L

R(X, p) =

l=1

l

R (Xl , p) =

L

nl

l=1

k=1

R l,k (x l,k , p).

(28)

The intuition of this result is the same as that of its analogue of (18) – it is its extension to the aggregation between the sub-groups into a larger group. The corresponding result about aggregation of revenue efficiency measures is summarized in the next corollary. Corollary 4. We have RE(X, Y , p) =

L 

l

l

RE (Xl , Y , p) × S l ,

(29)

l=1

where  l

S = pY / p l

L  l=1

 Y

l

, l = 1, . . . , L.

(30)

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Intuitively, this corollary says that the weighted sum of the revenue efficiencies of revenue-maximizing sub-groups of firms is the same as the revenue efficiency with respect to the aggregate technology (25) for the group that unites these subgroups (assuming all firms face the same output prices and the standard regularity conditions hold). That is, it is an analogue of (19)–(20). In turn, this corollary implies the following important result. Corollary 5. We have RE(X, Y , p) = OT E × OAE,

(31)

where OT E =

L 

l

OT E × S l ,

(32)

l=1

and OAE =

L 

l

l OAE × Sae ,

(33)

l=1

where  l

S = pY / p l

L 

 Y

l

, l = 1, . . . , L,

(34)

l=1

and l Sae

 L    l  l l l = pY × OT E / p Y × OT E , l = 1, . . . , L.

(35)

l=1

Intuitively, this last corollary provides a theoretically justified weighting scheme for an aggregation over sub-groups of the aggregate technical and aggregate allocative efficiencies into more aggregate analogues, and such that they decompose the aggregated revenue efficiency. Thus, this approach provides “internally consistent” aggregation within and between the sub-groups.

Price-Independent Weights In this section we summarize the method for converting the derived above pricedependent weights into the price-independent weight such that the same aggregation scheme based on and derived from the economic principles is preserved. This method was proposed by [12, 14, 48]. We first focus on the case of aggregating

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efficiency scores of the entire group. The key additional assumption here is the following:  pm Y m /

M 

 pm Y m

= αm , m = 1, . . . , M,

(36)

m=1

 k and α ∈ (0, 1) is a constant (known or estimated) for all where Y m ≡ nk=1 ym m  m ∈ {1, . . . , M} with normalization M m=1 αm = 1. Intuitively, (36) states that the weight of the industry revenue from the output m in the industry total revenue equals k /Y to be the weight of the k th firm in the αm . Furthermore, let us denote mk = ym m th group in terms of the m -output, and let us impose the condition (36) upon the weights for the aggregation of the revenue and technical efficiency scores derived above, to obtain Sk =

M 

αm mk , k = 1, . . . , n.

(37)

m=1

Intuitively, (37) says that the weight of a firm is the weighted average over all the output shares of this firm in its group, where the weights are the revenue shares of the industry for each output m in the total revenue of the industry. Next, use (36) and (34) to derive the weights for aggregating “between the sub-groups”

Sl =

M 

αm Wml , l = 1, . . . , L,

(38)

m=1 l

where Wml = Y m /Y m is the share of the l th sub-group in the entire group in terms of the mth output. Furthermore, with a bit more algebra we can derive the priceindependent weight for an individual efficiency of firm k ‘within a sub-group l’ to be S l,k = S k /S l , k = 1, . . . , nl ; l = 1, . . . , L,

(39)

i.e., we get an analogue of (37) which accounts for the weight of each particular sub-group in the entire group. On the other hand, the price-independent weights for aggregating allocative efficiencies are derived similarly as above but where the observed outputs are replaced with their technically-efficient analogues, i.e., S k × OT E k (x k , y k ) k , k = 1, . . . , n, = n Sae k k k k k=1 S × OT E (x , y ) where S k is given in (37).

(40)

25 Aggregation of Efficiency and Productivity: From Firm to Sector and Higher . . .

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Meanwhile, we can also employ the standardization in (36), along with (34), to get the weights for aggregating “between the sub-groups” to be given by M

l Sae

l

l

αm Wml OT E OT E × S l =  m=1 = , l = 1, . . . , L, L l l L M l l l=1 m=1 αm Wm OT E l=1 OT E × S

(41)

l

where Wml = Y m /Y m is the weight of l th sub-group in the entire group in terms of the mth -output. Note that (41) is analogous to what we obtained for the individual firms but for the sub-group level. Moreover, (41) can be used to derive the weight of an individual efficiency of firm k “within a sub-group l” to be l,k k l Sae = Sae /Sae , k = 1, . . . , nl ; l = 1, .., L.

Finally, note that all the derivations here were done for the case of the output orientation and analogous derivations can be made for the case of input orientation (and, potentially, for the joint input-output or profit-orientation), which we leave as exercises for the readers (see [7, 31, 32] for some related derivations).

Aggregation of Productivity Indexes Similarly as with the efficiency scores, applied studies involving productivity indexes usually need to present some aggregates of the estimated productivity indexes – to summarize the overall tendencies in a sample, to perform statistical inference about the population, etc. Typically, researchers use the simple or the equally weighted geometric mean for this purpose. The discussion above suggests that it would also be important to have some well-justified weights when aggregating productivity indexes. Such weights would help in accounting for the relative importance of each firm whose index is entering into the average. This question was first addressed by [56], who derived an aggregation scheme for the Malmquist productivity index (MPI), and we summarize this approach in this section.14 To simplify the notation, from now on we will consider just one group, i.e., drop the sub-group subscript l (but add the time subscript τ = s, t).

Individual Malmquist Productivity Indexes Let us first recall the definitions of the MPI. We will focus on measuring changes in productivity from a period s to a period t (s < t). Recall that the output-oriented MPI can be defined as

14 Also

see [31, 33] for extensions of this approach.

1056

V. Zelenyuk



M

k

(ysk , ytk , xsk , xtk )

Dsk (xtk , ytk ) Dtk (xtk , ytk ) × ≡ Dsk (xsk , ysk ) Dtk (xsk , ysk )

1/2 .

(42)

where Dsk (xtk , ytk ) is the Shephard’s output-oriented distance function that we now cast in the inter-temporal framework, characterizing technology of DMU k in period s and evaluated at the point (xtk , ytk ). Note that we dropped the subscript “o” to simplify our already intense notation.15 In the light of the duality between the distance function and the revenue function, one can also define the revenue (or dual) analogue of the MPI as RM k (·) ≡ RM k (ps , pt , ysk , ytk , xsk , xtk )

 −1 1/2 REsk (xtk , ytk , pt ) REtk (xtk , ytk , pt ) × , ≡ REsk (xsk , ysk , ps ) REtk (xsk , ysk , ps )

(43)

which, naturally, can be decomposed as RM k (·) ≡ M k (·) × AM k (·),

(44)

where M k (·) is defined in (42) and AM k (·) is the allocative component of the dual MPI, defined as AM k (·) ≡ AM k (ps , pt , ysk , ytk , xsk , xtk )

 −1 1/2 OAEsk (xtk , ytk , pt ) OAEtk (xtk , ytk , pt ) × . ≡ OAEsk (xsk , ysk , ps ) OAEtk (xsk , ysk , ps )

(45)

Aggregation Problem: Inter-temporal Perspective Here we adapt the aggregation concepts outlined above to the inter-temporal framework. As for the case of efficiency aggregation, a key stepping-stone for deriving the aggregation results for productivity indexes is to define a relevant group technology, and as before, here we admit the additive structure of aggregation of the output sets, i.e., P τ (X) ≡

n 

Pτk (x k ), τ = s, t,

(46)

⊕k=1

15 Again,

here we focus on the output orientation case and similar developments can be done for the input orientation case. See [31, 33] for some of these details.

25 Aggregation of Efficiency and Productivity: From Firm to Sector and Higher . . .

1057

and so the group revenue function at period τ is given by R τ (X, p) ≡ max{py y

:

y ∈ P τ (X)}, τ = s, t,

(47)

while the revenue efficiency at τ is given by RE τ (X, Y , p) ≡ R τ (X, p)/pY , τ = s, t.

(48)

Now, to measure changes in productivity between s and t, let the group (aggregate) analogue of (43) be ⎡

−1 ⎤ 12 RE s (Xt , Y t , pt ) RE t (Xt , Y t , pt ) ⎦ , RM(ps , pt , Y s , Y t , Xs , Xt ) ≡ ⎣ × RE s (Xs , Y s , ps ) RE t (Xs , Y s , ps ) (49) where the time subscripts indicate the particular values of efficiency measures for specific periods τ = s, t. Ideally, we want to find an aggregation function fRM (·) that can relate the aggregate measure (49) to the individual measures (43) in some “meaningful” way. Being unable to find such a “meaningful” way, [56] resorted to something that may seem “less than ideal,” yet feasible – find an aggregation function fRE (·) that can relate the aggregate measure (49) to all the components of all the individual measures (43), in a “meaningful” way, i.e., so that we have RM(ps , pt , Y s , Y t , Xs , Xt ) = fRE (REτ1 (·), . . . , REτn (·)), τ = s, t,

(50)

such that, preferably, the decomposition (44) is maintained at the aggregate level, i.e., RM(ps , pt , Y s , Y t , Xs , Xt ) = M(·) × AM(·),

(51)

where, in turn, one need to find some aggregation functions fD (·), fAE (·) that ensure that the aggregate primal MPI is related to all the components of all the individual analogues (42), i.e., M(·) ≡ M(Y s , Y t , Xs , Xt ) ≡ fD (Dτ1 (·), . . . , Dτn (·)), τ = s, t,

(52)

while the aggregate allocative-MPI is related to (45) or its individual components, i.e., AM(·) ≡ AM(Y s , Y t , Xs , Xt ) ≡ fA (OAEτ1 (·), . . . , OAEτn (·)), τ = s, t. (53)

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V. Zelenyuk

Such functions are found in the next subsection using, again, the Koopmans-type arguments that we described above.

Aggregation of the MPIs As for the aggregation of efficiency scores, the foundation here is the inter-temporal extension of the aggregation theory from [12], which in turn is an adaptation of aggregation theory from [25], which we restate below casting it in the inter-temporal framework, with the time subscript τ = s, t: R τ (X, p) =

n k=1

M Rτk (x k , p), x k ∈ N + , ∀k = 1, . . . , n, p ∈ ++ ,

(54)

and therefore, for j, τ = s, t, we have n

=

RE τ (Xj , Y j , pj )

k=1

REτk (xjk , yjk , pj ) × Sjk

(55)

where Sjk ≡ pj yjk /pj Y j , k = 1, . . . , n.

(56)

Thus, the decomposition is maintained at the aggregate level: for any j, τ = s, t, we have =

RE τ (Xj , Y j , pj )

OT E τ (j )

×

OAE τ (j ),

(57)

where OT E τ (j ) ≡

OAE τ (j ) ≡

n

[Dτk (xjk , yjk )]−1 × Sjk ,

(58)

k OAEτk (xjk , yjk , pj ) × Sae,τ,j ,

(59)

k=1

n k=1

and

k Sae,τ,j

  pj yjk /Dτk (xjk , yjk )   , k = 1, . . . , n. ≡  pj nk=1 yjk /Dτk (xjk , yjk )

(60)

Furthermore, applying (49), (55) and (56) we get a desired aggregation result – a solution to (50), given by:

25 Aggregation of Efficiency and Productivity: From Firm to Sector and Higher . . .

RM(ps , pt , Y s , Y t , Xs , Xt )  n REsk (xtk , ytk , pt ) × Stk nk=1 = k k k k k=1 REs (xs , ys , ps ) × Ss n 1/2 REtk (xtk , ytk , pt ) × Stk −1 × nk=1 . k k k k k=1 REt (xs , ys , ps ) × Ss

1059

(61)

Importantly, note that the decomposition at the aggregate level is preserved and given by RM(ps , pt , Y s , Y t , Xs , Xt ) = M(·) × AM(·),

(62)

where the solutions to (52) and (53) are given, respectively, by ⎡

OT E s (t) OT E t (t) M(Y s , Y t , Xs , Xt ) = ⎣ × OT E s (s) OT E t (s)

−1 ⎤1/2 ⎦ ,

(63)

and ⎡

OAE s (t) OAE t (t) AM(ps , pt , Y s , Y t , Xs , Xt ) = ⎣ × OAE s (s) OAE t (s)

−1 ⎤1/2 ⎦ ,

(64)

where, in turn, the four components inside (63) are given in (58) while the four components inside (64) are given in (59). The theoretical and practical importance of these results is that they give explicit formulas for aggregation of the MPIs. In particular, they give a way of obtaining a group productivity change score from the individual analogues, where the aggregation function and the aggregation weights are not ad hoc but derived from economic principles, besides being intuitive.

Geometric vs. Harmonic Averaging In earlier studies, noting on the multiplicative nature of the MPI, researchers often used not only the equal weights but also the geometric rather than the arithmetic or the harmonic averaging of the individual estimates when they wished to summarize the point estimates of MPIs (e.g., see [4]). That is, not only the weights were equal, but also the aggregating function used in previous practice was quite different from what the theoretical derivations in the previous subsection suggested. How can these different approaches be reconciled? From the discussions above, it must be clear that the weights can dramatically influence the results, whether quantitatively or qualitatively. A natural question is whether the functional form of the aggregation is

1060

V. Zelenyuk

critical and, in particular, can one use the geometric mean rather than the arithmetic mean? This question was also addressed by [56], who pointed out that (63) can be restated in terms of harmonic aggregations of individual distance functions, i.e.,  n M(·) =

k k k −1 k=1 [Ds (xt , yt )] n k k k −1 k=1 [Ds (xs , ys )]

n

× k=1

× Stk × Ssk

−1 −1

[Dtk (xtk , ytk )]−1 × Stk

n k k k −1 k=1 [Dt (xs , ys )]

× Ssk

−1 1 2

−1

.

(65)

and its geometric analogue can be defined as ⎤1/2 ⎡ n n  k k Dsk (xtk , ytk )ωt Dtk (xtk , ytk )ωt ⎥ ⎢ k=1 G k=1 ⎥ , M (·) ≡ ⎢ × n n ⎣  k k k ωk ⎦ k ω k k k s s Ds (xs , ys ) Dt (xs , ys ) k=1

(66)

k=1

for some weights ωtk , ωsk . The aggregation usually used in practice is a particular case of (66) that assumes equal weights across all k, i.e., ωtk = ωsk = 1/n. It must be clear that, in general, (65) is not equal to (66) and, in fact, no exact general relationship exists between n  k Dsk (xtk , ytk )St and the two. However, taking the first-order approximation of k=1 n  k (x k , y k )]−1 × S k −1 around unity (which is a natural point around of [D s t t t k=1 which and efficiency indexes can be approximated) in both cases we nproductivity get k=1 Dsk (xtk , ytk ) × Stk , meaning that one can conclude M(Y s , Y t , Xs , Xt ) ∼ = M (Y s , Y t , Xs , Xt ), for (ωtk , ωsk ) = (Stk , Ssk ). G

(67)

In words, (67) states that the first-order-approximation relationship exists between the aggregate MPI constructed with harmonic components derived above and the geometric aggregate of individual MPIs, if both use the same set of weights. This implies that, for anyone who prefers the geometric aggregation, this relationship gives a justification for choosing the aggregation weights (which are more influential) – the weights derived from economic principles, which account for the economic weight of each firm. A natural question is “How substantial is the difference between the geometric and harmonic aggregations?”16 Zelenyuk [56] presented some simulation results

16 From

theory, it is known that under the same weighting scheme, the geometric mean is larger than the harmonic mean but smaller than the arithmetic mean. Note however that the aggregate MPI in (65) involves products of ratios of the harmonic means and so it can be smaller or greater than

25 Aggregation of Efficiency and Productivity: From Firm to Sector and Higher . . .

1061

confirming that the difference is fairly small. For instance, if the scores of a productivity index come from uniform distribution around unity with the range of 50 percentage points (thus allowing a substantial change), then the square root of the mean squared difference between the harmonic and geometric means across various simulations was only about 1 percentage point. Thus, a practical implication that one can deduce from here is that the geometric-type and the harmonic-type aggregations of the productivity indexes (under the same weights) give similar aggregate scores for moderate variations of the scores being aggregated. In other words, the aggregation function per se (whether geometric, harmonic or arithmetic) is not as crucial – what is more important are the weights of aggregation, which needs to be justified on some theoretical grounds.17

Decomposition and Aggregation The aggregation results we summarized above can also be extended to the aggregation of components of various decompositions of MPIs. While there are many decompositions of MPI offered in the literature, here we focus on what seems to be the most popular decomposition in practice – the one proposed in the seminal work of [4], as the following M k (·) ≡ EF CH k (·) × T ECH k (·),

(68)

where the first component is referred to as the efficiency change, defined as EF CH k (·) ≡ EF CH k (ysk , ytk , xsk , xtk ) ≡

Dtk (xtk , ytk ) , Dsk (xsk , ysk )

(69)

and the second component is referred to as the technological change, defined as  T ECH k (·) ≡ T ECH k (ysk , ytk , xsk , xtk ) ≡

Dsk (xtk , ytk ) Dtk (xtk , ytk )

×

Dsk (xsk , ysk ) Dtk (xsk , ysk )

1/2 . (70)

The aggregation question then is to find appropriate group analogues to (69) and (70), i.e., some functions fEC (·) and fT C (·) that relate the aggregate measures

the aggregate MPI obtained via a geometric mean as in (66), depending on the relative magnitudes that appear in the numerators and denominators of (65). Both means are approximately equal (to the arithmetic mean) in the sense of first order approximation around unity. 17 One should however be careful aggregating when there are scores equal or very close to zero: both geometric and harmonic averages completely fail if at least one element is zero and may yield an unreasonably low aggregate score if at least one element is very close to zero (even if many others have large efficiency or productivity scores), unless they are “neutralized” by a very low weight in the aggregation, as can be done with weighted aggregates. In such cases, using arithmetic aggregation, which is less sensitive to the outliers, could also be a better solution.

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V. Zelenyuk

to individual ones. As above, a natural choice is to utilize the Koopmans-type arguments, as was done in [56], to arrive at  EF CH (·) =

OT E t (t) OT E s (s)

−1

n =

k k k −1 k=1 [Dt (xt , yt )] n k k k −1 k=1 [Ds (xs , ys )]

× Stk × Ssk

−1 −1 ,

(71)

and ⎡

OT E s (t) OT E s (s) T ECH (·) = ⎣ × OT E t (t) OT E t (s)

n =

−1 ⎤1/2 ⎦

 k k k −1 k −1 k=1 [Ds (xt , yt )] ×St n  k k k −1 k −1 k=1 [Dt (xt , yt )] ×St

 1/2 k k k −1 k −1 k=1 [Ds (xs , ys )] ×Ss . n  k k k −1 k −1 k=1 [Dt (xs , ys )] ×Ss n

(72) As before, the first-order approximation relationship can also be established between the harmonic-type aggregations in (71) and (72) and their geometric analogues. Moreover, these aggregation results can also be extended to aggregation across or over larger groups, in a similar manner as for aggregating efficiency scores that we discussed above, i.e., extending [48].

Aggregation for Scale Measures Measurement of economies of scale for an individual firm or for an industry has been one of the most frequently addressed research questions in economics and applied econometrics. This is usually done via estimating such measures as scale elasticity and/or scale efficiency. Here we will focus on the elasticity approach, following [15], while the aggregation for scale efficiency can be found in [59]. For analyzing economies of scale for a group (e.g., industry or sub-industry), researchers usually estimate the elasticity at some points of interest, e.g., the nonweighted mean or the median of the data or, alternatively, the non-weighted mean of the individual estimates of scale elasticities. Importantly, note that these different approaches do not give the same information, in general, and each has certain theoretical or practical appeals and caveats. Here we discuss another theoretical approach of measuring scale elasticity of a group, which is based on a similar aggregation result as that derived above. So far we considered the output-oriented framework, and in this section, because researchers often focus on elasticity of the cost function, we will consider the case of input orientation. To do so, first note that the technology set of firm k can be equivalently characterized by the input requirement sets, defined as

25 Aggregation of Efficiency and Productivity: From Firm to Sector and Higher . . .

L k (y k ) ≡ {x : x can produce y k }, y k ∈ RM +,

1063

(73)

so that x k ∈ L k (y k ), y ∈ RM ⇐⇒ (x k , y k ) ∈ T k . In turn, technology + can also be equivalently characterized by the input-oriented [45] distance function N Dik : M + × + → + ∪ {∞}, defined as Dik (y k , x k ) ≡ sup{δ > 0 : x k /δ ∈ L k (y k )}.

(74)

δ

A closely related concept is the input-oriented Farrell measure of technical efficiency: I T E k (y k , x k ) ≡ inf{θ > 0 : θ x k ∈ L k (y k )} = 1/Dik (y k , x k ). θ

(75)

In addition to assuming the main regularity axioms of production theory (A1A5), we also assume convexity of the input requirement sets, i.e., A7: Input requirement sets L k (y k ) are convex, ∀y k ∈ M +. As a result, due to duality theory in economics (see [10, 45, 47]), the technology N can be equivalently characterized by the cost function, C k : M + ×++ →+ ∪ {∞}, defined as C k (y k , w) ≡ inf{wx : x ∈ L k (y k )}, x

(76)

where w ≡ (w1 , . . . , wN ) ∈ N ++ is the vector of input prices. The related efficiency measure for a DMU k in the dual input-oriented framework would then be the cost efficiency (also referred to as the overall input efficiency), defined formally as CE k (y k , x k , w) ≡ C k (y k , w)/wx k .

(77)

From the duality theory for the cost function, we also have the so-called Mahler’s inequality: C k (y k , w) ≥ wx k /Dik (y k , x k ),

(78)

leading to the notion of the input oriented allocative (in)efficiency, defined as a multiplicative residual that turns (78) into equality, i.e., I AE k (y k , x k , w) ≡ CE k (y k , x k , w)/I T E k (y k , x k ), and so we have another useful decomposition:

(79)

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V. Zelenyuk

CE k (y k , x k , w) = I T E k (y k , x k ) × I AE k (y k , x k , w),

(80)

We will use these efficiency measures and their aggregate analogues later in the chapter, while for the framework of scale elasticity, we will focus on the distance function and the cost function characterizations. It is important to note that one can use both the primal and the dual characterizations to measure economies of scale via the scale elasticity. Specifically, with appropriate differentiability assumptions, for the dual framework the scale elasticity is defined as18 ec (y k , w) ≡

 ∇y k C k (y k , w)y k ∂ ln C k (y k θ, w)  , =  ∂ ln θ C k (y k , w) θ=1

(81)

and for the primal framework, the scale elasticity is defined as19  ∂ ln λ  ei (y , x ) ≡ = −∇y k Dik (y k , x k )y k . ∂ ln θ  Dik (y k θ,x k λ)=1, k

k

(82)

θ=1,λ=1

Now, suppose x ∗k is a solution to (76), then one can obtain equality between the dual and the primal measures, i.e.,20 ec (y k , w) = ei (y k , x ∗k ),

(83)

where x ∗k is a solution to (76), i.e., x ∗k ≡ arg inf{wx x

:

x ∈ L k (y k )}.

(84)

Intuitively, (83) states that the same information about the scale elasticity of an individual firm k can be obtained from the primal and dual approaches. Now, analogous to what we did in previous sections, let the group input requirement set be given by the Minkowski sum of the individual input requirement sets across all DMUs k (k = 1, . . . , n), i.e., L (y 1 , . . . , y n ) =

n 

L k (y k ).

(85)

⊕k=1

Note that L (y 1 , . . . , y n ) inherits the regularity conditions imposed on the individual input requirement sets and, in particular, convexity of the individual input

18 For

example, see [39]. [5, 10]. 20 See [5, 58] for more details on this. 19 See

25 Aggregation of Efficiency and Productivity: From Firm to Sector and Higher . . .

1065

requirement sets implies that L (y 1 , . . . , y n ) is also convex. Also note that the aggregation structure defined by L (y 1 , . . . , y n ) presumes no reallocation of outputs and no externalities across the individuals. In turn, the group cost function would be the aggregate analogue of (76), defined as C(y 1 , . . . , y n , w) ≡ inf{wx : x ∈ L (y 1 , . . . , y n )}, x

(86)

while the group input-oriented distance function would be defined as D i (y 1 , . . . , y n ,

n k=1

x k ) ≡ sup{δ > 0 : ( δ

n k=1

x k /δ) ∈ L (y 1 , . . . , y n )},

(87) as the aggregate analogue of (74). Therefore, one can measure the economies of scale for the group from the measures of scale elasticity defined for the aggregate technology – analogously to how it is done for the individual technologies, i.e., we have  1 θ, . . . , y n θ, w)  C(y ∂ ln  ec (y 1 , . . . , y n , w) ≡   ∂ ln θ =

∇Y C(y 1 , . . . , y n , w)Y C(y 1 , . . . , y n , w)

θ=1

(88)

,

where ∇Y C(y 1 , . . . , y n , w) ≡ (∂C(y 1 , . . . , y n , w)/∂y 1 , . . . , ∂C(y 1 , . . . , y n , w) /∂y n ) and Y ≡ (y 1 , . . . , y n ) . Meanwhile, for the primal framework, we get 1

n

ei (y , . . . , y ,

n  k=1

 ∂ ln λ  n x )≡  ∂ ln θ  D i (y 1 θ,...,y n θ, x k λ)=1, k

k=1

θ=1,λ=1

= −∇Y D i (y 1 , . . . , y n ,

n 

x k )Y.

(89)

k=1

Furthermore, let x ∗ be a solution to (86), then the dual and the primal measures of group scale elasticity would be equal, i.e., putting this formally, we have a desired result: ec (y 1 , . . . , y n , w) = ei (y 1 , . . . , y n , x ∗ ), where

(90)

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V. Zelenyuk

x ∗ ≡ arg inf{wx : x ∈ L (y 1 , . . . , y n )}. x

The reader shall notice that (90) is an aggregate analogue of (83). The main goal therefore now is to find a relationship between the aggregate and the individual scale elasticity measures that will enable getting the aggregate measures from the individual ones. As above, the fundamental step for reaching this aim is the following result. Theorem 2. The minimal cost of the group of DMUs with production plan Y = (y 1 , . . . , y n ) is equal to the sum of the minimal costs of all its member DMUs with the same production plan y 1 , . . . , y n , assuming all the member DMUs face the same input prices w, i.e.,

C(y 1 , . . . , y n , w) =

n 

C k (y k , w).

(91)

k=1

In words, this theorem states that whether the group of DMUs minimize the costs for their given output plans together via a “social planner” (and without reallocation of outputs across DMUs) or they minimize individually and then these costs are summed over, the result should be the same if they face the same input prices w. This theorem is the cost analogue of the theorem of [25] for aggregation of profit functions (see [7] for a proof). Now, for measuring the change in costs due to infinitesimal and equiproportional change of all outputs, we differentiate both sides of (91) along the ray from the origin through the point Y ≡ (y 1 , . . . , y n ) . Doing so for the l.h.s. of (91), we get  ∂C(y 1 θ, . . . , y n θ, w)    ∂θ

= ∇Y C(y 1 , . . . , y n , w)Y .

(92)

θ=1

while doing so for the r.h.s. of (91), we get

∂

 n  k=1

   C (y θ, w) /∂θ   

k

=

k

θ=1

n 

∇y k C k (y k , w)y k .

(93)

k=1

and combining the two, we get the following important equivalence results (originally derived by [15]), summarized in the next two corollaries.

25 Aggregation of Efficiency and Productivity: From Firm to Sector and Higher . . .

1067

Corollary 6. We have ec (y 1 , . . . , y n , w) =

n 

ec (y k , w) × S k ,

(94)

k=1

where S k ≡ C k (y k , w)/

n 

C k (y k , w).

(95)

k=1

This mathematical result is quite intuitive: In the dual framework, the scale elasticity of a group equals the weighted sum of the individual scale elasticity scores of all firms in this group, where the weights are the cost shares. As above, a strength of this result is that the weights are not ad hoc but derived from economic principles. Similar aggregation result can also be derived for the primal scale elasticity measurement. In particular, from (83) and (90), we get the following equivalence result. Corollary 7. We have ei (y 1 , . . . , y n , x ∗ ) =

n 

ei (y k , x ∗k ) × S k .

(96)

k=1

This important result tells us how to obtain the group scale elasticity measure from the individual scale elasticity measures in the primal framework. Specifically, note that (96) says that one can get the primal aggregate scale elasticity measure from the weighted arithmetic average of the individual scale elasticity scores of all firms in this group, where the weights are the individual cost shares, derived from economic theoretic reasoning. In case the researcher has no price information to calculate the weights, she/he may use shadow prices, estimated from the primal information, or, alternatively, impose additional assumption and help to derive the price-independent weights, similarly as discussed above and following [12, 14]. Specifically, the additional assumption here would be

wr

n k=1

xr∗k /



N r=1

wr

n k=1

xr∗k

= br ,

r = 1, . . . , N,

(97)

where br ∈ (0, 1) is a known or estimated constant. In words, (97) states that the share of the group expenditures on the r th input in the group total cost is given by k k br . Further, if we let r = xr / nk=1 xrk be the share of the k th firm in the group in terms of the r th input, then from (97) we get the price-independent weights given by

1068

V. Zelenyuk

Sk =

N r=1

rk br ,

k = 1, . . . , n.

(98)

In words, (98) states that a firm’s weight is the weighted average over all input shares of this firm in the group, where the weights are the shares of the industry expenditures on the r th input in the industry total cost. It is also worth noting that analogous developments can also be done for other “derivatives” of the cost function as well as of the revenue and profit functions. Moreover, such aggregation results can be generalized further to the case of aggregation within sub-groups (e.g., private vs. public, etc.) and then aggregation between these sub-groups into a larger group. Finally, similar analysis can also be done for the case of aggregation of scale efficiency scores, as was done in [55, 59].

Aggregation with Possibility of Reallocation In the discussion above, we restricted attention to cases where reallocation of inputs between DMUs are not allowed for the output orientation and reallocation of outputs between DMUs are not allowed for the input orientation. What if one of these or both restrictions are relaxed? This context was first considered in [36] in the context of aggregating Farrell-type efficiency scores, while [31] extended it to the context of aggregating MPIs. Both papers focused on the output oriented context, while the input-oriented context was outlined in [32] and refined further in [33], which also extended it to the context of aggregating Hicks-Moorsteen Productivity Indexes (HMPIs). In this section we briefly summarize some key results from these papers.

Aggregate Technology and Measures with Reallocation To measure the gains from allowing for the reallocation of resources among DMUs in a group, we need to allow for a more general structure of aggregate technology, which we will refer to as the group potential technology, and define it as the Minkowski sum of technology sets of all individual DMUs for a given period τ 21 : τ∗

≡

n 

τk .

(99)

⊕k=1

While aggregating technology sets rather than the output or the input requirement sets (as was done above), this type of aggregate technology allows for full

21 This

technology aggregation structure was earlier used in [1, 30] and goes back to [25].

25 Aggregation of Efficiency and Productivity: From Firm to Sector and Higher . . .

1069

reallocation of inputs and outputs among all the DMUs in the group.22 Other, and equivalent, characterizations of this technology can be given via the group potential input requirement set, defined as Lτ∗ (Y ) = {x : (x, Y ) ∈ τ∗ },

(100)

and via the group potential output set, defined as Pτ∗ (X) = {y : (X, y) ∈ τ∗ }.

(101)

Based on this aggregate technology, and following [36], let the group potential output-oriented technical efficiency be defined as23 OT Eτ∗ ≡ OT Eτ∗ (Xτ , Y j ) ≡ sup{θ : θ Y j ∈ Pτ∗ (X τ )},

(102)

θ

while the dual characterization of Pτ∗ (X τ ), the group potential revenue function is defined as Rτ∗ (Xτ , pj ) ≡ sup{pj y : y ∈ Pτ∗ (X τ )}.

(103)

y

The associated group potential revenue efficiency is then defined as REτ∗ ≡ REτ∗ (X τ , Y j , pj ) ≡

Rτ∗ (Xτ , pj ) pj Y j

, pj Y j = 0.

(104)

Due to duality between the revenue function and the output distance function, we have REτ∗ ≥ OT Eτ∗ , and so the group potential output-oriented allocative efficiency can be defined to turn it into equality, yielding the following decomposition: REτ∗ (X τ , Y j , pj ) = OT Eτ∗ (X τ , Y j ) × OAEτ∗ (X τ , Y j , pj ), ∀τ, j.

(105)

In words, (102) and (104) measure the group efficiency relative to the group potential output set (101) and the associated aggregate cost function, similar to the individual level. By the same token, and following [32, 33], let the group potential input-oriented technical efficiency be defined as

22 More

recently, another definition of aggregate technology, which involved the union of technology sets, was considered by [40,41], which later was shown to be equivalent to the Koopmans-type aggregate technology τ∗ , under standard regularity conditions of production theory (see [47]). 23 Here, note that we allow for different time subscripts for inputs and outputs for the framework to be compatible with the HMPI context.

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I T Eτ∗ ≡ I T Eτ∗ (Y τ , X j ) ≡ inf{λ : λXj ∈ Lτ∗ (Y τ )}, λ

(106)

while the dual characterization of Lτ∗ (Y τ ), the group potential cost function, can be given by Cτ∗ (Y τ , wj ) ≡ inf{wj x : x ∈ Lτ∗ (Y τ )}, x

(107)

and so, the related group potential cost efficiency is then given by CEτ∗ ≡ CEτ∗ (Y τ , X j , wj ) ≡

Cτ∗ (Y τ , wj ) wj X j

, wj Xj = 0.

(108)

As before, due to duality between the cost function and the input distance function, we have CEτ∗ ≤ I T Eτ∗ , and so the group potential input-oriented allocative efficiency can be defined to close this inequality, giving rise to the following decomposition: CEτ∗ (Y τ , X j , wj ) = I T Eτ∗ (Y τ , X j ) × I AEτ∗ (Y τ , X j , wj ), ∀τ, j.

(109)

In words, (106) and (108) measure group efficiency relative to the group potential input requirement set (100) and associated aggregate cost function, in a way similar to measurements done at the individual level.

Reallocation vs. No Reallocation A natural question at this stage is the following: What is the relationship between the group technology when the full reallocation is allowed to those we considered earlier (which did not allow for the full reallocation)? The following simple, yet important lemma clarifies this question. Lemma 1. We have P τ (Xτ ) ⊆ Pτ∗ (X τ ),

(110)

L τ (Yτ ) ⊆ Lτ∗ (Y τ ).

(111)

and

A proof of (110) is relatively simple and can be found in [36] and the proof of (111) is analogous [33]. This lemma affirms what is expected on an intuitive level: The aggregate technology characterizations where full reallocation across firms is allowed must always embrace, as a special case, the aggregate technology where the

25 Aggregation of Efficiency and Productivity: From Firm to Sector and Higher . . .

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full reallocation (of outputs in the input-oriented case and of inputs in the outputoriented case) is not permitted. As a result of this lemma, note that for any (Yτ , Y τ , wj ) we must have Cτ∗ (Y τ , wj ) ≤ C τ (Yτ , wj ),

(112)

and so for any (Yτ , Y τ , wj ), we must also have CEτ∗ ≤ CE τ .

(113)

Similarly, for any (Xτ , X τ , pj ) we have Rτ∗ (X τ , pj ) ≥ R τ (Xτ , pj ),

(114)

and so for any (Xτ , X τ , pj ) we must also have REτ∗ ≥ RE τ .

(115)

To measure the difference in efficiency between these different levels of aggregation, [36] introduced the concept of reallocative efficiency. Specifically, in the outputoriented context, we now also have the group revenue reallocative efficiency, as the multiplicative residual that closes the inequality (115), i.e., RREτ∗ ≡ RREτ∗ (Xτ , X τ , pj ) ≡ Rτ∗ (Xτ , pj )/R τ (Xτ , pj ),

(116)

and we obtain a useful decomposition of the group revenue efficiency REτ∗ = RE τ × RREτ∗ .

(117)

Meanwhile, in the input-oriented context (following [33]), we have the group cost reallocative efficiency, as the multiplicative residual which closes the inequality (113), i.e., CREτ∗ ≡ CREτ∗ (Yτ , Y τ , wj ) ≡ Cτ∗ (Y τ , wj )/C τ (Yτ , wj ),

(118)

and so we obtain a useful decomposition of the group cost efficiency: CEτ∗ = CE τ × CREτ∗ .

(119)

What is even more interesting is that both RREτ∗ and CREτ∗ can be further decomposed, as outlined in the following lemmas. Lemma 2. We have RREτ∗ = OT REτ∗ × OAREτ∗ , ∀τ,

(120)

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where group output-oriented technical reallocative efficiency is OT REτ∗ ≡ OT Eτ∗ /OT E τ ,

(121)

and group output-oriented allocative reallocative efficiency is OAREτ∗ ≡ OAEτ∗ /OAE τ .

(122)

This result is from [36], and its input-oriented analogue (from [32, 33]) is outlined next. Lemma 3. We have CREτ∗ = I T REτ∗ × I AREτ∗ , ∀τ,

(123)

where group input-oriented technical reallocative efficiency is I T REτ∗ ≡ I T Eτ∗ /I T E τ ,

(124)

and group input-oriented allocative reallocative efficiency is I AREτ∗ ≡ I AEτ∗ /I AE τ .

(125)

In words, these two lemmas say that the reallocative efficiency measures characterize the difference for the group between individual efficiency in each DMU and the collective efficiency, where outputs are allowed to be reallocated among DMUs in the input orientation or inputs are allowed to be reallocated among DMUs in the output orientation.

Aggregate vs. Individual Reallocative Measures of Efficiency What about the individual counterparts of the reallocative efficiency measures that appeared in the previous subsection? Using the path “from aggregate to individual,” [36] introduced the reallocative measures for individual output-oriented DMUs, which for k = 1, . . . , n are given by RREτk ≡ REτ∗ /REτk ,

(126)

OT REτk ≡ OT Eτ∗ /OT Eτk ,

(127)

OAREτk ≡ OAEτ∗ /OAEτk ,

(128)

and then established the relationship between individual and group reallocative measures, which we summarize in the next lemma.

25 Aggregation of Efficiency and Productivity: From Firm to Sector and Higher . . .

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Lemma 4. We have 

RREτ∗

n  = (RREτk (xτk , yjk , pj ))−1 × Sjk



OT REτ∗

n  = (OT REτk (xτk , yjk ))−1 × Sjk



OAREτ∗

k=1

−1 (129)

,

−1 (130)

,

k=1

n  k = (OAREτk (xτk , yjk , pj ))−1 × Sae,τ,j

−1 ,

(131)

k=1

where Sjk

≡

pj

pj yjk n

k k=1 yj

,

k Sae,τ,j

p(yjk × OT Eτk (xjk , yjk )) , k = 1, . . . , n. ≡ n p k=1 (yjk × OT Eτk (xjk , yjk )) (132)

Moreover, note that combining these results with the decompositions we derived above, we also get the following useful decompositions of group potential revenue efficiency REτ∗ = OT E τ × OAE τ × OT REτ∗ × OAREτ∗ .

(133)

Furthermore, [32, 33], using the same logic as [36], defined the corresponding reallocative measures for individual input-oriented DMUs (for k = 1, . . . , n) CREτk ≡ CEτ∗ /CEτk ,

(134)

I T REτk ≡ I T Eτ∗ /I T Eτk ,

(135)

I AREτk ≡ I AEτ∗ /I AEτk .

(136)

and then established the relationship between the individual and the group reallocative measures, which we summarize in the following lemma. Lemma 5. We have CREτ∗

 n −1  k k k −1 k = (CREτ (yτ , xj , wj )) × Wj ,

(137)

k=1

I T REτ∗

 n −1  k k k −1 k = (I T REτ (yτ , xj )) × Wj , k=1

(138)

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I AREτ∗

 n −1  k k k −1 k = (I AREτ (yτ , xj , wj )) × Wae,τ,j ,

(139)

k=1

where Wjk ≡

pj

pj yjk n

k , Wae,τ,j ≡ k

k=1 yj

pj (yjk OT Eτk (xjk , yjk )) , k = 1, . . . , n. n pj k=1 (yjk OT Eτk (xjk , yjk )) (140)

If we combine these results with the decompositions derived above, then we get the following decomposition of group potential cost efficiency: CEτ∗ = I T E τ × I AE τ × I T REτ∗ × I AREτ∗ .

(141)

These key results can be further used for extending various aggregation results discussed above to allow full reallocation: For MPI it was done in [31], while for HMPI it was done in [33]. Related extensions for the aggregation of scale efficiency and scale elasticity as well as for the directional distance functions are yet to be developed, which presents a fruitful field of research for the near future.

Remarks on Estimation of Aggregate Scores It is worth emphasizing here that our discussion so far was mainly theoretical, and we had not restricted our attention to any particular estimator. Indeed, the aggregation theories we summarized here are fairly general and can serve as a background for any suitable estimator, whether it is based on Data Envelopment Analysis (DEA), Stochastic Frontier Analysis (SFA), Free Disposal Hull (FDH) approach, or another appropriate paradigm.24 These well-established approaches can be used to estimate the individual efficiency scores which then can be aggregated to obtain the corresponding estimates of most of the aggregate scores we presented above. Note however that the group potential measures are not calculated from the individual efficiency scores, but require calculation directly from the group potential technology. Yet, after imposing two extra assumptions we can recover these measures from the individual scores as well. These two assumptions are actually very common for many methods in productivity and efficiency analysis, especially in DEA. In particular, we can assume that (i) the technology set τk is the same for all DMUs within each period and that (ii) it is also convex, then, following [30, 36] we get:

see [47], and the relevant  Chap. 8, “Stochastic Frontier Analysis: Foundations and Advances I”,  Chap. 9, “Stochastic Frontier Analysis: Foundations and Advances II”, and  Chap. 10, “Data Envelopment Analysis: A Nonparametric Method of Production Analysis” in this Handbook.

24 Example,

25 Aggregation of Efficiency and Productivity: From Firm to Sector and Higher . . .

τ∗ = nτ , ∀k = 1, . . . , n, ∀τ,

1075

(142)

which in turn, for any period τ , gives:

where x˜j ≡ n−1

n

k k=1 xj ,

Pτ∗ (Xτ ) = nPτ (x˜τ ),

(143)

Lτ∗ (Y τ ) = nLτ (y˜τ ),

(144)

and

 where y˜τ ≡ n−1 nk=1 yτk . Intuitively, Pτ (x˜τ ) and Lτ (y˜τ ) are, respectively, the output set and the input requirement set of the “average DMU” for the sample (i.e., a hypothetical DMU whose input-output allocation is the average of input-output allocations in the sample, in period τ ). Therefore, the output-oriented group potential efficiencies can be obtained as the efficiency measures of the average DMU in the group, i.e., we have: OT Eτ∗ (Y τ , X j ) = OT Eτ (x˜τ , y˜j ),

(145)

REτ∗ (X τ , Y j , pj ) = REτ (x˜τ , y˜j , pj ),

(146)

OAEτ∗ (X τ , Y j , pj ) = OAEτ (x˜τ , y˜j , pj ) = REτ (x˜τ , y˜j , pj )/OT Eτ (x˜τ , y˜j ), (147) where OT E, RE, and OAE are as defined in (5), (7), and (9), respectively, with superscript k dropped, and presented in the inter-temporal context. Similarly, the input-oriented group potential efficiencies are the same as the efficiency measures of the average DMU in the group, i.e., we have I T Eτ∗ (Y τ , X j ) = I T Eτ (y˜τ , x˜j ),

(148)

CEτ∗ (Y τ , X j , wj ) = CEτ (y˜τ , x˜j , wj ),

(149)

I AEτ∗ (Y τ , X j , wj ) = I AEτ (y˜τ , x˜j , wj ) = CEτ (y˜τ , x˜j , wj )/I T Eτ (y˜τ , x˜j ), (150) where I T E, CE, and I AE are as defined in (75), (77), and (79) respectively, which we cast in the inter-temporal context, with superscript k dropped. It is also worth reminding that (145) and (148) are the versions of aggregate efficiency measures suggested (without the theoretical developments as summarized here) and advocated by [19].

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Concluding Remarks In this chapter we briefly summarized some of the key results on aggregation in productivity and efficiency analysis. We mostly focused on the output orientation and pointed out that most of the results can be extended or generalized to derive analogous results for the input orientation as well as to various other contexts. Here we conclude by briefly mentioning a few interesting extensions and applications of these and other related aggregation results. First, the aggregation results for the directional distance functions were developed by [8].25 A similar theoretical framework for aggregating growth rates in the Solow’s growth accounting approach was derived by [57]. Meanwhile, the theory for aggregation of the scale efficiency was developed by [59]. Related aggregation analysis can be also found in [38, 43], and some interesting extensions can be found in [2, 9, 23, 24, 28, 29, 35]. Various applications analyzing real data for various economic questions can be found in [18, 21, 22, 34, 42, 52], to mention just a few. Finally, in terms of actual estimation, note that what we discussed is a pointmeasure and one may (and typically should) be interested in the corresponding confidence interval measures and related inference. The first theoretical foundation for this important aspect was laid out in [48], who proposed a practical bootstrap-based approach for constructing confidence intervals and performing related inference on the aggregate efficiency measures. More recently, [49] extended this framework, by deriving convergence rates and new central limit theorems (CLTs) for the aggregate efficiency scores estimated via DEA and FDH. With the help of the Monte Carlo study, they also confirmed that for statistical inference on aggregate efficiency, the standard CLTs work poorly even for very simple 1-input-1-output cases and do not work correctly at all for larger dimensions. Meanwhile, the new CLTs that they derived performed reasonably well, reaching the nominal levels when samples get large. While deriving their asymptotic results, [49] focused on aggregates of the Farrell-type efficiency scores, and similar developments are yet to be made for the other aggregates, which shall constitute key research questions in the area.

Cross-References  Data Envelopment Analysis: A Nonparametric Method of Production Analysis  Stochastic Frontier Analysis: Foundations and Advances I  Stochastic Frontier Analysis: Foundations and Advances II

25 Also

see [55] for this and other related results.

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Part II Applications

Choice of Inputs and Outputs for Production Analysis

26

Subhash C. Ray

Contents Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Inputs and Outputs: Some Basic Features . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Multistage Production . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Classification of Inputs: Variable and Fixed . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Fixed Inputs and Short Run Cost Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Bad Outputs: Zero or Negative Inputs and Outputs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Input Aggregation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Input Aggregation in Nonparametric Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Input Aggregation in Parametric Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Statistical Tests for Input Aggregation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A Statistical Test for Nested Radial DEA Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . DEA and Bootstrap . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Nondiscretionary Inputs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Second Stage Regression . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Truncated Regression in the Second Stage . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Contextual Variables in Parametric Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Choice Between Inputs and Contextual Variables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Input-Output Choice in Some Areas of Application . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Manufacturing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Banking . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Health Care . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Cross-References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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The paper has benefitted from valuable comments by Subal Kumbhakar on an earlier draft. S. C. Ray () Department of Economics, University of Connecticut, Storrs, CT, USA e-mail: [email protected] © Springer Nature Singapore Pte Ltd. 2022 S. C. Ray et al. (eds.), Handbook of Production Economics, https://doi.org/10.1007/978-981-10-3455-8_20

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Abstract

Production creates value by transforming inputs into outputs. Classification of variables as inputs or outputs depends on the scope of decision-making by the firm. Inputs enter the boundary of the firm from outside without any prior processing by the firm and once transformed into outputs exit its jurisdiction without any further processing. This chapter highlights the defining characteristics of inputs and outputs both for single stage and multistage production. The necessary conditions that must be met for a valid aggregation of several inputs into total expenditure are discussed both for nonparametric and parametric models of production. Several statistical tests of hypotheses related to aggregation of several inputs or exclusion of individual inputs in nonparametric models are discussed. The technology set of feasible input-output bundles invariably depends on many environmental or contextual variables that are outside the control of the producer. In parametric Stochastic Frontier Analysis, they can be directly included as determinants of the mean or variance the technical efficiency factor causing shifts in the production frontier. In nonparametric Data Envelopment Analysis, influence of such factors is measured through a second stage regression of efficiency scores on the contextual variables. The alternative approaches of a second stage least squares regression and a truncated regression are briefly discussed. The chapter ends with examples of input-output choice in several popular areas of application like manufacturing, banking, and health care. Keywords

Multistage production · Input aggregation · Contextual variables · Second stage regression

JEL Classification Numbers

D24, C44

Introduction Production is the act of transforming inputs into outputs creating value in the process. Enhancing human welfare is the ultimate goal of an economic system and welfare of an individual in the society is increased through consumption of goods and services. Thus, production of these goods and services is the means of improving welfare. As noted by Marx and Engels, men begin to “distinguish themselves from animals as soon as they begin to produce their means of subsistence, a step which is conditioned by their physical organization. By producing their means of subsistence men are indirectly producing their actual material life” (Marx

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and Engels The German Ideology, “Feuerbach” 1) [17]. Over the different phases of civilization, human societies morphed from hunter-gatherer tribes, into farming communities, and ultimately into the modern industrial economies supported by artificial intelligence, machine learning, and information technology. Despite all the technological progress, however, the essence of production remains conversion of inputs into outputs. In order to measure rates of technical progress or to evaluate productive performance of a firm, one must appropriately identify and measure inputs and outputs. The broad objective of this chapter is to highlight a number of important points that must be considered while classifying resources as inputs or outputs in any specific context. The rest of the chapter unfolds as follows. Section “Inputs and Outputs: Some Basic Features” starts with some defining characteristics of inputs and outputs. This is followed by a discussion of multistage production, the distinction between fixed and variable inputs, the role of fixed inputs in short run cost minimization, and dealing with bad outputs or zero and negative outputs and inputs. Section “Input Aggregation” considers the validity of input aggregation and use of total cost as a single aggregate input in both nonparametric Data Envelopment Analysis (DEA) and parametric Stochastic Frontier Analysis (SFA) models, statistical tests for aggregation and/or inclusion of inputs, and bootstrapping the estimated DEA efficiency scores. Various issues related to use of a second stage regression to measure the marginal effects of nondiscretionary contextual variables on efficiency measurement are discussed in section “Nondiscretionary Inputs.” Section “Input-Output Choice in Some Areas of Application” wraps up the chapter with examples of input-output selection for three frequently analyzed industries: manufacturing, banking, and health care. Section “Conclusion” is the conclusion.

Inputs and Outputs: Some Basic Features In production economics , the firm is the decision-making agent with a clearly defined boundary or jurisdiction of its decisions. It acquires its inputs from outside and, once production is complete, ships out the finished output. In most (if not all) cases, such inputs have been subject to prior processing by some other firm before it enters into its jurisdiction. For example, in agriculture, the farmer uses fertilizers and pesticides as inputs. These fertilizers and other chemicals had to be produced by other firms before they were available as inputs to the farmer. However, the farmer did not participate in the production of these chemicals. At the other end, wheat produced by the farmer is ground into flour by a flour mill. But once the bag of wheat has been shipped out by the farm, it plays no role in any further processing of its product. Thus, an input enters the jurisdiction of the firm from outside without any prior processing whereas its output, once it exits the boundary of the firm, is not subject to any further processing by the firm.

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In some cases, an input used by the firm may not be supplied from outside. A steel plant may acquire coal for its furnace from its own mines. But coal provided by the company’s captive mine as fuel for the furnace is no different from coal purchased from some other supplier in the market and has not been excavated by the steel plant. In classifying resources as inputs and outputs, the critical points to note are: • An input exists even before production starts. The truckload of coal was already there even before the steel plant started production. • An input is depleted in stock as production is carried out. The inventory of coal in the warehouse of the steel plant is reduced as more and more coal is used for steel making. • An input has not been subject to any prior processing by the firm. The truckload of coal has not been excavated from the mine by the steelmaking plant. • An output did not exist in its present form prior to production. Although the iron ore, coal, and other materials used by the steelmaker were in existence, there was no steel before production. • An output increases in stock as production is carried out. • Once the output is shipped out, it is not subject to any further processing by the firm. In some cases, the output may not be physically differentiable from the inputs. Making a bowl of salad, for example, requires slicing tomatoes, cucumbers, and lettuce, chopping onions, and adding salad dressing. The bowl of salad is the output while the various ingredients and labor are the inputs. In this case, the output does not look much different from the inputs and most of the ingredients (at least the vegetables) can be identified and physically separated. But the prepared bowl of salad is more valuable than the raw ingredients. The act of slicing, chopping, and mixing the ingredients is the production process that adds value to the ingredients. An extreme example can be found from the postal service industry where the accumulated mail (consisting of letters, magazines, and packages) at the post office are inputs and the delivered mail at the different addresses are the outputs. In this case, there is no physical transformation of the inputs at all. However, the letter carrier’s labor and the transportation service of the mailman’s vehicle are used to deliver the mail from the post office to the recipient’s addresses thereby creating value. Even though not physically altered in shape or size, the accumulated mail at the post office is turned into delivered mail at the recipient’s address and becomes more valuable to the addressee than the undelivered mail. In any empirical application, the relevant input and output variables need to be defined with care. There should be some obvious technical relationship between inputs and outputs. It is important to remember that definition of inputs and outputs will depend on the boundary of the firm.

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Multistage Production The process of producing the output typically involves various tasks – some to be performed simultaneously and others sequentially. Often, a number of tasks can be grouped together to define a specific stage of production. In many cases, one can identify multiple stages of production within the jurisdiction of the same firm. In such cases, the output from the earlier stage becomes an input for the next stage. For example, in furniture making, lumber, labor, and tools are the inputs and the fabricated parts are the outputs in the fabrication shop. These parts along with additional labor and tools are the inputs in the assembly shop while the assembled but unpainted furniture are the output at this stage. These unfinished furniture, more labor, and paints are the inputs and the finished furniture are the outputs at the final stage. One can think of each division within the firm as a decision-making unit with the inputs and outputs appropriately defined to measure efficiency at each stage. Alternatively, the entire firm can be viewed as the decision-making unit and only the lumber, labor, tools used at the different stages, and the paints are treated as inputs while the finished furniture are the outputs. A strand in the DEA literature described as network DEA (but more appropriately a multistage DEA) seeks to ascribe the overall (in)efficiency of a firm to the different stages of production in a multistage production setting.

Classification of Inputs: Variable and Fixed In production economics, a distinction is made between variable and fixed inputs. Variable inputs are those the quantities of which change with the quantity of output produced. These include raw materials, unskilled labor, and energy. By contrast, one does not see a change in plant size or the number of managers with every change in the output level. These are called fixed inputs. It is important to recognized that the fixed versus variable input designation is not a physical but an economic concept. When the output level increases, the firm can schedule production workers for additional hours and use more fuel and raw materials without any significant adjustment cost. Frequent changes in plant size (even when physically possible, by leasing extra facilities at short notice) would entail enormous amount of adjustment cost. Similarly, hiring and firing management personnel would result in large adjustment costs in the form of severance pay or hiring bonus. In such cases, the firm is better off by not altering the levels of these inputs. That is the reason why they are considered to be fixed. But this is only in the short run. In the long run, when the time comes to renew the lease on the premises or to renew the contract of management staff, the firm can change the levels of these inputs that would be appropriate for its planned level of output.

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Fixed Inputs and Short Run Cost Function For cost minimization in the short run, the firm will have to find the least expensive bundle of the variable inputs that can produce the targeted output when combined with the preset quantities of the fixed inputs. Note that because both the prices and the quantities of the fixed inputs are predetermined, the level of fixed costs is already set. Cost minimization in the short run, therefore, requires minimization of the variable costs only. Consider the production possibility set   n m T = (x, y) : x ∈ R+ can produce y ∈ R+

(1)

where x is a vector of inputs and y is a vector of outputs. Suppose that the firm’s input bundle is partitioned as x = (v; K) where v is the vector of variable inputs and K is the vector of fixed inputs. A corresponding partition of the input price vector is q = (w, r). The firm has a target output vector y0 and has the given bundle of fixed inputs K0 . The minimum variable cost would be     (2) V C w; K 0 , y 0 = min w  v : v, K 0 , y 0 ∈ T . In parametric econometric analysis, one can specify a suitable functional form of the variable cost and estimate the parameters of the model empirically. The regularity conditions that must be satisfied by any estimated variable cot functions are   (i) ∂V∂yC > 0; (positive marginal costs)  C (ii) ∂V > 0; (positive conditional input demands implied by Shephard’s ∂w lemma).  C  homogeneity in the variable input prices) and (iii) ∂V ∂w w =V C; (linear  ∂2V C (iv) The matrix ∂w.∂w is negative definite (concavity in the input prices).  C Additionally, (v) ∂V < 0. This last condition follows from the fact that ∂K because the marginal productivity of each fixed input is positive, any increase in a fixed input must be offset by a decrease in some variable input if the output is to remain constant. This causes the variable cost to decline. In fact, one can define the shadow price of the sth fixed input as ρs = −

∂V C . ∂Ks

(3)

In the single fixed input case, one can perform a statistical test for the hypothesis that the shadow price (ρ) of the fixed input is equal to its market price (r). When

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multiple fixed inputs are involved, one can apply the Mahalanobis D2 test to compare the mean vectors of shadow prices and the market prices of the fixed inputs. The corresponding DEA LP problem is V C ∗0 = min w  v N  λj v j ≤ v; s.t. j =1

N 

j =1 N  j =1 N  j =1

λ j Kj ≤ K 0 ; (4) λj y j ≥ y 0 ; λj = 1;

v ≥ 0; λj ≥ 0, (j = 1, 2, . . . , N ) . 

The short run cost of the firm is C(w, r, y0 , K0 ) = VC(w, K0 , y0 ) + r K0 . The shadow price of a fixed input is the negative of the optimal value of the dual variable associated with that particular fixed input. The marginal cost of any output is the dual variable associated with that output constraint. It is important to clearly distinguish between fixed inputs and outputs. Output is clearly a “cost driver” in the sense that an increase in the output quantity will result in an increase in the cost. For the fixed inputs, there are two opposing forces at work. Clearly, with prices of the fixed inputs given, any increase in the quantity of fixed inputs will increase the fixed cost. But to the extent that the fixed inputs have substitutability with any of the variable inputs, an increase in a fixed input holding the outputs unchanged will lower the quantity of some variable input and hence the cost of variable inputs. Hence, the short run total cost may rise or fall. Sometimes, fixed inputs are confused with outputs because there are variable inputs required for the maintenance of these fixed inputs. One requires labor and other resources for maintenance of a furnace in a steel plant. In this case, an increase in the level of a fixed input is actually increasing the quantity and the cost associated with variable inputs. None the less, the furnace remains an input and not an output because it is not produced in the steel plant. For accurate costing, the maintenance cost of the fixed input (like the furnace) should be included in the user cost of the fixed input and would constitute a part of the fixed cost of the firm.

Bad Outputs: Zero or Negative Inputs and Outputs Bad Output In some cases, production results in some unintended and undesirable bad outputs side by side with the desired or good outputs. Electric power generation also results in smoke emission from thermal plants. Cement production leads to toxic waste

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on the side. In production economics, bad outputs have been treated in various ways – as inputs, joint products with the good outputs, and as by-products of good output production. Of these, the input interpretation is the least satisfactory. There was no smoke before power generation started. The amount of smoke in the air is increased rather than reduced as more power is generated. Further, the smoke in the air is not used for producing anything by the power plant. Thus, it possesses none of three defining characteristics of an input listed above. As for the joint production vis-à-vis by product interpretation, the context should determine which one is more appropriate. In some cases, products are physically joint – like cowhide and beef or some flowers and allergenic pollens. There joint production is the obvious interpretation. Similarly, the same polluting input coal produces heat (for electricity) and carbon. The two outputs – power and carbon emission – are not physically joint but are ascribed to the same input. There by-production is more appropriate. In other cases, the conceptualized production technology depends on the analyst’s perspective. (For a detailed discussion of bad outputs, see the chapter by Murty and Russell in volume 1 of this handbook.) It is important to note in this context that good or bad characterization is related to an individual’s preferences and is not a characteristic of the technology. For example, while smoke is generally considered a bad output, in rural India villager’s often burn wet hay explicitly to create smoke to drive away mosquitoes. In that specific case, smoke is the intended good output.

Zero Input or Output In some cases, one finds that the firm is not using some input at all. For example, in agriculture in less developed countries, farmers use both chemical fertilizers and farmyard manure. But often farms are found to use no manure at all. The same is true for the use of bullocks or motor-powered tillers. This implies that the one input can be substituted for the other. Soil nutrients can be provided either by manure or by chemical fertilizers. In such cases, the two inputs should be aggregated as “Fertilizers and Manure.” The exact basis for aggregation may be their chemical composition or some other formula determined by experts. Similarly, different kinds of fuel (coal, oil, and natural gas) can be aggregated based upon their calorific values. In some other cases, substitutability between pairs of inputs is less obvious. One can think of chemical weed killers and labor. One may spray chemicals to kill weeds or use labor to root the weeds out. In this situation, aggregation is neither possible nor recommended and one must treat any zero value of the chemical herbicide input as it is. As for outputs, in multiproduct firms, some outputs may not be produced at all. This is a perfectly rational decision of a profit or revenue maximizing firm. In fact, even when all firms produce all of the outputs, one would like to compare the combined cost of standalone production of individual outputs with the production cost of a diversified firm producing all of the outputs together to measure economies of scope. This is particularly a problem when outputs are measured in log (as in a Translog cost function). Often the recommended solution is to use a Box-Cox

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transformation instead of the logarithm of the output quantity to circumvent the problem of defining the log of zero. In the nonparametric DEA models, a zero output poses no problem in a radial output-oriented model. Similarly, a zero input is easily accommodated in an inputoriented model. But with a zero input, in an output-oriented model, the benchmark can be constructed only from units that employ zero quantity of the relevant input.

Negative Inputs or Outputs In many papers (especially in the nonparametric DEA literature), authors have developed various mathematical models for evaluating production efficiency in the presence of negative inputs or outputs. One such model simply reverses the algebraic sign and treats negative inputs as outputs or outputs as inputs (when all observations of the relevant input or output are negative). When some observations have negative and others have positive values of an input or output, one adds a sufficiently large positive number to all observations so that the modified values are all positive. One appeals to the translation invariable property of the DEA models to justify this data transformation. (As explained in Ray [24], the output oriented radial DEA model is invariant to input translation but not to output translation. Similarly, in the inputoriented model is invariant to output translation but not to input translation.) But the fundamental point one needs to remember is that the neoclassical production m+n possibility set is a subset of R+ and for any feasible input-output bundle (x, y), n m x ∈ R+ and y ∈ R+ . Thus, negative inputs or outputs are precluded. One often regards the profit of a firm as an output and points out that it can be negative. But a firm does not produce profit. It uses inputs to produce output. Profit is neither an input nor an output but the difference between the revenue from the output and the cost of the inputs.

Input Aggregation Selecting the right number of distinct inputs and outputs for the specification of the underlying technology is often quite difficult in applied production analysis. In many cases, for a variety of reasons, several inputs (or outputs) may be aggregated into a single composite input (or output). In agricultural production, for example, the number of even the important crops may be too large to treat them as separate outputs. This requires aggregating them into a single index of crop output. Similarly, especially in developing countries, one constructs a measure of total cultivated area by adding up irrigated and unirrigated acreage, assigning a higher weight to irrigated land. Because there are farms that cultivate only unirrigated land, including irrigated land as a distinct input would create the usual problems associated with zero input levels. In manufacturing, quantities of different kinds of fossil fuels are sometimes aggregated in terms of their British Thermal Units (BTU) equivalent. In many cases, information about input use is available only in value terms. One may, for example, have to use the total wage payments instead of the numbers of different categories of employees to measure the labor input. This is especially true for applications

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where the input-output data are indirectly constructed from financial statements of firms. In applications of Data Envelopment Analysis (DEA), maintaining a large number of inputs (or outputs) in the model may result in all or most of the decisionmaking units (DMUs) to be rated as efficient. As explained by Leibenstein and Maital [16], this is a result of the dimensionality of the input/output space relative to the number of observations. Meaningful aggregation of multiple inputs or outputs, therefore, reduces the number of constraints in the linear programming model, thereby increasing the “degrees of freedom.” However, while input aggregation has some advantages when it is valid, it does pose problems in cases when it is not valid. The theoretical implications of such aggregation have been addressed in some previous studies. For instance, based on Monte Carlo simulations, Thomas and Tauer [40] argue that with linear aggregation of inputs, the technical efficiency measure becomes an economic measure of efficiency comprising of both technical and allocative components. As such, this introduces bias in the measurement of technical efficiency. In a subsequent study by Tauer [39], the results of simulations indicate that technical efficiency estimates computed by DEA are biased even when the exact aggregator function is used to aggregate inputs. Färe and Zelenyuk [12] and Färe et al. [13] demonstrate that if some inputs are introduced in the DEA model in price aggregated form, the technical efficiency measure will be biased downward as compared to if the inputs were included in disaggregated terms. Further, the bias is equal to the allocative inefficiency. They further illustrate that the DEA technique will yield unbiased technical efficiency scores even when inputs are price aggregated, if and only if there is no allocative inefficiency in the subvector of inputs that is aggregated. The above theoretical developments reveal that while including too many distinct inputs (or outputs) in the DEA model may lead to problems associated with overspecification, aggregating inputs into a fewer number of inputs to be included in the model may lead to biased measures of technical efficiency. The validity of input aggregation in any specific application is therefore an empirical question. However, the statistical side of this issue has been far less explored. Of the few studies that address this aspect of the problem, some are nonparametric, while others assume some parametric form of the statistical distribution. Simar and Wilson [36] discuss bootstrap procedures for testing several restrictions in nonparametric models, including tests for input or output aggregation. Sirvent et al. [38] use simulations to evaluate the performance of a number of statistical tests that can be used for the selection of variables in DEA efficiency models. In a recent paper, Banker et al. [7] propose a method of estimating allocative inefficiency when only aggregate cost or revenue data and quantity data (but not price data for individual inputs or outputs) are available. The idea is that the prices are not known to the analyst. But if all firms face identical prices and are allocatively efficient then the aggregated and disaggregated technical efficiencies are identical. It is a test for this equality. They provide a test of allocative efficiency through an application of Banker’s [4] F-test based on the DEA technical efficiency residuals from the aggregate and the disaggregated models.

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In an empirical application, even when price and quantity data are all available, one might prefer to aggregate subgroups of inputs (or outputs) to overcome the “curse of dimensionality.” There are, of course, various ways in which inputs may be aggregated. In some contexts, the aggregation weights may be guided by technical norms – like aggregating various kinds of fuels into a total energy input based on their individual Btu contents. In some other cases, the aggregation may be data driven – like using Principal Component Analysis where several inputs (or outputs) are replaced by a small number of their principal components, to reduce the problem of dimensionality. (For a discussion of the use of Principal Component Analysis (PCA) within DEA see Adler and Golany [1, 2]. An inherent problem with PCA is that typically there is no intuitive or economic interpretation of what these principal components represent.) An economically meaningful aggregation procedure is one where a subgroup of inputs (or outputs) is replaced by its total cost (or revenue). Using prices would be the valid way to aggregate when choice of the relevant inputs (or outputs) is allocatively efficient.

Input Aggregation in Nonparametric Models Consider an industry producing a scalar output, y, from a bundle of n inputs, x = (x1 , x2 , . . . , xn ). Let (xj , yj ) be the observed input-output bundle of firm j (j = 1, 2, . . . , N). (This section draws heavily from Ray and Mukherjee [28].) The technology is defined by the production possibility set   n T = (x, y) : y ≤ f (x); y ∈ R+ , x ∈ R+

(5)

where f (x) = max y : (x, y) ∈ T is the production function. Firms move from one point to another on the same isoquant when (relative) input prices change – either over time or across regions. But under competitive input market assumption (justifying price-taking behavior) firms in the same market will face same prices. The production function evaluated at the input bundle x0 is     f x 0 = max φy 0 : x 0 , φy0 ∈ T .

(6)

Obviously, every observed input-output bundle (xj , yj ) is feasible. Under the standard assumptions of convexity of the production possibility set and free disposability of inputs and output, an estimate of the set T is: ⎧ ⎨

S = (x, y) : x ≥ ⎩

N j =1

λj x j ; y ≤

N j =1

λj yj ;

N j =1

λj = 1; λj ≥ 0; j = 1, 2, . . . , , N

⎫ ⎬ ⎭ (7)

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A nonparametric estimate of the production frontier at the input bundle x0 is obtained by solving the following linear programming problem:   fˆ x 0 = max φy0 N  s.t. λj x j ρ and 0 otherwise. Stated differently, Tj = 1 indicates that exclusion of the input under consideration significantly alters the optimal value of the DEA problem for unit j. Viewed this way, the random variables Tj are binary outcomes of a series of independent Bernoulli trials with some constant probability p of “success” (i.e., Tj = 1). One may select some hypothesized probability p0 and test the null hypothesis H0 : p < p0 against the alternative H1 : p > p0 . If H0 is rejected, one may conclude that exclusion of the relevant input significantly affects the distribution of efficiency and, hence, it is a relevant input. The p-value for the test statistics is 1 − FB (N − 1, p0 ) where FB is the cumulative density function for the Binomial distribution B(N − 1, p0 ). (The p-value is the probability of getting T0 or more “success” in a series of N − 1 independent trials where the probability of success in any trial is p0 .)

DEA and Bootstrap Simar [32, 33], Simar and Wilson (1998, 2000) set the foundation for the consistent use of bootstrap techniques to generate empirical distributions of efficiency scores and have developed tests of hypotheses relating to returns to scale through bootstrapping. Following Simar and Wilson (1997a), we can describe the existing bootstrap techniques for the output-oriented technical efficiency measure from an output-oriented radial DEA model with the following algorithm:

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(i) Solve the DEA LP problem to obtain ϕˆj for each DMU j = 1, 2, . . . , N. (ii) Select the b-th (b =  1, 2, . . . , B) independent naive bootstrap sample  ∗ ∗ ∗ ϕ1,b , ϕ2,b , . . . , ϕN,b , which consists of N data values drawn with replacement from the estimated values ϕˆj s. 

∗∗ , ϕ ∗∗ , . . . , ϕ ∗∗ (iii) Construct the smoothed bootstrap sample ϕ1,b N,b , from the 2,b naïve bootstrap sample. Notice that all the ϕj s are greater than or equal to 1. Therefore, the smoothed bootstrap sample should be appropriately bounded. It will be computed according to:  ∗ ϕj + hεj if ϕj∗ + hεj ≥ 1 ∗∗   ; for j = 1, 2, . . . , N . (38) ϕj,b = 2 − ϕj∗ + hεj otherwise

As proposed in Silverman (1986), h = 0.9 A n-1/5 , where A = QR is the optimal bandwidth that minimizes the approximate min s.d.; I1.34 mean integrated square error of the distribution ofthe DEA  values of ϕ. (iv) Create the b-th pseudo-data set as {(xj* , yj * = yj

ϕˆ ϕj∗∗

); j = 1, 2, . . . , N}.

(v) Use the pseudo-data set to compute new φˆ j∗ s from the output oriented DEA LP problem. ∗ ; b = 1, 2, . . . , B} for each DMU (vi) Repeat steps (ii)–(iv) B-times to obtain {φˆ j,b j, j = 1, 2, . . . , N. (vii) Calculate the average of the bootstrap estimates of ϕ s for each unit j ϕ ∗j = B 1  ∗ ϕj,b . The bias corrected estimate of ϕj is ϕjbc = 2ϕˆj − ϕ ∗j .Use the B b=1

∗ to construct the relevant bootstrap empirical distribution function of ϕˆj,b confidence intervals for ϕj . (see Simar and Wilson [35] and Sickles and Zelenyuk [30] Sect. 9.3.1 for a detailed discussion).

It should be noted that an interpretation of the results obtained from the bootstrap procedure is not always clear. For example, in the bth replication using the pseudodata consisting of the actual input bundles coupled with the fictitious output levels of firms, the optimal solution φ ∗ shows the scalar expansion factor for the fictitious output quantity and its inverse is not a measure of the efficiency of the actual input output bundle. It is possible that the actual input-output bundle may lie above the production frontier constructed from the pseudo-data obtained in any one bootstrap sample. One may, of course, use the optimal solutions from the (bootstrap) DEA problems to construct measures of the frontier output level producible from the fixed input bundle of a firm. Thus, it is more meaningful to construct  ∗ a 95%  confidence ∗ . In principle, interval of the maximum output with lower and upper bounds y , y L U   the upper bound yU∗ may be used to derive a probabilistic measure of the technical efficiency of an observed input-output bundle. It is still possible that the actually observed output from a given input bundle may exceed its corresponding upper bound.

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Nondiscretionary Inputs Variable inputs are varied while fixed inputs are held constant (even as the level of production changes) at the discretion of the firm (see also the section on “contextual variables” in the chapter on DEA by Ray volume 1 of this handbook). There are, however, other factors that affect the output produced from the inputs chosen by the firm but are beyond its control. For a simple example, consider irrigation in farming. The farmer can increase the level of irrigation by choosing a bigger pump but may want to wait until its existing pump is sufficiently depreciated. Rainfall is an alternative source of water for crops. However, the farmer cannot alter the amount of rainfall at his discretion. These are treated as nondiscretionary inputs. One can think of the discretionary inputs as directly productive and the nondiscretionary inputs as facilitating production. (This description was suggested to me by Subal Kumbhakar.) As noted earlier, the production possibility set consists of all feasible inputoutput bundles and is thus defined in the input-output space. However, production takes place in a specific physical, social, and cultural environment. Differences in environmental conditions can play a decisive role in defining the feasibility of a particular input-output bundle. In measuring the efficiency of a decision-making unit, we assume that it can choose the input bundle it uses, or the output bundle it produces. Unlike inputs or outputs, the environmental factors cannot be chosen by the firm and must be treated as “nondiscretionary.” An obvious example of an environmental factor is rainfall in the context of agricultural production. The maximum output producible from a given bundle of inputs (say labor, fertilizer, and land) depends on the amount of rainfall. In that sense, rainfall contributes to the output much the same way as irrigation. However, while the farmer can choose the level of irrigation, the amount of rainfall is not within his control. Here, rainfall acts as a nondiscretionary input. Note that in defining the feasible set for a DEA LP problem, one has to include a constraint for the amount of rainfall. However, when the radial input-oriented technical efficiency is to be measured, the proportional scaling factor applies only to the discretionary inputs (like labor, fertilizer, and land) but not to rainfall. For another example, consider a secondary school where the average performance of its pupils in a standardized test in mathematics is one of the outputs and hours of classroom instruction in math is one of the inputs. An increase in this input is expected to improve the average test score in math. Now consider another variable – the median family income of the town where the school is located. There is ample evidence to conclude that students from more affluent families where the parents are professionals are better motivated and spend more time on homework and perform better in tests. In that sense, the economic status of the pupil acts like class time spent on math. However, the former is an input while the latter is a contextual variable. They are also referred to as nondiscretionary inputs. Another example of a contextual variable is the marital status of parents of a pupil. A child from a single parent family (irrespective of income) is unlikely to get the same level of parental attention as when both parents are present. Thus, an

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increase the proportion of pupils from single parent families will lower the average math score even when all other variables are unchanged. Two things emerge out of this illustrative example. First, unlike an input, a contextual variable may be either favorable (like family income) or detrimental (like single parent families). Second, a decision maker at an appropriate level of authority (the school superintendent or the Board of Education) can select the input bundle used. This is not true for the contextual variables. For yet another example, consider the efficiency of a water utility. The outputs are the number of customers served and the gallons of water distributed. The inputs are pumps, length of pipe lines, and hours of labor. Note that in an urban area the higher density of population implies that the same number of customers can be served and the same volume of water dispensed with a smaller network of pipelines than what is required in a rural area. Moreover, when many customers are located in the same building (as is the case in an urban community), the labor hours needed for meter reading will be lower than in a rural area where customers are located at distant points. In this case, density of population is an environmental variable. Where contextual variables are considered to be significant determinants of performance, an appropriate way to conceptualize the production technology is to define the production possibility set conditional on a specific vector of contextual variables z0 :     T z0 = (x, y) : y can be produced from x given the contextual variables z0 . (39) One then has a family of conditional production possibility sets. Efficiency is still evaluated at the inputs used and outputs produced. But the appropriate benchmark bundle depends on the applicable vector of contextual variables. The disposability and convexity assumptions about the technology apply to the input-output set but are not necessarily extended to the contextual variables z. Consider the revised transformation function F (x, y; z) = k.

(40)

The conditional production possibility set would then be:       T z0 = (x, y) : F x, y; z0 ≤ 0 .

(41)

There are two different ways to formulate the DEA problem depending on how the revised transformation function is conceptualized. One possibility is to consider the function multiplicatively separable between (x, y) and z as F (x, y, z) = a (x, y) .b(z) = k.

(42)

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In this case, any change in the contextual variables causes a neutral shift in the production frontier without altering the marginal rates of substitution between inputs or marginal rates of transformation between outputs. Such multiplicative separability permits a 2-stage analysis where a conventional DEA model is solved using only the inputs and outputs in the first stage followed by a regression of the DEA efficiency scores on the contextual variables in the second stage. It is important to recognize that this 2-stage analysis is permissible only if the environmental variables (the regressors) in the second stage are uncorrelated with the inputs and outputs in the first stage DEA. This second stage regression analysis was proposed by Ray [22,23] and is quite widely used in empirical applications to explain variations in measured levels of DEA efficiency. The other option is to include the contextual variables along with the inputs and outputs in a comprehensive DEA problem. In this case, favorable environmental factors are treated like inputs because they contribute positively towards outputs while unfavorable environmental factors are treated like outputs because they demand additional inputs. However, even though appropriate inequality constraints are included for these factors, the proportional scaling factors do not apply to them. This one step analysis would be recommended when the contextual factors are correlated with some inputs. In the water utility example, the population density in the area served affects the specific inputs like length of network or labor but not other inputs like the number of pumps. In the school example on the other hand, there is no reason to believe that socioeconomic factors (either good or bad) would be related specifically to some inputs and considering a neutral shift is more appropriate. There are two important points to note about a one-step DEA. First, the analyst must make a prior judgment about whether any relevant contextual variable is favorable or unfavorable because that would determine the nature of the inequality in the relevant constraint. Second, and more importantly, the convexity assumption about the input-output bundles may not be applicable for some contextual variables. This is particularly true for categorical variables. Suppose that in our water utility example, the service areas are classified as rural, urban, and metropolitan but the exact measure of population density is not available for each observation. In this case, all we know is that water delivery is most difficult in the rural areas and the least difficult in the metropolitan areas. Creating convex combinations of a categorical variable representing population density is not meaningful in this context. Following Banker and Morey [5], one may handle this by treating the conditional production possibility sets as nested in the sense that all input-output bundles that are feasible in a less favorable condition are also feasible in in a more favorable condition but not the other way around. In this case, we include only the rural observations to construct the frontier for evaluating utilities serving rural areas but all observations to construct the frontier that is to be used for evaluating the utilities serving the most densely populated areas. It should be noted though that for multiple contextual variables that are categorical such cross-classification may severely restrict the number of observations available for constructing the frontier for the less favored groups.

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Second Stage Regression Ray (1988, [24]) conceptualized a 1-output production function of the form y = g(x).h(z)e−η ; η ≥ 0; 1 ≥ h(z) ≥ 0

(43)

and argued that the DEA radial output-oriented efficiency score τy = ϕ1∗ is a y measure of g(x) = h(z)e−η when g(x) is additive. (Unless x and z are correlated and if the g(x) is additive, there should not be any bias. However, the DEA efficiency measure includes effects of difference in z (like soil type, public infrastructures, etc). The second stage regression is designed to appropriately shift the frontier downwards relative to the best possible environment. In the absence of CRS, the additivity property of g(x) does not hold and there will be a bias due to convexity.  Assume further that lnh(z) = z β. This leads to the second stage regression ln τy = z β + u; u ≤ 0.

(44)

One can estimate this regression by OLS to get consistent estimators of  the slope parameters. However, by construction, some of the fitted residuals uˆ will be positive. To overcome this problem, one can adjust the intercept upwards by the residuals  residual so that the so called “Greene corrected”  largest positive u∗j ∗ uj = uˆ j − uˆ max are all nonpositive. Ray suggests using e ≤ 1 to measure efficiency corrected for the contextual variables. This section is not designed to be comprehensive review of different approaches to measuring the impact of contextual variables on the measured efficiency through a second stage regression. The interested reader may refer to Chaps. 9 and 10 of Sickles and Zelenyuk [30]. Two specific problems should be highlighted. First, when g(x) is strictly concave, the DEA efficiency  measures  will be biased upwards. (Because in that case,  j     ˆ ˆ λj g x < g λj x j | λj = 1.) Also, the “fitted value” h(z) = ez β j

j

j

may exceed 1 for some observations. Moreover, as noted above, the second stage OLS regression will not have the standard statistical properties of the classical linear regression. Banker and Natarajan [6] and Banker, Natarajan, and Zhang [8] frame the second stage regression model differently. First, in the spirit of the parametric stochastic frontier analysis, they include a two-sided random noise along with the one-sided efficiency. Thus, their model is y = g(x).h(z)ev−u ; 1 ≥ h(z) ≥ 0

(45)

Further, while u ≥ 0, vM ≥ v ≥ − vM . Thus, the random noise, although two sided, is bounded both from above and below. Also, they specify h(z) = e−z β and

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only positive values of both the coefficient vector β and the contextual variables z. These non-negativity restrictions automatically ensure 1 ≥ h(z) ≥ 0. Thus, their model becomes   y = v − u − z β ln (46) g(x) For the simplest case of a single contextual variable, the regression with an intercept becomes  ln

y g(x)

 = v − u − β1 z

(47)

Define ε = v − u and ε˜ = ε − E (ε). Then the regression can be written as  ln

y g(x)

 = β0 − β1 z + ε˜ ; where β0 = E (ε) − v M .

(48)

One can use the log of the DEA score τy = ϕ1∗ as the dependent variable and estimate the second stage regression by OLS. While the OLS regression leads to consistent estimators of the coefficients of the contextual variables, the pure inefficiency measures (unrelated to these variables) cannot be derived directly. Banker and Natarajan [6] also consider maximum likelihood estimation of the second stage regression and derive conditional mean and mode of the distribution of the one-sided term (u) (comparable to the formulas in Jondrow, Lovell, Materov, and Schmidt [14]) but advise using the OLS residuals to rank order the units in terms of efficiency.

Truncated Regression in the Second Stage Simar and Wilson [37] extend their earlier models of smoothed bootstrap of DEA efficiency scores to incorporate contextual variables in a second stage regression. However, they object to using either OLS or a Tobit model (i.e., censored regression) and instead propose a truncated regression. Using the optimal value of the DEA output-oriented LP problem, they consider a second stage model ϕi∗ = zi β + ui ≥ 1 ⇒ ui ≥ 1 − zi β.

(49)

  One can specify a normal distribution N 0, σu2 for u and use MLE for truncated ˜ σ˜ u2 ) using only those observations regression to estimate the model parameters (β, ∗ for which the ϕ is strictly greater than 1. They advise bootstrap by drawing pseudo values (u˜ ib ) of the residuals from a Normal distribution with variance σ˜ u2 truncated ∗ = zi β˜ + u at 1 − zi β˜ and constructing a new bootstrap sample ϕib ˜ ib for truncated regression.

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In the computational procedure described above, bootstrapping is applied only in the second stage but the DEA efficiency scores are computed only once with the actual data for the initial estimation of the truncated regression. Simar and Wilson [37] also proposed an alternative and more elaborate computational algorithm that can be described as “double bootstrap” where the DEA results are also bootstrapped. (For details of the steps involved, see Simar and Wilson [37].)

Contextual Variables in Parametric Models In stochastic frontier analysis, a comprehensive maximum likelihood estimation is applied to estimate the production function along with the “efficiency function” nested into it. Typically, the specified model is ln y = β  ln x + v − u

(50)

  where the random noise v is distributed as N 0, σv2 while the inefficiency component u is some one-sided distribution either naturally non-negative (e.g., the exponential or he gamma distribution) or a two-sided distribution truncated from   below at 0. A popular choice for u is the truncated Normal distribution |N μu , σu2 |. To incorporate the contextual variables, one expresses the  mean of the  pretruncated  distribution as μu = α z. All parameters of the model β, α, σv2 , σu2 are estimated by maximizing the likelihood function. Estimates of efficiency at individual data points are obtained by the mean (or mode) of the one-sided component conditional upon the composite disturbance (see the chapters on SFA by Kumbhakar, Parmeter, and Zelenyuk in volume 1 of this Handbook for details). One can also make σu2 a function of the z variables. Wang [42] considers a framework where z affects both the mean and variance.

Choice Between Inputs and Contextual Variables Except in the obvious cases like rainfall, classification of factors affecting output as inputs or contextual variables is not quite straightforward. In principle, an input is something that the producer can choose. For example, the proportion of students from non-English language speaking families has an impact on the average test score of a school district. But the school superintendent cannot change this socio-demographic characteristic of its student population. In that sense, it is an exogenously determined (nondiscretionary) variable. However, the town’s Board of Education may adopt a policy of encouraging enrollment from such families. In that case, it becomes an input (like the number of teachers per student). In the end, classification of variables as inputs or contextual is a judgment call for the analyst. In general, however, the greater the scope of decision-making, the fewer are the variables deemed contextual.

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Input-Output Choice in Some Areas of Application Manufacturing A very popular subject of efficiency and productivity analysis is manufacturing (see also the chapter by Bhoumik in this volume).

Output In this sector, output can be defined in alternative ways. The three main choices are: (i) Value added For value added, the only two inputs included should be Labor and Capital. (ii) Sales/shipment Sales or shipment does not reflect total production. Should be adjusted for change in inventories. (iii) Gross output Gross output should be used only when intermediate inputs (materials and energy) are included. Inputs: The inputs included are labor, capital, energy, and materials. (Recently, purchased services are included as yet another input is the so called KLEMS (Capital-Labor-Energy-Materials-Services) data sets constructed at the 2-digit industry level for different countries. But there is hardly any KLEMS analysis at the firm level.) (i) Labor: Typically, one distinguishes between production workers and nonproduction workers. Because production workers are often scheduled for overtime during peak periods and fewer hours during slack times, a more accurate measure of this labor input is the number of hours worked. Nonproduction workers are usually full-time employees. Nonproduction labor is measured by the number of employees. (ii) Capital: Capital is, by far, the most difficult input to measure. Apart from the fact that the age and vintage differences of machinery make it hard to get an aggregate measure of the capital input, there is also the question of stock vs. flow. Usually, one can get the data only on the value of gross fixed assets. This can be appropriately deflated by a price index of machinery and transport equipment to get a real value of the capital stock. This can be used as a measure of the capital input. One major shortcoming of this approach is that it fails to take account of differences in capacity utilization across firms and/or over years. The true measure of the capital input is the actual flow of capital services that contributes to production of the output during the production period.

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A different approach is to construct a measure of the capital flow input indirectly using the “user cost approach .” The total user cost associated with the capital flow input can be measured by the sum of interest and amortization, depreciation, rent, and repair and maintenance expenses. Next one needs to construct a price of capital services. One can start with the wholesale price index of transport and machinery and compute the interest cost using an appropriate interest rate. For example, suppose that the machinery price index in a particular year is 250 and the interest is 5%. Next assume that the relevant depreciation rate is 7% and repairs and maintenance add another 3%. Hence, the user cost is 15% or 37.50 per year of capital service of one machine. If the total user cost of capital for a firm is 7500. We can measure the flow of capital 7500 service as 37.50 = 200 machine years. A note of caution here is that one should never use new capital formation to measure capital input. New capital formation is investment and neither the stock of capital nor the flow of capital services. (iii) Energy: In general, information on energy input is available mainly in terms of the expenditure on energy. The principal sources of energy are (a) coal, (b) oil, (c) electricity, and (d) natural gas. One needs to construct a price of energy as a weighted average of these four different energy inputs. The problem is that these fuels are measured in different units. For example, coal is measured in tons while electricity in kilowatt hours. One must, first, reduce these units of measurement to British Thermal Units (BTUs). Thus, price per ton of coal is to be transformed into the price per BTU of energy from coal. Similarly, for other fuels. Next a price of energy per BTU is constructed as the weighted average of these energy source specific prices using the expenditure share of industrial consumption of energy in the geographical region (e.g., the states of the USA) where the firm is located. Finally, a measure of the quantity of energy consumed (in BTU) by the firm is obtainable by the ratio of the expenditure on fuels and this price per BTU of energy. (iv) Materials: The materials input in most cases include too many different items and even though the total expenditure on materials may be available, it is difficult to construct a price and deflate the expenditure to get a measure of the quantity of materials consumed. The only reasonable way to measure materials is by the value (effectively setting the price per unit equal to 1). Of course, for inter temporal analysis, one can use the price index of industrial raw materials to deflate the nominal amount of expenditure on materials.

Banking Although an overwhelming share of empirical applications relate to measurement of efficiency in banking, there is no consensus on what constitute the inputs and

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outputs of a bank. (For a detailed analysis of the banking industry see the chapter by Miller in this volume.) There are two competing views about the technological character of a bank as a production decision-making unit. Berger and Humphrey [9] describe the two alternative views as production approach and intermediation approach. In the production approach, a bank is visualized as a provider of different kinds of services to deposit and credit account holders by performing transactions and processing documents like loan application, credit reports, demand drafts, and so on. In this view of a bank, labor, and physical capital (like building and equipment) are the inputs and the numbers of deposit and credit transactions and documents processed are the outputs. Because information on the number of deposit and credit transactions performed or the number of documents processed is not available, researchers use either the numbers of deposit and credit accounts or the monetary values of the balances in these accounts. In the intermediation approach (also known as the Asset Approach), a bank is seen to be a financial intermediary between savers with funds to lend and investors looking for funds to borrow. The principal act of value addition by a bank is to transform the deposits and other borrowed funds into loans and other investments that yield returns. As a financial intermediary, a bank uses labor and capital along with funds accepted from depositors (used as raw materials) to create income generating assets like loans and investments. In the intermediation approach based primarily on the model of the short run optimization problem of a financial firm (like a bank) developed by Sealey and Lindley [29], loans and investments are the outputs while labor, deposits, and (physical) capital are the inputs of the bank. A choice between the two different approaches is not merely a matter of interpretation of how a bank works but has important implications for selection of inputs and outputs. As Berger and Humphrey [9] argue, the production approach is more appropriate for evaluating performance at the branch level because branches primarily process accounts for the bank as a whole and branch managers have little control over bank funding and investment decisions. For the bank as the unit of evaluation, the intermediation approach is more appropriate because it explicitly includes interest expenses, which account for a major part of the total expenses of a bank. There are numerous studies, however, at the bank level which treat deposits as output. It is often argued that the bank as a financial firm does offer valuable service to the depositors by keeping their savings secure and deposits, therefore, should be treated as output. But these funds in deposit accounts generate no revenue to the banks unless transformed into credit given out to borrowers. Sealey and Lindley [29] argue that the deposited funds are themselves inputs that are turned into “loanable funds” as intermediate output which is ultimately transformed into revenue generating assets. Another argument often voiced against considering deposits as input is that it would imply that a bank becomes more efficient if it reduces deposits because

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economizing on the use of inputs enhances efficiency. But this line of argument is somewhat misleading in that it does not tell the whole story. One must recognize that acquiring more deposits cannot be an end in itself for the bank and if the funds in the deposit accounts are held idle (entailing interest expenses) without being processed into loans and investments generating revenue, an increase in deposits would indeed make the bank less efficient! A variable that has received little attention in the production efficiency literature on banking is the amount of equity capital of a bank. Very few studies have included it in the list of inputs and outputs. Berger and Mester [11] argue that within the intermediation or asset approach, the equity capital of the bank along with other liability items like core deposits and purchased funds are sources of funds and should be treated as inputs that enable a bank to produce the revenue generating assets. However, unlike deposits and other funds, the equity capital cannot be changed in the short run. Hence, it ought to be treated as a fixed input. Ray and Das [27] in their study of short run cost and profit efficiency of Indian commercial banks included a bank’s capital and reserve as fixed input. Finally, there is no consensus about how to treat the number of branches of a bank in the input-output classification. On one hand, increase in the number of branches amount to easier access for retail customers of the bank and can be interpreted as better service quality. In that sense, a branch is an output of the bank. On the other hand, a branch generates more business and can, therefore, be treated as an input. In fact, like deposits, number of branches have been treated as an output in some studies and as input in some others. An ingenious way to treat branches in the production technology of a bank is to include it as a conditioning variable in a cost function with unequivocal output quantities and input prices as regressors. If the coefficient of the number of branches is negative, then it can be treated as an input because a negative marginal effect implies that increase in this “input” permits reduction in the use of other inputs resulting in a lower cost without reducing the outputs. If, however, the marginal effect is positive, branches should be treated as output because in that case, with more branches, the cost of producing the unchanged quantities of the other outputs becomes higher.

Health Care In view of the fast-increasing cost of health care all over the world, it is not surprising that efficiency in delivery of health care has attracted considerable attention from both academic researchers and policy makers and DEA has been extensively used to evaluate the efficiency of hospitals, nursing homes, and other providers of health care. Conceptually the output of health care is the extent of improvement in health status or the physical quality of life of a patient. Just as in the case of education, the cognitive skills acquired by a pupil can only be indirectly (and imperfectly)

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measured by standardized test scores, the output of a provider of health care must be measured indirectly by a number of observable and quantifiable indicators. A hospital as the unit of analysis is regarded as a production decision-making unit combining different kinds of labor and capital inputs to provide treatment to different kinds of patients.

Inputs Three broad kinds of labor considered as separate inputs are numbers of physicians, nurses and paramedical staff, and other workers. Where detailed data are available, a further distinction is made between surgeons and general practitioners. Measuring the capital input is far more problematic. While the hi-tech equipment account for the bulk of the physical capital input of a hospital, lack of data makes it difficult to include them in a measure of the capital input used. Similarly, the size of the hospital building and the number of intensive care units available are measures of capital input that should be included if the data are available. In most cases, the number of beds is used as a “catch all” measure of the capital input. Basically, the number of beds is regarded as a measure of the size of a hospital and the implicit assumption is that other capital inputs are proportional to the number of beds.

Outputs Outputs are measured by the numbers of inpatient days and outpatient cases treated. In some cases, inpatient care is further broken up into emergency, maternity, and other kinds of treatment. In the case of health care, the preexisting condition of the patient at the time of admission is an important determinant of the resources need for treatment. Therefore, adjusting for complexity and intradiagnostic severity of cases is important while measuring output by the number of patients. In practice, however, even if detailed information on severity and complexity of cases may be obtained for one or two hospitals, it is unrealistic to expect that the analyst will have such information for all hospitals in the sample. A minimal adjustment for complexity may be made by grouping patients as seniors (needing geriatric care), children (receiving pediatric care), and other adults (requiring general care). Another important aspect of health care that should be taken into consideration in measuring efficiency is the quality of care provided. A provider may be able to treat a larger number of patients from the same bundle of labor and capital inputs if the quality of care is lower. Few studies of health care efficiency have taken explicit account of quality. Sometimes an index of quality is constructed from patient satisfaction survey. But constructing an objective measure of quality of care is difficult. Intuitively, a better quality of care restores the patient to normal health sooner and keeps the person healthy for a longer time period in the absence of any new and unrelated ailment.

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In the extreme case, negligence in critical situations may result in death of the patient. In some studies, the incidence of patient mortality is treated as a measure of the lack of quality. In other words, lower mortality rate ceteris paribus implies better quality of care. Another variable used to measure quality is the proportion of unscheduled readmissions. Here the presumption is that if a patient needs to be readmitted without a scheduled follow-up, the patient was given less than proper care in the hospital and was discharged even before treatment was complete.

Conclusion Inputs are related to the outputs by the technology. The role of any empirically constructed production or transformation function is to quantify this relationship. The technology can also be formulated through cost, revenue, and profit functions. But these require additional assumption about optimizing behavior by the producer. In principle, the production technology captures a physical relationship between inputs and outputs. In that sense, identical input bundles should produce identical outputs unless there are differences in the contextual variables – systematic or random. A second stage regression in DEA or incorporating such variables in the statistical distribution of the efficiency component of the error term in stochastic frontier analysis allows one to extract the impact of these contextual variables from the measured efficiency. Finally, appropriate choice of inputs and outputs is a precondition of any meaningful analysis of the production technology. Whether one wishes to evaluate productive performance through technical efficiency or simply wants to examine returns to scale properties, homotheticity, sub-/superadditivity or any other characteristic of the technology, improper selection of inputs and outputs will render any empirical analysis of production meaningless. This chapter ends with a note of caution. In some cases (especially among OR analysts), there is a perception that anything that the decision makers want to reduce is an input while anything that they want to increase is an output. An extreme example is one where in an international comparison of efficiency in emission control, fossil fuel is treated as input (because the author believes that nations should use less of it) while nonfossil fuel is treated as an output (because the author considers it desirable to use more of it). One can avoid such ill-conceived model formulation by paying attention to the underlying production economic theory (see the DEA example in Sect. 6.2.5 in Ramanathan [21]).

Cross-References  Bad Outputs  Data Envelopment Analysis: A Nonparametric Method of Production Analysis

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References 1. Adler N, Golany B (2001) Evaluation of deregulated airline networks using data envelopment analysis with principal component analysis with an application to Western Europe. Eur J Oper Res 132:260–273 2. Adler N, Golany B (2002) Including principal component weights to improve discrimination in data envelopment analysis. J Oper Res Soc 53:985–991 3. Aigner DJ, Chu SF (1968) On estimating the industry production function. Am Econ Rev 58(4):826–839 4. Banker RD (1993) Maximum likelihood, consistency, and data envelopment analysis: a statistical foundation. Manag Sci 39:1265–1273 5. Banker RD, Morey RC (1986) The use of categorical variables in data envelopment analysis. Manag Sci 32(12):1613–1627 6. Banker RD, Natarajan R (2008) Evaluating contextual variables affecting productivity using data envelopment analysis. Oper Res 56(1):48–58 7. Banker RD, Chang H, Natarajan R (2007) Estimating DEA technical and allocative inefficiency using aggregate cost or revenue data. J Prod Anal 27:115–121 8. Banker RD, Natarajan R, Zhang D (2019) Two-stage estimation of contextual variables in stochastic production function models using data envelopment analysis: second stage OLS versus bootstrap approaches. Eur J Oper Res 278:368–384 9. Berger AN, Humphrey DB (1991) Efficiency of financial institutions: international survey and directions for future research. Eur J Oper Res 98:175–212 10. Berger AN, Mester LJ (1997) Inside the black box: what explains differences in the efficiencies of financial institutions? J Bank Financ 21:895–967 11. Bhaumik S (2022, forthcoming) Technical efficiency and its determinants in the manufacturing sector: what we know and what we should know. In: Ray SC, Chamber R, Kumbhakar SC (eds) Handbook of production economics, vol 2. Springer Nature 12. Färe R, Zelenyuk V (2002) Input aggregation and technical efficiency. Appl Econ Lett 9: 635–636 13. Färe R, Grosskopf S, Zelenyuk V (2004) Aggregation bias and its bounds in measuring technical efficiency. Appl Econ Lett 11:657–660 14. Jondrow J, Materov IS, Lovell CAK, Schmidt P (1982) On the estimation of technical inefficiency in the stochastic frontier production function model. J Econ 19(2–3):233–238 15. Kumbhakar SC, Parmeter C, Zelenyuk V (2022 forthcoming) Stochastic frontier analysis: foundations and advances I. In: Ray SC, Chamber R, Kumbhakar SC (eds) Handbook of production economics, vol 1. Springer Nature 16. Leibenstein H, Maital S (1992) Empirical estimation and partitioning of X-inefficiency: a dataenvelopment approach. Am Econ Rev 82(2):428–433 17. Marx K, Engels F (1938) The German ideology (trans: Looch W, McGill CP). Lawrence and Wishart, London 18. Miller S (2022, forthcoming) Empirical analysis of production economics: applications to banking. In: Ray SC, Chamber R, Kumbhakar SC (eds) Handbook of production economics, vol 2. Springer Nature 19. Murty S, Russell RR (2022, forthcoming) Bad outputs in production economics. In: Ray SC, Chamber R, Kumbhakar SC (eds) Handbook of production economics, vol 1. Springer Nature 20. Pastor JT, Ruiz JL, Sirvent I (2002) A statistical test for nested radial DEA models. Oper Res 50(4):728–735 21. Ramanathan R (2003) An introduction to data envelopment analysis. Sage 22. Ray SC (1988) Data envelopment analysis, non-discretionary inputs and efficiency: an alternative interpretation. Socio-Economic Planning Sciences 22(4):167–176 23. Ray SC (1991) Resource-use efficiency in public schools: a study of connecticut data. Manag Sci 37(12):1620–1628

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24. Ray S (2004) Data envelopment analysis: theory and techniques for economics and operations research. Cambridge University Press, Cambridge, UK 25. Ray SC (2022, forthcoming) Data envelopment analysis: a nonparametric approach to production analysis. In: Ray SC, Chamber R, Kumbhakar SC (eds) Handbook of production economics, vol 1. Springer Nature 26. Ray SC (2022, forthcoming) Conceptualization and measurement of productivity growth and technical change. In: Ray SC, Chamber R, Kumbhakar SC (eds) Handbook of production economics, vol 1. Springer Nature 27. Ray SC, Das A (2010) Distribution of cost and profit efficiency: evidence from Indian banking. Eur J Oper Res 201:297–307 28. Ray SC, Mukherjee K (2005) The validity of input aggregation in DEA models: some statistical tests. University of Connecticut Economics working paper 2005-54 29. Sealey C, Lindley J (1977) Inputs, outputs, and a theory of production and cost at depository financial institutions. J Financ 32:1251–1266 30. Sickles R, Zelenyuk V (2019) Measurement of productivity and efficiency. Cambridge University Press, Cambridge 31. Silverman BW (1986) Density estimation for statistics and data analysis. London: Chapman and Hall 32. Simar L (1992) Estimating efficiencies from frontier models with panel data: a comparison of parametric, nonparametric, and semiparametric methods with bootstrapping. J Prod Anal 3(1):171–203 33. Simar L (1996) Aspects of statistical analysis in DEA-type frontier models. J Prod Anal 7(27):177–2185 34. Simar L, Wilson PW (1998) Sensitivity analysis of efficiency scores: how to bootstrap in nonparametric frontier models. Manag Sci 44:49–61 35. Simar L, Wilson PW (2000) Statistical inference in nonparametric frontier models: the state of the art. J Prod Anal 13:49–78 36. Simar L, Wilson PW (2001) Testing restrictions in nonparametric efficiency models. Commun Stat Simul Comput 30(1):159–184 37. Simar L, Wilson PW (2007) Estimation and inference in two-stage, semi-parametric models of production processes. J Econ 136:31–64 38. Sirvent I, Ruiz JL, Borrás F, Pastor JT (2005) A Monte Carlo evaluation of several tests for the selection of variables in DEA models. Int J Inf Technol Decis Mak 4(3):325–344 39. Tauer LW (2001) Input aggregation and computed technical efficiency. Appl Econ Lett 8: 295–297 40. Thomas AC, Tauer LW (1994) Linear input aggregation bias in nonparametric technical efficiency measurement. Can J Agric Econ 42:77–86 41. Varian HR (1984) The nonparametric approach to production analysis. Econometrica 52(3):579–597 42. Wang HJ (2002) Heteroscedasticity and non-monotonic efficiency effects of a stochastic frontier model. J Prod Anal 18:241–253

Airline Economics: A Survey of Applied Issues in the Performance of the US and International Airline Industry

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Levent Kutlu, Daniel Prudencio, and Robin C. Sickles

Contents Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Mergers, Alliances, Vertical Integration and Collusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Mergers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Collusive Behavior . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . On Collaboration and Vertical Integration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Financial Struggle in the Airline Industry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Pricing and Differentiation by Heterogeneous Characteristics . . . . . . . . . . . . . . . . . . . . . . . . Market Power and Price Premium . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Threat of Entry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Market Power and Efficiency . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Airline Efficiency . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Economic Impact of Delays . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Governance and Airport Efficiency . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Deregulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Prepared for Volume II of the Handbook of Production Economics, Robert Chambers, Subal Kumbhakar, and Subhash Ray (eds.), New York: Springer, in progress L. Kutlu () Department of Economics and Finance, University of Texas Rio Grande Valley, Edinburg, TX, USA e-mail: [email protected] D. Prudencio · R. C. Sickles Department of Economics, Rice University, Houston, TX, USA e-mail: [email protected]; [email protected] © Springer Nature Singapore Pte Ltd. 2022 S. C. Ray et al. (eds.), Handbook of Production Economics, https://doi.org/10.1007/978-981-10-3455-8_1

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Abstract

In this survey, we consider applied issues in the productivity and efficiency of the US and the international airline industry. We discuss the implications of mergers and alliances, vertical integration, collusive behavior, bankruptcy, pricing and differentiation by heterogeneous service and network characteristics, entry and competition, airport governance, inefficiency, and deregulatory dynamics. Due to legacies of regulations in the US, European, and international airline industry and the existence of subsidized national flag carriers, there are substantial differences in the level of productivity in the provision of airline services across different carriers and over time. Thus, we also examine various treatments in the applied airline economics literature that have modeled and measured the existence of time-varying inefficiency and its persistent, brought about in part by the longrun inefficiency of incumbent firms operating in non-contested markets. The formal rules and dynamics of regulation and deregulatory initiatives in the USA, Europe, and in international markets are also discussed, as well as the economic impacts of delays, the presence and exploitation of market power in projected niche market’s and the price premiums that go hand-in-hand with market power. Keywords

Airline industry · Competition · Mergers · Productivity · Efficiency · Innovation · Deregulation · Networks

Introduction Our survey of applied issues in the productivity and efficiency of the US and international airline industry will focus on a number of related issues. These include the formation of mergers and alliances, the exploitation by airline firms of the economies of vertical integration, as well as collusive behavior. We also will touch on issues of bankruptcy, pricing and differentiation by heterogeneous service and network characteristics, and how the threat of entry may have provided some competitive behaviors by firms as they acquired larger market sizes via contestability of those larger markets. Due to legacies of regulations in the US, European, and international airline industry and the existence of subsidized national flag carriers, substantial inefficiencies have also existed in the provision of airline services. Thus, we also will examine various treatments in the applied airline economics literature that have modeled and measured the existence of time-varying inefficiency and its persistent, as evidenced by the long-run inefficiency of incumbent firms operating in noncontested markets. The formal rules and dynamics of regulation and deregulatory initiatives in the USA, Europe, and in international markets will be discussed, as well as the economic impacts of delays, the presence and exploitation of market power in projected niche markets and the price premiums that go hand-inhand with market power. We also discuss recent work on the economics of airport

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governance, an issue that has had relatively little written on it in the economics literatures although it is probably the most important constraining factors in the ability of airline firms to be productive and efficient. In the course of our survey, we will at times provide a bit more analytic and modeling detail in order to provide the reader with technique as well as an historical overview. Our survey chapter begins with a discussion of the industrial organization of the airline industry, focusing on applied work that addresses mergers, alliances, vertical integration, and collusion. In section “Mergers, Alliances, Vertical Integration and Collusion,” we focus on the US industry in regard to such matters as its industry structure has been driven much more by market forces than by national carrier sovereignty, as has been the case in much of Europe and in Asian, Africa, and Latin America. Section “Financial Struggle in the Airline Industry” provides a brief discussion of entry and exit studies and those that focus on the puzzling record of financial instability in the airline industry. As a way to differentiate their product, legacy carriers have opted to expand networks to differentiate their product and they have done this largely by mergers and acquisitions. However, their cost structures post-merger and acquisition are not clearly lower. Our next section “Pricing and Differentiation by Heterogeneous Characteristics” explores issues of pricing and its differentiation in a setting of heterogeneous demand and supply characteristics. The impact of competition in a setting in which there is heterogeneity in consumers’ willingness-to-pay and brand loyalty may have different impacts (positive or negative) on different sets of consumers. The relationship between market power and efficiency is taken up in section “Market Power and Price Premium,” focusing on structural dynamic modeling of efficient super-game equilibrium of airlines competing in the same city-pair market, while section “Airline Efficiency” deals more explicitly with the form of and source of the inefficiency. The economic impact of delays is taken up in section “Economic Impact of Delays.” As air traffic grows and capacity is constrained by local zoning or other public policy priorities, delays are a natural by-product of growing airport congestion. This relates in part with the topic of governance and airport efficiency that is taken up in section “Governance and Airport Efficiency.” The section surveys studies that estimate the relative efficiency differences among US airports, controlling for airport specific heterogeneity, and speaks to efficiency impacts of local authority, hub size, and availability of multiple urban airports. Deregulation is taken up in section “Deregulation,” beginning with the 1978 Airline Deregulation Act and continuing through a series of transitional changes in regulatory policy in the USA and in Europe. A brief conclusion is provided in section “Conclusions.”

Mergers, Alliances, Vertical Integration and Collusion The airlines industry has experienced a rise in market concentration in the last two decades. Whereas in 2008 six legacy airlines served the market, mergers in the industry, decreased the number of legacy firms to three: American, Delta, and United. Furthermore, low-cost carriers have also experience mergers, both with

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other low-cost carriers or with legacy firms. As such, by 2015, Southwest Airlines, along with the three legacy firms, controlled 80% of the airline traffic in the continental US [91]. The recent consolidation of the market allowed firms to expand their networks and to increase efficiency, nevertheless, such a consolidation raises antitrust concerns. For the purpose of this review, hence, we are interested in, how mergers between large firms influence the structure of the market, how collusive behavior may take place in an oligopolistic market, and finally, how collaboration takes place in the form of code-sharing.

Mergers When antitrust authorities evaluate mergers, they evaluate how market power might be created, and whether barriers to entry may be enhanced by the merger. Both considerations are done for the short and long run. Early work by Nevo [76] has been used to evaluate the short run implications of a merger, and recent contributions in entry and dynamic games ([14, 36, 71], and [2, 3]) improved our understanding of the long-term effects of a merger. Since the early work on the short-term effects of a merger is well documented, we briefly describe the latter. Li et al. [71] analyze the endogenous service choice of a carrier to offer nonstop or connecting services on a route after a merger takes place. Their model has a standard two-stage structure where carriers choose their type of service and then choose equilibrium prices. Importantly, they assume complete information on the information of qualities and costs of the type of service provided (non-stop or connecting). The key insight the paper provides is to point out that the choice of service (to offer a non-stop or a connecting flight) reveals information about the firm, specifically about their efficiency to provide the service selected. The authors show that accounting for selection of the firms able to compete in the nonstop market is important, since we may otherwise overestimate the probability that other firms may enter a market where a new merger has taken place. Interestingly, Ciliberto et al. [36] reach to similar conclusions, although they do not focus on the type of service provided, but rather on the endogenous entry decision of the firms. The authors find that not accounting for endogenous entry to a market leads to overestimation of demand elasticities, which leads to biased markups, and therefore incorrect implications when analyzing the market power generated by a merger. When simulating a merger, the authors find that the new merged firm has strong incentives to enter new markets, and that it faces a stronger threat of entry from other legacy firms, rather than from low-cost carriers. Complementing the static analysis that measure the short run effects of a proposed merger, recent contributions by Benkard et al. [14] and Aguirregabiria and Ho [2, 3] in dynamic games, look to understand better how long increases in concentration due to a merger are likely to prevail, as well as their impact on prices in the medium and long run. These methods, rather than substitutes to the more used static models, are useful complements to them. For example, Benkard et al.

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[14], building on a two-step estimator by Bajari et al. [10], estimate policy functions based on historical premerger data, and then future industry outcomes are simulated both with and without the proposed merger. The authors find that low-cost carriers play an important role in creating offsetting entry, and that the size of an airline’s network at each end of the route is an important determinant of entry. Hence, the empirical model will predict offsetting entry after a merger takes place on routes where there is a potential entrant with a rich route network close to the route being contested. In a counterfactual analysis, they find that, when considering the dynamic aspects of a merger, some mergers that were not authorized may have been able to go through, as the attempted merger of United Airlines (UA) with US Airways. Given that the overlap of the network of UA and US Airways was in areas where other low-cost carriers were operating and expanding, the decrease in competition (due to the reduction of flights operated by both airlines) may have created incentives for the low-cost carriers to enter the less competitive routes mainly served by the new merged firm. Whenever a large network of a low-cost carrier does not overlap with the network of the proposed merger network, antitrust authorities can implement measures to promote the entry of low-cost carriers, so as to avoid monopolistic practices of the new merged company. Zhang et al. [91] evaluate one of such measures. The authors evaluate the effectiveness of forcing the merging companies to divest assets as a condition for merger approval. Specifically, they evaluate the divestiture of the control of gates, and find that such measure is effective at decreasing rates for both merging airlines (3% decrease), as well as non-merging ones (1% decrease) at affected airports, relative to airports where no forced divestiture took place. Understanding the long-term dynamics in a market is important to more accurately estimate the potential changes in the market due to a merger, and act accordingly. Likewise, understanding the sources of constraints to enter a market can aid antitrust authorities to implement measures that may avoid monopolistic behavior.

Collusive Behavior In oligopolistic markets, even if two large firms do not merge, collusive behavior can still have a corrosive effect on prices and consumer welfare. In an earlier work on collusion, Alam et al. [5] analyze the long run pricing behavior of carriers competing in the same rout. If firms sustain a price relationship, the series will be cointegrated when looking at the mean price, or stationary when considering the price distribution. Therefore, testing for stationarity and cointegration allow the authors to draw conclusions about parallel pricing. Using an index of price dispersion, the authors find long-run stability in approximately two-thirds of their sample. Similarly, Fang and Sickles [41] develop a dynamic model of collusion in airport-pair routes for selected US airlines and present an estimation procedure by

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Kalman-filtering techniques. Nevertheless, we should note that, as pointed out by Motta [75], parallel pricing can also be present in competitive markets. In a recent work, Ciliberto and Williams [34] aim to find what facilitates collusion. To test for collusion, the authors nest parameters into a standard oligopoly model with differentiated products, where firms compete in prices. The estimation of the conduct parameters is achieved using variation in multimarket contact across local airline markets. Intuitively, the authors find that multimarket contact facilitates collusion among airlines. Carriers with many markets simultaneously seem to sustain almost perfect coordination, and importantly, the cross-price elasticity seems to play a key role in determining the impact of multimarket contact on collusive behavior. Finally, Ciliberto et al. [37] formulate two additional tests for collusion based on the following insight. First, colluding firms will reduce pair-wise differences in prices within market, and second, in order to avoid informational costs, colluding firms will sacrifice efficiency in production by increasing price rigidity. The authors then test for contexts in the airline industry that may be consistent with the previous two implication, and find that greater multimarket contact leads to pricing patterns consistent with the previous two implications, whereas code-share agreements are consistent with only the second implication.

On Collaboration and Vertical Integration Another way that carriers have looked to strengthen their market power or differentiate their product is through formal collaboration with other carriers to broaden the set of services the carrier can provide, or to expand their networks. One of the most common alliances is code-sharing, which is an agreement between two carriers that allows them to sell seats on specific flights that are operated by one of the parties. Code-sharing is attractive for efficiency and competitive motives as it effectively widens the network of a carrier. Code-sharing may also eliminate the double marginalization that would exist had the airlines independently determined the price for different segments of a trip [25, 27]. Regarding the competitive motive, it may be used as a means toward preserving or increasing market power. As Bamberger et al. [11] argue, it may give the allied firms a broader reach in their advertising and service, which may broaden their consumer base while the advertising costs are shared with the allied firm. Others, as Goetz and Shapiro [51] argue, code-sharing may be used as a preemptive action when a legacy firm faces a threat of entry from a low-cost carrier. The gains in efficiency by the use of code-sharing may provide a larger network for consumers at lower prices; nevertheless, the benefits to the consumer may be hampered when one of the affiliated firms in the code-share agreement also provides a single-carrier product in the concerned market [49]. In such instances, doublemarginalization may not be eliminated by code-sharing, as the firm offering a competing product will have incentives to soften competition to their own product, and hence, it will negotiate higher prices in the segment where it has a code-share agreement.

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In an empirical application of the effect of code-sharing on consumer welfare, Amantier and Richard [8] study the impact of the 1999 domestic code-share agreement between Continental Airlines (CA) and Northwest Airlines (NW). The authors find that the consumer’s valuation of an airline product is not only significantly affected by the price of it, but also by the flight duration and the time spent in transit at an intermediate airport. Also, they find that the impact of the code-sharing did not have a significant impact on consumer welfare on average, nevertheless, this is not to say that the program had a null effect on all consumers. The program did increase the average surplus of connecting passengers but decreased it for non-stop passengers. This result may be attributed in large part not by changes in price, but by changes in the product characteristics. Apart from code-sharing, another frequent means of collaboration between airlines is the sub-contraction of services for specific routes. Frequently, major airlines sub-contract portions of their network to a local carrier, which may be owned by the major airline or it may be independent. Forbes and Lederman [45] study the effects of vertical integration on operational performance. Specifically, they analyze whether, in a particular airport, an airline’s use of an owned carrier rather than an independent one affects delays and cancellations of the airline’s own flights out of that airport. Using an IV estimation procedure, they find that integrated major carriers are systematically better than nonintegrated ones on the same day. Moreover, differences in efficiency seem to be more pronounced during days with extreme weather and when airports are more congested. These results are consistent with previous finding by the authors in Forbes and Lederman [44], who find that airlines are more likely to use owned regional carriers on city pairs with more complex schedules, and therefore, where administrative decision of the crew and delays need to be made more frequently.

Financial Struggle in the Airline Industry Although air transport is one of the most important means of transportation in the US, the historically poor financial record of the airline industry remains a puzzle. For a review of possible causes see Borenstein [17], who considers the following factors: exogenous cost drivers such as an increase in taxes and the price of fuel, demand shocks, and entry and expansion of low-cost carriers. During the last decades, legacy carriers have maintained higher costs then low-cost carriers and have seen their price premia consistently decline, hence, although the exogenous shocks affect all companies, legacy airlines have fared worse than low-cost carriers. To differentiate their product, legacy carriers have opted to expand their networks through mergers and acquisitions, nevertheless, it is not totally clear that their mergers have allowed them to decrease their costs gap. Berry and Jia [15] consider these causes and estimate a structural model of the industry to estimate the impact of demand and supply changes on profitability. On the demand side, the authors find that compared with the late 1990s, the price elasticity of air travel demand in 2006 increased by 8%. Also, passengers showed a stronger preference for direct flights. On the supply

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side, changes in marginal cost favored direct flights. Together with the expansion of low-cost carriers, the previous three factors account for more than 80% of the observed reduction in legacy carrier’s profits, of which 50% can be attributed to changes in demand. Recent studies have started to look into the impact that filing for bankruptcy has over the quantity of services provided, and over the fares charged. As expected, filing for bankruptcy decreases the level of services provided and the size of the network served [21, 31], nevertheless, there is less of a consensus over its effect on prices. In early work, Borenstein and Rose [20] find little evidence that bankruptcy filing affected pricing behavior, whereas Ciliberto and Schenone [31] find that, under bankruptcy protection, airlines lower their route-specific prices while under bankruptcy protection, and increase them after emerging from bankruptcy. Importantly, during the studied period, they did not find similar patterns in behavior by the airline’s competitors.

Pricing and Differentiation by Heterogeneous Characteristics In markets characterized by few competitors, the capacity of large airlines to influence the prices of the market has sparked an interest to understand better their pricing behavior. Since the early work of Borenstein and Rose [19], the focus of many studies has been to study the relationship between price dispersion and competition, with little consensus in the direction of such relationship. Whereas Borenstein and Rose [19] find that a positive relationship between price dispersion (as measured by the Gini Index) with competition, Gerardi and Shapiro [50] find the opposite. Recent work by Chandra and Lederman [30] and Chakrabarty and Kutlu [29] seem to explain the mixed in the literature. On the one hand, Chakrabarty and Kutlu [29] found an S-shaped relationship between price dispersion and market concentration. On the other hand, theoretically, the former show that if there is heterogeneity in the consumer’s willingness-to-pay and in their brand loyalty, then competition may decrease price competition between some consumers while increase it between others, hence, the ambiguous result on price dispersion and competition. More recently, Kutlu and Wang [66] examine the effect of market power and marginal cost efficiency on the US airline price dispersion. They use the same dataset that Kutlu and Wang [65] used so that the city-pairs originate from Chicago. They model price dispersion by the following equation: Yirt = β0 + β1 MP irt + β2 EF F irt + f (COST it ; δ) + g (P RODAT T irt ; γ ) + h (P OP AT T tr ; η) + Fi + Rr + Tt + εirt ,

(1)

where Y is a price dispersion measure (either Gini coefficient for ticket prices or difference of 90 and 10 percentiles of ticket prices); MP is a market power measure (either conduct parameter or Herfindahl-Hirschman Index); EFF is the marginal cost

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efficiency; f (.) is a function of cost related variables; g(.) is a function of product attributes; h(.) is a function of population attributes; and Fi , Rr , and Tt are firm, route, and time dummies; and εirt is the error term. The indices i, r, and t are for airline, route (city-pair), and time. In their analysis, their benchmark measure for price dispersion is the Gini coefficient, which reflects price inequality across the entire range of different prices. The estimation results of Kutlu and Wang [66] show that the marginal effect of conduct may be over-estimated if the marginal cost efficiency is omitted from the regression. The marginal cost efficiency has a negative effect on price dispersion and seems to be a relatively important determinant of price dispersion. This highlights another important aspect of efficiency as marginal cost efficiency not only directly affects deadweight loss through suboptimal behavior of airlines but also it affects welfare through its effect on price dispersion. Another finding that Kutlu and Wang [66] mention is that the marginal cost efficiency is more effective in leisure routes compared with big city routes. The data shows that the airlines are more successful in identifying different customer types in big city routes. The ability to identify customer types along with lower costs at big city routs may be giving airlines a room for more price discrimination. Other papers, rather than studying how the market structure affect prices, have instead looked at how the ability to price discriminate over time affects consumer welfare. Lazarev [69] propose a structural model where price discrimination can affect consumer welfare through three channels, by affecting the quantity and quality of the good, and as a result of misallocation of products among buyers. Compared to an ideal allocation that maximizes social welfare, the profit maximization allocation results in a welfare loss of 21% for the consumers. To understand how intertemporal price discrimination contributes to this loss, the author tests counterfactual market designs. When allowing a secondary market for the resale of tickets, the author finds that tickets for leisure travelers would decrease, while tickets for business travelers would increase. Social welfare would increase almost by 12%, driven by the welfare gains of the business travelers, but airline’s profits would decrease by 28%. Also considering the effect of price discrimination over the allocation products, Aryal et al. [9] develop a model of intertemporal and intratemporal price discrimination to analyze how the ability of discriminatory mechanisms could be used to remove sources of inefficiency. They focus on measuring the inefficiency in the allocation of products, and in identifying the portion attributed to two sources of inefficiency: asymmetric information and stochastic demand. The authors find that compared to the first-best welfare, the current pricing practices result in a 19% welfare loss. Of this loss, stochastic demand and asymmetric information account for 65% and 35%, respectively. Another method that enables airlines to price discriminate is the unbundling of services. In 2000, a flight ticket would generally include the possibility to check a bag along a carry on, and the choice of a seat and the provision of snacks were benefits taken for granted. In contrast, most airlines today charge fees for every additional service. In general, most airlines now provide a basic package with the

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least level of services, and then provide a set of optional purchases for various fees. In order to attract the more price sensitive passengers, the adoption of unbundling fares was pioneered by low-cost carriers (LCC), and later adopted by flagship carriers. Recently, Howell and Grifell-Tatjé [58] have studied this rather overseen phenomenon in the economics literature. The authors study both the history of the unbundling process and perform a case analysis of Frontier Airlines, an LCC based in Denver, Colorado. Frontier Airlines is an LCC operating under a hub and spoke framework. In 2008, their operating cost and passenger revenue were 11.2 and 10.5 cents per available seat mile (ASM), underperforming the average of low-cost carriers, with operating costs and passenger revenue of 10.5 and 11.7 per ASM. That year they averaged approximately $3.7 in ancillary revenue per passenger altogether they had an operating loss of $17 million. By 2016, the picture changed drastically. Although the operating cost and the passenger revenue had both dropped by 7.6 and 7.9 per ASM, the ancillary profits had risen to $48.33 per passenger, and their reported operating income to $317 million. Importantly for the study case, in 2014, Frontier announced a stripped-down cost structure, reduced their average economy fare by 12%, and unbundled the low-cost economy tickets. The observed change in strategy in 2014 allows the authors to study how the unbundling changed the Frontier’s revenue stream, and the structure of the markets in which it operated. More broadly, the authors study whether the segmentation of services allow for productivity change. Finally, given that the ability of a firm to price discriminate is interlinked with the market power it holds, the studies that look into price discrimination also consider how this affects the market power of a firm. In this line of thought, Kutlu and Sickles [64] propose a market power measure that is designed to capture price discrimination. This exercise is important because in industries where price discrimination is common, often mergers are analyzed under the framework of a single price. The lack of consideration of price discrimination by antitrust authorities when looking at market power could lead them to block socially beneficial mergers or to accept harmful ones.

Market Power and Price Premium Understanding the sources of market power are important both for antitrust authorities, to propose effective measures when the market power is over a threshold, and for firms, to better understand the obstacles of entry to a market. Recent literature has shown that firms may seek to strengthen their market power by their strategic control of airport facilities, specifically the gates. Others have pointed out the importance of the frequent flyer programs to strengthen the loyalty of consumers. Studying the former, Ciliberto and Williams [33] study the importance of the control of gates as a determinant of the hub premium, and find that when the percentage of gates controlled by the carrier in a hub increases from 10% to 30%, then fare prices increase by 3%. Similarly, if there are limits to the fees that the

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airline can charge for subleasing the gates, then prices decrease. Nevertheless, both the control of gates and restriction on sublease fees have an impact on fares when gates are scars relative to the number of departures from that airport. With respect to the importance of frequent flyer programs (FFP), Lederman [70] studies the impact of enhancements to an airline’s FFP on demand. Lederman shows that enhancements to an airline’s FFP is correlated with increases in demand on the routes that depart from the airports at which it is dominant. Also, the author shows that this upgrade in the FFP’s results in a new equilibrium with fewer passenger and higher fares, which is consistent with the idea that FFP’s are especially valuable for the passengers that already have a high valuation of the airline. The relationship between market power and both the control of gates and the use FFPs, highlights the importance of considering variables other than prices when studying market power. Consisting with this consideration, Röller and Sickles [83] study how sensitive is the measure of market power when competition is on both prices and capacity, rather than on prices only. The authors propose a structural model with a two-stage set up. In the first stage, firms make capacity decision, followed by a price setting game in the second stage. They study the European market for the period of 1976 to 1900, and find that in the two-stage set up, the market conduct is less collusive than in a one-stage specification. Overall, in order to avoid a biased measure of market power, it is important to understand the barriers of entry, and to consider variables other than prices to strengthen our understanding of the strategic behavior of firms.

Threat of Entry The consolidation of the airline industry has increased the network size of the remaining firms in the market, but as airlines seek to expand, we might expect that incumbent carriers at particular airports act strategically to deter entrance in order to avoid competition. To this end, Ciliberto and Zhang [35] estimate three different models for entry games in order to compare the fit of the models. They estimate a simultaneous game with complete information, as in Ciliberto and Tamer [32], and two sequential games with or without strategic entry deterrence. The authors show that the model with sequential games with strategic deterrence provides the best fit to the data, rendering some evidence to the strategic behavior to deter entrance of new firms. As to the specific actions that airlines may take to deter entrance, Goolsbee and Syverson [54] find evidence that incumbent firms under threat of entry cut fares significantly. Nevertheless, the authors find little support for strategic investment/excess capacity as a preemptive action. Consistent with deterrence, when the entrance of a competitor is guaranteed, firms seem to not lower prices prior to the entrance of the new firm. Although the authors do not find pre-emptive strategic investment, Parise [80] finds that when legacy firms are under threat of entrance of a low-cost carrier, they change their debt structure. Incumbents increase debt maturity before entry occurs, which allows firms to reduce rollover risk.

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Market Power and Efficiency Kutlu and Sickles [63] examine the relationship between market power and efficiency in two US city pairs (Chicago-San Diego and Chicago-Salt Lake City). For this purpose, they develop a dynamic conduct parameter game where in the full market power scenario the airlines play an efficient supergame equilibrium. As in Rotemberg and Saloner [85], they assume a full-information environment for the airlines. In the beginning of each time period, the airlines know the demand and cost shocks before making simultaneous decisions, which become common knowledge. Since the shocks are observable by the airlines, they can adjust their quantities and dampen the profits strategically. If the cost and demand shocks are such that the incentive to deviate is high, then the airlines adjust their quantities to lower their profits relative to the case in which the incentive to deviate is lower. While this may look like a price war, this is an attempt to prevent deviation from the equilibrium. In their model, the dynamic optimization of an airline i with full market power is given by: Qt ∗ (St , δ) = arg maxQt ,st



πit (sit Qt ; St ) s.t.    δ k Et πitr (St+k ) πitb (Qt ; St ) + k    δ k Et πit∗ (St+k ) ∀i, ≤ πit (sit Qt ; St ) + i

(2)

k

where s is the market share; Q is the total market quantity; π is the realized profit based on the actions taken; π b is the best response profit for retaliation period; π ∗ is the profit when the collusion is sustained; St is a vector of factors that represents the state of the word at time t; and δ is the discount factor. When we allow conducts that are less than full market power, the first order condition for the conduct parameter game counterpart of Eq. 2 would be given by:   θt P  Q∗t Q∗t + MK t − μ∗t = 0,

(3)

where θ is the conduct parameter; P(.) is the inverse demand function; MK is  the  market share weighted price-marginal cost markup (i.e., MKt = i sit MKit = i sit (Pit − MCit )); and μ∗ is a dynamic term that reflects the incentive compatibility constraint. If μ∗ = 0, this first order condition represents the static game scenario. Otherwise, we would have a dynamic conduct parameter game. Kutlu and Sickles [63] assume that μ∗ is a linear function of cost and demand shocks as well as (market share weighted) cost inefficiency of airlines for a city-pair. Since Kutlu and Sickles [63] do not have city-pair-specific airline cost data, they estimate an industry-specific stochastic frontier cost function for airlines. The quarterly dataset that they use for cost function estimation has 11 airlines and covers years between 1980 and 1993, which is collected from the Department of Transportation’s Form 41/T100. For the conduct parameter estimation, they only

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use data for top six airlines for the specific city-pairs that they use. The cost data has more airlines because it is for the whole US airline industry. Their cost function has four inputs: labor, energy, flight capital, and materials, which is a residual category that includes supplies, outside services, and non-flight capital. They calculate the price data using multilateral Tornqvist-Theil index number procedure. The output variables are the number of enplanements, and scheduled and non-scheduled (cargo and charter operations) passenger revenue mile. In order to control for different characteristics of the output, they include load factor and stage length variables. Also, they include average size (measured in seats) and fuel efficiency variables to describe flight capital. Finally, they use seasonal dummies and event dummies, which control for some important events occurred during the study period (Iran–Iraq war; Gulf war; air traffic strike; and dummies for airline mergers and strikes). They estimate the cost function using the distribution-free within estimator of Cornwell et al. [38]. In contrast to distribution-based models, the distribution-free model of CSS can be estimated by a fixed-effects type estimator, which allows fixedeffects to vary over time. The efficiencies in the CSS model can be correlated with the regressors. However, they capture both heterogeneity and efficiency. Among others, see Green [55, 56], Wang and Ho [90], and Kutlu et al. [67] who proposed models that can disentangle efficiency and heterogeneity. Another issue with the CSS model is that the efficiency estimates are not robust to outliers. Hence, some researchers avoid this by dropping observations with top and bottom 5% fixedeffects values. After estimating the cost function parameters, they calculate marginal costs specific to each airline and city-pair using corresponding enplanement and miles flown data for each airline. When calculating marginal costs, they utilized a distance of relevant city-pairs as well as airline fixed effects. In particular, they assumed that overall MC is the MC of passenger revenue output times miles flown plus MC of Enplanement. While Kutlu and Sickles [63] find evidence for a dynamic game for Chicago-Salt Lake City city-pair, they do not find such evidence for Chicago-San Diego. Their results accord with Hicks’ quiet life hypothesis, which argues that there is a positive relationship between market power and inefficiency. Finally, they argue that either ignoring dynamic environments or inefficiency may lead to serious miscalculation of deadweight loss. Duygun et al. [40] estimate a (stochastic frontier) production function for the US airlines using a dataset that covers years between 1999 and 2009, which is a time period that airlines faced serious financial troubles. Their main focus is to illustrate how the Kalman filter estimation method performs when estimating a production function and airline (firm) technical efficiency. Their dataset comes mainly from the International Civil Aviation Organization and includes 35 airlines and 298 observations. When constructing the output and input variables, they follow Sickles [86] and Sickles et al. [87]. Their dataset includes four input variables: the flight capital (K, quantity of planes), labor (L, quantity of pilots, cabin crew, mechanics, passenger and aircraft handlers, and other labor), fuel (F, quantity of barrels of fuel), and materials (M, quantity index of supplies, outside services, and non-flight equipment). However, their final models

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focus on value added from capital and labor by netting out from revenue output (RTK, revenue ton miles), the value of the intermediate energy, and materials. In order to control airline heterogeneity, their production function also includes load factor, stage length, jet engine aircraft percentage, and average plane size. Due to multicollinearity issues, they focus on a restricted translog model, which doesn’t include interaction terms between inputs and control variables. They estimate the value-added production function via three estimators: the Battese and Coelli [13] (BC) estimator, the within estimator of Cornwell et al. [38] (CSS), and a Kalman filter estimator (KFE), which is proposed in their paper. Since the KFE is the central part of their empirical analysis, we describe it below. Duygun et al. [40] consider the following Kalman filter model for production function: yit =

 j

βj xj it + μit + εit

(4)

μi,t+1 = μit + eit ,     where εit ∼ N 0, σε2 and eit ∼ N 0, σe2 are independently distributed error terms. Here μit is a state variable, which controls for unobserved time-varying heterogeneity. This method can be used for estimating both neoclassical production functions that doesn’t assume inefficiency and stochastic frontier production functions. In the stochastic frontier production function setting, Duygun et al. [40] interpret μit as a term that representsthe extent of efficiency. The efficiency  estimates are calculated by Eit = exp − maxi μˆ it − μˆ it where μˆ it is the estimate of μit . Compared to KFE, relatively inflexible stochastic frontier models (e.g., BC and CSS) will more likely fail to capture potentially complex time-varying patterns of inefficiency. Hence, KFE is especially useful when the efficiency patterns of airlines are suspected to have variation over time. Since μit is a random walk process, so that it is non-stationary, they use a diffuse prior for its initial value. In particular, they assume that the mean squared error matrix of the initial states are constant multiples of the identity matrix where the constant is a large number. They suggest that an alternative is using the exact diffuse priors. One of the difficulties of estimating Kalman filter models is that, due to numerical rounding errors, some of the matrices used in the Kalman filter recursive equations may end up being non-positive definite (when they should be positive definite). This may cause numerical instability in the optimization process. In order to avoid such issues, Duygun et al. [40] use a square-root Kalman filter, which is computationally slower but more stable. The estimate of the median returns to scale is 0.88 and the mean efficiency estimate is 58%. In the airline literature a common finding is that the firms operate in a constant return to scale environment. However, as Basu and Fernald [12] prove that the value-added estimate of returns to scale would be farther away from the unity compared to the corresponding gross output model. That is, it is smaller (greater) than the corresponding gross output model when there is decreasing (increasing)

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returns to scale. Therefore, they argue that their returns to scale estimate is within a reasonable range.

Airline Efficiency Kutlu and Wang [65] estimate firm-route-quarter specific conducts and marginal cost efficiencies of US airlines for Chicago based routes without using route-level cost data. Their main data sources are the Passenger Origin-Destination Survey of the US Department of Transportation (DB1B dataset), which is a 10% random sample of all tickets that originate in the US domestic flights. The quarterly dataset covers years between 1999 and 2009. In their study, they assume that a market is a city-pair (route). Following Borenstein [16] and Brueckner et al. [26], they assume that ticketing carrier is the relevant airline. In contrast to the conventional stochastic frontier cost function models, which estimate cost efficiency, they estimate what they call “marginal cost efficiency.” They argue that from the viewpoint of an antitrust authority who aims to analyze the welfare, marginal cost efficiency is a more relevant concept as the calculation of welfare utilizes marginal cost rather than total cost. Of course, marginal cost is not directly observed and thus their estimation method is unconventional as well. In particular, they estimate the marginal cost efficiency through a system of demand and supply relation equations, which is derived from a conduct parameter game. One advantage of this system of equations is that estimation of marginal cost efficiency does not require cost data under certain identification conditions. In line with the conduct parameter approach, the implied marginal cost and corresponding marginal cost efficiency is deduced from demand and cost shifters. Another advantage of this estimation strategy is that market power (i.e., conduct parameter) and marginal cost efficiency are estimated jointly. However, since what is being estimated is a system of equations with endogenous variables, their methodology requires instrumental variables in order to obtain consistent parameter and marginal cost efficiency estimates. For simplicity, for now we drop the route index. The demand-supply relation system that they estimate is described by:   Pit = P Qt ; Xd,t + εitd   ln Pit = ln cit∗ Xc,t + git + uit + εits ,

(5)

where P(.) is the inverse demand function; Qt is the total output in a route; Xd, t is a vector of variables that affect demand; cit∗ is the deterministic part of the marginal cost;Xc, t is a vector of variables that affect the marginal cost; git = g (θit , sit , Et ) =

sit − ln 1 − E θit ≥ 0 is a term that reflects the effect of market power on the price; t θ it is the conduct parameter; sit is the market share of airline i at time t; and Et = t Pt − ∂Q ∂Pt Qt is the (absolute value of) elasticity of demand. Three benchmark values for the conduct parameter that represent perfect competition, Cournot competition,

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and joint profit maximization are θ it = {0, 1, 1/sit }. While in theory marginal cost function can be chosen to depend on quantity, in their empirical application they assume that the marginal cost function is constant, i.e., it is invariant to the quantity. Note that the marginal cost still depends onXc, t . The identification of parameters of this model requires stronger assumptions compared to the parameters of a standard demand-supply system. In addition to the standard assumptions for identifying parameters in a demand-supply system, we need certain restrictions on the functional forms for demand and marginal cost functions. Bresnahan [23], Lau [68], Bresnahan [24], Perloff et al. [81], and Kutlu and Wang [65] exemplify some studies that provide detailed treatments of these identification conditions for conduct parameter models. Besides this, we also need to identify the parameters of the inefficiency term, which would require relevant identification assumptions from the stochastic frontier literature in the presence of endogenous regressors. Amsler et al. [6], Kutlu and Wang [66], and Kutlu et al. [67] discuss these identification conditions. Although it is possible to estimate inverse demand and supply relation equations simultaneously, Kutlu and Wang [65] estimate an inverse demand function first; and then plug in relevant parameters obtained from the first stage into the supply relation equation, which is then estimated by a variation of the control function approach for stochastic frontier models that is first introduced by Kutlu [61] and further developed by Karakaplan and Kutlu [59, 60]. For the inverse demand function, they choose the following functional form:   d ln Pitr = β0 + β1 ln Qtr + β2 ln P CI tr ln Qtr + fd Xd,itr + εitr ,

(6)

where fd (.) is a function of demand related variables Xd, itr ; Qtr is the total demand at time t in route r; PCItr is population weighted per capita income for the relevant d is the error term. They assume that lnQ and lnPCI ln Q route (city-pair); and εitr tr tr tr are endogenous; and are instrumented by logarithms of input prices (labor, capital, and energy), logarithm of number of passengers for other routes; as well as two other variables that they construct, which are inspired by the price dispersion literature (Geometric market share and its product with the number of firms). Using the parameter estimates obtained from this model, they predict the demand elasticity specific to a route. When estimating the supply relation, they assume that gitr is a function of concentration ratio for top four airlines (CR4 ), logarithm of city-pair distance, market share of airline, trend, and demand elasticity for the city-pair market. In particular, they model gitr as follows

gitr

  Bitr exp Xg,itr βg ,  =  1 + exp Xg,itr βg

(7)

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 where Bitr = − ln 1 − E1tr so that gitr ∈ [0, Bitr ] and Xg, itr is the vector of variables that explain gitr . Then, they estimate the conduct as follows:    θˆitr = Eˆ tr 1 − exp −gˆ itr ,

(8)

where gˆ itr is the estimate of gitr . Finally, they model the marginal cost as a (restricted) translog function of logarithms of output and input prices (labor price, capital price, energy price); logarithms of average number of flight segments, average stage length, average fleet size, and flight distance; online rate; low-cost carrier (LCC) dummy; and quarter, year, and airline dummies. They find that relative to the non-LCCs, the conducts of LCCs are closer to perfectly competitive values; and relative to LCCs, the conducts of non-LCCs are closer to Cournot competition values. When inefficiency is ignored in the estimations, the conduct parameter estimates are generally smaller and closer to perfect competition values. Hence, it is essential to control for inefficiency. They find that the conduct parameter is positively related with CR4 , which verifies that higher concentration helps increasing market power. The median efficiency estimates for the whole sample is 82.6% and for the non-LCCs is 84.4%, which indicates that LCCs and non-LCCs have similar marginal cost efficiencies. Finally, they argue that their results are in line with Hicks’ [57] quiet life hypothesis as the inefficiency is positively related with CR4 .

Economic Impact of Delays Air traffic delays are one of the most common complaints by airline passengers [43], and although the percentage of delayed flights has gone down in the last decades, the increase in demand and of the airline’s network complexity may hinder the achievement of on-time performance targets. As the demand for more air transportation grows, a growing congestion in airports may lead to greater delays. Moreover, as competition increases among carriers at an airport, departure times may seem to be less differentiated if the route is served by competing airlines rather than by a single firm [18], which further increases congestion. The delays do not only have a negative impact on consumers, as their time is lost and their flight experience deteriorated, but it has an impact on their willingness to pay for a ticket, and therefore it affects airline’s profits. In early work, Forbes [43] used a legislative change in the takeoff and landing restriction at an airport that resulted in higher delays, and thus used this exogenous variation to study its impact on prices. For connecting passengers, the author found that prices fell by $0.77 per additional minute of delay, while for direct passengers it fell $1.42. This reduction was larger at competitive routes, which showed a reduction of $2.44. To ameliorate congestion and delays, economists have long proposed the use of pricing mechanism (Morrison and Winston [74]; Aravena et al. [7]) rather than

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direct allocation of space, which would delegate the production decision from the airport to the airlines, and the better-informed agent may choose the production level. A complementary strategy taken by the Department of Transportation (DOT) since 1987 has been to inform the consumers about the relative on-time performance of the largest airlines, so as to incentivize airlines to reduce their delays. For this end, the DOT produces a ranking of airlines based on the fraction of their flights that arrive less than 15 min late. Although this program was created to encourage airlines to improve their on-time performance, it incentivized airlines to focus on the routes with performance just under the threshold. Forbes et al. [46] analyze this unintended incentive and find heterogeneous results in the response of the airlines to this incentive. The airline characteristics that matter for its response to such incentive seem to be their technology to compute the on-time performance (whether it is done manually or automatically), and whether it has bonus package for its employees that rewards efficiency. The heterogeneity in results point out that the effect of a disclosure program depends more than just the incentives created by the program, but also on the internal organizational practices of the firm affected. A second unintended consequence of the 15 min threshold is that airlines may increase their flight schedule in order to have more flexibility in meeting the performance target. In a recent study, Forbes et al. [47] show that flight schedules have lengthened over time. For flights by the same airline on the same route in the same month of the year, flight schedules in 2016 were around 8 min longer than in 1990. Thus, even though the official rate of delays has decreased, passengers are taking more time to reach their destination. Interestingly, this is not only true for the airlines that have long reported their on-time performance, but recent work by Forbes et al. [48] also shows that when an airline or their competitors become subject to a disclosure requirement for on time performance, they adjust their targets and lengthen their schedule. This research shows that it is not straight-forward to evaluate the operational performance of an airline, nevertheless, for future research, greater emphasis should be put to multiple dimensions of quality rather than just looking at one dimension. An index that could be used when considering more dimensions is the Airline Quality Ranking (AQR), developed by the Embry-Riddle Aeronautical University. This index includes information on: mishandled baggage, consumer complaints, ontime performance, and involuntary denied boarding.

Governance and Airport Efficiency Abrate and Erbetta [1] and Voltres-Dorta and Lei [89] provide excellent summaries of the literature on airport costs, efficiency, productivity, and type of ownership. In what follows, we briefly summarize papers related to airport governance, efficiency, and productivity; and present a study in more detail. Oum et al. [77] argue that managerial autonomy in airport efficiency is important. Oum et al. [78] and Oum et al. [79] find that airports that are privatized are more efficient compared to public sector airports, which in turn are more efficient compared to public-private

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structure airports. In line with these findings, Botasso et al. [22] find no productivity effect associated with mixed ownership structures for airports. Martin et al. [73] and Martin and Voltes-Dorta [72] find that multi-airport systems have higher unit costs as they operate under nonexhausted scale economies. While these studies illustrate that airport governance is a relevant factor, Vasigh et al. [88] argue that market structure and competition may be more relevant for airport productivity and efficiency. Craig et al. [39] analyze two types of public ownership: airport authorities and city-owned. Similar to Zhao et al. [92], they find that airport authorities governed airports are more efficient compared to city-owned airports. As an example of an empirical application, we review the work of Kutlu and McCarthy [62], who use stochastic frontier analysis to examine the effects of governance types for commercial airports on cost efficiency. They contribute to the literature in at least three ways. First, they examine relative efficiency differences across four US commercial airport ownership types (city, county, state, airport authority). Second, for the first time in the airport efficiency literature, they control for airport specific heterogeneity and illustrate the importance of controlling for heterogeneity when estimating airport cost efficiency. Third, they analyze the efficiency effects of local ownership, hub size, and multiple airport metropolitan areas via a series of counterfactual analyses. Their dataset includes 24 medium and 26 large hub commercial airports in the USA, and covers years between 1996 and 2008. Using data from Bureau of Transportation Statistics, the FAA, and the National Flight Data Center, they estimate a variable cost function with one output and three inputs where the output variable is airline departures; and the input variables are labor, contracting and repair/maintenance, and general airport operations. While they do not have input prices specific to airports, they have data on national and MSA price indices for each input, which they use to construct indices for input prices in a way that generates variation across geography and time. Their model also includes a quasifixed factor effective number of standard runways, and as control variables: the share of international departures, the share of freight, share of revenues from parking, share of revenues from retail activities, year dummy variables, and airport dummy variables. They include airport-specific dummy variables to control for airport specific heterogeneity, which is in line with Greene [55, 56]. In particular, they estimate the following stochastic frontier model for variable cost: ln V C it = f (xit ) + uit + vit Sit = Sit∗ + wit  uit ∼ N + 0, σu2 (zit )  vit ∼ N 0, σv2 wit ∼ N (0, w ) ,

(9)

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where VC is the variable cost; f (.) is a function that represents the deterministic part of the variable cost frontier; x is a vector of variables included in the frontier; u ≥ 0 is a one-sided error term representing inefficiency; v is the usual two-sided error term; S and S∗ are vectors that represent the observed and explained part of input shares; w is a vector of error term with multivariate normal distribution; z is 2 a vector  environmental variables that are explaining inefficiency; and σu (zit ) =   of exp zit δ . Assuming that u, v, and w are independent, the corresponding marginal log-likelihood function for panel unit i at time t is given by:

ln Lit = const. − ln σs + ln

εit λ σs

 −

1 1 −1 ln |w | − wit w wit , 2 2

(10)

where is the cumulative distribution function for the standard normal distribution, εit = uit + vit , σs2 = σu2 + σv2 , and λ = σ u /σ v . Note that this model differs from the conventional stochastic frontier models of cost function as it includes input share equations to improve statistical efficiency. Moreover, as mentioned earlier, the model controls for unobserved heterogeneity by including airport dummy variables in the frontier. Kutlu and McCarthy [62] argue that controlling for heterogeneity may be essential when estimating efficiencies of airports. When ignored, one potential problem would be that the inefficiency and heterogeneity may be confused in the estimation, furthermore, regularity conditions may be violated. When the authors ignored heterogeneity, 85 of 649 observations violated regularity conditions (concavity and monotonicity) for a cost function, while when they controlled for heterogeneity, only 3 of 649 observations did. A violation of regularity condition is more serious for a stochastic frontier variable cost function compared to its neoclassical counterpart that assume no inefficiency. The reason is that the efficiency measurements are invalid for those observations that violate regularity conditions. This situation would also contaminate the parameter estimates. Based on this model, the main findings of Kutlu and McCarthy [62] is summarized as follows. The median cost efficiency estimates for the US airports is 87.6%. For medium hub airports, airports owned by city or airport/port authorities have 9.6% higher variable cost due to cost inefficiency compared to county or stateowned airports. Moreover, for medium airports owned by the city or airport/port authorities, multiple-airport cities have 8.6% higher variable costs compared to single-airport cities. They did not find much difference in cost efficiency between medium and large hubs.

Deregulation Finally, we review the impact of the 1978 Airline Deregulation Act (ADA) on the market. Many studies focus on analyzing how the US market changed ex-post the ADA, and on how the US market progressed in comparison to other similar but more regulated markets, such as the European one. In early work on the former,

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Sickles et al. [87] develop a model of allocative distortions, and find that, during the period of deregulation, the airline industry decreased both the total cost of allocative distortions and the relative level of input and output allocative distortions. During the period of deregulation in the USA, Good et al. [52] compare the technical efficiency and productivity growth of the four largest European carriers with eight of their American counterparts. The period analyzed is between 1976 and 1986, which is particularly useful since it starts at the beginning of a period where we saw informal steps taken toward deregulation in America, and ends just prior to deregulation reforms in Europe. Based con Cornwell et al. [38], they find a rather large gap in efficiency which otherwise avoided would translate in savings of approximately of $4.5 billion per year. The subsequent studies in the literature focused on finding possible explanations for this gap. Prior to the deregulation reforms in Europe, the European carriers, mostly stated owned, where sheltered from competition, heavily subsidized, and the fares to international destinations where set by bilateral treaties. During this period, the average fares in Europe were consistently higher than the fares in the USA for routes of similar distance. To explain the price difference, Captain and Sickles [28] study European airlines in an oligopolistic structure with product differentiation, and find evidence of price/cost markups, mainly driven by the technical inefficient use of inputs, in particular of labor, which presented wages in excess of their marginal revenue product. Complementary, Captain and Sickles [28] and Postert and Sickles [82] test the extent to which market concentration also plays a role in explaining higher fares. Interestingly, despite inelastic demand, there was little evidence of large deviations of competitive pricing and monopolistic market concentration as an explanation of the higher fares in Europe. This result further strengthens previous findings that the gap in efficiency was mainly the result of differences in the cost structure of both industries. A more recent study that revisits the efficiency comparison of the USA and Europe is Good et al. [53]. The authors develop a dynamic industry model to test the counterfactual of how the European industry would have developed, had they deregulated in 1979 alongside the USA. Through simulation, they solve for optimal levels of employment, network, and fleet size for the period 1979–1990. The results reveal various sources of forgone earnings, mainly the need for European carriers to expand their networks in order to take advantage of returns to density. A first policy in this direction could be an increase in alliances and further use of code sharing. Now, when looking only at the US market, it is of interest to see how the industry in general developed after the deregulation and to empirically test the relationship between an increase in competition and technical efficiency. Throughout the deregulation period and after, we expect two patters in the times series of the airline’s efficiency scores; that they should be cointegrated and that they should converge. To see why we expect these patterns consider a market with two airlines. If efficiency-enhancing practices are made by one airline but not followed by the other, the efficiency scores of these two airlines would move apart. In time, the airline that fails to follow innovation is eventually driven out of the market. Hence, in the long run, the efficiency scores should be cointegrated among the remaining

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airlines in a market. Second, the efficiency scores should converge as increases in competition force airlines to close efficiency gaps. In line with this argument, Alam and Sickles [4] analyze the time series of technical efficiency of US airlines, using quarterly observations from 1970 to 1990. As an efficiency measure, the authors use an output-based distance function (OD). In order to define OD, let y denote a vector of outputs of length K, and x a vector of inputs of length L, then, the production technology is S = (x, y) |x ∈ RJ+ , y ∈ RK + , (x, y) is feasible . Then, OD is defined as OD(x, y) = min {λ  (x, y/λ ∈ S}. Notice that an output efficient firm has a score of one. Likewise, an output-inefficient firm has OD(x, y) < 1. The authors use two methods to estimate the model, first they use the data envelopment analysis (DEA), and then the free disposal hull (FDH) method. Since these methods are related, I will only explain the method DEA and point out to the differences when they exist. The DEA is a linear programming method where we solve: [OD (xnt , ynt )]−1 = max λnt , subject to :  wnt yknt , k = 1, . . . , K, λnt yknt ≤  n

n

t

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wnt xknt ≤ xj nt , j = 1, . . . , J,

t

wnt ≥ 0, n = 1, . . . . , N ; t = 1, . . . , T . Where wnt stands for weights and where the conditions imposed give constant returns to scale (CRS). The FDH frontier is obtained by replacing the last restriction   by n t wnt = 1, wnt ∈ {0, 1}, ∀ n, t. Using the estimates from the DEA or FDH, then, in a second stage, the results obtained are regressed on firm characteristics, along with firm and time trends. Here, the residuals will provide us with our appropriate performance measure. For the co-integration test, the authors use the Dicky Fuller Test, and for convergence, they make use of a Malmquist productivity index procedure. This method accounts for changes both in technical efficiency change (catching up effect) and changes in the frontier technology (innovation). For this last approach, we need to calculate output distance functions between periods. Letting  OD t (xt+1 , yt+1 ) = min {λ |( xt+1 , yt+1 /λ ∈ St , and  OD t+1 (xt , yt ) = min {λ |( xt , yt /λ ∈ St+1 , the Malmquist productivity index is then defined as:

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1/2     ODt xt+1 , yt+1 ODt+1 xt+1 , yt+1 ODt (xt , yt )   M xt+1 , yt+1 , xt , yt = ODt (xt , yt ) ODt+1 xt+1 , yt+1 ODt+1 (xt , yt ) 



= Et+1 × At+1 ,

where Et + 1 reflects the change in relative efficiency, and At + 1 the change in technology, between t and t + 1. The authors find evidence that the US airlines industry is consistent with both co-integration and convergence of the efficiency score. Nevertheless, although the effect of deregulation has been positive (the industry has shown lower fares, and an increase in non-stop flights and productivity [84]), the quality of the services has not fared as well. For example, an increasing reliance on a hub and spoke system can bring inconvenience to consumers. Färe et al. [42] consider service quality effects on consumers when estimating total factor productivity (TFP), and find that it lowered the rate of productivity growth of the TFP. This is not to say that a re-regulation of the industry would be advised, but rather that further care should be taken to consider the effects on the quality of services when analyzing the impact of deregulation on markets.

Conclusions Our Handbook Chapter has provided an overview of the empirical literature that speaks to the productivity and efficiency of the US and International Airline Industry. As with any survey of such a vast literature, coverage of all papers and of all topics was not possible. We have tried to focus our attention on issues and empirical findings that speak to the theme of this Handbook Volume II but leveraged by the modelling approaches and theoretical perspectives from Volume I of this Handbook. We trust we have succeeded in this endeavor.

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Globalization, Innovation, and Productivity

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Shunan Zhao and Man Jin

Contents Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . International Trade and Productivity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Modeling Productivity and Trade Effects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . FDI and Productivity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Learning and Spillovers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Persistent Benefits of FDI . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Empirical Models to Measure FDI Spillovers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Innovation and Productivity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Innovation Measures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . R&D Measure and Modeling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Financial Constraints, R&D, and Productivity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Cross-References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1146 1147 1150 1152 1152 1153 1154 1156 1156 1158 1159 1160 1160 1160

Abstract

We comprehensively review previous studies examining the effects of innovation and globalization on productivity. Innovation is measured by the R&D investment, patents, and novel business models and production practices. Globalization takes forms of the international trade in goods and services and foreign direct investment (FDI). In our review, we first present various mechanisms of how a form of innovation or globalization theoretically affects productivity, and then evaluate different model settings. Meanwhile, we summarize the empirical evidence, from both macro and micro analyses, such as the heterogeneous effects

S. Zhao () · M. Jin Department of Economics, Oakland University, Rochester, MI, USA e-mail: [email protected]; [email protected] © Springer Nature Singapore Pte Ltd. 2022 S. C. Ray et al. (eds.), Handbook of Production Economics, https://doi.org/10.1007/978-981-10-3455-8_28

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on productivity across different sectors and economic regions. Lastly, we discuss the related measurement, data source, identification, and modeling issues. Keywords

Productivity · Trade · FDI · R&D · Patents · Financial constraints

Introduction In 2017, the G7 countries, which are the seven largest advanced economies in the world, including Canada, France, Germany, Italy, Japan, the United Kingdom, and the United States, accounted for about 80% of the world’s research and development (R&D) expenditures, according to the data of United Nations Educational, Scientific and Cultural Organization. R&D is considered a key driver of these countries’ economic and productivity growth. In the same year, the World Bank reported that the top five countries/regions that have the fastest real GDP growth are Libya, Guinea, Ethiopia, Macao, and Maldives. The growth “star” in the past four decades – China – ranks 16th with an annual growth rate of 6.90%. Besides the traditional catch-up effect explaining the fast growth in these underdeveloped economies, perhaps more importantly, most of these countries are very export-oriented. For example, the export-to-GDP ratio is 67.6% in Maldives, which is two times more than the world average. Exporting along with other globalization initiatives provides these countries more opportunities to learn the cutting-edge technologies from abroad and the export-fueled “scale effect” primarily explains their fast pace of growth. Examining the relationship among globalization, innovation, and productivity is one of the most important topics in economics. Globalization is a process of growing interdependence of world economies, brought by cross-border trade – exports and imports – in goods and services, technology, and the flow of investments, people, and information. Innovation can be broadly defined as “production or adoption, assimilation, and exploitation of value-added novelty in economic and social spheres; renewal and enlargement of products, services, and markets; development of new methods of production; and the establishment of new management systems” [19]. According to the theory of production, total factor productivity is the part of output which cannot be explained by the inputs. It includes two components: one is the temporary productivity shock due to macroeconomic environment and the other is persistent productivity due to technological innovation, managerial efficiency, industrial organization, globalization, etc. And the growth in total factor productivity is largely explained by the growth in the persistent productivity [65]. In this chapter, we review studies examining the effects of innovation and globalization on productivity. Innovation is measured by R&D investment, patents, novel business models, and production practices. Globalization takes forms of international trade in goods and services and foreign direct investment. Theoretically, most economists agree that globalization in general has a positive effect

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on productivity, although the effect shows heterogeneity in different business sectors and economic regions. The ones that attract most economists’ attention are mechanisms through which globalization facilitates productivity growth. Here we discuss various mechanisms in the literature. Meanwhile, we summarize empirical evidence, from both macro and micro analyses, and discuss related measurement, identification, and modeling issues. While it is impossible to include all influential studies, on this long-standing topic, we structure the chapter according to our own knowledge and research interests, hoping that it will provide guidance for future research.

International Trade and Productivity The relationship between international trade/trade policies and productivity has been studied extensively for many years. Now, there are relatively more discussions and examinations, at the macro level, than in previous years. In the recent two to three decades, due to the increased availability of international trade data and firm-level datasets in various countries and industries, firm- or plant-level studies are becoming more common. In general, there are two opposite opinions on their relationship. Advocates of trade liberalization argue that international trade can increase productivity at the plant, industry, and country levels through multiple channels, but critics argue that the removal of inward trade barriers could be a double-edged sword which may hurt the domestic economy. Although the two views get almost equal attention, among public news and political debates, the former dominates the latter in academic studies. To be more specific, economic theories suggest that international trade has positive effects on productivity through the following channels: 1. Competition: exposure to international trade increases competition in the product market. This gives firms more incentives to improve existing production technologies through methods such as conducting more R&D investment or adopting better corporate management practices [6]. This implicit “challenge response” mechanism should work through both export expansion and import liberalization. 2. Reallocation: competition also reduces markups and decreases a firm’s profit margin [47]. Through competition, trade liberalization forces the least productive firms to exit and reallocates market shares toward more productive exporting firms (e.g., [52, 56]). Therefore, the aggregate productivity at the sector level increases with trade. 3. Learning: exporting firms may benefit from their foreign buyers’ technical and managerial expertise or the expertise of other foreign contacts (e.g., competitors, suppliers, or scientific agents) through the supply chain. Exporters can learn from product quality improvements, shipment size, or, even more directly, from specific investment requirements. This mechanism, whereby firms improve their productivity after entering export markets, is called “learning by exporting.” The

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importance of the learning effect goes beyond an academic issue and has been used to review government aid for the internationalization of firms. 4. Input access: international trade allows firms to get access to high-quality and cheaper equipment and intermediate inputs [5]. Therefore, firms can produce output of high quality at a lower cost. More importantly, imported equipment and intermediate goods may embody better production technologies and the import of specialized inputs might facilitate learning about the product, spurring imitation or innovation of a competing product. 5. Scale economies: exposure to international trade means a larger market for domestic firms. The existence of scale economy implies that the widened market, through trade, should lead to reductions in production costs and increases in productivity. The argument is usually made in terms of the benefits of expansion in demand through increased exports. 6. Scope economies: with rapid progress of communication and technology, production processes increasingly involve global value chains spanning across multiple countries, with different stages of the production taking place in several countries or regions. International trade provides opportunities for further deepening product specialization, both vertically and horizontally, which in turn increases productivity. According to the above channels, international trade can increase productivity of both traded and non-traded firms regardless of importing or exporting. Countries of all development statuses can benefit from international trade. However, some policy makers and researchers have also expressed their concern that, depending on its impact on competition and cost reducing incentives of producers in the medium to long run [54], international trade, even excessive export, may sabotage the development of an economy and decrease firm productivity. Import substitution industrialization (ISI), which advocates replacing imports with domestic production, is a trade and economic policy used widely in developing countries. It protects the “infant industries,” which, by definition, are in their early stages and are incapable of competing with established competitors abroad until they have grown up and become internationally competitive. Export expansion policies such as excessive export subsidies may distort incentives and lead to increasing inefficiency, such as the overuse of inputs. In addition, skeptics argue that domestic firms may not be able to realize productivity gains because they are unable to adapt foreign technologies to local methods of production or because domestic firms face binding credit constraints that prevent the expansion of efficient industries as well as investment in new technology. Empirically, to identify the effects of international trade on productivity, one needs to address the endogeneity issue of trade. Early studies, in development economics, consider a cross-country regression of a productivity measure (labor productivity or total factor productivity) on the export-to-GDP ratio. Such regressions typically find a moderately positive relationship. However, as pointed out by Frankel and Romer [28], Alcalá and Ciccone [4], and many others, this relationship may not reflect a causal effect of trade on productivity. The problem is that the trade

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share may be endogenous because the countries with higher productivity, for reasons other than trade, may trade more. Using measures, of countries’ trade policies, in place of, or as an instrument for, the trade share in the regression, does not solve the problem since trade policies may also correlate with aggregate productivity determinants omitted from the regression. Frankel and Romer [28] argue that countries’ geographic characteristics have important effects on trade and are plausibly uncorrelated with other determinants of productivity and income. They construct measures of the geographic component of countries’ trade and use those measures to obtain instrumental-variable estimates of the effect of trade on productivity. The results provide no evidence that ordinary least-squares estimates overstate the effect of trade. However, Irwin and Tervio [38] find that Frankel and Romer’s [28] result is not robust to the inclusion of latitude, or additional geographic variables, in the regression. Rodriguez and Rodrik [60] argue that the geographic attributes may be proxies of institutional quality. Therefore, the captured positive effect might not be due to trade. Alcalá and Ciccone [4] use a new measure of trade which eliminates distortions due to cross-country differences in the relative price of non-tradable goods, and simultaneously addresses the endogeneity of trade and missing control of institutional quality. They still find that international trade has an economically significant and statistically robust positive effect on productivity. To sum up, although most economists argue that international trade has positive effects on productivity, which has been demonstrated by different theoretical models [24, 52], it is not easy to find convincing empirical evidence to support the argument using macro-level data. Micro-econometric analyses of plant-level datasets provide us with better identification settings for the effects of trade on productivity. Early studies, such as Bernard and Jensen [9] and Aw et al. [7], explore firm/plant-level datasets from countries/regions including the USA, Taiwan, Colombia, Mexico, and Morocco, and find that more productive firms self-select into foreign markets. There is no evidence that exporting increases the growth rates of plant productivity. At the industry level, aggregate productivity benefits from the resource reallocations from the less productive firms to the more productive ones. To address the endogeneity issue of a firm’s trade decision, recent studies often rely on certain exogenous policy changes, since, in contrast to the nexus of international trade, trade policies, and aggregate productivity at the macro level, a firm’s trade behavior is more likely to be the only channel through which trade policy changes affect the firm’s productivity. For example, Pavcnik [56] investigates the effects of trade on plant productivity with the help of trade liberalization in Chile. She also distinguishes between tradedgoods and nontraded-goods sectors to separate productivity effects stemming from liberalized trade from the productivity variation stemming from other sources. Baldwin and Gu [8] examine how Canadian manufacturing plants have responded to reductions in tariff barriers between Canada and the rest of world between 1984 to 1996. Muendler [53] studies how reduced inward trade barriers affect productivity in Brazil from 1986 to 1998. Amiti and Konings [5] estimate the productivity gains from reducing tariffs on final goods and intermediate inputs in Indonesia. De Loecker [22] studies the productivity effects from reduced trade protection in

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the Belgian textile market. Topalova and Khandelwal [61] examine the causal link between changes in tariffs and firm productivity in India. Yu [64] explores how reduced tariffs on imported inputs and final goods affect the productivity of large Chinese trading firms. All these studies find significantly positive effects of trade on firm-level productivity, although the magnitude varies by countries and methods.

Modeling Productivity and Trade Effects In this subsection, we summarize common approaches to model productivity and the effects of international trade on productivity. These approaches are quite general in the sense that they can be easily modified to study the effects of other productivity determinants, such as FDI and R&D, which we will discuss later on in this chapter. Here, we describe a general framework for modeling the relationship between productivity and its determinants, leaving FDI- and R&D-specific estimation issues discussed in their specific sections. The standard approach to empirically examining the relationship between productivity and its determinants, such as globalization and innovation, usually has two steps: first, productivity is estimated or calculated from certain methods; second, the estimated/calculated productivity is regressed on a set of control variables including the proxies of globalization or innovation. For robustness checks, multiple productivity measures may be used. In the process, there are at least three widely used productivity measures. The first one is labor productivity, which is obtained by dividing the value added by the numbers of hours worked or the number of workers. Labor productivity is simple to calculate, but it does not account for other inputs in a value-added production process. Therefore, labor productivity may not be a good productivity measure for capital-intensive production processes [13]. The other two most common approaches, to measure productivity are residualbased. They begin by assuming a neoclassical production function: Y = F (X; t) ,

(1)

where Y is the single output in the production process, X is a K × 1 vector of inputs, and t is a time index that allows the function to be shifted by technological innovations or improvements. The total factor productivity (TFP) growth is the portion of growth in output not explained by growth in traditionally measured inputs (X). Without assuming a specific structure of F(·), we can write the TFP growth rate as: gT F P =

X˙ Y˙ K − θk k=1 X Y

(2)

Here a dot over a variable denotes its total derivative with respect to time, and θ k is the elasticity of output with respect to the k factor input. If one replaces the time derivatives with discrete changes and output elasticities with average factor shares

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of current and previous periods, then Eq. (2) leads to the benchmark of the second widely used productivity measure – the well-known Tornqvist TFP index. The traditional Tornqvist index could be restrictive, and some refinements can be made based on Eq. (2) to accommodate factors such as scale economies and imperfect market competition [62]. Finally, if we assume the production function is Hicksneutral, we can rewrite Eq. (2) as: Y = A • G(X).

(3)

Here, A provides the third productivity measure. Its growth rate, that is, gT F P = ˙ A/A, and itself can be calculated upon estimation of the function G(X). Despite its popularity, the above two-step approach often provides an estimated relationship, between productivity and its determinants, that is more appropriately interpreted as correlation, rather than causality. The reason for this is selfinconsistency between the first- and second-step estimation, in which the impact of trade, FDI, or R&D is ignored, or equivalently assumed nonexistent, in the productivity estimation and then the estimated productivity is regressed on its determinants in search for a connection. Due to the same reason, the two-step approach ignores the endogenous factor demands, which in turn leads to biased estimates of output elasticities, production function, and productivity. To see this clearly, consider the logarithm of Eq. (3): y = g (X) + ,

(4)

where the lower-case letters stand for the log values of their upper-case counterparts and A = e . Besides random shocks, the error term  also captures the effects of globalization and innovation on output, which may cause correlation between X and  since input choices of production may be affected by firms’/countries’ trade and innovation activities. Therefore, a simple ordinary least squares (OLS) regression based on Equation (4) will lead to biased estimates of g(X) and output elasticities, which in turn leads to biased productivity estimates. There are different methods to solve the above endogeneity issue, and the key idea here is to account for the effects of globalization and innovation on production when productivity is estimated. One can estimate an augmented production function (or input distance function) with productivity determinants incorporated. See Eberhardt et al. [25] for example. Within the framework of Olley and Pakes [55] and Levinsohn and Petrin [48], an alternative method is used to model firm productivity and its determinants, which is robust to the above issue. Assume At = eωt +ηt , where the log TFP is broke into two parts: ωt and ηt , and the subscript t stands for time. The term ηt measures the unpredictable random shocks to output, and ωt is persistent productivity. The effects of globalization and innovation on production can be incorporated through an assumption of a first-order Markovian process of ωt , that is, P(ωt | It ) = P(ωt | ωt − 1 , Gt − 1 , R & Dt − 1 ), where It represents information available at time t. Gt − 1 and R & Dt − 1 are some globalization and innovation measures at period t − 1, respectively. With the above augmented productivity

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evolution process, the production function as well as output elasticities can be consistently estimated following the methods proposed by Ackerberg et al. [1] or Gandhi et al. [29].

FDI and Productivity Learning and Spillovers The existence of productivity effects, stemming from FDI, is well-documented in the literature. Most economists, as well as policy makers, argue that FDI often involves the transfer of knowledge from one country/firm to another [43], and it facilitates gaining access to intangible productive “knowledge” assets from abroad, such as new technology, proprietary know-hows, more efficient and innovative marketing and management practices, established relational networks, and reputation, which can boost the productivity of domestic firms. Multinational enterprises (MNEs) and their main vehicle, FDI, are often studied together since, by definition, an MNE is an enterprise which owns, controls, and manages value-adding activities in more than one country and whose business activities are financed by FDI [14]. MNEs and FDI provide multiple direct and indirect channels through which domestic firms can improve their production technology. These channels include direct technology transfer across international borders by sharing technology among multinational parents and subsidiaries [50], labor training and turnover [27], and provision of high-quality intermediate inputs [59]. MNEs also transfer technology to local suppliers as part of a strategy to build efficient supply chains for overseas operations [39]. By doing so, MNEs enjoy a private benefit from lowering the cost of non-labor inputs. Moreover, to reduce the firm-specific risk, MNEs may have incentives to diffuse technology wider than a single upstream vendor, either by direct technology transfer to additional firms or by encouraging spillovers from the original recipient. Blomström and Kokko [11] provide detailed discussions of possible spillover channels associated with FDI. Since international trade is closely related to MNEs and FDI, it is not surprising to see that they share some common productivity-improvement channels. Empirically, there are a number of studies that provide mixed evidence of FDI effects: consistent with the above theoretical arguments, the majority of studies find robust evidence that FDI has positive effects on productivity for both immediate recipients, those who benefit from direct learning of foreign knowledge, and the other domestic firms in the region, who benefit from technology/knowledge spillovers [21, 39, 44, 49]. Others find evidence that FDI may have insignificant, or even negative, spillover effects on productivity due to reasons such as data limitation [43], heterogeneity across industries/countries [63], absorbing capacity [26], or “business-stealing” competition [3]. For example, consider the US-owned multinational company General Motor (GM), one of the largest manufacturers of highly specialized automobiles in the world. GM entered China in 1997, and since then it has established 11 joint ventures and two wholly owned foreign

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enterprises. GM has a complicated relationship with Chinese domestic automakers – it simultaneously collaborates and competes with local businesses. On the one hand, the presence of GM hurts domestic firms, such as SAIC motor, which is one of the “big four” automaker oligopolies in China, by “stealing” SAIC’s market share. On the other hand, SAIC benefits from GM’s mature production and management practices and its cutting-edge technology. The aggregation of the two opposite effects could lead to insignificant or negative FDI effects. Xu [63] uses the data on the US outward FDI into 40 countries between 1966 and 1994, and finds technology transfer provided by the US MNEs contributes to the productivity growth in developed countries but not in developing countries. He shows that a country needs to reach a minimum human capital threshold level to benefit from the technology transfer. Xu’s [63] analyses are at the country level, which may be subject to the aggregation bias because of heterogeneity across sectors and across firms. To explore the heterogeneity of spillovers at the firm level, Fons-Rosen et al. [26] investigate the types of domestic firms that overall benefit from FDI. They develop a sector-level measure of technology closeness based on whether firms within a sector hold similar patents. They construct measures of horizontal and vertical technology-weighted FDI, as a fraction of output produced by foreign-owned firms weighted by the technological closeness measure for the sector. Using these technology-weighted FDI measures, they find that domestic firms that are technologically close to MNEs become more productive following FDI, while firms that produce similar goods to MNEs, but are not technologically close, become less productive. Aitken and Harrison [3] find a negative relationship between FDI and productivity of domestically owned plants using Venezuelan plant-level data, which they interpret as a market-stealing or competition effect. Controlling for the increased competition through FDI, Griffith et al. [31] find a significant positive FDI spillover effect using the UK plant-level data. Keller and Yeaple [44] estimate international technology spillovers to the US manufacturing firms and find a positive and economically important spillover effect. According to their estimates, FDI accounts for about 11% of US manufacturing productivity growth. Blalock and Gertler [10] investigate the effect of FDI on local supplier productivity by estimating a production function using a rich panel dataset of local- and foreign-owned Indonesian manufacturing firms. Overall, the realized productivity gain is more than 2%. They also examine the market and welfare effects of technology diffusion from FDI and find that FDI increases the output and profits of both upstream and downstream firms. Meanwhile, FDI decreases prices in both upstream and downstream markets.

Persistent Benefits of FDI How persistent are the benefits of FDI? Is the superior performance of foreign affiliates due to a one-time know-how transfer? Or does it depend on the continuous flow of knowledge and headquarter services from the parent firm? These questions

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matter profoundly for policy making. Public policies, aimed at attracting FDI, are common in both developing and developed economies. Foreign investors are often given tax incentives/breaks in the hope that their affiliates will become a source of knowledge spillovers to domestic firms. How long they can remain such a source enters the government’s cost-benefit calculation of attracting FDI. The duration of tax break is usually prescribed by law, and tax break cannot be awarded after the foreign parent leaves. If foreign affiliates retain their productivity advantage even after their foreign parents leave, the value proposition of such tax policies would be much greater than that under the scenario where the advantage evaporates once the parent divests. Javorcik and Poelhekke [40] study these issues and examine the development of previous foreign affiliates that were sold by their parents to local owners. They use Indonesian plant-level data from 1990 to 2009 and focus on plants that were at least 50% foreign owned, whose foreign ownership dropped to less than 10% (a standard threshold used in the literature to denote FDI) and remained so for at least 3 years. They find that divestment is associated with a 0.038 log point productivity drop among divested plants. The decline starts in the year of ownership change and persists over time. They also find an increasingly large output drop, ranging from 0.35 log points in the year of divestment to 0.54 log points 2 years later, which is driven by the decline in export sales. Their results suggest that the benefits of foreign ownership, at least partially, depend on continuous injections of knowledge and access to headquarter services. Any externalities associated with the presence of foreign affiliates may fade away after foreign owners leave. Malikov and Zhao [49] study intra-industry FDI-facilitated productivity spillovers in China. They find the magnitude of productivity spillovers may not depend on the ownership of firms. The primary beneficiaries of the direct knowledge transfers generally appear to be the wholly domestically owned firms. This suggests that the bulk of a productivity boost is due to learning new knowledge and acquiring new technology, immediately after the domestic firm gains direct access to them, from its initial foreign investors. The indirect technology spillovers may be long-lasting, but the direct learning from foreign investors among immediate recipients is more likely to be a one-time knowledge transfer.

Empirical Models to Measure FDI Spillovers Keller [43] summarizes different empirical approaches employed to study international technology diffusion, including the case studies and the econometric methods such as association studies, structure studies, and general equilibrium analysis. Here, we pick out two commonly used regression approaches and briefly discuss their pros and cons.

Reduced-Form Model Generally, the reduced-form model tests whether a specific foreign activity (FA) leads to a particular domestic technology outcome (DTO) through a regression in

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the form of DT O = ψ (F A, X) + u,

(5)

where ψ(·) usually takes a linear form, X is a vector of control variables, and u is the noise term. For example, in Aitken and Harrison’s [3] analysis, FA is the industry share of employment in foreign-owned firms, and DTO is the growth of domestic firm productivity. Most studies routinely estimate FDI-driven technological spillovers by (linearly) regressing firm-level measures of productivity/output on various industry-level FDI aggregates, which reflects the overall extent of foreign presence in the sector. Since the reduced-form model has low data requirements and is easy to implement, it is probably the most commonly used approach to examine FDI externalities. The reduced-form model is subject to common issues in any regression models, such as individual/time heterogeneity and endogeneity. As in most empirical studies, finding good instrumental variables, for foreign activity variables, may be challenging. Moreover, such a coarse “reduced-form” formulation, of FDI spillovers, treats the foreign knowledge diffusion across firms as a black-box process without attempting to establish links between peer firms. It thereby obfuscates the measurement of positive “technological spillover” externalities, which is widely conceptualized as productivity flows from more efficient MNEs to less efficient domestic firms. It may confound such spillover effects with negative externalities like “business stealing effects.” This is, arguably, one of the reasons why positive horizontal FDI spillovers have been elusive for empiricists. Keller [43] also points out that the reduced-form model precludes a precise interpretation of the results since the black-box process can not reflect a particular mechanism of FDI spillovers.

Structural Model Different from the reduced-form approach, the structural model approach is more specific about the spillover mechanism, which helps in interpreting the results. Generally, the structural model can be written as DT O = ψ (M, F T , X) + u,

(6)

where the foreign technology variable, FT, replaces the foreign activity (FA) variable in equation (6), and the specification adds a specific channel or mechanism of diffusion (denoted by M). An example of applying this model is the study by De La Potterie and Lichtenberg [21], who estimate the following regression: logT F P = α + αd logR&D + α f logF R&D + u,

(7)

where logTPF is the logarithm of TFP, R & D represents the domestic R&D capital stock, and FR & D represents the inward/outward FDI-weighted foreign R&D capital stock. In equation (7), the coefficient α f can be used to test whether FDI serves as a channel for the international technology diffusion.

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De La Potterie and Lichtenberg [21] calculate the total fact productivity based on a constant-return-to-scale production function using the average share of capital income as the elasticity. They impose strong assumptions on the production technology. Malikov and Zhao [49] provide another example in which they apply the structure model to examine the FDI-facilitated productivity spillovers and estimate the production function, productivity and spillover effects jointly. The key component of their method is to model the productivity evolution process as follows: ⎛ ωit = h ⎝ωit−1 , Git ,



⎞ sij t−1 ωj t−1 ⎠ + ξit ,

(8)

j =i

where the subscript i is a firm index, t is a year index, ωit is a productivity measure, and G represents a firm’s foreign capital stock. {sijt − 1 } are the weights embodying peer connections of firms based on their exposure to foreign technology – the weight sijt − 1 for each firm i is constructed based on the foreign equity share of its foreign peers j in the industry (and the geographic  locality). Therefore, the weighted aggregate of other foreign firms’ productivity ( j =i sij t−1 ωj t−1 ) captures potential positive FDI externalities, giving rise to productivity spillovers from other firms with foreign capital in the industry.

Innovation and Productivity Innovation Measures Studies of growth, based on the aggregate production function, have revealed that growth in the traditional factor inputs, such as capital and labor, can only explain about half of the output growth in the USA and many other countries. The remaining portion is ascribed to growth in productivity or the broadly defined technological progress. As we see from the introduction, Crossan and Apaydin [19] give a comprehensive definition of innovation, which encompasses almost all channels of productivity growth. A large collection of literature has emerged to find more specific and trackable measures for innovative activities and to examine their impacts on productivity. Despite that, measuring innovation in a form that is useful for statistical analysis has proven to be challenging. The central problem is data availability. The previous work has mostly used two measures of innovative activities: R&D expenditures and patent counts. Doraszelski and Jaumandreu [23] use the Encuesta sobre Estrategias Empresariales (ESEE) survey to investigate the relationship of R&D and productivity. The ESEE survey of Spanish firms is a comprehensive and well-designed survey, which includes questions related to patents, innovation (product, process, organizational and marketing innovations), payments and income for licenses and technical assistance, salaries of R&D personnel, and complementary technological activities such as collaboration with customers and suppliers. Jin et al. [41] examine the nexus of financial constraints, R&D, and

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productivity using a survey of all large- and medium-sized Chinese manufacturing firms. This survey is conducted by China’s National Bureau of Statistics (NBS) and provides data on R&D expenditures after 2004. Different from ESEE, NBS’s questionnaires of R&D are not customized to take all R&D-related expenditures into consideration (see [41] for a detailed comparison). Lach [46] uses data on patent counts to assess the contribution of knowledge to productivity growth at the industry level. The output of Robert Evenson’s Yale-Canada Concordance Project, described by Kortum and Putnam [45], provides the patent data. Chen and Yang [17] employ both R&D stock and patent counts to investigate the relationship among technological knowledge, spillovers, and productivity of Taiwanese manufacturing firms from 1990 to 1997. Their patent data is collected from the computer file of patents from the Taiwan Intellectual Property Office. The Compustat and the China Stock Market and Accounting Research (CSAMR) databases also provide patent counts for the US and Chinese publicly listed firms, respectively [26, 42]. Hall [33] points out that, as measures of innovation, R&D expenditures and patent counts have both positive and negative attributes. They both pertain primarily to technological innovation and are more suited to measuring innovation in manufacturing firms than in firms in other areas, such as services. R&D expenditures have the advantage that they are denominated in comparable units and represent a firm’s decision variable. They reflect a firm’s appropriate level of innovative activities. For the same reason, R&D expenditures are only an input to innovation and cannot tell us about innovation success. In contrast, the patent count is a measure of invention success, and can be considered at least a partial measure of innovation output. But the counts are inherently very noisy – it is likely that a few patents are associated with very valuable inventions and most others are associated with little value. The extent of their innovation coverage varies by sector, with sectors like pharmaceuticals and instruments making heavy use of patents while other sectors use them sparingly. Recognizing the disadvantages of R&D expenditures and patent counts, economists have begun to examine innovation more broadly and search alternative measures including those more suitable for the services sector. Several such measures of innovative activities have been employed, including the use of information and communication technologies (ICT), business model patents, expenditure on organizational innovation, marketing expenditure related to new products, etc. Polder et al. [58] consider the ICT usage as an additional innovation input separated from R&D that increases firm productivity over time. Based on a data set merging different surveys (i.e., the Community Innovation Survey (CIS), business ICT survey, and Investment Statistics and Production Statistics), they find that ICT is the most important survey for innovation success in the services sector. ICT investment and the use of broadband connectivity and e-commerce positively affect all product, process, and organizational innovations. Martin and Nguyen-Thi [51] combine both CIS and annual ICT usage and e-commerce in an enterprises survey in Luxembourg and find the positive effect of ICT use on labor productivity in the services sector. A business model is a combination of technologies and markets, which requires strategies and resources to overcome mis-combinations

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[30]. Following an outdated business model can cause firms to miss a potentially valuable use of technology, and thus cause firms to miss a potential opportunity to increase productivity. To overcome the challenges of increasing development costs and shorter product life cycles, companies must be innovative in their business model. To measure innovation of business models, one can use the business model patent data, which is available in a worldwide patent database – the Global Patent Analysis Service System (G-PASS). It covers all major countries (e.g., the USA, Europe, China, Japan, Korea, Germany, France, the United Kingdom, and Canada) from 1960 to present. To overcome the noncomparability issue associated with using patent counts or other innovation counts to measure innovation – some patents create whole new markets whereas others are useful but trivial. One can, at least, use two other measures to quantify the quality of patents. First, as pointed out by Hall [33], innovation surveys typically have information on the share of the firm’s sales that are resulted from innovations introduced during a preceding period (usually the past 3 years). The share of sales of innovative products provides a good indicator of the overall importance of the innovations. However, it is useful only for innovations in goods and services and cannot be used to capture process or organizational innovations. Second, one can use the patent citations to measure the importance of a patent. Different from a simple patent count, citations can capture not only the quantity of ideas produced but also the quality of those ideas [18]. The National Bureau of Economic Research (NBER) patent database provides detailed information on almost 3 million US patents granted from 1963 to 1999, and it includes all citations made to these patents between 1975 and 1999. This database also has other original measures such as indices of originality and generality. See Hall et al. [34] for a detailed description. An updated version of this database includes data up to 2006.

R&D Measure and Modeling Among all the innovation measures mentioned above, R&D is the most intensively examined in previous studies. Firms invest in R&D to achieve productivity gains through innovations resulting from their investments, and there is a large body of literature estimating returns to R&D. The common approach used in these studies is to adopt the Griliches’ [32] knowledge production framework, which augments a standard production function with a measure of the current stock of technical knowledge (see a survey by [35]). In general, we can write the Griliches production function g(•) as: yt = g (xt , rt ) + t ,

(9)

where yt , xt , and rt are (log) output, standard factor inputs (such as capital, labor, and materials), and knowledge capital at time t, respectively. The knowledge capital is usually calculated through the perpetual inventory method from the observed R&D

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expenditures, that is, rt = (1 − δ)rt − 1 + R & Dt , where δ is the depreciation rate. Doraszelski and Jaumandreu [23] argue that the widely used Griliches framework has two potential problems: the depreciation rate is difficult to estimate, and the available history of R&D expenditures is short. they explore the law of

 Therefore, motion of (log) productivity, viz., t+1 = E t+1 t , R&Dt + error t+1 , in which calculation of knowledge stock is not needed. The effects of previous R&D investment on productivity are controlled by the contemporary productivity ( t ), and the current period R & Dt interacts with  t . They jointly determine the future productivity. However, neither approach accounts for the spillovers of R&D investment during estimation. Due to the non-excludability and inexhaustibility nature of knowledge, previous studies show that investment in R&D, by private corporations and countries, “spills over” for the third party to exploit [25]. To empirically measure the effect of spillovers on productivity, previous studies typically model the (prespecified) spatial correlation of the prior estimated productivity by applying spatial econometric tools, such as the spatial autoregressive (SAR) model and spatial error model (SEM), in a two-step approach. Nevertheless, spillovers lead to crosssectional dependence across units, which may lead to inconsistent productivity estimates in the first step [57]. Eberhardt et al. [25] and Malikov and Zhao [49] argue that spillovers should be accounted for during productivity estimation. They apply the interactive fixed-effects model and the dynamic law of motion of productivity to control the spillover effects, respectively.

Financial Constraints, R&D, and Productivity While R&D investment is a crucial driver of productivity growth, financing R&D tends to be difficult due to reasons such as lack of collateral value, uncertainty, and asymmetric information associated with R&D activities [16]. Intuitively, it seems natural that financial constraints would have a negative effect on firms’ R&D investment and thus decrease firm productivity. However, the results from empirical studies are mixed. There are studies that find a strong link between R&D and financial status measures, such as cash flow, internal and external equity finance [15, 37]. On the other hand, there are other studies that find a weak correlation or no correlation at all between R&D and cash flow [12, 36]. These inclusive results may be due to market/project heterogeneity, sample selection, and data limitation. For example, Czarnitzki and Hottenrott [20] find that R&D expenditures on cutting-edge projects are curtailed by financial constraints, while routine R&D investment is not. Brown et al. [16] find strong evidence that the availability of finance matters for R&D once (i) firm’s efforts to fund R&D with cash flow and (ii) firm’s use of external equity finance are directly controlled for. Recent studies also find a nonlinear relationship between financial constraints and productivity. Jin et al. [41] find that financial constraints have non-monotonic effects on productivity. Severely constrained firms cannot afford sufficient invest-

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ment in R&D activities, resulting in productivity loss. Unconstrained firms may not have incentives to invest in R&D, which also results in low productivity. Aghion et al. [2] argues that better access to credit makes it easier for entrepreneurs to innovate, but it allows less efficient incumbent firms to constantly remain in the market, thereby discouraging entry of new and potentially more efficient innovators.

Conclusion Globalization is one of the fundamental issues in production and development economics. International trade (import and export) and FDI are among the key components of the globalization process. Theoretical and empirical studies, based on different model settings and samples, have identified various mechanisms through which trade and FDI can increase productivity at both macro and micro levels, even though heterogeneity exists due to reasons such as competition and learning capacity. Innovation and productivity growth are intimately related. Different types of innovative activities have been studied with challenges of measuring innovation precisely. Commonly used innovation measures include R&D expenditures, patent counts, patent citations, business model patents, and the use of information and communication technologies. R&D expenditure is studied most intensively. The discussions of other measures are based mostly on empirical findings. We look forward to more theoretical frameworks on their connections to firms’ decisions in the field of economics. Finally, the consistent estimation of productivity and identification of productivity effects stemming from trade, FDI, R&D, and spillovers could be challenging due to various endogeneity issues (e.g., simultaneity problem associated with input choices and latent productivity, the omitting variable bias due to connection between globalization/innovation and firm input usage, and cross-sectional dependence resulted from technology spillovers). Fortunately, each of them can be solved by using the appropriate econometric tools, so future empirical studies of good quality, can, and should, take them into consideration.

Cross-References  Cost, Revenue, and Profit Function Estimates  Economics of Externalities: An Overview  Scale Elasticity and Returns to Scale

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29. Gandhi A, Navarro S, Rivers D (2018) On the identification of gross output production functions. Centre for Human Capital and Productivity (CHCP), University of Western Ontario 30. Gassmann O, Frankenberger K, Csik M (2014) The business model navigator: 55 models that will revolutionise your business. Pearson, Harlow 31. Griffith R, Redding S, Simpson H. (2003) Productivity convergence and foreign ownership at the establishment level. CEP discussion papers dp0573. Centre for Economic Performance, LSE 32. Griliches Z (1979) Issues in assessing the contribution of research and development to productivity growth. Bell J Econ 10(1):92–116 33. Hall BH (2011) Innovation and productivity. National Bureau of Economic Research, Cambridge, Massachusetts 34. Hall BH, Jaffe AB, Trajtenberg M (2001) The NBER patent citation data file: lessons, insights and methodological tools. National Bureau of Economic Research 35. Hall BH, Mairesse J, Mohnen P (2010) Measuring the returns to R&D. In: Handbook of the economics of innovation, vol 2, pp 1033–1082. Elsevier 36. Harhoff D (2000) Are there financing constraints for R&D and investment in German Manufacturing Firms? In: The economics and econometrics of innovation. Springer, New York, pp 421–456 37. Himmelberg CP, Petersen BC (1994) R&D and internal finance: a panel study of small firms in high-tech industries. Rev Econ Stat 76(1):38–51 38. Irwin DA, Terviö M (2002) Does trade raise income? Evidence from the twentieth century. J Int Econ 58(1):1–18 39. Javorcik B (2004) Does foreign direct investment increase the productivity of domestic firms? In search of spillovers through backward linkages. Am Econ Rev 94(3):605–627 40. Javorcik B, Poelhekke S (2017) Former foreign affiliates: cast out and outperformed? J Eur Econ Assoc 15(3):501–539 41. Jin M, Zhao S, Kumbhakar SC (2019) Financial constraints and firm productivity: evidence from Chinese manufacturing. Eur J Oper Res 275(3):1139–1156 42. Jin M, Tian H, Kumbhakar S (2020) How to compete and survive: the impact of information asymmetry on productivity. J Prod Anal 53:107–123 43. Keller W (2004) International technology diffusion. J Econ Lit 42(3):752–782 44. Keller W, Yeaple SR (2009) Multinational enterprises, international trade, and productivity growth: firm-level evidence from the United States. Rev Econ Stat 91(4):821–831 45. Kortum S, Putnam J (1989) Estimating patents by industry: part I and II. Unpublished Manuscript, Yale University 46. Lach S (1995) Patents and productivity growth at the industry level: a first look. Econ Lett 49(1):101–108 47. Levinsohn J (1993) Testing the imports-as-market-discipline hypothesis. J Int Econ 35(1–2): 1–22 48. Levinsohn J, Petrin A (2003) Estimating production functions using inputs to control for unobservables. Rev Econ Stud 70(2):317–341 49. Malikov E, Zhao S (2019) Cross-firm productivity spillovers in the presence of foreign investments. Working paper 50. Markusen JR, Maskus KE (2002) Discriminating among alternative theories of the multinational enterprise. Rev Int Econ 10(4):694–707 51. Martin L, Nguyen-Thi TU (2015) The relationship between innovation and productivity based on R&D and ICT use: an empirical analysis of firms in Luxembourg. Rev Écon 66(6):1105– 1130 52. Melitz MJ (2003) The impact of trade on intra-industry reallocations and aggregate industry productivity. Econometrica 71(6):1695–1725 53. Muendler M-A (2004) Trade, technology and productivity: a study of Brazilian manufacturers 1986–1998. CESifo working paper series 1148. CESifo Group, Munich 54. Nishimizu M, Robinson S (1984) Trade policies and productivity change in semi-industrialized countries. J Dev Econ 16(1–2):177–206

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55. Olley GS, Pakes A (1996) The dynamics of productivity in the telecommunications equipment industry. Econometrica 64(6):1263–1297 56. Pavcnik N (2002) Trade liberalization, exit, and productivity improvements: evidence from Chilean plants. Rev Econ Stud 69(1):245–276 57. Pesaran MH (2006) Estimation and inference in large heterogeneous panels with a multifactor error structure. Econometrica 74(4):967–1012 58. Polder M, van Leeuwen G, Mohnen P, Raymond W (2009) Productivity effects of innovation modes. MPRA paper 18893. University Library of Munich 59. Rodriguez-Clare A (1996) Multinationals, linkages, and economic development. Amer Econ Rev 852–873 60. Rodriguez F, Rodrik D (2000) Trade policy and economic growth: a Skeptic’s guide to the cross-National Evidence. Natl Bur Econ Res Macroecon Annu 15:261–325 61. Topalova P, Khandelwal A (2011) Trade liberalization and firm productivity: the case of India. Rev Econ Stat 93(3):995–1009 62. Tybout JR (1992) Linking trade and productivity: new research directions. World Bank Econ Rev 6(2):189–211 63. Xu B (2000) Multinational enterprises, technology diffusion, and host country productivity growth. J Dev Econ 62(2):477–493 64. Yu M (2015) Processing trade, tariff reductions and firm productivity: evidence from Chinese firms. Econ J 125(585):943–988 65. Zhao S, Jin M, Kumbhakar SC (2020) Estimation of firm productivity in the presence of spillovers and common shocks. Empir Econ, forthcoming

Empirical Analysis of Production Economics: Applications to Banking

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Contents Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Production Economics in Banking Research . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Organizing Production: How to Measure Output . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Productivity Growth . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Bank and Banking Profitability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Economies of Scale . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Economies of Scope . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Efficiency . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Bank Failure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Cross-References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Volume II of the Handbook of Production Economics provides surveys of empirical applications of the neoclassical production economics discussed in Volume I. This chapter examines empirical applications in banking that now enter what we can categorize as accumulated, accepted knowledge or wisdom. To begin, we consider how to measure output. Two basic approaches exist – the production and intermediation specifications. The treatment of deposits differentiates these two specifications, whereby the production approach takes deposits as an output and the intermediation approach takes deposits as an input. Then, this chapter proceeds to discuss various issues in bank production – bank

S. M. Miller () Department of Economics, Lee Business School, University of Nevada, Las Vegas, Las Vegas, NV, USA e-mail: [email protected] © Springer Nature Singapore Pte Ltd. 2022 S. C. Ray et al. (eds.), Handbook of Production Economics, https://doi.org/10.1007/978-981-10-3455-8_29

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productivity growth, bank and banking industry profitability, economies of scale and scope in banking, and bank efficiency. Bank efficiency includes efficiency as measured by the production function, the cost function, the revenue function, the profit function, and the efficient frontier between expected return and risk. The chapter concludes with an analysis of predicting problem banks and/or bank failures. Keywords

Banking-Productivity-Profitability-Scale · Scope-Efficiency-Expected return · Risk-Failure Prediction

Introduction The US banking industry experienced significant change over the last four decades. The financial reforms adopted due to the Great Depression promoted a tension between federal regulations and their implementation and enforcement. While regulation generally restricted geographic and product-line expansion of commercial banking activities, financial innovation reacted against these regulatory constraints. That is, bankers applied pressure for less regulation and more freedom of action. Such activity generally produced implementation and enforcement of existing regulations that loosened regulatory constraints. In addition, financial innovations aimed to avoid or work around existing regulations. Many prohibitions on geographic and product-line expansion of commercial banking activities proved less effective than the regulators originally planned. The process of deregulation from the regulatory environment implemented in response to the Great Depression began in the early 1980s with the Depository Institution and Monetary Control Act of 1980. The process continued through the passage of the Riegle-Neal Interstate Banking and Branching Efficiency Act of 1994 and essentially ended with the adoption of the Gramm–Leach–Bliley Act, also known as the Financial Services Modernization Act of 1999. This environment of deregulation or freeing up of commercial banking activities in the late 1980s and early 1990s also saw the largest number of commercial bank failures and mergers since the Great Depression. This reduction in the number of institutions through failure and merger (either voluntary or FDIC arranged) significantly lowered the number of commercial banks to under 8000 by 2008 from a previous peak of nearly 15,000 in the early 1980s. Today, the number of commercial banks falls below 5000. This process of deregulation and consolidation in the industry also accelerated a trend toward more complex banking organizations and activities. The conglomeration of different financial services under the umbrella of one legal organization, the bank holding company, increased the complexity of financial services offered by banks, such as derivatives and mortgage (and other credit) securitizations. These new financial innovations not only rightly received accolades for their ability to diversify risk, but they also brought new levels of risk to bank operations.

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For example, the subprime mortgage crisis of 2007 led to the bailout of US banks that had become “too big to fail.” This ultimately precipitated the passage of the Dodd-Frank Wall Street Reform and Consumer Protection Act in 2010 that was designed to address many of the apparent weaknesses within the US financial system that developed because of the deregulation process. In other words, a significant reregulation of the industry began to occur. In sum, the banking industry with its fundamental role in the operation of the economy and its experience with regulatory change provides a unique industry to study. In addition, the availability of income and balance sheet data on individual banks and/or bank-holding companies provides the grist for analysis of various aspects of production economics. Volume II of the Handbook of Production Economics provides surveys of empirical applications of the neoclassical production economics discussed in Volume I. This chapter examines empirical applications in banking that now enter what we can categorize as accumulated, accepted knowledge or wisdom.

Production Economics in Banking Research This section examines a variety of issues of production economics in the banking industry. Topics include productivity and profitability, economies of scale and scope, production, cost, revenue, and profit efficiency, and predicting (identifying) problem banks and/or failed banks. Most papers included in this chapter consider US banking data with a few exceptions. Typically, research projects using US banking data quickly translate into similar research projects using data from other developed and/or developing countries.

Organizing Production: How to Measure Output At the outset, important problems emerge in defining what a commercial bank produces or how to measure commercial bank output. In general, the existing literature in banking adopts two main specifications of how a bank operates in the banking industry: the production and intermediation specifications [1]. The production approach specifies the bank as producing loans, deposits, and other financial services, using inputs such as labor and capital. The intermediation approach specifies the bank as intermediating financial services between spenders (borrowers) and savers (lenders) and producing loans and other interest-earning assets using deposits and labor [2]. The production and intermediation specifications identify deposits, in turn, as an output and as an input, respectively. Other specifications also exist, although less frequently used in the literature. The asset specification aligns itself with the intermediation approach [3]. That is, deposits and other liabilities along with labor and capital produce output (assets), and only bank assets generate revenue such as loans or investments

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[4]. Alternatively, the value-added (revenue) specification in Data Envelopment Analysis (DEA) models identifies outputs as components of a portfolio that create value [5]. Therefore, this specification postulates deposits and loans as outputs. Further, the operating (income) specification models the bank as generating the revenue necessary to cover its total cost or expenses. This specification identifies interest and noninterest expenses as inputs and interest and noninterest income as outputs [6]. Finally, Holod and Lewis [7] specify a two-stage DEA model. In stage one, the bank employs labor, capital, and other inputs to produce deposits. Then, in stage two, the bank employs deposits with other inputs to produce interest-earning assets (output), treating deposits as an intermediate output (input).

Productivity Growth Solow’s [8] path-breaking work on growth accounting spawned an ongoing literature on measuring productivity and/or technological growth. Such measurement of productivity growth, using parametric and nonparametric methods, also exists in the banking industry. Several papers employ parametric models (e.g., [9–12]) to examine either total factor productivity growth or technological progress in the US commercial banking industry during the 1980s. Humphrey [10] calculates total factor productivity growth using growth-accounting models. He determines that the average annual productivity change varied between −0.07%, using the production method, and 0.6%, using the cost method, between 1977 and 1987. The lack of productivity growth is traced to several factors. First, before deregulation, US commercial banks competed significantly through “brick and mortar” investment, which offered convenience to bank customers through a large branching network (where permitted). Second, financial innovation and deregulation during the 1970s and 1980s pushed the cost of funds to higher levels for commercial banks due to more intense competition. Third, banks failed to adjust sufficiently or quickly enough on their investment in branching networks to compensate for the higher costs of funds. In sum, total factor productivity, measured either through the production or cost approaches, fell. Humphrey [11] and Hunter and Timme [12] estimate cost functions to measure technological change in the US commercial banking industry. Technological change excludes scale-economy effects. Both papers find small or negligible technological change in the 1977–1988 period, supporting the findings of Humphrey [10]. Finally, Bauer, et al. [9] implement both stochastic and thick-frontier specifications and estimate total factor productivity growth, using a sample of 683 banks. They discover that total factor productivity growth varied from −2.28 to 0.16% over the 1977–1988 sample period. Some papers employ nonparametric, rather than parametric, specifications to measure productivity and technology. Caves, et al. [13] employ the Malmquist productivity index, using the “proportional scaling” method introduced by Malmquist

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[14], to measure productivity growth.1 The Malmquist productivity index provides a nonparametric alternative to the prior parametric specifications. To derive the Malmquist index, Caves, et al. [13] adopt a translog model for the distance functions and assume that the firms operate efficiently. Färe et al. [16] allow inefficiencies in firm operation and model the technology as piecewise linear. They assume constant returns to scale and decompose the Malmquist index into efficiency change and technical change index components. A series of other papers develop further decompositions of the Malmquist productivity index (e.g., [17–20]). Since many DEA studies of bank efficiency employ cross-sectional data, they cannot measure productivity change. Using a sample from 1980 to 1985, Elyasiani and Mehdian [21] calculate technological change for large US commercial banks, but not their productivity growth. They identify significant technological advancement of about 2.6% per year over their 5-year sample period, classifying this technological progress as a non-neutral and labor-biased technical change. Berg et al. [22] employ the Malmquist index to measure productivity growth in Norwegian banking during its deregulation. The analysis examined the 1980–1989 period with the number of banks falling from 346 in 1980 to 178 in 1989. They used the value-added method to measure output. They consider productivity growth for the banking frontier as well as for individual banks, finding productivity decline prior to deregulation and productivity growth after deregulation. They also find less dispersion of productivity levels because of deregulation. Wheelock and Wilson [23] use a sample of all US banks between 1984 and 1993, omitting banks with missing observations or no loans, to construct the Malmquist index to determine productivity change. They measure the Malmquist index as the ratio of variable returns to scale (VRS) distance functions and decomposed this index into two factors: technical efficiency change and technical change. For example, they find that productivity generally declined for all banks. The subsample of banks that operated with more than $300 million in assets continuously from 1983 to 1993 experienced increased productivity from 1983 to 1989, but decreased productivity after 1989 that offset the productivity increase in the 1980s. Mukherjee, et al. [24] estimate productivity growth for 201 large US commercial banks in the postderegulation period from 1984 to 1990, using the DEA approach and Malmquist productivity indexes. They isolate the contributions of technical change, technical efficiency change, and scale efficiency change to productivity growth. They identify that overall productivity growth increased by 4.5% per year, on average. Productivity actually declined, however, by 7.61% and 0.33% between 1984 and 1985 and between 1988 and 1989, respectively. Second-stage panel regressions document that larger productivity growth associates with larger asset size and specialization of product mix, whereas lower productivity growth associates with higher net charge-offs.

1 Färe, et al. [15] provide a comprehensive survey of the theory and development of the Malmquist productivity index.

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Bank and Banking Profitability As a financial institution, banks generate profit in large measure by managing their portfolio of assets and liabilities, that is, their balance sheets. Of course, banks also hire labor and rent (own) capital to run their operations in addition to selling services, which also appear in the income statements of banks along with the income and expenses from managing their assets and liabilities. Nonetheless, managing the portfolio of assets and liabilities provides an important component of bank profitability. Using Consolidated Report of Condition and Income (Call Report) data, researchers have considered the important determinants of individual US bank profitability as well as the dynamic adjustment of industry profitability through the growth and structure of the industry.

Determinants of Bank Profitability2 Numerous studies directly attempt to identify the characteristics associated with bank profitability. Researchers frequently measure bank profitability by return on assets (ROA) or return on equity (ROE), where a measure of net income captures the profit from the income statement and total assets or equity comes from the balance sheet. Explanatory variables typically include bank-specific variables (internal factors) and macroeconomic variables (external factors). Bank-specific variables include the effect on bank profitability of some financial ratios that measure asset (lending and investment) management (e.g., total loans to total assets), liability (funding) management (e.g., total deposits to total assets), productivity and efficiency (e.g., total noninterest income to total income), and asset quality (e.g., provisions for loans losses to total loans). The role of macroeconomic variables in affecting bank profitability involves changes over time. Thus, studies of bank profitability that include both bank-specific variables as well as macroeconomic variables generally employ time-fixed effects in a panel data setting to control for macroeconomic variables. Several authors provide a mixture of findings on US bank performance with samples of various sizes of banks and over various periods of time (e.g., [25–28]. A few consistent explanations of good bank performance do surface frequently in these papers. High-performance banks possess low noninterest expense, and researchers frequently attribute good bank performance to quality management. Those factors that associate with high-performance banks do shift somewhat from study to study. Part of this diversity of findings reflects different samples of bank sizes and different sample periods. For example, Gup and Walter [25] examine the characteristics of top-performing small banks. Small banks differed significantly from the general results of the

2 Some

papers use other performance measures besides bank profit such as net interest margin to total assets, noninterest expenses to assets, nonperforming loans to total loans, or loan losses to total loans (e.g., [25–27]).

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other studies. High-performance small banks possess more securities, higher-quality loans, more capital, and more demand deposits. A large literature focuses on banks in countries other than the United States. Some of the more frequently cited papers include the following: [29–33].

Dynamic Adjustment of Banking Industry Profitability The dynamic adjustment of banks and their portfolios within that industry affects overall industry profitability. The US banking industry developed within an environment of strong aversion to concentrations of power and with significant regulation in the banking sector enacted in response to the Great Depression. Due to this history, US banking activity and regulation generated an industry encompassing many more banks than the norm around the world.3 During the 1970s, the banks developed financial innovations frequently to circumvent existing regulation. The ATM machine is the classical example. Commercial banks created ATMs to circumvent branching restrictions and reduce costs. (The ATM is not a branch but functions as a minibranch.) Those innovations gradually eroded the control of existing regulations, ultimately rendering much of the regulatory superstructure erected during the Great Depression much less effective. Thus, the last two decades of the twentieth century saw a series of deregulatory actions that significantly freed banks from the regulatory control enacted during the Great Depression. For example, the prohibition of intrastate and interstate banking slowly disappeared, first with a series of relaxations of regulation on a state-by-state basis, then with increasing state-level actions permitting interstate banking activity through multibank-holding companies, and finally with the adoption of full interstate banking with the passage of the Interstate Banking and Branching Efficiency Act of 1994. In sum, the geographic deregulation of banking activity at the state and national levels provides an unusual real-world experiment on the effects of such deregulation on banking behavior and performance.4 A noneconometric study by Duca and McLaughlin [42] develops a taxonomy of changes affecting bank profitability from 1985 to early 1990. Their discussion relies on balance sheet and income statement data for all insured commercial banks as well as for these banks separated into different size categories. Several general conclusions emerge from their analysis. First, they determine that variations in loan-loss provisions largely explain variations in bank profitability. In other words, the variation in net income drops dramatically after purging loan-loss provisions. Second, they note that total real estate loans grew as a fraction of interest-earning assets at all insured commercial banks and across all size categories reported. That

3 Kane

[34] provides an excellent historical account of the deregulatory movements in the US banking sector. 4 Existing work examines the effects of deregulation on various banking issues. For example, how did deregulation affect new charters, failures, and mergers [35–37] as well as bank performance [37–41]?

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is, the banking industry uniformly increased its exposure to real estate loans in the late 1980s. Decomposing industry performance measures typically adopts the approaches of Bailey, et al. [43] and Haltiwanger [44], which trace back to the work of Bennet [45] on decomposition. Bailey, et al. [43] provide an algebraic decomposition of an industry’s total factor productivity (TFP) growth into three effects – “within,” “between,” and “net-entry” effects. The within effect measures the contribution of surviving firms toward TFP growth. The between (or reallocation) effect measures the contribution of changing market share of surviving firms toward TFP growth, while the net-entry effect measures the contribution of firms’ entry into and exit from the industry toward TFP growth. Haltiwanger [44] extends the Bailey et al. [43] approach and separates the effects of firm entry and exit from the industry. Moreover, he also divides the between effect into two components – the “share” and “covariance” effects. The share effect measures the contribution toward aggregate TFP growth of the changing share of firms while the covariance effect measures the contribution toward aggregate TFP growth of the changing share of firms times the changing TFP growth of firms.5 Stiroh [46], using US banking data, further decomposes Haltiwanger’s [44] method by dividing banks into those that acquired other banks and those that did not. Such decomposition methods share a common index-number issue – the baseyear choice. Bailey, et al. [43], Haltiwanger [44], and Stiroh [46] all choose the initial year as the base. Thus, the within effect measures the change in performance measure at the firm level between the initial and final years weighted by the industry’s share in the initial year. Alternatively, another decomposition exists of within, between (reallocation), entry, and exit effects where the final year provides the base. That is, the within effect weights the change in the performance measure between the initial and final years for each firm by the firm’s industry share in the final year. Finally, a Bennet [45] dynamic decomposition combines these two dynamic decompositions into a simple average.6 Thus, the weighting of the within, between (reallocation), entry, and exit effects all employ simple averages of the initial and final year weights. In addition, the Bennet dynamic decomposition of the industry

5 As

illustrated below, the covariance effect emerges because of the decomposition method. The Bennet [45] decomposition method causes the covariance effect to disappear. 6 This discussion possesses an analogy to the price index literature. The Laspeyres [47] price index uses in the numerator the sum of the current prices times base-period quantities, and in the denominator the sum of base prices times base-period quantities. The Paasche [48] price index uses in the numerator the sum of the current prices times current-period quantities, and in the denominator the sum of base prices times current-period quantities. The Fisher [49] ideal price index, then, forms a geometric mean of the Laspeyres and Paasche indices. Pigou [50] also proposed the ideal price index. Bennet [45] specifies the analogy to the Fisher ideal price index for changes in revenue – the sum of prices times quantities. Diewert [51] provides an extensive discussion of the Bennet index, showing that the Bennet index equals the arithmetic average of the Laspeyres and Paasche difference index analogies.

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eliminates the covariance effect derived by Haltiwanger [44].7 Jeon and Miller [54] modify the Bennet [45] method to consider geographic (regional) effects. Their extension explores the effects, if any, of the deregulation of geographic restrictions on the state-level decomposition of bank performance (return on equity). In that analysis, they control for competition and the state of the economy in each state, employing fixed- and random-effects regressions in the panel database across the 50 states and the District of Columbia from 1976 to 2005. Researchers decompose average bank profitability in the industry, measured by ROA or ROE, into factors attributable to improved profitability of individual banks (the “within” effect), shifts of resources from less to more profitable banks (the “between” effect), entries of more profitable banks (the “entry” effect), and exits of less profitable banks (the “exit” effect). Jeon and Miller [54] implement their modification of the Bennet dynamic decomposition to the ROE in the commercial banking industry between 1976 and 2005 where the microeconomic unit equals the bank. They find that the between and exit (within and entry) effects contribute positively (negatively) and strongly to the banking industry’s trend ROE. Moreover, together the four components produce a negligible net effect over the 30-year period. Interestingly, although all four effects explain the cumulative long-run change in return on equity, they report that the within effect dominates the between, entry, and exit effects on a year-to-year basis. Thus, the within effect dominates the cyclical movements in bank performance, but the trend movement in bank performance reflects the between, entry, and exit effects. The growing market share of high-performance banks at the expense of lowperformance banks explains the trend movement in bank performance over the sample period.8 That is, the trend movement in industry performance reflects a process of creative destruction, whereas the cyclical, year-to-year movement primarily reflects the fortune of individual bank performance (within effect). Thus, creative destruction generally confines itself to the long-run trend adjustments but not to the short-run, cyclical adjustments. The 1993–2005 period after the banking crisis provides the exception to this rule whereby creative destruction played a more important role in short-run, cyclical movements in bank performance.

7 Griliches and

Regev [52] employ this decomposition method in their study of firm productivity in Israeli industry. Scarpetta et al. [53] briefly describe the Griliches and Regev [52] and Haltiwanger [44] methods of decomposition, noting how they differ. Jeon and Miller [54], however, link the differences to the base-year weighting issue. Finally, Bartelsman et al. [55] note that the covariance term disappears for their decomposition. Balk [56] also provides an extensive review of the Bennet [45] decomposition in terms of productivity changes. 8 Stiroh and Strahan [57], using a different methodology, reach a similar conclusion.

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Economies of Scale Research on banking as an application of production economics began in earnest with a series of papers that explored the existence of economies and diseconomies of scale around five decades ago. Numerous studies of scale economies in banking exist, generally tracing their roots to the work by Benston [58] and Bell and Murphy [59]. Clark [60] reviews most studies through the early phase of deregulation, including economies of scope. The original conclusion of these papers, using samples of smaller banks of under $1 billion in assets, generally finds that economies of scale quickly disappear and that further increases in bank size lead to diseconomies of scale. The extrapolation of findings for samples of smaller banks to large banks proved problematic. First, small and large banks provide different services to a different mix of markets and customers, suggesting that small and large banks probably face different cost structures. Second, the deregulation in banking probably rendered the results of prior studies moot since the initial studies occurred before the more recent significant deregulation. Several papers examine banks with deposits above $1 billion (e.g., [61–65]). These studies, however, except for Nelson [63] and Noulas et al. [64], treat output as a single composite commodity and do not disaggregate output. To aggregate, one must assume that the cost function is separable in output. That is, the relative marginal cost of any two outputs proves independent of input prices. If this separability assumption fails to hold, then the estimation may produce biased scale results. Noulas et al. [64] disaggregate output to test the separability assumption, finding that it does not hold. Since this original work on economies of scale in banking, the scale of banking organizations grew over time to much larger-sized banks. The experience of the Great Recession also ignited the debate on “too-big-to-fail” institutions. As such, the determination of when banking institutions exhaust economies of scale becomes an important public policy issue. As Humphrey [61] noted, estimates of scale economy based on 1-year’s cross section do not generalize to other years. Thus, Wheelock and Wilson [66] examine the extent of scale economies for 1986, 1996, 2006, and 2015, using cost, revenue, and profit functions. They derive estimates of returns to scale from a nonparametric, local-linear estimator. Their findings determine that for many institutions, increasing returns to scale continued to exist for larger size as time moved from 1986 to 1996 to 2006 to 2015. Their strongest findings relate to economies of scale derived from cost data, whereby the revenue and profit estimates proved more mixed.

Economies of Scope Economies and diseconomies of scale give information about optimal bank size. Scale effects, however, do not consider the growth of banking institutions by

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diversification of the financial products offered within the institution, that is, the multiproduct nature of bank production. (To examine this issue, researchers focus on economies and diseconomies of scope, considering the potential cost complementarities associated with multiproduct production.) Several authors develop the original research on economies and diseconomies of scope (e.g., [67–70]). These papers, however, fail to document evidence of economies of scope. They do find limited evidence of cost complementarities between some pairs of outputs. Ferrier and Lovell [71] report limited pairwise cost complementarities. Kim [72] and Buono and Eakin [73] find economies of scope. In contrast, Cebenoyan [74] documents diseconomies of scope. Mester [75] discovers evidence of significant economies of scope for savings and loan associations, estimating a translog cost function. She uses a sample of 1115 US savings and loans with positive equity in 1991. Mester [76], however, uncovers no evidence of economies of scope, estimating a translog cost function. She employs a sample of 214 3rd Federal Reserve District banks in 1991–1992. This initial research on scope effects in banking generally focuses on small banks with less than $1 billion in assets. Some research did consider large banks of more than $1 billion in assets. Rangan et al. [77] estimate scope effects for different sizes of banks, reporting diseconomies of scope for large banks. They actually measure cost subadditivity rather than scope economies since they do not permit specialized production. Hunter, et al. [78] also consider cost subadditivity for large banks, finding no subadditivity in the bank cost function and no measurable cost complementarities. Pulley and Braunstein [79] and Pulley and Humphrey [80] document widely varying scope estimates, using a translog cost function, when outputs take on zero values. They then employ a “composite function” that does not exhibit such instability and find economies of scope. Noulas et al. [81] determine that the ordinary translog cost function generates unreliable estimates of scope effects, showing that the scope measures change in size and magnitude considerably in a sensitivity analysis. Moreover, at the point of evaluation of scope effects, the regularity conditions fail to hold, suggesting that all studies of scope effects using a translog cost function are suspect. Instead, using a hybrid cost function (i.e., a Box-Cox transformation), Noulas et al. [81] discover economies of scope for the only specification where regulatory conditions hold at the point of scope evaluation. Furthermore, the results are stable, as revealed by evaluating the hybrid cost function at values of output close to zero. Mitchell and Onvural [82] employ the Fourier flexible functional form cost functions to estimate scope effects for large banks. Their main conclusion reports that banks experience little or no cost gains from producing an output as a single bank or as a part of a larger organization. That is, economies of scope do not seem too important. The prior work seeks evidence on economies of scope in the cost function. Economies of scope, however, may appear in the revenue function. Berger et al. [83] search for evidence of economies of scope in bank revenue function. They find that revenue economies of scope prove insignificant.

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The existing literature supports the conclusion that little, or unclear, evidence exists that commercial banks experience economies of scope. That is, the majority of studies find no, or unclear, evidence of such scope economies.

Efficiency Shortly after the search for economics and diseconomies of scale in banking began, another related line of research started on measuring the efficiency of bank production, using frontier-modeling techniques. Several authors develop the basic methods for measuring inefficiency in production (e.g., [84–86]), which involves deviations of actual from best-practice outcomes. That is, the best-practice frontier provides the benchmark of production against which the researcher measures actual production. Various mathematical programming and statistical techniques exist for the construction or estimation of the best-practice frontier. Two general categories of frontiers exist – deterministic and stochastic frontiers. Both techniques encompass the observed data, but in differing ways. Deterministic frontiers, once constructed, remain fixed in space and encompass all sample observations. Consequently, a small subset of data supports the frontier, making it more prone to sampling, outlier, and statistical-noise errors, which can distort the efficiency measure. Two different methods construct deterministic frontiers. Mathematical programming methods assume no statistical noise, an unreasonable assumption for large economic data sets [87], while the statistical techniques assert that random shocks, statistical noise, and firm-specific effects combined reflect inefficiency, also an unreasonable assumption [88]. Stochastic frontiers avoid some of the problems associated with deterministic frontiers by explicitly considering the stochastic properties of the data and distinguishing through a composite error term between firm-specific effects that relate to inefficiency and random shocks or statistical noise. Here, the frontier can shift from one observation to the next, being random rather than exact. Two types of stochastic frontier models exist – parametric and nonparametric. Parametric versions, on the one hand, own a longer record of development and accomplishment. Nonparametric versions, on the other hand, are newer approaches with research effort currently focusing on making the nonparametric methods more efficient [89]. Parametric stochastic frontiers confront researchers with other issues, however. First, researchers must adopt an explicit functional form to represent the production or cost function. Thus, researchers adopt flexible functional forms, such as the translog or the Fourier flexible functional form, to address such concerns to some extent. Second, researchers must make strong distributional assumptions on the onesided error term in cross section studies. Some evidence exists suggesting that the distributional assumptions do not exert a dramatic effect on the ranking of agents based on inefficiency estimates (e.g., [90, 91]). While the relative ranking of agents seems largely invariant based on inefficiency scores, the inefficiencies differ over alternative distributional assumptions on the one-sided error term, with “the single

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parameter models . . . providing a more pessimistic impression than warranted” [91, p. 158]. Berger and Humphrey [92] introduce the “thick-frontier” model, where the frontier depends on a group of “best-practice” firms and where weaker distributional assumptions support the estimation. For multiple-input/multiple-output models, however, this thick-frontier approach may prove problematic because the ordering criterion implies a different model from that estimated. Berger and Humphrey [93] survey 130 frontier efficiency studies of financial institutions in 21 countries. They evaluate the methods and estimates of the 130 papers, identifying similarities and differences in the modeling approaches and resulting efficiency estimates. Finding that differences existed in the results of these studies, they offer suggestions for improving the consistency, accuracy, and usefulness of future work. They conclude by recommending areas of future research. Sickles et al. [94] provide a summary of various methods to estimate productive efficiency for firms, sectors, and countries. They also provide references and internet links to data sources and R codes to implement the analyses. The authors describe the approaches used to estimate production frontiers using stochastic frontier analysis (SFA) and data envelopment analysis (DEA). The R codes estimate production efficiency with various methods, including “ . . . time invariant fixed effects, correlated random effects, and uncorrelated random effects panel stochastic frontier estimators, time varying fixed effects, correlated random effects, and uncorrelated random effects estimators, semi-parametric efficient panel frontier estimators, factor models for cross-sectional and time-varying efficiency, bootstrapping methods to develop confidence intervals for index number-based productivity estimates and their decompositions, DEA and Free Disposable Hull estimators” [94, p 267].

Production Efficiency At a most primitive level, we can consider the production efficiency of banks. That is, do banks produce the maximum output for given inputs or employ the minimum inputs for given outputs? The level of analysis focuses on the production function. Rangan et al. [95] compute a nonparametric frontier to estimate technical efficiency of a sample of 215 US banks. They find that banks could reduce their input use by 70% and still produce the same output, where most of this reflected pure technical inefficiency from wasted inputs rather than from scale inefficiency. Further, they show that larger banks exhibit more technical efficiency. Elyasiani and Mehdian [96] collect a random sample of 144 banks, including small (78) and large (66) banks, in 1985 and consider the total, technical, and scale efficiency of these banks. The analysis includes four inputs (labor, capital, deposits excluding large CDs, and large CDs) and one output (total revenue). They employ a deterministic statistical frontier method, imposing ray-homotheticity [97]. They shift the estimated production function by its intercept until no residuals are positive and at least one residual is zero. They conclude that larger banks prove more efficient and that much of the improved efficiency comes from

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improved scale efficiency, unlike Rangan et al. [95] who report the opposite finding. Elyasiani and Mehdian [21] also consider the determination of technical efficiency, using a nonparametric linear programming approach to calculating efficiency. They collect a sample of 191 large banks that match in 1980 and 1985. The production process includes four outputs (investments, real estate loans, commercial and industrial loans, and other loans) and four inputs (time and saving deposits plus CDs, demand deposits, capital, and labor). They conclude that significant technical progress occurred between 1980 and 1985. Miller and Noulas [98] consider the technical efficiency of 201 large banks (banks with assets over $1 billion) from 1984 to 1990, employing DEA analysis. They average the data for each bank over the 6 years of data. The analysis includes four inputs (total transactions deposits, total nontransactions deposits, total interest expense, and total noninterest expense) and six outputs (commercial and industrial loans, consumer loans, real estate loans, investments, total interest income, and total noninterest income). They conclude that large banks produced, with lowtechnical inefficiency, about 5%. Moreover, larger, more profitable banks exhibit lower-technical inefficiency. Elyasiani and Wang [99] examined the productive efficiency of Bank Holding Companies (BHC) in the United States. They calculate total factor productivity (TFP) using the Malmquist index, applying data envelopment analysis (DEA). They determined the change in TFP for their sample of BHCs using data from 1997 through 2007. The question focused on the effect of BHCs’ attempts to diversify their business across banking, securities, and insurance activities using Section 20 subsidiaries prior to the adoption of the Gramm-Leach-Bliley Act (GLBA) of 1999 and using financial holding companies after the GLBA. They concluded that technical efficiency correlated negatively with business diversification and that changes in diversification do not significantly affect TFP. The driving force behind their findings was whether BHCs experienced first mover advantage by employing Section 20 diversification prior to the adoption of the GLBA in 1999.

Cost Efficiency Initial work on bank efficiency concentrates on cost efficiency, which directly relates to the concept of economies and diseconomies of scale and scope. These costefficiency analyses generally report significant overall technical and/or allocative inefficiencies in commercial banking that generally decrease with bank size. These studies (e.g., [62, 70, 75, 76, 92, 100–102]), however, do not give much evidence across the range of banks of differing sizes, since they either exclude large commercial banks (i.e., banks with more than $1 billion in assets) or include a relatively small number of large banks in their samples. In other words, one cannot draw inferences about small-, medium-, and large-bank performance based on these studies. Berger and Humphrey [92], using the thick-frontier method, provide the exception, as they consider inefficiency for the sample of all commercial banks in 1984, finding, among other things, that differences in inefficiency dominate

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differences in scale and product mix when explaining differences in average cost across banks. Kaparakis et al. [103] extend the existing cost-efficiency literature in two directions. First, they produce inefficiency estimates for a large sample of commercial banks (i.e., 5548), including large banks with assets exceeding $1 billion, based on a flexible stochastic cost frontier. Thus, they directly compare the efficiencies of small, medium, and large banks, precluding unnecessary out-of-sample extrapolations. Second, they estimate the effects, if any, of external factors (e.g., state population density), enacted policies (e.g., state branching regulations), and managerial qualities (e.g., portfolio riskiness) on the variability of bank efficiency, finding that generally banks exhibit less efficiency with increasing size, reversing the finding noted above.

Revenue Efficiency As noted above, efficiency studies began with cost efficiency. Since the microeconomic theory of production begins with cost minimization and progresses to profit maximization, it makes sense to see a migration toward profit efficiency (see next section) as the next step from cost efficiency. Some researchers, however, take an intermediate step and consider revenue efficiency, where much of the work examines countries other than the United States. English et al. [104] employed a sample of 442 small banks that participated in the Federal Reserve’s Functional Cost Analysis program in 1982. They apply the output distance function to calculate output allocative and technical efficiency. The duality between the output distance function and the revenue function permits the derivation of the shadow prices of outputs for individual banks. Then the comparison of shadow and actual prices of outputs allows a determination of whether banks select an allocatively efficient output mix. They conclude that this sample of small banks exhibit significant technical inefficiencies and do not maximize revenue. Rogers [105] estimates cost, revenue, and profit efficiency using models that include and exclude a nontraditional measure of output – net noninterest income. He adopts stochastic translog specifications for the cost, revenue, and profit functions and uses a panel data sample of US commercial banks from 1991 to 1995 from the Sheshunoff, Ferguson and Co. Bank Source CD. He reports that cost and profit efficiency improve after including nontraditional output, whereas revenue efficiency declines. Profit Efficiency Applying the method from measuring cost efficiency to profit efficiency, Banker and Maindiratta [106] extend the approach of Farrell [85], who offers a procedure for calculating cost efficiency and its decomposition into technical and allocative components, generating multiplicative technical and allocative profit efficiency measures. Several authors measure profit efficiency by the ratio of actual to optimal profit, given input and output prices (e.g., [106–108]). Thus, the ratio of the firm’s actual profit to its input-oriented technically efficient profit measures technical efficiency, which creates several problems. First, prices affect the technical

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efficiency component in this multiplicative decomposition, which contrasts with the standard cost and revenue technical efficiency measures that do not depend on prices. Second, when the firm earns negative profit and optimal profit remains positive, then measured profit efficiency becomes negative. If optimal profit also becomes negative, then the profit efficiency ratio becomes positive, but greater than one. Banker and Maindiratta [106] note this latter problem and propose an alternative decomposition based on profit differences. In an earlier paper, Nerlove [109] measures profit efficiency as the difference between optimal and actual profit, which he calls lost profit. Nerlove [109] notes that this profit efficiency measure varies with proportional price changes. To address this issue, Chambers et al. [110] modify Nerlove [109] and normalize profit using the value of the reference input-output combination. Berger et al. [111] use the distribution-free method to consider profit inefficiency. They decompose profit inefficiency into technical and allocative components, using estimated shadow prices that can differ from actual prices. Technical inefficiency measures the movement from the actual production point to the frontier-based using the shadow prices; allocative inefficiency then measures the movement along the frontier from the shadow prices to the actual prices. They discover that large profit inefficiencies exist that include “half of all potential variable profit lost due to inefficiency” [p 328]. Technical inefficiencies dominate allocative inefficiencies, where the “technical components averaged between about two and five times as great as the allocative components.” [p 328] and that output inefficiency exceeds input inefficiency. Sahoo et al. [112] extend the work of Fukuyama and Weber [113], and Färe and Grosskopf [114] who likewise extend the work of Tone [115] on measuring cost and revenue efficiency as well as profit inefficiency. Sahoo et al. [112] develop new measures of directional cost- (DCE) and revenue-based (DRE) measures of efficiency that satisfy translational invariance and strong monotonicity. They then decompose the DCE and DRE measures into directional value-based technical (TE) and allocative (AE) efficiency components. Although this paper is primarily theoretical, the empirical application uses a sample of 50 US banks from 1996.

Nonperforming Loans and Bank Efficiency One strand of literature examines the effect of bad outcomes, typically nonperforming loans on bank efficiency. Two of the original papers in this literature are Berger and DeYoung [116] and Mester [117] that both use the stochastic frontier method of estimation. Berger and DeYoung [116] outline how nonperforming loans can affect bank efficiency measurement. They identify four channels of potential influence – “bad luck,” “bad management,” “skimping,” and “moral hazard.” The bad-luck hypothesis reflects external events to the bank. That is, the coronavirus leads to a series of bankruptcies for a number of a bank’s outstanding loans. The other three hypotheses all reflect internal events. The bad-management hypothesis refers to poor senior management practice in managing the portfolio of assets as well as the day-to-day operations of the bank. The skimping hypothesis captures the

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trade-off between allocating resources to the underwriting process and monitoring activities against future loan performance problems. That is, by reducing short-run costs of underwriting and monitoring loans will appear to improve bank efficiency, since the bank lowers costs for the same quantity of loans. But as time passes, the skimping activity will reveal itself in more nonperforming loans. The moral hazard hypothesis sees lower-capitalized banks taking on more risk. Berger and DeYoung [116] note that moral hazard does not directly affect back cost and efficiency, but it does contribute to nonperforming loans. Since the bad-luck hypothesis reflects external influences, researchers need to incorporate nonperforming loans into their estimates of cost, revenue, and profit efficiency. Doing so will eliminate the extra costs of dealing with nonperforming loans that reflect external factors and not managerial inefficiency. For the bad-management and skimping hypotheses, since they reflect internal influences, researchers should not incorporate nonperforming loans into their estimates of cost, revenue, and profit efficiency as this will tend to overstate efficiency as costs are artificially inflated. Berger and DeYoung find that bad luck and bad management both generate significant effects. Skimping also provides a significant effect, but its magnitude is smaller. Podpiera and Weill [118] reconsider the bad luck and bad management hypotheses in emerging markets, finding support for the bad management hypothesis. Koutsomanoli-Filippakia and Mamatzakis [119] slightly redefine the hypotheses as the “bad luck,” “bad management,” “skimping/moral hazard,” and “risk adverse” hypotheses, finding that most evidence supports the bad luck hypothesis with some evidence for the skimping/moral hazard hypothesis for a sample of EU banks. Further, Mamatzakis, et al. [120] follow the definitions in Koutsomanoli-Filippakia and Mamatzakis [119] and find evidence of skimping/moral hazard for bankrupt loans while evidence of bad luck for restructured loans, using a sample of Japanese banks.

Return and Risk Efficiency Implicitly, models of cost, revenue, and profit inefficiency assume risk-averse bank managers. That is, the decision rule maximizes or minimizes the expected outcome. No concern expresses itself about the level of risk in the bank’s operation. More recent research focuses on managing information asymmetries, adverse selection and moral hazard issues, and risk in addition to bank returns. This literature models the bank manager as maximizing utility, where the utility function includes expected return on bank operations and a measure of risk. Hughes and Mester [121] provide an overview of this literature. Hughes et al. [122–124] associate the risk of managerial decisions with the selection of production plans. Thus, the utility of management depends on profit and the choice of outputs and inputs (i.e., the production plan). A subjective probability distribution associates with each production plan, according to the beliefs of the managers about the relationship between future economic states and the production plans in the determination of profit. Thus, the managerial utility function of profit and its risk reflects how the production plans underlie the determination of profit and its risk. They employ the Almost Ideal Demand System (AIDS) developed

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by Deaton and Muellbauer [125], since this specification leads to the translog specification when bank managers are risk neutral. The frontier that emerges trades off expected return and risk, which represents the efficient frontier in portfolio theory. In a comment on Hughes et al. [122], Hunter [126] refers to their model as coming from the third generation. The first-generation models focused on the cost function of a single output that allowed the measurement of economies and diseconomies of scale. In second-generation models, banks produce multiple outputs and researchers employ modern duality theory to uncover measures of productive efficiency such as ray and expansion-path scale economies, economies of scope, subadditive cost functions, technological change, and measures of managerial inefficiency. Hughes et al. [122] develop a model that comes from the thirdgeneration and focuses on bank profit coupled with uncertainty and risk aversion. Hughes et al. [122] employ a sample of 443 bank-holding companies in 1994 whose assets ranged from $32.5 million to $249.7 billion. They calculate volatility of deposits as their risk measure. They find that bank managers do not operate in a risk-neutral manner and do not maximize profit. They also discover evidence supporting scale economies in all cases and significant inefficiency of operation. Finally, they conclude that more geographic and/or depositor diversification increases the bank’s expected return and that more branches move the institution toward the frontier reducing inefficiency. Färe et al. [127] use the directional distance function to derive profit inefficiency, normalizing by revenue plus cost. Technical efficiency improves with bank size, and profit inefficiency relates more to allocative than technical sources. They find that profit inefficiency and its allocative part decrease with bank size. They use 1000 banks ranging from $8 million to $150 billion in assets and 114 banks above $300 million in assets. A major conclusion from this literature relates back to the issue of economies of scale in banking. The history of measuring economies of scale led to different outcomes, depending on the range of bank sizes in the sample considered. That is, as researchers used samples of larger and larger banks, the range of economies of scale also expanded too much larger sizes. This new literature on estimating the expected return and risk frontier in banking also permits the estimation of economies of scale. Hughes et al. [128] determine that economies of scale depend on how the researcher models capital structure and risk taking. Larger banks associate with economies of scale that increase with bank size along the value-maximizing expansion path rather than the cost-minimizing path. In addition, lower risk through diversification associates with larger-scale economies. Hughes and Mester [129] argue that recent advances in the modeling of bank decision-making that accounts for the endogeneity of risk in the model affect the production, revenue, cost, and profit of banking institutions. Thus, for example, unless one accounts for the endogenous nature of risk taking in banking, one will mis-measure economies of scale. They state that banking studies of large banks that ignore the role of risk do not find evidence of economies of scale. Those studies of

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large banks, however, that do incorporate a role for endogenous risk find evidence of substantial economies of scale for large banks. Hughes and Mester [129] also report that modeling bank decision-making with endogenous risk leads to performance at large banks that may threaten financial stability. As such, this finding implies an important role for capital regulation of large financial institutions.

Bank Failure Bank failures generally occur during economic distress. Some studies link bank operating and capital characteristics and the likelihood of failure. Bank failures destroy banking relationships, lead to reductions in small business lending, and impair the public’s trust in the banking system. Bank failures prove costly to the US banking system, and during economic downturns bank failures can lead to further declines in output [130]. Bank failure prediction models provide regulators with a tool that aids in preventing bank closures and, thus, reduces the adverse effect of bank failure on communities. Jagtiani et al. [131] examine the Early Warning System (EWS) models used to predict problem banks in the Federal Reserve and the Comptroller of the Currency.

Economic Effects of Bank Failures Most bank failures occur during periods of economic decline. In his review of the aggregate banking data, Ashcraft [130] finds a statistically and economically significant relationship between bank failures and real declines in county income. Boyd et al. [132] find that bank failures lead to prolonged, significant declines in real GDP growth, especially for less developed countries. They conclude that the effect of failed banks on output may persist, be longer than previously thought, and in some cases be long-term for some local areas. Dell’Ariccia et al. [133] control for economic contractions and currency crises in their study of the effect of banking failures on real economic variables. They find that banking crises cause declines in real economic growth because of less lending. Driscoll [134] uses an instrumental variable approach, however, to determine that banking shocks do not have a statistically significant relationship with real output at the macroeconomic level. Early Warning System Models Bank failure prediction models, also called early warning systems, use a variety of econometric models. Crowley and Loviscek [135] review four methods by using annual data to determine which model would most accurately predict the failure of small banks in 1984. They find that logit and probit models deliver higher accuracy than linear probability and discriminant analysis models. Jagtiani et al. [131] also compare logit models to nonparametric models. Simple logit estimation techniques

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prove as valuable as more complex logit and nonparametric models in predicting future bank failures. Kolari et al. [136] use logit analysis to predict correctly large US commercial bank failures at 96% accuracy 1 year prior to failure, and 95% accuracy 2 years prior to failure. Their bank sample included quarterly data on 50 large banks with assets greater than $250 million that failed between 1989 and 1992. The authors also employ a trait recognition model with multiple variable interactions. Jagtiani et al. [131] develop a logit model with a primary focus on capital ratios. The basic logit model with two capital ratios appeared to predict capital inadequacy as accurately as more complex computer-based systems, especially 1 year prior to failure. Bank capital became a critical component of bank examinations during the savings and loan (S&L) crisis, specifically with the advent of the Basel I Accord in 1988, which established minimum capital requirements for banks. Researchers consider other determinants of bank failure, including Moody’s debt ratings, complex accounting ratios, and bond spreads. Jagtiani and Lemieux [137] find that bond spreads can signal trouble in the financial markets beginning up to six quarters before a failure. Bond spreads for banks that will probably fail exceed the bond spreads for healthy banks. Henebry [138] uses cash-flow spreads to develop a bank failure prediction model based on bank failures from 1986 to 1990. Jeon and Miller [37] discover from a failure regression that deregulation did not significantly affect bank failures and banks with high noninterest income were more likely to fail. DeYoung [139] uses hazard function analysis of annual data to model the likelihood of failure for new commercial banks. The author considers two major issues while shaping the model: banking conditions in the metropolitan market and different legal policies across states. Bank failure studies inform policy makers and federal regulators (e.g., [135, 140, 141]). Bank failures receive significant attention in the literature because bank failures are costly and compromising to the safety and soundness of the banking system. Early warning systems have been developed to predict the likelihood and timing of bank failures, and several variations have been evaluated to improve the accuracy and predictive power of these models.

Conclusion This chapter considers various applications of production economics to the banking sector. The chapter first considers the issue of how to measure output. Two basic approaches exist – the production and intermediation specifications. The treatment of deposits differentiates these two specifications, whereby the production approach takes deposits as an output and the intermediation approach takes deposits as an input. Then, this chapter proceeds to discuss various issues in bank production – bank productivity growth, bank and banking industry profitability, economies of scale and scope in banking, and bank efficiency. Bank efficiency includes efficiency as measured by the production function, the cost function, the revenue function,

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the profit function, and the efficient frontier between expected return and risk. The chapter concludes with an analysis of predicting problem banks and/or bank failures.

Cross-References  Bad Outputs  Cost, Revenue, and Profit Function Estimates  Data Envelopment Analysis: A Nonparametric Method of Production Analysis  Distance Functions in Production Economics  Index Numbers and Productivity Measurement  Stochastic Frontier Analysis: Foundations and Advances I  Stochastic Frontier Analysis: Foundations and Advances II

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Applications of Production Economics in Education

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Jill Johnes

Contents Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Cost Functions and Economies of Scale and Scope . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Background on Cost Concepts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Estimating Cost Functions in Education and Higher Education: Challenges and Methodology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Estimation Approach: SFA Versus DEA . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Findings from the Literature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Recent Developments in Estimating Cost Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Policy Implications and Future Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Production Functions, Distance Function, Shadow Prices, and Elasticities . . . . . . . . . . . . . . Background on Production Concepts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Estimating Distance Functions in Education and Higher Education: Challenges and Methodology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Findings from the Literature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Policy Implications and Future Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Efficiency, Productivity Change, and Analyses of Factors Underlying Efficiency . . . . . . . . Background on Efficiency Concepts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Findings from the Literature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Recent Developments in Efficiency Measurement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Policy Implications and Future Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Level of Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Individual-Level Analyses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Funding Area Analyses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . National-Level Analyses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Cross-References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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J. Johnes () Huddersfield Business School, University of Huddersfield, Huddersfield, UK e-mail: [email protected] © Springer Nature Singapore Pte Ltd. 2022 S. C. Ray et al. (eds.), Handbook of Production Economics, https://doi.org/10.1007/978-981-10-3455-8_30

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Abstract

This chapter provides a comprehensive survey of the existing literature on production economics from the education perspective, bringing together findings from the education costs, production, and efficiency contexts and relating to all levels of education including primary, secondary (both compulsory and noncompulsory), and higher education. Keywords

Education · Higher education · Costs · Production · Economies · Elasticities · Efficiency JEL Classification Numbers

D24, I20, I21, I23, L25

Introduction This chapter focuses on production economics in the context of the education sector. This sector makes an interesting case study because of its particular characteristics, which derive from the fact that returns to education can be both private (accruing to the individual in terms of higher salaries) and social (accruing to society in terms of increased productivity and economic growth). As a consequence of the benefits accruing to society as a whole from individuals being educated, education institutions are often publicly funded, although the extent of the public funding likely varies by level of education (and country). The public funding aspect of education affects costs, production, and efficiency in that sector, and these are all relevant in the production economics context. It is worth considering two broad components of education which we will term (a) higher or tertiary education, encompassing non-compulsory education for post18-year-olds often in universities, and (b) education, encompassing primary and secondary education. The latter is largely compulsory, at least up to the age of around 16 years, particularly in developed countries, and predominantly publicly funded. The former is not compulsory, but is also in receipt of substantial public funding, particularly in developed countries, since there are still considered to be some benefits of higher education accruing to society (as well as the individual). In addition to these categories, education can be provided at many levels to adults (typically 25 years plus). Adult education is provided very differently from the traditional primary, secondary, and higher education levels, with provision often being in the form of modules offered through a blended or distance medium [114]. Given the frequent lack of data on adult education provision, there will be only little reference to this particular sector in the paper below. To illustrate the importance of education, in 2015, across all OECD countries, the average spending on education institutions across the spectrum of education

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levels is some 5% of GDP, with a variation from around 3% to 6%. On average in the OECD, the majority of around 70% of the spending on education is to non-tertiary education institutions (as might be expected), and this is equivalent to 3.5% of GDP, with a variation of 3% to 4.5% (see Education at a Glance 2018 https://read.oecd-ilibrary.org/education/education-at-a-glance-2018_ eag-2018-en#page260, accessed June 14th 2019). The largely publicly funded nature of organizations in the education and higher education sectors therefore makes this an interesting focus in the context of production economics. Such organizations are generally non-profit making. Yet the amount of public funds received by schools and universities and the role these institutions play in driving growth in the economy make it imperative for them to be run efficiently and effectively. An empirical knowledge of concepts in production such as the size of economies of scale or scope, efficiency levels, and possibilities for substitution between inputs (or, indeed, between outputs) are all important in the education context. While the education and higher education sectors of many countries comprise largely publicly funded institutions, privately funded institutions also exist, to a greater or lesser extent, at all levels of education. The focus of this chapter is generally on the non-profit, largely publicly funded provision, but private sector examples will be reviewed as appropriate. This chapter is in six sections of which this introduction is the first. Section “Cost functions and economies of scale and scope” focuses on cost functions in education including concepts, estimation, and findings from the literature. The section ends with some recent developments, policy implications, and suggestions for future work. Output distance functions are the subject of Section “Production functions, distance function, shadow prices, and elasticities”, which examines concepts, estimation, and findings from the literature before concluding with policy implications and possible topics for future exploration. Section “Efficiency, productivity change, and analyses of factors underlying efficiency” turns to efficiency and productivity change including concepts, findings from the literature, recent developments, policy implications, and future work. Level of analysis is the focus of Section “Level of analysis” which examines and reviews the literature on various possibilities including individuals, funding areas, and countries. Final conclusions are drawn in Section “Conclusions”, which also suggests areas for future applications of production economics in the education and higher education contexts.

Cost Functions and Economies of Scale and Scope Schools and higher education institutions (HEIs) are multi-product organizations. School pupils, for example, are taught and attain qualifications in multiple subjects in schools; universities produce outcomes from teaching, research, and third mission activities. This leads to the estimation of multi-product cost functions [37] in the education and higher education contexts and permits the testing of a number of key production economics concepts such as:

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• Existence or otherwise of economies of scale • Existence or otherwise of economies of scope • Extent of substitution possibilities through evaluating elasticities of substitution Full details on the theory underpinning cost functions and economies of scale can be found in Chaps. 16 and 17 (respectively) of Volume I of this publication. Each of these concepts will be considered briefly in the context of the empirical literature in this section.

Background on Cost Concepts In a multi-product production situation such as we have in education and higher education, the cost relationship is C(y) = f (y; p)

(1)

where y is the vector of outputs p is the vector of input prices In order to estimate this function empirically, the researcher must select a functional form which should: • Be consistent with cost minimization given outputs and input costs, i.e., it must be a non-negative and non-decreasing function. • Provide predictions of costs when the value of one or more outputs is zero. This is particularly needed in order to derive estimates of economies of scale and scope and precludes cost functions in logarithms such as the Cobb-Douglas. • Allow for the existence of scale or scope economies or diseconomies, without enforcing their existence. Functional forms which fulfill these criteria and which have been used in empirical studies include the cross elasticity of substitution, quadratic, and hybrid translog. Each has been used in empirical studies and has various advantages and disadvantages in terms of estimation, a brief overview of which can be found in Johnes et al. [195], while a detailed comparison of the merits of the translog over the Cobb-Douglas can be found in Gronberg et al. [141]. In this multi-product case, there are two concepts relating to economies of scale [181]. Ray economies of scale are the savings in costs occurring when all outputs increase (while holding the output mix constant). Product-specific economies of scale are the cost savings which occur when one output increases and all other outputs remain at fixed production levels [197]. If we assume that we have k inputs (k = 1, . . . , K) and m outputs (m = 1, . . . , M), these concepts can be denoted for the general case as follows:

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SR = 

C(y) m ym Cm (y)

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(2)

where SR represents ray economies of scale ym is the mth output Cm (y) = ∂C(y)/∂ym is the marginal cost of producing the mth output Sm (y) = AI C (ym ) /Cm (y)

(3)

where Sm (y) denotes product-specific economies relating to product m (where m = 1, . . . , M) AIC(ym ) = [C(yM ) − C(yM − m )]/ym C(yM ) is the total cost of producing all M outputs C(yM − m ) is the total cost of producing all M outputs except output m Values above (below) 1 indicate the presence of economies (diseconomies) of scale in the estimated long-run cost equation. Evaluating these measures can be useful from a policy viewpoint in determining, for example, whether an expansion in provision is best effected through increasing the size of existing providers (schools or universities) or if diseconomies of scale are observed in the sector by introducing entirely new providers. Economies of scope, in contrast, occur when it is less costly to produce a number of outputs together rather than to produce each output independently in its own specialist production unit [181]. As with economies of scale, in this multi-product case, we have two concepts relating to economies of scope. Global economies of scope occur when the costs of producing all outputs together in a single firm are less than the sum of the costs of producing each output in a separate firm. Productspecific economies of scope for product m arise when the costs of producing all outputs together in a single firm are less than the sum of costs of producing output m in a separate firm and all outputs apart from m in another firm [197]. These can be denoted in the general case as follows:    SG = C (ym ) − C(y) /C(y) (4) m

where SG denotes global economies of scope C(ym ) is the cost of producing output m SC m = [C (ym ) + C (yM−m ) − C(y)] /C(y) where: SCm denotes product-specific economics of scope for output m

(5)

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Economies of scope can arise if it is possible to spread the costs of central services across an array of outputs. It is likely that both schools and HEIs benefit from producing their outputs in one production unit as they may be able to spread the costs of capital and administration across their different outputs whether it is teaching across different disciplines (schools and universities) or teaching and research (universities). The degree of scope economies in universities depends on the extent to which the products (e.g., research and teaching) are produced jointly as opposed to separately, and this issue is considered further below in the context of the empirical literature. The empirical evaluation of these measures can provide useful policy and managerial insights into the degree to which organizations should become more (or less) specialized in the outputs produced. In the higher education case, for example, economies of scope can indicate whether HEIs should be research-focused or teaching-focused or even whether they should specialize in a specific discipline (such as arts or medicine).

Estimating Cost Functions in Education and Higher Education: Challenges and Methodology Knowing the parameters of the estimated cost function in an education context can clearly offer useful insights to managers and policy-makers alike. But the implementation of the cost function methodology in education is not easy, and decisions regarding specification (e.g., of costs, outputs, and functional form) and estimation approach in particular can potentially affect outcomes and conclusions drawn from any cost function analysis. The first major challenge in estimating education cost functions is identifying what is meant by “costs.” Costs (or expenditure) can be allocated to various categories. For universities these might include administration versus academic expenditure; research versus teaching expenditures; and recurrent versus capital expenditures. For schools these might be instructional and noninstructional expenditures or total fee revenue. Many empirical studies are interested in total recurrent expenditure, and this is the typical definition of costs. There are, however, exceptions. Some studies, for example, have focused specifically on administration (rather than total recurrent) costs in the context of universities [64, 84]. When estimating cost functions, there is an underlying assumption that education providers are seeking to minimize their costs. Given that such organizations are typically in receipt of public funds, this assumption is open to debate. Indeed, an early examination of costs of universities in the USA suggests that universities do not minimize costs but rather spend all the income they receive [50]. This view is challenged [225] on the premise that providers with diverse sources of funding (such as universities) are more likely to adopt optimizing behavior than when receiving all funding from the public purse. More recently, the marketization of, and increasing competition in, higher education sectors across the world following the global financial crash and subsequent constraints on public funding have put increasing pressure on higher education providers to minimize their costs. Similar pressures can also be seen in the education sector where policies to increase competition

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among publicly funded schools in some countries would also lead to increasing cost minimization behavior. While the outputs of schools and HEIs may seem obvious, their precise measurement is not so clear-cut. Schools often use test or examination performance or graduate numbers (by subject) to reflect teaching outputs and their quality [34, 35, 41, 51, 59, 69, 109, 152, 165, 229, 233, 234, 239, 277, 278]. Where this is not available, student enrolment numbers might be substituted [31, 59, 62, 166, 167, 218, 230, 268, 271, 272, 294, 295, 326]. But the disadvantage of enrolment figures is that they fail to reflect output quality. Since schools’ outputs are affected by their environment, the background of the pupils who attend the school, and the quality of teachers, as well as other contextual variables relating to pupils, families, school, or the school location are often added to the cost equation to take these factors into account [33, 41, 51, 69, 107, 109, 141, 142, 152, 165, 167, 327]. Universities produce outputs which can be categorized as teaching, research, or third mission. Student numbers are commonly used to reflect teaching outputs [3, 18, 28, 39, 40, 158, 168, 183, 195, 197, 235, 249, 297, 301, 320], often categorized by level (e.g., undergraduate or postgraduate) and broad disciplines (such as science, non-science, and medicine), but various problems arise not least of which is the issue of quality of teaching output. Graduate numbers have been used in preference to student numbers in order to try to capture quality [27], but this ignores the quality of degrees obtained by different graduates. Quality is addressed in various ways including adding variables to reflect “quality” to the cost equation such as average entry qualifications of intake or a value-added measure [195, 197, 297, 309]. These outputs can be seen as the short-term outcomes of higher education. Longterm benefits from taking a higher degree might be measured using labor market metrics such as numbers of graduates achieving a job or graduates’ starting salary [7, 47, 176, 212, 222]. Measuring research is also problematic. Nationally organized research rating exercises (such as the Research Excellence Framework in the UK) have measures of both quantity and quality [136], but these are available only at intervals and therefore not always a reflection of current position. Citation and publication counts can also be used (as in, e.g., [101, 182, 248]) but can be difficult to obtain and may not reflect the current output. As a consequence, many studies resort to input measures, such as competitively won grant income, rather than output measures, in an effort to capture quality and quantity of current activity. Outcomes from third mission activities are the most difficult to measure in the higher education context with many empirical studies not even attempting it, although its omission will inevitably lead to problems of bias in the estimated cost function (some exceptions include [183, 195, 197]; and [301], where university income from other services rendered is included to reflect third mission activities). Input prices should also be included in the cost function if these vary across production units, with many studies including the price of capital and/or the price of labor [86, 134, 135, 226]. Average salary is commonly used to reflect the price of labor, but this, of course, may be more of a reflection of the distribution of staff across grades in their organization than an indication of the price of labor. If prices are not known precisely but are known to vary by broad geographical location of the

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university, a location indicator can be incorporated in the function as a proxy [183, 195, 197, 301]. Capturing price variations is typically likely to be more important in inter-country rather than within-country studies. Different organizations have different missions or objectives. In the higher education sector, some universities might choose to focus more on, for example, research as opposed to teaching, and vice versa. In the education context, some countries distinguish between schools with vocational as opposed to academic routes. An underlying assumption when estimating a cost function for an industry is that the firms within it all have similar objectives [131]. If this assumption does not hold, then cost functions might be estimated separately for different mission groups [86, 183, 195, 197, 301, 323, 325]. Recent advances in estimation techniques also provide alternative approaches for this situation (see Section “Recent developments in estimating cost functions”). A full and detailed examination of all these issues can be found in Johnes et al. [195]. In practice, the translog cost functional form provides further opportunities for estimating additional quantities of interest, namely, elasticities of substitution. We can write the translog cost function as [152] ln C = δ0 +

M 

δm ln ym +

m=1

+

K  k=1

M M 1  δmn ln ym ln yn 2 m=1 n=1

 1  μkl ln Pk ln Pl + γmk ln ym ln Pk + ε 2 K

μk ln Pk +

K

M

k=1 l=1

K

m=1 k=1

(6) We assume the following conditions: (a) Linear homogeneity of degree +1 in input prices K 

μk = 1

(7a)

k=1 K  l=1

μkl =

K 

γmk = 0

(7b)

k=1

(b) Symmetry δmn = δnm

(8a)

μkl = μlk

(8b)

We can derive a set of input share equations (Sk ) from Eq. (6) as follows:

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  ∂ ln C = Sk = μk + μkl ln Pl + γmk ln ym ∂ ln Pk K

M

l=1

m=1

(9)

Two possible measures of elasticity are the Allen and Morishima elasticities of substitution, respectively, which can be estimated from this set of equations. The Allen elasticity of substitution measures the impact of a change in the price of the kth input on the demand for the lth input with output held constant. This is estimated in this cost function context (denoted by superscript C) by AC kl = (μkl + Sk Sl ) /Sk Sl for k = l

(10a)

2 AC kl = (μkl + Sk (Sk − 1)) /Sk for k = l

(10b)

A positive (negative) value suggests that inputs k and l are substitutes (complements). The Morishima elasticity of substitution can be estimated in this cost function context (denoted by superscript C) by:   C C Mkl = Sl AC − A kl ll

(11)

The two measures are different when there are more than two inputs and the production technology is represented by the translog as here [152]. The Allen elasticities are symmetric, whereas the Morishima elasticities are not. As such, the Allen and Morishima elasticities of substitution may not provide consistent conclusions (see [152] for more details).

Estimation Approach: SFA Versus DEA The idea that production might vary by mission group leads on to a more general concern regarding production and technology and their implications for estimating empirical cost functions. Clearly concepts such as scale economies require estimation of a long-run cost function. The challenge for empirical researchers is that they must assume that their sample of production units (schools or HEIs) is operating on the long-run cost function in the time period under study [131]. In reality, this is unlikely to be the case, and the observations will be a mix of those operating in a long-run equilibrium, those operating in a short-run equilibrium, and those in either a short-run or a long-run position not operating efficiently. This might be a particularly pertinent consideration if the sector under study is going through a period of rapid change, in which case organizations may be in various short-run equilibria as they move toward their long-run positions [57]. Some of the earliest empirical cost functions in higher education are estimated using ordinary least squares (OLS) and a linear functional form [309, 310]; the latter therefore largely precludes the existence of economies of scale and scope, and the

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former simply estimates a line of best fit through all the data regardless of position (short run versus long run; efficient versus inefficient). The seminal work of Cohn et al. [86] incorporates the multi-product nature of production drawing on the work of Baumol et al. [37], but the estimation method (OLS applied to cross-sectional data) still does not address the issue of observations at different production points. This is also the case for many subsequent studies (see Section “Findings from the literature”). Early school cost functions use OLS and a quadratic function, permitting estimation of optimum size and scale economies, but precluding economies of scope [41, 60, 254]. The multi-product nature of school production is recognized in later studies by using a translog functional form, with applications in Bolivia, the USA, and Flanders [62, 166, 295]. But the estimation methods do not allow for inefficiencies or data points being at different production points (short run versus long run). There has been a growing recognition that using methods which preclude the possibility of inefficiency is a problem [95], especially as the non-profit nature of education does not naturally provide the incentives for efficiency which prevail in a private sector setting. The more widespread availability of frontier estimation methods which allow for inefficient operations, however, has led to frontier estimation methods increasingly being the customary approach when estimating empirical cost functions in education. A cost function estimated using frontier techniques envelopes the data; thus its position is determined by the outermost data points which are, in turn, likely to be those in a long-run equilibrium and/or most efficient in the sector. Frontier estimation techniques can be parametric, such as the family of estimation methods falling under the umbrella of stochastic frontier analysis (SFA) [20], or non-parametric, with data envelopment analysis (DEA) being a common approach in this context [74, 75]. Full details of SFA and DEA can be found in Chaps. 12, 13 and 14 of Volume I of this publication. SFA assumes an error comprising two components – one a normally distributed random error and the other a one-sided term, often following a half-normal or exponential distribution and attributed to inefficiency. The basic SFA model produces estimates of the cost function parameters which are identical for all organizations in the data set. The advantage of the approach is that the significance of the parameters can be tested [85, 283], and they can be used to produce estimates of economies of scale and scope as well as elasticities of substitution. As a consequence, SFA has been used in many empirical cost function studies (see Section “Findings from the literature”). DEA is a non-parametric approach often used to derive estimates of organizations’ efficiencies and makes no assumptions regarding functional form. This means that there are no problems with misspecification. Moreover, the linear programming method of estimation means that DEA easily accommodates multiple inputs and multiple outputs [229, 233]. Furthermore, it provides weights of inputs and outputs which vary by each organization (or decision-making unit – DMU) in the sample and which maximize the efficiency score subject to weights being

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positive and universal [84]. The inter-institutional variation in weights can be particularly advantageous in the context of education and higher education where we have noted that mission can vary by production unit. DEA can also provide useful benchmarking information for managers to help them improve performance. But estimates of economies of scale and scope, and of elasticities of substitution, are more difficult to derive in the non-parametric context (see [301] for an example). In the context of cost functions where there is a single input (expenditure) and multiple outputs, a parametric technique is typically preferred over a non-parametric one, although there are exceptions as will be discussed in Section “Findings from the literature”. As already noted, however, expenditure itself can be divided into different categories, thereby leading to a multi-input multi-output situation. DEA can therefore be advantageous where the underlying cost components are known and of interest [18, 19, 27, 39, 40].

Findings from the Literature We will focus in this section largely on literature which incorporates the multiproduct nature of education into the estimated cost functions [37]. This literature can be divided into parametric and non-parametric studies, and the former can be further divided into those using frontier estimation methods and those not doing so.

Parametric, Non-frontier Estimation The school context is complex with some studies examining scale at a funding area level (such as school districts in the USA or local education authority in the UK), others looking at the school level, and others still considering both. There is evidence of scale economies in funding areas [22, 60] and in funding areas up to a certain size [109, 327]; and a study of only large school districts finds no evidence of scale economies [273]. There is also evidence of scale economies in schools [41, 51, 107, 166]. When the two are considered together (schools and funding districts), economies of scale are found in schools [224] and in both schools and school funding areas [70]. In a rare study of secondary schools which incorporates multiple outputs (relating to education clusters), ray economies of scale are observed along with product-specific economies in six of the seven fields considered and also global economies of scope [295]. Economies of scale are confirmed in a study of school districts which are assumed to produce two outputs, namely, primary and secondary education, but in this case there are no economies of scope [62], and this aligns with findings from an earlier multi-product cost study by Jimenez [166] . In the context of higher education, and across various developed countries, cost functions estimated using parametric non-frontier estimation methods tend to find that there are ray economies of scale but that the evidence on economies of scope is more mixed [86, 99, 110, 134, 135, 213–215, 280, 319].

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Parametric, Frontier Estimation Much of the literature on estimates of scale economies in the context of schools is based on non-frontier estimation. There are some exceptions where both nonfrontier and frontier methods are applied [109], and these find that the coefficients for the frontier model are similar to those estimated using non-frontier methods. More recently, Gronberg et al. [143] use SFA in the school context to investigate the economies deriving from consolidation of school districts in Texas. They find that economies of scale can be gained from consolidating very small school districts (producing primary and secondary education). Other studies which use frontier methods have largely focused on efficiency aspects rather than scale economies, and so will be reviewed in Section “Efficiency, productivity change, and analyses of factors underlying efficiency”. There are many studies of economies of scale and scope in higher education using parametric, frontier estimation methods to estimate the multi-product cost function (e.g., [3, 10, 13, 164, 169, 183–185, 195, 197, 249, 297, 301]). While there is some variation in findings, typically the studies using these methods find that ray economies of scale are exhausted; an exception relates to Japanese private universities where many HEIs enjoy economies of scale [249]. Some also find there are product-specific scale economies relating to research and/or postgraduate outputs [14, 15, 187, 188, 249]. Most of the evidence on scope economies, however, points to diseconomies of scope both globally and (where calculated) for individual products [164, 170, 183, 195, 197, 249]. Non-parametric Frontier Estimation The need for cost function parameter estimates with which to derive measures of scale and scope economies means that cost function studies using non-parametric frontier estimation methods typically examine efficiency rather than the specific cost function concepts discussed above. As such, a review of these studies will be presented in Section “Efficiency, productivity change, and analyses of factors underlying efficiency”. An exception is Thanassoulis et al. [301] who examine costs in English higher education and find opportunities for expanding student numbers are possible through currently unexploited scale and scope economies [301, 302]. Summary Aside from the fact that the findings provide some mixed messages, there are some additional caveats. In the context of schools, most studies do not include externality costs of increasing school size. As a school increases its size, for example, student discipline issues increasingly arise, and the crime and violence which this may engender impose external costs on pupils, families, and society more generally, which are not taken into account in a standard school cost function [122]. This suggests that care should be taken when interpreting the results of standard cost functions. An area of future research which would be particularly useful to managers and policy-makers alike would therefore revolve around developing empirical cost functions for schools which incorporate these externalities.

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Another caveat is that if recommendations regarding scale economies are derived from cost functions which inadequately measure quality of outputs, there may be a detrimental effect on pupil outcomes from increasing school size. The relationship between size and outcomes can be examined more closely using a production function approach, and we will consider this further in Section “Production functions, distance function, shadow prices, and elasticities”, while the issue of increasing school size and pupil outcomes is investigated in depth by Schiltz and De Witte [281]. Caveats regarding inadequate measurement of quality apply equally in the higher education context.

Recent Developments in Estimating Cost Functions Education providers, whether at primary, secondary, or tertiary levels, can vary widely in terms of, for example, their mission, size, and history. Such diversity, if not taken into account, can potentially affect parameter estimates and hence the estimated economies of scale and scope. Inclusion in the cost function of contextual variables to reflect defined characteristics, such as mission or region, is one approach to addressing the diversity issue [323, 325]; another approach is to estimate cost functions within pre-defined groups based on perceptions about what characteristics ought to affect cost function parameters. These characteristics might be type of institution such as public or private [86, 165], or mission group [183, 195, 197, 301]. Such an approach can identify differences in estimated parameters across the specified groups. Thus, known characteristics of institutions affect costs, but there may also be unobserved characteristics which also affect costs. Random parameter (RP) SFA [140, 305] and latent class (LC) SFA [221, 253] allow both observable and unobservable characteristics to be taken into account in the estimation of parameters and efficiency scores – explored further in Section “Efficiency, productivity change, and analyses of factors underlying efficiency” [185]. In particular, RP SFA, which requires panel data for estimation, allows parameters to vary by each individual provider, while LC SFA, which can be applied to cross-sectional data, permits parameters to be derived for groups of HEIs – however the groups are not predefined by the analyst, but rather they are determined by the data. These methods not only lead to different parameters across (groups of) institutions but also allow the calculation of scale and scope economies by individual provider or by group. Studies adopting RP SFA have found evidence that ray economies of scale are typically exhausted or decreasing and diseconomies of scope are observed. There are some product-specific economies of scale, but these vary from study to study [14, 15, 183, 187, 188, 197]. When LC SFA methods are used, the findings on economies of scale and scope vary from group to group [184, 185]. While the LC approach is attractive in defining groups based on the data, the disadvantage is that the results can be difficult to interpret if the composition of the resulting groups does not align with any obvious patterns.

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Policy Implications and Future Work Empirical estimations regarding economies of scale and scope whether at school or university level are policy relevant as they can feed into considerations regarding potential consolidation – this might be at funding unit level (such as districts) or at organization level. Thus policies to amalgamate schools or to merge universities can be developed from such empirical work. However, the findings reported above at both school and university levels are often mixed or conflicting, and it is difficult to develop coherent policies on such a basis. Future work which examines the reasons why findings on economies of scale and scope vary by type of organization might therefore help to better inform policy [159]. A useful contribution to the literature undertakes a meta-analysis of cost function studies to identify reasons for the mixed findings in the higher education context [324]. It seems that estimates of scale efficiency vary according to model specification and functional form assumed and whether or not managerial efficiency is taken into account (a quadratic cost function in particular seems to lead to a conclusion of diseconomies of scale, as does a model which accounts for inefficiency). Estimates of scope efficiency are affected not just by model specification but also by period covered by the study, sample size, and type of data. In particular, estimates derived from older, cross-sectional data from small or developing country samples of universities are likely to lead to the conclusion that scope economies exist [324]. At school level, a meta-regression analysis of optimum school size based on 10 studies with 22 estimates finds the optimum school size to be around 1543 pupils [88]. These studies provide useful insights, and much more work of this kind would be welcome, particularly at the school level. A less explored area of research concerns the derivation of elasticities from parametric cost functions. Both Allen and Morishima elasticities (defined in Section “Estimating cost functions in education and higher education: Challenges and methodology”) can provide useful details about substitution possibilities between inputs. One example in higher education can be found [318] and suggests that within universities it is easier to switch into capital inputs than into academic or nonacademic labor; indeed the substitution possibilities between the two types of labor seem limited. Examples from the school context suggest that instructional, support, and administrative inputs are generally substitutable [62, 152, 166], although Allen elasticities imply less ability to substitute between the non-instructional input and others [152]. Much more work could be undertaken in this context to provide useful policy insights.

Production Functions, Distance Function, Shadow Prices, and Elasticities Concepts relating to the multi-product nature of education and higher education can also be examined in a production function context. Given both the multi-input and multi-output nature of production, estimation of production-related concepts leads

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to the output distance function approach which has numerous advantages: it does not assume any particular optimizing behavior on the part of the firms, which is an advantage in the non-profit context in which schools and universities often operate; it does not require a knowledge of prices of either inputs or outputs, the latter being particularly useful in education and higher education where teaching outputs, for example, are difficult to value; and it does not require prices to be exogenous [81, 82, 308]. In the context of education and higher education, the concepts of interest in the production setting include: • Existence or otherwise of returns to scale • Existence or otherwise of returns to scope • Extent of substitution possibilities in the production relationship through evaluation of elasticities of substitution (between inputs) • Extent of complementarity or substitutability between the outputs through evaluation of elasticities of substitution (between outputs) Full details of the theory underpinning production and the related concepts (including elasticities) can be found in Chaps. 3 and 22 of Volume I of this publication. Each of these concepts will be considered briefly in the context of the empirical literature in this section.

Background on Production Concepts We assume that schools or universities produce multiple outputs from a variety of inputs. Let us assume, as in Section “Cost functions and economies of scale and scope”, that providers – be they schools or HEIs – use a vector of inputs x ∈ RK + to produce a vector of outputsy ∈ RM + . We assume that providers focus on producing outputs relative to given inputs (an output-oriented approach) and hence define the production technology for a provider as  P (x) = y ∈ RM + : x can produce y

(12)

where y is already defined x is the vector of inputs The output distance function [287], denoted by D(x, y), is defined on the output set P(x) as D (x, y) = minθ {θ : (y/θ ) ∈ P (x)}

(13)

The output distance function is non-decreasing, convex, and positively linearly homogeneous of degree +1 and can be used to derive shadow prices and substitution properties. We define shadow prices of inputs as

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(14)

∂D (x, y) /∂xk

The marginal rate of technical substitution between inputs k and l (MRTSkl ) reflects the slope of the isoquant, provides a measure of substitutability between inputs k and l, and is derived from the ratio of input shadow prices: MRT S kl =

∂D (x, y) /∂xk ∂D (x, y) /∂xl

(15)

This statistic is affected by the units in which inputs are measured, and so it is conventional to calculate a normalized MRTSkl : subkl =

∂D (x, y) /∂xk xk . ∂D (x, y) /∂xl xl

(16)

If subkl > 1 (subkl < 1), it is difficult (easy) to substitute out of input k into input l [259]. An alternative measure of substitutability is the Allen elasticity defined in this output distance function context as Akl (x, y) =

D (x, y) Dkl (x, y) Dk (x, y) Dl (x, y)

(17)

If Akl (x, y) > 0 ( 1, we have increasing returns to scale as a 1% increase in xk results in a more than 1% increase in output expansion (with proportional changes in all outputs) [258].

Findings from the Literature Much of the literature regarding output distance functions focuses on deriving estimates of efficiency (the subject of Section “Efficiency, productivity change, and analyses of factors underlying efficiency”) rather than on the production concepts referred to in this section. Early efforts to model the higher education production function in a framework where multiple outputs are produced from multiple inputs employ canonical correlation estimation methods and find a degree of substitutability between inputs based on data on individual university students [78, 79]. Most recently a SFA translog output distance function suggests that returns to scale appear to be exhausted across the English higher education sector. Based on estimates of Allen and Morishima elasticities, substitution is difficult between

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academic and non-academic staff (a similar result to the school context reported in Section “Recent developments in estimating cost functions”) and much easier between academic staff and capital inputs [177]. Returns to scale can also be established using the non-parametric DEA approach. Where this has been used in the higher education context, the findings generally point to the prevalence of constant or decreasing returns to scale [38, 80, 177]. In the context of schools, the parametric approach taken has been to estimate a single output production function with multiple inputs typically to have a better understanding of returns to scale and optimal school size. The literature is somewhat simplistic in its approach with a surprisingly limited focus on functional form [22] especially compared to cost function studies (at school and higher education level) and production function studies in the higher education context. The output measures used are largely based on average test score, and efficiency is typically not included (exceptions are [106, 223]). The limited evidence from this arena suggests that returns are more often constant or decreasing [106, 123, 128, 298] than increasing [121, 209].

Policy Implications and Future Work The work on elasticities of substitution reveals some interesting differences in terms of opportunities to substitute between inputs between HEIs which subsequently merge, those which do not merge at all, and post-merger institutions [177]. Greatest opportunities for substitution are generally observed for HEIs which will subsequently merge. Institutional merger is sometimes considered as a policy initiative by governments (Cai and Yang [61] summarize merger activity across countries), and so this observation is important as it suggests that institutions which do not have the appropriate initial characteristics prior to merger may not reap the potential rewards (see Section “Policy implications and future work” for more on the efficiency effects of mergers in higher education). More work is needed to investigate these findings further and to confirm whether initial characteristics of providers are indeed important in determining success following merger.

Efficiency, Productivity Change, and Analyses of Factors Underlying Efficiency A by-product of the frontier estimation techniques applied in the costs or production contexts is that they also lead to the derivation of measures of efficiency for providers in the sample. By choosing a frontier estimation method, the researcher is therefore also able to undertake a detailed examination of efficiency and, if panel data is available, productivity of organizations. Such analysis is particularly important in the education and higher education contexts where the non-profit nature of the sector makes traditional financial ratios inappropriate for performance measurement [42], but yet the public funding aspect makes it crucial to understand that resources

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are being used efficiently. The interest in efficiency, and the availability of a wealth of data on inputs and outputs, in education and higher education sectors around the world has led to a large literature on education efficiency and productivity, a review of which can be found in Johnes [180]. Efficiency should not be confused with effectiveness: the latter relates to doing the right things – in education it means having the right quantity of outputs – while the former relates to doing things right, where in education it means using scarce resources to produce the highest possible outputs [76, 127]. Typically, efficiency receives the greater attention in the literature, and this will be the focus of this section. It should be noted, however, that one novel publication looks at, distinguishes, and provides comparative measures of both concepts (efficiency and effectiveness), with an application in the secondary schooling context, and this will be reviewed further below [76].

Background on Efficiency Concepts Efficiency work is rooted in the seminal contribution of Farrell [120], and the two main approaches used to derive and examine efficiency are SFA and DEA (already discussed in Section “Cost functions and economies of scale and scope”). These methods can be used to derive various measures of efficiency based on cost (or input distance) functions and output distance functions [200]. From a cost point of view, the parametric measure of efficiency is derived from the error term of, for example, Eq. (6), i.e., as ε = v + u where v is a stochastic error and u is the one-sided efficiency term. In the production context, the parametric estimate of efficiency is derived from, for example, Eq. (22). The distance measure, lnD(x, y), is the quantity of interest in Eq. (19) as this provides a measure of efficiency, and this is derived from the error term in Eq. (22), which is typically assumed to be split into two components, i.e., ε = v − u where v is a stochastic error and u is the one-sided efficiency term. The non-parametric measure of efficiency is often derived from the DEA approach such that D(x,y) is defined as [74, 75] M am ym D (x, y) = m=1 K k=1 bk xk

(25)

where ym and xk are as already defined, am is the weight applied to output m, and bk is the weight applied to input k. For each DMU, the weights are found by maximizing efficiency subject to the constraints that weights must be non-zero and universal. DEA can be applied in the context of constant returns to scale (CRS) or variable returns to scale (VRS). A DMU is fully efficient if D(x, y) = 1. In establishing the efficiency of an organization, we therefore examine its observed production/costs relative to best practice in the entire industry. As such, the frontier methodology provides a benchmark which an inefficient provider can

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use to help it to become more efficient and ultimately to move on to the best practice frontier. When we have a panel of data, bringing in a time dimension (denoted by t and by t + 1), we are able to perform an analysis of productivity change which can be measured using the Malmquist productivity index [228], developed by Caves et al. [66] and further by Färe et al. [119], which is derived as follows for the output distance function (where superscripts and superscripts denote the time period of the distance function):  M (xt+1 , yt+1 , xt , yt ) =

D t (xt+1 , yt+1 ) D t (xt , yt )



D t+1 (xt+1 , yt+1 ) D t+1 (xt , yt )

1/2 (26)

Notation is as defined earlier, and Dt (xt + 1 , yt + 1 ) denotes the distance of the period t+1 observation from the period t frontier. If the Malmquist productivity change index exceeds unity, there has been an improvement in productivity between periods t and t+1. Values less than 1 suggest the converse. The change in the production position of a provider over the two time periods has two underlying determinants: first, the provider can produce more because the output distance frontier for the sector has moved outward, and therefore the potential for production across all providers is expanded; second, the provider’s position relative to the time-relevant frontier can change. The Malmquist productivity index can be decomposed into two components as follows [118, 119]:

D t+1 (xt+1 , yt+1 ) M (xt+1 , yt+1 , xt , yt ) = D t (xt , yt )  t

t

1/2 D (xt+1 , yt+1 ) D (xt , yt ) D t+1 (xt+1 , yt+1 ) D t+1 (xt , yt )

(27)

 t+1  D (xt+1 ,yt+1 ) The first component, , measures the change in technit D (xt ,yt ) cal efficiency over the two periods (i.e., whether or not the unit is getting closer to its efficiency frontier over time), and the second component,

 t 1/2  t D (xt+1 ,yt+1 ) D (xt ,yt ) , measures the change in technology over the t+1 t+1 D (xt+1 ,yt+1 ) D (xt ,yt ) two time periods (i.e., whether or not the frontier is shifting out over time). Values of either of these components of greater (less) than unity suggest improvement (deterioration) in the measure.

Findings from the Literature Efficiency There is a huge literature reporting findings on efficiency in both education and higher education, and various reviews can be found (see, e.g., [53, 172, 179, 186,

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302, 317]) including a particularly detailed one [102]. This section provides a brief overview of that literature. In the context of schools and further education institutions, and taking a production perspective, mean efficiency varies from just under 0.6 to well over 0.9 using parametric estimation methods [71, 91, 92, 106, 124, 146, 148, 201, 210, 244, 296]. A similar spread of mean efficiency scores is observed when using nonparametric methods [5, 21, 43, 44, 48, 49, 53, 54, 59, 68, 72, 73, 118, 150, 153, 154, 157, 198, 202, 211, 230–232, 244, 246, 251, 252, 260, 262–266, 276, 279, 296, 303, 307]. For most of these studies, values are typically at the higher end of the range but depend on model specification, context of the sample, type of schools (e.g., public or private), and (in the case of DEA) whether constant or variable returns to scale are assumed, with the latter providing higher mean estimates. An exception to these studies is in the context of Australian schools [69] where mean efficiency is around 0.4 for primary schools and 0.5 for secondary schools – these results are discussed further later in this section. When a cost perspective is taken, mean efficiency is found to be relatively high with a range of 0.83–0.96 using parametric methods [33, 142, 278] and 0.664–0.95 using non-parametric methods [31, 34, 35, 156, 229, 230, 275, 277, 278]. Most studies at university level use non-parametric methods (often DEA) in a production context to estimate efficiency. Such studies, which cover an array of university sectors, find average efficiency to be relatively high. Mean values tend to fall in the range 0.5–0.97 [1, 3, 16, 38, 115, 178, 212, 257, 268], but there are some models which yield mean efficiency below 0.5 [108, 217, 242, 313]. Parametric estimation methods applied in a production context yield relatively low mean efficiency scores of the order 0.5–0.8 [178]. Mean efficiency derived from cost function studies falls in a similar range of around 0.5 upward with smaller, specialist institutions more likely to exhibit lower average efficiency [132, 164, 183, 185, 195–197, 297, 301]. Only a few studies have compared efficiency values of providers derived using alternative methods. While efficiencies from parametric and non-parametric estimations of cost or output distance functions are often significantly correlated [170, 178, 208, 240], these correlations are not always particularly strong suggesting that different estimation methods can lead to different conclusions. These findings on efficiency levels are interesting insofar as they lead to questions as to why one provider is substantially more (or less) efficient than another. It should be remembered, however, that they are only estimates; the possibility of providing standard errors around the efficiency scores allows the researcher to establish whether there are significant differences between providers. Where this has been done, the conclusion is that there are significant differences only between the best and worst performers [173, 178, 257]. It should be noted that the estimation methods assume that the units under examination are comparable – in terms of, for example, their production technology or environment. If such differences between institutions exist but are not allowed for, this might be captured in the efficiency score, and hence these scores should be interpreted with caution [184, 185].

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The differences between institutions and the subsequent questions raised by efficiency analyses often lead to a second-stage investigation as to what factors might actually influence how efficiently an institution can operate. There is a considerable literature examining the determinants of efficiency at both school and higher education levels. Methods of analysis vary. Early studies typically use DEA followed by a Tobit approach to accommodate the contention that the dependent variable (efficiency score) is a censored variable taking values between 0 and 1 [2, 49, 53, 54, 71, 91, 198, 202, 211, 216, 237, 240, 247, 265, 278, 279, 284]. Later studies argue that the dependent variable is not censored but fractional [238] and that the appropriate second-stage analysis should take an OLS estimation approach, with White heteroskedasticity-consistent standard errors, which produces consistent estimators for large samples [161, 238]. A regression approach (or suitable panel data methodology) is used in the second stage in a number of studies [59, 156, 157, 230, 241, 264, 266, 313]. Separate second-stage analyses, such as those referred to above, have been criticized. When using SFA to derive the efficiency scores, these scores are assumed to be independently and identically distributed. Yet in the second stage, they are assumed to be affected by factors relating to, for example, the DMU. Models which address this issue have been devised for both cross-sectional and panel data [36, 163, 219, 270], and such methods which simultaneously apply SFA and investigate the determinants of efficiency have been applied in the education context (see, e.g., [17, 240, 297, 328]). Analyses of efficiency have uncovered a vast array of determinants of efficiency. At the school level, school-related determinants including per pupil expenditures on teachers, teacher salary, physical resource expenditure, and scale (school or class) have all been found to be important, although direction of effect can vary from study to study [53, 59, 202, 211, 251, 264, 278]. Pupil discipline record, absenteeism, and having pupils with special educational needs also affect school efficiency [49, 91, 227, 230, 312] as does type of school such as selective and single sex girls’ schools [53, 59]. Factors relating to the pupils themselves are also highly important in determining the efficiency of schools. Such factors include ethnic background, socioeconomic status, and parental education [5, 91, 157, 211, 237, 251, 265, 266]. Variables relating to the wider region in which the school is located are also important in determining efficiency levels of schools. These include variables indicating the unemployment rate and the wealth and educational attainment of inhabitants of the area [5, 54, 157, 201, 251, 265, 278]. Direction of relationship between such variables and efficiency can vary according to study. Finally, political factors have also been found to play a part in determining school efficiency. These include the source of funding (particularly deriving from local sources) and political leaning of residents of an area, both of which can affect efficiency [4, 49, 312]. The array of variables affecting efficiency is therefore vast (more information can be found in Burney et al. [58]), and the variables vary in terms of what the school can control (such as resources) and what they cannot (such as characteristics of the pupils in the catchment area and the regional environment). Clearly this distinction

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is important in terms of developing policies to improve efficiency. As an initial step, the second-stage analysis can be used to compute a revised efficiency score which takes into account the variables. One study which does this finds that mean efficiency in primary schools rises from 0.4 to 0.9 and in secondary schools from 0.5 to 0.9 [69]. This demonstrates the effect these variables can have in explaining inter-institutional differences in efficiency, and managers and policy-makers should be aware of this. Similarly useful results are found in the higher education context. Universityrelated factors include provider size and composition, age, governance (such as public or private), source of funding, geographical location, as well as staff characteristics such as gender, age, and ethnicity [216, 240, 241, 284, 297, 313, 315, 328]. The influence of student characteristics on efficiency is less well investigated [297]. It is worth ending this sub-section with a quick note on effectiveness. Cherchye et al. [76] define a measure of effectiveness for organizations by assuming constant resources; in practice this means applying the CRS DEA framework with resources equal to unity for all DMUs in order to derive an effectiveness score. In applying this methodology to Flemish secondary schools, they find that performance can be improved more by improving efficiency (as there is unexploited production capacity) than effectiveness. It will be interesting to see this methodology applied to different sectors and countries.

Productivity Measures of productivity have typically been undertaken using non-parametric approaches in education and higher education. In applications to higher education sectors as diverse as the UK, Italy, Spain, China, Australia, Australasia, and Iran, productivity growth is found, and this appears to be more a consequence of technology change (the frontier shifting out) than of efficiency change (inefficient units getting closer to the frontier) [32, 126, 175, 178, 196, 197, 235, 250, 320]. There are, however, some exceptions where productivity has increased but due to efficiency rather than technology change [130, 269]. When samples are split, for example, by mission group, findings are more nuanced with some groups experiencing productivity decline, and this too is a consequence of shifting frontier [301, 302]. In the context of productivity improvement, it is hypothesized that recent innovations to higher education such as e-learning support for teaching and digital support enabling and supporting research networks may well be reasons for the frontier being pushed out. The inefficient universities may find it difficult to keep pace with the changing technologies. In the context of schooling, we find similar results regarding productivity change and the underlying cause being technology improvements at both the post-compulsory [54] and secondary school levels [113, 255] in the UK and Canada. Where productivity is found to decline [260, 303], this is also related to technological performance rather than efficiency decline. Johnson and Ruggiero [199] take the Malmquist decomposition one step further by adding in a component relating to environmental harshness. In a practical application to Ohio school

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districts, the approach reveals that while technological progress drives productivity change in top-performing school districts, it is the environmental harshness which is the most important driver for low-performing districts. A similar approach is applied to Dutch schools and also provides useful insights [55].

Recent Developments in Efficiency Measurement Many developments covered in earlier sections are relevant here. Heterogeneity among providers, and how it is addressed, is an important factor in efficiency studies. Some researchers choose to divide their sample based on a known characteristic, such as public or private funding [108, 212, 231], or by mission group. More recently, developments in the methodological approaches are used to address heterogeneity in the efficiency context. Thus LC and RP SFA, while providing different parameters by group or unit (respectively), also provide different efficiency scores by group or unit. We have referred throughout this chapter to the issue of institutional diversity in education and higher education sectors and considered ways in which diversity has been handled. Another emerging approach in the efficiency context (based on cost functions) is one which distinguishes between transient and permanent efficiency [89, 90, 125, 220, 306, 311]. The underlying premise is that some differences between organizations arise from a historical and geographical context which the education provider cannot alter. Inefficiency differences arising from such structural variations should be addressed differently from those arising from transient (or short-term) factors. There are some subtle differences in the precise approach, in this context. An SFA approach which allows for unobserved heterogeneity and incorporates the premise of transient and permanent inefficiency [220, 306] has been applied in the higher education context [11, 139]. It seems that for German and Italian universities, transient efficiency is relatively high, while persistent efficiency is much lower. Papadimitriou and Johnes [256] use an approach developed by Filippini and Greene [125] and also find that persistent efficiency is lower than transient efficiency in the English higher education sector. Clearly policies for improving efficiency likely need to be adapted in light of this finding: a low persistent efficiency value, for example, suggests a need for structural changes. An aspect of production analyses which we have not yet explored is that of complexities in the production process. So far we have assumed that all inputs go into a “black box” at the start of production, and all outputs come out of it at the end point. In reality, the “black box” may be hiding a more complex production process whereby some inputs may produce a set of outputs at one stage, and then (some of) these outputs, possibly along with other inputs, then become inputs into a second stage of production which produces more outputs. Where a production process can be divided into a series of sub-processes, a standard DEA fails to account for the efficiency of each sub-process. By ignoring such complexities, the standard DEA might lead to bias in efficiency estimates [203, 206] and conceals useful information

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about efficiency of each of the stages. Network DEA (NDEA) [117, 304] takes into account such complexities of production and provides estimates of efficiency at each stage. A number of studies have applied a network DEA approach mainly in the higher education context [176, 222, 322]. A network approach, whereby outputs such as student satisfaction and student achievement are assumed to happen in a first stage while employment outcomes happen in a second stage (where student achievement is an input into that second stage), reveals considerably more discrimination in terms of HEIs identified as efficient. Moreover, the second stage (production of student outcomes in the labor market) is less efficient than the first stage, thereby providing managers with useful information on where they should concentrate their efforts in terms of improving efficiency [176, 222]. Indeed, an analysis of the factors underpinning each of the sets of efficiencies (stage 1 and stage 2) indicates that there are different reasons for differential performance in each case and hence provides more information for managers and policy-makers [222]. More work of this type at both school and higher education level would be useful. We have noted in Section “Findings from the literature” above the many studies which employ a second-stage analysis to explore the variables which might impact efficiency scores. However, such studies are valid only if the separability condition between the input-output space of the first stage and the space of the external factors in the second stage holds. In the situation where the separability condition does not hold, then a conditional DEA model is the appropriate approach [67, 96, 97]. While it is important to check that the separability condition holds [288, 289], and a test of the validity of the separability assumption is available [98], studies which investigate the issue of separability and apply a conditional nonparametric approach are relatively rare to date (see, e.g., [45, 93, 94]). The early indication is that academic or school-related variables may be less important than economic and cultural indicators. A particularly novel and interesting application of the conditional efficiency model investigates efficiency of the provision of adult education programs in Flanders [282]. This work suggests that characteristics of the adult learners and homogeneity among the teachers on programs are important determinants of managerial efficiency in the adult education contact. Clearly more work using this approach is required at all levels of education. The Malmquist approach has been extended to allow comparisons of performance between groups rather than time periods [63], and this has further been extended to examine and compare patterns of change across groups over time [23]. For example, in the context of schools in the Basque country in Spain, this approach establishes that privately run schools have consistently better performance and that this is because of superior technological performance. The methodology can also be applied when there are more than two groups. When Ohio school districts are assigned to five groups based on environmental harshness, the Malmquist decomposition shows that productivity is largely explained by environmental harshness and that technological progress is also hampered by the harshness of the environment [199]. Distinctions are also found between public and private universities in Spain with private universities outperforming their public counterparts at the start of the

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study period, but the Malmquist decomposition reveals that the public universities catch up over the period [100].

Policy Implications and Future Work While average efficiency is generally found to be high in many education studies, there is typically a spread of performance across providers, and this means that the results can potentially be useful at a policy level. Efficiency-based funding [116], for example, is one aspect where there has been relatively little work, but the applications that exist suggest some potential for efficiency improvements by distributing resources based on efficiency. Sexton et al. [286] provide an example of an efficiency-based state funding scheme for HEIs underpinned by DEA. Such a scheme, which would encourage HEIs to behave in such a way as to be consistent with government or state objectives, would reap potential savings of 9% across the sector, with differential savings observed in each provider. A particular advantage of the approach is that, as efficiency improves relative to a given DEA frontier, any subsequent DEA will produce an improved frontier against which efficiency will be measured, and so there is a natural tendency of the approach for ongoing improvement [286]. A drawback of the approach is that efficiency estimates based on annual estimations can fluctuate from 1 year to another meaning that there is potential for instability in resource allocations [115]. A reduction in sensitivity might be achieved by using a moving average over several years. In addition, an efficiencybased funding scheme may not be appropriate if there is little significant deviation in efficiency across providers. In such cases, the studies should instead be used to provide institutions with useful information on benchmarking and examples of good practice [185]. Even where efficiency does apparently vary substantially across providers, we know from the second-stage analyses undertaken in previous studies that efficiency is affected by various factors and some of these are outside the control of the institutions. Strategies to improve efficiency must therefore be nuanced. For example, if efficiency is affected by the ethnic mix of pupils [54], providers can do little to alter that. Instead, they must focus on ways in which to improve outcomes of the at-risk groups, and this may then impact on efficiency. The importance of variables reflecting the conditions in the wider environment means that local and government policies to improve economic conditions in a catchment area can also impact school efficiency. The introduction of increased competition in school sectors has been a deliberate policy of some governments (e.g., in the UK) to improve school performance and efficiency. There are various studies which have specifically examined the impact of increased competition on efficiency in various state school sectors [6, 8, 52, 53, 157]. With one exception [148], these studies find that the larger the number of schools in a region, the higher the schools’ efficiency. Some studies find that competition from private schools impacts on efficiency in publicly funded schools [7, 8, 243],

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although the effect quickly diminishes as distance from the school decreases [201]. Competition has also been investigated as a driver of efficiency in higher education, where it has been found to have a positive effect in the Canadian higher education context, although not always significantly so [240, 241]. A final example of how efficiency analyses might inform policy arises in the context of mergers. Theoretically, a merger might be expected to have benefits in terms of increased efficiency accruing from returns to scale or returns to scope where the merging providers have complementary offerings [155, 293]. A suite of papers utilizing a sample of data relating to English higher education suggest that, typically, efficiency improves following merger but that the benefits accrue in the years immediately following the merger and do not continue indefinitely [178, 194, 257]. There is scope for more work into the evolution over time of the effects of merger on subsequent efficiency.

Level of Analysis In the preceding sections, we have made little reference to the level of the analyses undertaken. In many cases, the estimations, be they cost functions or output distance functions, are at provider level. There are some exceptions in the schooling context, where the level might equally well be the funding region (such as school district in the US context or local education authority – LEA – in the UK context). The review of efficiency in education by De Witte and López-Torres [102] confirms the provider (defined as organization, school, department, etc.) as the typical unit of analysis in such studies: of 223 papers relating to efficiency in the education context over the period 1977–2015, 147 are at the organization level (with 89 relating to HEIs and 58 to schools); 44 focus on the funding district, county, or city level, while 9 studies are at the level of the country, and 23 at the level of the individual student. A number of these studies are of note because they focus on a particular discipline or department [30, 65, 87, 105, 132, 205, 207, 236, 284, 292] or a support service [64, 204, 245, 267, 290]. In this section we take a brief look at the studies undertaken at individual-, funding area-, and national-level analyses to see what additional information they provide, and what challenges arise, in the context of production economics.

Individual-Level Analyses Individual-level studies are not uncommon in the schooling literature relating to education production functions, which has long recognized that pupils are nested within schools and hence the data are hierarchical in nature. As such, multi-level modelling (MLM) has been developed to estimate such functions while allowing for within-unit variations [137, 138, 316]. Recognition of the hierarchical structure avoids issues such as aggregation bias and mis-estimated parameters, and the MLM approach is sufficiently flexible that it can allow both intercept and slope coefficients to vary. An additional advantage of such an approach is that it is

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possible to disentangle the effects of both pupils and schools on their outcomes. The disadvantage is that MLM is not a frontier estimation technique, and so there is no allowance for inefficiency in the education production function. An alternative approach which allows for inefficiency is to apply DEA to individual-level data. Such an approach has been taken in a small number of studies in the schooling context [261, 299, 300]. By using a meta-frontier type of approach, it is possible to decompose overall efficiency for a pupil into that attributable to the pupil him-/herself and that attributable to the school (assuming just pupil and school levels – additional levels are possible). By careful aggregation of the pupil efficiencies [302], schools derive more information as to the source of their shortcomings (pupil or school) and can devise appropriate initiatives accordingly. Applications of individual-level DEA in universities are also relatively rare. Findings from such studies suggest that efficiencies derived from aggregate university level analyses incorporate both individual and institution performance components; an individual-level DEA, meanwhile, provides more detailed information about the source of the inefficiency, i.e., student or university [174]. A comparison of MLM and individual-level DEA applied to the same data set finds interesting differences in the performance rankings of universities based on the two approaches, and these are particularly relevant for the best- and worst-performing HEIs [173]. This is in contrast to findings at school level; De Witte et al. [103] find more alignment between their results from MLM and an individual-level non-parametric approach using a sample of school pupils.

Funding Area Analyses While not as prolific in number as organization studies, papers focusing on efficiency within funding areas in education are nevertheless reasonably numerous. They mostly relate to school-level education and are based on both parametric and non-parametric approaches. One of the earliest such studies utilizes maximum likelihood and corrected ordinary least squares to estimate efficiency among local education authorities in providing schooling in England, using a cost function approach [33]. The level of estimated efficiency depends on whether the approach is deterministic (with efficiency levels around 83% to 89%) or stochastic (with efficiency levels much higher at well over 90%). Experimentation with efficiency measurement continues in the context of funding areas with a comparison of ratios (comparing a single output to a single input, e.g., cost per student graduated) and efficiencies derived from a variety of DEA models [111]. There are significant inconsistencies between the ratios and DEA efficiency measures, which is not surprising as the ratios fail to take into account the multi-input multi-output nature of production. Subsequent studies largely use standard frontier techniques such as DEA and SFA (including conditional and network DEA), applied in cost or production settings, and generally establish similar levels of efficiency to the earliest studies [31, 129, 144, 147, 149, 199, 201].

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A non-frontier strand of literature employs a (modified) quadriform approach [160] to the identification of efficiency among school funding areas [162]. The modified quadriform approach is a means whereby performance of units can be displayed in a two-dimensional depiction. Specifically, costs are regressed on a set of uncontrollable school characteristics, and school output (such as graduation rate) is regressed on the same set of characteristics. The resulting residuals from each regression equation are plotted for each school district, and performance is examined in quadrants ranging from efficient (described as low input and high output) through effective, ineffective, and finally inefficient (described as high input and low output). While interesting and easy to interpret, such an approach does not adequately account for the multi-dimensional nature of production, is non-frontier, and relies on regression residuals which contain both unexplained variation and random error. An adaptation to provide a buffer around residuals which are low in magnitude (and therefore such districts can be assumed to be performing as expected) addresses the latter point to some extent, but other drawbacks remain. A comparison of the quadriform approach with frontier techniques can be found in Rolle [274]. Higher education studies rarely feature in the funding area context, mainly because higher education is often a national (not regional) responsibility – hence national-level analyses are more appropriate, and these are discussed in the next section. An exception is a study of Chinese higher education at the level of Chinese provinces which takes a production function approach [321] and where efficiency levels are found to be relatively low (with mean technical efficiency of under 40%). Such funding area studies can provide useful insights into efficiency or (in the rare cases where it is calculated, productivity [255]) for the funding providers. The relationship between the funding area and organizations within it is rarely utilized – a network approach by Grosskopf et al. [149] is an example where the relationship is adapted into the approach. A meta-frontier analysis of schools within funding areas might also provide a useful extension to this particular body of literature.

National-Level Analyses The benchmarking advantages of such tools as DEA are well known. As austerity measures have been introduced in various education and higher education sectors around the world in the last decade, there has been an increasing recognition that international comparisons are necessary to provide benchmarks of good practice which may be outside of national boundaries. Combined with this, the last decade has seen a constant improvement in the availability of data at all levels across countries meaning it is now increasingly possible to make such international comparisons and to identify exemplars of good practice across countries for national governments to emulate. Studies which make international comparisons – whether at school or higher education level – fall into two categories. There are those which use provider-level data across two or more countries and then frequently take a meta-frontier approach

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to make cross-country comparisons [11, 12, 16, 314, 315]; and there are those which use national-level data (i.e., the nation is the DMU) to derive their results [7, 9, 25, 26, 29, 47, 133]. Interesting differences between countries can be found. Agasisti and Pérez-Esparrells [16], for example, compare universities in Italy and Spain and find, using the Malmquist productivity index approach, that productivity has been rising in both countries over the study period. In comparing the countries however, it appears that technological change underpins productivity increases in Italy, whereas it is efficiency gains which underpin the observation for Spain. There is no doubt that such studies will proliferate as more data becomes available, and that is beneficial so long as results are treated with caution. There are various problems with cross-country comparisons and in particular the latter approach. It is extremely difficult to get comparable data on costs or inputs and outputs at the national level. The assumption that production technology and environment are the same across diverse sets of countries is open to serious doubt. Thus if a national-level study is to be undertaken, it is advisable either to use individual providers to seek useful insights into education provision across countries using a meta-frontier type of approach, or, if national-level data are to be used, then a parametric estimation approach which allows for unobserved heterogeneity should ideally be adopted. There is scope for much more work in this context.

Conclusions This chapter has examined empirical findings relating to production economic concepts in the context of education and higher education. Education is an important sector of any economy as the benefits (in terms of increased productivity) accrue to both the individuals who consume the education and also society as a whole. This is particularly the case for primary and secondary education, which are typically compulsory in many countries, and to a limited extent for tertiary education as well. As such, education and higher education are in receipt of publicly allocated funds, potentially making the incentives for efficient operation less compelling than in a private sector. The public funding of all levels of education, combined with the incentives and pressures which that imposes on the providers operating in the sector, makes education and higher education interesting sectors in which to examine concepts from production economics. This chapter examines findings relating to costs, production, and efficiency in education and higher education and contributes to the production economic literature by bringing together the findings of these diverse literatures, at all levels of education, into one repository. The review has uncovered a number of key areas for future research. The mixed findings emerging from all topics in both education and higher education contexts make it difficult for managers and policy-makers to take a consistent message on, for example, the existence (or otherwise) of economies of scale or scope, the degree of substitutability between different inputs, the extent of inefficiency in the sectors, and the identification of factors affecting efficiency. This

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points to a pressing need for more detailed analyses of the literature to provide a framework for why results vary and hence permit the users of the work to make informed decisions. A key contribution in this area is by Zhang and Worthington [324] who undertake a meta-regression analysis of the empirical cost function literature in higher education. They are able to identify reasons why the findings on economies of scale and scope vary across the studies. More studies of this type in the education context, or relating to output distance functions and efficiency (at both schools and higher education levels), are also needed. In terms of factors affecting efficiency, the conditional DEA approach offers a rigorous methodology for identifying those variables which are most important in affecting efficiency. This knowledge is essential in determining strategies for improving efficiency and hence getting more value for public funds and in particular in revealing whether institution-level or regional-level or national-level policies will be most effective. While economies of scale and scope (and returns to scale and scope in the production context) are relatively well researched, there is much less empirical research into elasticities of substitution between inputs (or between outputs). In times of public funding constraints, such information could be particularly useful to managers and policy-makers. Similarly, more work on the potential benefits of performance-based funding would be welcome. Finally, there is considerably more scope for education studies which make comparisons across countries. These might use national-level data, in which case appropriate methods which take into account unobserved heterogeneity should definitely be applied. But the increasing availability of large individual level data sets offers opportunities for findings from these sources. However, more work is required on the application of frontier methods to the individual-level context and using these results to derive insights into concepts, such as efficiency, relating to providers and even nations. Empirical applications of production economics to education and higher education have a long and fruitful history and are set to continue to provide useful information to both managers and policy-makers alike.

Cross-References  Elasticities of Substitution  Stochastic Frontier Analysis: Foundations and Advances I  Stochastic Frontier Analysis: Foundations and Advances II

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Dairy Farming from a Production Economics Perspective: An Overview of the Literature

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Boris E. Bravo-Ureta, Alan Wall, and Florian Neubauer

Contents Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Early Uncovering of Basic Relationships . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Technical Efficiency . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Output Growth and Total Factor Productivity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Cost Function Approaches: Efficiency and Economics of Scale, Size, and Scope . . . . . . . . Technology Adoption . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Supply Response and Government Intervention . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Risk and Uncertainty . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Sustainability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Concluding Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Cross-References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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The purpose of this chapter is to give a broad overview of the published empirical work on the production economics of dairy farming as well as an outlook on future challenges for this area of research. This chapter shows that the vast production economics literature on dairy farming has been used to address a wide variety of topics including efficiency and productivity, technology adoption, economies of size, scale and scope, the effects of government intervention policies in the sector, the effect of risk and uncertainty, and issues relating

B. E. Bravo-Ureta () · F. Neubauer Agricultural and Resource Economics, University of Connecticut, Storrs, CT, USA e-mail: [email protected]; [email protected] A. Wall Department of Economics, University of Oviedo, Oviedo, Spain e-mail: [email protected] © Springer Nature Singapore Pte Ltd. 2022 S. C. Ray et al. (eds.), Handbook of Production Economics, https://doi.org/10.1007/978-981-10-3455-8_31

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to sustainability including climatic effects, animal welfare, and environmental efficiency. Dairy farming faces important challenges, particularly with regard to environmental sustainability, animal welfare, structural changes, and input and output price volatility, all of which provide fertile ground for future production economics research in dairy. The conceptual frameworks and empirical analyses reviewed in this chapter show that production economists have several tools at their disposal to carry out studies related to these challenges and thereby contribute to policy analyses and formulation. Moreover, the role of production economists, working with scientists in various other disciplines, will be paramount in the search for avenues to improve the overall productivity of dairy farming while offering policymakers sound advice on sustainable technologies and tools to deal with greater risk and uncertainty. Keywords

Dairy · Milk · Production · Economics · Productivity · Efficiency · Cost · Profit · Supply · Technology · Size · Scale · Weather · Climate · Sustainability

Introduction Dairy farming has been and remains a very important agricultural activity around the globe. According to the Food and Agriculture Organization (FAO) [1], milk is one of the most valuable farm products and over 130 million farms have dairy cattle, with significant variability in cow numbers per farm across countries. Moreover, cows represent an important source of livelihoods, employment, food and wealth among rural households. In developing countries, dairy animals are an important mechanism for the empowerment of women. On the consumption side, fluid milk and dairy products are a significant source of energy, protein and micronutrients. These features can play a key role in decreasing hunger, and in enhancing food security, nutritional levels and diets. Worldwide demand for food is expected to double by 2050 as a result of growing population and incomes. To satisfy this rise in demand, farms around the world will have to produce significant quantities of additional food. This rise will present important opportunities and challenges to global agricultural systems and the dairy sector is no exception [2]. The objective of this chapter is to provide an overview of the production economics literature centering on the dairy sector, specifically on dairy farming and milk production rather than on the dairy products manufacturing sector. To provide some historical context, section “Early Uncovering of Basic Relationships” reviews early studies that sought to uncover basic production relationships, and this brings us to the 1970s. Section “Technical Efficiency” focuses on the literature analyzing technical efficiency, followed in section “Output Growth and Total Factor Productivity” by an exposition of studies dealing with output growth and total

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factor productivity using primal (production) approaches. Section “Cost Function Approaches: Efficiency and Economics of Scale, Size, and Scope” reviews studies using cost function models to measure efficiency as well as economies of size, scale and scope. Section “Technology Adoption” covers technology adoption, while Section “Supply Response and Government Intervention” considers studies that analyze supply response and government (intervention) policies in the dairy sector, especially quotas and price supports. Section “Risk and Uncertainty” tackles the literature on risk and uncertainty and section “Sustainability” reviews studies that have examined the sustainability of dairy production in a broad sense, including papers on weather and climate, animal welfare, and environmental efficiency. The final section concludes and comments on the role of production economics research with regard to future challenges facing dairy farming.

Early Uncovering of Basic Relationships How far to go back in time as well as the specific papers to include in a chapter like the present one is a matter of judgement and ultimately somewhat arbitrary. We have chosen the paper coauthored by Mordecai Ezequiel (1927), an eminent agricultural economist with a prolific professional career within the USA, as the beginning of our exposition [3]. Ezekiel, McNall and Morrison (1927) set out to provide a scientific understanding of why farm milk production records show considerable variability in milk per unit costs across farmers. The authors reasoned that this variability had two main sources: (1) the efficiency in the use of the milk production technology in terms of the mix of inputs used to generate the observed level of milk output; and (2) the effectiveness of the combination of factors considering their costs and output value under prevailing economic conditions. Using data gathered from Wisconsin dairy farmers, the authors documented “great variation from farm to farm in average production per cow” (p. 3). After conducting detailed statistical analyses, they reported that the essential problem in dairy farming was the technical relationship between feeding rations and milk output. A general conclusion was the consistency between their farm-level results with those obtained by other scientists in feeding experiments. In a 1932 report, Ezekiel, Rauchenstein and Wells focused on the response of milk production to changes in price [4]. These authors drew from earlier analyses and concluded that 53–88% of the nonseasonal changes in milk production in the winter season could be explained by changes in the milk-feed price ratio. The most elastic milk supply was found in Vermont, followed by Baltimore, and Saint PaulMinneapolis, although the authors concluded that farmers reacted in similar ways despite differences between the three areas. In a more theoretical contribution, Cassels (1933) stated that progress had been made on empirical investigations on the responsiveness of supply to price and highlighted several issues that deserved additional work [5]. The author argued that the market and long-run supply curves are difficult, if not impossible, to estimate

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statistically. In contrast, he concluded that the short-run curve was easier to estimate and closer to the studies that were being conducted at the time. Cassels emphasized that supply curves depend on the time period considered and are not reversible, where the latter implies a higher elasticity when prices go up compared to when prices decline. Cassels and Malenbaum (1938) expressed skepticism regarding statistical analyses of supply response [6]. They noted that Cassels (1937), contrary to Ezekiel, Rauchenstein and Wells (1932), found no evidence of a high correlation between milk-feed price ratios and output in Vermont, even though the methods and the data used were similar (the period of analysis, however, differed) [7]. Cassels and Malenbaum (1938) investigated this disparity and concluded that omitted variables that could not be included in the studies had to be responsible for the disparities and cautioned that results from statistical analyses could not be taken as economic laws [6]. The early work by Cassels and others was recognized by several authors in the literature on supply response in later decades. In particular, Halvorson (1958) highlights the importance of both the early Cassels contributions and the book Interregional Competition in Agriculture by Mighell and Black (1951) [8, 9]. As better data became available, many influential supply response studies appeared in the 1950s and 1960s, including Brandow (1953), Halvorson (1958), Schuh (1957), Cochrane (1958), Cowling and Gardner (1964), Wipf and Houck (1967) and Wilson and Thompson (1967), all of which used regression techniques [8, 10–15]. The econometric literature on supply response was influenced heavily by the seminal work of Nerlove in the mid-1950s and was carried on into the 1970s by Chen, Courtney and Schmitz (1972), and Prato (1973) [16, 17]. An extensive review of supply response papers in the period influenced by the Nerlove model in agricultural commodities, including milk, can be found in Askari and Cummings (1977) [18]. Linear programming models were also used to study supply response in the 1950s and 1960s by Faris and McPherson (1957), Cowling and Baker (1963) and Kelley and Knight (1965). Some drawbacks regarding the use of this technique in this setting were discussed by Barker (1965) [19–22]. Jensen (1940) focused on the estimation of “input-output relationships in milk production [in] an attempt to accomplish quantitative verification of theory and to obtain data that will be useful to economists, technical specialists and producers in determining the most economic organization and adjustment of production” (p. 249) [23]. He provided a critical overview of several research reports published in the 1920s and 1930s relying on data from farm records and argued that data generated from experiments designed specifically to investigate input-output associations were more desirable. He went on to assert that the determination of these physical relationships is not economic but purely technical. He then used linear regression to estimate parabolic production functions to explain the variation in milk output in terms of feed inputs where the unit of observation is a cow and the data came from several experiments conducted in different research stations in the USA.

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Jensen’s work inspired a number of agricultural economists to pursue quantitative production studies in dairy and other types of farming. It is interesting to observe that in the late 1930s an important policy concern was soil management, while governmental programs were being enacted to promote the conservation of natural resources. A net effect expected at that time was an increase in forage production that was suitable for dairy farming. Understanding how best to utilize an expanding availability of forage crops along with technological improvements and price changes became increasingly important in order to provide sound advice to farmers, extensionists and policymakers [24]. An early study using regression analysis along with records from operating farms was published by Herrmann (1943) who was interested in determining the most profitable feeding rate for commercial dairy herds in West Virginia [25]. The author concluded that his analysis provided good foundation for adjusting feeding rates in response to different input-output price ratios. In the 1950s and 1960s, considerable work using regression analyses of experimental data was undertaken to uncover the nature of the production function for milk, including alternative functional forms, and to analyze diminishing returns of different feeds, marginal rates of technical substitution, the geometry of isoquants and expansion paths, and related issues concerning dairy rations [26–30]. Heady and Dillon (1961) published a comprehensive examination of agricultural production function models and of empirical work done up to the late 1950s [31]. This publication appears to be the first to introduce a second-degree polynomial version of a standard two-input Cobb-Douglas model (p. 205), which over the years has become widely known as the translog model. The latter name comes from a research note published by Christensen, Jorgenson and Lau (1971) where they discuss the transcendental logarithmic function, abbreviating it as “Trans-Log” [32]. Among other early studies, Aune and Day (1959) used data from Minnesota dairy producers for 1956 and 1957 along with regression analysis to examine the relationship between labor used in different farm activities in herds varying in size [33]. Waugh (1951) used linear programming, a new technique at that time, to determine least cost combinations of feed inputs that would fulfill or exceed various nutritional requirements, while Weeks (1964) applied linear programming to find profit-maximizing dairy rations incorporating wheat as a feed source using parametric variation in wheat prices [34, 35]. Related dairy production work includes Coffey and Toussaint (1963), Heady, Madden, Jacobson and Freeman (1964), and Paris, Malossini, Pilla and Romita (1970) [36–39]. This brief review of early quantitative studies that examine production economic issues in dairy farming shows that emphasis was placed on the generation of information that had a direct bearing on farm management decisions. Another feature of this work was the close collaboration between agricultural economists and other disciplines involved in dairy science which explains the wide use of experimental data in these economic analyses.

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Technical Efficiency This section considers studies that examine the technical efficiency (TE for the remainder of the chapter) dimension of dairy farm productivity using primal models. Since many of the empirical papers included in the remainder of this chapter use frontier techniques, it is useful to provide an overview of the frontier methodology before moving on to dairy studies. The seminal 1957 article by Farrell sets the foundation for frontier function research, which has become a significant subfield in production economics [40–42]. Farrell specified a best practice constant returns to scale production frontier and then defined technical, allocative, and economic efficiency. Farrell and Fieldhouse (1962) extended the 1957 model to accommodate increasing returns to scale [43]. Estimation methods of the best practice frontier have evolved significantly over the past few decades and a rich menu is now available including parametric, nonparametric, stochastic, and deterministic formulations [44]. The origin of nonparametric frontiers is Farrell (1957), and this was followed by conceptual and empirical work by agricultural economists at Berkeley including Boles (1966 and 1971), Bressler (1966) and Seitz (1970) [40, 45–48]. However, the preceding work was largely ignored by agricultural economists, until it was brought to the forefront by Charnes, Cooper and Rhodes (1978), who introduced the data envelopment analysis (DEA) concept, and by Banker, Charnes and Cooper (1984) [49, 50]. Nonparametric measures of efficiency are now commonly obtained using DEA, which relies on mathematical programming techniques. This area of research, both in terms of new models and applications, has seen a vigorous evolution (e.g., Färe, Grosskopf and Lovell 1985; Simar and Wilson 2007; Färe, Grosskopf and Margaritis 2008) [51–53]. Aigner and Chu (1968) proposed the first parametric production frontier model, deterministic in nature and estimated with linear or quadratic programming [54]. Following them, Timmer (1971) formulated a probabilistic production frontier model, still estimated with mathematical programming, which he applied to US state-level agricultural data [55]. The parametric stochastic production frontier (SPF) framework was introduced around the same time by Aigner, Lovell and Schmidt (1977) and Meeusen and van den Broeck (1977) [56, 57]. The SPF has also seen rapid evolution in recent years, particularly in applications that make use of panel data (e.g., Schmidt and Sickles 1984; Battese and Coelli 1995; Greene 2005; Greene 2008; Tsionas and Kumbhakar 2014; Kumbhakar, Wang and Horncastle 2015; Filippini and Greene 2016) [58–64]. An initial shortcoming of the SPF was the ability to only measure average TE for the sample. This limitation was alleviated by Jondrow, Lovell, Materov and Schmidt (1982) who developed an approach to obtain individual TE scores for each observation given cross-sectional data [65]. Schmidt and Sickles (1984) presented various approaches for calculating individual scores for panel data models including fixed effects [58]. Battese and Coelli (1988) extended the Jondrow, Lovell, Materov and Schmidt (1982) approach to panel data and applied their method to study TE

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for a sample of milk producers from the Australian Dairy Industry Survey for New South Wales and Victoria for the years 1978–1979, 1979–1980 and 1980–1981 [66]. Another aspect of frontier models that has received considerable attention is the explanation of TE, or what has been called inefficiency effects. The initial approach was in two stages, where TE was first generated parametrically or nonparametrically and then the TE scores were regressed on a set of variables. This two-stage approach has been criticized by several authors for introducing statistical bias and this led Fried, Lovell and Schmidt (2008) to write: “We hope to see no more two-stage models” (p. 39) [44]. Several one-step SPF models have been introduced including Battese and Coelli (1992), which uses panel data where the temporal pattern of TE is the same for all units in the sample, and Battese and Coelli (1995), which allows for time-varying inefficiency effects and accommodates explanatory variables related to technical inefficiency [59, 67]. Two-step inefficiency effect models have also been applied and criticized in the DEA literature (e.g., Simar and Wilson 2008) and alternatives have been proposed (Simar and Wilson 2007) [52, 68]. Production frontier methods have been applied to a number of industries, sectors and subsectors and a primary interest has been the measurement of TE, which is the focus in this section [44]. An issue of substantial importance in the production frontier literature is the connection between managerial performance and TE, a link made by Farrell (1957) and subsequently by Martin and Page (1983) and Triebs and Kumbhakar (2018) [40, 69, 70]. A similar link between firm effects and managerial performance/ability and TE has been made in the nonfrontier literature by a number of authors including Hoch (1955), Mundlak (1961), Mundlak and Hoch (1965), and Hoch (1976) [71–74]. The remainder of this section will highlight applications of frontier methods that center primarily on the TE analysis of dairy production using primal (input and output quantity based) methods. This discussion begins with applications of nonparametric frontiers. The first published nonparametric study of dairy production seems to be Grisley and Mascarenhas (1985) who used the Boles (1971) approach to measure efficiency of Pennsylvania farms using records for 1981 and 1982 [46, 75]. Weersink, Turvey and Godah (1990) used the nonparametric methodology, based on Färe, Grosskopf, and Lovell (1985), to examine TE for a cross-section of Ontario farms for 1987 [51, 76]. Overall TE was decomposed into pure TE (producing below the frontier), congestion (input overuse), and scale efficiency (deviations from constant returns to scale). Cloutier and Rowley (1993) applied DEA to examine TE for a Quebec sample for 1988 and 1989 and that same year Tauer (1993) examined both short- and longrun TE and allocative efficiency (AE) for a sample of New York producers for 1990 [77, 78]. Fraser and Cordina (1999) used an input-oriented DEA model to assess TE for farms in 1994/1995 and 1995/1996 located in Northern Victoria, Australia [79]. Asmild, Hougaard, Kronborg and Kvist (2003) pointed out that TE scores derived from DEA were common in selecting benchmark firms [80]. They then used multidirectional efficiency analysis (MEA) to argue that it is desirable to separate benchmark selection from efficiency measurement. The authors compared MEA

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with DEA scores and found a high correlation but noted considerable differences between small and large farms in Denmark. Mugera (2013) introduced an innovation in dairy farm TE analysis where fuzzy set theory was combined with DEA to examine cases when both inputs and outputs are measured imprecisely [81]. Hansson, Manevska-Tasevska and Asmild (2018) explored another angle based on MEA [82]. They argued that the common interpretation of technical inefficiency as waste due to input overuse might instead represent a sensible handling of risk and uncertainty, a behavior referred to as rational inefficiency. Employing MEA, they found support for the rational inefficiency hypothesis and concluded that, for a sample of Swedish farms, what might appear as inefficiency represents rational management behavior. The following paragraphs provide an overview of applications of parametric frontiers, beginning with cross-sectional studies, followed by panel data applications. Müller (1974) appears to be the first to discuss dairy farm TE in the context of a production frontier [83]. Interestingly, Müller argued that TE is conceptually inadequate from a theoretical standpoint. He develops a model where proxy variables for information are introduced to account for observed efficiency differences. Using data from California’s San Joaquin Valley, he fitted a modified Cobb-Douglas production function using ordinary least squares; thus, this is a nonfrontier TE efficiency paper and the author concluded that his approach was “conceptually and analytically superior to the methodology of frontier production functions” (p. 730). It is interesting to note that the Müller paper was published three years before the seminal work by Aigner, Lovell and Schmidt (1977) and Meeusen and van den Broeck (1977) [56, 57]. The first published paper to apply a parametric production frontier approach to study TE in dairy farming seems to be Bravo-Ureta (1986), who implemented a Cobb-Douglas probabilistic production frontier à la Timmer (1971) based on data from New England, USA, for the year 1980 [55, 84]. Soon after, Dawson (1987) published what appears to be the first SPF study of dairy farming and they utilized data from England and Wales [85]. Bravo-Ureta and Rieger (1990) used farm records for New England and New York to evaluate the robustness of TE scores with respect to four different parametric frontier models estimated separately for 1982 and 1983 [86]. TE scores varied markedly across models but were highly correlated so that the ordinal rankings were similar. Bravo-Ureta and Rieger (1991) extended the Kopp and Diewert efficiency decomposition methodology from a deterministic to a stochastic framework [87]. Their method permitted the estimation of TE, AE, and economic efficiency (EE) using only estimates from an SPF model. Technical, allocative, and scale inefficiencies were analyzed by Kumbhakar, Biswas, and Bailey (1989) for Utah dairy farmers using a system of simultaneous equations comprising unconditional input demand and output supply functions that incorporate the three types of inefficiency [88]. Large farms were more technically efficient than small farms and most farms in all size categories were found to be scale efficient. Mbaga, Romain, Larue and Lebel (2003) provided another look at the robustness of TE measures using data from Quebec to compare the effect of the Cobb-

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Douglas, translog, and generalized Leontief (GL) functional forms, assuming different distributions of the inefficiency term and also included DEA measures [89]. The TE scores from all parametric models were highly correlated but correlations between the parametric and the DEA TE scores were low. Another evaluation across methodologies was provided by Balcombe, Fraser and Kim (2006) who compared a standard SPF, a Bayesian SPF, and DEA for dairy farms located in the River Murray region of Victoria and South Wales, Australia, for 1999/2000 [90]. Spearman rank correlation coefficients revealed a high positive and statistically significant association for the TE scores across all models considered. Moving on to panel data studies, Heshmati and Kumbhakar (1994) introduced a two-step model to separately identify time-invariant farm heterogeneity, farmand time-variant TE, and technological progress [91]. Kumbhakar and Heshmati (1995) extended the model developed in their previous paper by separating overall inefficiency into firm- and time-invariant components, as well as firm- and timevariant components [92]. Applications were provided in both papers to Swedish dairy farms. Cuesta (2000) generalized the 1992 Battese and Coelli approach by allowing the inefficiency term to vary across firms and over time and provided an application to Spanish dairy farms [93]. The Battese and Coelli (1995) panel stochastic frontier model was used by Kompas and Che (2006) to estimate TE for an unbalanced panel of Australian dairy farms observed in the years 1996, 1998, and 2000 [59, 94]. They found production to be characterized by constant returns to scale, with TE determined by the type of dairy shed used, feed concentration, and the number of dairy cows milked at peak season. The true fixed effects (TFE) and true random effects (TRE) models introduced by Greene [60] were used by Abdulai and Tietje (2007) to estimate time-invariant unobserved firm-specific heterogeneity for a sample of German dairy farms [95]. They also used the results from Mundlak (1978) [96] to account for possible correlation between heterogeneity and the regressors in the SPF. They found considerable variability in the results across models and concluded that unobserved heterogeneity needs careful consideration. Noting that SPF models typically utilize output-oriented (OO) TE measures, Kumbhakar and Tsionas (2008) used a nonhomogeneous SPF to calculate and compare OO and input-oriented (IO) measures and returns to scale based on balanced panel data for Spanish dairy farms for 1993–1998 [97]. The authors showed that the econometric models for the IO and OO measures are different for nonhomogeneous functional forms, so the resulting TE, returns to scale (RTS), and partial elasticities of production also differ. These differences become more pronounced the larger the deviation of RTS from unity. Input distance functions have also been used to study TE in dairy farming. Rasmussen (2010) used a stochastic input distance function based on the Battese and Coelli (1992) specification to examine scale efficiency for an unbalanced panel of Danish dairy farms for a 22-year period (1985–2006) [67, 98]. He found that average TE remained constant over time, but it was lower for older farmers and larger herds. Tsionas, Kumbhakar and Malikov (2015) estimated technical and allocative inefficiency for a panel of Norwegian dairy farms using a translog input distance

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function augmented by the set of independent first-order conditions, which makes it possible to address input endogeneity [99]. Much of the efficiency analysis in the economics literature has been based on static production models where firms are assumed to adjust their input levels instantaneously ignoring input fixity and adjustment costs over time. Dynamic models have been developed to deal with these limitations. Silva and Stefanou (2007) presented an intertemporal cost minimization nonparametric model to get short- and long-run dynamic measures of TE, AE and EE for each unit and year in a sample of Pennsylvanian dairy farms for the period 1986–1992 [100]. Serra, Oude Lansink and Stefanou (2011) implemented a quadratic specification of a parametric directional distance function to obtain TE, AE, and cost efficiency measures for a panel of Dutch farms and found that the dynamic efficiency rankings were consistent with those reported in the static literature [101]. Emvalomatis, Stefanou and Oude Lansink (2011) presented a stochastic distance function that accounted for possible persistence of TE over time (autocorrelation) [102]. In an application to German and Dutch farmers, they reported that technical inefficiency was persistent over time. A similar result was found by Skevas, Emvalomatis and Brümmer (2018), who implemented a dynamic stochastic output distance function to account for firm-level heterogeneity in long-run TE using a Bayesian estimation approach [103]. In an application to German dairy farms, they found that technical inefficiency was persistent over time and that average long-run TE was consistent with significant adjustment costs. The final papers considered in this section comprise dairy studies using metafrontiers and latent class stochastic frontiers, which are methodologies used to tackle the presence of different technologies within a sample. Battese, Rao and O’Donnell (2004) define the metafrontier as “an overarching function of a given mathematical form that encompasses the deterministic components of the SPF for the firms that operate under the different technologies” (p. 92) [104]. Refinements were incorporated by O’Donnell, Rao and Battese (2008) [105]. More recently, Huang, Huang, and Liu (2014); Amsler, O’Donnell and Schmidt (2017); and Amsler, Chen, Schmidt and Wang (2020) have developed alternative stochastic metafrontier models [106–108]. While the metafrontier approach has been employed in studies on several sectors, relatively few applications to dairy farming are available. The first such application of the deterministic metafrontier model to dairy farm data was by Moreira and Bravo-Ureta (2010) in their analysis of farms located in Argentina, Chile and Uruguay [109]. Latruffe, Fogarasi and Desjeux (2012) used the metafrontier approach to compare the productivity of Hungarian and French dairy farms (field crop farms were also included) and Jiang and Sharp (2015) used the deterministic approach to compare the performance of dairy farms located in the North and South Island of New Zealand [110, 111]. The only stochastic metafrontier study found for milk producers to date is Alem, Lien, Hardaker and Guttormsen (2019), who analyzed a rich panel dataset to evaluate dairy farm performance in five regions in Norway [112].

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An alternative to metafrontiers to capture intra-sample differences in technology is the latent class stochastic frontier model proposed by Orea and Kumbhakar (2004) [113]. This model was used by Álvarez and del Corral (2010) to estimate the technology of a sample of Spanish dairy farms that included intensive and extensive farms [114]. They found that intensive farms were more productive and more technically efficient than farms using the extensive technology. Another application of latent class stochastic frontiers to Spanish dairy farms was provided by Orea, Pérez-Méndez and Roibas (2015) who used this model to investigate the effects of land fragmentation on technology choice and TE [115]. They found that land fragmentation reduced the probability of adopting extensive production processes and had a greater effect on TE in extensive farms. Readers interested in a more detailed discussion of the TE literature focusing on dairy and other agricultural products can go to various meta-analyses that have been undertaken including Bravo-Ureta, Solís, Moreira López, Maripani, Thiam and Rivas (2007), Moreira and Bravo-Ureta (2009), and Ogundari (2014) [116–118].

Output Growth and Total Factor Productivity This section reviews papers that have analyzed output growth and total factor productivity (TFP) in dairy farming. It starts by providing a brief overview of key methodological contributions that have been made to decompose and better understand the sources of output and TFP growth. An early contribution toward TFP decomposition was Caves, Christensen and Diewert (1982a) who relied on Diewert (1976) to define “superlative” output and input indexes based on a constant returns to scale translog transformation function consistent with the Törnqvist index [119, 120]. Soon after, Caves, Christensen and Diewert (1982b) established the relationship between Törnqvist input and output indexes and the Malmquist index [121]. Contemporaneously, Nishimizu and Page (1982) stated that technological progress and TE have a common foundation on the production function; however, they argued “that applied work in these fields has evolved largely independently” (p. 920) [122]. These authors were pioneers in incorporating TE in addition to technological progress as a source of productivity change and they estimated a deterministic production frontier following the linear programming model introduced by Aigner and Chu (1968) [54]. Färe, Grosskopf, Norris and Zhang (1994) implemented a nonparametric mathematical programming framework to decompose Malmquist productivity indexes into changes in technological progress and TE for 17 OECD countries. They referred to the former component as shifts in the world frontier coming from innovation while changes in TE represent movements from individual countries toward the world frontier or catching up [123]. Kumbhakar and Lovell (2000) estimated a translog stochastic production frontier, calculated productivity change, and then decomposed the latter into three elements: technological progress, returns to scale, and time-varying TE [124]. They also presented various decompositions for dual

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cost and profit frontiers as well as multi-input multi-output technologies. More recently, O’Donnell (2016 and 2018) introduced the “proper” TFP index, which needs to satisfy a number of axioms and is consistent with measurement theory [125, 126]. O’Donnell decomposed TFP changes into various components including scale, technology, TE, environmental, and statistical noise. Moreover, he argues that commonly used TFP indexes (e.g., Fisher and Törnqvist) are not “proper” because they do not comport with measurement theory and violate important index number axioms. Turning now to empirical dairy studies that have focused on output growth and TFP, Ahmad and Bravo-Ureta (1995) used alternative fixed effects production functions and SPF models to measure TE and to decompose dairy farm production growth into TE, technological progress, and input growth for Vermont dairy farms using the Nishimizu and Page (1982) framework [122, 127]. Weersink and Tauer (1990) assessed the regional and temporal impacts of technological progress in the US dairy sector using a dynamic dual optimization model [128]. They found that the major milk-producing regions of the Northeast and Lake States benefited most from existing trends in productivity at that time. Weersink and Tauer (1991) applied multivariate Granger causality tests to investigate the relationship between productivity per cow and farm size using US data for the period 1964–1987, finding that the relationships varied considerably across states [129]. Parametric output distance functions have been used to decompose TFP growth by Brümmer, Glauben and Thijssen (2002); Newman and Matthews (2006); Emvalomatis (2012); and Cechura, Grau, Hockmann, Levkovych and Kroupova (2017) [130–133]. In the first of these papers, Brümmer, Glauben, and Thijssen (2002) used an output-oriented approach to decompose TFP growth into technological progress, TE change, AE change and scale components for Dutch, German and Polish farms over the period 1991–1994 [130]. They found that productivity growth in Germany and Poland was driven mainly by technological progress, whereas for the Netherlands the AE component was the most important driver. Newman and Matthews (2006) decomposed TFP into technological progress, TE change and changes in scale efficiency for Irish farms for the years 1984–2000 and found that productivity growth over the period was driven entirely by technological progress [131]. Emvalomatis (2012) used a random coefficients specification of an output distance function to measure and decompose productivity in German dairy farming [132]. TFP growth was decomposed into technological progress, TE change, and a scale effect. Average TFP growth over the period was estimated at 1.1%, which was driven primarily by technological progress. Cechura, Grau, Hockmann, Levkovych and Kroupova (2017) used a stochastic output distance function to analyze productivity for 24 EU member states and decomposed TFP into scale, TE, technological progress, and heterogeneity effects [133]. Among their results was that the impact of technological progress in Eastern Europe was lower than in the rest of the EU. Parametric input distance functions were used to analyze productivity change by Sipiläinen, Kumbhakar and Lien (2014), Sauer and Latacz-Lohmann (2015), and Singbo and Larue (2016) [134–136]. Sipiläinen, Kumbhakar and Lien (2014)

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analyzed the profitability and productivity dynamics of Finnish and Norwegian farms by estimating a translog input distance function and decomposing profitability change into output growth, output price change, input price change, technological progress, scale, markup, and TE change [134]. This decomposition permits TFP change to be obtained from profitability change. Sauer and Latacz-Lohmann (2015) analyzed productivity for a sample of German dairy farms where TFP change was measured with a Luenberger index and decomposed into efficiency change and technological progress [135]. The latter was found to be the main driver of TFP change while innovation, proxied by net investment, contributed to productivity change through its effect on technological progress. Singbo and Larue (2016) estimated a stochastic input distance function for a sample of Quebec dairy farms to decompose TFP into technological progress, TE change, scale efficiency change, and an input-mix component, finding that the farms in their sample were operating at suboptimal scale [136]. An application using a parametric stochastic production frontier as opposed to multi-output distance functions is Moreira and Bravo-Ureta (2016), who analyzed TFP change for a panel of Chilean dairy farms [137]. Decomposing TFP change into technological progress, TE change, scale efficiency change, and allocative change, they found that technological progress was the greatest contributor to TFP change. No relationship was found between farm size and productivity growth. DEA methods have also been applied in the analysis of TFP in dairy farming. Tauer (1998) examined the productivity change of New York dairy farms over the period 1985–1993 with Malmquist indices using DEA [138]. Productivity change was decomposed into TE change and technological progress and positive annual productivity growth was found to come mainly from technological progress. Latruffe, Fogarasi and Desjeux (2012), referred to briefly in the previous section, used a DEA metafrontier approach to compare the productivity of Hungarian and French dairy farms for the 2001–2007 period [110]. They found no differences in productivity between the countries based on their own frontiers. Using the metafrontier, however, they found that Hungarian farms are more productive. Finally, Jang and Du (2019) estimated productivity for dairy farms in the USA using recent developments in the literature on control functions to correct for biases induced by simultaneity and sample selection [139]. Computing time-varying productivity at the farm level, they found that surviving farms contribute more to regional productivity growth than entering and exiting farms. Herd size variation was negatively associated with productivity at the state level. To summarize, this section showed that stochastic input and output distance functions have been used widely for the measurement and decomposition of productivity in dairy farming, though DEA approaches have also been used. Regardless of the method used, the studies reviewed highlight the importance of technological progress as a driver of productivity growth.

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Cost Function Approaches: Efficiency and Economics of Scale, Size, and Scope The literature on TE and productivity reviewed in the previous two sections involved primal approaches (i.e., production and distance frontiers). This section turns to studies of efficiency and productivity that rely primarily on cost function models, providing an overview of selected articles that have dealt with cost efficiency and economies of scale, size, and scope. To clarify concepts, economies of size is a measure of the relative change in output with respect to changes in all inputs along the expansion path. In contrast, economies of scale measures how output responds when all inputs are changed in the same proportion, i.e., along a ray through the origin or the scale line. If the technology is homothetic then the expansion and scale lines are equivalent and thus economies of size and scale correspond to each other [140]. Economies of scope are present when “it is less costly to combine two or more product lines in one firm than to produce them separately” [141] (p. 268). Hoch (1976) is an early example of using a production function to investigate scale economies in dairy farming [74]. Before looking at papers dealing with cost efficiency per se, it is important to note that the issue of determining the most appropriate specification of technical inefficiency in a cost frontier was addressed by Orea, Roibás and Wall (2004) [142]. They estimated three versions of a translog cost frontier for a panel of Spanish dairy farms which differ according to the way technical inefficiency is introduced: input oriented, output oriented, and hyperbolic. The Vuong test was used to select the most appropriate model, and the authors found that the cost frontier with input-oriented TE was the best choice for their data. Cost efficiency in dairy was treated by, among others, Lund, Jacobsen and Hansen (1993); Cocchi, Bravo-Ureta and Cooke (1998); Maietta (2000); Hailu, Jeffrey, and Unterschultz (2005); and Álvarez, del Corral, Solís and Pérez (2008), which are now examined in turn [143–147]. Lund, Jacobsen and Hansen (1993) used linear programming methods to study the cost efficiency of a sample of Danish dairy farms, finding that pure cost inefficiencies were much more important than inefficiencies arising from suboptimal farm size [143]. Cocchi, Bravo-Ureta and Cooke (1998) derived cost efficiency indexes for US and Canadian dairy farmers and decomposed them into technological progress, regional competitive advantage, and economies of size [144]. A policy implication of their results is that the elimination of dairy support programs would accelerate the trend towards fewer and larger farms, with these changes varying across states. Maietta (2000) used a shadow cost model to decompose cost inefficiency into technical and allocative inefficiencies for a panel of Italian dairy farms [145]. Excess costs were found to stem primarily from technical inefficiency. Hailu, Jeffrey and Unterschultz (2005) estimated non-homothetic frontier cost functions with and without local concavity in input prices imposed to analyze cost efficiency of Alberta and Ontario dairy farms [146]. They found that imposing curvature had little effect on the cost efficiency estimates but did yield more plausible estimates of own-price and cross-price

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elasticities. Finally, Álvarez, del Corral, Solís and Pérez (2008) estimated separate stochastic average total cost frontiers for intensive and extensive dairy farms in northern Spain [147]. They found that intensive farms produced at lower average total cost and showed greater levels of efficiency than extensive farms. Economies of size and scale in dairy farming have received considerable attention. The relationship between farm size and productivity has important implications for farm policy, particularly in developing countries [148]. The structure of agriculture and the resilience of small-scale farms have been the focus of considerable discussion over the years and have regained popularity as a topic in recent times [149]. The sharp decline in the number of dairy farms motivated Matulich (1978), which is a good source for previous scale/size studies, to analyze the cost structure of large specialized Californian dairy farms (375–3600 cows) [150]. While generating short- and long-run average cost curves (LRAC) the results reveal that “significant economies of size were evident up to 750-cow herds” (p. 645), and the LRAC became quite flat thereafter. The influence of managerial ability on economies of size was analyzed by Dawson and Hubbard (1987) for English dairy farms. These authors use crosssectional data for 1980–1981 where they included “margin over feed cost per liter of milk” as a proxy for management [151]. Estimated output was incorporated into a translog LRAC function along with the management proxy and they found that the LRAC was U-shaped but exhibited more pronounced economies than diseconomies of size, with better managers producing any output amount at a lower average cost. In a similar formulation to Dawson and Hubbard (1987), Alvarez and Arias (2003) assumed that managerial ability is fixed and used a two-step procedure to estimate an average cost model [152]. Using Spanish dairy farm data, they found that managerial ability played a key role in the level of size economies achieved and recommended that policies promoting growth in farm size be complemented with actions that enhance managerial capacity, such as suitable extension programs. Mukhtar and Dawson (1990) also estimated a LRAC function for a sample of English and Welsh dairy farmers and reported considerable economies of size at small levels of production but less pronounced diseconomies at larger levels [153]. Moschini (1990) used nonparametric and semi-parametric methods to analyze scale economies for Ontario dairy farms [154]. Using a multiproduct cost function, he found that the technology exhibited substantial scale economies. Løyland and Ringstad (2001) examined economies of scale in Norwegian dairy farming by applying variations of the standard Cobb-Douglas form that allowed for U-shaped cost functions [155]. These authors found that exploiting scale economies fully would lead to markedly lower cost and a substantial drop in farm numbers. Mosheim and Lovell (2009) found substantial scale economies in an application using US survey data and stressed the importance of taking technical and allocative inefficiency into account when estimating scale economies [156]. The static concepts of scale and scope economies were extended into a dynamic adjustment-cost framework by Fernández-Cornejo, Gempesaw, Elterich and Stefanou (1992) [157]. In their empirical work, they estimated dynamic measures of

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scope and scale for a sample of German dairy farmers observed over the period 1981–1988 covering the period when production quotas were introduced. They found that as size increases, so does output specialization. Farm specialization, as well as large milk output, was associated with lower marginal costs by Wieck and Heckelei (2007) in a study of eight important European dairy farming regions from five countries (Denmark, France, Germany, the Netherlands and the UK) [158]. Alem, Lien, Kumbhakar and Hardaker (2019) analyzed economies of scale and scope for a panel of Norwegian farms using a system of equations including a flexible translog cost function and input shares and found that farm costs can be significantly lower by increasing scale, i.e., larger crop and dairy output, and scope, i.e., by producing both crops and milk in the same farm rather than in separate units [159]. Intertemporal cost-minimizing behavior among US dairy farmers was tested by Silva and Stefanou (2003) [160]. They developed a nonparametric dynamic dual cost approach to recover technological information from intertemporal costminimizing behavior without imposing a parametric functional form on the technology. In an application to a panel of Pennsylvania dairy operators, intertemporal cost-minimizing behavior was not supported and the joint hypothesis of constant returns to scale and dynamic cost minimization was rejected. Finally, behavioral objectives, and in particular the issue of whether farms behave in accordance with profit maximization or cost minimization, were addressed by Tauer (1995) who tested the weak axiom of profit maximization and the weak axiom of cost minimization for a sample of New York state dairy farms [161]. He found that farmers were not successful in maximizing profits but came closer to cost-minimization behavior. Stefanou and Saxena (1988), who developed a generalization of the dual (nonfrontier) profit function to test for the influence of training variables on allocative efficiency, reported that Pennsylvania dairy farmers allocated variable inputs to maximize production [162]. In sum, these cost-based studies show that the nature of the LRAC curve is far from universal, though several studies have pointed to U-shaped curves where cost per unit first declines (economies) as output increases, reaches a minimum, and then increases (diseconomies) as output continues to rise. Economies at low levels of output tend to be more important than diseconomies at larger levels. The empirical evidence on economies of scope is mixed, with some authors reporting that greater farm specialization is associated with lower costs and other findings that costs could be reduced by producing multiple outputs. A final common finding is that managerial ability has been found to play a crucial role in attaining cost advantages.

Technology Adoption The study of technology adoption, an area of enquiry pioneered by Griliches (1957), has a long tradition in agricultural economics [163]. This section provides an overview of technology adoption studies in dairy farming, covering animal-related

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technologies including growth hormones and genetic selection, the choice between conventional and organic technologies, and intensive and extensive technologies. Starting with animal-related technologies, Giesen, Oskam and Berentsen (1989) examined the expected profitability of adopting bovine somatotropin (BST) – or bovine growth hormone – in the Netherlands, a topic that received considerable attention in the 1980s [164]. The authors used representative farm and linear programming methods and found positive effects at the farm and national levels. A similar study is Marion and Wills (1990) who focused on Wisconsin farms and concluded that the expected economic effects would be significant but lower than reported in other studies, and sensitive to economic assumptions [165]. Stefanides and Tauer (1999) used econometric procedures to correct for selectivity bias in the adoption of BST for a panel of New York dairy farms and concluded that BST had a positive effect on output per cow but a negligible impact on profits [166]. The article includes a review of earlier studies on BST adoption aiming to identify the relation between socioeconomic characteristics of farmers and BST adoption intentions. Foltz and Chang (2002) studied the adoption and profitability of BST for a sample of Connecticut dairy farms surveyed in 1999 [167]. Adoption was estimated with probit and tobit models, and then adoption was endogenized in estimates of milk production and farm profit rates. In line with Stefanides and Tauer (1999), they found that adoption significantly increased milk production but had virtually no impact on profit per cow. The effects of adopting dairy management information systems (DMIS) were analyzed by Tomaszewski, van Asseldonk, Dijkhuizen and Huirne (2000) for Dutch dairy farms using regression analysis of panel data (1987–1996) for a group of adopters and a similar group of nonadopters (i.e., controls) [168]. They found that adopters of the technology achieved significant per-cow increases in milk and protein production per cow as well as a shorter calving interval compared to nonadopters. The effect of genetic selection on productive performance has also received attention in the literature. Roibás and Alvarez (2010) analyzed the impact of genetics on the profitability of dairy farming in northern Spain [169]. Employing an SPF translog model along with two auxiliary feed expense equations, they found that changes in the genetic index over time had a positive impact on profits and such effect was significantly higher for better managed farms (i.e., those with higher TE). Atsbeha, Kristofersson and Rickertsen (2012) examined the effect of genetics on the productivity of a panel of Icelandic dairy farms and found that 19% of the average annual productivity growth rate was contributed by the breeding technology [170]. For recent studies that measure the effects of genetic selection on efficiency and farm profits, see Whitt, Tauer and Huson (2019) for US dairy farms and Pérez-Méndez, Roibás and Wall (2020) for dairy farms in Spain [171, 172]. The comparison of organic and conventional technologies was studied by Kumbhakar, Tsionas and Sipiläinen (2009) who used an SPF model to jointly estimate the underlying technology and its adoption, considering that the latter is both endogenous and affected by TE [173]. In an application to Finnish farms, they found the conventional technology to be more productive. Mayen, Balagtas and

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Alexander (2010) also examined the impact of adopting organic dairy production technology on productivity and TE [174]. Using USDA data for 2005, they applied propensity score matching techniques to account for selectivity bias from observable variables. Organic and conventional producers exhibited different frontiers and again the former group was less productive than the latter while both groups exhibited similar average TE. In a study based on ARMS data for 2003, Nehring, Gillespie, Sandretto and Hallahan (2009) explored whether small dairy farms could compete with large ones using conventional and pasture-based technologies [175]. Results of a translog stochastic input distance function revealed that farms using a pasture-based system had more room to exploit scale economies than conventional operations. Ma, Bicknell and Renwick (2019) investigated the switch from a pasture-based to more intensive feeding technologies on TE using a fixed effects SPF model for New Zealand dairy farms [176]. The findings showed that more intensive feeding technologies, herd size, and milking frequency all had a positive effect on TE. Intensive systems had also been found to be more efficient for Spanish farms by Álvarez, del Corral, Solís and Pérez (2008) [147]. Overall, several of the studies considered in this section reveal the potential of various technologies such as BST, DMIS, and genetic selection to generate positive effects on production, profits, TE and scale economies. Moreover, conventional farming seems to be more productive than organic and intensive technologies to be more productive than extensive ones.

Supply Response and Government Intervention Supply elasticities have an important influence on the impact of government policies designed to support milk producers, a common feature of the dairy sector across many countries. This section reviews the production economics literature that has dealt with supply elasticities and implications of government intervention in dairy farming. There is a long literature on milk supply response and the estimation of the related elasticities. The early contributions have been reviewed in Sect. 2, so here the focus is on salient papers from the 1980s onwards. Levins (1982) stated that much of the econometric literature on milk supply response in the 1970s relied on the Nerlove (1956) partial adjustment model and the polynomial distributed lag model (Chen, Courtney and Schmitz 1972) [16, 177, 178]. The author argued that direct estimation of lagged price parameters in milk supply response models offered advantages relative to alternative models in terms of theoretical simplicity and the elimination of assumed a priori parameter restrictions. Early research on the factors determining milk supply was summarized by Buckwell (1984) who contended that concentrating on price-output relationships does not provide a satisfactory explanation of milk production [179]. He provided the first specific application to the dairy sector of the theory advanced by Kislev and Peterson (1982) and that changes in farm size can be explained by changes in relative factor prices, given

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fixed labor input [180]. Using UK data, he found that growth in milk output was determined mainly by growth in average herd size, which was in turn determined by the relative price of capital to labor. Howard and Shumway (1988) highlighted the role of herd size on the long-run milk elasticity of supply reported in various studies and applied a dynamic dual model to the US dairy industry to estimate the rate of adjustment of herd size and labor to their optimal values [181]. They concluded that dairy policies such as price support programs would need at least a 10-year duration to achieve fully their potential effects. An influential paper by Chavas and Klemme (1986) presented a dynamic model of herd composition and supply response in the US dairy sector, considering the herd as a capital good [182]. They modeled US milk production using yearly data over the period 1960–1982 by bringing together a dynamic model of the aggregate dairy herd’s size and age structure with a cow productivity equation. They found small short-run supply elasticities, implying that price supports would not lead to notable excess supply in the short run. As farmers react strongly to relative price changes in the long run, however, they concluded that it would be difficult to reduce excess supply once herd size has been expanded. Several subsequent papers build on the work of Chavas and Klemme (1986). Chavas and Kraus (1990) found that the short-run milk supply response to market prices was very inelastic for five US Lake States during 1950–1985 and that it would take at least seven years for this response to become elastic [183]. Adelaja (1991) conceptualized the relationships between long- and short-run supply elasticities for yields, farm size, and herd population for northeastern US dairy farms over the period 1971–1985 [184]. Price responsiveness was found to decrease with farm size in the short run, whereas in the long run large farms are more price responsive than smaller ones. Also highlighted were the implications of changes in price support on the distribution of revenues within the sector. Bozic, Kanter and Gould (2012) examined the evolution of the long-run US milk supply elasticity based on aggregate national data covering the period 1975–2010 [185]. They reported large differences between short-run and long-run responses to price changes, leading them to conclude that policymakers must take medium- and long-run policy impacts into account. Furthermore, the responsiveness to feed prices implies that dairy policy should focus on managing dairy farm profit margins rather than revenue streams. Short-, intermediate-, and long-run price elasticities of output and inputs were analyzed by Thijssen (1994), who estimated a dynamic factor demand model for Dutch dairy farms [186]. Investments were found to be sensitive to price and technological progress, but price elasticities of output and variable inputs were found to be small in the short run and even in the long run, limiting the effectiveness of price policies on output supply and variable input demand. A restricted profit function was used by Quiroga and Bravo-Ureta (1992) to analyze the structure of dairy technology for a sample of Vermont dairy farmers [187]. They found that observed levels of quasi-fixed inputs were significantly lower than their long-run optimal values and concluded that lower milk prices would threaten the viability of small- and medium-sized farms in the short run and decrease optimal farm size in the long run.

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Technological progress can counteract the effect of price changes on milk supply. This was explored by Blayney and Mittelhammer (1990), who used aggregate production and profit functions to decompose milk supply response into technology and price effects for the Washington state dairy sector and Munshi and Parikh (1994) for India [188, 189]. Blaney and Mittelhammer (1990) reported that technological advances outweigh lower supply due to decreases in milk prices, while Munshi and Parikh (1994) found that the dissemination of information may be more effective than strategies geared at providing technical inputs. The remainder of this section deals more explicitly with the effects of quotas and price subsidies. The impact of quota policies has been a popular topic of research. Stefanou, Fernández-Cornejo, Gempesaw and Elterich (1992) derived and estimated intertemporal cost-minimizing investment and variable demand functions to compare pre- and post-quota period producer responses of German dairy farmers in the early 1980s [190]. They found evidence of considerable excess production capacity and that the introduction of the quota changed variable input responses and investment behavior. The costs of quantity restrictions and transfer costs in dairy quota exchanges were examined by Boots, Oude Lansink and Peerlings (1997) for Dutch dairy farms [191]. Estimating a system of input demand and output supply, they found that the free trade of quotas would increase profits by 9%, and that small farmers would gain more than large farmers from quota trading in a context of trade restrictions. Similarly, Sauer (2010) analyzed the effect of quota transferability on the production structure of Danish conventional and organic farms [192]. He found an increase in overall market efficiency over time due to success in allocating quotas to the more efficient farms as well as an upward shift towards organic milk production. Colman, Burton, Rigby and Franks (2002) estimated a cost function for a crosssection of UK dairy farmers to simulate the adjustment in the sector to different dairy policy reform scenarios and concluded that incomes would be maintained only if herd size increased considerably and the number of producers declined [193]. Pierani and Rizzi (2003) examined the effect of the introduction of a milk quota in 1984 on the behavior of Italian farmers and provided evidence of a rigid productive structure during the pre- and post-quota period [194]. The relation between quota values and economic efficiency was analyzed by Álvarez, Arias and Orea (2006) for a sample of Spanish farmers [195]. They argued that efficient firms should be net purchasers while inefficient firms should be net sellers of milk quotas. Their results suggested that economic efficiency was far more important than farm size in explaining quota values, with the implication that authorities should take efficiency into account when allocating quotas. Kumbhakar, Lien, Flaten and Tveteras (2008) investigated the effect of quota policy on the output growth of a panel of Norwegian farms observed over the period 1976–2005 [196]. They showed that quota regulations had a negative effect on output growth and technological progress and suggested that a policy that does not have productive performance as a primary objective may have contributed to the increase in off-farm work captured in the dataset. The explanation of inefficiency under a quota regime was analyzed by Ang and Oude Lansink (2018) using a DEA framework where

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dynamic profit inefficiency was decomposed into technical and allocative inefficiency [197]. The authors reported considerable dynamic profit inefficiency among Belgian dairy farmers, which was primarily attributed to allocative inefficiency in the outputs and variable inputs. The effects of deregulation and milk policy reform in the EU have been investigated by several authors. De Frahan, Baudry, De Blander, Polomé and Howitt (2011) analyzed the farm-level supply and income effects of removing milk quotas and reducing producer prices while increasing direct compensatory payments for a sample of Belgian dairy farmers covering the period 1996–2006 [198]. They found that quota removals with a 20% drop in milk prices would maintain aggregate milk supply and farm income at its reference level. Latruffe, BravoUreta, Carpentier, Desjeux, and Moreira (2017) analyzed the relationship between agricultural subsidies and dairy farm TE in nine Western European countries [199]. Estimating a stochastic production frontier using a method of moments estimator that addressed input endogeneity, they found that decoupling following the 2003 CAP Reform weakened the link between subsidies and TE but did not change the direction of such link except for Italy. Frick and Sauer (2018) analyzed the relationship between deregulation, represented by the phasing out of the EU milk quota, and efficient resource allocation [200]. Using data for German dairy farms for the period 2000–2014, they found evidence that deregulation led to higher productivity through greater reallocation of resources towards more productive farms but found no effect of output price risk on this reallocation. While the papers just mentioned were centered on the EU, several authors have studied the effects of intervention policies on efficiency for other regions. Thus, Bezlepkina, Oude Lansink, and Oskam (2005) analyzed the effect of subsidies in dairy farming in the Moscow region of Russia [201]. They concluded that while subsidies had a distorting effect on the input-output mix, they relieved credit constraints while improving AE. Slade and Hailu (2016) studied the cost efficiency of dairy farms in Ontario and New York State between 2005 and 2007, which operated under two different regulatory regimes [202]. Farmers in Ontario received higher government support and were found to be more cost inefficient than those in New York mainly due to greater AE, leading the authors to conclude that managers in more competitive environments make better decisions. Larue, Singbo and Pouliot (2017) evaluated the impact of supply management policies on Quebec farms [203]. They reported that a fall in the volume of production quota traded had large effects on TE and established that exchange regulations, such as price ceilings, should be removed in order to make production quota available to farms willing to make technological innovations and herd size adjustments. Komaki and Penzer (2005) used a structural time series model to estimate price elasticities for two Japanese regions over the period 1970–1997, which covered major dairy policy changes in the form of price supports and quotas [204]. Overall, the supply response literature generally points to relatively inelastic short-run milk supply and much higher long-run elasticity, with adjustment in the long run potentially taking several years. The role of relative factor prices and herd size has been emphasized in long-run adjustment and technological progress has

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been shown to be able to counteract the long-run impact of price adjustment on output. Results indicate that tradeable quotas can increase market efficiency while contributing to technological progress and herd size adjustment. Studies have also supported the notion that deregulation can have a positive effect on dairy farm productivity.

Risk and Uncertainty Risk and uncertainty are inherent to agricultural activities and have been the subject of a large theoretical and empirical literature in agricultural economics (see, for example, the review by Moschini and Hennessy 2001, [205]). The dairy sector is no exception, with producers facing multiple risks including production risk, volatile output qualities, as well as institutional and market price risk. Finger, Dalhaus, Allendorf and Hirsch (2018) provide relevant references on these sources of risk [206]. As argued by Antle and Goodger (1984), the stochastic properties of production technologies may have important effects on producer behavior so that an important task for economists is to estimate and test stochastic technologies and use the estimates to evaluate alternative decision models [207]. In the dairy farming literature, the most common models used to estimate stochastic technologies have been the heteroskedastic Just-Pope production function ([208]) and variants of the moments-based approach of Antle (1983) [209]. In recent years, state-contingent production models, which consider that producers can manage uncertainty through the allocation of productive inputs to different states of nature, have become increasingly popular in agricultural economics in general, though applications to dairy have been scarce [210, 211]. The flexible moment-based approach of Antle (1983) was used by Antle and Goodger (1984) to estimate the stochastic structure of large-scale dairy farms in California and highlighted the importance of generalizing decision models to incorporate third and potentially higher moments [207]. Their results indicated that capital-intensive farms were riskier. Comparing a mean-variance (MV) criterion with the mean-variance-skewness (MVS) criterion yielded a particularly interesting result: the MV criterion indicated that the effects of uncertainty were unimportant for decision-making, whereas the MVS criterion showed that uncertainty had important effects on optimal decisions. Finger, Dalhaus, Allendorf and Hirsch (2018) followed Antle (1983) by assuming that producers choose inputs to maximize utility taking into account the moments of the different outputs produced [206]. They explored the determinants of German dairy farmers’ risk exposure in an analysis combining downside risk, the effect of climatic extremes, and animal health in a multi-output stochastic production framework. They found that animal health affects average revenues and production risk. An interesting policy implication was that income stabilization measures may encourage riskier production decisions at the expense of animal health. The Just-Pope approach was used by Tveteras, Flaten and Lien (2011) and Orea and Wall (2012) [212, 213]. Tveteras, Flaten and Lien (2011) estimated a multi-

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output version of the Just-Pope production function for a panel of Norwegian dairy farms, specifying mean, variance, and covariance production functions. They found that inputs primarily increase output variance, with the risk-reducing effects of inputs mainly present in the covariance functions. They also found that technological progress shifted the profit distribution, increasing mean profit but also the variance of profit, so that no welfare improvement was found for risk averse farmers. Orea and Wall (2012) used the Just-Pope approach in an analysis of how technological progress, production risk, and risk attitudes interact to affect producer welfare. Using a sample of Spanish dairy farms, a welfare index was presented comprising total factor productivity (TFP), production risk, and risk preferences. The study showed how producer welfare may fall despite positive TFP growth when this growth is associated with an increase in production risk and farmers are risk averse. An alternative approach was used by Lien, Kumbhakar and Hardaker (2017), who estimated a translog input distance function incorporating three risk-related indices as variables along with those defining the technology [214]. In an application analyzing the effects of risk on productivity for Norwegian farms they found that decreases in risk aversion, increases in optimism of risk perceptions, and increases in risk management skills all lead to higher productivity. The impact of price uncertainty has also received attention in the production economics-oriented dairy farming literature. Melhim and Shumway (2011) used the flexible mean-SD utility approach introduced by Saha (1997) to estimate risk preferences and derive dual estimates for scope, product-specific, and multi-output scale economies under price uncertainty for a sample of US dairy farms for 2000 [215, 216]. Dairy producers were found to be risk averse and the authors asserted that ignoring these risk preferences would lead to underestimation of the effect of scope economies for large farms, highlighting the importance of incorporating risk preferences when estimating scope and scale economies. Pieralli, Hüttel and Odening (2017) analyzed how milk price uncertainty and TE affect the decision of farmers to abandon milk production in a real options model [217]. Using German data, they found that abandonment was related to lower efficiency in milk production but that greater price volatility reduced the probability of abandonment. Finally, the state-contingent production approach was used by Mallawaarachchi, Nauges, Sanders and Quiggin (2017) to estimate a stochastic technology for a panel dataset of Australian irrigated dairy farms [218]. Defining two states of nature to reflect favorable and unfavorable conditions of water availability, they found that the production technology was consistent with state-contingent technology. Thus, when seasonal conditions were variable, milk production was relatively stable, with farmers using state-allocable inputs to manage their exposure to unfavorable conditions. This implies that producers responded to information by reallocating inputs towards states of nature that appear more likely in the light of new information. While there is a sizeable literature on risk and uncertainty in agriculture in general, applications to dairy have been surprisingly scarce. What little literature there is seems to point to risk aversion being negatively related to productivity and

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highlights that technological progress may not improve farmers’ welfare if they are risk averse.

Sustainability This section reviews dairy production economics papers that deal with sustainability. A broad view of sustainability is considered, encompassing the effects of weather and climate change, animal welfare, and environmental efficiency. There is a small but growing literature on the effects of weather and climate change on production performance. Mukherjee, Bravo-Ureta and De Vries (2013) appears to be the first paper to use an SPF framework to study the effects of climatic variables on dairy production [219]. Using the Battese and Coelli (1995) approach they found a significant negative effect of heat stress on farm productivity and a robust positive return to investing in fans combined with sprinklers as an adaptation mechanism [59]. Key and Sneeringer (2014) also applied the Battese and Coelli (1995) SPF approach to examine the relationship between heat stress, output losses, and TE [220]. Using data for US conventional dairy farms they reported a negative association between TE and expected heat stress levels. The influence of weather on milk production was examined for a panel of Spanish dairy farms by PérezMéndez, Roibás and Wall (2019) [221]. In their model, weather can affect milk output through direct effects on cow and forage performance, and indirectly through other inputs. They found that warm weather had a substantial positive impact on milk output due to better forage production. The consequences of climatic conditions on farm productivity were assessed by Qi, Bravo-Ureta and Cabrera (2015) using panel data for Wisconsin dairy farms [222]. Their results showed that higher summer and autumn temperatures had a negative effect on milk output. Warmer winters and springs had a positive effect whereas higher precipitation led consistently to negative outcomes. The combined climatic effects on milk output over the 17-year period studied were negative. In a related paper, Njuki, Bravo-Ureta and Cabrera (2020) used a random parameters SPF model and examined the differential effects of weather and climate on TFP change [223]. Annual TFP growth was 2.16%, coming mainly from technological progress. Animal welfare has been a theme of increasing attention among consumers, producers, and policymakers in recent years [224], and there is a small but growing production economics literature dealing with milk production. Measures to improve animal welfare generally come at a cost, and the cost-effectiveness of measures to alleviate disease and heat stress has been addressed by several authors. For example, Chi, Weersink, Van Leeuwen and Keefe (2002) designed a model to determine optimal cost-minimizing strategies for managing four infectious diseases that can significantly diminish milk production [225]. Gunn, Holly, Veith, Buda, Prasad, Rotz, Soder and Stoner (2019) studied the cost-effectiveness of alternative heat abatement strategies to reduce heat exposure in US dairy [226].

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The focus of the following paragraphs are studies relating animal welfare to productive performance (fundamentally TE), beginning with parametric studies and then turning to nonparametric studies, which have been far more numerous. Beginning with parametric studies, Lawson, Agger, Lund and Coelli (2004) used stochastic frontier analysis to study the TE outcomes associated with various illnesses and reproductive disorders for Danish dairy farms [227]. They found that the adverse effects of reproductive disorders on milk production were compensated by good managerial decisions in efficient farms. Atsbeha, Kristofersson, and Rickertsen (2012) estimated a cost system to explore the connection between genetic progress, animal health and fertility characteristics, and variable costs for a panel of Norwegian dairy farms observed over the period 1999–2007 [170]. Their results pointed to significant variable cost savings from genetic progress in production, health, and fertility traits. Finally, Pérez-Méndez, Roibás and Wall (2020) assessed how health, reproductive conditions, and genetic selection can affect TE and profits using a stochastic frontier approach for a panel dataset of Spanish dairy farms [172]. DEA methods feature far more prominently in the animal welfare literature. Hansson and Öhlmér (2008) used DEA to evaluate the effects of animal breeding, health, and feeding decisions on the efficiency of Swedish dairy farms and found that breeding practices were important [228]. Hansson, Szczensa-Rundberg and Nielsen (2011) analyzed the effects of preventative measures against mastitis on dairy farm efficiency, again using DEA for Swedish dairy farms [229]. Barnes, Rutherford, Langford and Haskell (2011) reported a negative association between lameness and TE for a sample of British dairy farms [230]. Another strand of the literature explores the possibility that farmers may deliberately trade off use value (i.e., priced) and nonuse value in the management of livestock. This so-called ”rational inefficiency” literature applied to dairy farming assumes that some farmers may supply animal welfare beyond levels that would be optimal from a purely financial perspective due to the existence of nonuse value. Hansson, Manevska-Tasevska and Asmild (2018) provide a discussion of this literature and an application to Swedish dairy farms using multidirectional efficiency analysis (MEA) [82]. The last group of papers reviewed in this section addresses environmental efficiency, which can be defined in various ways. Reinhard, Lovell and Thijssen (1999) is an early study that applied econometric procedures to examine environmental efficiency in dairy farming [231]. Using data on Dutch dairy farms, environmental efficiency was measured using an input that captured the excess nitrogen derived from overuse of manure and chemical fertilizer. Reinhard and Thijssen (2000) used similar data to estimate input-oriented TE and mean nitrogen efficiency, where the latter was an aggregate measure incorporating the nitrogen content of all relevant inputs and output [232]. Shortall and Barnes (2013) investigated the connection between TE and environmental efficiency for Scottish farms where they defined greenhouse gas (GHG) emissions as an environmental bad [233]. They reported that farms that exhibited higher TE were larger or had higher output per cow and performed at a higher efficiency regarding GHG emissions. The concept of eco-efficiency to measure environmental performance was used by Pérez-Urdiales,

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Oude Lansink and Wall (2016) to evaluate the environmental performance of a sample of Spanish dairy farms [234]. Analyzing the role of farmers’ socioeconomic characteristics and attitudes in explaining eco-efficiency, it was found that younger farmers, farmers that plan to continue operation in the foreseeable future, and those that participate more in training schemes were found to be more eco-efficient. The consideration of environmental goods as outputs has been addressed by several authors. Peerlings and Polman (2004) argued that agriculture produces both commodity and noncommodity products, where the latter in some cases are public goods [235]. They formulated a symmetric normalized quadratic profit function system to study milk and wildlife/landscape services as joint products. Using Dutch data, they found that wildlife/landscape services and milk are substitutes, and that most farms in the sample exhibited diseconomies of scope and would benefit from specializing. Gullstrand, De Blander and Waldo (2014), following Peerlings and Polman (2004), examined the connection between providing for biodiversity and the cost structure of dairy farms in Sweden [236]. They found milk and beef output to be substitutes of biodiversity provision. In contrast, they reported a complementary relationship between biodiversity provision and crop production. Areal, Tiffin and Balcombe (2012) redefined TE by incorporating the provision of environmental goods as one of the outputs [237]. Using data on dairy farms in England and Wales, they determined that farm efficiency rankings change when provision of environmental outputs by farms is incorporated in the analysis. In two related studies, Njuki and Bravo-Ureta (2015) and Njuki, Bravo-Ureta and Mukherjee (2016) applied directional distance functions along with the generalized true random effects estimator to analyze the trade-off between milk production (a good output) and an undesirable or bad output [238, 239]. Their results indicated that policies designed to curb emissions would have heterogenous effects across space and farm size and thus need to be evaluated and implemented with care to minimize adjustment costs. The possible effects of environmental regulation on the location of dairy production in the USA were investigated by Isik (2004) [240]. After developing a behavioral model of location and production, spatial lag models were estimated using US county-level agricultural and economic data. Differences in state environmental regulation were found to have contributed to dairy farms relocating to states with less strict regulation. A policy implication is that harmonization of environmental rules by increasing regulation in states with laxer regimes would increase concentration of dairy production in states with more stringent levels of existing regulation. This section finishes with two studies relating compliance with environmental standards and production. Samson, Gardebroek and Jongeneel (2017) set up a microeconomic model to estimate separate production functions for milk, feed, and roughage, as well as a manure production function derived from agronomic standards [241]. The focus was on analyzing the costs and benefits of dairy farm size growth in a context where policy changes such as milk quotas and environmental constraints on manure production and handling needed to be accounted for. Zhang (2018) examined the compliance of large confined animal facilities (CAFs) with practice-based air quality regulation in California’s San Joaquin Valley and the asso-

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ciated costs and economic performance on regulated farms [242]. The regulation under study was found to have a nonsignificant effect on the total costs of producing milk for the regulated farmers relative to the control group. Summing up, the tools from production economics have proven useful in the study of a wide range of topics related to the fundamentals of sustainability in dairy farming. The studies have documented the impact of weather and climate on productive performance and have shed light on the relation between animal welfare and output. The production of bad outputs such as pollution has also been studied, as well as the impact of regulatory reform. These studies constitute a valuable body of work whose results provide well-informed advice and recommendations to stakeholders on the increasingly important issue of sustainable production.

Concluding Remarks As shown in this chapter, the production economics literature on dairy farming is vast and has been used to address a wide variety of topics including: efficiency and productivity; technology adoption; economies of size scale and scope; the effects of government policies; the effect of risk and uncertainty; and issues relating to sustainability including the weather and climate change, animal welfare, and environmental efficiency. Given the importance and changing nature of the dairy sector, this literature can be expected to continue to grow. Dairy farming and milk consumption offer many societal benefits but there are important challenges that will deserve attention from the research community. A major growing concern has to do with the adverse environmental impact of dairy production. Dairy farming offers the possibility to increase human welfare, but also contributes to environmental degradation [2], the effects of which are far-reaching and include GHG emissions and soil and water contamination. Consequently, a significant challenge for the dairy sector will be to reduce its adverse environmental impact while meeting the increasing demand for dairy products of societies around the world. One would expect that the methodological arsenal that has been developed in the production economics literature will be increasingly applied to examine the environmental effects of dairy farming along with potential policy responses. The role of production economists, working with scientists in various other disciplines, will be paramount in the search of avenues to improve the overall productivity of dairy farming while offering policymakers sound advice on sustainable technologies and tools to deal with greater risk and uncertainty. On the other hand, the sector not only contributes to environmental degradation but is increasingly being affected by changing environmental conditions. Rising average temperatures, droughts, and floods jeopardize forage and grazing, imposing additional heat stress on animals and added burden from disease [243]. Another important issue is animal welfare, with increasing pressure from consumers and policymakers towards more stringent animal welfare standards. The challenges from climate change and animal welfare coupled with globalization and a fast-changing

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technological environment are putting additional pressure on the industry’s sustainability [243]. The future evolution of input and output prices is also a matter of concern. Increasing pressure from climate change on the availability of feed, as well as higher demand for animal products and grains used for biofuels are all expected to put upward pressure on feed and land prices [244]. In addition, farmers often face highly volatile dairy prices, leading to a risky environment for investors and farm operators. Another challenge for the dairy sector concerns the structure of the industry. In some developed countries dairy farms are highly specialized. While this can provide cost advantages, dependence on a single commodity exposes producers to economic shocks and the ensuing price and income variability. Furthermore, an ageing farmer population generally and the small size of dairy farms found in many developing countries also present challenges to dairy farming. In addition, farmers often occupy a relatively fragile position in the food supply chain with a weaker bargaining muscle than the processing or retail links in the value chain, thus making them vulnerable to asymmetric market power. These challenges provide fertile ground for future production economics research in dairy. The conceptual frameworks and empirical analyses that have been reviewed in this chapter show that production economists have several tools at their disposal to carry out studies related to these challenges and thus contribute to policy analyses and formulation.

Cross-References  Data Envelopment Analysis: A Nonparametric Method of Production Analysis  Dynamic Analysis of Production  Neoclassical Production Economics: An Introduction  Stochastic Frontier Analysis: Foundations and Advances I  Stochastic Frontier Analysis: Foundations and Advances II Acknowledgments Boris E. Bravo-Ureta acknowledges partial support from USDA-NIFA Grant #2016-67024-24760. Alan Wall is grateful for support from the Spanish Ministry of Economics, Industry and Competitiveness grant ECO2017-85788-R.

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Performance Evaluation of Mutual Funds Using Frontier Methods

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Subrata Sarkar

Contents Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Brief Methodology of DEA and SFA . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . DEA and Performance Evaluation of Mutual Funds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Early Attempts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Introducing Additional Variables in the DEA Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . Network DEA . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Introducing Stochasticity into DEA Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Other Nonparametric and Partial Frontier Measures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Other Nonparametric Measures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Partial Frontier Measures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Stochastic Frontier Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Abstract

The mutual funds industry in an economy plays an important role in channeling savings into investments. Given the limited amount of savings, economic efficiency requires that such savings be allocated to firms with the highest return on investments, adjusted for risk. Performance evaluation of mutual funds,

I thank Koustuv Saha for excellent research assistance in preparation of this manuscript. The usual disclaimer applies. S. Sarkar () Indira Gandhi Institute of Development Research, Mumbai, India e-mail: [email protected] © Springer Nature Singapore Pte Ltd. 2022 S. C. Ray et al. (eds.), Handbook of Production Economics, https://doi.org/10.1007/978-981-10-3455-8_32

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therefore, becomes important. In this chapter, we survey the extant literature on this subject outlining the important works that have been carried out till date. Keywords

Performance evaluation · Mutual funds · Data envelopment analysis · Stochastic frontier analysis

Introduction Background The mutual funds industry in an economy plays an important role in channeling savings into investments. Given the limited amount of savings, economic efficiency requires that such savings be allocated to firms with the highest return on investments, adjusted for risk. Performance evaluation of mutual funds, therefore, becomes important. Early attempts at appraising the performance of managed portfolios were confined to comparing returns of mutual fund returns against the returns from a randomly chosen unmanaged “market” portfolio or the return of a weighted average of many market portfolios. However, since higher expected returns are invariably associated with higher risks, the average returns of a portfolio as a benchmark measure proved to be an incomplete yardstick of evaluating mutual fund performance. Therefore, the second-generation measures that were introduced in the 1960s evaluated the performance of managed funds like mutual funds, based on returns maximization subject to the risks involved. However, the lack of consensus in the literature about the appropriate measure of risk led to a similar lack of consensus about the appropriate risk-adjusted measure for evaluating mutual fund performance [40]. The most popular measures that were used were (i) the Sharpe ratio that measured the excess returns earned by the portfolio per unit of its total risk, (ii) the Treynor ratio, which measured the excess returns earned by the portfolio per unit of nondiversifiable market risk, and (iii) the Jensen’s alpha which measured the excess return earned by a portfolio over an unmanaged portfolio with identical market risk [25]. These indices are still much in use both in academia and the mutual fund industry. It is important to understand the theoretical underpinning of these three ratios to set the context for the evolution of the frontier methods in assessing the performance of mutual funds. The Treynor ratio and Jensen’s alpha are both based on the capital asset pricing model (CAPM). These methods appraise the performance of individual mutual funds against a benchmark portfolio, which is the efficient market portfolio. Roll [47] criticized this method of portfolio evaluation on three counts, namely (i) these measures require the identification of a market portfolio which includes all assets, both marketable and nonmarketable, and is, therefore, impossible to observe, (ii) these measures use the market portfolio as the benchmark, and, therefore,

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require a test of the joint hypotheses that the market portfolio is mean-variance efficient and that CAPM is valid, which is impossible to carry out due to the problem of measuring the true market portfolio, and finally (iii) if an ex-post inefficient benchmark is used for performance evaluation, than any ranking is possible depending on the benchmark used and slight changes to the benchmark can completely reverse the performance rankings. Sharpe’s ratio circumvents these problems as it does not necessarily use the market portfolio as the benchmark, but nevertheless runs into the problem of arriving at a consensus of what the benchmark portfolio should be. Notwithstanding the theoretical challenges faced by these second-generation measures for evaluating mutual fund performance, all these measures had a common shortcoming, namely, that in evaluating portfolio performance exclusively in the risk-return space, they systematically ignored the role of the fund manager in achieving returns while minimizing risks. Put it differently, these measures ignored managerial effort as a crucial input in return discovery. If collection and implementation of information are costly, better-informed managers should be able to generate higher returns than less-informed managers. In addition, portfolio managers with superior abilities may be able to charge higher fees to obtain economic rent [52]. If so, measures that recognize the connection between returns and cost of information and managerial fees are likely to outperform those which do not [50]. To overcome these shortcomings, in the late 1990s new methods of appraising mutual fund performance were introduced which relied on frontier analysis. In frontier analysis which has its origin in economics, the production frontier represents the theoretical limit of output that can be produced given the amount of inputs and technology. The aim of a production unit is to achieve an input-output combination that lies on the production possibility frontier, though in reality it may fall short of it due to any number of reasons. The notion of shortfall gave a theoretically grounded concept of inefficiency which can be measured [25] and which can now be used to construct a ranking of production units in terms of their efficiency with the lowest deviation being associated with the most efficient unit. The two most popular frontier-based methods that were developed were the nonparametric data envelopment analysis (DEA) and the parametric stochastic frontier analysis (SFA). The key differences between the two approaches lay in the implicit assumptions set on the functional form of the efficiency frontier, allowance or nonallowance of random error which may produce transitory positive or negative deviations in outputs, inputs, costs, or profits, and in cases where the random error was allowed, the distributional assumptions imposed on it to distinguish the effect from the inefficiencies and random disturbances [5, 15]. The development of these two frontier techniques opened the floodgates for performance evaluation in a variety of industries which now had a strong theoretical basis. The earliest attempts at applying frontier methods for evaluating mutual fund performance were published over two decades ago [42]. Since then a considerable body of work on mutual fund appraisal using frontier methods has developed in the literature. However, these studies have been overwhelmingly based on the DEA technique. Basso and Funari [12] list 76 papers written between 1997 and

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2014 which studied mutual fund performance based on the DEA. Whereas there have been only three published studies of mutual fund performance that employed stochastic frontier analysis (SFA) [4, 5, 48]. The reason for the dominance of the DEA technique in the performance evaluation of the mutual funds industry and indeed of many service industries may lie in the relative flexibility it offers in constructing the benchmark or the most efficient production unit that does not require a parametric specification as is required under the SFA.

Brief Methodology of DEA and SFA The DEA method was introduced by Charnes et al. [18] in the context of a production unit operating with a constant return to scale (CSR) production function. The DEA generates an index of performance for each decision-making unit as the ratio of its weighted outputs to its weighted inputs. Unlike parametric methods, the weights for the output-input ratio are not derived from any underlying model of the preference structure of the decision-maker. Rather the weights are determined by solving a fractional programming problem. Each decision-making unit is assigned a distinct set of weights which maximizes its output-input ratio. Thus, if a decision-making unit turns out to be inefficient using the most favorable weights, it can be safely concluded that it will be inefficient for all other combinations of weights as well. The SFA was proposed by Aigner et al. [3] and Meeusen and van Den Broeck [39] in independent studies. This approach estimates a flexible functional form, whether cost, profit, or the production function, as the frontier along with an error term to signify that the frontier itself is random due to exogenous shocks which are beyond the control of the manager, or more generally, a decision-making unit. Given inputs, the deviation of actual output from the frontier output is the sum of this random shock and another error term that represents the inefficiency of the decision-making unit. Since no manager can produce more than the frontier, the inefficiency-related error term is always nonnegative and orthogonal to the estimated frontier. The orthogonality ensures that the estimated inefficiency scores are uncorrelated with the regressors and any scale economies [5, 24]. The addition of the stochastic error term in the SFA makes it more robust to noise and measurement errors which are potential problems for DEA. In fact, using a deterministic technique like the DEA to determine the efficient portfolio frontier has been criticized on the grounds that the phenomenon being studied is strongly stochastic in nature [16]. Notwithstanding these concerns, the relative popularity of DEA as a method for evaluation for mutual funds, and indeed for evaluation of most service industries, is due to the fact that it does not require any a priori specification of the form of the production frontier other than convexity or any assumptions about the distributions for the error term as required in SFA. This is a great advantage for empirical measurement. For manufacturing units that produce clearly identified outputs with clearly identified inputs and a well-specified technology, explicit specification of functional forms of the frontier that exploit the theoretical properties of production (cost) is relatively easier. This is not the case for service industries like mutual funds

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where the technology that transforms inputs to outputs itself may not be clearly defined let alone the identification of outputs and inputs where a lot of subjectivity may be involved.

DEA and Performance Evaluation of Mutual Funds The basic idea behind applying the DEA to evaluate mutual fund performance is that fund performance is characterized by multiple attributes which can be broadly classified as outputs or benefits, like mean returns, and inputs or cost, like risk, transaction costs, administration fees, loads, and minimum initial investment. Thus the first study to apply the DEA to mutual funds was motivated by the need to include transaction costs and management fees in the analysis [42]. Since then DEA applications to mutual funds have evolved and been extended in several directions. These advancements can be broadly classified into the following three categories: (i) Addition of new variables like stochastic dominance indicators on the output side ([8, 19, 36]), higher moments like skewness and kurtosis on the input side [28, 43], ethical indicators on the input side [9], and quantile-based measures like VaR and CVaR on the input side [19]. (ii) Development of network DEA models with multiple stages (Premachandra 2012). (iii) Introduction of stochasticity to the deterministic DEA approach [1, 35, 44]. We cover each of these advancements in the following subsections. We begin by describing the DEA approach as applied in the early attempts to evaluate mutual fund performance.

Early Attempts Murthi et al. [42] identified three major issues which the erstwhile measures of portfolio evaluation like the Jensen’s alpha and the Sharpe ratio were unable to address: the appropriate benchmark for comparison; the role of market timing; and the endogeneity of transaction costs. They developed their new DEA-based index to address two of these concerns. First, in their DEA portfolio efficiency index by comparing the performance of individual funds with the performance of an endogenously created best-performing fund from the sample of all available funds, they circumvented the problem of validating the choice of an exogenous benchmark. Secondly, by incorporating transaction costs like the turnover into the constructed index as inputs, they successfully avoided endogeneity issues. Murthi et al. [42] modeled the DEA portfolio efficiency index (DPEI) based on the Sharpe index. The Sharpe index is given as the ratio between R and σ where R is the difference between the actual return and the risk-free return, and σ is the standard deviation of the portfolio. They defined the DPEI as:

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DP EI = 

R ω X i i i + υσ

(1)

where Xi refers to transaction costs like expense ratio, which accounts for management fees, marketing expenses, and other operational expenses; load, which measures the amount that investors have to pay at the time of entering or exiting the fund; and turnover, which captures the trading activity of the fund manager. The weights ωi and υ associated with Xi and σ were determined through by solving an optimization problem. Murthi et al. [42] based their optimization problem using the DEA model proposed by Charnes et al. [18]. In particular, Murthi et al. [42] derived their DPEI measure by solving a fractional programming problem that maximized the weighted output-input ratio of a mutual fund, and can be represented as: Maximisevi ,ur h0 = subject to

t ur yrj r=1 m i=1 vi xij

≤1

t ur yrj0 r=1 m i=1 vi xij0

j = 1, . . . , n,

(2)

ur ≥ ε, r = 1, . . . , t, vi ≥ ε, i = 1, . . . , m, where ε is a small positive number (non-Archimedean constant) which prevents the weights from disappearing, n is the number of mutual funds indexed by j, m is the number of inputs indexed by i, t is the number of outputs indexed by r, xij is the amount of input i for unit j, yrj is the amount of output r for unit j, vi is the weight assigned to input i, and ur is the weight assigned to output r. The subscript 0 refers to the particular fund being evaluated. The optimal values of ur and vi obtained from the above optimization problem indicate the position of the fund relative to the Pareto-efficient frontier. Thus, the above program finds the weights that maximize the ratio of the weighted sum of the outputs to the weighted sum of inputs of a fund subject to the condition that all such ratios are less than or equal to one. The efficiency measure thus obtained is the best that the fund can achieve for any given value of weights, thereby giving the benefit of the doubt to the mutual fund under evaluation. The problem assigns the value 1 to h for the bestpracticed fund in terms of the output-input ratio. The weights of the best-practiced fund determine the slope of the straight line connecting the fund and the origin in the mean-variance or return-cost space (Fig. 1). This straight line is the DEA efficiency frontier. The slack variables obtained from the programming problem, if nonzero, indicate the extent to which each input can be reduced to achieve unit level of relative efficiency. Thus the optimization process also identifies the particular areas in which management is inefficient [21, 42]. For the actual optimization exercise the fractional problem in equation system 2–2 is converted into a more convenient linear programming problem (LPP) by  allowing Ii wi xi0 + vσ0 = 1.

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Fig. 1 DEA efficiency frontier is the straight line connecting origin with best-performing fund under CRS [21]

Maximise

t 

ur yrj0

r=1 m 

vi xij0 = 1 i=1 t m   ur yrj − vi xij ≤ 0 r=1 i=1

subject to

(3) j = 1, . . . , n,

− ur ≤ −ε r = 1, . . . , t, − vi ≤ −ε i = 1, . . . , m.

This LPP has t + m variables (the t output weights and m input weights) and n + t + m + 1 constraints. The LPP thus described is the input-oriented Charnes et al. [18] model. It is one of the simplest and most widely applied DEA models in mutual fund performance analysis [2, 8, 26, 31, 42, 50]. Murthi et al. [42] used the DPEI to evaluate 731 US mutual funds across seven categories, namely aggressive growth, asset allocation, equity income, growth, growth income, and balanced income, using data from the Morningstar database. They found strong evidence that mutual funds are mean-variance efficient and that efficiency is not related to transaction costs. However, in a later study, Tarim and Karan [50] using the DPEI, but with additional restriction on the weights, were unable to validate the mean-return efficiency observed in Murthi et al. [42] for a sample of 191 Turkish mutual funds. The additional restrictions on weights used by Tarim and Karan [50] prevented extreme variations in the input and output weights. They pointed out that since DEA assesses the efficiency of decision-making units using the most favorable weights, it may lead to outlier decision-making units being

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classified as efficient, or a factor that turns out pertinent with nontrivial weights for one fund and may come out with a zero weight for another. Comparing the results from the no-bounds DEA and the DEA with added restrictions, they showed for their sample of funds that the presence of outliers may lead to false hypothesis that funds are mean-variance efficient under no-bounds DEA. The mean-variance efficiency hypothesis remains much contested in empirical literature and a number of DEA-based studies have found evidence in favor of and against the hypothesis. Published in the same year as Tarim and Karan [50], Basso and Funari [8] illustrated the usefulness of the cross-efficiency matrix as an alternative strategy to cope with extreme variations in input-output weights for a given sample of funds. The element in row i and column j of the cross-efficiency matrix represents the efficiency ratio of the mutual fund i obtained with the weights that are optimal for mutual fund j. Thus, a series of efficiency scores are generated for each fund and comparisons between them take into account the average efficiency score based on the different input-output weights.

Variable Returns to Scale A major drawback of the Charnes et al. [18] DEA is that it ignored the effect of returns to scale of a decision-making unit performance since it assumed a CRS efficiency frontier. Choi and Murthi [21] were the first to apply the Banker et al. [7] DEA model, which allowed variable returns to scale (VRS) to mutual fund evaluation. The general Banker et al. [7] DEA model is given as the following fractional programming problem: t u y −u r=1 m r rj0 0 i=1 vi xij0 t u y −u r=1 m r rj 0 < 1 j i=1 vi xij

Maximiseur ,vi ,u0 subject to

= 1, . . . J

(4)

where the restrictions and notations from Eq. 1 still hold and u0 is the additional free variable introduced to characterize returns to scale for the mutual fund. The modified output-input ratio, I, proposed in Choi and Murthi [21] was given by I=

R0 − u0 i ωi Xi + υσ

(5)

Depending on whether u0 is positive, zero, or negative it is possible to infer increasing, constant, or decreasing returns to scale for the fund under consideration. The DEA efficiency score, I, had the interpretation as the ratio of the fund’s Sharpe index over that of the best performing fund, assuming u0 and υ are zero. The optimal u∗0 gives the intercept of the tangent line on the facet of the efficiency frontier. The slope of the tangent line is given by the optimum weights (Fig. 2). Note that Fig. 1 assumes constant returns to scale among mutual funds, and, therefore, the DEA efficiency frontier, which is determined by the best-practiced fund, is a straight line connecting the origin and the risk-adjusted return of the bestperforming fund. In contrast, Fig. 2 depicts that the efficient frontier consists of three

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Fig. 2 The DEA efficiency frontier with VRS fund performance

linear segments with each segment depicting different returns to scale. Beginning from increasing returns to scale, then constant returns to scale (45-degree line) and decreasing returns to scale. Choi and Murthi [21] used the modified DEA performance index to evaluate the same sample of mutual funds considered in Murthi et al. [42]. They found that about 90% of aggressive growth funds showed increasing returns to scale with these funds generally also exhibiting the highest gross returns. This, they argued, implied that the superior performance of the aggressive growth funds in terms of returns was due to increasing returns to scale over and above that could be attributed to taking higher risks. The relative ranking of the funds using the new index did not significantly change from the rankings obtained using DPEI and the authors found that mean-variance efficiency property continued to hold in the relative sense. Another property of the Banker et al. [7] DEA which made it particularly suited for mutual fund evaluation is its translation invariance. For the Charnes et al. [18] DEA model to yield accurate results the output and input variables have to be nonnegative. However, in empirical applications to mutual fund evaluation, it is often possible to have negative returns for the output variable. The translation invariance property of the Banker et al. [7] DEA means that a suitable constant (greater than the absolute value of the negative output) can be added to the negative output value without affecting the results of the DEA exercise. The input-oriented Banker et al. [7] model is suitable for this purpose since it is translation invariant with respect to its outputs, whereas the output-oriented Banker et al. [7] model is translation invariant with respect to its inputs.

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The input-oriented Banker et al. [7] DEA [20, 28–31, 38, 43] is specified as an input minimization problem given a fixed level of outputs. On the other hand, the output-oriented Banker et al. [7] DEA [13, 33, 38, 53] maximizes the outputs given a fixed level of inputs. Among the two methods, the input-oriented Banker et al. [7] DEA is more the popular of the two. However, recently Basso and Funari [12] argued that the input-oriented DEA is not the most suitable method to evaluate mutual funds. They reason that since the average investor is mainly interested in maximizing the mean returns and other output variables, without increasing the costs, the output-oriented model is more suitable for the analysis. To overcome the issue of negative mean returns, Basso and Funari [12] use the final value of the fund as output in their output-oriented DEA. Since the final value of investment is always positive, by doing so, Basso and Funari [12] successfully sidestep the problem of having negative output variables. They admit, however, that using the final value of the fund could potentially make the DEA outcome sensitive to the choice of the holding period. Other techniques adopted in the literature to deal with the possibility of negative mutual fund returns include using an additive DEA model instead of a multiplicative model [36]; using slack-based models [34]; and using directional distance-based DEA [49]. But as Basso and Funari [12] contend that while these approaches are technically consistent, they tend to make the interpretation of these efficiency measures difficult from a financial point of view.

Multihorizon DEA Yet another interesting use of DEA to evaluate mutual fund performance is that presented in Morey and Morey [41]. Whereas Murthi et al. [42], and almost all other studies thereafter, used DEA primarily because it allowed the explicit incorporation of transaction costs into the evaluation process, Morey and Morey [41] used the DEA to develop a multihorizon approach by incorporating the average monthly returns and risks over three different time horizons as outputs. They justified their methodological approach by drawing an analogy to the “typical investor” who, when deliberating over which mutual fund to invest in, is available to the information on the fund’s performance over multiple time horizons rather than a single time horizon. Morey and Morey [41] use their DEA model to evaluate 26 “aggressive growth” mutual funds from the Morningstar database. They used monthly percentage return data to calculate the mean monthly returns, variances, and covariances for each mutual fund and each of three horizon lengths – 3 years, 5 years, and 10 years. They employed both the input-oriented and output-oriented DEA approaches and compared the fund rankings with a ranking order based on the Morningstar rating system. Through their empirical exercise, they showed that the DEA-based method ranked the funds in a significantly different order to that of the Morningstar rating system.

Introducing Additional Variables in the DEA Models Over the years the basic DEA models for evaluating mutual fund performance have been extended to include various other measures to incorporate the varying nature

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of risk preference of investors. These risk measures have been introduced as new output and input variables in the traditional DEA models.

Stochastic Dominance in DEA Models: CARA, IARA, and DARA Basso and Funari [8] constructed a DEA mutual fund performance index with a stochastic dominance indicator and returns as the two outputs. The stochastic dominance indicator was included to reflect the investor’s preference structure and the time occurrence of the returns. Stochastic dominance techniques allow the partial ranking of random variables, assuming that investors prefer those alternatives that maximize their expected utility. The assumptions underlying stochastic dominance rules concern the signs of the successive derivatives of the investor’s utility function. The first-order stochastic dominance relation between two random portfolios X and Y follows from the principle of non-satiety and is defined as follows: X is said to dominate Y according to the first-order stochastic dominance criterion if for all nondecreasing utility functions U we have E(U (X)) ≥ E(U (Y)) and there exists a nondecreasing utility function U* such that E(U*(X)) > E(U*(Y)). The secondorder stochastic dominance between the random portfolios X and Y arises from the principles of non-satiety and risk aversion; X is said to dominate Y according to the second-order stochastic dominance criterion if for all nondecreasing and concave utility functions U we have E(U (X)) ≥ E(U (Y)) and there exists a nondecreasing and concave utility function U* such that E(U*(X)) > E(U*(Y)). Another widely accepted stochastic dominance criterion concerns the assumption of decreasing absolute risk aversion (DARA) of the investors’ utility functions; X is said to dominate Y according to the DARA dominance criterion if for all utility functions U satisfying the DARA assumption we have E(U (X)) ≥ E(U (Y)) and there exists a DARA utility function U* such that E(U*(X)) > E(U*(Y)). The stochastic dominance rules get weaker with successive orders, so a prospect which dominates another under order n will be dominant under every successive order after n. Thus the higher the order of the dominance rule, the higher is the number of dominance rules observes. However, the higher the number of dominance rules observed, the more are the restrictions on the utility function of the investor implying that the efficient set will be suitable for a smaller group of investors and involve a loss in generality. For the purposes of their DEA analysis, Basso and Funari [8] used the decreasing absolute risk aversion (DARA) rule which is based on a widely accepted hypothesis, is more selective than the first three orders of stochastic dominance, and can be computationally tested using a convenient dynamic programming algorithm. To define the stochastic dominance indicator, they began by taking the past returns of the mutual fund over a convenient time period and dividing that period into subperiods. They then constructed the stochastic dominance indicator for fund j as the relative number of periods in which fund j was not dominated by any of the other funds in the sample. dj =(number of nondominated subperiods for fund j )/(total number of subperiods)

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They defined the two-output Charnes et al. [18] DEA portfolio performance index IDEA_2 as the optimal value of the objective function of the following fractional programming problem: u1 oj0 +u2 dj0 k i=1 vi qij0 + i=1 wi cij0 u1 oj +u2 dj h k i=1 vi qij + i=1 wi cij

Maximiseur ,vi ,wi h subject to

≤ 1, j = 1, . . . , n

ur ≥ ε, r = 1, 2 vi ≥ ε, i = 1, . . . , h wi ≥ ε, i = 1, . . . , k

(6)

where j = 1 . . . n denotes the n mutual funds, r = 1, 2 denotes the two output variables, dj denotes the relative number of periods in which fund j was not dominated by any of the other funds in the sample, oj denotes the expected returns of the fund, q1j . . . qhj denote the h risk measures for fund j, c1j . . . ckj denote the k subscription or transaction costs for fund j, and u1 , u2 , vi , and wi are the weights to be determined through the fractional programming process. Basso and Funari [8] carried out two rounds of DEA calculations for a sample of 47 Italian mutual funds over the years 1997–1999, once with the IDEA _ 2 index and once with the IDEA _ 1 index, which is the same as the IDEA _ 2 index except that it excludes the stochastic dominance indicator. They considered the standard deviation of the portfolio returns, the square root of the half variance, and the β coefficient as the risk measures and included subscription cost and a redemption cost as the two transaction costs. These measures served as inputs to the evaluation process. Comparing the results from the two DEA analyses revealed that the inclusion of the stochastic dominance indicator as an output produced a significantly different ranking among the funds and generated a larger set of efficient funds. Tavakoli Baghdadabad et al. [51] extended the stochastic dominance approach presented in Basso and Funari [8] to both constant risk aversion and increasing risk aversion. They applied the three DEA models (corresponding to DARA, CARA, and IARA stochastic dominance rules) to a sample of 17,555 US mutual funds. Their findings show that the average fund efficiency values are quite different for each of the stochastic dominance rules. The calculated average efficiencies were the highest under the DARA rule, whereas the average efficiencies under CARA and IARA were quite negligible. Lozano and Gutiérrez [36] posited six different DEA models which used return, risk, and safety measures that had previously been proved to be consistent with second-order stochastic dominance when used in portfolio optimization models. They did this to better account for the effect of diversification on the overall risk of holding a mutual fund. They argue that conventional DEA models that compute the risk measure of the benchmark portfolio as a linear combination of the risk measures of the intervening mutual funds do not take into account the diversification effects of mutual funds. This can result in significant overestimation of the risk measure and lead to underestimation of the efficiency scores for some mutual funds.

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Risk Measures Chen et al. [20] used the IDEA _ g index to rank Chinese mutual funds from the year 1999–2001. The IDEA _ g has the same set of inputs as IDEA _ 2 , but in addition to returns and stochastic dominance as outputs, it also included several traditional measures of mutual fund performance like Sharpe ratio, reward to half-variance, Treynor ratio, and the Jensen’s alpha. Chen et al. [20] were one of the first to try to address a controversy surrounding the choice of risk measures for DEA mutual fund appraisal. Most DEA studies on mutual funds only included the first moment (mean) and the second moment (variance) among the input-output variables. This was fine as long as asset returns were normally distributed. However, a large number of empirical studies have shown that this assumption was not true. Other scholars have shown that investor’s utility functions are not quadratic, and that that they prefer higher skewness and lower kurtosis. A more skewed returns distribution means a higher probability of aboveaverage returns and a higher kurtosis means a higher chance of the fund undergoing a major change. So skewness is to be preferred and kurtosis is to be avoided [20, 28]. Chen et al. [20] introduced value at risk (VaR) and conditional value at risk (CVaR) as additional inputs into the IDEA _ 2 index of Basso and Funari [8] to accommodate the skewness and leptokurtosis observed in the distribution of returns of actively managed funds. VaR and CVaR are quantile-based measures which make them suitable to measure risk in case of asymmetric distributions, as the investor is more likely to be concerned about the risk of a loss than of a gain. A portfolio’s VaR, given a prespecified level of confidence and a particular time horizon, is the maximal loss that one expects to suffer at that confidence level holding that portfolio over that time horizon. The CVaR is the conditional expectation of losses exceeding VaR in a specified period at the given confidence level. Alongside the quantile-based measures, Chen et al. [20] also included the conventional measures of risk like alpha, beta, and the square root of the lower semi-variance. They argued this was necessary as quantile-based measures focused only on the lower tail of the distribution of returns and, therefore, they need to be combined with the other risk measures so that all the risk characteristics of the fund’s returns can be simultaneously modeled. They applied their inputoriented index on a sample of Chinese mutual funds, once with VaR and CVaR included and once without them. They found that the proper combination of VaR and CVaR with other risk measures can more comprehensively reflect mutual funds’ risk properties and thus better measure the overall performance of mutual funds. For reasons similar to Chen et al. (2006), Guo et al. [28] and Pendaraki [43] each constructed an input-oriented Banker et al. [7] type DEA models where they incorporated skewness directly as one of the outputs and kurtosis as one of the inputs. They both found that inclusion of the higher moments to the DEA measure changed the ranking of the mutual funds, and increased both the average efficiency scores and the number of efficient funds.

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DEA with Ethical Measures Of late ethical and socially responsible investment has come into focus while evaluating mutual fund performance. Ethical or socially responsible funds allow investors the opportunity to support social causes or companies and nongovernmental organizations which are sensitive to social, cultural, and environmental issues through their investments. However, these ethical features could involve a financial penalty for the investors as socially responsible investment could be possible only in a subset of economic activities, those that avoid activities that could be considered morally dubious such as weapons manufacturing, gambling organizations, alcohol production, and activities which may pollute the environment. Since socially responsible investing fulfills two different needs, namely the need to obtain satisfactory returns as well as the need to be ethical, performance indicators that take into account only the financial objective will tend to be biased against the funds committed to socially responsible investment. To address this problem, Basso and Funari [9] proposed using a two-output DEA model, consisting of financial returns and an indicator for ethical behavior as the two outputs. Two issues arose while incorporating ethical behavior in the DEA model. First, usually the only information available on ethical behavior was binary in nature or at most categorical. In this case, a basic DEA model which uses continuous variables is inappropriate. Second, the ethical level is chosen by savers a priori and cannot be arbitrarily modified, so that any proxy for ethical behavior needs to be considered as an exogenously fixed variable. This lead Basso and Funari [9] to develop three DEA models: first, a simple two-output Charnes et al. [18] DEA model which included an output indicator for ethical behavior; second, an intermediate model which took the ethical level as an exogenously fixed variable, and third, the most appropriate model which was a categorical model with an exogenously fixed output. For exposition purposes, we illustrate the third model which is based on the DEA model developed by Banker and Morey [6]. For the simple case where dj is a binary variable that takes the value 0 to indicate ethical investing and 1 to indicate otherwise, the DEA model is the dual of the linear programming problem in output orientation. Assuming a set of n mutual funds j = 1, . . . , n with risky returns Rj , the following notations are introduced: oj a return measure of fund j q1j , . . . , qhj h risk measures of fund j c1j , . . . , ckj k subscription costs for fund j z0 a dual variable associated with the equality constraint λj dual variables associated with the mutual fund constraints dj0 the ethical measure variable for the mutual fund being evaluated s1+ dual variable associated with the input weight constraint si− dual variables associated with the output weight constraints ε a non-Archimedean constant.

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The associated fractional programming problem can be written as: h 

Max z0 + ε subject to n  j =1

i=1

n  j =1

si− + ε

k  i=1

− sh+i + εs1+

qij λj + si− = qij0 , i = 1, . . . , h

− cij λj + sh+i = cij0 ,

z0 oj0 − n  j =1

n 

i = 1, . . . , k

oj λj + s1+ = 0

i=1 q1j q1j0 dj λj

(7)

≤ dj0

λj ≥ 0, j = 1, . . . , n si− ≥ 0, i = 1, . . . , h + k s1+ ≥ 0 The constraint associated with the binary variable for the ethical status of the fund n  q1j dj λj ≤ dj0 (8) q1j0 j =1

is redundant for nonethical funds (dj0 = 1) but for ethical funds (dj0 = 0) it requires that the multipliers associated with the nonethical funds be zero (λj = 0 if dj = 1). This model could be easily adapted to cases where ethical funds are measured on a scale instead of a simple binary rule. Basso and Funari [9] tested the applicability of the DEA models empirically using randomly generated data. Since then, several other studies have extended this approach to capture the effect of variable returns to scale (for example, [13]) and to be robust to negative returns (for example, [10, 11]).

Network DEA Conventional DEA models evaluate mutual fund performance as a single-stage production process and, therefore, fail to consider the internal structure by which the mutual funds are managed [27]. Recently, network DEA models have been developed that explicitly consider the different activities that make up a mutual fund’s management process as separate stages, thereby allowing better modeling and better identification of the sources of inefficiency. Network DEA models are in the early stages of their development, and till the year 2018, there have been three studies that have used a network DEA approach to appraise mutual fund performance [27, 45, 46].

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Premachandra et al. [45], who were the first to implement a network DEA model in the context of mutual funds using a two-stage DEA model, decomposed overall efficiency into (i) operational efficiency which reflected how well a fund family has managed its resources in securing or generating funds for the family (ii) portfolio efficiency which measured how well a mutual fund family manages its investment portfolio to realize returns subject to a chosen set of factors that may influence them. They modeled fund management as a two-stage process. In the first stage, the operational management stage, the fund tries to attract monies from the investors and incurs management fees (X11 ) and marketing and distribution expenses (X21 ). In the first stage, the output variable is the net asset value generated (z1 ). The second stage is the portfolio management stage. Here the net asset value (z1 ), the fund size (X12 ), the net expense ratio (X22 ), the turnover ratio (X32 ), and the standard deviation of the returns over the past three years (X42 ) are the inputs that go into producing the mean returns of the portfolio (y1 ). The net asset value which is the output of the first stage becomes an intermediate input into the second stage and forms the connecting link between the two stages. It is possible to solve the two stages of the problem separately as two standalone DEA problems by assuming that they operate independently of each other. However, ignoring the fact that the outputs from the first stage are the inputs in the second stage can lead to various inconsistencies in the evaluations. In an earlier work in the literature on production economics, Kao and Hwang [32] offered a possible solution by pointing out that an overall measure of efficiency can be expressed as the product of the efficiencies of two stages under constant returns to scale assumption. Later, Chen et al. [20] extended this approach by using additive efficiency decomposition under the constant and the variable returns to scale assumptions. Premachandra et al. [45] employed the two-stage DEA model under the variable returns to scale assumption illustrated in Chen et al. [20]. The choice of the model was motivated by the translation invariance property of the variable returns to scale DEA model which makes it better suited to situations where some of the variables (e.g., mutual fund returns) take negative values. The DEA-based procedure used in Premachandra et al. [45] can be described using a general two-stage DEA network structure  for the mutual fund family j 1 1 , x1 , . . . , x1 with i1 inputs to the first stage denoted by Xj = x1j i1 j ; i2 inputs to  2j  2 , x2 , . . . , x2 the second stage denoted by Xj2 = x1j i2 j ; D intermediate measures 2j denoted by zdj (d = 1, . . . , D), and s outputs from the second stage denoted by yrj (r = 1, . . . , s). With respect to the empirical example in Premachandra et al. [45] (illustrated in Fig. 3), X1 has two input variables, X2 has four input variables, z has one variable, and y has one variable. Following Banker et al. [7], the VRS efficiency score of a decision-making unit at the first and second stages can be calculated using models (1) and (2), respectively. The first stage is given by:

32 Performance Evaluation of Mutual Funds Using Frontier Methods Fund Size ( Management Fees ( )

Operational Management Function (First Stage)

Marketing and Distribution Fees ( )

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)

Portfolio Management Function

Net Asset Value ( ) Net Expense Ratio ( ) Turnover (

)

Standard Deviation (

)

Average Returns ( )

(Second Stage)

Fig. 3 Structure of the network DEA model in Premachandra et al. [45]  1 1 d ηd zd0 +u  1 1 i vi xi 0

Maximise

1 1 1  1 η zdj +u1 d d 1 1 ≤ 1, j = 1, 2, . . . , n i 1 vi 1 xi 1 j vi11 , η1d ≥ ε; u1 is free

s.t.

(9)

where, vi11 , η1d are the weights for the inputs and intermediate variables to be obtained through the optimization of the first stage. u1 is a free variable associated with the returns to scale for the DEA in stage 1. The second stage can be expressed as follows: 

Maximise 

2 r0 +u r ur y 2 2 2 d ηd zd0 + i2 vi2 xi2 0



2 r ur yrj +u ≤ 1, 2 z + v 2 x 2 η dj d d i2 i2 i2 j vi22 , ur , ηd2 ≥ ε; u2



s.t.

j = 1, 2, . . . , n

(10)

is free

where, vi22 , ur , ηd2 are the weights for the inputs, intermediate measures, and outputs of the second stage, to be obtained through optimization of the second stage. u2 is the free variable associated with returns to scale for the DEA in the second stage. Overall efficiency is then be calculated as the weighted average of the efficiency scores from the first and the second stage as:  w1 ∗

 ηd1 zd0 + u1 ur yr0 + u2 + w2 ∗  2 r  1 1  2 2 i1 vi1 xi1 0 d ηd zd0 + i2 vi2 xi2 0 d

(11)

The weights w1 and w2 sum up to unity and are defined by the user to reflect the relative importance of each stage to overall performance. Premachandra et al. [45] used the share of the total resources devoted in each stage to determine the weights.

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For their empirical exercise, they used data from 66 large US mutual fund families consisting of 1296 individual mutual funds. Using the two-stage decomposition process they demonstrate that in addition to identifying the efficient funds they are also able to quantify the relative contributions of management efficiency and operational efficiency to overall efficiency. Galagedera et al. [27] extended the two-stage approach developed by Premachandra et al. [45] by adding a third stage to the management process. In the first stage of the two-stage model in Premachandra et al. [45] management fees and marketing and distribution fees were the inputs that went into producing net asset value (NAV). Galagedera et al. [27] divided this stage into two. In their model, the first stage, the operational management stage, had the same inputs as in Premachandra et al. [45] but now its output was the fund size. The new second stage, the resource management stage, took the intermediate output fund size and the two inputs, turnover ratio and expense ratio, to produce the output, the net asset value (NAV). The NAV is then used as an input to the final stage, which produced returns as the output after combining NAV with various risk measures. They argued that though NAV and fund size were distinct measures, both cannot be included either as outputs or as inputs since they were related to the scale of operations and, therefore, highly correlated. By considering NAV as total funds transformed through a resource management process, NAV can be included as an output while total funds can be considered as an input to the second stage.

Introducing Stochasticity into DEA Models Mutual fund returns generation is a stochastic process. Studying stochastic processes using a nonparametric technique like the DEA leaves one at the risk of confusing chance events with what is actually a product of the manager’s efficiency or lack thereof. Parametric techniques like stochastic frontier estimation (SFE), which we cover later in the review, are immune to this shortcoming though they have their own set of drawbacks. Several studies have therefore attempted to incorporate elements of stochasticity into the DEA framework itself. One of the earliest studies in this respect was by Premachandra et al. [44], who constructed a spreadsheet-based stochastic DEA (SDEA) based on the Monte Carlo simulations using the @RISK program of Microsoft Excel. They used the SDEA to model an asset allocation decision facing a fund manager who must choose a stock and money market investment portfolio that will perform efficiently relative to a number of important alternative portfolios selected from the same potential investments. The money market investments were a risk-free debt market investment and the potential stock market investments consisted of market indexing investment as well as small, medium, and large company investments. The risk-free market, small company, medium company, and large company investments were combined in different ways to form alternative portfolios so that all major asset allocation portfolio strategies were covered. The two inputs were the total dollar value initially invested in the risky holdings (X1 ) and the dollar value of the initial risk-free

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investment. The single stochastic output was the portfolio’s market value at the end of a time period minus the comparison benchmark return which would have been generated by a market index investment strategy. The @RISK simulation package produced summary statics for each of the alternative portfolios which were used to compare their relative efficiencies. The numerical SDEA model predicted future underperformance by identifying portfolios that were grossly inefficient and did a better job than the standard Markowitz mean-variance portfolio ranking approach. Lamb and Tee [35] used bootstrapping, a technique similar to the Monte Carlo simulations to produce estimates of fund efficiency. Whereas Premachandra et al. [44] modeled the DEA problem as the decision facing the mutual fund manager, Lamb and Tee [35] used the SDEA to model the decision-facing investors deciding between alternative mutual funds to invest in. The two input variables used were max (CVAR, 0) and max (SD, 0), where CVAR is the conditional value at risk and SD is the difference between the lower semi-deviation and the mean. The two output variables were the mean returns and max (−CVAR, 0). The choice of the input and output variables was motivated by the desire to make the DEA model sensitive to diversification [36]. Lamb and Tee [35] used the bootstrap method to estimate the biases and subsequently constructed the bias-corrected estimates of the DEA efficiency measures. They found that the efficiency estimates had significant biases and, more crucially, the magnitude of the bias was fund specific. They also noted that the rank ordering of the funds was also markedly different when using the biascorrected efficiency estimates than when using the raw efficiency measures. The confidence limits were estimated using a bootstrap method and the mutual funds were partially ranked using matched pairs test. They also estimate diversificationconsistent DEA efficiency measures using the bootstrap methods. Hu and Chang [31] presented an approach different from the above two studies in terms of identifying stochastic elements using the DEA framework. Rather than employing an SDEA, they designed a three-stage model to decompose mutual fund performance and obtained pure managerial performance. In the first stage, they ran an output-oriented CRS DEA model using around 156 funds from the Morningstar database for the period 2005–2006. In the second stage, the output slacks obtained in the previous stage were used as the dependent variable in a stochastic frontier regression with various attributes of the fund and the investor as the independent variables. This was done to decompose the output slacks obtained in the first stage into three components: one component reflecting environmental influences, one component reflecting luck, and the final component reflecting managerial inefficiency. The stochastic frontier regression estimated was of the following form:     si = f (zi ; β) + νi − ui , νi ∼ N 0, σν2 and ui ∼ N + 0, σu2 ∀i = 1, . . . , I (12) where si is the output slack from the first stage DEA, zi is N observable environmental variables like fund characteristics and manager attributes, f (zi ; β) is the deterministic component of the slack variable with parameter vector β, ui is the

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nonnegative managerial inefficiency term, and ν i is the stochastic term representing statistical noise. After estimating the sources of inefficiency, this information was used to adjust the output data to better reflect pure managerial effort. This was done by adjusting upwards the output variables of each fund in proportion to the quantum of disadvantage faced by the fund on account of their relatively unfavorable environment or relatively bad luck. In the third and final stage, the adjusted output variables were used to rerun the CRS DEA from the first stage. They found that the number of efficient funds significantly falls in the third-stage DEA as compared to the first-stage DEA, implying that the variations in original underperformance could almost be explained by fund characteristics, manager attributes, and luck.

Other Nonparametric and Partial Frontier Measures Other Nonparametric Measures There are some nonparametric methods other than the DEA to evaluate mutual fund performance, though such methods are scarce. Among them is the Free Disposal Hull (FDH) technique, which is the nonconvex counterpart of the DEA. It shares the same underpinnings as the DEA, however, unlike the DEA it does not rest on the hypothesis of convexity of the attainable set. While the fact that it does not require convexity makes the FDH more flexible than the DEA, it also makes the technique susceptible to what is known as the scarcity bias. Because the FDH drops the convexity assumption in DEA, if a given fund cannot be compared with the other funds due to their input-output combinations, it is classified as efficient by default. Thus, despite being more flexible than the DEA, the FDH has trouble in discriminating and ranking alternatives in the absence of a sufficient number of similar mutual funds [37]. There are no studies that have exclusively applied the FDH to mutual fund evaluation, it has been used in conjunction with other methods like the DEA and partial frontier methods [23, 37, 49]. What is more noteworthy is the applicability of new partial frontier techniques like order-m [17] and order-α [22] estimators, to mutual fund evaluation and the advantages these methods offer over full frontier methods like the DEA and the FDH.

Partial Frontier Measures There are two major limitations of full frontier methods like the DEA and the FDH, which partial frontier methods can overcome, namely (i) their susceptibility to outliers and extreme values, and (ii) the curse of dimensionality. Full frontier methods like the DEA and FDH measure efficiency and inefficiency using the absolute minimum achievable level of inputs (given the output) as the benchmark. Partial frontier techniques (specifically the order-m frontier) on the other hand use the expected minimum achievable level of input which is a less extreme benchmark. As a result, partial frontier methods are more robust to outliers than

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full frontier methods since they do not try to envelop all observed points, no matter how extreme. Another way in which partial frontier methods offer an advantage over full frontier methods is by avoiding the curse of dimensionality. Due to the curse of dimensionality, for nonparametric methods like the DEA and FDH, as the dimensions of the input-output space increase, the number of observations needed to maintain the original level of precision also increases rapidly. Partial frontier techniques like order-m [17, 37] and order-α [37] on account of being √ n- consistent are less susceptible to the curse of dimensionality. Since the methodologies of these techniques are quite involved, we do not describe them in the present review. Matallín-Sáez et al. [37] provide an informative discussion of the methodology of partial frontier techniques. The next section provides a summary of studies applying SFA to the context of mutual funds.

Stochastic Frontier Analysis Though nonparametric methods such as the DEA have come to dominate the literature of mutual fund evaluation, the stochastic frontier approach offers a theoretically sound alternative to model and evaluate the efficiency of mutual funds. Indeed, there is reason to prefer the SFA over the DEA when evaluating any decision-making units which operate under uncertain conditions, all of which are not within the control of the “managers” of these units. The advantages of the SFA technique, its robustness to outliers and extremes values, are due to the incorporation of a stochastic error term. Notwithstanding this, the need to specify explicitly the functional form for the frontier and the distributional assumption of the two error terms related to noise and efficiency makes the SFA analysis rather challenging, especially for service industries like mutual funds. As a result, examples of studies that take an SFA approach to mutual fund analysis are very limited. At the time of writing this review, there were only three such published studies. A brief discussion of the three studies follows. Annaert et al. [4] applied SFA to evaluate a sample of European mutual funds. As a preliminary indication of fund performance, they computed the Jensen’s alpha of each of the 179 funds. Jensen’s alpha (α p ) is given as the difference between the excess return earned by the mutual fund portfolio and the expected excess return.  αp = R p − E Rp = R p − βp E (Rm )

(13)

where R p is the actual average excess return of fund p; β p is its beta coefficient, which is a measure of the fund’s volatility relative to the market portfolio, and E(Rm ) is the expected excess return on the market portfolio. The second equality in Eq. 13 follows from the CAPM (though other asset pricing models can also be assumed to hold). The alpha for each fund was obtained as the intercept term in a time-series regression of the fund’s excess returns (Rpt ) on a proxy for the market portfolio return (Rmt ). The estimated beta coefficients centered around one, indicating the relevance of the benchmark portfolio. The average alpha coefficient was close

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to zero, indicating that the individual funds did not outperform the benchmark portfolio. According to the efficient market hypothesis, alpha, which measures the expected abnormal returns for a mutual fund, should be zero, or even negative in case a fund incurs excessive transaction or management costs. However, in a given sample some funds will likely be found to exhibit positive abnormal returns due to sampling noise. Hence, in the cross-sectional regression of mutual fund returns on their beta coefficients, given by: R p = γ0 + γ1 βˆp + αp

(14)

the expected abnormal returns (α p ) will be zero or negative depending on the fund’s transaction or management costs. Significantly positive abnormal returns (α p ) for any fund in the sample is purely due to sampling error. The regression coefficient γ 0 should be zero and the coefficient γ 1 should be the average excess return on the market portfolio as long as the CAPM holds. The CAPM says that the returns on a mutual fund are proportional to its systematic risk. To increase the statistical power of the test, the author’s augmented Eq. 14 with a composed error, consisting of a symmetric disturbance capturing measurement error (vp ) and a nonnegative disturbance term, modeling the level of efficiency (ξ p ): R p = γ0 + γ1 βˆp − ξp + vp

(15)

Both error terms are independent of each other and across funds. The nonnegativity of ξ p follows from the assumption that funds cannot systematically outperform the benchmark portfolio. Recognizing that one of the major concerns against the parametric techniques is that the assumption made about the distribution of the error terms can be restrictive, Annaert et al. [4] used three different gamma distributions for ξ p , and pooled them together in a Bayesian framework. The error term vp was assumed to follow the usual normal distribution with mean zero and variance σ 2 . The βˆp for each fund was estimated by regressing fund excess returns on Rmt for the previous 24–36 months. The estimated slope coefficients were taken as the βˆp ’s. Using this framework, Annaert et al. [4] estimated five models for mutual fund efficiency: a baseline model based on Eq. 14; three models based on Eq. 15 and the three assumed distributions of ξ p ; and the pooled model. Their results showed that individual mutual fund efficiencies exhibited large variability. While the most efficient mutual funds were located very near to the efficiency frontier, the least efficient funds were located approximately at a level 50% below the frontier. They used combinations of Spearman and Pearson correlation tests to investigate the relationship between fund efficiency fund characteristics like fund size, its age, and historical performance. Though they failed to find any significant relationship between efficiency and age both size and past performance were revealed to have a significant positive effect on efficiency.

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Building on the work of Annaert et al. [4], Santos et al. [48] implemented a similar SFA approach based on the CAPM functional form to evaluate Brazilian stock mutual funds. They assumed a seminormal distribution for the efficiency term ξ p . Since their approach was very similar to the earlier paper it will not be discussed in this review, in the interest of brevity. Babalos et al. [5] offer the only example of the SFA technique being employed to investigate mutual fund performance in the post-financial crisis era. Through their analysis, they attempted to investigate the veracity of two hypotheses: one, if an increase in a fund’s risk causes an increase in its efficiency, and two, if an increase in the fund size results in its efficiency increasing. Post-2008, the question of how fund managers will respond to a rise in the degree of risk associated with a mutual fund has gained a special significance. Fund efficiency was measured using a distribution-free approach based on Berger [14]. This was done to avoid the usual pitfall of SFA which comes from its heavy reliance on the assumed underlying error distribution. They carried out their analysis for a panel dataset of US no-load mutual funds observed over the years 2002–2010. The choice to only model no-load funds meant that they did not have to deal with any confusion regarding whether to add sales cost to the mutual fund’s operating cost since no-load funds are directly distributed to investors and do not charge sales costs. They did not rely on the CAPM relation to benchmark fund efficiency. The basic functional form for the frontier they estimated is given by: Rit = f (Nit , Zit ) + ϑit + uit

(16)

where Rit is the observed returns for fund i in year t, N is a vector of fund-specific variables like expense ratio and turnover ratio affecting its return, Z is a vector of control variables like volatility index and bond quality spread, ϑit corresponds to random fluctuations and follows a symmetric normal distribution, and uit represents the fund’s efficiency compared to the best practice level. The selection of inputs and outputs was based on earlier nonparametric studies like Murthi et al. [42], Basso and Funari [8, 9], Daraio and Simar [23], and Matallín-Sáez et al. [40]. In the empirical estimation, Eq. 14 was fitted in a flexible translog specification to consider for nonlinearities. The stochastic frontier model was estimated using a seemingly unrelated regression (SUR) framework. They estimated the mean efficiency scores for US no-load funds for each of the years 2002–2010 and efficiency scores for different categories of funds such as small-cap, large-cap, financial fund, and technology funds. The average efficiency scores across all funds were quite high at 81% and the dispersion of the scores was found to be highest in 2002 and 2008, which is to be expected considering the effect of the credit crunch on the global financial market. Among the different mutual fund classes, large-cap fund categories had the highest efficiency score and financial sector funds were also found to have performed well considering the effects of the GFC. Following the estimation of the efficiency scores, they employed a panel VAR (vector auto-regression) to explore the causal relationship between efficiency and some of its determinants like fund size, risk, flows (as measured by percentage asset

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growth rate net of appreciation), and Morningstar ratings. Predicting the risk of endogeneity bias in using a simple OLS in this context, they used a VAR framework instead since it allows all the variables in the analysis to be entered as endogenous variables. Interpreting the results from the IRFs (impulse response functions) and the variance decomposition they showed that there was a double-sided relation between risk and fund performance. Higher risk-taking led to better performance, while a decline in the fund’s efficiency was associated with a rise in riskiness. Contrary to Annaert et al. [4], they found the relationship between fund efficiency and size to be negative.

Conclusion The motivation behind the present paper was to provide a comprehensive review of the diverse body of literature on mutual fund performance evaluation. The focus was on displaying the variety of methodological approaches adopted by the studies so far. The earliest attempts at ranking mutual funds used simplistic measures which relied on comparing mutual fund returns with benchmark market returns. Later measures like the Sharpe ratio, Treynor ratio, and Jensen’s alpha, which are still among the most popular measures for mutual fund evaluation, also took into consideration the trade-off between returns and risk associated with financial assets like mutual funds. However, it was only in the last decade of the twentieth century that advancements in frontier analysis techniques allowed for a truly holistic approach to measuring mutual fund efficiency. The two best-known methods of frontier analysis are the DEA and the SFA. Among the two, the DEA has been more popular among researchers studying mutual fund performance, though both these methods have their advantages and shortcomings. Namely, the DEA by not requiring any prespecified functional form for the frontier can be easier to apply in case of mutual funds evaluation, since mutual funds do not involve a traditional production process. Whereas the SFA is more robust to outliers and sampling errors due to the inclusion of the stochastic error term. Another caveat of the DEA which is particularly relevant in the case of developing countries is the need for the sample of funds to include at least some efficient funds for the DEA rankings to provide accurate indications that scarce financial capital is being allocated to its most efficient use. Nonparametric methods like the DEA will always identify at least one fund as “efficient” irrespective of whether the fund is actually investing in the best available projects in the economy. While this may not be a concern in developed economies where capital markets are well developed and many efficient funds exist, this concern could be real for emerging markets with less-developed financial markets. In such cases, parametric methods like the SFA can be useful as these models specify the benchmark frontier exogenously based on a normative concept of what constitutes efficient investment using theoretical models of corporate finance. It is clear that both of the techniques have their own strengths and weaknesses, and there is evidence of these two strands of mutual fund research informing each other.

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It is hoped that future research into the mutual fund industry will demystify the process of fund management and thereby catalyze mutual fund evaluation literature, especially parametric methods of evaluation.

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Performance of Microfinance Institutions: A Review∗

33

Christopher F. Parmeter and Valentina Hartarska

Contents Introduction and Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Production/Cost Environment of MFIs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . What Efficiency Means to MFIs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . How Efficiency Has Been Measured Across MFIs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Two-Stage Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Data Availability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Data on Subsidies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Key Modeling Issues . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Selection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Measurement Error . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Loans Versus Savings and Loans . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . How to Quantify Outputs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Main Findings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Returns to Scale . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Economies of Scope . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . MFI Heterogeneity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Women’s Impact on MFIs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Governance and Performance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Outreach and Mission Drift . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The Role of Risk . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1310 1311 1313 1314 1315 1316 1318 1318 1318 1319 1320 1321 1322 1322 1323 1324 1325 1325 1326 1327

∗ We

would like to thank Erika Schutt Pardo for comments on an earlier version of this chapter. All errors are ours alone. C. F. Parmeter () Department of Economics, University of Miami, Miami, FL, USA e-mail: [email protected] V. Hartarska Auburn University, Auburn, AL, USA e-mail: [email protected] © Springer Nature Singapore Pte Ltd. 2022 S. C. Ray et al. (eds.), Handbook of Production Economics, https://doi.org/10.1007/978-981-10-3455-8_33

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Competition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Future Directions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Cross-References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Abstract

This review covers current issues that applied researchers assessing the performance of MFIs are likely to encounter and should be cognizant of. Keywords

Efficiency · Productivity · Microfinance · Economies of scope · Returns to scale JEL Classification

C10, C13, C14, C50

Introduction and Overview The primary goal of this chapter is to introduce the wide audience of this Handbook to the empirical literature that has developed over the past 20 years or so relating to the productivity and efficiency of microfinance institutions (MFIs). One may question why an entire chapter needs to be devoted to such a topic. While MFIs operate quite broadly in much the same way that traditional banks do, making loans to clientele and in some cases offering to collect deposits, they face an entirely different set of objectives and hurdles that traditional banks do not (see [11]). MFIs typically lend to the poorest of the poor and cannot use standard screening tools given the lack of credit record or collateral; the loan amounts are typically of such a small magnitude that standard banks would have a difficult time capitalizing from said loans. In addition, while most banks offer both savings and intermediate loans, MFIs traditionally have serviced loans and have only recently begun holding deposits which are mainly viewed as an additional product, rather than a strict source of capital. Moreover, while the primary objective of a traditional bank is to turn a profit, in many instances MFIs promote outreach to the unbankable at the expense of increased profit. All of this taken together suggests that while methods of assessing productivity/efficiency of MFIs may overlap with those used for traditional banks, there are a host of specific issues that require direct attention and detail when studying MFIs. This is also true for data pertaining to MFIs. While nearly every country has a central bank that collects, aggregates, and reports on the banking sector, no such country-level body exists for MFIs. Thus, how and where data are collected and reported is an important issue for the applied researcher interested in studying MFIs. We pay special attention to data sources and threats to credibility and external validity for said data sources here.

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It is important to understand the performance of MFIs broadly as the industry has changed substantially over the recent past. While MFIs have had a solid focus on providing credit, the nature of the industry has evolved. This is due to the fact that providing credit to the poor is generally a costly activity and may impinge on the financial sustainability of MFIs. The origins of MFI were such that NGOs and donors would provide financial support to MFIs by offering loans at below-market interest rates, helping them lend. More recently, however, a shift from this model of subsidizing MFIs to a model of financial sustainability and cost-efficiency has taken over. This model focuses on MFIs having the ability to cover their internal costs of lending through the interest income generated from the corresponding loan portfolio and by minimizing costs in the delivery of these loans. A business model of cost-efficiency and financial stability is appropriate given the many challenges that MFIs face. Since they operate under significant financial constraints with a higher-risk clientele without collateral, coupled with intensive labor use, to eschew complete reliance on subsidies, high interest rates on loans are inevitable. These high interest rates can, unfortunately, run counter to many MFIs’ mission statements of outreach and poverty alleviation. Rosenberg et al. [118], van Rooyen et al. [130], Dehejia et al. [42], and Sinclair [126] all conclude that high interest rates are one of the central reasons why MFIs may fail to meet their outreach platform. Other studies have found that direct or indirect subsidies help keep interest lower, and recent studies (including productivity) have shown that continued subsidies play an important role in controlling costs and affect measures of scale economies in MFIs [36]. The remainder of this survey will touch on the myriad empirical issues that have arisen as academics have probed and studied the operating environment of MFIs, their efficiency levels, the factors which impact these levels, and future issues that deserve more attention.

Production/Cost Environment of MFIs At the heart of performance evaluation of MFIs is whether they should be viewed as profit-oriented, self-sustaining businesses or socially minded, nonprofit organizations [27, 102]. As [27, pg. 28] ask, “Should MFIs be compared based on their profitability or based on their outreach . . . .” How a researcher chooses to answer this question lies at the heart of performance evaluation of MFIs and will shape the policy narratives derived from it. As different MFIs may have different targets, modelling how MFIs produce output is a key challenge for empiricists. One can envision a setting where MFIs produce outputs that maximize financial revenue (yield, for instance) as well as outputs that maximize outreach (minimum average loan size, number of loans, number of clients, etc.). In this instance a standard production environment is unlikely to allow proper benchmarking of MFIs. Rather an output distance function [27] or cost minimization framework [77] will be necessary.

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For our purposes, we will focus on modeling of costs for MFIs. The cost function is a natural fit for MFIs as they typically take outputs (loans/deposits) as given and minimize input costs in the production of these outputs. The cost function is also an empirically accessible tool to model multiple outputs, which as noted above is important for MFIs when comparing across MFIs who may have different objectives (profit vs. outreach). The basic cost function framework is ln Cit = m(y it , w it , zit ; β) + αi + εit ,

i = 1, . . . , n;

t = 1, . . . , T ,

(1)

where y it is the vector of outputs for MFI i in period t, wit is the vector of input prices (cost of labor, capital carrying costs, etc.), zit is a vector of controls which may affect the cost environment of the MFI (clientele served, operating area, region, type of MFI, etc.), αi is an MFI-specific term which captures time-constant heterogeneity across MFIs, and εit is an idiosyncratic shock. When cost-efficiency is being studied, εit can be decomposed into a pure noise effect, vit , and a one-sided inefficiency component, uit , which serves to raise costs above their minimal level: εit = vit + uit . The parameters of the model can be estimated once the cost function, m(y it , w it , zit ; β), is parametrically specified.1 A common empirical specification is the translog functional form. To write this in matrix form, we let ln x it = (ln y it , ln wit ) denote the vector of outputs and input prices in logarithmic form. We then have m(ln x it , zit ; β) = ln x it β 1 + ln x it β 2 ln x it + β 3 zit ,

(2)

where β = (β 1 , vec(β 2 ), β 3 ) and β 2 an appropriately sized matrix of coefficients that respects linear homogeneity and symmetry of the cost function. A serious empirical concern is how to deploy the translog functional form when some MFIs do not produce all outputs. For example, in the study of scope economies of MFIs, some MFIs may only offer loans, while others offer both loans and deposits. In this case the pure translog functional form is insufficient for this. A common, though empirically dubious, approach is to add a small number to the corresponding 0 for the MFI-specific output prior to taking logarithms. While many empirical studies uncover “reasonable” estimates of various metrics using this approach, it does not parry the initial concern, and there is no consensus as to how small a number should be added to the 0 output value prior to taking logarithms. Going forward authors are counseled to avoid this practice, and if they do engage in this behavior, they should be explicit as to which number they added, how robust their results are to minor changes in this small number, and how many observations in the overall dataset this impacts. A better alternative is to model separate translog functional forms as in [96] or to use a different functional form such as generalized quadratic or Leontief.

1 Nonparametric

methods can also be deployed.

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What Efficiency Means to MFIs Efficiency improvement is meaningful to any business or sector. Efficiency has perhaps more important implications for MFIs given their (potential) reliance on subsidies [36, 40] and the dichotomy between outreach and operational performance [27]. The majority of MFIs opened initially with the goal to serve the poorest clientele. However, “mission drift” [100] may set in given the absence of substantial scope and/or scale economies [96]. This is illustrated with the financial sustainability of the Grameen Bank, a flagship MFI, whose performance improved materially after changing its business model to start offering micro-savings (in addition to loans). These performance improvements coincided with a simultaneous abandonment of its poorest clients. This is not surprising as once an MFI is licensed to collect savings deposits, they become subject to banking regulations, and the additional stringent supervisory environment incentivizes MFIs who are profitoriented to curtail outreach to costly-to-reach-customers [39]. Understanding the efficiency of a single MFI or of the industry as a whole has a wide-ranging importance. Given the gradual withdrawal of subsidies, this promotes the need for MFIs to demonstrate long-term viability and sustainability. There are concerns that subsidies reduce MFI incentives to perform efficiently. As an example, [29] show that MFIs in Central Asia and Eastern Europe that rely more heavily on deposit funding and less on subsidies are more efficient than similar MFIs over time. Using different methods, [79] find the opposite effect: “smart subsidies” [11], which allow MFIs to build their infrastructure and develop institutional know-how, result in MFIs which are more efficient than those that do not receive these subsidies. Regardless of the type of subsidy, they need to be accounted for when assessing MFI performance; as noted by [104, pg. 98], “. . . [MFIs’] performance on the basis of traditional financial ratios without unearthing the degree of subsidy dependence provides only a partial and often meaningless or misleading picture of the social cost of maintaining the MFIs. . . .” Cost-efficiencies are also important to MFIs given the increase in competition over the past several decades. The entrance of a variety of for-profit MFIs has led to concerns over higher than necessary interest rates, which can undermine outreach objectives. This is characterized by [118, pg. 1]: “An interest charge represents money taken out of clients’ pockets, and it is unreasonable if it not only covers the costs of lending but also deposits ‘excessive’ profits into the pockets of an MFI’s private owners. Even an interest rate that only covers costs and includes no profit can still be unreasonable if the costs are excessively high because of avoidable inefficiencies.” Thus, MFIs need to operate efficiently if they are to provide microloans at rates that actually help their clientele and remain financially stable. To illustrate this point more deeply, consider recent research on six prominent randomized controlled trials (RCTs) [14], which found only a small average impact of microcredit access on new marginal borrowers. These modest impacts could have the potential to produce sizable effects if MFI costs are proportionally small; thus the ability to maintain cost-efficiency becomes all the more apparent. More recent work

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has revisited these six randomized controlled trials (and others as well), see [41, 103, 111], and raised questions concerning the role of MFIs on poverty reduction, arguing that RCTs are best viewed as one of the myriad approaches to evaluate the impact of MFIs. Regardless which side one falls one, it is apparent that MFIs need to be cost-efficient to remain effective, regardless of the wider-reaching effects of offering access to loans/deposits to the unbankable (see also [38]).

How Efficiency Has Been Measured Across MFIs The spectrum of empirical studies investigating technical and allocative efficiency of MFIs has used a host of statistical methods. However, a majority of this research have deployed either data envelopment analysis (DEA) or stochastic frontier analysis (SFA). Both methods have their merits and criticisms. DEA offers the ability to model the production or cost environment in a fully nonparametric setting and does not require distributional assumptions to identify inefficiency. Unfortunately, these advantages require an assumption that idiosyncratic noise does not exist in the model. In Equation (1) this is captured as εit = uit . This makes DEA susceptible to outliers which are likely to exist in (especially cross-country) MFI data [72, 73]. Methods do exist to seamlessly handle potential outliers when using DEA. We direct the curious reader to [125] for more details. Another empirical concern with the deployment of DEA is the fact that the nonparametric nature of the methods means that potentially large finite sample biases may exist given the dimensionality of the data. In this instance one may consider dimension reduction methods [136]. SFA relies more heavily on parametric methods. It is common that the researcher will have to specify both the cost function (translog, quadratic, etc.) along with imposing distributional assumptions on the composed error term. The most common specification is to assume that vit ∼ N(0, σv2 ) and uit ∼ N+ (0, σu2 ), the ubiquitous normal-half-normal specification, popularized by [3]. The benefit of using SFA over DEA is that random fluctuations can be more easily handled in one’s analysis. Naturally one may ask if the potentially erroneous parametric assumptions compensate for this benefit, and a large literature has developed studying this question. Recent advances in SFA allow the production/cost technology to be estimated in a nonparametric fashion [108] and to dispense with distributional assumptions on the composed error [107,128,139]. We direct the reader to [106] and the chapter of Kumbhakar et al. [91] in this Handbook for more detailed treatments on various aspects of SFA. One important aspect of applying either SFA or DEA to study efficiency of MFIs is the fact that the microfinance literature has reached a consensus that the external operating environment impacts how MFIs behave and, more importantly, that these factors need to be accounted for in studies of MFI efficiency [2, 56]. Hartarska et al. [72] were the first to include these impacts in an efficiency study of MFIs. Moreover, the vast majority of studies on MFI efficiency typically find moderate levels of technical efficiency, indicating substantial room for improvement. See [52] for a meta-analysis of this literature.

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Notable studies relying on DEA to estimate MFI level technical efficiency include [62, 93, 109, 122]. Across the spectrum of studies exploring efficiency with DEA, many focus on MFIs in a particular country/region: [54] study MFIs in Peru; [62] focus on MFIs across Latin America more broadly; [8] study agricultural co-ops in Bulgaria; and [93] study Vietnamese MFIs, while Efendic and Hadziahmetovic’s [49] work looks at MFIs in Bosnia and Herzegovina. One of the more geographically comprehensive studies deploying DEA is [63] who look at MFIs across Africa, Asia, and Latin America. A key finding is that nongovernmental microfinance institutions are the most efficient, a result consistent with this type of MFIs’ fulfillment of their (competing) dual objectives: poverty alleviation and maintaining financial sustainability. An early study of MFI efficiency using SFA is [75], focusing on the Grameen Bank. Their findings suggest that female-only branches are the most efficient among the various Grameen Bank branches, but the average inefficiency across all branches is in the 3–6% range, suggesting that the Grameen Bank is highly efficient to begin with. Other recent studies include Servin et al. [123] who focus on technical efficiency of Latin American MFIs, Hermes et al. [77] who use SFA to investigate if a trade-off between outreach to the poor and technical efficiency of MFIs exists, Gregoire and Tuya [60] who study efficiency of Peruvian MFIs, and Pal and Mitra [105] who explore the linkages between number of borrowers per loan officer and MFI asset quality (measured as portfolio value at risk) using data on 1,575 MFIs spanning the period 2006–2013. Kendo [86] studies MFI efficiency in African countries, using a panel of 163 MFIs over the 2004–2011 period. Bensalem and Ellouze [21] study how the current wave of commercialization of MFIs impacts both their financial and social efficiency using a sample of 162 MFIs over the period 2007–2013 across Africa, Asia, Latin America, and the Middle East. Until recently, many SFA efficiency studies of MFIs, while having access to panel data, did not fully embrace the panel nature of the data. That is, the time constant nature of MFI-specific heterogeneity was ignored. A step in this direction is the “true random effects” estimator [59] deployed by [68] who study if the presence of a female CEO at the MFI is related to efficiency. They find MFIs with female CEOs have significantly higher “outreach efficiency” (as measured by the number of clients) than similar MFIs with male CEOs.

Two-Stage Analysis In several studies of MFI efficiency, researchers have deployed what is known as the two-stage approach [134].2 This approach estimates the cost function in Equation (1) (or some equivalent); recovers technical efficiency scores, usually by following the methodology of [85]; and then regresses these technical efficiency scores on a set of MFI characteristics. This approach should be avoided at all costs.

2 Prominent

examples include [21, 66, 75, 86].

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Empirical results derived in this fashion are not to be trusted. The reason is twofold. First, any set of characteristics that are omitted when the cost technology is estimated will produce an omitted variable bias. Depending on the correlation between traditional input prices and outputs and the omitted MFI characteristics, these biases can be severe. This will undoubtedly impact not only the estimates of the cost function parameters but also the subsequent residuals that are used to produce the technical efficiency scores that feed into the second stage. Second, even assuming away any impact of omitted variable bias, [134] have shown that the estimated technical efficiency scores are underdispersed. This suggests that the dependent variable in the second stage, the estimated technical efficiency scores, has less variation than it should, which will adversely affect the quality of the estimates in the second stage. Given both of these concerns, studies attempting to discern the characteristics of MFIs that improve technical efficiency should do so in a single-stage setting. This can be done in a straightforward manner using either maximum likelihood (by modeling the parameter σu2 as a function of said determinants) or nonlinear least squares. The use of nonlinear least squares is appealing as in this case parametric distributional assumptions are not necessary, just invocation of the so-called scaling property [7] which multiplicatively decomposes inefficiency into deterministic and stochastic components; the multiplicative form of the decomposition allows estimation of the model eschewing distribution assumptions since only the mean of technical efficiency enters the model and can be captured as a single parameter. Readers are directed to [106] for a textbook discussion of this issue.

Data Availability There are several main sources of data that are used in empirical studies of MFI performance and efficiency. First, many papers use hand collected data from field experiments (one prominent example is [14]).3 These data are not typically available unless made public by either the authors of the study or via replication policies of the journals (see the American Economic Review or Quarterly Journal of Economics data policies, for instance). Curated databases of various metrics for MFIs do exist, with the most prominent being the MIX Market database (https://www.themix.org/). A free version exists through the World Bank’s Data Catalog. The website provides information on over 2,000 microfinance institutions across all regions of the globe, dating back as far as 1996. MIX, often referred to as the “Bloomberg of microfinance,” represents a huge open data and transparency win for the entire microfinance industry. One existing issue is that data at the MFI level are self-reported and some MFIs either do not report or report incomplete information. This engenders potential concerns

3 Almost

no studies of MFI efficiency collect data by hand from MFIs. Hartarska [64] is one example that looks at performance of MFIs based on governance where the data is hand collected.

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over endogeneity and self-selection when deploying this data. While a selection bias is anticipated and MFIs reporting to the MIX Market likely represent the better performers, [17] find that in the early years of MIX Market reporting, MFIs’ patterns of reporting to the MIX Market are correlated with institutions’ region of operation, mission, and size, which makes the data adequate for many purposes.4 MIX Market data are used in numerous microfinance studies, including on productivity and efficiency of MFIs and of gender-related aspects. Another database on MFIs is housed at the Center for Research on Social Enterprise and Microfinance (CERSEM) at the University of Agder in Norway.5 Compared to the MIX Market data, which are used in most studies on the performance of MFIs, the CERSEM dataset includes a representative sample of much smaller MFIs, which are typical examples of firms working with both financial and social goals. The dataset includes 660 MFIs around the globe. The dataset is an unbalanced panel ranging from 1998 to 2015. CERSEM possesses several advantages over MIX Market: (i) data are verified by professional external rating agencies; (ii) data have less of a large-firm bias; and (iii) accurate information on more variables is available in the CERSEM dataset allowing greater control for important MFI characteristics. A more specialized database, focusing exclusively on MFIs operating in Europe, is available through the European Microfinance Network (EMN).6 This dataset is compiled on a biennial basis, based on a survey of microfinance institutions (MFIs) in Europe (see [28]). The MFIs surveyed by EMN self-report data mainly about their mission, target, and social and financial impact. Currently the data covers key institutional characteristics, outreach, social performance, and financial performance across 444 MFIs for the period 2006–2015, 34 of which are observed for at least 8 years in the reference time interval. We note here that the MIX data is the dominant dataset that is used in the literature when focusing on broad comparisons of MFIs globally. Many authors collect data on specific MFIs in specific regions and conduct various surveys with clientele or run field experiments to discern the impact that the MFIs are having (e.g., [88]). However, this requires specialized skills, institutional knowledge, and boots on the ground. The broad databases just listed here abstract from MFI-specific issues. Moreover, recently many of the criticisms against MIX have begun to ebb as the dataset remains up to date and publicly available, whereas both EMN and CERSEM capture fewer underlying operational variables and are not as current as MIX. In fact, to our knowledge, the last reporting year for CERSEM currently is 2017.

4 This

is true for earlier versions; in later years of data collection as more MFIs understood that reporting to MIX is a good way to attract investors and soft credits and grants, it has become less so. 5 https://cersem.uia.no/dataset/ 6 https://www.european-microfinance.org/

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Each of the regional/global datasets discussed here has their own unique flavor, and different researchers may have specific questions that one of the databases is more apt to help answer. Thus, we view these datasets as complements to one another rather than as direct competitors.

Data on Subsidies Primarily due to data limitations, the role of the subsidy-efficiency trade-off is still poorly understood, even though subsidies are present in about half of MFIs. Monetary subsidies are pure or conditional grants and soft loans, i.e., preferential debt issued at below-market conditions and donations. In-kind donations include paying part of labor costs (usually senior management and board of directors), buildings, equipment, and sponsored management information system. Datasets accessible to researchers contain accumulated donated equity data (from the balance sheet), and much of the literature uses donated equity (contained in the MIX Market data) to assess subsidies flowing to MFIs [79] or determines the subsidization ratio as donated equity over total assets [25]. An alternative definition of subsidies stems from the donations reported in the income statement [48]. Aside from hand collecting data designed to measure subsidies, to understand their effect one would need to use the proprietary component of the MIX Market database and a variety of adjustments made by Mix Market personnel. Such an example is the work by [40] who find relatively low levels of median subsidies suggesting that even modest benefits of microcredit could yield impressive cost-benefit ratios. Surprisingly, the authors report that subsidies are skewed and most go to MFIs serving fewer poor borrowers. Using subsidy data is important because many empirical studies find that ignoring subsidies leads to misleading results. It seems that subsidies have different implications in deposit-collecting and in loans-only MFIs especially in nonprofit institutions, where lack of subsidies may be associated with socially harmful consequences [70, 88]. Studies that do not distinguish between business models find contrasting results in that subsidies are negatively related to outreach and sustainability, may worsen efficiency, increase costs in time, and may crowd out deposit collection [5, 27, 29, 36].

Key Modeling Issues Selection Selection bias is a common ailment of many empirical microeconometric studies, and microfinance is not immune to this. We see two main areas where selection is potentially an issue for the researcher: First, are the databases that collect data representative? Second, if we study both MFIs that offer strictly loans and those MFIs that offer loans and deposits, are we properly accounting for the potential

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choice the MFI made prior to operation to offer a specific product mix? Both of these issues need to be seriously taken into account in one’s study of MFIs. For example, as section “Economies of Scope” will detail for studying economies of scope, MFIs are not randomly assigned to offer strictly loans or a mix of loans and deposits but self-select. How this impacts the findings is an open question. Both [72] and [44] estimate the degree of scope economies in microfinance institutions but do not consider selection across product mix, estimating a single cost function for all MFIs regardless of financial services. Malikov and Hartarska [96] go a step further and directly model the selection of MFI’s financial services they offer using the approach of [92]. Cozarenco et al. [36] build on the insights of [96] by modeling selection in the technology of MFIs to vary across subsidization status as well as the decision to offer strictly loans or loans and deposits. This is an important step in understanding the performance of MFIs.

Measurement Error An issue that has received only marginal attention, but is important nonetheless, is the quality of the data being used in the empirical literature. As we noted in section “Data Availability”, the majority of data for studies of MFIs are collected either through surveys and field experiments on the ground or through various agencies, with MIX Market being the most popular database. MIX Market data has been self-reported in the past, and this leads to potential concerns over the accuracy of the data that is reported. One aspect of the database that can be exploited to investigate concerns over data quality is the diamond rating.7 MIX Market classified MFIs into diamonds according to the availability of data provided by the MFIs. Diamonds range from 1 to 5 with higher numbers indicating greater transparency and (potential) reliability of data. Al-Azzam and Parmeter [6] conduct their analysis of competition among MFIs using all of the available data (diamond ratings 1–5) against a subset using only MFIs with diamond ratings of 4 and 5. Their qualitative findings were identical, thus allaying potential concerns that measurement issues were impacting their findings. We note that the original purpose of the diamond rating was not to capture measurement error via the quality of the data but whether the MFIs provide complete information on all indicators that they were asked to provide. Thus, for example, an MFI that only provided data on the number of borrowers and dollar value of their loan portfolio but not specific cost data would be rated as 1, while an MFI that provided all necessary data would get a 5 rating. It does not necessarily mean that the data reported are correct. However, MFIs that take the time to report on

7 We

note that the Diamond data are no longer available via the world bank MIX Market database.

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all aspects that were asked of them could be assumed to more reliably report those numbers than MFIs that only report a limited set of measures. Using historical MIX Market data with the diamond ratings is a strategy that could be deployed in future analyses of MFI performance. While removing observations may lead to important reductions in the sample size, this is useful to determine how much influence data quality/reporting may have on one’s empirical results. A related issue that has yet to receive attention in the performance evaluation literature of MFIs is that of missing data. The common approach is to drop observations where any of the key variables are missing (costs, outputs, determinants of inefficiency, etc.). However, this may have an undue influence on the analysis if these variables are not missing at random.

Loans Versus Savings and Loans A more recent issue that has arisen in the study of MFI performance is the (potential) difference in those MFIs that offer purely microcredit and those that both offer microcredit and take deposits. Inherently there are differences in these types of MFIs both from the outreach perspective and their governance. While it is widely acknowledge that savings are important to allow consumption smoothing over income shocks that are likely to arise, the poor find it hard to save, chiefly because resource scarcity often combines perversely with behavioral biases. Cozarenco et al. [37] is one of the first studies to look at the separate characteristics of MFIs that offer micro-savings and those that do not, while previous work has evaluated the savings-collecting subgroup independently, for example, estimating efficiency of cooperatives [69] or of municipal banks [94]. Armendariz de Aghion and Morduch [11] note that micro-savings are often considered as the “forgotten half,” hence the lack of academic attention. However, the regulatory environment also plays a key role in either the availability or lack thereof of micro-savings. To protect clients of microfinance services, regulators typically create barriers that make it costly for many MFIs to provide micro-savings accounts. Christen et al. [32] point out that this lessens the accessibility to these products resulting in a much smaller fraction of the impoverished to be reached by MFIs. Micro-savings deposits come in two forms: compulsory and voluntary. Compulsory savings represent “hidden collateral” [9], while voluntary savings represents the true demand for micro-savings. To understand more of the underlying factors that drive MFIs to take voluntary micro-savings, [37] run random effects probit models coupled with MIX Market data. Their summary statistics suggest that MFIs taking voluntary savings are not significantly different in terms of financial performance with those MFIs that do not take voluntary micro-savings but perform worse in both outreach and the percentage of women served. One of their main (unsurprising) findings is that MFIs collecting voluntary micro-savings are older. This is due to the fact that micro-savings stem from regulatory compliance, which is typically not feasible for new/young MFIs. Another important finding is that subsidies crowd out micro-savings hampering outreach efforts of MFIs.

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The study of [37] is important for future research for several reasons. First, the findings are useful to understand how better to promote micro-savings generically. Second, as we previously discussed, selection issues related to product diversification are likely to arise in empirical work, and understanding what drives the decision to offer micro-savings is key in specifying selection mechanisms. Third, the performance of MFIs is likely dependent upon the product mix, and so understanding decisions of both voluntary and compulsory micro-savings is prudent prior to making comparisons across different MFIs. Fourth, the impact of subsidies is an important area within MFI performance, and their study documents an important result on the effects of subsidization on MFIs.

How to Quantify Outputs When considering a financial institution, both number of accounts and volume of accounts could be used as measures of output. However, the prevailing banking literature uses volume of loans and deposits as banks’ outputs. This is for several reasons. First, it has been argued that the intermediation approach [121] better describes what a banking institution does and presumably achieves a more apt description of the operating environment [55]. Second, variables which measure the number accounts are difficult to come by in existing banking databases, and, when available, these data are viewed as unreliable. It has been argued, however, that in some financial firms, output is better measured by the total number of accounts and transactions as opposed to volume or value [129]. Given many MFIs stated outreach mission to reach more clients, microfinance studies have argued that the number of clients or even the overall level of clientele poverty, rather than the monetary equivalent of accounts and profit, is an appropriate measure to gauge MFI performance [29]. A goal of outreach for MFIs is to service the largest number of borrowers with small loans; thus the volume of loans and the number of loans can/should affect the productivity of these institutions. Caudill et al. [29] report that their results do not differ substantially from those when only the number of clients were used to measure lending. As such, future studies have used number of loans and clients to measure MFI performance through efficiency [65] as well as scale and scope economies [72, 74].8 A detailed comparison of the impact of using volume versus number on MFI performance was conducted in [73]. Their empirical work studies possible differences in both the mean and the overall distribution of estimated scope economies using either total accounts held by the MFI or the total dollar value of these accounts as output measures within a semiparametric quadratic cost function. They also conduct several tests to determine how different their estimates of scope economies are not only at the mean (median) but also on the overall distribution of estimated scope

8 Hartarska

et al. [68] use the number of clients (borrowers and savers) which equals the number of borrowers in lending-only MFIs but captures better the output of savings-and-loans MFIs.

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economies. Their main conclusion for future empirical work is that qualitatively their estimates of MFI performance (scope economies) are similar both globally and within specific regions. Some caution with this result is needed as their tests of distributional equality suggest statistically meaningful (though economically small) differences. While [73] found that differences in their estimates exist, no clear-cut measure of output emerged as the definitive measure. There are two implications of this for empirical work. First, it may be preferable to use the active number of clients as the appropriate measure of output for an MFI since this better reflects the outreach mission statement. Second, given that access to reliable data on the number of active borrowers or active savers is not typically available for the majority of datasets, or in the case of the MIX Market database, for a select subset, their findings suggest that the use of dollar values should still provide similar insights. Researchers assessing performance of MFIs should attempt to use number of clients or loans and not the dollar value if at all possible (though recognizing that data limitations may prevent this in all cases).

Main Findings Returns to Scale Generically, returns to scale can be computed directly from the cost function −1  M  ∂ ln C parameter estimates using RT S = for the M distinct outputs (here ∂ ln ym m=1

this would most likely be volume or number of loans and possibly volume or number of deposits).9 The importance of RTS as a performance measure is that this can yield important insights into the overall health of the industry (either globally, within a region, or within a given state/country). Increasing returns to scale suggest that (proportional) cost reductions can be had while simultaneously increasing output (which has connections to outreach). The majority of existing studies find increasing returns to scale (IRS) suggesting that the industry is progressing towards optimal size. This also presages future cost savings as MFIs continue to grow. However, the finding of IRS is not fully robust. For example, when outputs are measured as the number of active clients rather than the volume of loans (or both), or when subsidies are properly accounted for, constant and even decreasing returns to scale have been found for MFIs in both Eastern Europe and Latin America [30, 36, 74].

9 When

one includes quasi fixed inputs, e, then RT S =

1−∂ ln C /∂ ln e . M  ∂ ln C m=1

∂ ln ym

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Economies of Scope Another important metric that has been assessed broadly is economies of scope (also known as economies of diversification). The industry has trended away from solely lending activities into one that offers both savings and loans. As noted in [96, pg. 162], in the early 2000s roughly one third of MFIs offered both savings and loans, whereas a decade later, over half of all MFIs were offering both products to their clientele, a near 100% increase. This reflects the growing demand of the poor for expanded financial services [34]. As mentioned earlier, with deposits comes additional oversight, which can be costly. Thus, MFIs need to be sure that the regulatory burden is offset by offering deposits. This is most commonly achieved through economies of scope. Economies of scope can emerge from two different sources. First, cost reductions can be had by distributing fixed costs across the product mix. Allocating fixed costs over the services an MFI offers can contribute to scope economies when excess capital capacity is reduced by providing both savings and loans rather than just by offering loans. Second, cost complementarities may exist between different products. These complementarities are derived when consumer information developed in the production of either savings or loans is used to reduce the monitoring requirements of the other product. Scope economies for an MFI that offers either loans (y1 ) or deposits (y2 ) exist, in the traditional sense [18], if C(y1 , 0) + C(0, y2 ) > C(y1 , y2 ). Here we have simplified our notation from Equation (1) to make the discussion simpler. Pulley and Braunstein [113] suggest the estimation of quasi scope economies, while [114] define a normalized version of this criterion to assess scope economies. This measure does not restrict the calculation of scope economies to the case of perfectly specialized output, the counterfactual situation where one output is set to zero. In settings where all firms offer both products, this is a useful tool to have to approximate the appropriate curvature of the cost function. However, for MFIs, where many MFIs do not offer deposits, and virtually no MFIs are specialized to offer only deposits, this is not an issue. In fact, a more pressing concern for the estimation of scope economies for MFIs is the opposite: the required counterfactual estimation of C(y1 , y2 ) to the case of a single output, known as the “excessive extrapolation” problem [51, 80]. Malikov and Hartarska [96] deploy a modified version of the traditional measure of scope economies accounting for excessive extrapolation. They do this in two ways. First, rather than estimate a single cost technology for all MFIs, they estimate separate technologies for loan-only and savings-and-loan MFIs. The estimation of the loanonly cost technology yields C(y1 ) as opposed to C(y1 , 0). Second, rather than shift y1 to 0 in the counterfactual analysis of the cost technology for the savings-and-loan MFI, they down weight y1 by an amount such that each counterfactual MFI does not produce less of each output than what appears in the data. Thus, the weight, ω, is such that ωy1 ≥ min y1 (for loan only MFIs) and (1 − ω)y1 ≥ min y1 (for savings and loan MFIs). Economies of scope are said to exist if C(ωy1 )+C((1−ω)y1 , y2 ) > C(y1 , y2 ). Following [114], economies of scope can be calculated as

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ES(ω) =

C(ωy1 ) + C((1 − ω)y1 , y2 ) − C(y1 , y2 ) . C(y1 , y2 )

(3)

The natural question here is what value to select for ω as there will exist an admissible region of values that ensure above minimum production of y1 . Malikov and Hartarska [96] suggest a measure of global economics of scope, minES(ω). ω When the smallest value of global scope economics is positive, one can still conclude that economics of scope exists in the production of savings and loans. One issue that the approach of [96] cannot deal directly with is the further decomposition of economies of diversification into separate components for fixed and complementary costs. The seminal work of [72], while potentially suffering from excess extrapolation, can investigate economies of scope broadly, as well as decompose scope economies into separate pieces due to fixed and complementary costs. This is due to the specific functional form that they specify and their use of a semiparametric smooth coefficient model [76, 95]. This model is also used in [44]. Knowledge of the magnitude of both of these components of economies of diversification are important as [72, pg. 391] note, “. . . cost complementarities accrue to MFIs if the account information that is developed in the process of creating deposits is subsequently used to help monitor and gather credit information on loans for the same customer base. Spreading fixed costs over an enhanced product base produces scope economies if the same set of tools required to manage deposits can also be used to produce and monitor loans.” As far as results from various applied papers focusing on economies of scope exist, the first in this area were [72, 73]. More recent studies using a variety of sophisticated econometric tools include [44, 96, 97]. Nearly all of the studies investigating scope economies have found that they indeed exist; however, there is substantial heterogeneity in the range of estimates, and earlier studies, using standard methods (i.e., not accounting for excessive extrapolation) ,find much higher levels than newer studies that take this into account. More specifically, it appears that estimated economies of scope are modest for most small-size MFIs but can be quite substantial for large-scale institutions.

MFI Heterogeneity Many studies of MFI performance have documented substantial heterogeneity across MFIs, which is not surprising. Differential performance has been observed regionally [96], across MFI type [72], by area served (namely, rural versus urban versus both as in [44]), whether MFIs are subsidized [36], and composition of the board of an MFI [68]. The work of [61] is also a recent example illustrating the diverse heterogeneity that exists when assessing MFI performance. They study the impact of government ideology on MFI sustainability, documenting diversity across left-wing and rightwing ideologies of the governments in places where the MFIs operate. Gul et al. [61] find that MFIs operating in a left-wing regime have higher portfolio growth rates and

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lower funding, operating, and default costs. It appears that the electoral incentives of left-wing governments impair the capacity of MFIs to increase financial revenue. Returning to the issue of scope economies, [72, Table 2] find substantial differences in economies of scope (generally) as well as fixed and complementary costs which vary heavily by region. Average scope economies are 0.245 in MFIs operating in Africa, whereas those in Latin America have scope economies on average of 0.105. These stark differences suggest that regional heterogeneity is an important component to understanding MFI performance.

Women’s Impact on MFIs A majority of an MFI’s clientele are women, who typically benefit from smaller loans than men [11]. A large literature has documented links between the gender of the microfinance borrowers and the performance of the MFIs. Agier and Szafarz [1] compare denial rates and loan sizes for male and female applicants in an MFI located in Rio de Janiero with similar expected creditworthiness to check for disparate treatment linked to taste-based discrimination. This is an important first step in understanding how MFIs perform based on gender given that MFIs commonly offer fixed-interest loans and loan sizes are tailored to the expected creditworthiness of the applicant [101]. This lending approach is derived from the need to keep operating costs at a minimum and to help paper over any inefficiencies that may exist. D’Espallier et al. [46] find that a higher percentage of female clients in MFIs is associated with lower portfolio risk, fewer write-offs, and fewer provisions, all else equal. More recently, [33] finds evidence of a trade-off between sustainability and outreach depth, suggesting that, as women are poorer, fewer women would be reached within a sustainable model. Explicit research looking at potential differential effects of gender on performance include [16, 23, 46, 50, 68, 87]. Specifically, [46] find that a focus on women in MFIs is significantly related to smaller loan sizes within the portfolio, the use of collective lending methods, nonprofit status, and a broader orientation. Again, the smaller loans lead to higher operating costs, but women also have lower default rates, which may offset these costs. This balance is an important component in the overall assessment of gender on MFI performance, and it connects to Hartarska and Parmeter’s [71] recent work documenting economically meaningful differences in cost elasticities from loan provision based on gender.

Governance and Performance Another important area of focus on MFI performance is the role that leadership plays. A variety of studies have evaluated the role of gender on the composition of the board of directors and upper management and MFI performance. At present this research has produced mixed results [66, 68]. Recent work points out that female managers and credit boards’ ability to deliver depend on the gender composition and

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matching of credit boards and managers [110]. Research also suggests differences in efficiency in MFIs run by female CEOs [68], while female loan officers have been documented to achieve distinctively different results with men and women clients [19]. An early study on MFI performance and governance is [99]. They investigate the impact of various governance mechanisms on both of the key tenets of MFIs’ mission platform, outreach and financial sustainability. They identify three main channels: vertical, horizontal, and external. The vertical dimension consists of the link between owners and management, the horizontal dimension captures the MFI, and the customer and the external dimension represents that of governance itself. Mersland and Strøm [99] present an array of econometric evidence: (i) financial performance improves when an internal auditor is present and also when the directors are from the local area as opposed to international directors; (ii) ownership type has no effect on performance; (iii) female CEOs lead to better financial performance of the MFI; and (iv) few governance variables appear to influence MFI performance, either statistically or economically. One finding that deserves more attention, but has yet to receive it, is why a female CEO of an MFI does not lead to broader outreach (see Table 6 of [99]). Hartarska and Mersland [66] show that efficiency increases with board size initially but then reaches a saturation point (their estimate is nine members) and harms efficiency thereafter. They also demonstrate that donors’ presence on the board is harmful for efficiency in reaching many poor clients. Lastly, [66] do not find strong evidence of competition or regulatory environment on MFI performance/efficiency. Looking specifically at governance and gender, many studies have found gender effects on performance. Examples include Boehe and Cruz [24] who find that female membership improves an MFI’s performance through enhanced debt repayment, Strøm et al. [127] who find that enhanced female presence on the board to be positively related to MFI performance, and [23] who estimate a positive effect of female loan officers on technical cost-efficiency of an MFI.

Outreach and Mission Drift One benefit of subsidization of MFIs is that it can alleviate financial sustainability concerns, promoting greater attention towards meeting social goals. As an example [45] show that when NGOs transform into commercial microfinance banks, their reliance on subsidized funds declines in favor of deposits and commercial debt. While these MFIs’ long-term profitability improves, it comes at the expense of sharp increases in average loan size, which directly corresponds to diminished outreach. The forward facing side of all MFIs is to fight against poverty (outreach). However, empirically, one pertinent issue when assessing MFI performance and outreach is how to measure both poverty and access to microcredit. The standard in the literature [10, 38] is average loan balance per borrower or the total number of

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loans or deposits taken. The smaller the average loan size, the deeper the outreach; the higher the number of loans given out, the deeper the outreach. Bos and Millone [27] find that disbursing larger loans implies a lower yield on the gross loan portfolio and is correlated with higher personnel and financing costs. Further, for their data, NGOs have lower costs per loan. Bos and Millone [27] document that some MFIs can indeed combine the depth and breadth of outreach and operate with above average levels of technical efficiency. Yet, they also find that efficiency quickly decreases as the loan portfolio becomes larger. More broadly, the literature points to the trade-off between outreach and financial efficiency [39, 77, 116], and this suggests that there may be linkages to the loan officers. Indeed, [119] focus on the role of microloan officers in the provision of microfinance services by studying the preferences of microloan officers over loan allocation. They examine whether, when given a choice, microloan officers select the less advantaged client which is consistent with the stated outreach mission of an MFI. This allows them to study the (potential) trade-off between better financial results and achieving social goals. Similarly, [20] find that more experienced loan officers serve fewer vulnerable clients. The news is not all bad however. Bos and Millone [27] also document that MFIs which specifically target the poor, lend at a higher frequency to women, and provide educational programs are more technically efficient. This finding runs contrary to several earlier studies looking at outreach and MFI performance. This result also piggybacks off of the work of [100] who find no evidence of mission drift, suggesting that the profit motive incentivizes MFIs to seek out new markets and to become more efficient [31]. See also [98] for more insights on mission drift of MFIs. Quayes [115] highlights that during periods of financial distress, there is a potential for trade-offs between outreach and financial sustainability. Wagner and Winkler [133] find credit growth and thus MFIs’ ability to reach the poor dropped sharply after 2008. MFIs in countries with better institutional quality (more advanced financial systems) were more resilient to the global financial crisis [124], and MFIs in general were more resilient than traditional banks [132]. In line with that, [135] finds differences in the reaction of MFIs’ productivity to the shock of the 2008 financial crisis based on ownership type and organizational structure, with microfinance banks and microfinance non-bank financial institutions suffering the most, while non-government organizations (NGOs) and cooperatives were least affected.

The Role of Risk In the microfinance industry, the focus has been on default risk (or credit risk), measured by the proportion of loan portfolio overdue more than 30 days (also known as portfolio at risk overdue more than 30 days or PAR30) since loans are mostly short term in nature. Within the structural approach to efficiency, such risk is essential to control for when modeling the cost structure of financial institutions, because

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lower-quality assets (reflected in a higher non-performing loans ratio) require more resources to manage a higher-level risk exposure, thereby raising the costs for MFIs. Failure to account for riskiness/quality of loans may therefore produce misleading results (e.g., see [81]). Thus, credit risk has been used consistently in most efficiency and productivity studies (starting from [27, 29, 44, 65, 68, 74, 96, 97, 99]). Credit risk is essential to monitor because repayment problems among a few microfinance clients may quickly spread to many clients leading to “borrowers’ run” [26] such as the example of Bolivian MFIs between 1996 and 2000 [131]. The role of credit risk has been explored in the context of geographic diversification, showing that geographic diversification comes with more credit risks, attributable to difficulty of monitoring remote operations especially in NGOs and in cooperatives [138]. The overall risk level of the MFIs themselves has been evaluated in the context of ratings of MFIs. Ratings of MFIs are done by five microfinance rating agencies and cover a wider range of categories than is common for traditional banks, including outreach, ownership, regulation, governance, clients, and financial products as well as financial information. MFIs obtain a rating to signal to investors and donors their quality and to raise more funds. Yet, rating by only a few selected MFI rating agencies was found to help MFIs raise funds [67]. In addition to credit risk, MFIs also have financial risk (e.g., liquidity risk, market risk – including interest rate risk and foreign exchange risk, investment portfolio risk, and capital adequacy risk) and the risk of overall failure. These risks have been evaluated in relation to MFI transparency and governance. Better transparency and governance reduce financial risks in nonprofit MFIs; transparency seems to be associated with increased credit risk, while the presence of insiders and international directors is associated with increased failure risk [57].

Competition While microfinance has operated as a mechanism to alleviate poverty, it has received substantial support from donors, social investment funds, NGOs, and subsidies. Over time, the increase in opportunities to turn profits has gradually moved microfinance into the provision of financial services to the poor on a commercial basis. This commercialization has brought with it competition following the entry of for-profit MFIs and the transformation of many NGO-backed MFIs into forprofit MFIs. As such, competition has gradually become an important facet of the microfinance industry [112]. A variety of studies have begun to investigate the performance of MFIs based on competition as well as their profit status. The existing body of empirical studies that examine the impact of competition on interest rates has reached inconclusive and counterintuitive results. Baquero et al. [15] examine the impact of competition on interest rates and portfolio quality for both nonprofit and for-profit MFIs using the Herfindahl-Hirschman Index. The authors find that in less concentrated markets, for-profit MFIs charge lower interest rates and have better portfolio quality and

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that nonprofit MFIs are insensitive to changes in market concentration. Mersland and Strøm [99] construct a measure of competition based on the challenge of competition as perceived by the MFIs and conclude that interest rates respond positively to higher levels of competition. Depending on the profit status of the MFI and using the number of MFIs active in each country in 2009 as a measure of competition, [117] finds that competition among nonprofit MFIs reduces interest rates, while competition among for-profit MFIs increases interest rates. See [12] for a theoretical discussion of interest rates and over-indebtedness for MFI clients. Al-Azzam and Parmeter [6] study 1997 MFIs between the years 2003 and 2016 using three different measures of competition (a Lerner Index, a Herfindahl Index, and a geographical indicator) to evaluate the impact of competition on the interest rate(s) charged by MFIs. Their results for the geographic indicator and the Lerner Index display an accordant impact of competition on interest rates: increased competition reduces interest rates charged by both for-profit and nonprofit MFIs. This reduction in interest rates implies a greater concern for cost-efficiency as longterm viability of MFIs is impacted if returns decrease (as measured through interest rates) and inefficiencies remain. As [118, pg. 1] note, “. . . there is widespread agreement, within the industry at least, that in most situations MFIs ought to pursue financial sustainability by being as efficient as they can and by charging interest rates and fees high enough to cover the costs of their lending and other services.” This ties in with the work of Ghosh and Van Tassel [58] who, assuming that MFIs vary in their operations, find that competition over external funds can lead to higher aggregate poverty reduction. This arises since the payment of higher returns for external funds forces higher interest rates; it also redirects funds from inefficient MFIs to efficient MFIs. See also [47].

Future Directions The business model for an MFI is quite challenging by definition. If success were possible with standard approaches to banking, there would be no need for microfinance to begin with. As with any industry, there is evolution. As firms evolve, their performance is an important issue to focus on. The microfinance industry has certainly evolved, migrating from a heavily subsidized industry which offered microcredit to one with many banks being self-sufficient, offering loans and taking deposits.10 As the dual profitability-social outreach mission comes into conflict with one another, efficiency of MFIs is paramount to understand. For example, a new frontier in the study of MFI performance is the impact of digital payments and mobile money, both of which have the potential to augment the standard method of business

10 This

is not to say that subsidization does not exist however; [40] find that subsidies are still pervasive in the industry, representing on average 13 cents per dollar lent across all MFIs.

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for MFIs, allowing greater reach to the poorest customers. This area has yet to be rigorously studied. Another important aspect of the business model of MFIs is the impact that women have. A large majority of MFIs have clientele, be it through savings or loans, composed primarily of women. The poverty alleviation mission of many MFIs is consistent with specifically targeting women because more women than men are poor and because women are poorer than men [137]. While research exists on the impacts of microfinance on a variety of aspects of women’s lives (empowerment, health status, education of children), as well as on how adeptly MFIs themselves are able to meet their missions to serve women [98], little is known about the cost consequences of serving more women versus men. An important question to ask is how costly it is to serve women versus men? Would MFIs elect to serve more women because women have better repayment rates or stronger social networks? If this is true, how might it be reflected in MFIs’ cost structure? These are insightful but unanswered questions. A crucial issue to study moving forward is understanding how the returns to the investments undertaken by women are, in general, much lower than that to the investments undertaken by men, yet millions of poor women successfully repay high interest rate loans (for one study of this, see [50]). Properly embracing the panel nature of the available MFI data represents another important direction for empirical research. This entails breaking down efficiency into two distinct components, one a persistent effect that is constant over time and another that is time-varying. These two disparate components can shed light onto various performance aspects of MFIs and are important from a policy perspective as each yields different implications. Colombi et al. [35] refer to time-varying inefficiency as short-run inefficiency and mention that it can arise due to failure in allocating resources properly in the short run. A variety of methods exist to estimate models of this nature; see [90] for a simple OLS application, [35] for maximum likelihood, and [53] for a simpler simulated maximum likelihood approach. These new methods should be of interest to applied researchers studying the performance of MFIs. The implications from economies from (geographic) market and product diversification are yet to be identified. Consider rural-urban market diversification. Where MFIs continue to serve more clients in urban than in rural markets [68], we can ask if specialization is less costly or if there are cost benefits from serving both markets. Lending only in rural areas may have higher transaction, screening, and monitoring costs associated with lower population density, underdeveloped infrastructure, and limited entrepreneurial opportunities [29, 43, 56]. Dependence on agriculture leads to mismatch between borrower repayment capacity and the frequent repayments structure of a typical microfinance contract [11]. Yet, diversifying by serving both rural and urban markets could capture important remittances flows and even lower costs if seasonality of loans and savings in rural markets is compensated for. Previous work on scope economies from savings and lending finds economies associated with sharing fixed costs (i.e., infrastructure), likely very important in low-density rural markets. Hartarska et al. [72] find no cost complementarities associated with learning from savers that can be useful to lending and vice

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versa. If future work finds diseconomies of market diversification, when costs are lower for MFIs specializing in rural-only or urban-only markets, this would have important policy implications for donors, investors, and the larger microfinance community. A related issue is the question about (dis)economies from product market diversification. This question can be addressed only now as the MIX Market dataset has started to collect such information. The market and product cost economies can be addressed separately or together. The nature of the lending technologies effective in urban and rural areas may be sufficiently different so that the overall costs of an MFI operating in both rural and urban markets do not decrease. For example, to address adverse selection and moral hazard issues in rural markets, MFIs have successfully used joint liability – group loans and village banks [4, 13, 120]. In both rural and urban markets, MFIs also use dynamic incentives – individual contracts with progressively increased loan size [83, 89]. Thus, with the improved scope and scale economies’ methods and specifications described in this chapter, research can identify the cost implications from product and market diversification. In other related financial industries such as banking and insurance, benefits from specialization in one output over joint production of outputs are found and attributed to the external environment in which MFIs operate which is also an important issue in the measurement of productivity as this chapter shows [22, 81, 82]. We hope this chapter serves to illustrate the large literature that exists exploring the performance of microfinance as well as to demonstrate that more work can and should be done moving forward. Microfinance is a key avenue to mitigate poverty worldwide, and the success of this industry is paramount for bringing individuals and families out of poverty and expanding their ability to engage with the greater economy. One unstudied area on MFI performance is adverse incorporation. Howson [78] highlights the impact that MFIs can have on adverse incorporation, emphasizing the causal interaction of unequal power relations and the terms of access to state, market, community, and household resources in perpetuating rural poverty. Howson’s [78] study does not investigate performance of MFIs, only their potential impact and place in adverse incorporation. More work in this area may help to further pin down various outreach impacts of MFIs and overall financial performance. Finally, the ideas in [84] represent another dimension which to explore MFI performance. They note that the majority of empirical work studying MFIs cannot assess general equilibrium effects. While their research does not focus on MFI performance, building in the dual outreach/sustainability platform into their work would be a useful extension.

Cross-References  Aggregation of Efficiency and Productivity: From Firm to Sector and Higher

Levels

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 Cost Assessment of (Un)bundling: Separation of Vertically Integrated Public

Utilities  Modeling Technical Change: Theory and Practice  Performance of Microfinance Institutions: A Review  Stochastic Frontier Analysis: Foundations and Advances I  Stochastic Frontier Analysis: Foundations and Advances II

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The Economics of Production in Marine Fisheries

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Dale Squires and John Walden

Contents Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Vessel-Level Production . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Capital . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Labor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Management or Skipper Skill . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Nonrivalrous Inputs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Resource Stock . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Dual Representations of Technology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Product Transformation and Substitution Possibilities . . . . . . . . . . . . . . . . . . . . . . . . . . . . Structure of Multiproduct Costs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Multiproduct Joint Production . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Separability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Distance Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Technical Efficiency and Stochastic Production Frontiers . . . . . . . . . . . . . . . . . . . . . . . . . Rationing and Quotas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Le Chatelier Principle, Quotas, and Product Transformation Possibilities . . . . . . . . . . . . . . Fishing Time . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Technological Change . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Productivity Growth . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Bioeconomic Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Effort as an Input . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Concluding Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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D. Squires () NMFS, Southwest Fisheries Science Center, La Jolla, CA, USA e-mail: [email protected] J. Walden NMFS, Northeast Fisheries Science Center, Woods Hole, MA, USA e-mail: [email protected] © Springer Nature Singapore Pte Ltd. 2022 S. C. Ray et al. (eds.), Handbook of Production Economics, https://doi.org/10.1007/978-981-10-3455-8_34

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Abstract

Production economics is important to the economic analysis and public regulation of fishing industries in order to address the market failure associated with a common renewable resource stock. Application of production economics arose out of bioeconomic analysis of the aggregate fishery production framework. Production economics gained in importance as it, along with econometrics and mathematical programming, developed as fields. Coupled with an industrial organization orientation and public regulation focus, production economics contributed to analyses of fishing industries, and addressed the underlying market failure. Compared to bioeconomics, this reorientation shifted the focus to shorter time periods and to the individual firm – usually the vessel – and to multiproduct, multi-input production. Production economics now contributes to further development of the bioeconomic model, and addresses additional sources of market failure in fishing industries arising from pure and impure public goods associated with new technology and biodiversity, as well as ecosystem services impacted by fisheries. The chapter reviews the historical developments of production economics applied to fisheries, and concludes with potential future directions forward. Keywords

Economics · Production · Marine fisheries · Common resources · Industrial organization · Productivity · Efficiency

Introduction Beginning in the 1950s, the problem of the “commons,” particularly in relation to fish resources, started to generate interest among economists. Economists sought to answer questions centered on resource depletion, dissipation of economic rents due to absent or ill-structured property rights, the optimal number of fishing vessels, and appropriate harvest rates. Subsequent empirical work showed the accumulation of physical capital starting to occur in fisheries [21, 186]. These studies were among the first that used data collected from a specific fishery to show a result that policymakers could use to restrict capital in a fishery. The introduction of economic tools and resulting policy advice was timely because the biological models used by regulators at the time were insufficient to provide coherent policy advice in the presence of increasing capital, and perhaps more importantly, technological change. Into this void, economic models and thought, which had slowly been developing, were well positioned to provide policy advice to fishery managers. At the heart of this guidance were models grounded in modern production economics. The introduction of production economics into questions surrounding fishery management began with seminal work by Gordon [90], Scott [197], and Smith [202]. Gerhardsen [88] also wrote another very early economics paper. These papers

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established the theoretical basis for rent dissipation and overcapitalization under open access and how the rent-maximizing fishery would entail lower levels of capitalization (or effort). They assumed a fleet of homogeneous fishing vessels with the same underlying production technology and cost functions, and a fish stock assumed to be under a stationary condition. The surplus production framework and aggregate production function from the biologist Schaefer [193] were typically adopted. The total number of vessels that could fish was usually viewed as the control variable in the system, and the objective was to understand the relationship between the number of vessels, harvest levels, and remaining stock size, within an objective of maximizing economic rent. Overall, these aggregate models provided policymakers with advice that could be used to limit fishing effort within the constraints of the productivity of the underlying fish stocks. They also improved the biological models of the time, which were void of economic thought. The aggregate production framework was limiting, and by the late 1960s and early 1970s researchers started to specify models based on modern production theory focused on individual vessels as firms. In one of the first studies, Comitini and Huang [46] estimated production functions for a panel of 32 fishing vessels. This was followed by Carlson [39, 40], who used cross-sectional data from the New England trawl fleet along with the tropical tuna purse seine fleet to estimate generalized production functions for the two fleets. To move away from effort as an aggregate input, which had been the dominant approach in the aggregate production framework, Huang and Lee [113] and Anderson [9, 10] developed models that recognized fishery production as a two-stage process, with an intermediate output (effort) being used as an input for the final output (landings). This period firmly established modern production theory as a tool that could be used to model the fishing production process and provide relevant policy advice for managers. Hannesson [99], the first to introduce modern concepts of production economics and empirical analysis, applied separability to fishing effort specified as a composite input. He also introduced (disembodied, exogenous) technological change, (deterministic) frontier functions, technical inefficiency, and functional forms beyond the Cobb-Douglas, notably the homothetic frontier of Zellner and Revankar [261] and the translog. Influential work by Squires [205–211], Kirkley [122], and Kirkley and Strand [126] introduced dual-based methods and other flexible functional forms to examine the underlying multiproduct costs and production technology (especially input and output substitution possibilities) and the nature of joint production for multioutput, multi-input commercial fishing vessels and public regulation of groundfish trawlers in New England. The dual-based approached readily allowed disaggregation of aggregate output and input (i.e., effort) into individual products and inputs, specifying exogenous prices as regressors, and testing for various types of separability and joint production. Bjørndal [24], Dupont [56, 57], and Bjørndal and Gordon [26] further developed the dual approach in fisheries, including the impact of restricted inputs, while examining practical problems in the British Columbia salmon, Scandinavian herring, and other fisheries and further introducing contemporary empirical analysis.

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In short, the empirical models developed during the 1980s and early 1990s ushered into fisheries the era of modern production economics. These models shifted analysis beyond the popular aggregate bioeconomic models of the time to empirical firm-level analysis applying rigorous microeconometric techniques to cross-sectional and panel data on individual fishing vessels, flexible functional forms, duality, concepts of multiproduct costs and production, homotheticity, separability, and aggregation, joint production, technological change, and more advanced econometric estimation techniques that were emerging in production economics and econometrics. These empirical analyses explicitly recognized the multiple-output, multiple-input joint production nature of the fishing firm’s production process, which had not been fully explored at that time, although clearly recognized by Comitini and Huang [46, 47], Carlson [39, 40], Huang and Lee [113], and many of the other early production economists. Applying production concepts to multiproduct revenue- and profit-maximizing firms with endogenous products in the face of public regulation addressing common resource market failure led to a number of extensions of production economics to topics relevant to fishing industries. Squires [206, 207, 209], Segerson and Squires [198, 199], Kirkley and Squires [124], Färe et al. [70], and Lindebo and Vestergaard [145] extended the economics of capacity and capacity utilization to maximization of short-run profits or revenues with endogenous multiple outputs and accounting for a second capital stock (in addition to physical capital), the natural capital stock. Squires [206, 207, 209, 211] and Squires and Kirkley [216] extended the short- and long-run multiproduct cost structure from firms minimizing the costs of exogenous or predetermined single and multiple products to revenue- and profitmaximizing firms with multiple endogenous products. Segerson and Squires [198] further develop the ray measure of multiproduct returns to scale plus two original alternatives. Herrick and Squires [105] and Squires [210, 212] extended total factor productivity and index numbers to account for multiple species of the natural capital stock using consistency of multistage aggregation and superlative index numbers. Segerson and Squires [199], Squires [213, 215], Squires and Kirkley [216, 218], Dupont and Gordon [58], Asche et al. [16], Ekerhovd [66], and Hansen and Jensen [103] extended virtual price [161] and virtual quantity theory [160] to rations and quotas and the related shadow price approach to individual nontransferable and transferable quotas on catch and effort. Asche [13] developed a dynamic revenue function with adjustment costs. Dupont [57] developed the relationship between Diewert’s [54] elasticity of intensity and the conventional price elasticity of input demand. Squires [215] extended the elasticity of intensity to allow for adding or removing quantity controls (using virtual prices and quantities). By the early 2000s, input and output distance functions were applied to commercial fishing fleets at the vessel level [76, 124, 125, 247, 257]. Distance functions were used in response to concerns about capacity and excess capacity in commercial fisheries [60, 130, 131, 172, 250]. (See Squires and Segerson [220] for a review of capacity and capacity utilization including fisheries.) Kirkley and Squires [124] and Kirkley et al. [127] first applied data envelopment analysis (DEA) with output distance functions and the stochastic production frontier (SPF) to analyze technical

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efficiency and fishing capacity. Distance functions have subsequently been used to answer questions about discards, product transformation possibilities, and to value fishing vessel capital [73, 75, 173, 175, 194]. Distance functions are far less restrictive than a production function based upon a specific functional form. The balance of this chapter discusses vessel-level production models and a deeper discussion of capital, labor, management, or skipper skill, and effort as inputs in the vessel production function. Next, dual-based methods using revenue, profit, and cost functions are reviewed. Distance function and directional distance function models follow. We then discuss multiperiod production models and dynamics, productivity measurement, and technical change, along with examples of economic growth models applied to fisheries. Using the theory of production concepts developed in the chapter, we conclude with a discussion of the aggregate production framework typically found in bioeconomic models.

Vessel-Level Production Comitini and Huang [46], in one of the first examples of vessel-level production, specified and estimated Cobb-Douglas and constant elasticity of substitution (CES) production functions. Results showed differences in vessel productivity, which they attributed to the managerial ability of the captains (“skipper” effect). Carlson [39] specified production functions for New England trawl and the tropical tuna purse seine vessels. The study was the first to extend capital input beyond a single characteristic, such as vessel horsepower. Instead, capital input was defined as gross tons, horsepower, hull construction (i.e., steel vs. wood), and vessel age. Labor was still defined as crew size, and the study recognized the “skipper” effect on vessel productivity. Productivity of the fishing grounds was accounted for by home port dummy variables. Finally, although data were not available to test for technological change, it was the first to recognize technological change as a component of the production function. Other early empirical production function studies included Comitini and Huang [47], MacSween [148], Liao [144], Buchanan [32], Comitini [45], Hussen and Sutinen [116], Taylor and Prochaska [234], Strand et al. [232], Holt [110], Hannesson [99], Kirkley [122], Staniford [230], Greenberg and Herrmann [95], Agnello and Anderson [4], Bjørndal [25], and Campbell and Lindner [36]. These early studies, consistent with the applied economics of the time, did not consider the potential endogeneity of inputs (or simply appealed to the maximizing of expected profit discussed by Zellner et al. [262]), areas fished, and ports, or fully employ panel data techniques (one-, two, or three-way fixed and random effects, mixed effects). They largely preceded flexible functional forms for the production function, generalized approaches to addressing error terms for heteroscedasticity and serial correlation, or quasi-experimental methods (such as difference in differences) for causal inference on policy or other issues. Other influential production function studies include Grafton et al. [94], who evaluated the impact of individual transferable quotas in the earliest “modern”

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microeconomics approach of evaluating natural experiments. Wolff et al. [260] utilized three-way fixed effects to distinguish vessel and skipper effects. Horrace and Schnier [112] specified fixed effects for area, and Natividad [159] utilized difference in differences to evaluate the causal effect of individual transferable quotas (ITQs) on catch and productivity of the Peruvian anchovy fishery. Zhang and Smith [263] specify a two-stage estimation method to address the latent stock problem with errors in the production function and stock dynamics. Fousekis and Kolonaris [82] and Weninger and Strand [258] estimated the first multiproduct production (distance and ray) functions.

Capital Nøstbakken et al. [165] lists natural, physical, human, and immaterial capital as distinct categories of capital in a fisheries framework. Natural capital refers to the fish biomass St . Except in the case of a sole owner fishery, natural capital (St ) is never under the control of an individual fishing vessel. Human capital refers to the labor input, and has been studied by some in the context of skipper skill as discussed below. Immaterial capital refers to assets such as fishing rights. In this section, we will focus on physical capital Kt , which is the vessel and equipment that is needed to conduct fishing operations. Aggregate industry Kt in a fishery is the sum of all the capital on individual vessels participating in the fishery. The perpetual inventory method provides a more comprehensive measure of Kt than simply vessel counts by inventorying entering and exiting vessels, differences in productivity between newer and older vessels, and depreciation [150]. This method, however, requires detailed data collection that is often not collected regularly. The value of capital (Kt ) can also be estimated through the use of insurance surveys, surveys of secondhand prices, book values, or prices of newly constructed vessels [183]. Kirkley and Squires [123] specified a hedonic model to estimate vessel value for a fleet of vessels operating off the eastern US coast. Färe et al. [75] estimated capital values for a group of fishing vessels in the eastern USA based on secondhand sales advertisements from commercial vessel brokers and an input distance function. Mean values for the capital inputs based on the distance function model were used to construct a Lowe capital quantity index. The vessel stock Kt needs to be turned into a flow of capital services. The dual approach is through the user cost of capital or the Christenson-Jorgenson capital services price [42]. Squires [206–208, 210–212], in the first estimates, assumed an opportunity cost of Kt equal to the rate of return on a BAA rated bond, which is considered a “risky” bond, and an economic depreciation rate of 7%. Fisheries studies typically assume there is no “unanticipated revaluation” of physical capital (capital gains or losses), but could apply to an asset like an ITQ share. Aggregate capital services costs aggregate over individual asset types and ages. Primal measures of capital services are through multiplying Kt by a measure of time, such as days at sea or days fishing.

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Labor The crew (i.e., labor) is typically rewarded using a “lay system” rather than an hourly wage rate. These arrangements have existed for well over 100 years. Crew is compensated with a percentage of the revenue from a fishing trip either before or after trip costs are deducted. There are a wide variety of lay systems in place, and who pays for the trip costs is an important part of each system, as well as the split of revenue. When crew are paid through shared remuneration systems, the sharing within crew members is often not homogeneous, highlighting differences in marginal productivity and payment of labor quasi rents. Shared remuneration systems vary across the globe [96], including four remuneration systems commonly used in fisheries: (1) fixed remuneration systems, (2) shared remuneration system: proportional to catch or revenues, (3) shared remuneration system: proportional to revenues minus operational costs, and (4) shared remuneration system: proportional to profits. Sometimes fishermen’s remuneration includes two or more features of these classifications. The lay system, similar to crop sharing, is usually explained as a means of sharing risk [231], and is widely used in fisheries worldwide [153]. Vestergaard [244] recognized risk sharing in lay systems, but also characterized these arrangements as a principal-agent issue. This is particularly evident if supervision of workers’ effort is unobservable, costly, or ineffective [231]. Vessel owners often do not participate in fishing, but instead hire a captain and crew to fish their vessel for extended periods of time. The share system allows the owner to both share risk and lower their monitoring cost. McConnell and Price [153] suggest that moral hazard and team agency can explain the share system. Moral hazard exists because individual effort is usually unobserved by the vessel owner, which leaves stochastic harvest as the only output of crew effort. Team agency conflicts come about because individual fishing crew independently allocate effort which is both costly and unobservable [153]. The role of the lay system in commercial fisheries, and how it impacts vessel production, has not been as extensively studied as other topics in fisheries production. Early studies include Sutinen [233], Plourde and Smith [182], Craig and Knoeber [48], and Matthiasson [151]. A lay system changes some of the behavioral assumptions which underlie the usual neoclassical production model. Since the share system allows the crew to share in revenue earned on a fishing trip, they may be earning more than their opportunity cost, meaning they are earning an economic surplus (i.e., economic rent) from their participation in the fishery [96]. Revenue maximization may then be a better assumption than profit maximizing, or cost-minimizing behavior when modeling fishing vessel behavior [13]. Moreover, incentives for investment in an individual transferable quota, or ITQ system, may become distorted, so that the presence of a share system could lead to either overinvestment or underinvestment in vessels depending on the share which goes to the boat owner [100]. McConnell and Price [153] addressed whether the lay system distorts empirical fishery production model results. They cautioned that the presence of a share system may undermine econometric results (e.g., create biased and inconsistent parameter

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estimates) from random utility models or dual-based production models, in which input demand for labor cannot be derived. This model limitation is overcome when the analysis is an economic one that uses the opportunity cost of labor as a shadow wage rate and shadow prices for other inputs when appropriate and capital services prices rather than a private or financial analysis that uses observed prices [238–240].

Management or Skipper Skill The firm’s management in fishing industries is known as the skipper effect or the “good captain hypothesis.” The firm’s management can also be viewed as part of the larger issue of unobserved heterogeneity between firms and unobserved inputs in general, which extends beyond the individual firm’s management to include multiple unobserved factors that influence production [157]. When management is specified as a residual, it includes the effects of factors that do not depend on management, but rather on the firm’s particular environmental conditions. Vessel management and this residual have been addressed in several different ways. Authors looked for skipper effects by examining the size of the residual variance remaining in the analysis of vessels’ catch rates after accounting for vessel characteristics and other inputs [2, 23, 87, 106, 168, 169, 235]. Researchers have specified proxy variables for the age, education, and experience of managers or principal components analysis of personal characteristics of managers to derive a proxy variable for management. Comitini and Huang [46] and Campbell [35] employed dummy variables based upon one or more knowledgeable experts’ subjective evaluations of ordinal levels of skipper skill. Holt [110] specified a cardinal measure of fishing skill, based on the proportion of successful pursuits adjusted for vessel characteristics and days of effort, designed to distinguish between professional and nonprofessional vessels. Del Valle et al. [53] used the ratio between the number of small landings and total landings as a measure of skipper skill. As with all proxy variables, measurement error and bias can follow, although the asymptotic bias expected from inclusion is generally smaller than from exclusion. Endogenous regressors with biased and inconsistent parameter estimates are also a danger. Kirkley et al. [128] specified managerial efficiency or skipper skill as technical efficiency measured through a stochastic production frontier. Extending the technical efficiency approach, Kirkley et al. [128], Sharma and Leung [200], Vishwanathan et al. [246], and others included an additional equation explaining the technical inefficiency term. Such studies were often unsuccessful or inconsistent when attempting to explain technical inefficiency identified as skipper skill by variables such as years of education or experience. Squires and Kirkley [219] applied the panel data approach of fixed and random effects for the combined effect of the vessel and skipper to distinguish productive performance between vessels. Wolff et al. [260] specified a three-way fixed effects model, distinguishing between the vessel and skipper fixed effect (along with time

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effects). They further allowed for potential time-varying firm management, whereby the skipper can learn through acquiring additional experience with the production process over time and through length of job tenure with a vessel or firm. Tingley et al. [238] utilized data envelopment analysis, along with the stochastic production frontier, to evaluate skipper skill.

Nonrivalrous Inputs A vessel’s production function depends not only on its rivalrous and excludable (i.e., private) inputs but also its nonrivalrous and (partially) nonexcludable inputs. The nonrival and (partially) excludable public good knowledge or ideas – new and accumulated technology – can be embodied in both physical and human capital of the fishing firm [221, 224]. Knowledge has an accompanying externality and social learning external to individual producers [12, 187]. A firm’s production function then depends on the level of knowledge in the economy or fisheries sector. One firm’s innovation, adopted by other firms, enhances all firms’ productivity and innovation. Nonconvexities arise with knowledge embodied in accumulated and new technology. Once the high fixed cost of creating new and better knowledge is incurred, the public good knowledge can be repeated at little or no additional cost [12, 187]. This generates increasing returns to scale external to the individual producer over all inputs – both rivalrous and excludable (private) and nonrivalrous and (partially) nonexcludable (public) – in the production function. New sources of knowledge – new technology – can either arise exogenous to the sector, such as with information and communications technology embodied in electronic and other equipment to find fish, or can arise endogenously through research and development, such as biased technological change to reduce bycatch by reducing the bycatch-target catch ratio. In either case, this new technology becomes endogenous within the fisheries sector due to the producer’s investment decisions required to implement it and the knowledge externality accompanying it. Knowledge can be accumulated indefinitely without diminishing returns to physical and human capital, leading to ongoing, endogenous economic growth in effort and pressures upon the natural resource stock. Knowledge embodied in Kt intensifies this process.

Resource Stock The resource stock S, measured in biomass or numbers of fish, is not under the control of the individual firm, and hence is an exogenous technological constraint in a positive firm-level analysis rather than normative industry-level analysis [126, 206, 207, 209, 212]. In the fisheries stock-flow production technology, ∂π (W, P; K, S)/∂S ≥ 0, i.e., an increase in S does not reduce restricted profits π (W, P; K, S) and ∂π (W, P; K, S)/∂S = WS , i.e., the firm’s shadow value for S.

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A similar interpretation can be given to environmental parameters, such as sea surface temperature, wind speed, etc. Similar considerations hold for the full static equilibrium profit function and the full and partial static equilibrium cost and revenue functions. S is treated like other quasi-fixed or fixed stocks such as K, except in a dynamic approach in which K (with adjustment costs) is treated differently than S. Several specifications are possible for S [8, 15]. In cross-sectional studies, S is treated as fixed, common to all vessels and equally distributed spatially, and not explicitly specified. In time series, S times a series of cross-sections (pseudo-panel data) or panel data has been treated with indices of abundance or actual measures from population assessments [24, 56, 99]. In multispecies fisheries, aggregate biomass is typically specified, if it is specified at all, due to the multicollinearity that would otherwise arise. Dummy variables or time fixed effects are often specified, especially in short panels [26, 198, 199, 206, 207, 211–242]. Time dummy variables can capture not just S but also changes in disembodied technology, state of the environment, regulations, and other factors that change over time. Squires [212] and Pascoe et al. [175] used revenue shares to aggregate individual species measures of abundance into a composite index of abundance. When population assessments are unavailable, indices serving as proxy variables (with attendant issues of endogeneity and measurement error) are sometimes specified [8, 15]. Comitini and Huang [46] used catch per skate (a flatfish). Kirkley et al. [127] used a vessel’s last tow’s trips using a scallop dredge. Eggert [63] used overall average landings value and Pascoe and Coglan [171] specified average catch value per hour fished. Kirkley et al. [130, 131] used lagged average fleet-wide landings per unit effort. Similarly, Pascoe et al. [175] used average fleet-wide catch per unit effort during the season’s first week. Andersen [8] showed that production estimates (here from DEA) differ whether a stock index is specified for each primary species based on catch per unit effort, one stock index is obtained from independent stock assessments for each of the primary species, or inclusion of one composite stock index for each observation is based on the independent stock measures and relative importance of the primary species. If such catch indices are not properly specified (e.g., lagged and/or fleet wide), endogeneity and identification and proxy variable issues can arise along with other limitations discussed by Andersen [8]. All approaches implicitly assume that S is constant (not appreciably depleted) over the specified time period. The stock elasticity, ∂ ln Yt /∂ ln St , measures the impact of changes in St upon Yt in a production function. Conventional wisdom holds that the stock elasticity is close to zero for pelagic stocks due to their schooling behavior [99, 240] and closer to one for demersal stocks due to their more even spatial distribution [193]. Limited empirical studies find cod and saithe’s stock elasticity lies between zero and one, pelagic herring and albacore at unity, and anchovy at 0.39. Gordon and Hannesson [91] find that the presence and size of the stock effect depend upon the time period and overall state of technology.

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Dual Representations of Technology Squires [206–211], Kirkley and Strand [126], Dupont [56, 57], Bjørndal [24], and Bjørndal and Gordon [26] introduced the dual approach to econometrically analyze the vessel-level technology, using cost, revenue, and profit functions rather than the primal approach of the production function. The dual approach was accompanied by the introduction of flexible functional forms, such as the translog, normalized quadratic, and generalized Leontief, which allow less restrictive input and output substitution possibilities and biased technological change. The disaggregated dual approach opened up the possibility of many types of analyses consistent with other areas in industrial organization and production economics that were under development at the time. The dual approach greatly facilitated examining the multiproduct firm by allowing disaggregated outputs and inputs and exogenous prices as regressors. Specifications were generally partial static equilibrium or short run, conditional upon Kt . The dual approach has been used to estimate optimal vessel size [26, 207, 210, 211] and optimal engine power and headrope length [175]. It also allowed researchers to examine product supply and product transformation possibilities [122, 126, 205–207, 209–211], input demand and input substitution possibilities [56, 206, 207, 209–211], the specification and testing of various types of joint production, output and input-output separability, quantity controls (including ITQs) through virtual prices and quantities and quota shadow prices (discussed below), capacity and capacity utilization, and the multiproduct cost structure for revenue and profit-maximizing firms [208, 210, 211, 216].

Product Transformation and Substitution Possibilities The dual approach provides short-run Hicksian (net, compensated) and longrun Marshallian (gross, uncompensated) output supply and derived input demand price elasticities. Due to local Le Chatelier effects from the expansion effect, long-run or Marshallian elasticities are more elastic than short-run or Hicksian elasticities. Measures can also be obtained from directional distance functions equal to the difference between strong and weak output disposal with efficient production frontiers [194]. Most dual-based models show that own- and cross-price elasticities of output supply and variable input derived demand are typically inelastic across gear types, and cross-price elasticities indicate a mixture of substitutes and complements and inelasticity for both inputs and outputs. Some elastic responses have been found in the long run using Morishima elasticities. Output transformation possibilities reflect “selectivity” and “targeting” ability [37, 38, 194]. Complementarity and substitutability and the degree of elasticity reflect vessels’ ability to change product or input mix as they change when, where, and how they fish, and ex ante they indicate how vessels might respond to

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changes in input or output controls and whether or not vessels might discard quota overages. Elasticities and product transformation and input substitution possibilities change with changes in S and environment, and change when, where, and how fishing occurs. Morishima elasticities of substitution are occasionally used [13, 142, 173, 174]. Differences in output disposability indicate limited output substitution possibilities [194]. Substitution between unrestricted inputs (outputs) and restricted inputs (outputs) can be evaluated by the elasticity of intensity [54]: ∂ ln Xi (W, P; K)/∂lnKk , where k denotes a type of capital (or output) and a negative (positive) elasticity shows a substitute (complementary) relationship. Studies examining the relationship between currently restricted inputs (such as a limited allowable fishing days) and unrestricted inputs using the elasticity of intensity include Dupont [57] and Deacon et al. [51] – who along with Dupont [59] pay particular attention to the dissipation of rents, Pascoe et al. [175], Hansen and Jensen [103], and Squires [215] who evaluated the impact upon the elasticity of intensity from adding or dropping quantity controls under the virtual quantity framework. Dupont [57] shows inelasticity and complementarity between the restricted and unrestricted input (which limits rent dissipation that would otherwise occur with input substitution), while Hansen and Jensen [103] show slightly elastic substitution between restricted days and fuel and no relationship between restricted days and vessel (capital).

Structure of Multiproduct Costs The firm’s (vessel’s) multiproduct cost structure is central to analyses of multiproduct industry structure and the impact of public regulation [20]. The multiproduct cost structure was developed under the behavioral hypothesis of cost minimization of a given, exogenous output vector. However, because multiproduct fishing vessels’ products are endogenous, Squires [207, 211] and Squires and Kirkley [216] retrieved the multiproduct cost structure from the information contained in the revenue and profit functions under the behavioral assumptions of revenue or profit maximization with endogenous outputs, where the costs are shadow costs with the revenue function. Economies of scope measure the cost savings from producing multiple outputs rather than producing each separately when production is (almost) joint in inputs [20]. Scope economies derive from weak cost complementarities or fixed costs that do not depend on the quantities of outputs produced but do vary on which outputs are chosen (since that affects local cost complementarities or anticomplementarities) [20, 93]. Squires [207, 210, 211] and Squires and Kirkley [216] extend the measurement of scope economies, transray convexity, incremental and average incremental costs, and weak cost complementarities to revenue- and profit-maximizing firms with endogenous outputs using the revenue and profit functions. Empirical results find both economies and diseconomies of scope in fishing vessels, either by directly

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estimating from the cost, revenue, or (restricted) profit function or from cost complementarities [7, 15, 119, 207, 208, 210, 211, 216, 256, 258]. Economies of scope are largely found in trawl vessels (as expected) and diseconomies of scope are found in the surf clam and ocean quahog fleet (reflecting the spatial stock separation). Product-specific returns to scale Si [Y] measure the change in costs through variation in the quantity of one product while holding other products’ quantities constant [20]. Firms with increasing product-specific returns to scale have a cost incentive to expand the scale of production of this product and may become specialized in its production. Squires and Kirkley [216] develop and show how to measure Si [Y] and incremental and average incremental costs for the revenue or profit-maximizing firm. A sufficient condition for Si [Y] can be obtained by examining incremental marginal shadow costs or costs found from the diagonal elements of the Hessian submatrix for outputs from the estimated parameters of the profit or revenue function [207, 210, 211, 216]. Empirical results indicate that different pelagic and demersal species are produced under conditions of both increasing and decreasing product-specific economies of scale [7, 15, 55, 119, 199, 207–211, 216, 256]. Some species, such as those long lived and slow growing, which are subject to increasing product-specific returns, can be vulnerable to overharvesting due to the decreasing marginal production costs. Multiproduct economies of scale are typically measured along a ray in output space that keeps outputs in fixed proportions, although other measures exist [20, 198]. The revenues exceed, are less than, or equal to (long-run) costs as there are decreasing, increasing, or locally constant long-run ray returns to scale. Increasing multiproduct returns to scale are found by Hannesson [99], Bjørndal [24], Asche et al. [17], Weninger [256], Bjørndal and Gordon [27], Felthoven and Paul [76], Nesbøkken [164], and Lazkano [142] and decreasing multiproduct ray returns to scale are found by Squires [206–209], Squires and Kirkley [216], Alam et al. [7], Horace and Schnier [112], and Hoff and Frost [107]. Increasing multiproduct ray economies of scale are sometimes found in output-regulated fisheries that prevent vessels from increasing the scale of production. A cost function is subadditive at an output vector Y if and only if it is lower cost to produce Y than to produce the outputs comprising Y individually, i.e., C(Y) ≤ C(Y1 ) + C(Y2 ) [20]. Evans and Heckman [68, p. 615] stated: “Thus an industry is a natural monopoly if a single firm can produce all relevant output vectors more cheaply than two or more firms.” Cost subadditivity would suggest that some form of fishermen’s monopoly is appropriate on private efficiency grounds [167, 210, 211]. Squires [207, 211] develops local sufficient conditions for cost subadditivity using revenue or (restricted) profit functions. Onofri and Francesc [167] devise an additional test for cost subadditivity in the fishery sector. Squires [211], Alam et al. [7], and Onofri and Fransesc [167] reject cost subadditivity in fishing industries. Decreasing multiproduct ray economies of scale would explain the absence of cost subadditivity.

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Multiproduct Joint Production The nature of joint production, including the production possibilities frontier (PPF), impacts model specification, spatial management, rights-based management, and fisheries management in general. Changes in St shift the PPF in or out and twist it in the stock-flow production process. Area fished does not alter the PPF (except in a disaggregated model in which different areas contain different resource stocks). Different areas, aggregations by age and size, etc. are readily accommodated under block joint production as discussed below. Here we only discuss desirable products and the most relevant types of joint production for desirable two species (products) Y1 and Y2 . Joint-in-input quantities production arises when all inputs are used to produce all outputs [98, 140]. The PPF implicitly assumes either a reasonably homogeneous distribution of both species across all fishing grounds or aggregation across such areas. Many of the ex ante analyses discussed above specified and tested this type of PPF. This PPF could also be applied to outputs specified as species area and to completely different resource stocks (and hence areas) as in Holzer and De Piper [111]. If only some of the species are regulated by transferable quotas, substitution from regulated to unregulated species may occur [16, 58, 66, 189]. Squires and Kirkley [218] developed the two-price, two-quantity direct elasticity of transformation for two ITQ-regulated species and this type of jointness. Almost joint-in-input quantities arises when the production process uses not only standard inputs such as fuel and labor but also quasi-public inputs, especially the vessel Kt that cannot be explicitly allocated among nonjoint production processes [141, 146]. Many of the ex ante analyses were short run in Kt and hence almost joint-in-input quantities. Pascoe et al. [173] applied this approach in a multiproduct distance function, and Hutniczak et al. [118] specified a restricted profit function. Hansen and Jensen [103] recognized the issue of almost jointness, but specified a distinct and unique model using days as an input and a multistage production process. Nonjointness-in-input quantities [98], also called output independence [138, 139], arises when: (1) there are separate production processes for each harvested species or areas and (2) inputs are allocated between the different production processes. Each production process can be separately regulated without affecting production of the other processes because there are no technological or cost tradeoffs between the output of one activity and that of another [126, 205, 207]. No empirical study has found nonjointness-in-input quantities in fisheries throughout the species set. Different multiproduct production processes can even be scalar multiples of one another when there is either Leonteif aggregation of inputs and outputs, so that they are in fixed proportions, or a single input and output (homothetic input-output separability and nonjointness-in-input quantities). This is the specification of linear programming, which Reimer et al. [184] utilized for different areas. Each product

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combination is produced in fixed proportions, and hence there are not changes in product-species compositions for an individual production process. Instead, changes in species mix arises through shifting to a different production process (which could pertain to different area or gear type). Block jointness in inputs [33, 34, 134], also known as block output independence [138, 139] and first discussed in fisheries by Squires [204], arises when there are multiple but distinct and separate production process that are joint among a range of production of products but nonjoint between these processes. Squires and Kirkley [216] found block jointness for Pacific coast groundfish trawlers. Two forms of joint-in-input quantities production that directly addresses area and area-specific species combinations or simple species aggregations for interrelated species or sequential production of different species, species groups, or gear configurations is a block structure and almost jointness [146]. Hence, individual species (or groups of species) could be partitioned into joint blocks (i.e., groups) that capture product-species combinations and or product-species groups and areas (which can be a simultaneous decision, since area often defines species composition and density) or different gear configurations. For example, two separate fishing grounds for species groups Y1 and Y2 , in which there are product transformation possibilities within each block but not between blocks. The block joint, almost joint, and nonjoint-in-input quantities specification could be combined with a multinomial logit/probit model [35] or switching regression, or random utility model [29] for choice of area, depending upon the vessel’s behavioral assumption. Alternatively, products species could be explicitly defined by area species and three-way panel data specification to also allow for endogeneity in both area and species [112]. Area dummy variables as regressors could potentially be biased and inconsistent since they are potentially endogenous due to the choice of fishing ground. Almost joint-in-output quantities [140, 141] arises when there are multiple production functions for each type of variable input with the exception of sharing the fixed inputs. Such jointness can imply the sequential use of different inputs or different fishing strategies as found by Hutniczak et al. [118]. Allocated fixed inputs can create product interdependence that differs from technical interdependence. Hansen and Jensen [103] develop a restricted profit function with allocable days which is also discussed by Reimer et al. [184]. Empirical tests for gill net, trawl, purse seine, longline, and dredge vessels almost always reject nonjointness in inputs or almost nonjointness-in-input quantities [7, 13, 15, 55, 118, 126, 173, 189, 199, 206, 207, 236]. Exceptions include Campbell and Nicholl [38], who find nonjointess for generalist firms and nonjointess for purse seine vessels (specialized firms), Squires and Kirkley [216] who find nonjointness for one product and reject nonjointness for all others in a trawl fishery (giving a block joint production process), Weninger [256] for the ocean quahog and surf clam fishery, and Alam et al. [6] who find both input-output separability and nonjointness in a Malaysian gillnet fishery, which implies that all supply equations are scalar multiplies of one another [98].

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Separability Aggregation of individual outputs, notably individual species, into a single output and inputs, such as capital, fuel, and labor (L) into a composite input effort (E) is the structure of the bioeconomic model. Aggregation occurs through either Leontief-Sono separability, Hicks-Leontief aggregation, or the generalized composite commodity theorem [28, 143]. Input-output separability is the implicit structure of the bioeconomic model, in which there is a single composite input and output, and the marginal rate of transformation between outputs is independent of changes in inputs, and the marginal rate of substitution between inputs is independent of changes in outputs [99, 207]. Only the levels of catch and effort require regulation, and regulation of the input (species) mix does not adversely affect the optimal product (factor) combinations [207]. Input-output separability has largely been empirically rejected [7, 13, 37, 38, 55, 118, 126, 173, 189, 207, 210, 211, 216, 256]. Two notable exceptions are Alam et al. [6] and Squires [206] who did not reject input-output separability.

Distance Functions The distance function starts from a set-theoretic foundation and does not require a specific functional form in contrast to the production function. Distance functions also allow specifying disaggregated outputs and inputs in the primal rather than the dual specification. Distance functions have been used to estimate technical efficiency [127, 128, 258, 180, 181], capacity [60, 124, 125, 126, 129, 145, 247], productivity [76, 166, 252, 255], vessel valuation [75, 130, 131], targeting and bycatch problems [72, 173, 174, 194], vessel buyback programs [250] and optimal fleet size and the basis for vessel buyout programs [132, 250]. Dupont et al. [60] and Herrero et al. [104] included slack variables into DEA models to allow for nonradial changes in input and output mix. Walden and Tomberlin [249] introduced an “order-m” frontier and free disposal hull to estimate fishing capacity [251]. Technical efficiency refers to the individual firm or vessel’s level of production given its bundle of rivalrous inputs, and states of technology, environment, and resource stocks, relative to the best-practice frontier established by the highest achieving firms or vessels. Technical efficiency from an input orientation is TEI (y, x) = 1/DI (y, x) ≤ 1, where DI (y, x) denotes an input-oriented distance function. The TEI value indicates the amount a vessel will have to scale their inputs downward it be technically efficient and operate on the bestpractice production frontier. If an output orientation is desired, TE is given by TEo (x, y) = [Do (x, y)]−1 ≥ 1, where Do (y, x) denotes an output-oriented distance function [85]. A natural extension of measuring TE was the estimation of vessel capacity using the Johansen [121] plant capacity definition. The introduction of data envelopment analysis (DEA) [41] and the stochastic production frontier (SPF) [5] were pivotal in using distance functions to model vessel-level production.

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Although these methods were introduced before 1980, it took some time for fisheries researchers to integrate them in their work. The directional distance function, a generalization of the traditional distance function, can be used to account for unintended outputs, such as bycatch. This modeling of TE measures a vessel’s ability to expand its intended catch and contract its bycatch given their input use. Such models to estimate vessel productivity and TE include Weninger [257], who modeled the efficient production frontier for vessels operating in the mid-Atlantic surf clam and ocean quahog fishery. Scheld and Walden [194] examined TE for multispecies fisheries where the ability to catch one species may be hindered due to regulations regarding catch of other species. The stochastic multiproduct distance function has also been estimated to evaluate output transformation possibilities in fisheries [81, 173, 174, 176, 184].

Technical Efficiency and Stochastic Production Frontiers The stochastic production frontier is the most widely used specification in fisheries to measure technical inefficiency or deviation from the best-practice frontier. The stochastic production frontier relates a vessel’s maximum output given inputs, X1it, X2it , . . . , XNit while allowing for stochastic events. A second, simultaneously estimated equation can explain the technical inefficiency according to exogenous or predetermined variables. Hannesson [99] applied the first production frontier, a deterministic one and to Norwegian cod fisheries, in which the one-sided deviation from the frontier captures both stochastic shocks and technical inefficiency. Kirkley et al. [127, 128] followed with the stochastic frontier, identifying technical efficiency with skipper skill (see the skipper skill section). These early papers specified only the production frontier. Sharma and Leung [200], Vishwanathan et al. [246], and Squires et al. [226, 228] first included the second equation to explain technical inefficiency. Grafton et al. [94] first accounted for economic inefficiency, including both technical and cost inefficiency, in a study of the impact of ITQs in the British Columbia fishery for Pacific halibut. Kompas et al. [133] first related technical inefficiency to input controls, showing that technically efficient fishers substituted unregulated for regulated inputs and that technical efficiency declined with increasing restrictions on production. These and many other studies that followed find different degrees of technical inefficiency, and those relating technical inefficiency to measurable attributes of skippers and crew find a wide range of results but typically do not find a statistically valid relationship. Asche and Roll [14], using a shadow revenue function, estimated revenue inefficiency and its decomposition into technical and allocative inefficiency in the Norwegian groundfish fishery. Horace and Schier [112] introduce time-varying technical inefficiency through nonparametrically identifying time-varying technical efficiency by exploiting the spatial variation of vessels in three-dimensional (cross-sectional, time, and area) panels in Berring Sea flatfish fisheries, where each cross-sectional vessel can move across space and time.

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Recent analyses focused upon the impact of rights-based management upon technical efficiency. New [162] found that structural adjustment reducing fleet size did not impact vessels’ technical efficiency in the Australian Eastern Tuna and Billfish Fishery. Schnier and Felthoven [196] found that a vessel’s measure of technical inefficiency is a significant and positive factor in explaining whether it exits a fishery following ITQs. Huang et al. [114] found that participation in the collective rights-based management system (“sectors”) of the New England groundfish fishery impacted behavioral responses rather than technical efficiency even though sector participation led to shifts in the production frontiers for trawl and gillnet vessels. Estrada et al. [67] examined the impact on technical efficiency of cooperative catch shares for artisanal vessels in anchovy and sardine fisheries in south-central Chile using a difference-in-differences causal inference framework. The cooperative catch shares reduced average technical efficiency, although the impact on heterogeneity depended upon the characteristics of fishermen’s organizations, so that greater cooperation among members increased technical efficiency. Mainardi [149] specified and estimated two stochastic frontier semiparametric models for a panel of Falkland Island fisheries over 2003–2014 that treat unobserved heterogeneity as a finite mixture or discrete approximation to continuous parameter variation, by adjusting for sample selection and latent classes, respectively. The hypothesis of frontier-enhancing effects of the new ITQ/ITE regime is supported for most, albeit not all, fishing companies. Evans et al. [69] estimate a cost frontier with time-varying inefficiency to allow for spatial variation in unobserved productivity effects and measure changes in technical efficiency, capital investment/divestment incentives, and resource rent following ITQs. Other approaches can measure TE. Salvanes and Steen [190] specified a thick frontier, in which the best-practice frontier is determined by grouping together vessels with the smallest estimated disturbances. Holloway et al. [109], Holloway and Tomberlin [108], and Tomberlin and Holloway [239, 240] apply the Bayesian approach to composed error models under alternative, hierarchical characterizations, and demonstrate the Bayesian approach to model comparisons using recent advances in Markov Chain Monte Carlo methods. Collier et al. [44] evaluate a California multiple-input, multiple-output fishery using a hybrid DEA stochastic frontier model in which DEA is used in a first stage to measure aggregate output used in the second stage, the stochastic production frontier. Pascoe et al. [175], estimating a restricted multiproduct profit function, applied the fixed effects approach of Schmidt and Sickles [195].

Rationing and Quotas The microeconomic theory of quotas, rations, and other quantity controls allows better understanding of their impact upon fishing vessels. These quantity controls include nontransferable individual vessel quotas and individual transferable quotas (ITQs) for catch and effort and limits on gear, fishing time, vessel size, and inputs in general.

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The microeconomic theory of rationing and quotas for firms, initially developed in consumer theory [161] and international trade [160], was extended to production theory by Fulginiti and Perrin [86], Squires and Kirkley [216–218], Segerson and Squires [199], Squires [213, 215], Squires et al. [225], Vestergaard [243], and Vestergaard et al. [245]. Using the virtual price framework of Neary and Roberts [161], Squires and Kirkley [216–218], and Squires [215] showed that the unit rent of an individual transferable quota, or ITQ, is the difference between the output price and virtual price of the quota and forms the firm’s inverse derived demand function for the ITQ. Firm inverse demand is horizontally summed to form the market ITQ demand. In equilibrium, the horizontally summed aggregate inverse ITQ demand curve equated to the exogenous aggregate supply curve (typically a total allowable catch or effort) gives the market ITQ price. The Antonelli matrix of changes in endogenous unit rents in response to exogenous marginal quota changes gives this unit rent and forms the basis of ITQ price flexibilities. Squires and Kirkley [218] also calculated the gains from trade, estimated the ITQ product transformation frontier, and developed the direct elasticity of substitution (two pricetwo output) between ITQs. Extending Neary’s [160] ex post framework using virtual quantities – the dual to virtual prices, Squires [214, 215] evaluated the effects of adding, subtracting, or changing existing quotas. The microeconomic theory of rationing and quotas addresses the substitution of unregulated inputs for regulated inputs in input-regulated fisheries, first described by Pearce and Wilen [179] and Wilen [259], by Squires [206, 213–215] and Dupont [57] and to the spillover effects between quota (including ITQ)-regulated species and unregulated species [16, 58, 66, 117, 215].

Le Chatelier Principle, Quotas, and Product Transformation Possibilities The Le Chatelier principle as applied to economics by Samuelson [191] shows that there are behavioral implications of rationality which are only exhibited when extra constraints are imposed or withdrawn. The Le Chatelier principle applies when transitioning from short-run to longer-run production or adding or subtracting or adjusting quantity controls, such as quotas or trip limits, or property rights or other direct regulations that impact production. The local Le Chatelier principle states that if variables in a system are chosen to optimize a function, then as a result of an infinitesimal (i.e., marginal) change to the system, e.g., an extremely small change in prices or quotas, the responsiveness of the chosen variables will be reduced (increased) when extra constraints are added to (dropped from) the optimization problem. The Le Chatelier principle can be local, corresponding to marginal changes, or global, corresponding to nonmarginal changes [155]. The key question is whether the change in production is due to a nonmarginal (discrete) rather than marginal (infinitesimal, local) policy shock. The Le Chatelier principle can explain the failure of ex ante analyses in fisheries to consider the full range of a vessel’s ability to adjust its catch and/or input mix

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when there are nonmarginal (discrete) changes in policy, production constraints, industry structure, markets, biology, environment, or prices. Squires and Kirkley [216–218], Squires et al. [227], and Pascoe et al. [173, 174] ex ante analyzed the introduction ITQs in multispecies fisheries. Appealing to the local Le Chatelier principle, they observed that ITQs may be ineffective due to limited substitution possibilities between species, leading to potentially high discards or difficulties in quota balancing. However, Sanchirico et al. [192], Branch and Hilborn [30], Abbott et al. [1], Reimer et al. [184, 185], and Scheld and Walden [194], through ex post empirical evaluation of multispecies fisheries with ITQs compared to production prior to ITQs, found that vessels can frequently adjust their species mix far easier than the ex ante analyses incorrectly anticipated. A global meta-analysis of 345 stocks showed that for many fisheries, management controls improve under ITQs in terms of reduced variation in catch around quota targets [154], although counterevidence also exists [135]. ITQs can also reduce bycatch [62]. Reimer et al. [184, 185] posit that the estimated model must be “structural” with respect to that shock, as discussed by Haavelmo [97] and Lucas [147] for economics in general and for rights-based management in particular. That is, the ex ante analyses’ assumptions of invariant behavioral equations are inconsistent with dynamic maximizing behavior and changed incentives. Reimer et al. [184, 185] observe that revealed production possibilities are frequently constrained and confounded by regulatory incentives, and that the empirically revealed production set strongly depends on the institutional, economic, and biological setting in place when fishing was observed. The Le Chatelier principle more rigorously explains the failure of ex ante models to anticipate vessels’ responses to introducing rights-based management. When rights are introduced and existing direct regulations are reformed in marginal ways, then the local Le Chatelier principle applies. Rights, however, are often introduced into a deteriorated fishery and replace existing and highly restrictive direct regulation or lead to its substantial modification. This situation constitutes a nonmarginal policy shock and a change in complementary institutions, regulations, and business practices, whether or not the property rights as transferable output controls are simply marginal extensions of the existing regulations. The global Le Chatelier principle framework then fully explains the observed results of greater flexibility and is consistent with economic theory.

Fishing Time A tension can exist between the standard economic approach to production, which specifies variable inputs by some physical capital, notably gear and equipment, and labor, materials, and energy, and the standard biological specification of days as steaming, search, and fishing time. (A few studies examining fisheries regulated by days, in which case days were specified as quasi-fixed. These studies include Dupont [56, 57] and Hansen and Jensen [103]. The use of days to represent variable inputs represents use of a proxy variable as a flow of energy and services from stocks of

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labor, gear, and equipment. Proxy variables can introduce measurement error, in turn leading to biased and inconsistent parameter estimates. Days as a variable input in an econometric model can also be subject to endogeneity and identification issues, in turn also leading to biased and inconsistent parameter estimates. The use of days, depending upon its application, can implicitly assume either Leontief aggregation or homothetic input separability. Another tension exists between the stock of physical capital K and labor L and flow of capital and labor services. When K and L are specified as fixed or quasi-fixed factors, then production is conditional upon these stocks and issues do not arise. When K and L stocks are specified as variable inputs, then the implicit assumption is made that flows are proportional to stocks. When this assumption is invalid, then biased and inconsistent estimates potentially arise due to the measurement error.

Technological Change Technological change is one of the main driving forces behind the historical development of fishing industries. Along with investment in Kt , technological change impacts the status of the resource stocks and the extension of fishing grounds by depth and geographical range, broadening of species harvested, and habitat impact. Population biology addresses technological change through time-varying catchability and changes in selectivity. Technological change can be: exogenous, endogenous, or both, oriented on the process (inputs) through process innovation (and factor augmentation) or outputs through product innovation (and product augmentation); disembodied or embodied; and centered on target species or bycatch species. Technological change can be oriented to the target species or desirable outputs, the conventional approach. Technological change can also be oriented to bycatch or habitat impact or undesirable outputs, which is biased technical change. More generally technological change can be either neutral (typically Hicks neutral) or biased in the outputs or inputs. Technological change can lead to lower costs per unit of effort (input augmenting) or increased catch rates per unit of effort (output augmenting) given St . Technological change can also lead to new species caught through expanding the range of production and introducing new areas and depths to fish with new gear and equipment. Technological change on target species and input usage is largely exogenous to the fisheries sector [221, 222, 224]. Endogeneity arises through any investment in physical capital, by which the technological change is embodied, or for research and development to adopt the external sources of new technology to fisheries. Technological change to reduce bycatch and habitat impact is largely endogenous to the fisheries sector. Endogeneous technological change arises because technological change typically requires research and development. Further endogeneity arises if the new technology is embodied in new physical capital requiring investment. Measurement of technical change in fisheries is typically based on primal specification of technology, notably the production function, or cost diminution given output (dual), over time [46, 130, 131]. Economic analysis of technological change essentially

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treats it as a residual after all other inputs and control variables (technological constraints) have been included in the model [204], including the resource stock [212]. The most common specification of technological change in fisheries production functions is disembodied and accounted for by a linear time trend, which implicitly assumes that technological progress is Hicks neutral, exponential, and progresses at a constant rate [46, 65, 80, 99, 110, 156, 158, 175, 201, 221, 224]. Pascoe et al. [175] find a positive linear time trend and negative squared linear time trend, indicating increasing Hicks-neutral technological change but at a declining rate. Hannesson et al. [102] adopted the approach of Baltagi and Griffin [18], which allows technology to progress at a variable rate for each individual year, to evaluate technical change over 100 years in the Lofoten cod fishery and extended by Kvamsdal [137]. Gordon and Hannesson [91] applied an ARMAX model to the Norwegian winter herring fishery. Banks et al. [19] and Kirkley et al. [130, 131] analyzed embodied technical change in the Sète trawl fishery using dummy variables. Kvamsdal [136] applied a structural time series model with a stochastic trend to measure technological change in a Cobb-Douglas production function with both single equation and multivariate models to the Norwegian Lofoten cod fishery. Fissel and Gilbert [79] specified a compound Poisson process incorporated into the catchability coefficient of the Schaefer production function. Gilbert and Yeo [89] examined technology adoption patterns and productivity differences in a Malaysian artisanal fishery to evaluate whether technology is a substitute or complement for managerial skill, i.e., examining skill-diluting and skill-augmenting technological change.

Productivity Growth Measurement of total factor productivity (TFP) growth in fisheries requires accounting for all the sources of growth to disentangle changes in the TFP residual from changes in St and the environment. This section surveys measuring TFP growth using the growth accounting framework of Solow [204] and economic index numbers as opposed to econometrically estimating technological progress using a production function to then infer productivity growth. Walden et al. [254] give an additional survey of productivity change in fisheries. Bell and Kinoshita [22] measured labor productivity and Kirkley [122], Norton et al. [163], and Davis et al. [50] estimated TFP in a growth accounting framework but without accounting for changes in St . Squires [208–213] and Herrick and Squires [105] specified a growth accounting framework using economic index numbers and recognized the stock-flow nature of the production technology to disentangle  changes in St from inputs changes, while adjusting for St s elasticity of output and variations in capacity utilization under both open access and the economic optimum (sole owner). Squires [212, 213] developed superlative index number approaches to consistently aggregate the resource stocks of individual species into an aggregate S. Jin et al. [120], Arnason [11], Hannesson [101], Torres et al. [241], Eggert

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and Tveterås [64], Pan and Walden [170], and Walden et al. [255] analyzed TFP growth in different fisheries. Brandt [31], Walden et al. [252, 253], Färe et al. [74], Solís et al. [203], and Thunberg et al. [237] evaluated the impact of ITQs upon TFP growth. Felthoven and Paul [76], Squires et al. [229], and Paul et al. [178] accounted for changes in the state of the environment. Squires and Vestergaard [223] developed the impact of TFP growth on optimum resource use within the context of the bioeconomic model. Pascoe et al. [177] further discuss the impact of productivity growth (and TE) on maximum economic yield. Norton et al. [163] developed an index of profits or economic health that incorporated TFP, but not accounting for changes in St . Fox et al. [83] fully decomposed a profitability index, which included price, productivity, and capacity utilization indexes and changes in St , to determine whether changes in productivity or prices have the largest impact upon profitability change. Subsequent profitability index analyses include Dupont et al. [61], Fox et al. [84], and Walden and Kitts [248]. Two broad approaches have been used to construct economic index numbers [254]. The first approach used constructed superlative economic index numbers using prices as indicators of production elasticities in the Lowe, Törnqvist, and Fisher ideal index numbers. Economic index numbers can also be constructed using distance functions and linear programming. These methods construct a production frontier based on observed values of inputs and outputs in different time periods, which also allows decomposing productivity change into change in technical, allocative, and scale efficiency. Some of the different indices that can be constructed include the Malmquist, Hicks-Moorstein, and Lowe.

Bioeconomic Models Effort as an Input Early bioeconomic models specified an aggregate production function coupled with a biological model [90, 197, 202]. These early models, which grew out of biological production models, typically specified a single composite rivalrous input “effort,” E [49, 99, 126, 188, 206, 207]. Effort in biological models usually involved a time component, such as days absent from port, or time the gear was in the water. Huang and Lee [113] and Anderson [9, 10] were among the first economists to question the biologists’ specification of effort. Anderson [9, 10] specified effort as the output of a two-stage production process, although not founded upon theory (Leontief-Sono homothetic separability or Leontief aggregation, exact separability, and aggregate production functions). In contrast, Huang and Lee [113] discussed effort within the context of separability and aggregate inputs and the conditions for an aggregate production function. They specified the aggregate production function in terms of individual inputs K and L rather than composite E. Anderson [9, 10], interested in individual firms’ production functions but retaining E, constructed cost curves based on E. Hannesson [99] provided a theoretically consistent framework of

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separability and aggregation to composite E and defined a more general production framework that integrated E with traditional neoclassical production theory. Multiproduct technologies have a comparable two-stage optimization process in which revenues are optimized in each stage of aggregation and with allocative efficiency to form the composite output Yt [206, 207, 221, 224]. The separability is exact, rather than approximate, since Yt = f (q, Et , St ) is an exact representation of technology. Weak (and therefore also strong) Leontief-Sono separability is not a sufficient condition for the first stage of aggregation. Homothetic separability provides both a necessary and sufficient condition, which requires a linearly homogeneous aggregator function [28, 92, Lemma 3.3a]. Linear homogeneous Et aggregator functions for the rivalrous inputs, Et = g(X1t , Kt ), satisfies Fisher’s factor reversal test, in which the cost of the rivalrous inputs equals the product of g(•) and the corresponding implicit price index [77]. (To simplify notation, let X1t denote a scalar composite of variable rivalrous and excludable variable inputs and let Kt denote the scalar composite of rivalrous and excludable nominal physical capital stock in natural units, i.e., not in efficient units.) Kt is aggregated over different individual units of capital of different vintages and levels of embodied capital according to specific conditions [78]. Linear homogeneity of g(•) in rivalrous inputs also satisfies the replication argument of production functions with rivalrous inputs [187]. Linear homogeneity in g(•) in rivalrous inputs gives the familiar Grahamβ Schaefer production function in which the exponent of Et is linear: Yt = qEt1 St . Leontief aggregation requires: Et = min (AX1t , BKt ), where A and B are fixed coefficients and in which one of X1t , Kt is the limiting factor [3, 113]. ∼

Allowing for embodied technical change in Kt , E t = min (AX1t , Bt Kt ) = Et = min (AX1t , J ) [221, 224]. Either X1t or Jt (Jt =  t Kt ), i.e., Kt in efficiency units, where the average embodied technical efficiency,  t , is defined as the weighted average level of best-practice efficiency associated with each past vintage of investment, will be partially idle in the sense that a small change in one input will not affect output or factor prices. There will be historical partial surplus of either X1t or Jt . The more general Hicks-Leontief composite commodity theorem requires that the ratio of input prices or quantities of individual rivalrous inputs comprising Et to the composite effort price or quantity is independent over time [143]. The standard fisheries bioeconomic model often assumes the more restrictive Leontief aggregation [43]. Specification of days as effort implicitly assumes one of these forms of aggregation (homothetic Leontief-Sono, Leontief, or the Hicks-Leontief composite commodity theorem) and forms a proxy variable. With homothetic Leontief-Sono separability for rivalrous inputs and embodied technological change, E¯¯ t = f (X1t , t Kt ) = f (X1t , Jt ) and Jt =  t Kt are themselves aggregates in full static equilibrium [221, 224]. As a linearly homogeneous function, E¯¯ t can be written as E¯¯ t = t f ( 1t X1t , Kt ) = t E¯¯ t . Assuming a constant rate of embodied technical change ψ and constant capital share of income M2 with Cobb-Douglas functional form for production function f (•) implies that t = eM2 ψt , where the growth rate of  t is ψ t [115]. When the production technology is Cobb-Douglas or its Graham-Schaefer form (exponents of one for Et , St ), and the production technology is an exact

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representation of technology, then separability inflexibility, a restriction on the technology, is required (Proposition 1 of Denny and Fuss [52] and Blackorby et al. [28]). That is, the technology must be a Cobb-Douglas production function f (•) with a translog effort aggregator function g(•) or f (•) must be translog and g(•) CobbDouglas. Hicks-neutral, disembodied technical change independent of the rate of rivalrous Kt formation can be specified as growing at a constant exponential growth rate λ. Nonconvexities arise with knowledge embodied in accumulated and new technology, creating dynamic increasing returns to scale external to the individual vessel over all inputs, both private and (partially) public, in the production function (as discussed above). Endogeneity is created because the dynamic increasing returns to scale are external to the individual production unit. This dynamic positive externality continously lowers unit costs with knowledge adoption, which incentivizes further adoption of new technology, and thus the endogeneity. The aggregate production function can be specified with disembodied and embodied technical change, knowledge spillovers accompanying this technical change, and homothetic exact Leontief-Sono separability [221, 224]. Effort is then an unobserved composite, private (rivalrous) input. Endogenous nonrival knowledge β spillovers, measured by θ > 0, gives aggregate effective effort Ktθ Et 1 , which in turn gives β 1 + θ > 0 and increasing returns to scale external to firm i in  time t. Firm i s output in time t with homothetic exact Leontief-Sono separa β2 1 bility is Yit = (Jt , t) it f it X1it , Kit St . Assume additive seperability across all firms i to give an aggregate technology. Let (Jt , t) = (t)Jtθ = (t)(t Kt )θ = (t)tθ Kitθ . Letting t = eM2 ψt and (t) = qeλt − μ(t, Z) , where q β β denotes the catchability coefficient, gives Yt = qKtθ Et 1 St 2 e(λ+M2 (θ+1)ψ)t−μ(t,Z) . For the Graham-Schaefer specification, β 1 = β 2 = 1. Absence of the external effect, θ = 0, gives only rivalrous (private) inputs, giving Ktθ = 1 and Yt = β β qEt 1 St 2 e(λ+M2 ψ)t−μ(t,Z) .  Firm i s output in time t with Leontief aggregation builds off of Clark [43] specifying rivalrous Et as the rivalrous stock of physical capital Kt formed under Leontief aggregation and Kt as the limiting factor [221, 224]. This allows for explicit, intentional, endogenous net investment in Kt , and β 1 = θ + 1, θ > 0, is the positive knowledge spillover, 1 is the rivalrous or private effect (each producer operates under the assumption of constant returns to the inputs that the producer controls), and β 1 is the aggregate effect. This specification assumes full capital and capacity utilization. The aggregate production frontier under Leontief aggregation β for Kt as the limiting factor is written Yt = qKtθ+1 St 2 e(λ+M2 (θ+1)ψ)t−μ(t,Z) = β1 β2 (λ+M2 β1 ψ)t−μ(t,Z) qKt St e . Both specifications allow for Debreu-Farrell economic (technical, allocative, and scale) inefficiency and disembodied technical change, learning by doing that can be exogenous and endogenous to net investment, and embodied technical change with accompanying knowledge spillovers and learning [224]. The Leontief aggregation specification of effort, in contrast to the homothetic Leontief-Sono separable specification of effort, explicitly explains, through endogenous and intentional net investment in Kt , how embodied technological change that originates from sources

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external to the sector and accompanying endogenous knowledge sillovers are endogenously introducted, through intentional investment in Kt , into the production process in the fishery sector. The homothetic Leontief-Sono separable specification provides a comprensive specification of rivalrous Et using rivalrous X1t , Kt and allows for input substitution as a source of endogenous growth, but does not account for intentional net investment. The Leontief aggregation specification assumes that rivalrous Kt is always the limiting factor without any input substitution as an endogenous source of growth.

Concluding Remarks Production economics is increasingly important to the economic analysis and public regulation of fishing industries. Fisheries economics arose out of population dynamics and a focus upon the long-term dynamics of the natural resource stock and an aggregate production technology. These bioeconomic models aimed to obtain the optimum fleet size and resource stock. Production economics gained in importance as it developed as a field and was able to contribute to analyses of fishing industries with an industrial organization orientation and public regulation to address the market failure stemming from the common resource. This reorientation shifted the focus to shorter time periods and to the individual firm – usually the vessel – and to multiproduct, multi-input production. Other factors also contributed to the increased application of production economics to the fishing industry: (1) the development of large-scale data bases at the vessel level and econometric software and estimation procedures and (2) rapid development of production economics to address issues in other economic sectors and the general economy. Empirical production analyses show that the individual vessel’s multiproduct production process is typically joint-in-input quantities, that a consistent aggregate output or input seldom exists, and that input substitution and output transformation possibilities are typically inflexible or limited. Moreover, the longer the time period and adjustment of fixed or quasi-fixed inputs or even outputs, and notably the physical capital stock, the greater the input substitution and output transformation substitution possibilities due to the local Le Chatelier effect. Technical inefficiency is pervasive, sometimes ranging widely between vessels, allocative inefficiency exists, and technological progress is important. There is indeed skipper skill, but it cannot be readily explained by measurable factors. Production economics in fisheries has come full circle to refine the specification of the original dynamic renewable resource economics (bioeconomic) models from which fisheries economics originally emerged. The concept of “effort” as an aggregate input used in early models has evolved substantially due to the general production economics literature. The theory of homothetic separability, aggregation, and index numbers makes clear that effort as a composite of rivalrous and excludable (private) inputs (capital, labor, energy, materials, etc.) is an index that requires consistent aggregation according to either Hicks-Leontief aggregation or homothetic Leontief-Sono separability. The homothetically separable effort

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aggregator function of nonrivalrous and excludable inputs is linear homogeneous and the exponent of effort is one. Allocative efficiency of the rivalrous inputs is accounted for in this effort aggregator function. The exponent of effort can exceed one when there is the nonrival input knowledge (due to technological progress) and fall short of one when there is congestion. Technological change has often been overlooked in fishery economics and bioeconomic models, but is now more often incorporated, almost invariably as Hicks-neutral and exogenous. When the public good knowledge, in the form of disembodied and embodied technical change, is incorporated into the production technology, the resulting knowledge spillovers require may create dynamic endogenous economies of scale in rivalrous and nonrivalrous inputs. Fisheries economics now routinely addresses the stock-flow production technology through either specifying biomass, time trends, or dummy variables. Studies of total factor productivity growth disentangle growth in the productivity residual from changes in the resource stock. Productivity analyses are estimated from either econometrics and estimates of technical change or more likely growth accounting and economic index numbers. Many fairly recent studies have examined changes in total factor productivity due to the introduction of individual transferable quotas. The impact of rights-based management, notably before and after and occasionally with counterfactuals of individual transferable quotas, is a major focus of analysis of production economics. The analytical approach has varied from estimation of multiproduct cost functions and frontiers, or stochastic or deterministic production frontiers, or measurement of total factor productivity growth through economic index numbers. One major finding is that product transformation possibilities become more flexible, due to local or global Le Chatelier effects as various previous direct regulations are lifted, thereby unbinding production possibilities. Technical and scale efficiency and total factor productivity tend to increase in most fisheries after the introduction of rights, although not in all cases, due to retirement of redundant capital, more efficient use of retained capital and other inputs, and quota transfers from less efficient to more efficient vessels. Local or global Le Chatelier effects clearly contribute. Most studies could not attribute causal inference since they were “before-and-after” rather than “with-and-without,” where the “without” is modeled by a counterfactual. The length of industry adjustment following the introduction of rights-based management depends in part upon how many and to what extent and timing direct regulations, forming binding constraints, were relaxed or eliminated entirely, thereby impacting the local or global Le Chatelier effects. Accounting for full economic efficiency, accumulated and new technology, and knowledge spillovers in bioeconomic models leads to a broader concept of the dynamic economically efficient equilibrium (i.e., maximum economic yield) than the original steady state and hence static scale efficiency. Since much of the ongoing technological change is in the form of information and communications technology originating external to the fisheries sector, when combined with knowledge spillovers, such a dynamic economically efficient equilibrium leads to dynamic

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increasing returns to scale, ongoing endogenous growth, and an optimum resource stock that is less than that of maximum sustainable yield. Best-practice econometric estimation in fisheries production is improving over time. Attention is increasingly paid to identification issues, notably testing for endogenous regressors (such as effort), and when found instrumental variable estimation. Heteroscedasticity of a general nature is now routinely corrected through Eicker-Huber-White methods. Serial correlation receives less attention. The use of heteroscedastic-and-autocorrelation consistent standard errors, such as NeweyWest or Driscoll-Kray, should be considered, given pervasive heteroscedasticity and serial correlation due to searching behavior. Spatial autocorrelation seldom receives attention. Panel data methods, notably fixed or random effects and their testing, are now well recognized. Addressing sample selection bias requires further attention, where this bias arises due to vessel entry and exit and the almost universal use of fishery-dependent data and nonrandom vessel search for catch. Moreover, fisher search is nonrandom or there can be gear saturation or density-dependent gear avoidance behavior. Sample selection bias is particularly important for bioeconomic models specifying an aggregate production function, since otherwise the results are not representative of the population and maximum economic yield is biased. Unit root and if necessary cointegration tests of time series of data, including panel data, for stationarity and degree of integration should become routine when the time series is sufficiently long; otherwise regression may be spurious and differencing may be necessary before further regression analysis. Where do fisheries vessel-level production studies consistent with production economics head in the future? In the past, such studies largely followed new developments in production economics and econometrics, although the application of production economics to fisheries also contributed to further development of production economics, or by responding to the unique feature in fisheries of transferable property rights – essentially an application of quota and rationing theory. One topic that has received considerable attention is the spatial nature of production due to the area-based nature of fisheries production. Future analysis requires consistency with production economics and in particular the nature of joint production and separability. Block jointness-in-input quantities (block output independence) definitely requires further investigation and can contribute to incorporating area fished. When such a block structure is found, then a discrete choice model in the vein of Campbell [35] can be coupled with a primal or dual specification of technology. A comparable development arose in agriculture with allocable inputs. Random utility models of location can be made consistent with econometrics through identification and accounting for potentially endogenous inputs (such as distance or cost) as regressors, but face difficulty in pairing with production technologies (as in Campbell [35]) due to the objective function of utility maximization while production technologies with endogenous catch assume maximization of catch, revenue, or profit rather than utility. Causal inference, including evaluating natural experiments, will grow in importance to evaluate policy. Explicit application of the local and global Le Chatelier principle can evaluate the impact of the transition from direct regulation to rights-

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based regulation. Accounting for bycatch and broader ecosystem impacts, which requires proper specification of joint production, is promising. Fisheries economics evolved from production functions with an aggregate output – a primal specification of technology – to dual-based methods and has shifted back to the primal problem but relying upon the directional distance function for a primal specification in part due to its accommodation of multiple outputs and joint production, including undesirable ones or public bads. Analysis of technological change is yet another area. Most studies to date have specified Hicks-neutral disembodied and exogenous technological change at a constant rate measured through a linear time trend. Insufficient research has been conducted on technological change that is biased in either (desirable or undesirable) outputs or inputs. Endogenous technical change that is bycatch and habitat saving (directed technical change) has received little attention, and can draw from the considerable progress that has been made in general economics in this area. Little is empirically known about the factors that induce or direct biased technological change. Insufficient attention has been given to embodied technological change and the impact that different vintages of the physical capital stock, embodying different levels of technology, have upon harvesting. Similarly, research on investment, economic depreciation, and capital accumulation could receive additional attention. Production economics can contribute to standardization of effort as practiced by population biologists. Standardization in fisheries refers to combining disparate technologies, each with different levels of productivity, into a single aggregate technology and composite input, effort, that in turn enters into a population model [152]. Topics within production economics that can contribute to standardization include consistent aggregation across technologies, firms, and inputs, frontier functions and economic efficiency (especially technical efficiency), multilateral and bilateral economic index numbers, joint production, and technological change that impacts time-varying catchability and could be biased. The related discipline of econometrics can contribute to identification strategies for potentially endogenous regressors in the catch-effort production technology and selection bias for fishery-dependent data. Econometrics can also contribute to standard errors when standardizing. In sum, production economics in fisheries can both open up new areas of empirical analysis and contribute to theoretically consistent specification of production technology in both fisheries economics and population dynamics.

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Luis Orea and Inmaculada C. Álvarez

Contents Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Knowledge Production Function and Spatial Economic Growth Models . . . . . . . . . . . . . . . Network Inputs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Transport Infrastructures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ICT and R&D Activities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Local Versus Global . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Agglomeration Economies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Spatial Returns to Scale . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Individual Elasticities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Mean Elasticities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Internal, External, and Total Returns to Scale . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Economy-Wide Returns to Scale . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Returns to Scale in Heterogeneous Coefficient Models . . . . . . . . . . . . . . . . . . . . . . . . . . . Spatial Econometric Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Spatial Stochastic Frontier Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Distribution-Free Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Distribution-Based Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Estimating Efficiency in Spatial Frontier Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Spatial TFP Growth Decomposition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Final Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Cross-References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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L. Orea () Oviedo Efficiency Group, Department of Economics, University of Oviedo, Oviedo, Spain e-mail: [email protected] I. C. Álvarez Oviedo Efficiency Group, Department of Economics, Universidad Autónoma de Madrid, Madrid, Spain © Springer Nature Singapore Pte Ltd. 2022 S. C. Ray et al. (eds.), Handbook of Production Economics, https://doi.org/10.1007/978-981-10-3455-8_35

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Abstract

This chapter summarizes the empirical literature that uses a spatial analysis framework in production economics. This literature takes advantage of the spatial dimension of the data to capture the spillover effects of neighboring production units. In the first three sections, we outline standard spatial extensions of the neoclassical production models aiming to measure knowledge spillovers, the effect of network inputs, and economies of agglomeration. The next four sections outline the literature that on one hand examines returns to scale and productivity growth from both internal and external inputs, and on the other hand summarize the spatial econometric techniques used in frontier and non-frontier analyses of firms’ production. The last section includes a set of final remarks regarding the application of spatial econometric techniques in production analyses. Keywords

Spatial econometrics · Stochastic frontier models · Production economics

Introduction This chapter summarizes the empirical literature that uses a spatial analysis framework in production economics. Overall speaking, this literature incorporates external returns from other (nearby) production units, extending the set of production inputs in neoclassical production models. Most empirical models in this field are estimated using individual (e.g., firms) or aggregate (e.g., regions) production units. In some settings, these production units can be associated with locations and, therefore, can be placed on the map. The spatial dimension of the data is used in this literature to compute overall marginal products, returns to scale or productivity growth measures from both internal and external factors, or simply to get better parameter estimates. Other researchers have used the spatial information in frontier production analyses to control for unobserved but spatially correlated variables that or to analyze interesting features of firms’ economic performance. This chapter is organized as follows. Section “Knowledge Production Function and Spatial Economic Growth Models” introduces the so-called “knowledge production functions” and formalizes the existence of knowledge spillovers in this setting. We take advantage of this discussion to introduce the three most popular production specifications in spatial econometrics: the spatial lag of X model (SLX), the spatial autoregressive model (SAR), and the spatial Durbin model (SDM). As firms’ productivity might be spatially correlated due to the existence of knowledge spillovers, the standard neoclassical growth models (built from a previously defined production function) can also be estimated using some of the above spatial specifications. Section “Network Inputs” presents several papers that argue that some inputs have network characteristics and generate external effects on neighboring production units (regions). This is the case of transport infrastructure and information and communication technologies (ICT). We introduce

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in this section a discussion on the selection of local versus global spillovers. Section “Agglomeration Economies” outlines the empirical literature that examines agglomeration economies, i.e., external returns from the concentration of economic activity, via localization effects or urbanization economies. The next four sections are more methodological. Section “Spatial Returns to Scale” discusses how to measure internal, external, and total returns to scale once a spatial specification of production units’ technology has been estimated, either using homogeneous or heterogeneous coefficient models. This section also discusses how to compute aggregate or economy-wide returns to scale from a set of observations. Section “Spatial Econometric Models” provides a brief discussion on the general estimation techniques that are typically used to estimate the spatial models. Section “Spatial Stochastic Frontier Models” offers a brief discussion of the small but evolving literature on spatial stochastic frontier modeling. This literature is resuscitating the interest in spatial error models (SEM), because the stochastic frontier models have two different random terms and controlling for spatial spillovers in both noise and inefficiency terms does matter due to the significant and different economic consequences of such correlations. Section “Spatial TFP Growth Decomposition” is devoted to the decomposition of spatial measures of firms’ total factor productivity (TFP). We outline the main features of two papers that have extended the standard TFP growth decomposition to include both direct (own) and indirect (spillover) components. We conclude this chapter with a set of final remarks in section “Final Remarks.” It is worth mentioning that, for notational ease, we have developed this chapter for panel data. We also confine our discussion to the estimation of (frontier) production functions due to other primal and dual representations of the technology deserve similar comments.

Knowledge Production Function and Spatial Economic Growth Models As the spatial terms in a standard neoclassical growth model appear due to the existence of knowledge spillovers, we first introduce in this section the concept of “knowledge production function” (KPF). We next summarize several empirical papers that have estimated neoclassical growth models using spatial econometric techniques. The knowledge or ideas production is crucial in the theory of innovation and in the definition of optimal public policies. Pakes and Griliches [100] defined the KPF as a function intended to represent the transformation process leading from innovative inputs (e.g., R&D) to commercially valuable knowledge or innovative output (e.g., patents). Most of the studies aiming to estimate a KPF depart from a Cobb–Douglas functional form where the level of technological knowledge (Ait ) depends on the amount of physical resources allocated to R&D activities (zit ). This function can be enlarged to include learning ideas processes [68] that depends on neighbor’s knowledge (see, e.g., [12]):

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lnAit = α + β lnzit + λ

N j =i

wij lnAj t + vit

(1)

where subscript i(=1, . . . , N) stands for regions, t(=1, . . . , T) stands for periods, zit is a vector of knowledge determinants, vit is the traditional noise term, and the weight term wij formalizes the connectivity between firm i and firm j. According to this, the learning ideas process drives to knowledge spillovers if λ is positive. Cunha and Neves [32] review the empirical literature on KPFs, identifying a handful of papers that estimate KPFs using spatial econometric techniques. For instance, Bottazzi and Peri [19] estimate a KPF where the regional (spatial) spillovers decreases with the geographical distances between regions. The spatial specification of this model is justified by theories of localization that argue that geographic proximity reduces the cost of accessing and absorbing knowledge spillovers ([57], p. 19). This explains why knowledge spillovers and spatial spillovers are different but related concepts. In the early articles, it was common to find spatial interdependence in the determinants of knowledge, while more recently researchers consider also spatial spillovers in knowledge itself. These considerations result in different spatial specifications of KPF that correspond with different models widely known in the spatial econometric literature [127]. On the one hand, the model that incorporates the spatial lag of the dependent variable or the weighted average of neighboring values of the dependent variable is the well-known spatial autoregressive model (SAR).1 Equation (1) is an example of a SAR model. On the other hand, the model that incorporates the spatial lag of the explanatory variables is the well-known spatial lag of X model (SLX). For instance, Álvarez and Barbero [2] assumes that the level of technological knowledge depends on physical and human capital of both the own region and neighboring regions. Their KPF can be thus written as: lnAit = α + βlnzit + θ

N j =i

wij lnzj t + vit

(2)

where zit stands now for physical and human capital variable. There is not a consensus about the most preferred specification (SLX vs. SAR) and whether the spillovers are local or global. Although section “Local Versus Global” provides a critical discussion on the nature of the spillovers generated by these two spatial specifications, it is germane to mention here that the spillovers induced by the SAR model in (1) are global in the sense that shocks disturbing a firm might affect all other firms. In contrast, the SLX model in (2) yields more local spillover effects, because they do not involve endogenous feedback effects from neighbors to the neighbors and so on. Empirical results indicate that both type of spillovers might play an important role in the production of knowledge [23, 61]. If so, the model that should be

1 The

[6].

“spatial lag” terminology used in spatial econometrics was originally introduced by Anselin

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estimated is the well-known spatial Durbin model (SDM), which can be written as follows: lnAit = α + λ

N j =i

wit lnAj t + βlnzit + σ

N j =i

wij lnzj t + vit

(3)

Notice that, as customary in spatial econometrics, this model can be rewritten in a simpler fashion using matrix notation as follows: lnAt = α + λW lnAt + βlnzt + θ W lnzt + vt

(4)

where lnAt , lnzt , and vt are Nx1 vectors, and the set of spatial weight terms in (3) are now written using a spatial weight NxN matrix W, where diagonal elements are equal to zero, and the off-diagonal elements are nonzero if the firm i is assumed to be correlated with firm j. Quite often, the off-diagonal elements are equal to one if both observations are located in adjacent locations. In many applications, the weight matrix is also row-standardized with the number of adjacent units (i.e.,  N j =i wij = 1). In this case, WlnAt in (4) can be interpreted as the average values of the technological knowledge of adjacent firms. The choice of a proper spatial weight matrix is contentious. For instance, Tiefelsdorf et al. [121] point out that this standardization procedure may emphasize the prevalence of the spatial dependence on those units with fewer connections. Plümper and Neumayer [104] demonstrate the enormous influence that choosing the functional form of the weighting matrix can exert on the empirical results. Given the relevance of geography in the diffusion of knowledge and R&D, Fingleton and López-Bazo [44], Ertur and Koch [43], and Fischer [45] have augmented the Solow neoclassical model by including both global and local spatial autocorrelation on growth and convergence. The technology of the whole economy is characterized by a Cobb–Douglas production function with constant returns to scale in per worker terms: lnyit = ln Ait + α lnkit + vit

(5)

where yit = Yit /Lit is output per worker, kit = Kit /Lit is capital services per worker, and Ait captures the level of technological knowledge. As in (2), the abovementioned authors define the technological knowledge term using a SLX specification, i.e.: ϕ

Ait = kit

N

ϕρw k it j =i j t

(6)

where the technological parameter 0 < ϕ < 1 reflects the size of the home externalities and p allows formalizing spatial interdependence by means of the spatial weight terms wij . If we plug (6) into the neoclassical production function (5), we get a (per worker) production function with spatial interactions:

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ln yit = ln  + βlnkit + θ

N i=i

wij ln kit + vit

(7)

where β = α + ϕ and θ = ϕρ. Therefore, the existence of knowledge spillovers explains why output per worker in region i depends on its own capital investment but also on neighbors’ capital investment. If we now introduce the production function (7) into a neoclassical growth model, we can obtain the output per worker at the steady state and the speed of convergence to the steady state. Interesting enough, the obtained convergence equation includes spatial lags in both the dependent and independent variables: y˙it = α + λ

N j =i

wij y˙j t + βzit + θ

N j =i

wij zj t + vit

(8)

where y˙it is the annual rate of growth of output per worker, zit is the logarithm of the sum of the labor rate of growth, the rate of depreciation and the rate of technical change, and the speed of convergence can be obtained from the estimated λ parameter. For notational ease, we have omitted in this equation the initial output (real income) per worker and the fraction of output that is saved. The spatial convergence equation in (8) thus follows a SDM model and it predicts convergence if output (per worker) growth is a negative function of initial output (not shown), after controlling for the determinants of the steady state (i.e., labor rate of growth, the rate of depreciation, and the rate of technical change) and the possible existence of spatial interdependence among nearest economies. More recent papers also include other determinants of economic growth and convergence in regions and countries, such as human capital and public sector (e.g., [5]). To finish this section, it is worth mentioning that the spatial specifications of the above models come from economic theory. In general, one of the main criticisms regarding the spatial econometric models is the absence of theoretical basis [31]. Therefore, a remarkable exception is the spatial neoclassical growth model in which the spatial specification relies on the existence of knowledge spillovers and learning processes.

Network Inputs This section presents several papers that argue that some inputs have network characteristics and generate external effects on neighboring production units (regions). For instance, Munnell [93] points out that the transport infrastructure localized in a region could benefit other regions. Stiroh [115, 116] and Griliches [59] also stated that ICTs and R&D activities can be treated as network inputs because they can generate externalities to other firms as well. In most cases, the “network” nature of these inputs is because they indeed are public goods, i.e., they are inputs that one firm can use without reducing their availability to others and from which no one is excluded.

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Transport Infrastructures Transport network or transport infrastructure has been considered one of the public policy decisions that has the greatest impact on economic development, both for its effect on the structure of the population, and its capacity to reduce costs and increase production. This explains why many studies have tried to quantify the effect of transport infrastructure on private production. For a summary of this literature, see Cohen and Morrison [25] and Pereira and Andraz [103]. More specifically, after the seminal paper by Aschauer [10], there is a wide literature that has extended the traditional (aggregate) production function where the provision of infrastructures (KPit ) is complementary to labor (Lit ) and private capital (Kit ). Using a Cobb– Douglas specification and assuming constant returns to scale in private inputs, this production function can be written after taking as follows: lnyit = ln Ait + α ln kit + β ln KP it + vit

(9)

where yit and kit are again output and capital services per worker, vit is the noise term, and Ait can be interpreted as a total factor productivity index. Note that the provision of infrastructures enters the production function as a standard factor of production. Straub [117] states that the inclusion of the infrastructure variables as simple inputs is questionable because, despite the increasing market mediation of infrastructure, this type of capital is not completely remunerated according to its marginal productivity in the real world. This has prompted several authors (see, e.g., [34]; and [67]) to instead consider infrastructure as part of the total factor productivity term (Ait ), i.e., as an efficiency-enhancing externality specifically linked to the accumulation of infrastructure capital. As the Cobb– Douglas production function in (9) does not allow researchers to distinguish the direct effect of infrastructure (i.e., through the production of specific services) from the indirect effect (i.e., the efficiency-enhancing infrastructure externalities), Orea et al. [99] have pointed out that this problem can be addressed if we first use a frontier specification of the production model and then we treat the set of infrastructure variables as efficiency determinants. In addition, transport infrastructures generate (spillover) effects outside the geographical place where they are located, given their network characteristic. In other words, firms located in a region not only use the infrastructure of its own region but also the infrastructures located in neighboring regions. Therefore, firms use two public infrastructures, not only one as it is implicitly assumed in (9). For this reason, the majority of the literature examining spillovers effects attributed to public infrastructure adopts a similar strategy, i.e., the addition of spatial lags of KPit as a standard input. Therefore, the production function that is estimated is the well-known SLX model: lnyit = lnAit + α ln kit + βlnKP it + θ

N j =i

wij lnKP j t + vit

(10)

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Depending on the strategy followed to define the spatial weights (Wjy), some papers have confirmed the existence of positive spillover effects derived from investment in transport infrastructures (e.g., [25], and [102]), while other studies find negative spillover effects (e.g., [3, 4, 18]). The latter studies conclude that negative spillover effects from transport infrastructures are found due to factor migration or when the set of neighboring regions are defined in economic terms (e.g., regions that are competitors or have similar characteristics), while positive spillovers are generally found when the neighboring firms are defined using geographical criteria. The spillover effects also vary with the set of countries or regions examined and with the specification of the model in levels or in rates of growth. Indeed, in his revision of the literature, Straub [117] finds that specifications using a standard production function in levels are generally more supportive of a positive effect of infrastructure than those using output (productivity) growth rates. He interprets this result as an indication that transitory effects are more often observed than long term effects. This authors also points out that in most cases, growth-accounting studies find lower levels of infrastructure externalities for more developed countries or regions than for developing ones. Other authors analyze the impact of infrastructures provision on economic performance using more comprehensive spatial models. Yu et al. [129] summarize this specific literature and attribute their SDM production model to variations in the rate of capital (capacity) utilization, an unobservable production driver in many applications. To address this issue, Gajanan and Malhotra [47] suggest modeling the rate of capital utilization as a function of the economic performance of the neighboring provinces. This empirical strategy is supported on the basis that each region accommodates its rate of capital utilization to meet the output increases in other regions [21]. In this sense, Arbues et al. [8] and Álvarez et al. [5] assume that the flow of capital services per worker in (10) is kit = CU it · kit∗ , where kit∗ is the stock of capital per worker. They next define their capacity utilization rate as  N

CU it = eλY j =i wij lnyj t +τit . Substituting this spatial specification of CUit into (10), they obtain the following SDM model: lnyit =lnAit + λ

N j =i

wij lnyit + αlnkit∗ + βlnKP it + θ

N j =i

wij lnKP j t + εit (11)

where εit = vit + ατ it and λ = αλY . Notice that the production function (11) depends on the capital stock kit∗ and the spatial lag of the dependent variable, while the production function (10) only depends on capital services that were (implicitly) assumed to be proportional to the capital stock.

ICT and R&D Activities ICT can be considered as a network input because it may enable a process innovation itself (see, e.g., [15, 20]). The main debate in this literature has to do with the

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empirical techniques used to link ICT and productivity growth. For instance, while Jorgenson and Stiroh [70], Gordon [56], Van Ark et al. [123], and Van der Wiel [124] use growth-accounting techniques with observed factor shares, other papers (e.g., [116]) have questioned the use of growth-accounting techniques because the estimated factor shares (from an estimated production function) do not often coincide with their observed counterparts. Strobel [118] suggests that the divergence in observed and estimated factor shares can be interpreted as evidence of the existence of ICT spillovers. As ICT can generate externalities to other firms, van Leeuwen and van der Wiel [125, 126] among other authors introduce spatial lags of the ICT variable into their production functions or growth accounting models. Using Dutch firm-level data, they find that the ICT spillovers are an important source of TFP growth. They also corroborate that the production function approach yields more significant and plausible results than the growth accounting approach. Most recent papers include more sophisticated specifications of the ICT spillovers. For instance, Strobel [118] include the ICT spillovers as an intermediate input. While Bloom et al. [16] use the degree of product market proximity to compute the weight matrix W, Lychagin et al. [88] use the geographical distance. Interestingly, the latter weight matrix seems to be more relevant for R&D spillovers than for ICT spillovers due to the network effects associated to ICT are not confined to a limited geographical space. Since the seminal contribution by Griliches [58], many empirical studies have provided solid evidence about the impact of R&D activities in firms’ production using either firm or regional level data. For firm level applications, see Hall and Mairesse [62], Klette and Kortum [77], and Rogers [108]. Regarding the second set of papers, Prenzel et al. [105] highlight the relevance of regional and geographical characteristics in the impact of R&D investment on productivity. It should be noted that R&D is an input that create new ideas and innovation [89] that other firms can copy, and thus an important aspect is the possibility of externalities and knowledge spillovers [11]. In this case, as pointed out by Gråsjö [57], the knowledge spillovers can be viewed as a pure externality, i.e., as an unintended side effect of firms’ ordinary activities. Alternatively, knowledge can be transmitted by explicit agreements of transaction of knowledge. To capture these externalities, Bloom et al. [16, 17] and other authors extend the production function with variables measuring R&D spillovers based on spatial and technological proximity, and find remarkably spillovers associated to R&D activities. Eberhardt et al. [37] find as well that the estimated returns of private R&D are seriously biased if the R&D spillovers are ignored.

Local Versus Global As shown before, the spatial production models allow us to examine the existence of spatial spillovers associated to public infrastructures. The issue here is the selection of an appropriate spatial specification in order to produce valid estimates and to

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infer accurate predictions of the scope of the existing spillovers. In most of the abovementioned papers, the authors use a specification of the production function that corresponds with the SLX model. In this sense, LeSage [83] states that most spatial spillovers are local in applied regional science modeling, but remarks that a network input, as for example a highway, represents a resource shared by numerous locations and it thus could also cause global spillovers. If so, the SDM model should be chosen instead of the SLX model. However, as pointed out by this author, the distinction between local and global spillovers based on estimated SLX vs. SDM models could be somehow artificial , since it relies on the implicit assumption that the local spillovers in a SLX model only involve adjacent neighbors, but not higher-order neighbors. However, this is not true if the W matrix is defined in (very) broad terms, e.g., using the inverse of the distance between the totality of regions or adopting economic concepts of distance. In this case, most spatial observations will be involved, as it happens in the SDM model, but now using a SLX model. Despite the abovementioned discussion, LeSage and Pace [87] state that most papers in this literate still maintain this artificial distinction because it facilitates to test for local versus global specifications. However, this test is only informative if the W matrix is defined in very narrow terms (using, e.g., first-order neighbors) because the difference between the SLX and SDM models is larger. The same applies for SLX vs. SAR models. In this sense, Gibbons and Overman [48] show that the reduced forms of these two models are very similar if the W matrix is broadly defined. In summary, the above discussion shows that it is necessary to pay much attention to the spatial specification of the model when we aim to capture spillover effects. However, this does not take place in practice. Indeed, as pointed out by Gibbons and Overman [48] and Vega and Elhorst [127], many empirical applications lack a proper justification for the selected spatial specification. More thoughts about this can be found in the last section of this chapter.

Agglomeration Economies A vast literature confirming the relation between productivity and economies of density has appeared since the seminal paper by Ciccone and Hall [24] on agglomeration economies. The theory of agglomeration economies proposes that firms benefit from the concentration of economic activity, via localization effects or urbanization economies [46]. Krugman [78] and Fujita et al. [46] show that the presence of agglomeration or concentration economies in geographical space might explain the existence of increasing returns to scale in many empirical applications. The empirical literature measuring the effect of agglomeration economies on firms’ productivity often are based on estimated production functions, which contain

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some representation of agglomeration economies.2 This literature is focused on the relative importance of two different agglomeration economies. While localization economies are caused by industrial concentration [26], urbanization economies are associated to city size [36]. A survey of this literature can be found in Rosenthal and Strange [112]. In general, there is no consensus in this literature about the production effect of the different agglomeration measures [92], although these studies usually find a positive productivity gains from urban agglomeration [26, 38, 111]. This lack of consensus has generated a heat debate on the level of disaggregation, the specification of the model, the econometric methods, or the measurements of agglomeration. Regarding the econometric issues, Combes and Gobillon [26] propose several strategies to deal with potential endogeneity problems. Other source of differences is the existence of missing production drivers positively correlated with agglomeration, as for example, land, local public infrastructures [38] or natural advantages of some locations [42]. Selection biases in location choice are also expected because productivity of firms can be conditioned to the density of their locations. In this sense, Combes et al. [29] developed a formal test that allows examining whether firms’ selection does not explain spatial productivity differences. Other differences are likely caused by the use of different spatial concentration indexes that try to measure inequalities in the spatial distribution of economic activity. In this sense, Combes and Overman [27] and Combes et al. [28] identify six properties that an ideal index of spatial concentration should fulfill. As most concentration indexes based on clusters of firms do not satisfy all properties, Duranton and Overman [35] and Arbia et al. [7] suggest using distance-based spatial concentration indexes. It is worth mentioning that the abovementioned papers have to do with the spatial location of the economic activity, but they do not use the SLX, SAR or SDM models introduced in previous sections because this literature ignores the existence of spillovers between “neighbors.” A remarkable exception is Han et al. [63] who follow Ertur and Koch [43] and propose estimating an augmented version of the production function (5), where the TFP term (Ait ) is modeled as a function of two indicators of urban agglomeration and their spatial lags, and the TFP term of neighboring cities: wij N  β Ait = Zit Zjθ t Aλjt (12) j =i

where Zit is a vector of two agglomeration measures. In particular, Han et al. [63] assume that technological interdependence among cities operates through spatial externalities, and the external effect of technology generated by specialization and diversification agglomeration of manufacturing in one city extends across its borders. They next plug (12) into (5) and estimate the following production function:

2 If

real wages are proportional to labor productivity, this issue can be also examined using wage functions.

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ln yit = βlnZit + αlnkit + λ +θ

N j =i

N j =i

wij lnZj t − αλ

wij lnyj t (13)

N j =i

wij lnkj t + vit

For notational ease, we do not include any dynamic term in (13) as well as other production drivers measuring urbanization, human capital, government intervention, foreign direct investment, and energy consumption. The specification of knowledge spillovers in the TFP term yields an SDM model. Han et al. [63] estimate this model using spatial econometric techniques and find evidence of the existence of spatial spillovers that are influenced by the use of economic and spatial definitions of proximity of the weighted matrix.

Spatial Returns to Scale Glass et al. [53] introduce the idea of spatial returns to scale (RTS) by adapting wellknown concepts introduced by LeSage and Pace [86] in applied spatial econometrics to the measurement of classic technology characteristics, such as elasticities and returns to scale. They suggest computing three different returns to scale measures (i.e., internal, external, and total) once a spatial SAR and SDM production function is estimated. As the authors pointed out, these three RTS measures can also be calculated using other primal and dual technology representations, such as cost, profit, revenue, and input and output distance functions. To catch the differences among these three measures, assume that we have already estimated the following single input SDM production function: lnYit = α + β lnXit + λ

N j =i

wij lnYj t + θ

N j =i

wij lnXj t + vit

(14)

where Yit and Xit are the output and input levels, A is the spatial autoregressive parameter, and wij is an element of the spatial W matrix, which describes the strength of the spatial interaction between the units. Note that the spatial autoregressive parameter in (14) is common to all units. Therefore, the above equation is a homogeneous coefficient spatial model. Notice that (14) can be rewritten using matrix notation as follows: lnYt = (I − λW )−1 [α + β lnXt + θ W lnXt + vt ]

(15)

where lnYt , lnXt , and vt are Nx1 vectors, and W is a NxN matrix of spatial weight terms. If the off-diagonal elements are equal to one if both observations are in adjacent locations and the weight matrix is row-standardized, WlnXt can be interpreted as the average input level of adjacent firms or regions.

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Individual Elasticities Differentiating (15) with respect to own factor inputs (i.e., lnXt ) and the inputs of all the other units in the sample (i.e., WlnXt ) yields the following matrix of direct and indirect elasticities for each unit: ⎡

e1 e12 ⎢ e21 e2 ⎢ ⎢ . .. ⎣ .. . eN 1 eN 2

⎤ ⎡ β ω12 θ e1N ⎢ ω21 θ β e2N ⎥ ⎥ −1 ⎢ .. ⎥ = (I − λW ) ⎢ .. .. ⎦ ⎣ . . . · · · eN ωN 1 θ ωN 2 θ ··· ··· .. .

⎤ · · · ω1N θ · · · ω2N θ ⎥ ⎥ . ⎥ .. . .. ⎦ ··· β

(16)

where ei = ∂ ln Yit /∂ ln Xit and ein = ∂ ln Yit /∂ ln Xnt are direct and indirect elasticities, respectively. The direct elasticity in a spatial production function is the rate of increase in a unit’s output following a (proportional) increase in its own input variable(s). The indirect elasticity is the rate of increase in a unit’s output following an increase in the factor inputs of another unit in the sample. Both direct and indirect elasticities include a feedback effect, which pass through other units via the spatial multiplier matrix and back to the unit which initiated the change. It is worth mentioning that this feedback effect has also to do with our previous discussion on the existence of local vs. global effects in spatial models. Notice in this sense that, as pointed out by LeSage [83], the global multiplier (I − λW)−1 can be expressed as I + λW + (λ2 W2 + λ3 W3 + · · · ). This implies that any elasticity in turn can be decomposed into a nonspatial elasticity via the identity matrix, plus a local effect that produces impacts on production of firstorder neighboring units via λ W , and a global effect that arises from impacts on production of second-order neighboring units via λ2 W2 , and so on. More accurately speaking, the indirect elasticity does not have a proper nonspatial effect because the departing elasticity that later on will be multiplied by (I − λW)−1 already includes the W matrix and thus the first component can be interpreted as a local effect, and the next components as global effects. Unlike the simple SLX model which only includes local effects, the above SDM production function will have both the local and global effects (see [41, 86]; and [55]). Notice however that, as pointed out before, the SLX model might also involve distant production units if the W matrix is defined in broad terms. As et and ein are linear functions of the elements of the global multiplier (I − λW)−1 and these elements are unit-specific, both direct and indirect elasticities vary across units even though we have estimated common β and θ parameters for all observations. Notice also that if the estimated spatial production function uses a SAR specification and there are not local spatial spillovers associated to the input variables (i.e., θ = 0), we get that both direct and indirect effects are simple adjustments of the original parameter. However, even if a SAR specification is used, both elasticities still exhibit a nonlinear relationship with the underlying model parameters.

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Mean Elasticities LeSage and Pace [86] suggest reporting mean values of direct and indirect concepts to facilitate interpretation. While the mean direct elasticity (hereafter DE) is computed asa simple arithmetic average of the diagonal elements of (16), i.e., DE = 1/N N i=1 ei , the mean indirect elasticity (hereafter IE) is computed as the cumulative sum of the off-diagonal  Nelements of (16) from each row, averaged over all rows, i.e., I E = 1/N N j =1 n=i ein . The mean total elasticity (hereafter TE) is next computed as the sum of the mean direct and mean indirect elasticities, i.e., TE = DE + IE. To compute the t-statistics for the mean direct, mean indirect and mean total elasticities, LeSage and Pace [86] and Elhorst [41] propose conducting Monte Carlo experiments that simulate the distribution of the mean elasticities using the variance-covariance matrix associated with the ML estimates. Interesting enough, while Glass et al. [51] and Glass and Kenjegalieva [50] use Bayesian simulation techniques to compute the t-statistics for the mean effects, they are calculated using the delta method in Glass et al. [52, 54].

Internal, External, and Total Returns to Scale The main contribution of Glass et al. [53] is noticing that the above elasticities can be interpreted as measures of the technology’s returns to scale in a spatial setting. As the spatial effects in (14) are related to inputs, this allows extending a classical characteristic of production to the spatial case. They proposed three returns to scale measures: internal, external, and total. Using the standard terminology in spatial econometrics, they can be alternatively labeled as direct, indirect, and total returns to scale. The internal returns to scale is defined as the rate of increase in a unit’s output following a proportional increase in its own input variable(s). The unit-specific internal RTS can be simply computed as et due to only a single input has been considered. The external returns to scale are defined as the rate of increase in a unit’s output following an increase in the inputs of all the other units in the sample. Glass et al. [53] propose computingthe unit-specific external RTS using the simple sum of indirect elasticities, that is, N n=i ein . Finally, total returns to scale is defined as the rate of increase in a unit’s output following a simultaneous increase in its own inputs and the inputs of all the other units in the sample. Therefore, the calculation of total returns to scale is based on all N units in the sample simultaneously changing their inputs. The unit-specific total RTS can  be computed as N e . Glass et al. [53] also examine the concavity of the spatial in n=i production function and find that all definitions of concavity (i.e., internal, external, and total) in a spatial setting depend on the specification of the spatial weight matrix. Glass et al. [53] also used simple arithmetic averages to summarize  their RTS results. While mean internal returns to scale is computed as 1/N N i=1 ei , mean  N external returns to scale is computed as 1/N N e . Finally, mean total i=1 n=i in returns to scale is the sum of the mean internal returns to scale and the mean

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external returns to scale. Using these three mean values, the spatial production function exhibits decreasing returns if RTS < 1. Constant returns appear if RTS = 1. Finally, the spatial production function exhibits increasing returns if RTS > 1. The above authors find positive labor and positive capital spillovers in their empirical application to a set of European countries over the period 1990–2011. While they cannot reject constant internal and external returns to scale, they reject constant total returns to scale in favor of increasing total returns. Their findings thus provide some empirical support for the endogenous growth theories which are based on the assumption of increasing total returns to scale, although their increasing total returns to scale are not caused by knowledge spillovers as in Romer [109, 110].

Economy-Wide Returns to Scale The above mean RTS measures mimic the approach suggested by LeSage and Pace [86] to summarize their marginal effects. Similar, but not the same, expressions can be found if we aim to compute aggregate or economy-wide returns to scale from the whole set of basic production units or regions. First notice that the aggregate or economy-wide technology can be defined as: Yt = G (X1t , . . . , XN t ) =

N i=1

Yit =N i=1 fi (Xit , X−it )

(17)

where X–it is the vector of inputs of all the other units (regions) in the sample, and fi is the production function of unit i , which depends on its own inputs and the inputs of other units under the presence of spatial spillovers. Notice that G(x1t , . . . , xNt ) is not separable in individual inputs because all regional outputs depend on own and neighboring inputs. In contrast, the aggregate technology in  a nonspatial model is separable in individual inputs as it can be written as Y = N i=1 fi (Xi ). Differentiating the above economy-wide technology with respect to all inputs and assuming that all individual inputs increase in the same proportion, we get after some straightforward algebra that economy-wide returns to scale (hereafter, E) can be measured as: E=

N i=1

si ei +

N i=1

si

 N n=i

 ein

(18)

It is now worth mentioning that, if the estimated model is a SAR model, the economy-wide measure  of returns to scale is a simple weighted average of all direct elasticities, i.e., E = N i=1 si ei . In summary, we should use a weighted not a simple arithmetic average of individual elasticities if we are willing to compute economywide RTS. In other words, LeSage and Pace [86] and Glass et al. [53] mean measures of total, direct, and indirect effects cannot be used to measure aggregate (economywide) RTS except all units (regions) are of similar size.

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Returns to Scale in Heterogeneous Coefficient Models The increasing availability of large (panel) data sets explains why important contributions have been made in recent years to estimate spatial models with autoregressive coefficients that vary across units. For instance, Malikov and Sun [90] and Sun and Malikov [120] compute the spatial autoregressive values from an unknown smooth function of a set of environmental factors. Gude et al. [60] do so using a parametric function and in a frontier setting. They use a heteroscedastic version of the spatial stochastic frontier models introduced by Glass et al. [54] as they allow for province-specific degrees of spatial dependence. Allowing for unit-specific spatial coefficients not only will lead to less biased conclusions but also to richer conclusions (see, e.g., [60]), especially when the spatial data represent firm-level rather than regional observations (see, e.g., [84]). However, Sun and Malikov [120] state that such models are also useful in the estimation of growth models as it is expected in this literature that the intensity of knowledge spillovers greatly depend on institutional and cultural compatibility of neighboring countries. As mentioned in section “Mean Elasticities,” LeSage and Pace [86] and Elhorst [41] proposed using arithmetic averages of either diagonal or off-diagonal elements of (16) to simplify the task of interpreting estimates of direct and indirect effects from the model. However, LeSage and Chih [84] stated later that scalar summary measures are not consistent with the notion of parameter heterogeneity. In this case, we should report observation-level effects estimates. For the case of the heterogeneous coefficient SAR panel model, the N diagonal elements of the matrix should be provided to produce direct effects estimates for each of the units (regions). As estimates of unit-specific indirect effects, it is recommended to follow the proposal of LeSage and Chih [84] and use the cumulative sum of off-diagonal elements in each row of (16).

Spatial Econometric Models We provide in this section a brief discussion on the general estimation techniques used to estimate spatial econometric models with cross-sectional and panel data sets. Earlier developments in testing and estimation of SAR models in the context of cross-sectional models has been summarized in Anselin [6]. Ord [95] first proposed to estimate the SAR models by the method of maximum likelihood (ML). Kelejian and Prucha [72, 73] later introduced the spatial two-stage least squares (2SLS) and methods of moments (MM) estimators. There are numerous extensions that have been made in more recent articles. In this sense, Kelejian and Prucha [74] extended the model to a system of equations spatially interrelated, Kelejian and Prucha [75, 76] introduced a method robust to heteroscedasticity and autocorrelation in disturbances in a spatial autoregressive model, and Lee [81] introduced a spatial

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quasi-maximum likelihood estimator (QMLE), which is more appropriate when the disturbances in the model are not truly normally distributed. The availability of panel data sets has allowed to observe individuals characterized by spatial features over time. As the panel econometric models are less vulnerable to multicollinearity issues and they allow to control for unobserved but time invariant variables, many researchers have recently adapted (and extended) the previous spatial econometric estimators to panel data settings. Elhorst [41] provides a nice introduction to this literature. For instance, Kapoor et al. [71] and Mutl and Pfaffermayr [94] have adapted the MM methods from cross-section to panel data. Elhorst [39, 40] and Lee and Yu [82] introduced respectively the ML estimators of the SLX and SEM models for panel data with fixed and random effects. Baltagi and Liu [13] extended the traditional panel data error components estimator to SAR model. In parallel, there is a literature that has focused on nonparametric techniques, as is the case with the Bayesian spatial econometrics (see, for instance, [86]). In recent years, we have noted open debates regarding the spatial econometric techniques. One of them affects the decision about the W matrix representing spatial linkages. Recent developments in spatial econometrics introduce new techniques to automatically select the spatial weight matrix [130]. Another hot debate has to do with the heterogeneous nature of the spatial autoregressive coefficients. As anticipated in section “Returns to Scale in Heterogeneous Coefficient Models” the so-called heterogeneous spatial models allow measures of spatial dependence specific to each observation, information that can be useful for policymaker’s decisions. Bayesian and Markov Chain Monte Carlo (MCMC) mixture estimation methods [85] have been mostly proposed to achieve this flexibility in the spatial autoregressive parameters.

Spatial Stochastic Frontier Models Although there is extensive spatial econometric literature dealing with spatial interactions across spatial units, the literature on efficiency analysis has not generally taken spatial effects into account. Several studies have found that failure to account for spatial correlation effects in SF models results in biased estimates of efficiency scores (e.g., [114]). For this reason, it is important to use an econometric framework that allows controlling for the presence of cross-sectional dependence when measuring the efficiency performance of spatially distributed production units. It should be stressed that not only it is important to control for spatial spillovers in efficiency analyses but also for the existence of heterogeneous spatial dependence parameters. Indeed, Gude et al. [60] find that the standard efficiency estimates (i.e., the estimated “u” term) change a lot when they (incorrectly) use a common spatial dependence parameter for all observations. The efficiency results are expected to change even more if the total efficiency scores proposed by Glass et al. [54] are computed, because these efficiency measures include direct and indirect spatial effects that not only depends on the abovementioned “u” term but also on the

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estimated spatial dependence parameters, which might vary significantly across observations. This section offers a brief discussion of the small but evolving literature on spatial stochastic frontier modeling.3 This literature tries to overcome this issue by including spatial autoregressive terms in their models. Generally speaking, this literature can be split into two groups, depending on whether distributional assumptions are made for the inefficiency term.

Distribution-Free Models The first group of papers estimate panel spatial models based only on the distribution of the noise term and without making any distributional assumption for the inefficiency component of the error term. Examples of papers that belong to this group are Druska and Horrace [33] and Glass et al. [51, 52]. The model estimated in these papers can be summarized using the following single-input production function: lnYit = αit + β lnXit + λ

N j =i

wij lnYj t + θ

N j =i

wij lnXj t + εit

(19)

where Yit and Xit are respectively the output and input levels of unit i, λ is the spatial autoregressive parameter, α it is a unit-specific effect that can be defined as time-invariant (i.e., α it = α i ) or as an individual-specific parameterized function of time (i.e., α it = α 0i + α 1i t + α 2i t2 ), and εit is a noise term that might also be spatially correlated as in Druska and Horrace [33]. The individual efficiency scores are simply computed from the cross-sectional specific effects using the approach in Schmidt and Sickles [113] (hereafter SS) and Cornwell et al. [30] (hereafter CSS). In this setting, the observation with the largest individual effect in each period is placed on production frontier, and the efficiency estimates are the exponential of the difference between the best performing individual effect and the corresponding effect for each of the other observations in the sample. That is, efficiency is measured as EFit = exp (α it − maxj (α jt )). Druska and Horrace [33] implicitly assumed in eq. (19) that λ = θ = 0. They ignored any spatial correlation in the frontier as because they interpreted the production function as a purely deterministic (engineering) process where the production units control all the inputs. This assumption allowed them to focus their application on spatial correlations associated with the noise term as they developed a spatial error (SEM) model with time-invariant fixed effects, which were used later to calculate unit-specific efficiency scores using the SS estimator. The SEM models have not been very popular in non-frontier settings, because the spatial dependence that is accounted for in these models is not a representation of substantive economic

3 The

content of this section is highly inspired in Orea and Álvarez [97].

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spillovers [86]. The residual  term in this model is assumed to follow a SAR process, that is: εit = vit + ρ N j =i wij εj t . The error term in this model consists of two components, an idiosyncratic noise term (vit ) and a spatial component that relates a unit’s random shocks to the random shocks of neighboring units. Glass et al. [51] is a similar type of study as they use the fixed effects from a SAR stochastic frontier model to estimate time-varying efficiency using the CSS approach. They use maximum likelihood techniques as Glass et al. [52], who extended the CSS methodology to the spatial autoregressive case and estimate direct, indirect, and total efficiencies for each production unit.

Distribution-Based Models The second group of spatial stochastic frontier models follows most of the nonspatial stochastic frontier literature by making assumptions about the distribution of both the noise and inefficiency terms. This group is not entirely homogenous as some papers allow the frontier to be spatially correlated across production units (e.g., [1]; and [54]), while other papers allow the error terms to be spatially correlated (e.g., [9, 114]; and [122]). Most of these models can be summarized using the following single-input production function: lnYit = α + β lnXit + λ

N j =i

wij lnYj t + θ

N j =i

wij lnXj t + vit − uit

(20)

This equation includes two error terms, vit and uit . While the former term is a symmetric error term measuring pure random shocks, the latter term is a nonnegative error term measuring unit-specific inefficiency. The above equation describes the so-called spatial Durbin frontier (SDF) model proposed by Glass et al. [54] that accounts for both local and global spatial interactions. If we assume in (20) that θ = 0, we get the so-called spatial autoregressive stochastic frontier (SARF) model. If in addition we assume that λ = 0, we get the traditional nonspatial stochastic frontier model. If we, however, assume that one or both error terms in (20) are spatially correlated, we get a spatial error stochastic frontier (SEF) model. The latter model can be viewed as a vehicle to resuscitate the interest of the scientific community, policymakers, and regulators in SEM models because, unlike the traditional SEM model, we have two random terms in a frontier analysis framework and controlling for spatial spillovers in both random terms does matter due to the (different) economic consequences of such correlations. This conclusion is motivated later in section “Final Remarks.” Another restricted specification of (20) is provided by Adetutu et al. [1] who propose a stochastic SLX frontier model because they only include spatial lags of the exogenous variables as frontier determinants (i.e., they assumed λ = 0, but allowed 0 to take nonzero values). As they limit their analysis to local spatial dependence, their model can be estimated using the standard procedures for the nonspatial stochastic frontier. This model, however, overlooks any global spatial

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dependence as they omit the endogenous autoregressive variable. Glass et al. [54] estimate a SARF model that only accounts for global spillovers, and a SDF model that accounts for both global and local spatial dependence. To minimize issues relating to convergence, these authors adopt a pseudo maximum likelihood estimator and estimate their model in two steps, first estimating a non-frontier SDM model and then splitting the first-stage residuals into the idiosyncratic error and timevariant efficiency. Gude et al. [60] generalize the above SARF and SDF models in two aspects. First, they allow for heteroscedastic specifications of the inefficiency term. Second, both models allow the researchers to identify the determinants of the spatial dependence among the Spanish provinces. A parallel paper focusing on the EU regions is Ramajo and Hewings [107] that explicitly consider (common) spatial spillover effects by including a spatial lag of the dependent variable at the frontier. Another set of papers allows the inefficiency term to be spatially correlated. In these papers, the one-sided error term consists of two components, an idiosyncratic one and a spatial component that relates a unit’s inefficiency to the inefficiency of neighboring units. Standard maximum likelihood techniques are not used here because the addition of spatial lagged inefficiency terms does not yield a closed form for the likelihood function, and several computational algorithms are proposed to conduct simulation-based inference and efficiency measurement. For instance, Areal et al. [9] avoid this issue by using a Gibbs sampler and two MetropolisHastings steps to estimate the spatial dependence of firms’ efficiency. A similar model is proposed by Tsionas and Michaelides [122], who develop a Bayesian estimator for a model that allows for spillover effects in inefficiency. Schmidt et al. [114] also adopt a Bayesian approach to estimate a variety of spatial stochastic frontier models. In contrast to the previous papers, their inefficiency term does not follow a spatial autoregressive process, but it depends on a latent (unobserved) local effect. In several specifications, they assume that the local effects follow a conditional autoregressive distribution which depends on its neighbors. Similarly, Herwartz and Strumann [64] estimate a frontier model with region-specific random effects in the inefficiency term that allows for spatial dependence. As their likelihood function does not attain a closed-form solution, the model is estimated by simulated ML. Previous spatial stochastic frontier models have focused solely on spatial spillovers in either the inefficiency term or the noise term. Thus, they have tended to neglect one or the other of these sources of spatial correlation. Orea and Álvarez [97] have recently proposed a new stochastic frontier model that permits separate but simultaneous analyses of the spatial correlations of both noise and inefficiency terms, which are likely to be of a different nature. Their model can be written as: lnYit = β lnXit + v˜it (ρ) − u˜ it (τ )

(21)

where now the noise and inefficiency terms are spatially correlated using spatial moving average (SMA) or spatial autoregressive (SAR) stochastic processes, and the coefficients ρ and τ measure the degrees of spatial correlation between firms’ noise and inefficiency terms, respectively. In order to get a closed form for the

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likelihood function, Orea and Álvarez [97] assumed that the basic inefficiency term uit possesses the scaling property in the sense that the idiosyncratic inefficiency term can be written as a function of exogenous variables times an industryspecific or economy-wide inefficiency term. The above specification implies that the distribution of the inefficiency term is not affected by the spatial transformation. This is the crucial aspect of the model that enables them to get a tractable likelihood function that can be maximized using standard software. The above authors also estimated a portion of the model using nonlinear least squares (NLLS) with no distributional assumptions, except exogeneity of all explanatory variables.

Estimating Efficiency in Spatial Frontier Models4 The presence of the endogenous autoregressive variable in the spatial frontier model requires correcting the individual efficiency estimates, i.e., ξ it = exp (−uit ), that have been obtained using Jondrow et al. [69] or Schmidt and Sickles [113]. Glass et al. [54] and Kutlu [80] have suggested two alternative methods to carry out this adjustment. While Glass et al. [54] use the following technical efficiency measure TEit = (I − λW)−1 ξ t , Kutlu [80] propose estimating the total efficiency as TEit = exp [−(I − λW)−1 ut ]. Note that Kutlu’s efficiency calculation has the global multiplier (I − λW)−1 inside the exponential operator, whereas Glass et al.’s efficiency calculation has the global multiplier outside the exponential operator. This subtle difference has important practical consequences. If we use the global multiplier after the exponential operator, TEit might be larger than unity, which is a necessary condition for total efficiency being well-defined. In order to address this concern, Glass et al. [54] adapt the Schmidt and Sickles [113] method and compute relative efficiencies by normalizing the above (absolute) efficiencies with the most efficient observation. As pointed out by Kutlu [80], this approach is, however, sensitive to the best performance in each period being an outlier. His proposal is in line with the distribution-based methods as he does not carry out any posterior normalization because (I − λW)−1 is always nonnegative as long as 0 ≤ λ < 1.

Spatial TFP Growth Decomposition Interest in the analysis of productivity at regional level has grown considerably in recent years as productivity growth is one of the most important drivers behind regional income. Thus, analyzing how regional productivity evolves over time is essential to provide insights for the promotion of productivity growth in the future. In the recent literature analyzing spillover effects on productivity at regional level, there is a general consensus about the importance of spillover effects. A relevant 4 This

subsection is inspired in Kutlu [80].

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contribution to this research topic using production functions is Baltagi et al. [14]. In addition to capturing spatial correlations in the model using a modified HausmanTaylor approach, this chapter allows for inter-sectoral spillovers that affect firms’ productivity in China’s chemical industry. An estimated frontier production function can constitute the building block for the measurement of TFP growth and its decomposition into its basic sources. The traditional nonspatial TFP growth decomposition (see, e.g., [96]) includes three components: changes in technical efficiency (EC), technical change (TC), and a scale effect (SE) that relies on scale elasticity values and on changes in input quantities, and therefore it vanishes under the assumption of constant returns to scale or constant input quantities. Glass et al. [51] extend the standard TFP growth decomposition to include direct (own) and indirect (spillover) components using a spatial autoregressive production frontier model. In particular, they estimate a fixed-effect SAR spatial panel model using maximum likelihood techniques. Once the model is estimated, they compute time-varying efficiency scores from the cross-sectional specific effects using CSS. They next use the so-called quadratic identity lemma to obtain the following TFP growth decomposition: T F˙ Pit = T E˙ it + ηDir it +

K

Dir SF kit X˙ kit k=1



K I nd I nd ˙ + ηit + SF kit Xkit (22) k=1

where a dot over a variable stands for rate of growth, a line over a variable stands for arithmetic averages in t and t — 1, ηit and eit are output elasticities with respect to time and input levels, SFkit is a scale factor that vanishes under constant returns to scale, and Dir and Ind denote elasticities and scale factors which are calculated using the relevant direct and indirect marginal effects. The first three terms in (22) are respectively direct (own) EC, TC, and SE productivity components that can be found in nonspatial TFP growth decompositions. However, the above direct components differ from the standard nonspatial ones because they contain feedback effects, i.e., effects which pass through other units via the spatial multiplier matrix and back to the unit which initiated the change. The last two components above are indirect components associated respectively to technical change and the size effect, which do not appear in a nonspatial setting. A more compact decomposition can be obtained if we aggregate the direct and indirect components above into two total effects, that is: T F˙ Pit = T E˙ it + ηTit ot +

K k=1

T ot SF kit X˙ kit

(23)

Later on, Glass and Kenjegalieva [50] extended their previous TFP growth decomposition by adding spatial spillovers associated to the change in technical efficiency. Their extension, however, relies on a different spatial stochastic frontier model because they compute firms’ efficiency using the spatial SAR and Durbin

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stochastic (cost) frontiers introduced by Glass et al. [53, 54] that are estimated using (pseudo) maximum likelihood techniques. Once individual efficiency scores are obtained, they estimate the direct, indirect, and total efficiencies using the method outlined in “Estimating Efficiency in Spatial Frontier Models”. That is, direct efficiency for a unit is interpreted in the same way as own efficiency from a nonspatial model but, in contrast, comprises own efficiency plus efficiency feedback. The indirect efficiency is the sum of the efficiency spillovers to a unit from all the other units in the sample. Total efficiency is the sum of its direct and indirect efficiencies. They next extend the growth of these three efficiencies as part of their new spatial decomposition of TFP growth. In addition, they include the growth in direct, indirect, and total allocative efficiency growth components, which can be viewed as an extension of the allocative efficiency growth component introduced by Kumbhakar and Lovell [79] in a nonspatial setting. They propose a four-component spatial TFP growth decomposition: T F˙ Pit = ηTit ot +

K k=1

T ot SF kit X˙ kit + T E˙ itT ot +

 K  s Tkitot − eTkitot p˙ kit (24) k=1

where skT ot is the total input expenditure share weight and pk stands for input prices. The first two components in (23) are the TC and SE productivity components that already appeared separated into direct and indirect effects in eq. (22). The third component captures the impact of a rise or fall in total efficiency, which Glass and Kenjegalieva [50] in turn decomposed into its direct and indirect parts. The last term captures the effect of a change in total allocative efficiency (AE), which again can be decomposed into direct and indirect changes in allocative efficiency.

Final Remarks Spatial spillovers can be defined as the impact of changes to input (explanatory) variables in a unit on the output (dependent) variable values in other units. Units could be firms, cities, regions, and so forth depending on the nature of the study. As Vega and Elhorst [127] point out, a valuable aspect of spatial econometric models is that the magnitude and significance of spatial spillovers can be empirically assessed. To achieve this aim, several spatial specifications have been proposed in the literature that rely on imposing model structure in the form of a spatial weight matrix W, which reduces the number of parameters to be estimated. However, this literature has been criticized due it often lacks theoretical background and it suffers from non-negligible identification problems, because it is generally difficult to distinguish different spatial models from each other without assuming prior knowledge about the true data-generating process, which is often not possessed in practice (see, e.g., [101]; and [31]). The same applies to the weight matrix as the true W is generally unknown. Several papers have tried to address this issue by combining several spatial weight matrices that are often used in the spatial econometric literature to capture spatial spillovers. To achieve this aim, Case et al.

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[22] and Qu and Lee [106] used a known function of geographic and economic distance between units. While Sun [119] does so using nonparametric techniques, the generalized spatial stochastic frontier models introduced by Gude et al. [60] also use a parametric function to estimate a combination of spatial weight matrices. Another identification problem highlighted by Gibbons and Overman [48] and Vega and Elhorst [127] occurs when the unknown parameters of a model cannot be uniquely recovered from their reduced-from specification even if the spatial econometric model and W are correctly specified. Gibbons and Overman [48] propose the use of natural experiments and microeconomic data sets, a solution that often is not possible in standard applications in production economics where it is compulsory use real data. Vega and Elhorst [127] suggest taking the SLX model as point of departure, unless the researcher has an underlying theory or coherent economic argument pointing toward a different model. They show that the SLX specification not only is more flexible in modeling spatial spillover effects than other specifications but also it is the simplest one. Moreover, in contrast to other spatial econometric models, standard instrumental variables (IV) approaches can be used to investigate whether (part of) the input variables and their spatially lagged values are endogenous. The authors of this chapter share this view: applied researchers in production economics are encouraged to find sound economic arguments to first justify the existence of spatial spillovers, and second to select the appropriate spatial specification of their production (cost, profit or distance) functions when spillovers are expected. However, as the economic arguments in production economics are of different nature than in other research fields, the preferred specifications in each field may differ. In this sense, it is worth mentioning that the SAR, SLX, or SDM specifications are the preferred specifications in standard (i.e., non-frontier) spatial econometric settings, because the spatial dependence that is accounted for in the SEM model is often not a representation of substantive economic spillovers. Notice as well that the spatial spillovers in the SAR, SLX, and SDM specifications are treated as determinants of the estimated production/cost function, i.e., as technological drivers. This treatment is, however, more difficult to justify in production analyses using firm-level data. For instance, Orea et al. [98] did not use a frontier-based spatial specification in their application to electricity distribution firms because the Norwegian regulator did not see major systemic technical reasons for the cost of an electricity distribution firm to be affected by those of an adjacent firm to any significant degree. Similarly, Druska and Horrace [33] point out that we do not need a model with spatial correlations in the frontier if the technology is viewed as a purely deterministic (engineering) process where the firm controls all the inputs. Moreover, in a stochastic frontier analysis of firms’ efficiency, we have two random terms. Orea and Álvarez [97] state that controlling for spatial spillovers in both noise and inefficiency terms does matter due to the significant and different economic consequences of such correlations. They argue that while the spatial specification of the noise terms is likely capturing an environmentally induced correlation, the spatial specification of the inefficiency term will likely capture

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a kind of behavioral correlation. Both spatial specifications of the error terms are important, although for different reasons. On one hand, a model specification with spatial correlation in the noise term is useful as it accounts for unobserved but spatially correlated variables that if ignored might result in biased estimates of efficiency scores [98]. Thus, the effect of spatially correlated error terms in a stochastic frontier model is not as benign as in a standard spatial econometric model. On the other hand, a model with spatial correlation in the inefficiency term is useful when firms tend to “keep an eye” on the decisions of other peer firms trying to overcome the limitations caused by the lack of information [91], firms are regulated using benchmarking techniques [97], or they simply emulate each other [9]. As these issues provide interesting information on firms’ performance, the recent spatial stochastic frontier literature is resuscitating the interest of the scientific community, policymakers, and managers in spatial error-based models, which have not been very popular so far. The spatial production models are also useful when some firms benefit from (best practices implemented in) adjacent firms. As Vidoli et al. [128] pointed out, this could especially be the case if local firms belong to communitarian networks (e.g., cooperatives) characterized by a collaborative environment, exchange of technical advice and continuous interaction, or common technicians (consultants) are advising all local firms. In this sense, the literature on spatial production economics summarized in this chapter is highly related to emerging literature on network production functions, where the network structure is endogenous. For instance, Horrace et al. [66] develops a model where worker’s productivity is a function of the productivities of the co-workers on her team or, in our spatial framework, where firm’s production is a function of another firms’ production. Horrace and Jung [65] propose a similar model but in a stochastic frontier framework, where worker-level inefficiency is correlated with a manager’s selection of worker teams. As the endogeneity of the network structure (i.e., the W matrix in a spatial setting) is of primary concern in this literature, various estimation techniques have been recently developed in the econometrics of network literature to address this issue. As Gibbons et al. [49] point out, these methods are likely very helpful in spatial settings in other to deal with the endogeneity of some popular economic-based weigh matrices.

Cross-References  Economics of Externalities: An Overview  Neoclassical Production Economics: An Introduction  Scale Elasticity and Returns to Scale  Stochastic Frontier Analysis: Foundations and Advances I  Stochastic Frontier Analysis: Foundations and Advances II Acknowledgments This chapter was supported by the Spanish Ministry of Economics, Industry and Competitiveness (Grant MINECO-18-ECO2017-85788-R). The authors would like to thank the two “Salvador de Madariaga” grants obtained from the Spanish Ministry of Science, Innovation

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and Universities (Grants PRX19/00596 and PRX19/00589). We also wish to acknowledge helpful suggestions from an anonymous reviewer.

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Technical Efficiency and Its Determinants in the Manufacturing Sector: What We Know and What We Should Know

36

Sumon Kumar Bhaumik

Contents Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Estimating TE/Efficiency . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Factors Affecting TE/Efficiency . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . External Factors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Internal Factors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Discussion and Concluding Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Cross-References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Appendix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Abstract

There is a larges literature on technical efficiency (TE) of manufacturing sector firms in developing country contexts, much less so for developing country contexts. Most of empirical research use either data envelopment analysis (DEA) or stochastic frontier analysis (SFA) to estimate TE. Available evidence suggests that TE of firms is low in developing countries, and there is considerable interfirm heterogeneity in TE. Firm-level TE is correlated with (or influenced by) external factors such as liberalization of trade and foreign direct investment (FDI) and overall economic liberalization, as also by internal factors such as firm size, ownership, whether or not they operate in the formal sector, and use and adoption of ICT. However, there is mixed evidence about the extent to which

S. K. Bhaumik () Sheffield University Management School, University of Sheffield, Sheffield, UK IZA – Institute of Labor Economics, Bonn, Germany Global Labor Organization, Geneva, Switzerland e-mail: [email protected] © Springer Nature Singapore Pte Ltd. 2022 S. C. Ray et al. (eds.), Handbook of Production Economics, https://doi.org/10.1007/978-981-10-3455-8_36

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liberal economic policies and firm characteristics such as size and ownership are associated with higher TE. Future research should focus on the impact of the institutional environment in which firms operate and their managerial capability on firm-level TE. Keywords

Technical efficiency · Manufacturing sector · Data envelopment analysis · Stochastic frontier analysis · Developing countries

Introduction Broadly speaking, a firm can increase its productivity in two ways. If there is slack in the production process such that factor inputs are not used efficiently, then it can work on increasing its efficiency, i.e., generate more output for a given combination of inputs or use less inputs to generate the same amount of output. If it is already fairly efficient in the use of inputs, however, productivity increase may require (disembodied or embodied)1 technical progress.2 Traditionally, much of the discussion about productivity in the context of the manufacturing sector in the developed economies (DEs) has been about technical progress, or about changes in total factor productivity (TFP) and labor productivity3 . The genesis of it possibly lies in the (implicit) assumptions that (a) DEs are contexts where there are functioning markets for factor inputs and (b) profit maximizing firms operating in these contexts would necessarily use an efficient input bundle to produce the optimum (or profitmaximizing) output level, such that technical efficiency would be less of an issue. Both these assumptions are brought into question in the context of less developed economies (LDEs), of which emerging market economies (EMEs) are a muchdiscussed subset. To begin with, these contexts are characterized by missing factor markets, such that it may be difficult for firms (and their managers) to choose the optimal input bundles. In some cases, this is on account of paucity of appropriate factor inputs – chronic shortage of capital and skilled labor in LDEs has been much discussed in the development economics literature – and, to the extent that these factors are available, market failure may occur on account of high information and transactions costs in these factor markets.4 At the same time, firms in LDEs are

1 See

Intriligator [1] and Jorgenson [2] for a discussion about disembodied and embodied technical progress (change). 2 In principle, one can also talk about scale economies contributing to productivity growth. Coelli et al. [3], for example, decompose total factor productivity (TFP) into three components, namely, efficiency change, technical progress, and scale change. However, in the literature, much of the focus is on technical efficiency and technical progress, and much less on scale economies. 3 Some papers that focus on TE (or change there of) in DEs include [72]-[75]. 4 The challenge posed by informational and transactions cost is perhaps most evident in the market for financial capital that is necessary to acquire physical capital, especially in contexts where

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characterized by weak management practices that are arguably correlated with state and founder–family ownership of firms in these contexts [4]. Alternatively, these contexts are characterized by missing markets for managerial capital [5], whether on account of paucity of such capital or on account of the lack of a functioning market for managerial talent. In addition, developing countries are often characterized by traditional industries such as textiles and food processing, and a large proportion of firms in these contexts are small and far removed from the global technological frontier. For these industries and firms, it is often meaningful to talk about the efficiency of the production process, i.e., the efficiency with which inputs are converted into output, rather than technical progress. That is not to say that technical progress is inconceivable in such contexts, e.g., a move from handloom-based production to machine-based production in the textiles industries can lead to technical progress because of the technology embodied in the machines (or physical capital). But, by and large, the discussion about efficiency is much more relevant in the context of LDEs than in the context of DEs, and this is reflected in the large literature on efficiency – often, specifically, technical efficiency (TE) – of firms in the literature on firm and industry performance in LDEs.5 Much of this literature concludes that the average firm in most LDE industries has average TE of about 0.6–0.7, and that the dispersion of TE among firms can be quite large, e.g., 0.3–0.8. The relatively low average TE levels are consistent with the aforementioned discussions about factor markets and managerial abilities in LDE contexts. The large dispersion in TE levels, on the other hand, confirms the absence of competitive markets, arguably on account of weak institutions, and other factors such as family and state ownership of firms that rule out exit of inefficient firms over time. More importantly, this literature identifies factors that are correlated with – correlation is easier to establish than causality – high (and hence also low) levels of TE.6 Broadly speaking, the literature identifies two sets of factors that are correlated with TE/efficiency levels, namely, external factors such as the business environment

internal accrual of firms is insufficient for acquisition of physical capital. Factors such as weak property rights, absence of rule of law, and weak enforcement of contracts make it difficult for creditors to overcome the twin problems of adverse selection and moral hazard. At the same time, investment in equity/shares of firms may be hindered by weak corporate governance that is often associated with high levels of ownership concentration and, consequently, entrenchment of incumbent managers. 5 There is, of course, a literature on TE in the context of DEs as well (e.g., [6]), but the corresponding literature in the context of LDEs is arguably much larger. 6 Some of the relevant empirical papers estimate TE from data envelopment analysis (DEA) or from the econometric estimation of a Cobb–Douglas or a translog production function – we call this stochastic function analysis (SFA) – and, in the second stage of the analysis, regress TE on its potential correlates. Some other papers simultaneously estimate a production function and the relationship between firm characteristics and external conditions such as the business environment on firm-level inefficiency. Hence, while there is, strictly speaking, a difference between firm-level efficiency and TE, we shall use these terms interchangeably.

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(and changes thereof, by way of policy changes and reforms) and internal factors such as the size and ownership of companies.7 In the rest of this chapter, our focus would be on these correlates of TE/efficiency. We start with a discussion about the ways in which TE/efficiency is estimated in the literature. Next, we discuss the empirical evidence reported in the literature and discuss the mechanisms through which the aforementioned external and internal factors affect efficiency. Finally, we reflect on the way forward for the TE literature for the manufacturing sector.

Estimating TE/Efficiency In the early literature on TE/efficiency in the manufacturing sector, inferences about the efficiency of a certain type of firm relative to others was made on the basis of simple regression models. Consider, for example, Tyler [7], who attempts to shed light on whether, among other things, TE of foreign-owned or controlled firms is different, on average, than that of domestic firms. His empirical strategy involves estimating the extended Cobb–Douglas production function ln Xi = ln A + γ DF i + α ln Ki + a1 (DF i . ln Ki ) + β ln Li + a2 (DF i . ln Li ) (1) where γ captures the TE of foreign firms relative to domestic firms and where a1 and a2 capture the differences in “the respective output elasticities for foreign firms in relation to domestic firms” (pp. 369). The model is estimated using ordinary least squares (OLS), even though Tyler acknowledges the limitations of the OLS estimator. Later in the paper, he tests for the adequacy of the Cobb–Douglas specification – the alternative being the CES production function – and concludes that “the Cobb-Douglas is an adequate model for explaining the production behavior of foreign firms but not for domestic firms” (pp. 372). By the 1980s, the empirical modeling of TE had become more sophisticated. For example, Page Jr [8] considers a translog production function ln Xˆ s = αˆ 0 +

 m

αˆ m ln Zm (s) +

1  βˆmn ln Zm (s) ln Zn (s) m n 2

(2)

where Xˆ s is an index of maximum potential output for firm s (s = {1, 2, . . . ., S}), Zm (s) are indices of input levels, and m, n = {1, 2, . . . ., N}. The model parameters are estimated using the optimization program

7 It

can be argued that internal factors such as (family) ownership can themselves be determined by external factors such as institutions, but we shall abstract that from discussion for the purposes of this chapter.

36 Technical Efficiency and Its Determinants in the Manufacturing . . .

min

S s=1

 αˆ 0 +

 m

αˆ m ln Zm (s) +

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  1  βˆmn ln Zm (s) ln Zn (s) − ln Xˆ s m n 2 (3)

and constraints are imposed to ensure that the production function is concave and monotonic and that, for each firm s, the observed input–output combinations lie on or below the frontier. The parameters of the model are obtained using linear programming. It is easy to see that Xs = Xˆ s only if a firm operates on the production frontier. Page Jr., therefore, is able to generate a firm-specific efficiency (or Farrell) index [9] which is given by Xs /Xˆ s ≤ 1 and “the value of the index provides a measure of relative technical efficiency” (pp. 133). If the production process follows constant returns to scale, the Farrell index also gives us the percentage by which the firm’s unit cost can be reduced, given the current output level.8 Since the 1990s, the empirical literature on TE/efficiency has been dominated by (nonparametric) DEA models and (parametric) SFA models. The DEA approach involves using linear programming to solve a constrained optimization problem (e.g., [11]), and this optimization problem is given by uy0 maxu,v g0 = m i=1 vi xi0

(4)

subject to uyj m ≤ 1, i=1 vi xij m vi xij = 1 i=1

ui , vi > 0

(5) (6) (7)

where y0 , x0 , and g0 are output, input(s), and technical efficiency of firm 0, whose TE is being measured, j = {1, 2, . . . ., n} is an indicator for firms, and i is an indicator for the inputs. The first constraint ensures that a firm is either on or below the efficient frontier, the second constraint ensures that there is an unique combination of u and v for which the objective function is maximized, and the third constraint ensures that any increase in input will necessarily increase output. Further, the assumption of constant returns to scale that is implicit in the optimization program can be easily

8 Variations

of this approach has been used in other papers, without the linear programming element. For example, Blomstrom [10] estimates an industry-level efficiency index as follows: “First the efficiency frontier is obtained by choosing the size class within each four-digit industry showing the highest value-added per employee. Value added per employee in this size class is denoted y+ . Them the industry average (denoted y) is calculated as the ratio of total value-added in each industry to the total number of employees. The efficiency index, e, for each industry i, [is then defined] as ei = yi+ /y i .” (pp. 102)

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relaxed. The solution to this program results in a TE score of between zero and one for each firm. An example of the SFA approach, on the other hand, is Sena [12] who estimates a translog production function 1 1 ln y it = αk ln xkit + αkl ln xkit ln xlit + αl ln xlit + γ t + γ t 2 2 2 + μk ln xkit t

+ μl ln xlit t

+

(8)

it

where y is output, k and l are capital and labor, respectively, t is a time trend, i = {1, 2, . . . ., I} denotes the number of firms, and t = {1, 2, . . . ., T} is the number of time periods. The error term  it can be decomposed into a one-sided (in)efficiency term uit and an iid noise term vit ,  it = uit − vit . In this setup, TE is given by TE = exp (−uit ). Papers that examine the determinants of TE, after estimating firm-year level estimates of TE using the DEA or the SFA approach, regress the estimated firmyear estimates of TE on the relevant firm characteristics and characteristics of the business environment within which these firms operate. An example of this approach is Chirwa’s [13] examination of the relationship between TE and the following variables: state–ownership in a firm (state), dummy variable indicator of privatization (priv), the Herfindahl–Hirschman index of market concentration (hhi), import competition (imps), capital-intensity of technology (kint), dummy variable indicator of multinationality (mnc), dummy variable indicator of structural reform programs (saps), and a dummy variable indicator of post-privatization enterprises (psepriv). The regression model is given by     TEj t = α0 + α1 statej t + α2 priv it + α3 (hhi it ) + α4 imps it + α5 kint j t    + α6 mncj t + α7 saps t + α8 psepriv t + εj t (9) where i is an indicator of the firm, j is an indicator of the industry, and t is the time period. As such, such models can be estimated using fixed or – if there are time invariant variables of interest (e.g., ownership status) – random effects models. Some recent papers, however, also correct for potential endogeneity by using variations of the generalized method of moments (GMM) estimator (e.g., [14]). Others have used the Tobit estimator [15]. A variation of this empirical strategy is sometimes used by researchers who adopt the SFA approach to estimate TE. It involves an estimation of the production function yit = f (Xit ) + vit − uit

(10)

 where y is the output, X is a vector of inputs, vit ∼ N 0, σv2 iid error term, and  u ∼ N μit , σu2 nonnegative inefficiency term is to characterize the inefficiency term as follows:

36 Technical Efficiency and Its Determinants in the Manufacturing . . .

μit = δ0 + δ1 Z1it + δ2 Z2it + · · · + δn Znit

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(11)

The production function and the inefficiency equation can then be jointly estimated to explore the relationship between inefficiency and the Z variables that can include firm-level characteristics such as size (e.g., [16]) as well as environmental variables such as institutional quality [17]. While a number of methodologies have been used in the literature to explore TE in the manufacturing sector – some of which have been described above –the current literature almost exclusively used DEA or SFA to estimate TE, and thereafter uses a second-stage regression model to explore the relationship between TE and its potential determinants. In other words, there is a large degree of agreement about the methodological approach to estimating TE and examining its determinants and the debate, therefore, is about the conceptual issues and the evidence about the aforementioned determinants. We discuss these issues and the evidence in the next section.

Factors Affecting TE/Efficiency In light of the discussion earlier in this paper, it is not surprising that much of the literature on TE/efficiency is in the context of LDEs and EMEs. Much of this literature focuses on the relationship between specific aspects of an economy (e.g., openness to trade and foreign direct investment (FDI)) and specific firm characteristics (e.g., ownership and formal/informal status of companies) on efficiency. Some of the literature, however, focuses on “reforms” which can include a number of different things. For example, in the case of India, reforms during the 1980s and 1990s included elimination of the requirement to obtain government permission to start businesses (the so-called license raj), liberalization of both current and capital accounts of the balance of payments, reform of credit and capital markets, and part privatization of some state-owned financial and nonfinancial enterprises. Hence, it is difficult to attribute the mixed evidence about whether or not broad-based “reforms” contributed to TE improvement – Ray [18] and Din et al. [19] find that reforms did improve TE in India and Pakistan, respectively, while Bhaumik and Kumbhakar [20] did not find evidence of a significant change in TE in India9 – to specific aspects of these reforms. However, this research highlights interesting caveats that merit deeper exploration. For example, Mukherjee and Ray [21] find that economic reforms in India did not change the relative efficiency ranking of the states and that there is no evidence of convergence in the distribution of (state-level) efficiency subsequent 9 Specifically, while Ray [18] finds that economic reforms in India, whose center piece was the elimination of licensing requirements to facilitate entry, led to greater productivity growth largely on account of TE improvement, Bhaumik and Kumbhakar [20] find that median TE of all but one industry (examined in the paper) declined between 1989–90 and 2000–01 and that change in TE explains a very small proportion of the change in gross value added.

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to the reforms. This raises questions about the role of initial conditions and path dependence in determining the impact of reforms on TE/efficiency. Similarly, Chirwa [22] finds that, in Malawi, structural changes improved TE/efficiency of only those sectors which were inefficient before the reforms. As such, this is not surprising; inefficient sectors do have the greater scope to improve. However, it is certainly something about which policymakers should be mindful. Next, we discuss the literature on the relationship between specific aspects of the business environment (e.g., infrastructure quality), government policies (e.g., trade liberalization), and firm characteristics (e.g., ownership) on TE/efficiency (and its change). We first take a look at factors that are external to a firm, and then at firm characteristics themselves. Note that we use the word “impact” – which implies causality – somewhat loosely. In much of this literature, it is easier to establish correlation than causality.

External Factors Both cross-country and single-country studies have confirmed that factors such as infrastructure quality, availability of factor inputs, competition, and government regulations/policies are correlated with TE/efficiency levels of firms and industries. Some of these relationships are easy to predict, e.g., TE improves with infrastructural quality and availability of factor inputs. Mitra et al. [23] find that the infrastructure elasticity of TE in the context of the Indian manufacturing sector is 0.12 on average, and much higher for some industries. However, they also find that some elements of infrastructure such as power shortages matter much more than other elements such as transportation quality. Similarly, Hailu and Tanaka [24] find that TE in Ethiopia, which is low on average and with a high level of dispersion, is adversely affected by shortage in supply of raw materials. Wang and Wong [25] argue that, in addition to having a direct effect on TE, in some contexts, factor inputs such as skilled labor may have a moderating effect on the relationship between TE and its correlates such as trade and foreign direct investment (FDI). From a conceptual perspective, TE should also be affected by competition – competition is expected to induce profit-maximizing firms to become more efficient, irrespective of whether the source of the competition is domestic or international. In one of the few studies that directly link market structure with TE, Setiawan et al. [26] finds that industry concentration is negatively correlated with sector-level TE. Evidence linking the drivers of market concentration, namely, entry and exit of firms, with TE is found, interestingly, in a low productivity region of a developed country. Harris [27] finds that the average level of TE is lower in Northern Ireland than in the other parts of the UK, in part because there is little churning to facilitate replacement of inefficient firms (which exit) with more efficient firms (that enter) and, in part because the new firms that enter are not much more efficient than the

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incumbent firms. Interestingly, however, there is little discussion in the literature about how institutions that underpin entry, exit, and market concentration are related to TE.10 The literature on the impact of trade and FDI on TE is much more extensive but it is difficult to disentangle the competitive effects of trade and FDI from factors such as learning-by-doing and spillover effects of FDI. In addition, this literature has had to grapple with the possibility that higher productivity and TE of exporting firms and those with foreign ownership may reflect self-selection, whereby more productive firms are more likely to export and attract overseas investment, rather than any causal impact of export and overseas investment on TE and productivity. Evidence reported in the literature suggests that TE is positively associated with export orientation [28, 29], but that efficiency gains from exporting are largest for firms that are new to exporting, on account of both learning-byexporting and self-selection of the most efficient firms into the exporting cohort [30]. Correspondingly, TE (and productivity) are positively associated with trade liberalization [31–33], with TE gains for both import-substituting and exportpromoting industries [34], and negatively associated with high level of protection [32, 35]. Similarly, available evidence suggests that TEs have positive association with foreign ownership and outward FDI activities (i.e., overseas investment) of firms [36]. However, the benefits of sector-level inward FDI may only be significant for industries that are regionally concentrated [37], suggesting that aspects of economic geography may moderate the relationship between FDI and TE. Neither trade liberalization nor FDI may, however, be a panacea for TE improvement. Trade liberalization, for example, may reduce efficiency of domestic firms, on average, if they are not flexible enough to adjust to this change in their business environment [38]. Similarly, while FDI may improve the TE of local firms who are part of the multinational enterprises’ (MNE) supply chain, there may be a decline in the TE of the domestic competitors of these foreign firms (and their suppliers) [39]. More generally, the reaction of domestic firms to trade liberalization and FDI may depend on whether or not they (or the sectors to which they belong) are close to the global technological frontier [40],11 such that there can be considerable variation in interindustry and intra-industry impact of trade/openness and FDI on TE.

10 Indeed,

in perhaps the only notable study that examines the relationship between formal institutions and efficiency [17], the focus is on labor market institutions that directly matter more for factors such as motivations of workers than for market concentration. 11 The broad intuition is that firms that are close to the global technological frontier would be induced by overseas competition to become more efficient and/or invest in better technology, while firms that are far from this frontier are unlikely to be willing to make these adjustments to their production process.

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Internal Factors In the literature, the single most robust relationship between a firm attribute and TE involves firm size; in nearly all relevant studies, larger firms have been associated with higher levels of TE (see Appendix). This is as true in contexts in which micro and small firms are not intrinsically inefficient but have lower TE levels than larger firms [41], as in contexts where the average TE is low but larger firms are relatively more efficient [42]. In an interesting deviation from this stylized observation about the relationship between firm size and efficiency, Patibandla [43] finds that medium-sized Indian firms are more efficient than both small and large firms. This is consistent with an earlier study in the Indian context [8], which found that firm size was positively associated with TE in only one of the four industries that were examined in the study. Patibandla argues that this is on account of low levels of organizational efficiency in larger firms, which is consistent with more recent evidence that suggests that larger firms in developing countries are badly managed, and that the challenges associated with poor management practices may be particularly acute in organizational forms such as family firms. The importance of managerial capability is also highlighted in the research of Page Jr [44] who finds that TE in Ghana is positively associated with education and industry experience of managers, and also with the ratio of expatriate managers to total managers. Available evidence also provides links between TE and factors such as investment in research and development (R&D) and information and communication technology (ICT) that are correlated with a firms organizational structure strategic organization (e.g., Bayo-Moriones and Lera-Lopez [78]). Both R&D investment [45, 46] and ICT investment [47, 48] are positively associated with TE. Similarly, while deterioration in TE over time may sometimes be attributable to poor investment planning and implementation [49], higher TE is associated with firms that implement state-of-the-art management practices such as just-in-time purchasing of materials and components that are used in the production process [50]. Management, however, is a complex process that is difficult to view through the prism of single factors such as R&D investment and supply chain management. Good management requires ensuring that investment in any one firm-level resource or capability is accompanied by investment in complementary resources that are necessary to exploit the former. For example, Mahadevan [51] finds that TE may be negatively associated with capital intensity, on account of unavailability of workers with commensurate skills, implying that investment in capital without commensurate investment in training may not lead to an improvement in TE.12

12 Indeed, the capital intensity of firms itself might not be optimal. If larger firms pay more for labor

than their smaller counterparts, for example, their use of capital may be more than what would have been optimal if they paid the same per unit of labor as the smaller firms [14].

36 Technical Efficiency and Its Determinants in the Manufacturing . . .

1421

The relationship between ownership and TE is also not clear-cut. We have already noted the challenges associated with organizational forms such as family firms. Available empirical evidence suggests that, in some contexts, TE is positively associated with foreign ownership and negatively associated with state ownership [15].13 Correspondingly, privatization has been associated with an increase in industry-level TE is some developing economies [13], while access to government subsidies, which is generally much more relevant for state-owned firms than for private and foreign firms, has been found to be positively associated with inefficiency [52]. Evidence also suggests that increased concentration of ownership in the hands of the largest shareholder and inclusion in (pyramidal) business groups, which reduce Type I or principal–agent agency problems (albeit replacing it with Type II or principal–principal agency problems), are also associated with higher (or increase in) TE [53]. However, the difference in efficiency between state-owned and private/foreign firms is not always significant [7]. Further, in some contexts, the positive impact of certain types of ownership, e.g., that of foreign ownership in Northern Ireland [54], decreases over time.14 In LDEs, the (in)formal status of firms is strongly correlated with their efficiency levels, largely on account of the unfavorable business environment within which these firms operate [55]. Specifically, TE of informal sector firms can be low on account of factor market imperfections experienced by these firms [56], and transactions costs for these firms can be particularly high in the credit market. In the specific context of the capital market, however, the evidence is mixed. Evidence suggests that while firms that have access to credit from banks and other sources such as clients and suppliers have higher TE than those that are reliant on friend, family, and their own financial resources [57] – TE is highest for firms that have access to bank credit, the incentive to increase TE may be higher for firms that experience financial constraints [12]. This literature, however, does not take into consideration the possibility that operating in the formal sector may itself be a strategic decision of the firm which trades of the benefits of easier access to credit and other factor inputs with the higher direct costs of doing business in the formal sector.

13 This

is consistent with a much wider literature on the relationship between private and foreign ownership of firms and their performance. 14 In this context, one may draw on the international business literature which posits that foreign firms may have “ownership advantages” in the form of better technology, managerial skills, and even country-specific factors such as greater access to key resources, but that these firms also face challenges when they operate overseas. In the context of LDEs and EMEs, the main challenge may be the high transactions cost of operation in weak institutional contexts.

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Discussion and Concluding Remarks Since the notion of TE/efficiency follows from the microeconomic theory about production, it is not surprising that much of the empirical literature about it is concerned about issues such as firm characteristics such as ownership, factor market imperfections, and competition. Concerns about the impact of ownership and market structures on firm performance are a natural extension of the neoclassical theory of the firm. The essential argument is that private ownership and competition, whether from domestic or foreign sources, give a firm the incentive to become efficient, among others, in its use of factor inputs. This process of efficiency gain may be aided by increased x-efficiency, which itself may be a consequence of greater competition, and access to well-functioning factor markets. More recent extensions to this literature involve focus on factors such as training (in addition to x-efficiency) and agency problems (in addition to public versus private ownership). While the empirical evidence is by no means unanimous, these core propositions of the drivers of TE/efficiency find a fair amount support in the literature; the caveats are discussed above and further highlighted in the Appendix. The question, therefore, is where the literature on TE (in the context of the manufacturing sector) can go from there. Further estimations, using data from different countries and time periods, and for different industries, can doubtless be undertaken, using ever more sophisticated SFA estimators [76], for example, but it is unclear as to whether these new estimates of TE would provide us any new insights unless we extend the literature in one of two directions. First, building on the work of not only Bloom et al. [4] but also a large management literature, future research should consider examining the role of managerial capability in influencing TE/efficiency (and its increase). In LDE and EME contexts, a specific aspect of management capability might be political connections of key managers such as the chief executive officer (CEO) [58]. Second, building on the work of Bhaumik and Dimova [17], future research may want to further explore how formal and informal institutions affect TE. In other words, future research should perhaps unpack the (neoclassical) firm further, on the one hand, and be more mindful about the role of institutions, which are more structural in nature than policies and regulations, in influencing the choices and decisions of the different stakeholders of the firm, the aggregation of which is captured by performance measures such as TE.

Cross-References  Data Envelopment Analysis: A Nonparametric Method of Production Analysis  Modeling Technical Change: Theory and Practice  Neoclassical Production Economics: An Introduction  Stochastic Frontier Analysis: Foundations and Advances I  Stochastic Frontier Analysis: Foundations and Advances II

36 Technical Efficiency and Its Determinants in the Manufacturing . . .

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Appendix

Country Business environment Cross-country

Method Reference

Empirical finding

SFA

[59]

Cross-country

SFA

[79]

Cross-country

SFA

[60]

Cross-country

SFA

[25]

Ethiopia

SFA

[24]

Efficiency distribution for most sectors unaffected by explanatory variables; with weak positive association between high efficiency levels and the expansionary phase of the business cycle. Considerable intercountry and intra-country dispersion in efficiency levels, and inefficiency is mostly associated with managerial experience, infrastructural quality, and competition. Inefficiency is lower in countries that have more sophisticated production processes and higher capacity for innovation. TE improvement is positively associated with infrastructure, political stability, and urbanization. Human capital matters as well; benefits from trade and FDI-driven technology transfer increases a country’s TE only after human capital level reaches a minimum threshold. TE is low and with high dispersion level. The main reason for low efficiency levels across sectors is shortage in supply of raw materials but low infrastructure quality and unfavorable government rules and regulations play a role as well. Infrastructure elasticity for TE is 0.12, on average, and higher for some industries. However, some elements of infrastructure such as power shortage matter much more than others such as transportation quality.

India

Competition Indonesia Northern Ireland

Economic policies and reforms Bangladesh

[23] and [81]

DEA

[26]

SFA

[27]

[34]

Industry concentration is negatively associated with sector-level TE. Average levels of TE is lower in Northern Ireland than in other parts of the UK, in part because there is little churning to facilitate replacement of inefficient firms (which exit) with more efficient firms (that entry). Where entry does take place, the entering firms are not much more efficient than the incumbents.

There was an increase in the overall TE of most industries over time, for both export-promoting and import-substituting industries. (continued)

1424 Country Chile

S. K. Bhaumik Method Reference [32]

China

[33]

Hungary

SFA

[52]

India

DEA

[18]

India

DEA

[21]

India

SFA

[20]

Malawi

SFA

[22]

Malawi

SFA

[13]

Pakistan

SFA & DEA

[19]

Firm size and formality Chile

[41]

Empirical finding There is little evidence of productivity improvement on account of trade liberalization but a greater reduction in protection levels is associated with a larger improvement in the average efficiency level. Average TE declined in the mid-1980s but has increased since 1992. There is considerable regional variation in TE and inefficiency is impacted by a variety of reforms related to privatization and trade and FDI liberalization, and infrastructure development. Inefficiency is higher for firms that are recipient of government subsidy and inefficiency also increases with the level of inefficiency. Annual rate of productivity growth is higher in the post-reform period, in part, because of improvement in TE. Economic reforms did not change the relative efficiency rankings of the states, and there is no evidence of convergence in the distribution of efficiency subsequent to the reforms. Median TE of all but one industry declined between 1989–90 and 2000–01 and that change in TE explains a very small proportion of the change in gross value added. Structural changes did not significantly affect the TE of sectors that were relatively efficient before the reforms, but reforms did have a positive impact on firm efficiency in the sector that was inefficient before the reforms. Privatization increased industry-level TE but the direct impact of privatization on firm-level TE was negative; efficiency of firms declined following privatization. TE of most industries increased over time and can possibly be attributed to the reforms that were initiated in the late 1980s, aimed at increasing competition and improving the business environment.

Micro and small firms are not intrinsically inefficient; there are considerable variations across industries. On average, however, medium firms have higher efficiency levels than micro and small firms. Inter-firm differences in efficiency are explained by factors such as capital intensity. (continued)

36 Technical Efficiency and Its Determinants in the Manufacturing . . . Country Cote d’Ivoire

Method Reference SFA [56]

Cote d’Ivoire

SFA

[42]

Cote d’Ivoire

SFA

[55]

Ghana

SFA

[14]

India

SFA

[8]

India

SFA

[43]

India

SFA

Raj SN (2011)

Kenya Korea

SFA SFA

[16] [61]

Philippines Firm capability China

SFA

[82]

China

DEA

[62]

Ghana

DEA

[44]

India

SFA

[80]

[46]

1425

Empirical finding TE is low on account of input market imperfections, and larger firms, those operating in the formal sector and those that are part of international networks, are better able to overcome the challenges posed by these imperfections. TE is low, on average, but larger firms are more efficient and the lower TE of smaller firms can perhaps be attributed, at least in part, to their informality. TE is lower for informal firms than for formal sector firms, mainly on account of the unfavorable business environment within which the former operate. Large firms pay more for labor and hence use more capital than would be optimal if they paid the same for labor as the smaller firms. Average TE is higher and dispersion of TE is lower in the more modern industries, but firm size is positively associated with TE in only one of the four industries that were examined. TE is low among both small and large firms, relative to the medium-sized firms. The inefficiency of large firms can be attributed to low levels of organizational efficiency. TE is low among informal sector firms and this can largely be attributed to frictions (or high transactions costs) they experience in the credit and labor markets. TE is positively associated with firm size. TE is positively associated with firm size in every sector examined in the paper. TE is positively associated with firm size. TE is positively related to firm capability as measured by capital intensity and R&D. TE is low for SOEs and productivity growth is driven largely by technical progress. Best practice SOEs are significantly different from average SOEs in terms of technology, human capital quality, and managerial capacity. Average TE level is high, and TE is positively associated with education and industry experience of managers. The ratio of expatriate managers to total managers is positively associated with TE as well. TE is higher for firms that undertake R&D and those that collaborate with foreign partners. (continued)

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S. K. Bhaumik

Country Italy

Method Reference SFA [12]

Italy

DEA

[53]

Italy

SFA

[47]

Mexico

SFA

[57]

Singapore

SFA

[51]

Taiwan

SFA

[45]

Tunisia

SFA

[48]

USA

SFA

[50]

Yugoslavia

[49]

Multiple factors China

[63]

India

SFA

[64]

Indonesia

SFA

[65]

Indonesia

SFA

[28]

Empirical finding Firms that experience financial constraints (i.e., difficulty in accessing external capital) have an incentive to increase their TE over time. TE increases with percentage of shares owned by the largest shareholder and is higher for firms that belong to a pyramidal (business) group. ICT investment by firms is negatively associated with their inefficiency levels. Firms that have access to credit from banks, moneylenders, clients, and suppliers have higher TE than firms that are reliant on family, friends, and their own financial resources. TE is highest for firms that have access to bank credit. TE is negatively associated with capital intensity. One plausible explanation is that increase in capital intensity has not been accompanied by availability of workers with commensurate skills. Efficiency is positively associated with firm-level investment in training and R&D. TE is low, on average, for Tunisian firms but intensive use of ICT can increase efficiency levels by around 5%. TE is higher for firms whose management is committed to implementing just-in-time purchasing. TE deteriorates over time, quite possibly on account of poor investment planning and implementation which, in turn, points at poor management and coordination of intermediate input supply.

On average, TE is lower for SOEs than for privately owned companies, and highest for foreign firms. TE varies positively with firm size and newness of fixed capital assets. Efficiency is higher for consumer goods industries than for capital goods and intermediate goods industries. Efficiency is positively associated with factors such as skill and profit, but negatively associated with capital intensity. Efficiency is higher for younger firms than for older firms, for larger firms than for smaller firms, and for domestic firms than for foreign firms. There is considerable inter-firm variation in TE, and TE is positively associated with export orientation and financial integration of firms, as well as on female participation in the workforce. (continued)

36 Technical Efficiency and Its Determinants in the Manufacturing . . . Country Indonesia

Method Reference SFA [66]

Nepal

SFA

[35]

Pakistan

DEA

[11]

Thailand

SFA

[67]

Thailand

SFA

[68]

Turkey

SFA

[69]

Turkey

SFA

[70]

UK

SFA

[6]

Vietnam

SFA

[83]

Ownership and organizational form Brazil

[7]

1427

Empirical finding Larger firms are more efficient than smaller firms, and private firms are, by and large, more efficient than the public sector firms. The Asian crisis had a negative impact on the growth rate of TE across all sectors examined in the analysis. Efficiency is positively associated with firm size and negatively with capital intensity. It is also adversely affected by high levels of protection. Efficiency is higher for newer firms, those that are managed by entrepreneurs with at least primary education, and those that are involved in subcontracting. TE varies inversely with firm size and is generally higher for firms in urban areas. State ownership is associated with declining TE over time, perhaps because the more efficient firms are privatized. TE is influenced by a number of factors such as firm size, firm age, access to skilled labor, ownership, and location. TE is higher for larger firms and those located in the metropolitan areas or their hinterlands. On average, TE is also higher for private enterprises than for public sector enterprises, but this is mostly relevant for the post-1982 period. There are considerable inter-sectoral differences in efficiency, and efficiency is influenced by a number of factors such as legal status of firms, firm size, forms of contracting, and location. Market concentration has a curvilinear relationship with efficiency. Higher export and import intensity are associated with higher spreads of efficiency within industries. Capital intensity is negatively associated with efficiency, possibly because large sunk costs make it difficult for firms to alter their behavior as demand etc. change. TE is similar, on average, for state- and private-owned domestic firms, but is lower for foreign-invested sectors. TE also increases with greater compliance with the labor code, and export orientation of firms.

TE of foreign firms not significantly different from that of domestic firms but the former enjoys greater returns to scale and has greater elasticities of substitution. (continued)

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S. K. Bhaumik

Country China

Method Reference [15]

India

DEA

[71]

Northern Ireland

SFA

[54]

Trade and FDI China

Cross-country

[29]

SFA

India

[30]

[31]

Indonesia

SFA & DEA

[39]

Spain

SFA

[38]

Taiwan

SFA

[36]

UK

SFA

[37]

Empirical finding Within the public sector, TE is highest for relatively large TVEs and lowest for SOEs; COEs are less efficient than TVEs but more efficiency than SOEs. Cooperative firms are more efficient than their counterparts but the result is influenced by the choice of the sample. TE of Northern Irish firms increased over time. The positive impact of foreign ownership on TE, however, decreased over time, and the largest increases in TE were observed among domestically Northern Irish firms. TE is positively associated with export orientation and FDI intensity of industries, even though it is also associated by factors such as firm size and capital intensity. There is also regional variation in TE. Efficiency gains from exporting are large and are largest for new entrants to exporting. This is driven by both learning by exporting and self-selection of the most efficient firms into exporting. Trade liberalization was positively associated with productivity in four out of the six industries examined. Foreign firms are more efficient than domestic firms, and the former may increase the inefficiency of the latter. In particular, FDI may have negative impact on TE changes of domestic competitors and positive impact on TE changes of domestic suppliers. Trade liberalization was negatively associated with efficiency, in part, on account of the lack of flexibility on the part of the firms to adjust to the resultant change in the environment in which they operate. TEs of manufacturing firms are positively associated with their OFDI activities. TE is positively associated with the extent of foreign ownership in the domestic industry, but the spillover effect is only significant for industries of above-average regional concentration.

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78. Bayo-Moriones A, Lera-Lopez F (2007) A firm-level analysis of determinants of ICT adoption in Spain, Technovation, 27(6–7):352–366 79. Chaffai M, Kinda T, Plane P (2012) Textile manufacturing in eight developing countries: Does business environment matter for firm technical efficiency, Journal of Development Studies, 48(10):1470–1488 80. Kalirajan K, Bhide S (2004) The post-reform performance of the manufacturing sector in India, Asian Economic Papers, 3(2):126–157 81. Mitra A, Sharma C, Vegazones-Varoudakis M-A (2012) Estimating impact of infrastructure on productivity and efficiency of Indian manufacturing, Applied Economics Letters, 19:779–783 82. Mini F, Rodriguez E (2000) Technical efficiency indicators in a Philippine manufacturing sector, International Review of Applied Economics, 14(4):461–473 83. Pham HT, Dao TL, Reilly B (2010) Technical efficiency in the Vietnamese manufacturing sector, Journal of International Development 22:503–520

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Contents Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Regulatory Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Overview of Regulatory Frameworks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Efficiency Studies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The Decision Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Estimating Levels of Efficiency . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Review of Selected Studies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Economies of Scope and Scale . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Overview of Studies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Empirical Comparison . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Productivity and Productivity Change . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Measures of Productivity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Empirical Comparison . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Cross-References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Abstract

The companies in the electricity distribution sector are mainly natural monopolies. Thus, in most countries they are regulated. It is therefore of interest to investigate the efficiency and productivity of electricity distribution firms, and to examine how the regulation of these firms affects their efficiency and productivity over time. It is also of interest to examine economies of scope and scale in the electricity distribution industry, since these potential economies

Ø. Mydland () · G. Lien Inland School of Business and Social Sciences, Inland Norway University of Applied Sciences, Lillehammer, Norway e-mail: [email protected]; [email protected] © Springer Nature Singapore Pte Ltd. 2022 S. C. Ray et al. (eds.), Handbook of Production Economics, https://doi.org/10.1007/978-981-10-3455-8_37

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may be constrained under some regulatory regimes. In this chapter, we examine empirical findings reported in earlier studies of the electricity distribution sectors, with a focus on efficiency, economies of scope and scale, productivity, and productivity change. Additionally, we report evidence on how regulatory regimes affect and constrain the performance of firms, leading to the identification of implications for policy settings. For future research, it would be desirable to have more reliable and relevant data available to fortify conclusions and policy recommendations. Keywords

Regulatory regimes within the electricity distribution sector · Productivity and efficiency analysis · Economies of scope and scale · Productivity change

Introduction In this chapter, we summarize the empirical findings reported in earlier studies of the electricity distribution sectors. The research literature on production economics within the electricity distribution sector consists largely of research focusing on efficiency, economies of scope and scale, and productivity and productivity change. Additionally, regulatory regimes are often part of the focus within these themes. The electricity sector can be divided into three main activities: generation of electricity; distribution and transmission of electricity; and sales/trading of electricity. While generation and power trading are market-oriented and exposed to competition because customers can buy electricity from different providers, the customers of network services generally cannot choose between networks and are charged by the service provider (distribution company) located in their specific area. This means that the distribution companies have monopoly power within each concession area, given that it would not be expedient to build parallel power lines to introduce competition in this part of the electricity industry. There are two reasons for this: first, society would not benefit from cities or landscapes with many more power lines; and second, investment in building power grids is very large, and therefore it is not possible for a second provider to compete effectively with the established provider. Hence, the electricity distribution companies are regarded as natural monopolies, and we expect average and marginal costs to be decreasing with scale. As will be discussed later, this is the reason why this part of the electricity industry is being regulated, to ensure that companies do not take advantage of their monopoly power at the expense of consumers. By regulating the companies, governments can improve the incentives for companies to increase productivity and efficiency, thereby reducing costs and making it possible for prices paid by customers to also be reduced. In this chapter, we focus on four themes. In section “Regulatory Systems”, we present an overview of different regulatory systems. “Efficiency Studies” reports

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on efficiency studies, section “Economies of Scope and Scale” outlines studies of economies of scope and scale, and section “Productivity and Productivity Change” covers studies of productivity and productivity change. Section “Some Final Remarks” concludes with some final remarks.

Regulatory Systems As electricity distribution companies have the characteristics of natural monopolies, they are generally regulated to prevent or restrict their use of their monopoly powers. Hence, an important aspect in studying the performance of electricity distribution industries in different countries is the particular regulatory framework the companies face. Because of the regulation of the electricity distribution companies, efficiency and productivity analysis and other related issues have been subject to increased interest by researchers working on electricity distribution companies, providing examples of “real-world” applications of production economics of interest to both consumers and policy makers. In this section, we review the similarities and differences in regulatory frameworks across Europe and in the USA.

Overview of Regulatory Frameworks To prevent electricity distribution companies from taking advantage of their market power in order to set prices far above costs, thereby earning abnormal profit, regulatory authorities have been established in most countries to regulate distribution companies. Because it is also important that returns to investors are sufficiently high to attract the needed levels of investment in power distribution networks, it is necessary for the authorities to construct the regulation system such that investors can earn an adequate rate for return on their capital, given the risks involved. However, getting the balance right between the interests of consumers and investors can be difficult.1 Regulatory systems are very diverse such that, if we were to study the systems across all countries, we suspect that we would not find any two of them to be identical. However, we can classify the systems into two main categories: “costplus” and “incentive” regulation, with many variations within these two categories. Below we describe the two systems in turn and give some examples of how regulation can reach its intended goals.

1 One

possible approach is for the power grid to be wholly owned by the government. However, the huge investment required may be too high for some countries to meet all costs of building and maintaining the power grid. Quiggin [76] provides a discussion from an economic point of view of the pros and cons of government ownership, as well as the costs and benefits of reforms of electricity industries in past decades.

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Cost-Plus Regulation Cost-plus regulation has been widely used in the twentieth century [20]. Under this system, the regulated companies are allowed to set consumer prices equal to actual costs plus an extra amount representing a reasonable rate of return on the invested capital. The problem with this system is that it gives no incentives for companies to increase productivity to reduce costs or limit cost rises. It is therefore likely that cost-plus regulation will lead to higher prices for customers. Incentive Regulation As the name indicates, an incentive regulatory system is meant to give the electricity distribution companies incentives to increase productivity. This type of regulation can take many forms. The regulator can set a price cap or a revenue cap for a company, based on the performance of the company. For example, a price cap can be designed so that the price increases that consumers face cannot exceed changes in the consumer price index minus the expected productivity growth of the company. Price cap regulation is often called “CPI-X regulation,” formerly known as “RPI-X regulation,” and was first introduced by Stephen Littlechild in 1983 (see Littlechild [59] and Amundsveen and Kvile [7] for more on this).2 Such a system will give the company a clear incentive to increase productivity. The crucial and difficult issue under such a regulatory system is to estimate the appropriate expected productivity growth for the company. If the assumed productivity growth is set too low, the company will have less incentive to increase productivity because it can earn above normal profit. If expected productivity growth is set too high, the company may face financial problems (see Coelli et al. [20] for further details). Under the revenue cap regulatory system, instead of setting a price cap, the regulator can decide the maximum revenue the company can receive from its operations. As a revenue cap system will affect the prices the company can charge their customers, some call this approach an indirect price cap system [20]. Within the revenue cap system, there are also many ways of setting the revenue cap. The Council of European Energy Regulators [18] gives an overview of the regulatory systems in most European countries, and also describes how price caps and/or revenue caps are decided in each case. CEER [18] includes data from 26 European countries (Austria, Belgium, Croatia, Czech Republic, Denmark, Estonia, Finland, France, Germany, Great Britain, Greece, Hungary, Iceland, Ireland, Italy, Latvia, Lithuania, Luxembourg, Netherlands, Norway, Poland, Portugal, Romania, Slovenia, Spain, and Sweden) submitted by the various national energy regulators. Only three of these countries (Estonia, Greece, and Latvia) reported that they had a cost-plus regulatory system. The majority of countries thus reported incentive-based regulatory systems, with either a price cap, a revenue cap, or a combination of both. As noted, in implementing such regulatory systems, it can be difficult to determine the appropriate assumed increase in performance/productivity growth. However, regulators can use benchmarking as a basis for making the companies 2 Retail price index (RPI) is an older measurement of inflation used in the UK. Today, the consumer

price index (CPI) serves as the main measure of inflation.

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compete. By comparing productivity across companies, regulators can measure the efficiency of each company, and then decide the revenue cap and/or price cap for each company in the industry, incorporating a specified productivity improvement based on the benchmarking information. There are several ways to apply a revenue cap or price cap system in practice. For example, in Norway, where a revenue cap (RC) is the main system used, the cap for a given firm is calculated as: RC = (1 − ρ) × C + ρ × C* , where C is the cost base (calculated from the company’s own costs), C* is the estimated cost norm (from the benchmarking model), and ρ is a scalar between 0 and 1. The ρ term dictates the strength of the incentives. As 0 < ρ < 1, by deciding the strength of the incentives, the Norwegian model can be said to be a “hybrid” between costplus and revenue cap systems. The last step in the Norwegian regulatory model is a calibration procedure, where the regulator calibrates the estimated revenue cap so that the sum of the revenue caps of all firms equals the sum of the costs in the industry. Then, a firm with an average benchmarking score will receive a revenue cap intended to give it an average rate of return (RoR), and a firm that is higher (lower) than the benchmarking average will receive a higher (lower) RoR [5]. The advantage of this method is that, by measuring all firms, there is no need for the regulator to set a target for the performance of each company. The “best” company will receive the highest RoR, and the “worst” company will receive the lowest RoR. However, one problem with benchmarking in an incentive regulation context is that it is difficult to construct a model that gives correct measures of the performance of each company, and one that is suitable for comparing all the companies. Even if all electricity distribution companies in a country belong to the same industry and thereby mainly have the same operations, there will be some heterogeneity among the firms. To make the benchmarking model fair and to persuade the electricity distribution companies to accept the regulatory model, the heterogeneity among the firms must be taken into account.3 Furthermore, there is an issue concerning strategic behavior in an incentive-based regulation system, as discussed, for example, by Agrell and Teusch [3]. The strategic behavior of the regulated firm can consist of deliberately reporting incorrect data to increase the efficiency results in the regulation model. The firm can also make strategic investment decisions based on how this will make the firm perform in the regulatory model. As illustrated by Agrell and Teusch [3], a firm may find it is a good strategy to merge with one or several of the most efficient firms, thereby making the merged business appear relatively more efficient than the remaining unmerged firms. A general problem for most regulators in the electricity distribution industry is that often the electricity distribution companies are vertically integrated with the generation of electricity. As the latter is in a competitive market, it might be tempting to transfer operation costs to the regulated part of the company, to increase the “cost base” used in the regulation model. 3 There

are several cases where distribution companies have sued the regulatory authority. In some cases, such action leads to rejection of the current regulation model, leading to reversion to costplus regulation.

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Efficiency Studies The Decision Problem Efficiency is an ex post measure of the extent to which managers have managed their firms in an optimal way. In the literature, it is normal to define efficiency such that it takes values of 1.0 if the firm is managed in a fully efficient way, and a value of zero in the unlikely case that it is managed totally inefficiently. There is a range of different measures of efficiency, depending on the ways firms operate or are regulated. Options include output-, input-, revenue-, cost-, profit-, and productivity-oriented measures of efficiency (see O’Donnell [68], PP 175–218 for a detailed explanation of these efficiency measures). An output-oriented measure is relevant in a situation where inputs are predetermined and the manager can choose either the quantity of outputs or both the quantity of outputs and the mix of outputs. When outputs are predetermined, an input-oriented measure is relevant. If the input mix is predetermined, the relevant measure is input-oriented technical efficiency. If the manager can choose both the input quantity and the input mix, the relevant measure is input-oriented technical and mix efficiency. If the aim of the manager is to minimize costs, and we assume outputs are predetermined, efficiency is maximized by finding the optimal input mix. The input mix should accommodate any economies of input substitution. For the case of cost minimization, the relevant measure is cost efficiency, which can be decomposed into input-oriented technical efficiency and input-oriented allocative efficiency (i.e., the degree of under- or overutilization of inputs when the objective is to minimize costs). Several measures of outputs of electricity distribution companies have been used in different studies and by different regulators around the world (e.g., [6, 42]). A thorough discussion of choice of inputs and outputs for empirical analysis is given in  chapter “Empirical Analysis of Production Economics: Applications to Banking”. The choice of output used depends on data availability. Among the most frequently used output variables are size of the network (km), electricity delivered, and total number of customers served; among the most frequently used inputs are either total costs or capital, labor, and operational costs. Then, electricity distribution is a production process involving multiple outputs and multiple inputs, implying that estimating a standard production function is not an option. In that case, it is usual to assume that the outputs are more or less given, and the company managers decide the level and mix of the inputs. The objective of an electricity distribution company may be presumed to be to minimize costs given that their outputs (electricity delivered, customers served, etc.) are exogenous, whereas their inputs (the cost elements related to capital, labor, etc.) are endogenous. It follows that the relevant performance measure is cost efficiency and, typically, either a cost function or an input distance function has been estimated. In the case of the input distance function, use is typically made of the duality between the input distance function and the cost function.

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Estimating Levels of Efficiency The concept of efficiency measurement began with Farrell [29], who drew upon the work of Debreu [24] and Koopmans [47].4 To estimate a firm’s level of efficiency requires a benchmark or best practice for comparison. The standard technique is to specify and estimate frontier models to identify the efficient frontier in terms of input/output efficiency, cost efficiency, profit efficiency, etc. Frontier models may be parametric or nonparametric. Parametric models are based on an assumption that the data distribution can be defined in terms of a given set of parameters, while for nonparametric models it is assumed that an infinite set of parameters is needed to define a data distribution [77]. The most commonly used nonparametric model is data envelopment analysis (DEA). DEA is an estimation approach that involves enveloping a scatterplot of data points as tightly as possible by forming multiple linear segments of the frontier. Application of DEA depends on a few assumptions about the production technology. The main assumption is that the technology function is convex. This kind of model is typically estimated using mathematical programming. Estimation of parametric models using econometric methods, typically a stochastic frontier model, is based on some a priori selected parametric functional form. A further classification is to distinguish between deterministic and stochastic models. Stochastic models allow for individual observations to be somewhat affected by random noise, generating an estimated frontier stripped of the effect of random noise. In deterministic models, the noise is ignored and any variations in data are considered to contain information about the efficiency of the firms. Examples of deterministic and stochastic parametric models are corrected ordinary least squares (COLS) and stochastic frontier analysis (SFA), respectively. Examples of deterministic and stochastic nonparametric models are DEA and stochastic DEA, respectively. There are many nonparametric DEA models and also many parametric SFA models. Examples of the state of the art in the application of DEA and SFA models in the literature on electricity distribution firms are the conditional DEA of Bjørndal et al. [15] and Nieswand and Seifert [66], the 4-component SFA models of Kumbhakar and Lien [49], Filippini et al. [34], and Badunenko et al. [12]. In addition, between nonparametric and parametric models lies a range of alternative model specifications. In the electricity distribution industry, the semiparametric stochastic nonparametric envelopment of data (StoNED) model of Kuosmanen [56] is probably the most widely applied. In the choice between DEA and SFA, a key question is whether one wants flexibility in the frontier structure or precision in noise separation. DEA gives the best fit to the empirical data, but the most robust estimation method is SFA, because 4 For

reviews of models used and recent applications, see, for example, Kumbhakar and Lovell [50], Coelli et al. [21], Bogetoft and Otto [16], Kumbhakar et al. [54], O’Donnell [68], and Sickles and Zelenyuk [83].

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it gives results that are not so sensitive to random noise in the data. The choice may also depend on sample size, that is, the number of regulated companies. In practical applications, the typically small sample sizes favor the choice of DEA. Furthermore, the reliability of efficiency scores hinges on the ratio of variation in efficiency to the variation in noise [11]. Flexible models that are robust in the presence of random noise are generally seen as superior, and recent advances in this direction include semi- and nonparametric SFA models (e.g., [73, 83]). Badunenko et al. [11] analyze well the underlying assumptions of SFA and DEA models and provide useful recommendations for regulators. The methods of efficiency analysis used in the regulation of electricity distribution companies vary between countries and sometimes also between states in a country. Among OECD countries, deterministic methods such as DEA and COLS are used in Australia, Netherlands, Norway, Sweden, and the UK. In Finland, StoNED is applied. In some countries, the parametric method SFA and related frameworks are applied as complementary methods. In the US states of California and Maine, total factor productivity (TFP) is used for benchmarking as an input into the regulation process. There is also variation in how the regulators use benchmarking results. In some countries, such as Norway, the UK, and Netherlands, the benchmarking results are used in a rather “mechanical” way, as an explicit part of the regulatory process. In countries such as the USA, Australia, and Finland, the benchmarking results are used only as an additional instrument for regulatory decisions (Farsi et al. [30]). The increasing use of frontier analysis in the electricity distribution sector has raised concerns among regulators and companies regarding the reliability of efficiency estimates (see, e.g., [82]). The empirical evidence suggests that the efficiency estimates are sensitive to the benchmarking approach used (e.g., [34, 43, 49, 66]). The choice of frontier model and access to reliable data sets are two obvious challenging issues. Other issues include how to handle noise in the data, how to account for risk, and how to account for heterogeneity such as differences in environmental factors between the distribution companies. These and related challenges imply that the regulators and practitioners should take extra care in choosing and applying efficiency models used to develop and apply regulations intended to reward or punish companies to achieve better performance.

Review of Selected Studies In this review of the efficiency literature, we focus only on modeling of efficiency in the electricity distribution sector. Other parts of the energy sector are not included; nor are analyses using SFA models of energy efficiency (e.g., [33]). We also exclude models of market power (e.g., [52]). Hence, this is not a complete literature review of efficiency. It mainly covers frequently cited efficiency studies of the electricity distribution industry over the past two decades, along with a sample of some of the most recently published studies within this field. In what follows, we first give a

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chronological summary of the studies; second, methods applied are discussed; and finally, the empirical findings in these studies are reviewed.

Chronology of Studies Most of the studies of efficiency in the electricity distribution sector published before about 2010 applied the nonparametric deterministic DEA method (Table 1), while some applied COLS, SFA, or other frontier models. Among those published after 2015, we found a higher proportion using the parametric SFA framework or related approaches. As indicated in Table 1, while there is considerable variation in the countries examined in previous studies, quite a number of the newly published studies have used data from and analyses of Norwegian distribution companies. This is mainly because, for many years, the Norwegian regulator has made data used in the regulation available to the public, whereas regulators in other countries generally have not. This clearly represents a challenge because it limits the opportunity to compare companies across different countries. In the future, easier access to reliable and relevant electricity distribution data from a range of countries would increase the scope for more useful comparative studies, potentially leading to better regulation models. Estimation Methods In this section, we review the use of estimation methods in studies selected from the literature. Tables 2 and 3, which include more details of the studies listed in Table 1, together provide an overview of the estimation methods applied, function forms used, and the main empirical findings in these studies. As noted, the nonparametric DEA method is widely used by regulators. It is also the most frequently used method in the academic literature. Mathematical programming is used to estimate the best practice frontier, which is then used to evaluate the relative efficiency of different firms. The best practice frontiers are typically the best possible technology or production possibilities set from among the analyzed firms. The best possible technology is constructed according to the minimal extrapolation principle. That is, DEA finds the smallest production possibilities set consistent with the data (i.e., input-oriented, meaning use of the least possible amount of inputs for a given output) while also satisfying certain technological assumptions such as some form of convexity. The estimated frontiers are comprised of multiple linear segments and, accordingly, are also known as piecewise frontiers. DEA is based on the implicit assumption that there is no noise in the data and that no information on the technology is missing. If the data used are random, because of exogenous shocks, bad reporting practice, or ambiguity in accounting practices, the results will not be valid. Consequently, firms would neither be evaluated against the best possible standard, nor against a cautious standard [16]. Most of the earlier DEA studies listed in Tables 2 and 3 used the standard DEA briefly described above. Several attempts have been made to deal with noise in data in a DEA framework. One approach is to use the bootstrapping technique, where step one involves

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Table 1 Reviewed papers on the technical efficiency of electricity distributors Author(s) Selected publications frequently cited up to 2015 Pacudan and De Guzman [72]

Country

Sample

Method

Philippines

DEA

Resende [79]

Brazil

Edvardsen and Førsund [26]

International

Jamasb and Pollitt [43]

International

Korhonen and Syrjänen [48] Farsi and Filippini [31]

Finland Switzerland

Jamasb et al. [44]

USA

Agrell et al. [4]

Sweden

Giannakis et al. [38]

UK

Pombo and Taborda [75]

Colombia

Thakur et al. [86] Sadjadi and Omrani [80] Growitsch et al. [39] Cullmann [22] Kuosmanen [56]

India Iran International Germany Finland

Çelen [19] Dai and Kuosmanen [23]

Turkey Finland

Cross-sectional data. 15 firms Cross-sectional 1997/98. 24 firms 1997, 5 European countries 63 utilities, 1997, 1998, and 1999, depending on country 1998. 102 firms Balanced panel data, 1988–1996. 59 firms Cross-sectional data 2000. 28 firms Unbalanced panel data 1996–2000. 238 firms 1991/92 to 1998/99. 12 utilities Balanced panel data 1985–2001. 12 firms 2001–2002. 26 firms 2004. 38 firms 2002. 499 firms 2001–2005. 200 firms 4-year averages over 2005–2008. 89 firms 2002–2009. 21 firms 6-year averages over 2005–2010

Selected publications from 2015 Hafezalkotob et al. [41] Kumbhakar et al. [53]

Iran Norway

Mullarkey et al. [63] Orea et al. [70] Ervural et al. [28] Agrell and Brea-Solís [2]

Ireland Norway Turkey Sweden

Arcos-Vargas et al. [8] Kumbhakar and Lien [49]

Spain Norway

2008. 38 firms Unbalanced panel, 1998–2010. 128 firms 2008. 26 firms 2001–2004. 128 firms 81 regions Balanced panel 2000–2006. 118 firms 2011. 102 small firms Unbalanced panel 2000–2013. 134 firms

DEA DEA DEA, COLS SFA DEA SFA DEA DEA DEA DEA DEA DEA, SFA SFA SFA StoNED DEA StoNED, DEA

DEA SFA DEA SFA DEA Latent class SFA DEA SFA (continued)

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Table 1 (continued) Author(s) Li et al. [58]

Country Japan

Orea and Jamasb [69]

Norway

Bjørndal et al. [15]

Norway

Deng et al. [25] Filippini et al. [34] Makieła and Osiewalski [60] Orea et al. [71]

China New Zealand Poland Norway

Silva et al. [84]

13 European companies

Kumbhakar et al. [55]

Norway

Musau et al. [64]

Norway

Badunenko et al. [12]

Germany

Sample Unbalanced panel data 1980–2010. 10 firms Unbalanced panel 2004–2011. 129 firms Average data for 2008–2012. 123 firms 1999–2013. 31 firms 2000–2011. 28 firms Balanced panel 2004–2011. 129 firms Cross-sectional over 2012–2014 Unbalanced panel 2000–2016. 146 firms Unbalanced panel 2000–2016. 149 firms Unbalanced panel 2006–2012. 242 firms

Method Metafrontier SFA DEA, StoNED SFA SFA SFA SFA DEA, SFA with entropy estimation SFA SFA SFA

obtaining the efficiency scores from the original DEA, and then, in step two, the standard errors of the DEA estimators are obtained by bootstrapping [85]. An alternative approach to bootstrapping for dealing with noise in data, called robust optimization, was proposed by Sadjadi and Omrani [80]. However, as pointed out by Coelli et al. [21], these DEA techniques for dealing with noise in data are designed to deal with sampling variability, meaning that they indicate how the DEA estimates would vary if a different random sample were to be selected. These approaches do not account for random noise in the modeling of efficiency. DEA results are influenced by operational heterogeneity among the electricity distribution firms. To account for this heterogeneity, regulators often use second-stage regressions to control for the observed differences in the operational environments of firms on the estimated frontier (e.g., [4, 19, 48, 75]). Environmental or z-variables are regressed on the DEA estimates (from the first step) to determine the effect of the operational environments on efficiency. The efficiency scores are then adjusted to compensate for the effects of the z-variables. A recent development to deal with observed heterogeneity is the conditional DEA approach, as demonstrated by Bjørndal et al. [15] and Nieswand and Seifert [66]. They used a kernel estimation to restrict the reference sets (i.e., from the whole sample to a restricted sample) with respect to z-variables, prior to measuring actual performance, meaning that firms are compared only with others with similar distribution environments.

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Table 2 Selected publications frequently cited up to 2015. Estimation method, estimated technical efficiency, and main findings/implications Author(s) Pacudan and De Guzman [72]

Estimation method DEA

Function Inputs and outputs

Technical efficiency 0.82

Resende [79]

DEA

Inputs and outputs

0.78–0.84

Edvardsen and Førsund [26]

DEA

Inputs and outputs

0.81

Jamasb and Pollitt [43]

DEA, COLS and SFA

Inputs and outputs + input distance functions

Korhonen and Syrjänen [48] Farsi and Filippini [31]

DEA-VRS

Inputs and outputs Cost function

DEA 0.54–0.79 COLS 0.60–0.63, SFA 0.62–0.72 0.77

Jamasb et al. [44]

DEA

Agrell et al. [4]

DEA

Giannakis et al. [38]

DEA, technical and service inefficiency DEA CRS + VRS

Inputs and outputs

Thakur et al. [86]

DEA, both TE and scale efficiency

Inputs and outputs

Sadjadi and Omrani [80]

DEA with uncertain data, SFA

Growitsch et al. [39]

SFA

Inputs and outputs with random component Input distance functions

Pombo and Taborda [75]

Four SFA models

Inputs and outputs Inputs and outputs

Inputs and outputs

Findings/implications Demand-side management incentives slightly improve efficiency Access to reliable and relevant data is a challenge Should apply a common technology for identifying multinational peers Estimated efficiency scores are sensitive to the applied frontier model and model specification

The ranking of firms varies significantly across models

0.68

CRS 0.76–0.90 VRS 0.83–0.05 0.57–1.00

DEA-CRS 0.68 DEA-VRS 0.84 SFA 0.79, DEA 0.80

Small utilities 0.45, medium 0.49, large 0.80

Focus on the theoretical foundation of the analysis It is important to include service quality in the analysis Regulatory reform in 1994 created improved efficiency

Efficiency estimates are smaller when uncertainty is accounted for Quality of services should be an integrated part of the efficiency analysis (continued)

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Table 2 (continued) Author(s) Cullmann [22]

Estimation method Latent class SFA

Kuosmanen [56]

StoNED

Çelen [19]

DEA + TOBIT

Dai and Kuosmanen [23]

StoNED + DEA

Function Input distance functions

Cost function + inputs and outputs Inputs and outputs Cost function + inputs and outputs

Technical efficiency Mean 0.91 Class 1 0.90 Class 2 0.92

0.92

0.91–0.94

Findings/implications Allowing for different technologies results in more robust individual efficiency estimates Accounts for noise in the data and for heterogeneity Private ownership affects efficiency positively Cluster-based analysis used to account for heterogeneity

All parametric SFA approaches deal with random noise, caused by exogenous shocks, bad reporting practice, or ambiguity in accounting practices. There also exists a range of SFA panel data models, accounting for unobserved firm-specific or time-specific heterogeneity (also called “common errors”) (e.g., [2, 31, 53, 70]. However, using such models comes at a cost because, compared with DEA models, parametric SFA models require more a priori assumptions about the structure of the production possibility set. Using the SFA approach, it is presumed that the structure of the production possibility set is known a priori through the specification of some chosen functional form. In addition to estimating the efficiency of firms, it is often of interest to learn about factors that affect efficiency. In recent decades, there has been discussion about the use of either a “one-step” model to simultaneously estimate the SFA model and the way in which the efficiency term (u) depends on z-variables, versus using a “two-step” procedure, where the first step is to estimate a standard SFA model, and the second step is to estimate (via regression) the relationship between (estimated) u and z. However, this issue is now resolved, and the efficiency determinants should be estimated simultaneously as an integrated part of the SFA model [81, 89]. When including determinants of efficiency or z-variables in the SFA framework, one can explain systematic differences in efficiency within and between firms. For example, of the SFA studies listed in Table 3, Orea et al. [70], Agrell and BreaSolís [2], Kumbhakar and Lien [49], Li et al. [58], Orea and Jamasb [69], Deng et al. [25], Orea et al. [71], Kumbhakar et al. [55] and Badunenko et al. [12] all included efficiency determinants in their analyses. In the 4-component model (see below), Badunenko et al. [12] included determinants of both transient and persistent inefficiency.

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Table 3 Selected publications from 2015. Estimation method, estimated technical efficiency, and main findings/implications Author(s) Hafezalkotob et al. [41] Kumbhakar et al. [53] Mullarkey et al. [63]

Orea et al. [70]

Ervural et al. [28] Agrell and Brea-Solís [2] Arcos-Vargas et al. [8] Kumbhakar and Lien [49]

Estimation method Robust data envelopment model (RDEA) “True” FE SFA DEA (6 alternative models) SFA combined with supervised dimension reduction. DEA Latent class SFA DEA 4-component SFA

Function Inputs and outputs Inputs and outputs Inputs and outputs

Cost function

Inputs and outputs Input distance functions Inputs and outputs Input distance functions

Li et al. [58]

Metafrontierbased analysis

Metafrontier cost function

Orea and Jamasb [69]

Nested latent class SFA

Cost function

Bjørndal et al. [15]

Conditional DEA, StoNED

Inputs and outputs, Kernel function

Deng et al. [25]

Bayesian SFA

Input distance functions

Technical efficiency 0.82–0.91

Findings/implications Deals with discrete uncertain input and output data

0.82–0.87 Vary from 0.65 to 0.91. Rural 0.91, urban 0.83 0.90–0.96

Accounts for influence of heterogeneity on the DEA score Higher efficiency when controlling for environmental variables

Critique of latent class SFA 0.63–0.84 Long-run TE 0.53–0.93. Short-run TE 0.89–0.96 Group frontier 0.80 (1996–2010), Metafrontier 0.72 (1996–2010) ZISF 0.87, NLCSF 0.91 Unconditional DEA 0.67, conditional DEA 0.70 0.21–0.33

A flexible model to explain firms’ observed practice and efficiency

Distinguishes between fully efficient and nonefficient firms Accounting for heterogeneity in a DEA framework Distinguishes between service quality and technical efficiency (continued)

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Table 3 (continued) Estimation method 4-component SFA

Function Cost function

Technical efficiency 0.78–0.94

Makieła and Osiewalski [60]

Bayesian SFA

Cost function

0.90

Orea et al. [71]

Spatial SFA

Cost function

0.92

Silva et al. [84]

DEA, SFA with entropy estimation A system approach, SFA + cost minimization

Input distance functions

0.12–1.00

Production function + first-order conditions of cost min Production function + first-order conditions of cost min.

Transient TE 0.76–0.93. Persistent TE 0.95. Overall TE 0.76–0.93 At mean: overall TE 0.85, transient TE 0.98, persistent TE 0.86 At median: overall TE 0.68, transient TE 0.95, persistent TE 0.72

Author(s) Filippini et al. [34]

Kumbhakar et al. [55]

Musau et al. [64]

A system approach, SFA + cost minimization

Badunenko et al. [12]

4-component SFA with determinants

Input distance functions

Findings/implications Important to disentangle transient and persistent efficiency Framework where the model specification is determined based on information in the data How to control for unobserved environmental conditions For small sample size, incomplete, and noisy data The costs of input misallocation of Norwegian electricity distribution firms are not negligible Includes determinants of transient and persistent inefficiency, and determinants of allocative input misallocation Firms in East Germany achieves better persistent efficiency on average than those in West Germany

Orea et al. [70] and Makieła and Osiewalski [60] also demonstrated how to apply a form of machine learning to deal with a large set of environmental variables and heterogeneity in an integrated panel data model. Orea et al. [71] presented a spatial SFA model, which accounted for spatial heterogeneity. An important recent development of SFA models is the 4-component model. In this model, the first component captures the unobserved heterogeneity of firms, the second component captures persistent inefficiency, the third component captures transient inefficiency, and the last component captures random shocks. Application of the 4-component model to electricity distribution has been demonstrated by Kumbhakar and Lien [49], Filippini et al. [34], and Badunenko et al. [12]. Filipini et al. [34] also developed a theoretical model that shows that an imperfectly informed regulator may not be able to disentangle the persistent and transient parts

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of cost efficiency. The empirical results of Filipini et al. [34], Kumbhakar and Lien [49], and Badunenko et al. [12] show that the transient and persistent parts of efficiency are relatively different, and differ from efficiency measured by previous approaches. From a regulatory point of view, these results imply that both persistent and transient technical efficiency measures should be used in regulation. Applications of SFA to study the efficiency of the firms that distribute electricity have overwhelmingly focused on technical inefficiency. The implicit assumption is that all firms are either fully allocatively efficient (i.e., there is no input misallocation), or that their level of allocative inefficiency is negligible and can be ignored. The reason input misallocation has been ignored or not modeled in earlier studies could be because of the unavailability of data. Moreover, modeling input misallocation is complicated. In any event, studies that use the cost function tend to ignore allocative inefficiency and focus on technical inefficiency, because the cost function (or input distance function) typically used in the literature does not allow for the separation of technical inefficiency and allocative inefficiency. Kumbhakar and Wang [51] used a Monte Carlo study to illustrate that allocative efficiency cannot be lumped together with technical efficiency in the estimation of a cost function. Their results show that failure to include the cost of allocative inefficiency or input misallocation in a cost function framework biases the cost efficiency estimates. One modeling strategy to overcome this problem, illustrated by Kumbhakar et al. [55], is to use a system approach, consisting of a production function and the first-order conditions for cost minimization. Input misallocation for each pair of inputs in that framework is modeled via the first-order conditions of cost minimization. Their findings, based on data for the Norwegian electricity distribution sector, show that the costs of input misallocation are not negligible. In fact, the costs to the industry arising from input misallocation can be high, and ranged on average for the firms analyzed from 9.4% to 10.9% in their analysis. Kumbhakar et al. [55] included determinants of both transient and persistent technical inefficiency, while Musau et al. [64] extended that model framework to also include determinants of allocative inefficiency (input misallocation). For a more thorough and technical description of SFA and DEA models see  Chaps. 8, “Stochastic Frontier Analysis: Foundations and Advances I”,  9, “Stochastic Frontier Analysis: Foundations and Advances II”, and  10, “Data Envelopment Analysis: A Nonparametric Method of Production Analysis”.

Empirical Comparison Figure 1 presents plots of estimated technical efficiency from the studies frequently cited up to 2015 (Table 2), and Fig. 2 depicts the efficiency scores for those studies published from 2015 (Table 3). The average technical efficiency of the studies plotted in Figs. 1 and 2 is 0.77. Compared with the studies cited up to 2015 (Fig. 1), there is, on average, a somewhat higher efficiency score with less variability in the studies published from 2015 (Fig. 2). By our own calculations, we found that the estimates based on DEA estimators are on average about 0.75, while estimates based on SFA are on average about 0.84. Of course, these variations in estimated efficiency levels are probably

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Fig. 1 Comparison of reported estimated technical efficiency in papers frequently cited up to 2015, using data from different countries/regions

Fig. 2 Comparison of reported estimated technical efficiency in papers cited from 2015, using data from different countries

caused by a combination of differences between the countries investigated, the time period analyzed, variable specifications, and estimation methods. Nevertheless, the difference between the two averages highlights the variability of efficiency estimates in published studies.

Economies of Scope and Scale The concept of economies of scale dates back to Adam Smith and is a technical concept that tells us what happens to output if we increase inputs. If we double our inputs and output doubles, more than doubles, or less than doubles, then we have constant, increasing, or decreasing returns to scale (RTS), respectively. Economies of scale are considered to be a local measure, meaning that RTS can differ for different parts of the production function. Hence, a measure of economies of scale gives a good description of the local properties of the production function. Furthermore, economies of scale are a useful tool in finding “optimal scale” (i.e., RTS = 1). A thorough treatment of scale elasticity and returns to scale can be found in  Chap. 17, “Scale Elasticity and Returns to Scale”.

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The concept of economies of scope dates back to the work of William J. Baumol, starting in 1974, and the work of John C. Panzar and Robert D. Willing who independently started similar work around the same time. Their main work is collected in Baumol et al. [14]. The concept is to examine whether one multi-output firm can produce two or more outputs more cheaply than several different firms producing one output each. The existence of economies of scope means that there are some positive synergies between different outputs produced by a single firm. In the electricity industry, the two outputs can be present in a company that both produces and distributes electricity. If we find the existence of economies of scope, it means that the company is operating more efficiently than would be the case if the two functions were completely separated. This section about economies of scale and scope is rather brief; for a thorough treatment of both the theoretical concepts and ideas and a critical review of the literature, see Chap. 45, “ Cost Assessment of (Un)bundling: Separation of Vertically Integrated Public Utilities”.

Overview of Studies Table 4 gives a chronological summary of most of the studies of economies of scope and/or scale in the electricity distribution sector. As can be seen, most studies have used a parametric estimation approach. Only one study of electricity utilities in Spain [9] used the nonparametric DEA approach to analyze economies of scope and scale. In the group of studies using parametric estimation approaches, a large proportion used a quadratic cost function. The frequent choice of this functional form is attributable to the problems that stem from the existence of zero value outputs; zero values cannot be handled in a translog cost function because the log of zero does not exist. There are several examples, albeit not within the electricity sector, where the translog cost function has been used in scope studies, achieved by replacing all the zero values by arbitrary small numbers. However, Fraquelli et al. [36] showed that this is not an appropriate approach because the value chosen for the arbitrary small number can have a substantial effect on the results. There are also some problems with quadratic cost functions. Battese [13] showed that if the number of zero values represents a large proportion of the total number of observations, the parameter estimates may be biased because the zero values are not really zero, they are nonexistent. In Triebs et al. [88], the flexible dummy variable approach was applied. This approach has advantages in estimating economies of scope and scale in industries with some multi-output firms when using quadratic, translog, or any functional form because a different set of parameters is estimated for the nonexisting/zero-value observations. It is also possible to test for different technologies (e.g., one technology for the power producers and one technology for the electricity distributors) when applying the flexible dummy variable approach. Among the parametric studies, ordinary least square (OLS), seemingly unrelated regressions (SUR), and random effect models have all been used. Arocena [9] showed that when the DEA approach is used, there is no need for assumptions on functional form.

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Table 4 Summary of previous empirical scope and scale studies of combined generation and transmission/distribution electricity companies Author(s) Kaserman and Mayo [46]

Data Cross-sectional (1981, US)

Functional form Quadratic cost function

Established method OLS

Kwoka [57]

Cross-sectional (1989, US)

Quadratic cost function

OLS

Jara-Díaz et al. [45]

Panel data (1985–1996, Spain)

Seemingly unrelated regressions (SUR)

Arocena [9]

Panel data (1989–1997, Spain)

Quadratic cost function together with cost share equations Cost function

Piacenza and Vannoni [74]

Panel data (1994–2000, Italy)

Multiproduct & multistage Box–Cox transformed cost function

Nonlinear SUR

Fetz and Filippini [32]

Panel data (1997–2005, Switzerland)

Quadratic cost function

Random effects GLS and random coefficient model

Arocena et al. [10]

Cross-sectional (2001, US)

Quadratic cost function together with cost share equations

SUR

Data envelopment analysis (DEA)

Economies of scope and scalea Economies of scope (EOS) = 0.12 (at mean) EOS = 0.27 (at median). Reports substantial costs of vertical integration and highest for the smallest utilities EOS = 0.065–0.28. Returns to scale (RTS) = 1.07 EOS = 0.11–0.49. RTS: Reports that the scale efficiency of the largest units in the sample could improve by dividing them into smaller units EOS = 0.24. RTS = 0.96. Reports findings of both vertical integration gains and horizontal economies of scope EOS = 0.50–0.60 (at median). RTS = 1.40–1.70 (at median). Presence of considerable economies of vertical integration and economies of scale for most companies EOS = 0.04–0.10. RTS = 1.01–1.03. Reports positive sample mean estimates of both vertical and horizontal economies (continued)

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Table 4 (continued) Functional form Quadratic cost function

Established method OLS

Panel data (2000–2003, US)

Flexible technology translog cost functions with different specifications

SUR

Gugler et al. [40]

Panel data (2000–2010, 16 European countries)

Multistage quadratic cost function

Nonlinear SUR

Mydland et al. [65]

Panel data (2004–2014, Norway)

Flexible technology translog and quadratic cost functions

Maximum likelihood, random effects

Author(s) Meyer [61]

Data Panel data (2001–2008, US)

Triebs et al. [88]

a Estimates

Economies of scope and scalea EOS = 0.19–0.26, when separating generation from distribution and retail. Reports that, if generation and transmission remain integrated but are separated from distribution and retail, EOS = 0.08–0.10 EOS = 0.04 (0.40 when zeros are replaced by small numbers in the common cost function model). RTS = 1.10–1.13. Reports evidence of economies of scale and vertical economies of scope EOS = 0.14–0.20. Reports that, at the median, integrated utilities have EOS = 0.14, while large-scale utilities have EOS = 0.20 EOS = 0.10 (at median) RTS = 1.70, 1.15, 1.02 (at median), integrated, distribution, and generation, respectively

of economies of scale (measured by RTS) are for integrated firms

Five of the studies presented in Table 4 used data from the USA, and five from European countries: two from Spain, and one each from Italy, Switzerland, and Norway. In addition, Gugler et al. [40] examined economies of scope in 16 European countries (Austria, Czech Republic, Finland, France, Germany, Greece, Hungary, Italy, Latvia, Norway, Poland, Portugal, Spain, Sweden, Switzerland, and the UK).

Empirical Comparison Estimates of economies of scope from all of the studies reported in Table 4 are shown in Fig. 3, with results varying from 4% to 60%. The results are quite

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Fig. 3 Comparison of the estimated economies of scope reported in studies using data from different countries

Fig. 4 Comparison of the estimated returns to scale reported in studies using data from different countries

consistent over the studies estimated with different methods and functional forms. There is some variation in estimates between countries. The highest estimated economies of scope reported are for Switzerland. The five studies from the USA yielded some differences in estimates of economies of scope, ranging from below 10% to almost 30%. In Fig. 4, the average estimated economies of scale are shown. Only Italy was found to have decreasing RTS. All other studies estimated RTS above one, implying a potential for cost reduction by increasing the size of the distribution firms.

Productivity and Productivity Change Productivity is basically a measure of a firm’s performance in terms of the ratio between outputs and inputs. The inputs here are the variables used to produce an output (products and services), and they are controlled by the manager of the production process. For variables that are controlled by the manager, it is possible to add or subtract inputs or increase and decrease the amount of one or more inputs. Variables that affect the production process but are not controlled by the managers are often referred to as environmental variables, which can be weather

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conditions, infrastructure, or legislation. A main part of the productivity theory is the concept of production technology. Leaving environmental variables aside, production technology can be said to be a technique, ability, or recipe to transform inputs into outputs. In a historical context, we can say that the industrial revolution represented a major shift in production technology when, for example, farmers went from using animals to plow their fields to using tractor-drawn plows. If a farmer goes from taking 2 weeks to plow the fields on his land, to taking only 2 days, because of the replacement of animals by machinery, we would say this represents a change in production technology. Although the concept of productivity is quite simple and the studies on productivity and productivity change include the same kinds of variables, there is debate about how to construct indices to measure changes in the variable(s) over time. O’Donnell [68] argued that indices must be consistent with measurement theory, and that they should satisfy axioms Q1–Q8 listed in O’Donnell [67].5 Q1: Weak monotonicity, Q2: Homogeneous of degree one, Q3: Identity, Q4: Homogeneous of degree zero, Q5: Proportionality, Q6: Time–space reversal, Q7: Transitivity, and Q8: Circularity. Two of the most important axioms are the transitivity and proportionality axioms. The transitivity axiom is that a direct comparison of two firms should give the same index number as an indirect comparison through a third firm. The proportionality axiom is that, if firm A produced λ times as much as firm B, then the index that compares the outputs of firm A with outputs of firm B must take value λ. Indices satisfying axioms Q1–Q8 are called “proper indices” by O’Donnell [67, 68]. For a further discussion of index numbers and productivity measurement see  Chap. 19, “Index Numbers and Productivity Measurement”.

Measures of Productivity The Malmquist total factor productivity (TFP) index, introduced by Caves et al. [17], is widely adopted in the studies of productivity and productivity change in the electricity distribution industry.6,7 Total factor productivity can be decomposed into several measures to explain the productivity results and changes over time. One frequently used step is to decompose the Malmquist TFP index into the following economically meaningful sources: 1) a technical efficiency change component, measured relative to the best practice technologies; 2) a technical change component, characterizing the shift in the best practice technologies; and 3) a scale component, measuring the contribution of scale economies (see, e.g., [37]). Several other ways

5 Here Q refers to the output index. The arguments and interpretations are analogues for TFP, or for

input indices. Malmquist TFP index is also sometimes called any one of “Hicks–Moorsteen TFP index,” “Hicks–Moorsteen index,” “Bjurek index,” “Bjurek productivity index,” or “Bjurek TFP index.” See Sickles and Zelenyuk [83] for more on this. 7 Note that the Malmquist index does not satisfy all axioms Q1–Q8 listed in the previous subsection. 6 The

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to decompose the TFP index are possible. It is also possible to include some environmental or z-variables into the TFP index in these kinds of analyses and to decompose the TFP index to show the environmental effects (see, e.g., [68]). Construction of the components requires some estimation procedures. The most common methods for estimation of the decomposed components are variants of either DEA or SFA. Table 5 gives a chronological summary of most of the studies of productivity and productivity change in the electricity distribution industry. As shown, most studies have used a nonparametric approach by applying DEA; only Tovar et al. [87] used the parametric SFA method. Furthermore, all the studies used Malmquist TFP indexes or some variation thereof. Three of the studies used data from Norwegian electricity distribution companies, although for different time periods [27, 35, 62]. All three studies reported some productivity growth. Two of the studies in Table 5 used data from Brazil [78, 87] from the same period (1998–2005). Both reported productivity growth mainly caused by technical change, even though the methods chosen were different; the former used DEA while the latter used SFA. Table 5 also includes one study using data from Australia for the period 1969–1999 [1].

Empirical Comparison Figure 5 shows the TFP results for some of the studies presented in Table 5. There are two studies from Norway for two different time periods. Førsund and Kittelsen [35] reported a 2% increase in productivity using data for the period (1983–1989), while Edvardsen et al. [27] reported an average increase in productivity of 5% over two periods (1996–1997 and 1996–2003). Using data from Australia, Abbott [1] reported an annual growth rate in productivity of 2.5% for 1969–1999. Using data from Brazil, Ramos-Real et al. [78] reported an annual growth rate of 1.3% for 1998–2005.

Conclusion In this chapter, we have described different regimes in the regulation of electricity distribution companies, and discussed the two main regulatory regimes of cost-plus and incentive regulation. Furthermore, overviews of empirical studies on efficiency, economies of scope and scale, and productivity and productivity change have been provided. The studies presented are from 1991 to 2020/2021, a period during which there have been substantial developments in the methods and knowledge relating to the production economics of electricity distribution. In most cases, we see that results regarding scale, scope, and productivity differ more between the data used in each study (data from different countries) than between different models, methods, and technical applications. However, this is not entirely the case for the efficiency estimates. In addition to variation in efficiency estimates between countries, the results are sensitive to how efficiency is modeled

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Table 5 Chronological summary of most of the studies of productivity and productivity change within the electricity distribution industry Functional form/technology set Piecewise linear frontier technology

Established method DEA, Malmquist TFP index

Author(s) Førsund and Kittelsen [35]

Data Panel data (1983–1989, Norway)

Abbott [1]

Panel data (1969–1999, Australia) Panel data (1996–2003, Norway)

DEA technology

Malmquist TFP index

DEA technology

Bilateral Malmquist cost productivity index

Ramos-Real et al. [78]

Panel data (1998–2005, Brazil)

DEA technology

Tovar et al. [87]

Panel data (1998–2005, Brazil)

Translog input distance function

DEA Malmquist approach, Decomposing TFP using Malmquist TFP index SFA, Malmquist index

Miguéis et al. [62]

Yearly average panel data (2004–2007, Norway)

DEA technology

Edvardsen et al. [27]

DEA, decomposed Malmquist index

Productivity/productivity change Overall positive development in productivity of 2% per year, mainly caused by technical change On average TFP increased with annual growth rate of 2.5% Increase in productivity of 0.2% 1996–1997. Average productivity increase of 8.0% 1996–2003 TFP index gives yearly growth rate of 1.3% 1998–2005. Technical change as main component with average growth of 2.1% per year Mainly caused by technical change, TFP increased in 1998–2003 and decreased in 2003–2005 On average the companies’ productivity increased between 2004 and 2005, did not change from 2005 to 2006, and decreased between 2006 and 2007

and interpreted. We recommend that future empirical research and applications should pay more attention to modeling and interpreting efficiency as well as to the assumptions underlying each chosen model. The studies reviewed in this chapter give estimates of levels of efficiency, economies of scale, economies of scope, productivity change, and components thereof. These estimates of performance should be useful for regulators, policy makers, and perhaps also for individual firm managers and researchers. However, the

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Fig. 5 Comparison of TFP results from studies using data from different countries

results would likely be more useful if there were more agreement on and adoption of standard practices across studies. Such standards might include determinants of reasons behind the estimates of efficiency, scale, scope, and productivity change. Of course, for future research in this direction, access to relevant and reliable data is also a requirement. It is highly likely that developments in business analytics will influence future research. It would be interesting to consider how developments in areas such as artificial intelligence, machine learning, and big data might open up new possibilities for the collection and control of data. We expect these developments to have significant effects on the efficiency of the collection, control, analysis, and reporting of data by researchers, regulators, and companies.

Cross-References  Cost Assessment of (Un)bundling: Separation of Vertically Integrated Public

Utilities  Envelopment Analysis: A Nonparametric Method of Production Analysis  Empirical Analysis of Production Economics: Applications to Banking  Index Numbers and Productivity Measurement  Scale Elasticity and Returns to Scale  Stochastic Frontier Analysis: Foundations and Advances I  Stochastic Frontier Analysis: Foundations and Advances II Acknowledgments We would also like to thank participants in the project ElBench for providing funding to complete this work.

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Production and the Environment

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Moriah Bostian and Tommy Lundgren

Contents Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Modeling Production and the Environment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Externalities, Efficiency, and Productivity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Radial Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Non-radial Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Network and Multi-function Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Valuation and Substitution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Valuation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Substitution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Environmental Policy and Firm Performance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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We would like to thank Rolf Färe and Shawna Grosskopf, as well as our reviewer. This work has greatly benefited from their comments. M. Bostian () Department of Economics, Lewis & Clark College, Portland, OR, USA Department of Economics, Centre for Environmental and Resource Economics (CERE), Umeå University, Umeå, Sweden e-mail: [email protected] T. Lundgren Department of Economics, Centre for Environmental and Resource Economics (CERE), Umeå University, Umeå, Sweden e-mail: [email protected] © Springer Nature Singapore Pte Ltd. 2022 S. C. Ray et al. (eds.), Handbook of Production Economics, https://doi.org/10.1007/978-981-10-3455-8_38

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Abstract

Production theory offers a mathematical framework for modeling important relationships between production activities and the environment. These include the generation and valuation of production-related environmental effects, environmental contributions to production processes, and production effects of environmental management practices. In this chapter, we review the seminal and recent empirical work in each of these areas. We anchor our review to multi-input/multi-output production processes, as these make up a large share of environmental applications in the field, and their associated models offer the practitioner considerable flexibility in terms of specification and estimation. Keywords

Pollution-generating technology · Externalities · Valuation · Competitiveness · Environmental performance

Introduction Production theory offers a mathematical framework for modeling important relationships between production activities and the environment. These include the generation and valuation of production-related environmental effects, environmental contributions to production processes, and production effects of environmental management practices. In this chapter, we review the seminal and recent empirical work in each of these areas. We anchor our review to multi-input/multi-output production processes, as these make up a large share of environmental applications in the field, and their associated models offer the practitioner considerable flexibility in terms of specification and estimation. The chapter unfolds as follows: In section “Modeling Production and the Environment,” we summarize the relevant theory for modeling multi-input/multioutput production and the environment and the duality theory underlying production and valuation. We intend this first section to serve as a unifying reference for modeling pollution generation from production activities, the economic costs of environmental protection, and efficiency and productivity. We also provide additional references for the interested reader who wishes to delve deeper into the theory. In the remaining sections, we explain how this basic modeling framework has been adapted and applied to the environment in practice. We begin in section “Externalities, Efficiency, and Productivity” with externalities. This includes some consideration of what constitutes an externality in the production context, applications to better understand how the production process generates externalities, and how to incorporate externalities into measures of productivity and efficiency. Then in section “Valuation and Substitution” we turn to the topic of valuation, distinguishing between the benefits and costs of pollution abatement. Here we review how others have used production theory to estimate prices for externalities,

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as well as other non-marketed environmental goods and services. For our final application area in section “Environmental Policy and Firm Performance,” we examine firm competitiveness in relation to the environment. This includes a review of the empirical evidence for and against the well-known Porter hypothesis and the growing field of what is commonly termed “corporate social responsibility (CSR).” We conclude in section “Conclusion” by considering the current trajectory of empirical work in the field.

Modeling Production and the Environment To examine production and the environment through the lens of production theory, we begin with a general model of the production technology as a multi-input/multioutput process. For notation, let x ∈ N + represent a vector of inputs and y ∈ M a vector of intended production outputs. In practice, producing y from x + can (and often does) also generate unintended environmental effects, the focus of this chapter. Let u ∈ J+ represent these unintended environmental effects (e.g., pollution). A common approach in the literature is to incorporate u into (1) as an undesirable output, in order to model the production of both intended and unintended outputs. The resulting technology is often labeled the “environmental production technology,” “joint production technology,” or the “pollution-generating technology.” We will use the latter here, abbreviated as PGT, as it speaks more directly to the undesirable nature of u. We define the PGT, T , as T = {(x, y, u) : x can produce y and u},

(1)

which can also be represented in terms of the feasible output set, P (x), P (x) = {(y, u) : (x, y, u) ∈ T },

(2)

or the input requirement set, L(y, u), L(y, u) = {x : (x, y, u) ∈ T }.

(3)

Distance functions offer a functional representation of the PGT and are widely applied in the field to incorporate environmental effects into estimates of productivity and efficiency, valuation, and policy analysis. These models take two general forms, radial [92] distance functions and non-radial [70] or directional distance functions [25]. The Shephard output and input distance functions, DO and DI , are defined, respectively, as

DO (x, y, u) = inf{θ : (y/θ, u) ∈ P (x)}, 0 ≥ θ ≥ 1,

(4)

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and DI (x, y, u) = sup{λ : x/λ ∈ L(y, u)}, λ ≥ 1,

(5)

where DO measures the feasible radial expansion of intended outputs from the origin and DI measures the feasible radial contraction of inputs to the origin. A distance value θ = 1 or λ = 1 implies that an observation is operating on the corresponding output or input frontier. Numerous studies have also adapted the radial input distance function to undesirable outputs, which we denote as DU , DU (x, y, u) = sup{λ : u/λ, x/λ ∈ L(y)}, λ ≥ 1.

(6)

 O and D  I , are defined, The directional output and input distance functions, D respectively, as  O (x, y, u; gy , gu ) = sup{β : (y + βgy , u − βgu ) ∈ P (x)}, β ≥ 0, D

(7)

 I (x, y, u; gx ) = sup{β : x − βgx ∈ L(y, u)}, β ≥ 0, D

(8)

and

 O measures the feasible joint expansion of intended output in the direction where D  I measures the gy and contraction of unintended output in the direction gu , while D feasible contraction of inputs in the direction gx . A distance value β = 0 implies that an observation is operating on the corresponding output or input frontier, and the distance value increases with inefficiency.  O in the context of pollution resulting from Chung et al. [29] first introduced D production. One important distinction between the directional and radial models for environmental applications is that while both models can be used to contract undesirable production outputs, radial distance functions treat the bad output as an input, as opposed to the directional distance function which explicitly models u as an output. While the radial approach has been commonly used to model the PGT, the treatment of bads as inputs can imply substitute relationships between pollution and other inputs that may not exist in practice. Both types of distance models and their corresponding production set representations satisfy key axioms and desirable mathematical properties from production theory. Given the empirical focus of this chapter, we omit their discussion here and instead point the reader to [25], as well as [46] for more theoretical detail. Bostian et al. [18] also provide summary discussion of functional form and parameterization of distance functions. Dakpo et al. [34] review recent nonparametric developments for modeling the PGT. To aid in interpreting the empirical work that follows, we illustrate the Shephard and directional distance functions, along with their corresponding technology sets in Fig. 1. In the left panel (i), the radial Shephard output distance function projects observation A to point B on the production frontier, expanding both goods and bads.

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(i) PGT Output Set

(ii) PGT Input Set

y

x2 A

B

C

C B L(y, u)

A (gy , gu )

P (x)

gx u¯

u

x1

Fig. 1 Radial and directional distance function models of the PGT. (i) Output distance, (ii) input distance

The directional output distance function instead projects observation A to point C on the frontier, expanding goods and contracting bads. In the right panel (ii), both models contract inputs to the frontier. The Shephard input distance function contracts A radially to point B while the directional input distance function to point C. Two important concepts follow from the production technology. The first is technical efficiency, which measures the extent to which an observation attains maximal output production and/or minimal input use, relative to the technology frontier. The second is productivity, which measures the change in overall production over time, accounting for both change in technical efficiency and change in the production technology. Distance function values provide a measure of technical efficiency, or inefficiency in the case of the directional distance function, by modeling distance to the frontier. Using Fig. 1 to illustrate, the radial Shephard OA efficiency measure is the ratio of OB in panel (i) and its inverse in panel (ii), while the directional inefficiency measure is given by AC in both panels. Productivity indexes (or indicators for additive models) can then be constructed from distance and technology estimates across time. Two widely applied productivity measures are the Malmquist productivity index [24, 43], in the case of Shephard distance functions, and the [70] productivity indicator [29], in the case of directional distance functions. Both the Malmquist and Luenberger can be used in an environmental context to incorporate unintended outputs into composite productivity measures and decomposed into separate measures of efficiency and technology change. We note that the output set in panel (i) of Fig. 1 also illustrates two commonly made assumptions in the empirical literature for the PGT P (x), null jointness and weak disposability for u and y. The null joint condition implies that for (y, u) ∈ P (x), if u = 0, then y = 0. Related to this, the unintended output also entails some cost of disposal. While y and x are generally assumed freely disposable (i.e.,

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the firm could always simply use fewer inputs to produce less output), u and y are only weakly disposable. More formally, if (y, u) ∈ P (x) and 0 ≤ θ ≤ 1, then (θy, θ u) ∈ P (x). Weak disposability imposes a physical trade-off between y and u at the margin, where any reduction of the bad requires some proportional sacrifice of the good. The weak disposability assumption has however generated some debate in the literature, particularly concerning the materials balance condition and other physical laws governing mass and energy flows [30, 53, 55]. However, [89] shows that the model can be specified to make weak disposability consistent with materials balance. Bostian et al. [18] also note that for empirical work, physical laws such as materials balance are presumably borne out by the observed data, obviating the need to impose the materials balance condition ex post. The framework above models the PGT in quantity space, in terms of the physical relationships between production, abatement, and pollution. We review related empirical applications to externalities, efficiency, and productivity in section “Externalities, Efficiency, and Productivity.” Duality theory connects these physical production relationships to price space, allowing for valuation of nonmarketed environmental effects, the focus of section “Valuation and Substitution.” Namely, both the radial and directional input distance function are dual to firm cost, while the output-oriented counterparts are dual to firm revenue. A number of empirical studies exploit these dual relationships to estimate shadow prices for generated pollution, abatement, and changes to environmental quality [16, 42]. For a detailed exposition of the relevant duality theory, refer to Färe and Primont [52] for radial models and [46] for directional models. Färe et al. [51] include a comprehensive review of both the theory and shadow price applications. We review related empirical applications in section “Valuation and Substitution,” for both costs and benefits of generated pollution, abatement, and environmental quality.

Externalities, Efficiency, and Productivity In practice, externalities frequently result from production activities. Examples include soil and water pollution from agriculture and mining, as well as both localized air pollution and greenhouse gas emissions from manufacturing and energy sectors. Unlike other outputs of production, externalities are often unintended and non-marketed. Reducing them requires some reallocation of resources away from intended output, so that in the absence of regulation or appropriate policy incentives, firms have little reason to factor externalities into their production decisions. However, the production theoretical framework outlined in section “Modeling Production and the Environment” can be used to better understand how externalities are generated by the production process, the potential for their reduction, and to incorporate externalities into composite productivity measures. That said, the framework provides little guidance in determining whether or not production outputs, intended or otherwise, actually constitute externalities. For example, waste generated from livestock might be considered an externality if allowed to run off the farm into surrounding waterways, or an internalized cost if

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the farm is charged a fee for any runoff that occurs. That same waste might even be considered a desirable intermediate output/input if it is collected and subsequently used to fertilize crops. Thus, for empirical application, the practitioner must first identify the relevant production and environmental processes.

Radial Models One of the first, and most widely cited, studies to include externalities in the production technology is that of [41], who specify a hyperbolic distance function to radially expand and contract desirable and undesirable outputs, respectively. This hyperbolic specification can be considered a pre-cursor to the more general directional distance function, which can also be used to model joint expansion and contraction of production inputs and outputs. In their application to US pulp and paper production, the authors find that introducing the environmental objective to contract effluent emissions generally decreases estimates of firm inefficiency. In other words, without having to also reduce emissions, firms could produce far greater levels of intended output with a given set of input resources. Other hyperbolic distance applications include [78], who incorporate toxic release data into efficiency estimates for US oil refineries; Yang et al. [98], who incorporate energy use and emissions reductions into productivity measures for provinces in China; Mamardashvili et al. [74], who incorporate nitrogen pollution into efficiency estimates for Swiss dairy farms and estimate shadow prices (discussed more in section “Valuation and Substitution”) for nitrogen production; Duman and Kasman [40], who estimate potential CO2 reductions for EU member and candidate countries, finding some evidence of convergence between the two groups; and Peña et al. [82], who incorporate both desirable and undesirable externalities, forest preservation and degraded land, into efficiency measures for ranching in the Amazon. We now turn to the more widely applied Shephard and directional distance literature. Given their radial nature, Shephard distance functions are often used to construct Malmquist-type ratio measures of good to bad output, sometimes termed “environmental performance.” Färe et al. [47] introduce the use of Shephard-based Malmquist indexes to construct an environmental performance index as the ratio of two multilateral Malmquist productivity indexes, one measuring expansion of desirable outputs and the other measuring contraction of undesirable outputs. They apply this framework to emissions of CO2 , SO2 , and NOX for a cross section of OECD countries, finding an average environmental performance index value of roughly 1.10, which indicates higher proportional increases of good to bad outputs. Building on this static case, Kortelainen [63] introduces a dynamic Malmquist index for environmental performance, along with its associated decomposition into environmental efficiency change and environmental technical change. He applies this framework to data on production value added and air pollution for the EU member states from 1993 to 2000, finding an overall improvement in environmental performance, driven mainly by environmental technical change. Zhou and Ang

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[102] apply a similar Malmquist framework to CO2 emissions for OECD countries, further decomposing productivity into multiple aspects of efficiency and technology change, including the change in carbon factor, energy intensity, emissions, and GDP. Färe et al. [44] apply a dynamic Malmquist index to air pollutants for US coalfired power plants, from 1998 to 2005. They find average overall improvements but also a large amount of variation, with more than 25% of plants exhibiting deteriorating performance for the study period. Barnhart et al. [12] apply a multilateral Malmquist framework to agricultural nitrogen and phosphorus pollution from crop and livestock production in the US Mississippi-Atchafalaya river basin. They find that for many of the corn- and soy-producing regions that are most targeted for nutrient reduction, the production increases proportionally outweigh increases to pollution, while overall environmental performance tends to be lowest for marginal low-production lands. Bostian et al. [17] extend the Malmquist index approach to a network technology setting, to incorporate investments in environmentally friendly practices into measures of environmental performance, energy efficiency, and overall productivity for Swedish pulp and paper firms. They find that these three aspects of change in production generally move together for firms in the industry.

Non-radial Models Chung et al. [29] introduce the use of the non-radial directional distance function to model the joint production of intended outputs and unintended externalities. They also develop an additive productivity indicator, which they term the “Malmquist-Luenberger” index, with associated decomposition of efficiency change and technology change. This represents the first empirical environmental application of the directional distance framework, using a panel of Swedish pulp and paper firms to incorporate multiple water pollutants into the production technology. They find sustained average improvements to productivity over the study period, with technology gains outweighing efficiency losses. The early work by [29] has led to a proliferation of non-radial, directional distance function approaches in the empirical literature. Zhang and Yongrok [101] review this progression more comprehensively. We provide selected references here. Weber and Domazlicky [95] provide an early application, adjusting US manufacturing productivity estimates to also include toxic releases. Picazo-Tadeo et al. [83] apply the directional distance framework to estimate both the regulated (assuming costly disposal of bad outputs) and unregulated production technologies (assuming free disposal of bad outputs) for firms in the Spanish ceramic tile industry. They interpret the difference in frontier output between the two as the regulatory impact, finding an aggregate regulatory impact of roughly 4.8% foregone production value. Kumar [64] uses a similar technology comparison, estimated with and without free disposal of bad outputs, to consider the potential for productivity convergence in the face of regulation for CO2 emissions. Application to a panel of both developed and developing countries reveals that on average the difference in productivity

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growth between the two technologies is positive for developed countries and negative for developing countries, suggesting a lack of convergence for environmentally sensitive productivity. Focusing just on the OECD countries, [81] estimates both country/year-specific technologies and a global reference technology which pools the panel in order to also estimate a technology gap component of productivity. Contrary to Kumar [64], Oh [81] finds that environmentally sensitive Luenberger productivity indicators fall behind those not accounting for pollution emissions, when restricting the analysis to developed countries. Zhang and Yongkrok [100] employ a similar global, or metafrontier, approach to fossil fuel-powered energy production in China. So also do [99], in their case to China’s provincial industrial sector. Wei et al. [96] extend this non-radial metafrontier approach to the 2015 Paris Agreement signatory countries, finding technology progress to be the main driver of improvements to CO2 emissions productivity. Numerous applications consider agricultural water use and related non-point source pollution. Färe et al. [49] use the directional output distance function to estimate shadow prices for runoff from fertilizer and pesticides for the US agricultural sector. Piot-Le and Le Moing [85] apply the Luenberger indicator to examine the productivity effects of participation in agri-environmental programs for French agriculture, finding support for the well-known Porter hypothesis (discussed more in section “Environmental Policy and Firm Performance”) that participating farms were more productive. Azad and Ancev [6] use the Luenberger indicator to estimate productivity of water use in the Australia agricultural sector, while [7] use the non-radial Russell graph measure to estimate a global efficiency index for irrigation in Australia. Bostian et al. [20] incorporate the directional output distance function into an integrated production-biophysical model to consider the optimal spatial allocation of nitrogen fertilizer reduction in a US agricultural watershed. More recently, [93] use the directional output distance function to estimate shadow prices for reductions to deforestation in the Brazilian Amazon. The authors then connect these shadow price estimates to a biophysical model for carbon sequestration to estimate trade-off values for CO2 emissions and agricultural production.

Network and Multi-function Models The relationship between production and the environment often depends on a series of processes across both space and time. The extension of standard single-stage production technology estimation methods to a multi-function, network technology framework can allow for more realistic structural representation of important underlying processes. Førsund [53] outlines a theoretical justification for this approach, while [18] review the subsequent development of network methods to estimate the PGT. Empirical work related to production and the environment increasingly makes use of network methods in order to better assess efficiency, productivity, and environmental performance. We review recent advances here.

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P1

(y 1f , u1f )

x1 x

y

(y 1i , u1i ) x

2

P2

u (y 2f , u2f )

Fig. 2 General two-stage network technology for goods and bads

To begin, we use Fig. 2 (adapted from [18]) to illustrate a basic two-stage network production technology for goods and bads, which can be extended to multiple stages in practice. The network framework separates the single-stage production technology into two underlying subtechnologies, P 1 and P 2 ; aggregate inputs, x, into subtechnology inputs x 1 and x 2 ; aggregate final outputs, (y,u), into subtechnology final outputs, (y 1f , u1f ) and (y 2f , u2f ). Intermediate inputs, (y 1i , u1i ), connect P 1 to P 2 , forming the network. The individual subtechnologies can be modeled using the distance function methods outlined in section “Modeling Production and the Environment.” The general network framework in Fig. 2 can be tailored to more detailed subtechnology processes for empirical application. Perhaps most prominent among these, [79] decompose the technology into separate goods- and bads-generating subtechnologies, using bads-generating and non-bads-generating inputs, respectively. They term this approach “by-production” and include an empirical application to US coal-fired power plants to illustrate. Chambers et al. [26] apply the byproduction approach to estimate shadow prices for nitrogen pollution from Catalan farms. In a related application, [90] incorporate risk by estimating the by-production technology within a state-contingent production framework. Dakpo et al. [35] apply a similar framework to estimate shadow prices for greenhouse gas emissions from French sheep farming, adding an additional materials balance constraint for badsgenerating inputs across the subtechnologies, which is consistent with the critique of [34]. Dakpo and Lansink [36] extend the byproduction framework to the dynamic case. Though not explicitly applying by-production, [54] develops a similar multiequation framework, termed “factorially determined multi-output production,” to also satisfy the materials balance condition. Ray et al. [88] apply a by-production model to country-level CO2 emissions, relaxing the weak disposability assumption for bads-generating inputs and pollution outputs to joint disposability instead. Both [73] and [66] estimate the by-production model econometrically, using Bayesian methods. The former use the results to construct an environmental productivity

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index, with decompositions for environmental efficiency and technology change, while the latter impose cost-minimization constraints to correct for endogeneity of production inputs. Arjomandi et al. [5] use the by-production framework to construct a Luenberger-Hicks-Moorsteen productivity indicator, along with associated decompositions, with application to global airline emissions. Murty and Russell [80] show that by-production satisfies the standard production axioms for PGTs and provide guidance for selecting functional form in estimation. Another strand of the environmental network technology literature addresses pollution abatement, both instead of and along with pollution generation. Indeed, [79] include abatement as an intended output in the intended production subtechnology, as does [54]. Färe et al. [45] develop a two-stage network, with a joint production PGT in the first stage, where generated pollution and intended outputs from the first stage serve as intermediate inputs in the second stage (similar to Fig. 2). They apply this approach to US coal-fired power plants. In a similar application, [56] adds a materials balance condition to the abatement subtechnology. Lozano [68, 69] builds on this to also consider allocative efficiency of abatement, in two related applications to US and EU power generation. Bi et al. [15] apply a similar production-abatement network model to power generation in China. Bostian et al. [18] extend the production-abatement network approach by decomposing total abatement into “beginning of pipe” prevention activities in the first stage and “end of pipe” treatment activities in the second stage, with application to Swedish pulp and paper firms. The authors also consider the optimal allocation of investment in abatement between prevention and treatment. Network and multi-function approaches can also be used to model environmental production processes over time and space, incorporating dynamic relationships between production, abatement, and emissions. Färe et al. [48] apply a dynamic model termed “time substitution” to estimate the optimal time path of GHG emissions reductions for the Kyoto Protocol signatory countries, in light of technology change. Zhou et al. [103] extend this approach to allow for spatial-temporal substitutions of CO2 emissions reductions in China. Bostian et al. [20] and Whittaker et al. [97] use bi-level optimization methods [11] to integrate the environmental production framework with a biophysical model to estimate the optimal spatial allocation of nitrogen reductions from agriculture in the USA. Bostian et al. [18] model the optimal dynamic allocation of environmental investments and expenditures to reduce GHG emissions in Swedish manufacturing. Cheng et al. [28] model the optimal time path for emissions reductions in China, under increasingly stringent emissions targets.

Valuation and Substitution In this section we look at applications where the environment is treated as an input or as an output in the production function. We focus on valuation and substitution. We mention briefly shadow pricing/marginal productivity of environmental factors. The previous sections have primarily considered the costs of abatement or benefits of

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pollution from the firm’s perspective. In this section we also consider the benefits of environmental protection and distinguish these from abatement costs in the context of production. The production theoretical background on the valuation of benefits or costs resulting from a change in the environment can be found in [76], which is based on the standard case – without explicitly considering the environment – as described in [60]. The theoretical background for empirical applications on the environmental input substitution part stems from the seminal work on mass/material balance production functions by [8] and by more recent elaborations on that work, e.g., [94]. A well-cited paper on undesirable output substitution and shadow pricing undesirables is [42], where both deterministic and stochastic directional distance function methods are used to estimate and illustrate the axiomatic models discussed earlier in this chapter. See also [51] for more shadow pricing applications.

Valuation The environment affects the production opportunities of firms. Think of the simplest case: a single firm selling its output and buying its inputs on competitive markets. Output y depends on inputs x and an environmental quality input q (generally a “good”). Now, suppose we can write the transformation function t (y, x; q) = 0. The associated production function is assumed to be concave and non-decreasing in inputs, including q. Let us suppose some environmental damage occurs, so that q decreases from q0 to q1 . The resulting cost to the firm, given no price changes, is simply the change in producer surplus (quasi-rents) or profits, π(p, w, q 1 ) − π(p, w, q 0 ), where p and w are prices of output and inputs, respectively. When more than one firm is affected, the total change in producer surplus is simply the sum of changes to profits across firms. We note that a given change to the environment may benefit some firms while harming others. The classic laundryfactory example illustrates this point, where pollution from the factory harms the laundry but benefits the factory; the reverse holds for the case of abatement by the factory. If complete data on profits before and after an environmental change is available, then calculating the change in producer surplus becomes a simple exercise. But this is rarely the case, and we have to attempt recovering essential parts of the profit function or approximations. Note that here if the resulting change in profits from a decrease in q is positive (e.g., due to avoided abatement costs), then we are referring to the costs of environmental protection; if the resulting change is negative, then we are referring to a benefit of environmental protection. Assume a profit function π(p, w, q) = max [py − C(y, w, q)]. Figure 3 y

illustrates the change in costs – and consequently producer surplus – for a given change in the environment, q, from q 0 to q 1 holding the output price constant. Think of this change as going from bad to worse in terms of pollution. Less resources spent on internal pollution control means lower costs for the firm, ceteris paribus.

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Monetary units Cy (y, w, q 0 ) Cy (y, w, q 1 )

p0

A

B

D C y(p0 , w, q 0 )

y(p0 , w, q 1 )

Output

Fig. 3 Production cost shift due to changing environmental quality

The cost savings are represented by the area ABCD, i.e., the increase in producer surplus. The approach illustrated in the figure would not be appropriate if our interest lies in looking at the benefits of environmental improvements, since here we are focusing solely on the costs. If the opposite shift in costs were to occur, where a decrease in q from q 1 to q 0 resulted in an increase in costs (e.g., the case where more pollution reduces the productivity of other inputs), then this same area ABCD would represent a benefit of pollution control to the firm. However, this benefit to the firm may not represent the full benefit of pollution control to society; that would depend on whether the pollution is fully internalized. The framework says little about the negative externality since it is in the context of a single firm and is focused on production possibilities rather than behavioral choices, which in turn hinge on whether the costs of environmental degradation are internalized or not. A comprehensive welfare measure would include both the benefits and costs of a decrease in q, which cannot be generally represented by the framework in the figure. One obvious way to recover the production costs or benefits of changes in the environment is to estimate a profit function and calculate the difference in value evaluated at different levels of the environmental input. But adequate data on profits are not generally available. Alternative approaches involve using the envelope properties of profit, cost, and revenue functions to estimate behavioral supply or demand functions which incorporate the environment. With the proper restrictions on the profit function, these behavioral functions can provide a valuation measure for production effects due to changes in the environment. In the absence of adequate data to estimate output supply, another option can be to instead estimate input demand functions. McConnell and Bockstael [76] show

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that if the input is essential (zero amount of that input means zero output), then it is possible to relate the change in input demand to producer surplus and changes in welfare due to a change in environmental quality. The environment only affects a firm’s production if the input is positive. Assume, for example, that one watershed is polluted by industrial activity. The producer surplus associated with changes in the pollutant can be estimated as the change in the area under the derived demand for the contaminated water. The researcher needs to be able to estimate supply and/or demand functions which are continuous in prices and that include the environment explicitly. There are examples in the literature where this approach has been used. One of very few examples is [61], who derive a fishery supply function which is modeled as depending on sub-aquatic vegetation levels. Another example is [77], where benefits from a fishery are studied when derived demand and supply functions depend on water quality. Most applications are in developed countries where markets are “well-behaved” and an ample amount of alternatives exist to reasonably expect supply and demand function to be continuous in prices. Applications in developing countries with limited data or not well-functioning markets often resort to practical approaches that attempt to approximate producer surplus effects. McConnell and Bockstael [76] chronicle these practical approaches, which include (i) valuing changes in output through a damage function, (ii) valuing only changes in output, and (iii) assessing changes in costs. More recent dual production economics approaches in the spirit of [61] are scarce. A few can be found in [9], who reviews studies on valuing ecosystem services as inputs in a fish habitat setting. Another more recent example also looking at a supply function and estimating the value of changing ocean habitats is [57]. In sum, we see some interest in the literature in using production economics for valuation when the firm is subject to changes in environmental factors. However, even though the theoretical underpinnings are quite robust – as outlined in [76] – the use of production economics for comprehensive welfare analysis is limited to instances in which the environmental effect is borne solely by producers. This holds if the environmental effect is fully internalized to the firm, or if the profit measure evaluated includes the combined production benefits and costs to all firms, so that the change in profit represents the change in total net benefits. In comparison to using production economics to value change in producer surplus, using primal or dual representations of technologies and looking at substitutability, marginal productivity, or shadow pricing of environmental factors are more common in the literature. We first turn to studies looking at marginal productivity and shadow pricing, since those concepts are connected to valuation.

Marginal Productivity and Shadow Pricing Marginal productivity and shadow price estimates of environmental factors can be useful in evaluating policy measures or economic assessment of changes in the environment. A number of authors and studies have taken this more primal approach to evaluate the marginal productivity of environmental factors as inputs

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in a production function. Note that these studies on marginal productivity provide estimates of the benefits of environmental improvements, while shadow pricing generally provides an estimate of the costs in terms of foregone production. The production function does not have to be that of a firm, but can instead pertain to any type of economic agent or decision-making unit amenable to analysis within this framework. For example, the marginal productivity of quality of ocean fish habitats has been studied by, e.g., [1, 10, 75]. See [9] for more references on similar studies. These studies analyze the potential increase in marginal productivity from improving the ocean fish habitats. In a similar vein, [62] examines Brazilian forests and the value of fallow ecosystem services in shifting forest management cultivation, including hydrological externalities that may affect other farms. The author estimates a production function to assess the value of forest fallow and test whether it provides local externalities to agricultural production. Applications that shadow price bad outputs are quite common in the literature, especially after the widely cited work of [42]. This influential paper uses a quadratic directional output distance function to estimate the shadow price of SO2 for 209 electric utilities before (1993) and after (1997) implementation of Phase I regulations of the US acid rain program. They find that the shadow price of pollution increases over the study period, reflecting the increased stringency of regulations and, as a consequence, the increased cost. While the majority of shadow pricing examples pertain to quantitative changes in pollution levels, [16] apply this framework to qualitative changes in the environment, estimating shadow prices for improved wetland condition in an agricultural watershed.

Substitution This section describes substitution relationships between the environment and production, both from the input side and the output side. We start by looking at the environment as an input. This input is not to be seen as an input that is used up in the process, that is, as a productive input such as soil or water. Rather it is – in what is described below – defined as emissions or pollutions that serve as an input by enabling more production (cutting emissions thus restricts production). It is important to understand the distinction between these two concepts of inputs. The actions of firms usually involve the direct or indirect use of environmental resources that are altered from a natural state to a more or less degraded state. Ayers and Kneese [8] developed the material or mass balance principle to measure this transformation, arguing that material inputs should be defined more broadly to include water and air in addition to fuels and conventional material inputs. Van den Bergh [94] claims that neoclassical production functions are not inherently inconsistent with mass balances, but notes that empirical models based on production economics often do not explicitly explore input substitution possibilities for improving/degrading the environment.

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In an attempt to close this gap in the literature, [31], building to some extent on [23], develop a model for US electricity production that includes conventional inputs, along with environmental resources as factors of production. This study examines the case of sulfur emissions trading, where emissions serve as environmental resources. The authors employ a dynamic factor demand model developed originally by [84]; that is, quasi-fixed stocks potentially induce adjustment costs, and firms minimize the expected sum of discounted (translog) costs so that factor demands and Euler equations (for quasi-fixed factors) are given by the solution to a dynamic optimization problem. Quasi-fixed factors are the stocks of capital and permits, respectively. The user cost for permit stocks is proxied by the pollution allowance price. Considine and Larson [31] motivate their specification of a mass balance production function by referring to [8] and go on to argue that according to the material balance relationship, the amount of environmental pollutants should approximately equal the weight of energy and raw materials inputs, which include minerals, water, air, and other environmental resources. Førsund [54] critiques this setup, which we explain in more detail below. Results in [31] suggest that considerable substitution possibilities exist between environmental emissions and other inputs in US electric power generation. While emissions prices are significant, relative fuel prices are more important in determining factor substitution than emission permit prices. Yet, the substitution elasticities between labor, capital, and emissions are significant. Considine and Larson [32] – using a similar approach as [31] – study fossil and fossil-free fuel substitution in electricity production, following the introduction of the European Union’s Emissions Trading System for greenhouse gas emissions. Despite low emission permit prices, this study finds statistically significant substitution between fossil fuels and fossil-free energy in electric power production. Førsund [54] finds treating pollution as an input to some extent problematic, arguing that this option is proposed without proper motivation in, e.g., the influential textbook of [13]. In motivating this approach, it has been argued that good outputs increase when pollution increases; less resources are used on pollution abatement, and these resources are then reallocated to output production (e.g., [33]). However, in the standard production economic setting, inputs are given, so there is no leeway to use inputs for abatement without extending the model. Another explanation is that generation of pollution needs services from the environment (nature) to take care of these residuals and that such services working as inputs can be measured by the volume or weight of the residuals (this is the motivation that [31] is embracing). However, measuring environmental services this way, an increase in the use of the environment cannot increase desirable output for given resources, because this is impossible keeping inputs constant due to the materials balance condition. Førsund [54]: “A partial increase in a residual as input cannot technically explain that a good output increases by reasoning that inputs are reallocated from abatement activity to the production of goods. Again, by definition, the inputs that are explicitly specified in this relation must be kept constant. Having sort of additional inputs behind the scene is not a very satisfactorily way of modelling.”

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Now let us turn our attention to the case when the undesirable is treated as an output or by-product in production. A string of studies have looked at substitution among pollutants when they are considered as non-separable or joint outputs. These include [2, 42, 48, 65]. Färe et al. [42] use a quadratic directional output distance function to measure the Morishima output elasticity of substitution between US electricity production and SO2. They suggest that the elasticity of substitution shows that the ability to trade reductions in electricity production for reductions in emissions became more difficult after the implementation of Phase I regulations of the acid rain program. The elasticity is estimated using both deterministic (DEA) and stochastic (SFA) techniques. Kumar and Managi [65] test the implicit assumption in the empirical literature that (i) production of desirable output, undesirable output, and abatement are separable and (ii) that different undesirables can be abated separately. Using a unique plant-level data set from India, they find sufficient evidence to reject the null hypotheses of separability between marketable output and pollutants and between different pollutants. Firms must incur abatement costs for reducing pollution levels. In addition, they find statistically significant complement and substitute relationships between water pollutants. Färe et al. [48] use a directional distance function to estimate the Morishima transformation elasticity between SO2 and NOx, as well as the desirable output in US electricity-producing utilities during a period when they were subject to regulations associated with the Clean Air Act Amendments of 1990. The main finding is that SO2 and NOx are substitutes, which implies that any gains in benefits from the reduction of one undesirable are being partially offset by any costs due to increases in another undesirable. Agee et al. [2] use a 10-year panel for 77 US electric utilities to estimate a multiple-input, multiple-output directional distance function, combining good inputs and a bad input to produce good outputs and bad outputs (SO2, NOx, and CO2). They find that considerable jointness (substitutability) exists between SO2, NOX, and CO2 emissions. They conclude that failure to account for this jointness increases the cost of pollution control.

Environmental Policy and Firm Performance Climate change and other rising environmental problems have motivated governments to introduce or plan various types of environmental policies to mitigate or adapt to these problems. One concern is that strict environmental policies will set back growth and hinder development, at least for standard goods. The conventional or neoclassical view is that stricter policies related to the environment lead to an increase in cost, lower productivity, and decrease in profitability. The literature on competitiveness, economic performance, and environmental and/or energy and climate policy is vast; there are at least five comprehensive

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reviews published in the general area of environmental policy and firm performance [4, 21, 38, 39, 58]. Interest in this line of research dates back to the early 1990s, when Harvard professor M. Porter challenged the conventional wisdom about the impact of environmental regulation on firms. Two influential works, [86] and [87], introduce the argument that well-designed or “right-kind” of regulation could actually increase competitiveness, widely known today as the Porter hypothesis. In the empirical literature on the impacts of environmental regulations on firms’ performance, competitiveness is typically measured by trade, industry location, employment, productivity, or innovation (or some combination of these variables). Here we review some of the recent studies focusing on production measures such as total productivity, efficiency, and technological change. The review by [58] directly responds to Porter’s claim. The authors find no systematic evidence supporting the revisionist hypothesis that environmental regulations stimulate innovation/productivity and improved competitiveness. Brännlund and Lundgren [21] conclude that the theoretical literature can identify the (rather non-general) mechanisms that must exist for a Porter effect to occur, but the empirical literature gives no general support for the Porter hypothesis; in terms of productivity, the impact is typically negative (see also [22]). Ambec et al. [4] reach a similar conclusion as that of [21], i.e., that the empirical evidence of the Porter hypothesis in terms of productivity enhancements is mixed, however, with more recent studies suggesting somewhat more clear support. Dechezleprêtre and Sato [38] conclude that evidence indicates that environmental regulation has both negative, short-term impacts on productivity in some sectors and for some pollutants and positive productivity impacts in others; again the main message is that results are mixed. Dechezleprêtre et al. [39] observe that recent interest in a causal relationship between environmental and economic performance is indicative of a new wave of research; that is, being more “green” – be it because of internal or external policy – not only affects firm costs but also potentially affects firm revenues. This connects to some extent to the literature on corporate social responsibility (CSR), which we do not include in this chapter. Dam et al. [37] provide a comprehensive exposition of the theory underlying CSR and include recent applications for reference. Before proceeding to recent empirical studies, we illustrate the Porter hypothesis from a production economics perspective. How should we think about environmental policy and competitiveness? Figure 4 illustrates the basic Porter argument using the relationship between the firm’s desirable and undesirable output (figure recreated from [21]). According to Porter, a regulation will highlight inefficiencies in a company. One way to illustrate this in the pre-regulation period is to assume that a company is not producing on the production possibilities frontier but rather at point C. The “rightkind” of regulation of emissions from z0 to zR would highlight inefficiencies, which would allow the company to move (outward) to the production possibilities curve. At point B, all inefficiencies are neutralized and the company increases production (from q 0 to q R ), earns higher profits (from π 0 to π R ), and, at the same time, reduces emissions. The hypothesis implicitly assumes that it is cost-free to move toward the

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P roduction, q

q

q = fR (z) π1

E

1

A qR

q = f0 (z) πR

D B π0

q

0

C

zR

z0

Emissions, z

Fig. 4 Illustration of the Porter hypothesis

frontier. There could, of course, be several reasons why a company might not be producing efficiently, or on the frontier. The Porter hypothesis stipulates that environmental regulations have a “dynamic” effect in that they stimulate innovation and new processes. Figure 4 illustrates this by showing how the production possibilities frontier shifts upward, representing a “new” technology f R (z). The production and emission levels at point B are now inefficient under the new production technology, but the regulations also make this inefficiency visible to the company. Ultimately, this means that the company will move itself from point C to B and then move even further as a result of the new technology to a point between D and E. Given stable prices (in both the product itself and the “emissions” input), the firm maximizes profit at point E. We note that this result of the Porter hypothesis relies on the implicit assumption that the development of the new technology does not make use of the company’s alternative productive resources; i.e., there are no crowding out effects, or at least they are very small. Below we look at the main results from some of the most influential papers concerning the Porter hypothesis and its connection to productivity and/or efficiency. Also, we provide a sample of more recent studies on the subject. Additional references can be found in the reviews cited above. In an important early contribution to the empirical literature, [59] divide the Porter hypothesis into three distinct testable versions: (1) the “weak” version posits that environmental regulation will stimulate environmental innovations; (2) the “narrow” version asserts that flexible environmental policy regimes give firms greater incentive to innovate than prescriptive regulations, such as technology

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standards; and (3) the “strong” version stipulates that properly designed regulation may induce cost-saving innovation that more than compensates for the cost of compliance. Using US survey data on pollution costs and expenditures (PACE), they find evidence supporting the weak and narrow versions but not for the strong. The remaining studies included here mainly concern testing the strong version within a productivity or efficiency modeling setup. Berman and Bui [14] examine the effect of air quality regulation on productivity in oil refineries of the Los Angeles Air Basin. They use local air pollution regulation to estimate the effects on the demand for abatement capital during a period of sharply increased regulation, 1979 to 1992. They construct measures of total factor productivity (TFP) using data on physical quantities of inputs and outputs. Despite high costs associated with regulation, TFP rose sharply between 1987 and 1992, a period of decreased refinery productivity in other regions. The authors conclude that abatement costs may grossly overstate the economic cost of environmental regulation as abatement may very well increase TFP; this conclusion would not rule out support for the strong version of the Porter hypothesis. Shadbegian and Gray [91] investigate the impact of abatement expenditures on productivity, using plant-level data for 68 pulp and paper mills, 55 oil refineries, and 27 steel mills for the 1979–1990 period. They estimate a Cobb-Douglas production function to measure the contribution of capital, labor, and material inputs to output. Their access to data on abatement expenditures allows them to distinguish between productive and abatement expenditures for each input. They find that abatement expenditures contribute little or nothing to production or productivity, a result that somewhat contradicts [14]. Lanoie et al. [67] test the significance of all three different variants of the Porter hypothesis. Their analysis draws upon a large database that includes observations from approximately 4,200 facilities in 7 OECD countries. In general, and consistent with [59], they find support for the weak and narrow version but no support for the strong version. Lundgren et al. [71] analyze productivity effects of the Swedish CO2 tax and the European Union Emissions Trading System (EU ETS) for the Swedish pulp and paper industry 1998–2008. They compute a Luenberger TFP indicator using DEA. The results indicate that these climate policies had a modest impact on technological development in the pulp and paper industry, and when significant, these effects were negative. They conclude that when designing policy to mitigate CO2 emissions, it is vital that the policy generates a carbon price that is high enough to put pressure on technological development. Lundgren et al. [71] investigate how firm-level environmental performance (EP) – a measure described earlier in this chapter – affects firm-level economic performance, measured as profit efficiency (PE) in a stochastic frontier setting. Analyzing firms in Swedish manufacturing from 1990 to 2004, their results show that EP induced by environmental policy does not determine PE, while voluntary or market-driven EP seems to have a significant and positive effect on firm PE in most sectors. The evidence generally supports the idea that good EP is also good for business (CSR), as long as EP is not brought on by external policy measures, in this

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case a CO2 tax. Thus, the results provide no general support for the strong version of the Porter hypothesis. However, the results also show no significant negative effect of policy on efficiency. Using the same data set and modeling framework, similar results are found in [50, ch. 5]; i.e., the effects of a CO2 tax on efficiency are mixed, depending on what industry sector is analyzed; but some evidence is found for a positive dynamic effect of the CO2 tax on those sectors that exhibit the largest inefficiencies. Albrizio et al. [3] investigate the impact of environmental policy on industryand firm-level productivity growth in a panel of OECD countries. To test the strong version of the Porter hypothesis, they use a productivity model to allow for effects of environmental policies. They find that a tightening of environmental policy is associated with a short-term increase in industry-level productivity growth in the most technologically advanced countries. This effect diminishes with distance from the global productivity frontier. For the average firm, no evidence of the Porter hypothesis is found. However, the most productive firms see a temporary boost in productivity growth, while less productive firms experience a productivity slowdown; this directly contradicts the findings in [50]. Rødseth [89] looks specifically at environmental regulations and their implications for allocative efficiency. He establishes a model framework that allows disentangling managerial and regulatory-induced allocative efficiencies. Applying DEA for estimation to a sample of 67 coal-to-gas substituting power plants from 2002 to 2008, he calculates Nerlovian profit efficiencies and their technical and allocative efficiency components. The empirical results illustrate that failing to control for environmental regulations leads to overestimation of managerial allocative efficiencies by ignoring compliance costs. From this meta-review of earlier reviews and considering the more recent results discussed above, the jury is still out on the Porter hypothesis; while it is possible to find cases and circumstances where it seems to hold, even the strong version, the literature lacks more general support. We see that the literature associated with analyzing environmental policy and firm performance has many applications using the toolbox of production economics, some of them mentioned above, especially at the micro level. The Porter literature runs parallel with the literature on selfregulation (CSR), as noted by [39], where the holy grail is to find a positive relationship between doing good and doing well; green business is good business. These two strands in the research on environmental policy (internal or external) and economic performance have, to a large extent, merged with an increasing focus on the effects of voluntary self-regulation and effects on firm performance (see, e.g., [37] for a theoretical discussion or, e.g., [72], or [27], for empirical applications).

Conclusion We intend for this chapter to serve as a starting point for those interested in environmental applications of production theory. Importantly, we have by no means provided a comprehensive survey. For instance, we omit the closely related topic of

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energy and the environment, as one could easily fill another chapter on this alone. The same is also true of agriculture. Instead, the selected topics reflect emphasis in the field of environmental economics, as well as our own areas of expertise. With this in mind, we conclude by considering the trajectory of empirical research in the field. Beginning with externalities and efficiency/productivity, there is growing work to incorporate better structural representations of the production-pollution-abatement technology nexus, largely through the use of multi-equation network models (see section “Network and Multi-function Models”). Related to this is the incorporation of environmental processes into the production model framework. This can include specifying a materials balance condition (à la Førsund [54]), as well as, and with the advent of recent computational advances, integrating more complex biophysical models [20, 97]. Turning now to production, the environment, and valuation, we see only limited efforts to value environmental changes by treating the environment as an input. While the theory is well-founded [76] and application seems straightforward, still we see surprisingly few empirical studies, most likely due to data limitations. The majority of existing applications also relate to fisheries production, mainly in the tropics. Studies on substitution possibilities between the environment as an input and conventional inputs are also quite scarce. The reference that stands out in the literature is the [31] application of a dynamic factor demand model based on a dual cost function. However, substitution studies treating the environment as an output are more common. The interest in assessing substitutability between different bads/pollutants (non-separability) and between intended and unintended outputs increased significantly after [42], who investigate substitutability and shadow pricing of unintended outputs, using both deterministic DEA and stochastic frontier analysis. The empirical literature on the relationship between environmental policy, economic performance, and environmental performance is now colossal and still growing. As noted by [39], the field is merging to some degree with the research on CSR. That is, both external policies and internal management policy decisions affect firm performance – both economic and environmental. We also see some interest more recently in using production economics in this context, e.g., [27, 71, 72].

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Phill Wheat, Kristofer Odolinski, and Andrew Smith

Contents Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Competition and Governance in the Transportation Sector . . . . . . . . . . . . . . . . . . . . . . . . . . Approaches to Production Analysis in Transportation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Key Features of Transport . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Outputs Used . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Input Prices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Variable Cost Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Cost and Efficiency Studies for Railways . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Infrastructure Studies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Passenger Train Operations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Cost and Efficiency Studies in Other Transport Sectors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Road Infrastructure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Local Public Transport . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Air Transport . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Cross-References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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P. Wheat () Institute for Transport Studies, University of Leeds, Leeds, UK e-mail: [email protected] K. Odolinski Institute for Transport Studies, University of Leeds, Leeds, UK Society, Environment, and Transport, The Swedish National Road and Transport Research Institute (VTI), Stockholm, Sweden e-mail: [email protected] A. Smith Society, Environment, and Transport, The Swedish National Road and Transport Research Institute (VTI), Stockholm, Sweden e-mail: [email protected] © Springer Nature Singapore Pte Ltd. 2022 S. C. Ray et al. (eds.), Handbook of Production Economics, https://doi.org/10.1007/978-981-10-3455-8_39

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Abstract

In this chapter, the experience of production and cost analysis in transportation is reviewed. The key production and cost analysis needs of the sector can be identified providing strategic operational insight, establishing evidence as to which market and regulatory structures yield best outcomes and providing the evidence base for regulatory scrutiny either through yardstick competition or more formal price cap regulation. The upshot of this is that transportation has provided the motivation and illustration for many innovations within production, cost, and efficiency methods, and this chapter brings to life the issues and solutions found in the sector. Keywords

Transport infrastructure · Public transport · Marginal cost · Cost efficiency · Railways · Roads · Air transport

Introduction Analysis of production and cost in transportation has a long history and the sector has provided motivation for many innovations in the economist production and cost analysis tool box. For example, the pioneering work of Caves et al. [1, 2] in the US railroads and airlines introduced the need to disentangle economies of density from economies of scale in network industries. Transportation has also been a key sector for illustration of methods used to analyze efficiency, including Swiss railways to illustrate panel data stochastic frontier models [3, 4] and models involving nonnormally distributed noise errors [5]. The richness of data and applications in transportation arises for a number of reasons. Firstly, some transportation could be considered a public good (for example, non-tolled highways); however, transport supply often requires subsidy or exclusivity to make provision viable and efficient (such as provision of infrastructure). This motivates a strong public interest in the sector, either through direct government provision, subsidy to private providers and/or a system of economic regulation. Secondly, the study of the economics of transportation is nontrivial given the joint production of transport services (for example, a bus service provides many journey possibilities along a route) and the temporal nature of transport services implies non-storability. Ultimately the public interest dimension, coupled with the complexity of the sector, means datasets are often potentially rich – in terms of input and output variables that are available for different locations and time periods – and therefore transportation cost and production analysis has become a major area of research. Such rich data enables a relatively complex characterization of outputs and inputs in the analysis and critically an analysis of the influence of quality. The structure of this chapter is the following. A brief overview of competition and governance in the transportation sector is provided in section “Competition and Governance in the Transportation Sector”. Section “Approaches to Production Analysis in Transportation” reviews approaches to production and cost analysis in

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transportation. It is important to recognize the variety of approaches beyond the econometric approach – including alternative top-down approaches and bottomup modelling approaches – which are used within transportation for both research purposes and to inform operational decisions in the sector. Section “Approaches to Production Analysis in Transportation” then considers the features of transportation which makes production analysis challenging. These include multiple outputs and multiple inputs, arising from the fundamental non-storability of the transportation product. Section “Key Features of Transport” presents empirical findings from the rail mode, where substantive econometric works have been undertaken, to act as an illustration of the analysis undertaken in transportation. Section “Cost and Efficiency Studies for Railways” then briefly surveys work across other transport modes. Section “Cost and Efficiency Studies in Other Transport Sectors” concludes.

Competition and Governance in the Transportation Sector1 Transformations of the transport industry and the differing approaches taken by the public sector was an early core subject of economic analysis, where researchers tried to answer questions such as why the public sector owns and/or regulates the production and pricing of certain goods and services, and whether the public sector should be involved in this matter or not. Ownership and regulation of the transportation sector has changed over the years, where large shifts have been triggered by innovations or even new modes of transport being introduced. For example, up until 1980, railways were often seen as a typical example of a natural monopoly where the government needed to intervene and reduce market inefficiency by the means of regulation. However, the benefits of public ownership and regulation of railways can be questioned when competition from other modes increases and more subsidy from the public are required to cover costs. This can explain the vertical separation of the railway sector in many European countries around the 1990s, where Sweden was the first to do so in 1988, mainly due to the growth of required subsidies to the state-owned railway company [6]. Other countries such as the USA and Japan chose a different path by keeping a vertical integration between infrastructure management and train operations. This can partly be explained by a reliance on intermodal competition, as well as parallel competition (e.g., two railway lines serving the same areas) and source competition (i.e., a producer/customer can choose another source and customer/producer of a certain good). Still, the vertically integrated railways are often regulated where for example the railway firms in Japan are subject to yardstick competition, as originally formulated by Shleifer [7]. Contestability is one explanation for the railway reforms chosen in Europe where train operation was, to various degrees, subject to competition while infrastructure management was kept as state-owned monopolies (apart from Great Britain where

1 This

section is partly based on Odolinski [77]

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the rail infrastructure company was privatized in 1996). Undoubtedly, rail infrastructure has a high level of sunk costs and is less contestable than train operations with rolling stock that may be used on another market – that is, train operations do not have the same entry and exit barriers as infrastructure provision. It was argued that the efficiency gains from competition between train operators could outweigh any potential losses caused by economies of scale not being fully exploited. However, the move towards open competition for train operations was slow in Europe, with mainly competitive tendering being used. Still, Sweden introduced open access to freight services in 1996 and to passenger services in 2011, while Germany at least nominally opened up for entry and on-the-tracks competition right from the start of the vertical separation [8]. Infrastructure management has used competition for the market to increase their cost efficiency, but compared to other industries, the tendering of infrastructure maintenance poses some challenges on its own. One example is the interdependence with renewals, where maintenance activities are carried out on a structure with a long service life, and both maintenance and renewals may affect future performance of the assets. However, future contingencies make it costly to write long-term maintenance contracts, that nonetheless have the benefit of inducing more investments in quality as the producer can recover its investment costs. Specifying and monitoring quality is thus an important aspect in infrastructure provision. This can in turn impact the choice between competitive tendering and in-house production. As pointed out by Hart et al. [9], in-house production can be preferable to contracting out to private firms when important quality aspects are ex-ante non-contractible and innovations in quality are not important. This is the case even though an in-house production unit has weaker incentives for investments in cost reductions, as well as for investments in quality, compared to the contracted firm. The detrimental effect in this case is that the firm will tend to focus too much on cost reductions compared to quality. Overall, incentive structures that are beneficial in the short run need not be so in the long run, which can especially be the case when managing an asset with a long service life. The internal organization of the infrastructure manager is also an important aspect in creating a cost-efficient production. A large organization may well have different working procedures between different units even though there is a central planning unit or a manager giving (more or less clear) instructions. Estimates of the relative cost inefficiency between units are therefore useful for internal benchmarking. A related issue is the size of the contract areas chosen by the infrastructure manager, which will determine whether economics of scale are fully exploited or not. There are clearly many different elements to consider when regulating or reforming the transport sector, and the effects of reforms and incentive structures are not obvious. Understanding the production and cost structure of this industry is therefore vital. To summarize, key production and cost analysis needs of the sector can be identified as:

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• Strategic operational insight: understanding how costs change with output to inform operational and investment decisions. Essential here is the allowance for the quality of output as well as a simple scale metric given the jointness and non-storability of production. • Establishing evidence as to which market and regulatory structures yield best outcomes given the complexity in the transport sector and the inevitable tradeoffs between maintaining network coordination and avoiding useless duplication of resources against potential inefficiency introduced by reducing competition for or in the market. • Providing the evidence base for regulatory scrutiny either through yardstick competition or more formal price cap regulation. In general, these issues would tend towards a direct analysis of costs. Indeed in many transport settings, public or private firms can be stylized as delivering a set of transport services (such as bus or train km) at a given quality for minimum cost, with public authorities being tasked with design of the transport system. Thus it makes sense to compare firms through the cost minimizing paradigm even if in reality there is some inefficiency relative to this ideal.

Approaches to Production Analysis in Transportation2 In this section of the chapter, we review the context in which econometric methods contribute to the broad understanding of production in transportation. There are three main approaches for production and cost analysis in transportation: • An econometric approach • An accounting (cost allocation) approach • A bottom-up approach using mechanistic models The approaches used need not be in the either of these extremes; there are, for example, hybrids or combinations of the different approaches. Moreover, the modelling approaches can be either deterministic or stochastic, where the former makes predictions that excludes random variation, while the latter predict distributions of potential outcomes. The econometric approach tries to establish relationships with factors that may explain variations in costs. This approach is often applied to analyze economies of scale, scope, or density, or the impact of organizational changes such as (de-) regulation. It is also a common approach when establishing relationships between output and (aggregate or disaggregate) costs, such as the impact traffic has on costs for providing rail infrastructure. This is the primary method used by economists and is the primary subject of this chapter.

2 This

section is partly based on Odolinski and Wheat 51

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The accounting approach has been used to allocate costs to different activities carried out in the provision of transport services. It is often used when the aim is full cost recovery, where the users of the transport services bear the full cost. Hence, the fees are usually set at average costs. This has long been the case within the civil aviation industry, where States have agreed on international policies on airport costrecovery. The International Civil Aviation Organization has, for example, developed a manual on airport economics [10] to assist in the management of airports and describes the accounting method at length. There is also a tradition of using the accounting approach in determining bus operating costs. Early reviews on these costing models are made by, for example, Cherwony et al. [11] and Stopher et al. [12], and Sinner [13] is a recent review and application of the cost allocation method for bus and train lines. The accounting approach also has a history in railway transportation. As noted by Braeutigam [14], the Interstate Commerce Commission (regulatory commission in the USA) applied the accounting approach to railways from the 1930s to the 1980s to determine the cost of transport services and the revenue required to cover their costs. Moreover, the British rail regulator ORR3 used a mix of the accounting approach and an engineering bottom-up approach prior to 2008 in order to determine the cost for infrastructure services. The allocation of costs to different activities can be carried out in numerous ways. Essentially it uses some set of allocation factors or principles to determine the allocation of fixed and joint costs or identify incremental or avoidable costs. Total costs are thus split into different categories or activities, in which the allocation factors depend on the nature of the cost category or performed activity. For example, time can be used to allocate costs for staff, while distance travelled, weight of the vehicle, or consumed capacity may be used for operation or maintenance costs. Link et al. [15] provide a list of different allocation procedures and allocation factors in studies using cost allocation in road infrastructure analysis and in air transport. The accounting approach, however, has a set of weaknesses. As noted by Waters [16], this approach struggles in determining the opportunity cost of the assets, as well as in distinguishing between fixed and variable costs. Costs in the accounting system are also often aggregated making it impossible to determine a cost for an output that may be important for decision-making and efficient management, for example, the cost of transport in peak hours. Another downside is that the accounting approach calculates average costs for different activities and not the marginal cost. From an efficiency perspective, the latter can be preferred in decreasing cost industries as a basis for price regulation, and transport regulatory bodies have often sought estimates on marginal costs, especially for railways. In fact, the use of cost allocations for pricing purposes has been disapproved by economists since the late 1800s, partly due to its rather arbitrary cost allocation procedures, cf. Taussig [17] and Griliches [18].

3 As

of 2021, Road; previously the Office of Rail Regulation and prior to that the Office of the Rail Regulator.

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Both the econometric and accounting approaches are known as top-down approaches as they allocate total cost to several drivers of costs. The benefit of these approaches is the simplicity and that they use actual costs. However, the resolution available on the underlying process relationships through the top-down approach is limited, as the underlying mechanisms are highly correlated or there are (unknown) mediating variables. The bottom-up (engineering) approach can be particularly useful in these cases as it starts at a lower level to model the physical mechanisms behind the cost relationships. This approach has been used to estimate the relative cost impact of different vehicle types or specific characteristics of a vehicle. In short, simulations are performed based on a set of engineering models to provide estimates of the damage on the infrastructure caused by a certain characteristic of a vehicle or train operation. Another application concerns damages done to the vehicle using an infrastructure with a set of characteristics. These damages are then linked to maintenance activities or costs that can be implemented in order to reduce the need for future maintenance and minimize costs over the whole life cycle of the transport system in question or its subsystems (i.e., vehicles, infrastructure, etc.). In doing this, a cost function is useful (and required). The strength of the bottom-up approach is that it explicitly models the underlying mechanisms behind the cost relationships. However, the use of this approach is limited by data availability. For example, cost data are usually more aggregated than the predicted mechanisms. The bottom-up approach thus needs to link mechanisms with costs at a more aggregate level. In doing this, the elasticities of production need to be acknowledged. Using simple unit costs might, for example, be problematic as it ignores factors such as economies of scale and/or scope. Such aspects can be modelled with a statistical model. Hence, the bottom-up and top-down approaches can complement each other and a combination of them can be useful in transportation cost analyses.

Key Features of Transport4 In this section, the key features and issues in transport are reviewed. To illustrate, we utilize the literature in railways. Table 1 summarizes the key studies and states what function was estimated and the key inputs and outputs used. In the remainder of this section, the key advances in the methodology associated with econometric analysis of railway performance are surveyed. The multi-output nature of transport has motivated recent studies to use either the cost frontier or distance function. Distance functions are related to a multi-output generalization of the production function (the transformation function) and yield estimates of technical inefficiency through considering feasible radial expansions (contractions) of outputs (inputs) with respect to the production set.

4 This

section is based on an updated review by Wheat 91

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Table 1 Summary of the characteristics of parametric cost studies in railways Function estimated Translog cost function. Returns to scale and productivity studied but not explicit allowance for inefficiency Input specific technical inefficiency via Generalized McFadden cost function Deterministic production function

Study Andrikopoulos and Loizides [19]

Sample 1969–1993 European rail companies

Christopoulos et al. [114]

1969–1992 European rail companies

Coelli and Perelman [20]

1988–1983 European rail companies

Coelli and Perelman [21]

1988–1983 European rail companies

Deterministic production and input and output distance functions

Couto and Graham [22]

1972–1999 27 European railway companies

Short-run variable cost function with first-order cost shares to separate out technical and allocative inefficiency

Cowie and Riddington [23]

1992 European rail companies

Deterministic production functions

Deprins and Simar [24]

1970–1983 Europe + Japan rail companies

Deterministic production function

Inputs or prices used Total cost. Includes capital costs (historic cost depreciation + interest)

Outputs used Sum of passenger-km and Freight tonne-km

Total cost. Includes capital costs (historic cost depreciation + interest)

Total train-km

Number of employees Rolling stock capacity Route kilometers Number of employees Rolling stock capacity Route kilometers Input prices for labor, service rendered by third parties, equipment (variable inputs) and measure of capital stock. Also some network characteristic variables

Passenger-km and Freight tonne-km

Number of employees Capital (financial measure) Number of employees Number of coaches / wagons Energy consumption Route kilometers

Passenger km Freight tonne km

Two models: (1) Passenger-km and freight tonne-km (final outputs) (2) Passenger train-km and freight train-km (intermediate outputs) Passenger train-km Service provision index Total train-km

(continued)

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Table 1 (continued) Study Farsi et al. [3]

Sample 1985–1997 50 railway companies in Switzerland

Gathon and Perelman [25]

1961–1988. European rail companies

Gathon and Pestieau [26]

1961–1988 European rail companies

Ivaldi and McCullough [27]

1978–1997 25 US Class 1 Railroads

Kumbhakar [28]

1951–1975 13 US Class 1 Railroads

Kumbhakar [29]

1951–1975 42 US Class 1 Railroads

Function estimated Various stochastic total cost frontier specifications examining the effect of controlling for time invariant characteristics Stochastic factor requirement function

Inputs or prices used Input prices: energy labor and capital

Stochastic production function (also second stage regression) Translog variable cost function

Number of employees Number of rolling stocks Route kilometers Prices: indexes of labor, equipment, fuel, and materials

Cobb-Douglas stochastic distance function with demand system to separate out technical and allocative inefficiency Stochastic distance function with demand system to separate out technical and allocative inefficiency

Quantities of labor, energy, and capital

Passenger train-km Freight train-km Route km Sum of passenger tonne-km and freight tonne-km Car miles of (a) bulk, (b) high value, (c) general traffic and replacement of ties installed (infrastructure output) and also average length of haul and length of road miles Passenger-km and Freight tonne-km

Quantities of labor, energy, and capital

Passenger-km and Freight tonne-km

Number of employees

Outputs used Passenger-km and freight tonne-km

(continued)

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Table 1 (continued) Function estimated Output and Input distance function in a latent class framework Two distance functions one modelling technical efficiency the other modelling service effectiveness

Inputs or prices used Quantities of labor, energy, and capital

Short-run cost function (not frontier) with coefficients which vary by firm or year Short-run variable cost function with first-order cost shares to separate out technical and allocative inefficiency Stochastic cost function

Operating costs Capital stock (financial measure)

1970–1990 Europe rail companies

Stochastic cost and revenue functions – operating costs, revenue

1969–1992 European rail companies

Stochastic production frontier with firm environmental variables as determinants of mean inefficiency

Labor price, energy price, material price – price of passenger and freight outputs (rev model) Number of employees Energy consumption Capital (financial measure)

Study Kumbhakar et al. [30]

Sample 1971–1994 Europe rail companies

Lan and Lin [31]

1995–2002 39 international railways

Loizides and Tsionas [32]

1969–1992 Europe rail companies

Parisio [33]

1973–1989 8 European Railway companies

Cantos and Villarroya [34]

1970–1990 Europe rail companies

Cantos and Villarroya [35]

Tsionas and Christopolous [36]

Efficiency model: number of passenger rolling units, number of employees Effectiveness model: passenger train-km and freight train-km

Outputs used Passenger-km and Freight tonne-km Efficiency model: passenger train-km and freight train-km Effectiveness model: passenger-km and freight tonne-km Passenger-km and Freight tonne-km

Input prices: labor, energy, materials. Length of track is the measure of the fixed input

Passenger-km and freight ton-km

Variable cost (excludes capital cost)

Passenger train-km Freight train-km Passenger-km and Freight tonne-km

Passenger-km and Freight tonne-km

39 Applications of Production Theory in Transportation

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For cost frontier models, both variable cost frontiers (in railways, [22, 27, 33]) and total cost frontiers [3, 34, 35] have been estimated; the difference depending on whether the infrastructure is deemed quasi-fixed or variable. This decision is partly determined by the robustness of the available capital stock level variable(s) versus the capital price variable. For example, one challenge of the cost function is the difficulties in developing data on input prices, particularly infrastructure capital. Instead, he estimates a variable cost function which requires data on the levels of capital and not their associated price.

Outputs Used Network industries can be viewed as producing many different heterogeneous outputs. Transport networks in particular, given the non-storability of the product, the large number of origin and destination combinations, and the many different trip purposes, produce a very large number of outputs. In the limit, transport could be thought as producing individual travel opportunities, by time, space, and purpose. For public transport, such as railways, where the transport service serves many points at many times of day, such a disaggregation of outputs is likely to be too extreme to undertake meaningful econometric analysis. As such, a more pragmatic approach has to be taken in specifying outputs. Several common features of the output specification can be considered. First, two general classifications of outputs are common [37]. One set are termed “available outputs” which are measures of the service that the railway (or other transport modes) produces (capital supplied) which are available to customers to consume. Examples include train-km, vehicle-km, and seat-km. The second set are termed “revenue outputs” which are measures of consumed outputs. Examples include passenger-km and tonne-km of freight hauled. These two sets could also be thought of as intermediate versus final outputs of the transport system, although it must also be borne in mind that the demand for public transport services is often a derived demand. When choosing whether to use available or revenue outputs, it is important to consider what is required to be measured in the analysis and whether the implicit assumptions on what is under the firm’s control versus what is exogenous is reasonable. For example, using available outputs can be justified when considering the performance of a railway manager where the required outputs from the railway are heavily prescribed by a regulator or government. As such, the railway manager does not have much discretion as to how many train-km, vehicle-km, etc. can be run. This is instead set by the regulator. However, if analysis of the effect of government policy is the aim of a study, then it is more appropriate to adopt revenue output measures as policy makers have discretion in the specification of railway services to best

1502

P. Wheat et al.

meet demand. Of course, available outputs might be used in this context, alongside revenue outputs as a measure of characteristics (quality) of the revenue output. Any measured inefficiency from models reflects both inefficiency of the managers and of policy makers or regulators [37]. Lan and Lin [31] cite Fielding et al. [38] who define specific terms for these concepts. They define the degree of suboptimal transformation of inputs into intermediate outputs as “technical inefficiency,” while they define degree of suboptimal transformation of inputs into final outputs as “technical ineffectiveness.” They define a further concept, “service ineffectiveness,” as the degree of suboptimal transformation of intermediate outputs into final outputs. They point out that it is the non-storability property of transport outputs which requires such distinctions. Often of immediate concern is the “technical inefficiency” concept, since the public transport undertakings (at least in the short run) has to take its outputs as given. “Technical inefficiency” is bounded by quotation marks in order to distinguish the Fielding et al. concepts from the definition of technical efficiency in production theory. In particular in this chapter, cost inefficiency is considered which includes allocative as well as technical inefficiency even though this applies to the transformation of inputs into intermediate outputs (and not final outputs). This is appropriate given the chapter is considering the cost characteristics and performance of different parts of a vertically separated industry. It is important to emphasize that network size is viewed in public transport as a characteristic of transport outputs, since the size of the network affects the scope of travel opportunities available to users. This is in contrast to the use of network size as a proxy for the capital stock for which empirical estimation of related coefficients has yielded counter-intuitive signs (see the discussion about inputs below). Therefore, empirical evidence suggests that network size has a strong relation to the output of the railway rather than as a measure of the stock of capital of the railway. The second general distinction that has been made is the need to distinguish between scale and density effects. Density effects comprise the effect on costs of increasing all outputs (in equal proportion) while holding network size constant. Scale effects comprise the effect on costs of increasing all outputs and network size in equal proportion. This distinction is important since it is often argued that marginal costs in network transport industries are below average costs and this is a problem in terms of opening such markets to competition. Specifically, there is strong reason to suggest that the marginal cost of accommodating an additional consumer using the current network size through greater utilization is very small, while it is not clear that the marginal cost of expanding the network to accommodate the marginal consumer (here marginal O-D pair) is less than average cost. This was one argument for choosing a vertical separation between train operations and rail infrastructure management in Europe, where the former was exposed to competition and the latter was kept as state-owned monopolies. Caves et al. [1, 2] outlined expressions for returns to scale and returns to density in cost functions. Caves et al. showed returns to scale (RtS) and density (RtD) can be computed as follows:

39 Applications of Production Theory in Transportation

 m−1 1  RtS =

1503

 εyi + εS

(1)

i=1

 RtD = 1

m−1 

εyi

(2)

i=1

where εyi is the elasticity of cost with respect to the ith output (i = 1, . . . , m-1) and εS is the elasticity of cost with respect to the network size variable.5 The need to distinguish between scale and density effects or the choice between revenue versus available outputs is only part of the wider issue of how to account for the heterogeneity of railway outputs, as introduced at the start of this section. One way to deal with the heterogeneity in outputs is to group outputs into m groups and include a further set of r variables which characterize the outputs. C (y1 , . . . , ym , q1 , . . . , qr , p1 , . . . , pn )

(3)

The move from potentially hundreds or thousands of outputs to a more manageable number of m outputs is obviously a simplification. However, the inclusion of output characteristic variables is an attempt to reintroduce heterogeneity in outputs back into the model. Such variables may include revenue measures (such as passenger-km and freight tonnes-hauled) where available measures are adopted as output and vice versa. As such it can become ambiguous as to what variables represent outputs versus output characteristics versus network size. By implication it also means that in practice, the distinction between the “technical inefficiency” and “technical ineffectiveness” of Fielding et al. [38], discussed earlier, is far from clear (e.g., if train-km and passenger load factor enter the model). The inclusion of characteristic variables in the cost function specification has prompted new definitions of returns to scale and density to be proposed to allow for the possibility of characteristics of outputs changing along with the outputs or network size themselves. (See Oum and Zhang [39] for a discussion.) The ideas are similar to the discussion in Caves et al. [40] regarding the need to consider changes in unobserved network effects in RtS described above; however, in Oum and Zhang [39], these relate to changes in observed rather than unobserved variables. These ideas are applied to the analysis of train operating company (TOC) costs in section “Input Prices”, where several scale and density measures are proposed taking into account variations in output characteristics as well as “primary” outputs. While this formulation does simplify the problem to a tractable level, the resulting function may be very complicated, given the number of variables and

5 For

notational convenience and consistency with other equations which do not distinguish between the network size variable and other outputs, the network size variable is treated as the mth output and so only the first m-1 output elasticities are used in the RtD equation (which excludes this output).

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possible interaction and higher order terms for each. As a result, the cost function may still not be suitably parsimonious. An alternative is the hedonic cost function developed by Spady and Friedlaender [41] and applied, for example, by Wheat and Smith [113] to train operating company data as described below.

Input Prices The measures of the price of inputs should reflect the opportunity cost of a unit of those outputs. For example, the opportunity cost associated with 1 h of labor is the wage rate. Less obvious is the price of capital. It should reflect the hourly rental of the capital. This is problematic to measure because of heterogeneity in capital (see below) but also due to the fact that capital tends to be owned rather than leased. Methods such as the perpetual inventory method (see Bishop and Thompson [42]) have been developed to better capture a measure of capital price. A further issue with the price of capital is the relationship between this and the network size which could be viewed as a measure of capital. In particular, because of a positive coefficient on miles of railroad, a negative marginal product of capital is suggested [43]. However, it is clear that in a railway cost function, network size is much more related to the scale of output of operation than a measure of the capital stock of the network. In practice, there is a similar problem to defining input prices as in defining outputs, i.e., the problem of heterogeneity in inputs. For example, average salary is likely to be a poor measure of the labor price as workers may work a different number of hours across observations. Likewise, there is the possibility of a different mix of workers across different observations. One firm may thus face higher labor costs because it utilizes more expensive but higher skilled labor. This is likely to distort coefficient estimates (and indeed estimates of inefficiency) due to endogeneity of explanatory variables. The usual way to remedy this is to disaggregate further the input prices in the model (such as wage rates per staff type), but this adds to the number of coefficients to be estimated and the data may simply not exist.

Variable Cost Function As stated earlier, the transport sector is to a large extent a regulated industry, and in many regulatory settings, it is often not reasonable to assume that the firm can adjust the levels of all inputs. For example, the size and configuration of railway infrastructure is often fixed. In these circumstances, the variable (short run) cost function is appropriate. It can be derived using duality from a production function under the assumption of cost minimization, a level of the fixed input(s), and prices of the variable inputs. The resulting function for m outputs, n inputs (o of them fixed) is given as:

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V (y1 , . . . , ym , p1 , . . . , pn−o , z1 , . . . , zo )

(4)

where y and p are as before and zi represents the level of the ith fixed input. The measure of variable cost in the function should only include the costs associated with the variable inputs and not those associated with the fixed input(s). The issues raised in the discussions above on output, functional form, and inputs are applicable to the variable cost function in addition to the total cost function, but the measurement of RtS and RtD are subtly different. The reason is that there is a need to consider the effect of the fixed factor(s) when computing RtS and RtD. Caves et al. [1, 2] give the expressions as:   o   m−1 1 − ε zi  RtS = i=1

 εyi + εS

(5)

i=1

  o  m  RtD = 1 − i=1 εzi ε

yi

(6)

i=1

It is not entirely clear from the subsequent literature when (Eqs. 5 and 6) should be employed vis-à-vis (Eqs. 1 and 2). For example, Wilson [43] has two variables to capture the fixed factor. First, length of railroad and second, average speed rating. He defines RtS as (Eq. 5) except εzi only includes the variable cost elasticity with respect to length of rail road. This seems intuitive given the line speed measure is a characteristic of the track which may a priori not be expected to change with size of network. However, RtD is given as (Eq. 2) rather than (Eq. 6) (Wilson [43], footnote 20), which seems odd given the definitions in Caves et al. [1]. Caves et al. [2] compute RtD for their variable cost specification as (Eq. 6) where εzi is the cost elasticity with respect to capacity (defined as the sum of the annual service flows (measured in constant 1977 dollars) from flight equipment and from ground property and equipment – footnote 19). Clearly, either (Eq. 1/2) or (Eq. 5/6) could be valid measures of RtS and RtD in a variable cost function; ultimately the two sets of measures are aimed at answering subtly different questions. Equations (Eq. 1/2) are measuring how variable cost is impacted on by changing scale and density, while (Eq. 5/6) are measuring how total cost are impacted on by changing scale and density. What is important in any analysis is to clearly state to what costs RtS/RtD relate. To some extent, this point is mute for this chapter, since it is debatable whether what is being estimated is a total or variable cost, especially in vertically separated industries such as railways. In particular, we can consider the cost function (Eq. 3) to be a total cost function in the sense that cost comprises all costs under direct control of train operating companies. Similarly in (Eq. 4), the sum of maintenance and renewals cost is all that is in control of the infrastructure manager, so again it is a total cost function. However, from the perspective of a country’s railway, each cost set is only a part of the wider system.

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Related to this discussion is whether network length is viewed as a fixed factor in the variable cost function or simply an output to distinguish RtS from RtD (as it is described in the total cost function). Clearly, this decision affects the appropriate decision as to which expression for RtS and RtD to adopt. Lee and Baumel [44] point out that a model with a fixed factor included alongside a capital price in a cost function violates the properties of both long-run and short-run cost functions. All in all, it is not clear which measure of RtS and RtD to adopt in practice.

Cost and Efficiency Studies for Railways In this section, we discuss in detail the application of production and cost econometrics in railways. This mode has been chosen due to the extensive and diverse work undertaken, which is effective to illustrate the breadth of analysis undertaken within the transport sector.

Infrastructure Studies The railway system has long been considered a natural monopoly in which the market equilibrium results in one firm producing railway services. In Europe, however, many countries have separated train operations from infrastructure management as evidence has suggested that train operators could lower their average costs. This made it possible to introduce competition between train operating companies by running train services on one or several lines while maintaining the natural monopoly case of infrastructure management with one owner, the state. Many European countries used in-house resources to maintain the infrastructure, yet there are examples where these services have been contracted out. In Britain, this resulted in concerns over the quality of the track, while Odolinski and Smith [45] found that the use of competition for maintenance contracts in Sweden reduced costs by around 11% without any measurable falls in quality. Many other countries have chosen a vertically integrated railway system and have introduced regulations to reduce the negative effects of market power. Irrespective of the solution used, the public sector needs to decide on the price to be set (or allowed to be set), either through ownership or (de)regulation. There has therefore been a large amount of work to understand railway infrastructure maintenance and renewal cost from the perspective of quantifying the wear and tear by traffic on the infrastructure network. In Europe in particular, this has been to inform marginal cost-based pricing.

Marginal Cost Studies Beginning with research by Johansson and Nilsson [46], there have been several studies that have estimated variable cost functions for rail infrastructure maintenance and for the sum of infrastructure maintenance and renewal costs. Studies have utilized either track section or regional data. Most studies make use of

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observations over a number of years. The time dimension makes it possible to consider intertemporal effects. For example, maintenance costs in 1 year can have an impact on the maintenance carried out in the subsequent year(s). The reason is that, for example, a (sudden) change in traffic can imply a (temporary) deviation from the original cost-minimizing plan. It can take time to adjust to the new situation due to, for example, budget and/or planning restrictions. Indeed, intertemporal effects have been found in rail infrastructure cost studies (see Odolinski and Nilsson [47]) as well as road infrastructure cost studies (see Haraldsson [48]). There are two commonly defined (high level) cost categories relevant in determining infrastructure marginal wear and tear costs: maintenance cost and renewal cost. Maintenance generally contains expenditures on activities associated with day-to-day upkeep of the infrastructure, while renewals contain expenditures on activities on replacement of assets whose life is expired, on a like-for-like basis. Both cost categories contain substantial elements that are variable with traffic and so both should be analyzed in econometric modelling of marginal costs. However, most studies in this area have considered maintenance expenditure only as the dependent variables and the limited number of studies that consider the sum of the two categories suffer from poor fit. This is because renewals expenditure tends to be lumpy (discrete in nature) and also depends on past, as well as current, traffic levels. In terms of the choice of the sum of maintenance and renewal cost versus analysis of maintenance cost only, there is the obvious benefit of using the sum of these cost categories as the dependent variable since this considers the majority of the infrastructure manager’s activity that can (non-arbitrarily) be allocated to individual track sections or areas. It also avoids problems associated with different definitions of what exactly comprises maintenance versus renewal which can differ from zone to zone within an infrastructure manager and particularly from one infrastructure manager to another. However, there is less certainty that the cost functions for the maintenance and renewals combined have all of the appropriate variables within it due to the dynamic and lumpy nature of renewals expenditure. This could bias any efficiency estimates derived from the model. As such, a model for maintenance only expenditure is still a useful complement to a model with both cost categories as the dependent variable. Another approach is to model the dynamics between maintenance and renewals, acknowledging that these activities are input substitutes in most life cycle asset management. Wheat [49] and Odolinski and Wheat [50] are examples of top-down approaches on infrastructure costs, while Gaudry et al. [52] provides estimates based on an (bottom-up) optimization model for maintenance and regeneration of rail infrastructure. Models for renewal costs only have also been considered in the literature, using either corner solution models or survival analysis to capture the impact traffic has on this cost category (see Andersson et al. [53, 54] for railway examples; see [55] for a survival analysis on road infrastructure reinvestments; and Odolinski and Nilsson [112] for a comparison of the corner solution and survival model approach in this context). The logic behind the cost impact is that a (temporary) increase in traffic implies that future renewals will be carried out earlier than originally planned.

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Note that a permanent change in traffic may also result in a change in the length of the renewal intervals. The type of outputs used are intermediate outputs. The primary reason for this choice is that the motivation for the costing exercise was to derive marginal costs with respect to intermediate outputs. This also corresponds to the type of output which the infrastructure manager perceives, and so intermediate outputs are appropriate for measuring the efficiency (as opposed to effectiveness, Lan and Lin [31]) of this decision-making unit. The most popular output measure is gross tonne-km as, relative to commonly available alternatives, this seems to be most aligned to the true physical driver of damage and thus explains infrastructure costs. There may be a priori reason to believe that there is benefit from distinguishing between gross tonne-km of passenger and freight traffic as this would be more cost reflective. While there has been some success in doing this (see [56] for a synthesis of research), there is concern in the plausibility of the relative magnitudes of marginal costs for the two traffic types. In particular, freight traffic seems to do less damage per gross tonne-km than passenger traffic by up to seven times (on average across the network) which seems implausible. Therefore, most studies have preferred to work with a single measure of output. Much work has gone on into trying to better characterize the nature of the infrastructure. There seems to be three distinct measures of this input. First, measures of what the infrastructure actually is, i.e., its characteristics. Second, what the capability is of the infrastructure, given its composition, in terms of what quality of train service it can support. Third, there are measures that describe the condition of the infrastructure, although these are often interrelated with the second category. Table 2 gives examples of measures for each category through review of those used in several European studies. There are a limited number of condition variables used in these studies. Potentially, the condition measures adopted by Kennedy and Smith [63] (number of broken rails and infrastructure manager caused delays) could be useful to incorporate into these cost functions. In particular, this can be used to analyze the trade-off between costs for producing infrastructure services and costs for unreliability. Finally, Table 3 presents RtD and RtS from a selection of studies in the literature that have examined infrastructure maintenance cost. The focus is on infrastructure maintenance cost (as opposed to maintenance and renewal together). These studies have all found increasing RtD, with elasticities of cost with respect to traffic density of the order of 0.2–0.4 at the sample mean [56]. Less clear is the evidence on RtS, with some studies finding large increasing RtS while other studies find only small increasing RtS. However, the usefulness of the RtS measure here has to be questioned, especially for studies that utilize observations by track sections (such studies are Johansson and Nilsson [64], Tervonen and Idstrom [65], Munduch et al. [60], Gaudry and Quinet [61], Andersson [59]). In particular, the length of a track section has little to do with the organization of maintenance and renewal activities, because typically maintenance/renewal teams are responsible for a number of track sections. Thus when analyzing track section data, a more appropriate measure of RtS would relate to the overall track-km maintained/renewed by each operational

Sweden Odolinski [57]

Infrastructure Track length characterisNumber of tics tracks Length of switches Tunnels Bridges

Country Study

Track length Route length Length of switches

Great Britain Wheat and Smith [58]

Track section distance Route length Tunnels Bridges Rail weight Rail gradient Rail cant Curvature Lubrication Joints Continuous welded rails Frost protection Switches Switch age Sleeper age Rail age Ballast age

Sweden Andersson [59]

Track section length Length of single-railed tunnels in meters Length of double-railed tunnels in meters Track radius Track gradient Length of the switches Station rails (as percentage of track length)

Austria Munduch et al. [60]

Table 2 Infrastructure variables used in previous railway infrastructure cost studies

Number of track apparatus Whether the track is electrified Route length Number of tracks Automatic traffic control included or not

France Gaudry and Quinet [61] Track length Track distance (route length) Length of switches Length of bridges Tunnels Level crossings Track radius Track gradient Noise/fire protection Number of switches (by type) Shafts Platform edge

Switzerland Marti and N’schwander [62]

Sweden Johansson and Nilsson [46] Track length Switches Bridges Tunnels

(continued)

Finland Johansson and Nilsson [46] Track length Switches

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Sweden Maximum axle load Track quality class Rail weight

Great Britain Continuously welded rails Maximum line speed Maximum axle load Rail age

Switch age Sleeper age Rail age Ballast age

Sweden Rail weight Continuous welded rails Track quality class Rail age

Austria

France Maximum line speed

Source: Work carried out by Phil Wheat, ITS, University of Leeds. Reproduced from Link et al. [15]

Condition

Country Capability

Table 2 (continued)

Rail age Sleeper age

Switzerland Maximum line speed

Sweden Track quality index Secondary lines

Finland Electrified Average speed

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Table 3 Estimates of returns to scale and density from infrastructure maintenance cost studies Study Johansson and Nilsson [64] Johansson and Nilsson [64] Tervonen and Idstrom [65] Munduch et al. [60] Gaudry and Quinet [61] Andersson [59] Wheat and Smith [58] Smith et al. [66] Marti et al. [67] Smith and Wheat [68] Odolinski [57] NERA [69]

Country Sweden Finland Finland Austria France Sweden Britain International study Switzerland Britain Sweden US

Returns to scale 1.256 1.575 1.325 1.449–1.621 Not reported 1.38 2.074 1.11

Returns to density 5.92 5.99 5.74–7.51 3.70 2.70 4.90 4.18 3.25

Not reported 1.13 1.07 1.15

4.54 3.29 2.47 2.85

Source: Amended from Wheat and Smith [58]

crew which is likely to be greater than the track section-km and invariant across track sections within each operational area. In this chapter, no instances of any such variables being used within these cost functions has been found. Further, more recent studies of rail infrastructure maintenance have been conducted using French data for the purpose of setting track access charges – these broadly support the results above (see Smith et al. [70]). This recent body of work has also focused on establishing asset by asset elasticities within rail infrastructure and expanding the literature on rail renewals marginal cost, building on the work of Andersson et al. [54].

Efficiency Studies Published research in the academic literature on performance of railway infrastructure managers is also limited. As with the train operating company research, all (published) studies relate to the British infrastructure manager, but some do involve international comparisons with other infrastructure managers. At the 2003 regulatory review of the British infrastructure manager’s efficiency performance, ORR commissioned LEK [71] to undertake internal benchmarking of Network Rail. This looked at potential efficiency savings for various expenditures categories based on comparisons across Network Rail’s operating areas (seven in total). Some of the work involved statistical analysis but the analysis was far from a top-down econometric efficiency study. Efficiency techniques employed were limited to OLS adjusted by either a COLS shift or lower quartile shift. A more rigorous econometric study aimed at measuring disparity between the performances of individual geographical areas within the British infrastructure manager was undertaken by Kennedy and Smith [63]. This internal benchmarking study adopted both deterministic and stochastic input orientated distance function models and utilized relatively robust data, sourced directly from the industry.

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They considered, in two separate models, maintenance only cost and the sum of maintenance and track renewal cost as inputs, combined with delay minutes and broken rails as the two other inputs. The levels of these inputs were then assumed to be endogenously determined given a set of outputs (hence the input orientation of the distance function). The outputs were track-km and two traffic density variables – freight tonne-km and passenger train-km both per track-km. Their findings suggest that the infrastructure manager Railtrack (now replaced by Network Rail) made substantial improvements in efficiency from privatization to 2000/2001, but then their efficiency deteriorated post this period. There was a key event in October 2000 (the “Hatfield accident”) which for various reasons prompted a revision in the behavior of the infrastructure manager and ultimately led to it going into administration and being replaced by Network Rail. In particular, they find that most of the earlier gains in efficiency were wiped out by the determination post Hatfield. They conclude that the substantial variation in efficiency between the geographical areas means that there were substantial opportunities to improve performance going forward. One issue with the modelling at geographical area level, as opposed to the track section level analysis discussed earlier in respect of marginal cost estimation, is that this can lead to a more limited number of variables being included in the function to characterize the infrastructure. This is partly because of data challenges in computing geographical averages from more disaggregate data and partly due to smaller sample sizes restricting what can plausibly be done – this being typical with regulatory applications of efficiency analysis across the network industries. A key contribution of Kennedy and Smith [63] is that it represents an early attempt to incorporate quality into the analysis alongside other variables – in this case, quality being represented by number of broken rails (asset condition) and infrastructurecaused delay minutes. Overall Kennedy and Smith demonstrated that suitable data existed within infrastructure and could be used to find evidence of inefficient practice through internal benchmarking of geographical units. More recent work has sought to incorporate more variables into efficiency analysis of rail infrastructure using geographical data. ORR [72] included measures of speed and also criticality (impact of infrastructure failure on rail operations). This was enabled by utilizing more disaggregated (maintenance units) data than the more aggregate zonal data used in Kennedy and Smith [63], thus increasing the number of units for estimation. In a similar way, Smith et al. [73] used maintenance unit data within France and were able to include a richer specification of the technology, for example, track age, number of bridges and tunnels, and track curvature measures. Moving to country-level studies, econometric efficiency analysis of Network Rail formed a very important part of the regulatory efficiency determination process, starting with the 2008 Periodic Review determinations. This comprised two pieces of analysis, both benchmarking studies utilizing international comparators. The primary piece of analysis utilized a data set collected by the UIC (International Union of Railways) and previously analyzed for the Lasting Infrastructure Cost Benchmarking (LICB) project [74]. This was data for a selection of railway infrastructure managers who were members of the UIC. The original LICB project

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was based on adjusted average cost calculations. Thus, unit costs were computed, but adjustments were made based on the characteristics of railways (see Smith and Wheat [75] for a review of the adjustment factors). However, the subsequent work sponsored by ORR undertook econometric efficiency analysis of the dataset (1993– 2006) [66, 76]. The preferred model utilized a time varying inefficiency model which estimated firm specific paths of adjustment. The model found Network Rail to be 60% efficient. This analysis demonstrated that international comparisons of railway infrastructure managers could be made using econometric techniques. A supporting piece of econometric analysis was using a bespoke dataset collected by ORR comprising five infrastructure managers. This dataset included observations for regions within each infrastructure manager and, in some cases, data over time. At the 2008 Periodic Review, this dataset was relatively new and so analysis was limited to verification of the inefficiency estimates from the main LICB data analysis (which were confirmed). Smith and Wheat [68] develops this analysis further. In particular, models are proposed which best exploit the multi-level structure of the data. A similar approach was used in Odolinski [77] who compare the cost inefficiency between different infrastructure maintenance regions in Sweden, as well as between contract areas within the regions. Given the nature of the datasets, the international studies adopted as part of the 2008 Periodic Review made use of a more limited set of explanatory variables than has been the case with marginal cost literature, where track section data permits a very rich configuration of the cost structure. That said, the studies are more in line with the wider cost modelling literature across other industries which typically would include a small number of output variables in the cost function. One issue facing rail, and transportation more widely as discussed earlier, is the problem of accurately measuring input prices. In the rail infrastructure literature, these issues have partly been addressed through using PPP exchange rate adjustments in international work, or relying on the commonality of input prices within countries for geographically based internal benchmarking work. As noted, the recent impulse has been towards richer characterizations of the technology through utilizing more disaggregate maintenance unit data for efficiency analysis purposes. However, the richness of the rail infrastructure cost functions estimated for cost efficiency purposes still does not match that of studies used to estimate marginal costs, where the latter utilize much larger datasets at track section level .

Passenger Train Operations There has been limited published work on the performance of passenger train operating companies (TOCs). Papers on British TOCs have used a variety of methods including nonparametric DEA [78–81] and index number approaches [82, 83], as well as parametric estimation of cost functions ([84]; Smith and Wheat [68]), production functions [85], and distance functions (Affuso et al. [78, 79]). Clearly, the former methods can only consider cost or technical efficiency and produce no estimates regarding the actual cost structure.

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The papers by Cowie consider three inputs: staff, rolling stock, and network. This is deficient given that the network input is fixed and difficult to characterize. Specifically, in Cowie [84, 85], route-km are used as the input for network; however, as Cowie [80] acknowledges, this is a poor proxy for the true network input. Cowie [80] replaces this with the cost of access for each TOC as measured by the charges paid to Network Rail. Given the arbitrariness of the allocation of the fixed charge to individual operators, the usefulness of this measure has to be questioned. Also, post 2002, the infrastructure manager was not fully funded by TOC access payments. Instead, the Network Grant (direct payment from government) was introduced alongside access charges. This further distorts any “price” for network access post 2000 (affecting the Cowie [80] study). Given the regulatory regime, network access can easily be thought of as a passthrough with respect to franchised TOCs since TOCs are compensated directly with respect to changes in access charges as a condition of the franchise contract. Thus, a cost function which considers TOC cost less access charges as the dependent variable seems most appropriate, rather than the cost function estimated by Cowie. Affuso et al. [79] do not include any network inputs into their distance function, but this simple exclusion does not seem optimal since the network may affect the transformation function. It is considered that a better way to deal with this is to estimate a variable cost function with infrastructure held fixed. Turning to the results on TOC performance, all the studies ([78–85]; Smith and Wheat [68]) report improved performance over the period from privatization to the period 2000/2001. A consistent finding is that this improvement in performance, as measured by a Malmquist total factor productivity measure, has tended to be driven by positive technical change with only a small improvement in average technical efficiency over the period. Thus, while the best performing TOCs seemed to be improving up to 2000/2001, there was little evidence that all firms were converging, i.e., that franchising was successfully driving out poor performance. Cowie [80] and Smith and Wheat [86] are the only studies to have considered the period following the Hatfield accident in October 2000. Cowie’s study covered the years 1996/1997–2003/2004, while Smith and Wheat extended the sample to 2005/2006. Cowie found that, following Hatfield, there was a deterioration in TFP and this was across all TOCs, i.e., was found to be as a result of negative technical change growth rather than a deterioration in technical efficiency of a subset of firms (see Figs. 2 and 3 in Cowie [80]). In fact, Cowie finds that average technical efficiency improves over the post-Hatfield period. This suggests that, even with the distribution of some firms moving to renegotiated contracts, franchising had still begun to proliferate best practice across the industry. This finding has to be moderated however by the finding that, overall, TFP was not found to be substantially different at the end of the period than at the first year following privatization. Smith and Wheat [86] also found that technical change was, in the early years of their sample, beneficial in terms of lowering costs; however, following the Hatfield accident, not only was there a statistically significant upward shift in costs but the direction of technical change shifted, such that costs began to increase

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over time. These observations are the same with respect to overall TFP in the Smith and Wheat [86] model. For the parametric studies, it should be possible to derive returns to scale and density results from the models. In Cowie [85] and Affuso et al. [79], these properties of the models are not discussed in the chapter. Furthermore, the fact that the data does not appear to be normalized at the sample mean, coupled with the adoption of Translog functional forms, means that the results in the papers cannot be used to derive these results. Of the nonparametric research, Merkert et al. [81] did estimate a variable RtS model and found that British and Swedish TOCs were below minimum efficient scale, while the large German operators were above. Only Cowie [84] and Smith and Wheat [86] provide an explicit discussion of the returns to scale properties of the models. Cowie defines returns to scale simply in relation to his single output train-km (there are of course different possible measures of RtS in this context such as returns to network size, train-km, and train length). His results seem to suggest decreasing returns to scale at low train-km, but then increasing RtS at higher train-km. Smith and Wheat [86] put forward a model which yields estimates of the extent of both returns to scale and returns to density, where the primary usage output is train-km rather than train-hours. They found constant RtS and increasing RtD. A subsequent study Wheat and Smith [113] applied a hedonic cost function approach and was also able to include train hours (and thus train speed) into the modelling for the first time, and the method permitted estimates of returns to scale and density to vary with the heterogeneity characteristics of output. This study reported increasing returns to density for all types of operator but importantly found that some operators were operating beyond efficient scale; the latter finding having important implications for the optimal size of rail franchises. Overall, the received studies on passenger train operations have concentrated on technical change, cost efficiency, and overall TFP trends, and the cost structure in respect of returns to scale and density. The motivation for concentrating on these issues were, firstly, studies focus on Britain and, secondly, at the time the railway in Britain suffered from a substantial cost shock which resulted in several franchises getting into financial difficulty, and where cost pressures therefore prompted increased focus on the optimal size and structure of rail franchises.

Cost and Efficiency Studies in Other Transport Sectors Road Infrastructure Within the literature on road infrastructure costs, the seminal work by Small et al. [87] has been followed by a number of empirical studies on the cost impact of road traffic. This research aim is analogous to the work in railways to determine the marginal wear and tear costs associated with running more traffic. In addition to top-down econometric approaches, the dominant approach in this mode, and piloted by Small using simplifying assumptions, is to use engineering

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(bottom-up) approaches, be them through survival analysis or through direct laboratory experiment. Survival analysis is a statistical method using real data on asset replacement times, to establish a deterioration elasticity, based on the suggested model transformation as outlined in Lindberg [88]. There are also many examples of more standard top-down approaches analyzing different types of road infrastructures (motorways, paved roads, gravel roads) and types of activities (maintenance, renewals, winter road operations) (see, for example, Link et al. [15]; Link [89]; Yarmukhamedov and Swärdh [90]). A growing literature has developed concerning the efficiency of road infrastructure managers. Studies include Wheat [92] for the UK, Massiani and Ragazzi [93] for Italy, Welde and Odeck [94] for Norway and Fritzsche [95], and Kalb [96] for Germany. All studies use an econometric approach, including using stochastic frontier analysis, while some have also considered a DEA approach (usually with a second stage regression): Welde and Odeck [94] and Kalb [96]. In terms of the outputs considered in the road efficiency studies, these include measures of both scale and density. Scale is the size of the road network (sometimes broken down into different classes of roads) and density is the traffic usage intensity. Particular consideration tends to be given to the extent of usage by heavy goods vehicles as these are found to have a disproportional impact on road damage. Other outputs tend to be the quality of the infrastructure. This can be measured by a road condition index or number of defects, but Wheat [92] also proposed using an index of public satisfaction with highways. The logic for including this alongside a road condition index was that quality of roads involves not just the engineered quality of the road but also how responsive the road authority is in response to defects emerging. A further area topic of investigation is the extent to which the introduction of competitive tendering for road maintenance services has reduced costs. Yarmukhamedov et al. [115] present evidence for Sweden using a panel data cost function and include controls for contracting arrangement. They find that competitive tendering to private sector organizations results in costs between 8% and 20% lower than contracts to the state-owned companies, all other things equal. This is in line with other evidence that they cite which suggests competitive tendering reduced road maintenance costs (by around 20–30% in Australia [97], 10–35% in Canada [98], and 22–27% in Sweden [99]).

Local Public Transport Local public transport includes bus services, tramways, and metros. Smith et al. [100] synthesize the approaches as being very diverse. The local nature of provision permits more comparisons within a nation or region. There is a large volume of research on the efficiency of public transport operations. This is because there is a great variety of operating models such as public ownership, private provision, tenders with revenue risk, or tenders with no

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revenue risk. As such, there is a research need to understand which organizational structure is best in which circumstances. Usually researchers use “available outputs,” as opposed to “revenue outputs,” as the measure of output. This helps to separately identify operator inefficiency from local authority decisions to provide some loss-making but socially important services. Since operators have little influence on utilization of capacity and only provide services according to a specified timetable and tariff, “available outputs” are a more appropriate measure. If the operators have more influence over what services to run, or we have to compare possibly complex efficiency of organizational models and not firms, “revenue outputs” tend to be better, especially if we can capture the effects of differences in operating conditions through additional independent variables (such as speed, peak to off-peak kilometers ratio, yearly mileage of vehicles, etc.). In some cases, total cost can be expressed as a function of outputs and operating conditions variables and benchmarked. For example, Farsi et al. [101] evaluated Swiss rural bus companies using vehicle-kilometers as an output. The usage to “available output” was extensively justified by the authors – they state that frequency is regulated by public authorities, thus leading to oversupply of capacity (which also varies between firms). In such conditions, revenue output may be misleading. On the other hand in India, where operators have much more freedom, Bhattacharyya et al. [102] used number of passenger-kilometers (a “revenue output”) as the dependent variable. Accounting for heterogeneity is very important. This is because difference external factors (such as network structure) on costs. In many cases, such differences may be stronger than differences in efficiency. The approach is to include variables that characterize the difference in output. Drawing again on Bhattacharyya et al. [102], load factor (passenger-km to vehicle-km), vehicle utilization (average number of kilometers travelled daily by a bus), number of breakdowns (per 10,000 vehiclekilometers), and fleet utilization (proportion of the number of vehicles on the road to the total fleet) are candidate metrics to be included alongside measures of overall output. Peak vehicle requirement is also an important measure to differentiate between firms operating in different demand conditions [103]. There is a broad consensus on a basic set of input variables across studies. These include rolling stock, labor, and fuel inputs – this approach has been used since the beginning of contemporary research and was used inter alia by Viton [104]; however, it may be extended .

Air Transport Airport operations is another part of the transport sector that have been increasingly privatized, with commercialized entities that rely on non-aeronautical services. Government regulations have thus changed in this sector, moving from rate of return and price cap regulation towards a more deregulated environment [105]. Still, many airports are government owned with regulated charges and research on marginal costs have been conducted.

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Bottasso and Conti [106] note that recent papers consider the multi-output nature of the airports, producing both aeronautical and non-aeronautical services, which in turn can be categorized into various outputs. For example, a distinction between passenger and freight services can be made, and sometimes the passenger and freight variables are used together with measures on air transport movements (landings and take-offs of aircrafts). Similar to railway cost functions, there are examples of hedonic output functions for air transport: see, for example, Martin and Voltes-Dorta [107] who use a hedonic specification for different aircraft operations (or Gillen et al. [108], who use a hedonic output function for an airline cost function). Regarding inputs, most studies use number of runways, airport surface area, and number of check-in desks. Similar to the discussion for railway inputs, the use of the perpetual inventory method has been suggested to better reflect the capital price, since, for example, physical measures of capital may not properly reflect this price in regulated airports (e.g., the gold plating phenomena is often found in industries of with rate of return regulation). Moreover, Bottasso and Conti [106] notes that the increased commercialization of the airport industry implies that the exogeneity assumption of outputs in econometric cost estimations can be questioned and needs more research. Oum and Yu [109] and Yu [110] lists examples of outputs and inputs used to study the production and cost structure of airlines. Common outputs for scheduled passenger services are number of passengers and revenue passenger miles, while scheduled freight and mail services use revenue ton-miles. The so-called incidental services (noncore activities of an airline such as catering and aircraft maintenance) are often represented by an output quantity index. Input examples are labor, fuel, flight equipment, and ground and property equipment. Yu [110] notes that some studies use available seat miles and available ton-kilometers, either as outputs or inputs. Jara-Díaz et al. [111] summarize the airline market literature (covering 30 years of research) and report the cost functions used and estimates on RtS and RtD. Most studies find increasing RtD, and nearly constant RtS. Jara-Díaz et al. also find evidence of economies of spatial scope, stating that there are advantages of jointly serving markets (e.g., domestic and international services) that imply larger networks.

Conclusion In this chapter, the experience of production and cost analysis in transportation has been reviewed. Transportation is an active application of econometric techniques with a significant public interest given that many public transportation services require government subsidy and a degree in exclusivity in supply to make their provision viable and efficient. Key production and cost analysis needs of the sector can be identified as: • Strategic operational insight: understanding how costs change with output to inform operation and investment decisions. Essential here is the allowance for

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the quality of output as well as a simple scale metric given the jointness and non-storability of production. • Establishing evidence as to which market and regulatory structures yield best outcomes given the complexity in the transport sector and the inevitable tradeoffs between maintaining network coordination and avoiding useless duplication of resources against potential inefficiency introduced by reducing competition for or in the market. • Providing the evidence base for regulatory scrutiny either through yardstick competition or more formal price cap regulation. The upshot of this is that transportation has provided the motivation and illustration for many innovations within production, cost, and efficiency methods. This chapter acts as a survey of the transportation literature, bringing to life the issues and solutions found in the transportation sector. Some of the challenges of understanding the cost structure of transportation and the optimal industry structures remain, coupled with the introduction of new technologies, and the ability to model cost-quality relationships, and so we expect research in transportation to continue to be vibrant into the future. This will be supported by ever increasing datasets which critically enable construction of measure of heterogeneity in output. This in turn allows for the complex features of transportation networks to be better characterized, which improves the robustness of findings to the transportation sectors analysis needs.

Cross-References  Airline Economics: A Survey of Applied Issues in the Performance of the US and

International Airline Industry  Cost Assessment of (Un)bundling: Separation of Vertically Integrated Public

Utilities

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33. Parisio L (1999) A comparative analysis of European railroads efficiency: a cost frontier approach. Appl Econ 31:815–823 34. Cantos P, Villarroya J (2000) Efficiency, technical change and productivity in the European rail sector: a stochastic frontier approach. Int J Transp Econ 27(1):55–76 35. Cantos P, Villarroya J (2001) Regulation and efficiency: the case of European Railways. Transp Res A 35(5):459–472 36. Tsionas EG, Christopoulos DK (1999) Determinants of technical inefficiency in European Railways: simultaneous estimation of firm-specific and time-varying inefficiency. KONJUNKTURPOLITIK 45:240–256 37. Oum TH, Yu C (1994) Economic efficiency of railways and implications for public policy: a comparative study of the OECD countries’ railways. JTEP 28:121–138 38. Fielding GJ, Babitsky TT, Brenner ME (1985) Performance evaluation for bus transit. Transp Res 19A:73–82 39. Oum TH, Zhang Y (1997) A note on scale economies in transport. JTEP 31:309–315 40. Caves DW, Christensen LR, Tretheway MW, Windle RJ (1985) Network effects and the measurement of returns to scale and density for U.S. Railroads. In: Daughety AF (ed) Analytical studies in transport economics. Cambridge University Press, Cambridge, pp 97– 120 41. Spady RH, Friedlaender AF (1978) Hedonic cost functions for the regulated trucking industry. Bell J Econ 9(1):159–179 42. Bishop M, Thompson D (1992) Regulatory reform and productivity growth in the UK’s public utilities. Appl Econ 24:1181–1190 43. Wilson WW (1997) Cost savings and productivity in the railroad industry. J Regul Econ 11:21–40 44. Lee T, Baumel CP (1987) The cost structure of the U.S. Railroad industry under deregulation. J Transp Res Forum 27(1):245–253 45. Odolinski K, Smith ASJ (2016) Assessing the cost impact of competitive tendering in rail infrastructure maintenance services: evidence from the Swedish Reforms (1999 to 2011). JTEP 50(1):93–112 46. Johansson P, Nilsson JE (2002) An economic analysis of track maintenance costs, deliverable 10 Annex A3 of UNITE (UNIfication of accounts and marginal costs for Transport Efficiency), Funded by EU 5th Framework RTD Programme. ITS, University of Leeds, Leeds. Online: http://www.its.leeds.ac.uk/projects/unite/ 47. Odolinski K, Nilsson J-E (2017) Estimating the marginal maintenance cost of rail infrastructure usage in Sweden; does more data make a difference? Econ Transp 10:8–17 48. Haraldsson M (2007) Essays on transport economics. Doctoral thesis, Uppsala University 49. Wheat P (2015) The sustainable freight railway: designing the freight vehicle-track system for higher delivered tonnage with improved availability at reduced cost SUSTRAIL, Deliverable 5.3: access charge final report annex 4, British Case Study 50. Odolinski K, Wheat P (2018) Dynamics in rail infrastructure provision: maintenance and renewal cost in Sweden. Econ Transp 14:21–30 51. Odolinski K, Wheat P (2021) Rail cost functions. In: Vickerman, Roger (eds.) International Encyclopedia of Transportation, 1:425–430, United Kingdom: Elsevier Ltd. https://doi.org/ 10.1016/B978-0-08-102671-7.10080-6 52. Gaudry M, Lapeyre B, Quinet E (2016) Infrastructure maintenance, regeneration and service quality economics: a rail example. Transp Res B 86:181–210 53. Andersson M, Björklund G, Haraldsson M (2016) Marginal railway renewal costs: a survival data approach. Transp Res A 87:68–77 54. Andersson M, Smith A, Wikberg Å, Wheat P (2012) Estimating the marginal cost of railway track renewals using corner solution models. Transp Res A 46: 954–964 55. Nilsson J-E, Svensson K, Haraldsson M (2015) Estimating the marginal costs for road infrastructure reinvestment. CTS working paper 2015:5. CTS

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56. Wheat P, Smith ASJ, Nash CA (2009) CATRIN (Cost Allocation of TRansport INfrastructure cost), deliverable 8 – rail cost allocation for Europe. Funded by Sixth Framework Programme. Coordinated by VTI, Stockholm 57. Odolinski K (2019) Estimating the impact of traffic on rail infrastructure maintenance costs: the importance of axle loads. JTEP 53(3):258–274 58. Wheat P, Smith A (2008) Assessing the marginal infrastructure maintenance wear and tear costs for Britain’s Railway network. JTEP 42(2):189–224 59. Andersson M (2006) Marginal railway infrastructure cost estimates in the presence of unobserved effects. Case study 1.2D I Annex to Deliverable D 3 Marginal cost case studies for road and rail transport, Information Requirements for Monitoring Implementation of Social Marginal Cost Pricing, EU Sixth Framework Project GRACE (Generalisation of Research on Accounts and Cost Estimation) 60. Munduch G, Pfister A, Sögner L, Stiassny A (2002) Estimating marginal costs for the Austrian Railway system, working paper 78. Department of Economics, Vienna University of Economics and B.A, Vienna 61. Gaudry M, Quinet E (2003) Rail track wear-and-tear costs by traffic class in France. Universite de Montreal, Publication AJD-66 62. Marti M, Neuenschwander R (2006) Case study 1.2E: track maintenance costs in Switzerland, annex to GRACE (Generalisation of Research on Accounts and Cost Estimation) Deliverable D3: marginal cost case studies for road and rail transport. Funded by 6th Framework RTD Programme. Ecoplan, Berne 63. Kennedy J, Smith ASJ (2004) Assessing the efficient cost of sustaining Britain’s rail network: perspectives based on zonal comparisons. JTEP 38(2):157–190 64. Johansson P, Nilsson J (2004) An economic analysis of track maintenance costs. Transp Policy 11(3):277–286 65. Tervonen J, Idstrom T (2004) Marginal rail infrastructure costs in Finland 1997–2002. Report by the Finnish Rail Administration. Available at http://www.rhk.fi. Accessed 20 July 2005 66. Smith ASJ, Wheat PE, Nixon H (2008) International benchmarking of network rail’s maintenance and renewal costs, joint ITS, University of Leeds and ORR report written as part of PR2008, June 2008. Presentation available at http://www.rail-reg.gov.uk 67. Marti M, Neuenschwander R, Walker P (2009) CATRIN (cost allocation of transport infrastructure cost), Deliverable 8 Annex 1B – rail cost allocation for Europe: track maintenance and renewal costs in Switzerland. Funded by the Sixth Framework Programme. VTI, Stockholm 68. Smith ASJ, Wheat P (2012b) Estimation of cost inefficiency in panel data models with form specific and sub-company specific effects. J Prod Anal 37:27–40 69. NERA (2000) Review of Overseas Railway Efficiency: A Draft Final Report for the Office of the Rail Regulator. NERA, London 70. Smith ASJ, Walker P, Wheat PE, Guiraud L, Silavong C (2017) Estimating the marginal maintenance cost for the French railway network: a comparison of models. ITEA Conference, Barcelona, p 20 71. LEK (2003) Regional benchmarking: report to network rail. ORR and SRA, London 72. Office of Rail Regulation (2018) PR18 econometric top-down benchmarking of Network Rail, a report. http://orr.gov.uk/__data/assets/pdf_file/0011/27875/pr18-econometrictop-down-benchmarking-of-network-rail.pdf 73. Smith ASJ, Thiebaud JC, Wheat PE (2020) Efficiency analysis of the rail network in France. mimeo 74. International Union of Railways (UIC) (2008) Lasting infrastructure cost benchmarking (LICB). December 2008. Available at http://www.uic.org/IMG/pdf/li08C_sum_en.pdf. Accessed 20 Dec 2013 75. Smith ASJ, Wheat P (2010) Sensitivity analysis on the UIC harmonisation factors. Report for Rail Consult 76. Smith ASJ (2012) The application of stochastic frontier panel models in economic regulation: experience from the European rail sector. Transp Res E 48:503–515

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101. Farsi M, Filippini M, Kuenzle M (2006) Cost efficiency in regional bus companies. JTEP 41(3):345–361 102. Bhattacharyya A, Kumbhakar SC, Bhattacharyya A (1995) Ownership structure and cost efficiency: a study of publicly owned passenger- bus transportation companies in India. J Prod Anal 6:47–61 103. Pickerl DH (1985) Rising deficits and the uses of transit subsidies in the United States. JTEP 19(3):281–298 104. Viton PA (1981) A translog cost function for urban bus transit. J Ind Econ 29(3):287–304 105. Bitzan JD, Peoples JH (2017) The economics of airport operations. Advances in airline economics, vol 6. Emerald Publishing 106. Bottasso A, Conti M (2017) The cost structure of the airport industry: methodological issues and empirical evidence. In: Bitzan JD, Peoples JH (eds) The economics of airport operations. Advances in airline economics. Emerald Publishing, pp 181–212 107. Martin JC, Voltes-Dorta A (2011) The econometric estimation of airports’ cost function. Transp Res B Methodol 45(1):112–127 108. Gillen DW, Oum TH, Tretheway MW (1990) Airline cost structure and policy implications: a multi-product approach for Canadian airlines. JTEP 24(1):9–34 109. Oum TH, Yu C (2001) Winning airlines: productivity and cost competitiveness of the world’s major airlines. Springer Science + Business Media, LLC 110. Yu C (2017) Airline productivity and efficiency: concept, measurement, and applications. In: Bitzan JD, Peoples JH, Wesley WW (eds) Airline efficiency. Advances in airline economics, vol 5. Emerald Group Publishing Limited, pp 11–53 111. Jara-Díaz SR, Cortés CE, Morales GA (2013) Explaining changes and trends in the airline industry: economies of density, multiproduct scale, and spatial scope. Transp Res E 60:13–26 112. Odolinski K, Nilsson J-E, Yarmukhamedov S, Haraldsson M (2020) The marginal cost of track renewals in the Swedish railway network: Using data to compare methods. Economics of Transportation 22:100170 113. Wheat PE, Smith ASJ (2015) Do the usual results of railway returns to scale and density hold in the case of heterogeneity in outputs: a hedonic cost function approach. J Transp Econ Policy 49(1):35–57 114. Christopoulos DK, Loizides J, Tsionas EG (2000) Measuring Input-Specific Technical Inefficiency In European Railways: A Panel Data Approach. International Journal of Transport Economics 27(2):147–171 115. Yarmukhamedov S, Smith ASJ, Thiebaud J-C (2020) Competitive tendering, ownership and cost efficiency in road maintenance services in Sweden: A panel data analysis. Transportation Research Part A: Policy and Practice 136:194–204

Productivity in Global Aquaculture

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Frank Asche, Ruth Beatriz Mezzalira Pincinato, and Ragnar Tveteras

Contents Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Bioeconomic Modeling of Aquaculture Production . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Bioeconomic Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The Rotation Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Risk and Risk Aversion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Biological Shocks and Price Dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Productivity in Aquaculture . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Empirical Analyses of Productivity and Efficiency in Aquaculture . . . . . . . . . . . . . . . . . . . . Analyses of Production Risk and Economic Risk . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Analyses of Environmental Externalities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Analyses of Agglomeration Economies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Cross-References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Abstract

This chapter provides insights on productivity in aquaculture based on production economic studies of several farmed species in different countries. We first survey studies of bioeconomic modeling of aquaculture production, and the farmer’s optimization problem. Next, we look at empirical studies of productivity and efficiency. Aquaculture is an industry with considerable production and

F. Asche School of Forest, Fisheries and Geomatics Sciences, Institute for Sustainable Food Systems and Fisheries and Aquatic Sciences, University of Florida, Gainesville, FL, USA e-mail: [email protected] R. B. Pincinato · R. Tveteras () UiS Business School, University of Stavanger, Stavanger, Norway e-mail: [email protected]; [email protected] © Springer Nature Singapore Pte Ltd. 2022 S. C. Ray et al. (eds.), Handbook of Production Economics, https://doi.org/10.1007/978-981-10-3455-8_41

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price risk, and we survey econometric studies which estimate the structure of risk and farmers’ risk preferences and behavior. Studies of relationships between productivity and environmental factors, which are today one of the main barriers for further production growth, are discussed separately. We also present studies of agglomeration economies in aquaculture, including economies and diseconomies of geographic farm density. Finally, we discuss the challenge of growing aquaculture sustainably through productivity growth and lower external environmental footprints. Keywords

Aquaculture · Bioeconomic modelling · Production risk · Productivity · Efficiency · Agglomeration economics

Introduction

Aquaculture production (million tonnes)

Aquaculture has in recent decades been the world’s fastest growing food production technology. Global production increased from 0.6 million metric tons in 1950 to 120 million metric tons in 2019, as shown in Fig. 1, implying that production increased 200 times. Productivity growth has been a central driver of this increase, caused by innovations in, e.g., fish genetics, feed, fish health, and production equipment. Aquaculture is farming in water. It takes place in fresh, brackish, and marine waters. Aquaculture involves the release of fish, shrimp, or other aquatic species

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Fig. 1 Global aquaculture production (million tons) from 1950 to 2019 by species groups. (Source: FAO [1])

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into a confined water environment, feeding, and harvesting. The production process also includes management of animal health and environmental risks through regulation of water quality, medication, etc. Fish farmers will typically have some kind of ownership rights to the farm locations, implying that they have some degree of control with the production environment. The degree of control with the biological production process from release of fingerlings into the water to harvesting distinguishes aquaculture clearly from the production process in fisheries, discussed by Squires and Walden in another chapter of this book, where there is typically very limited control with the biological production processes in lakes and oceans. Most aquaculture production technologies are not closed. Many aquaculture technologies depend on inflow of water, and the qualities of that water in terms of, e.g., temperature, oxygen content, acidity, salinity, and pathogens. Furthermore, most aquaculture production technologies emit water to the surroundings and can influence the environment through emissions of, e.g., nutrients, escaped aquatic species, aquatic diseases, and antibiotic residue. Hence, aquaculture farms can both be affected by biological and environmental external effects and cause external effects on other farms and economic agents. Why should we be interested in the productive performance of aquaculture? First, a growing global population will benefit from an increasing supply of healthy seafood, as it is rich in micronutrients and contain high levels of healthy omega-3 fatty acids. Compared with terrestrial farm animals, farmed fish are more efficient converters of energy and protein. Global fisheries have plateaued out at production levels (Fig. 2) which are difficult to increase in a sustainable manner, as the world’s

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Fig. 2 Global seafood production (million tons) from 1950 to 2019 by production technology. (Source: FAO [1])

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stocks of wild fish by and large are fully exploited or overexploited. In fact, as shown by Squires and Walden in this book, many of the studies on bioeconomic modeling focus on the maximum sustainable/economic yield, which limits fleet size and how much of the finite natural resource stock one can exploit. Thus, only aquaculture can provide a significant increase in seafood supply. An expansion in aquaculture production depends on the sector’s ability to sustain or increase productivity levels such that consumers can afford aquaculture products. Furthermore, aquaculture sectors face environmental and sustainability challenges related to input use in production processes and environmental emissions [2]. Production growth depends on innovations and productivity growth leading to more efficient use of scarce natural resources in fish feed and lower levels of harmful emissions to the environment. In many countries, particularly in freshwater aquaculture, the potential to increase aquaculture production by expanding the present aquaculture area and increasing water consumption is limited. Consequently, the most sustainable way to increase aquaculture production is through intensification of aquaculture by producing more seafood using the same area and water resources [3]. Aquaculture also represents an opportunity for poverty alleviation and economic growth in developing countries [4]. Both supply and demand side factors have driven growth in aquaculture production over time [5, 6]. We can observe this when we examine price and production development of individual farmed species. Markets for aquaculture species are generally competitive. When prices fall significantly over a longer period accompanied by an increase in production volume, this is typically an indication of productivity growth contributing to lower production costs driving expansion of the market. On the other hand, when prices increase substantially, and production still increase over a longer time period, this can be an indication of positive shifts in demand for the species. This chapter will shed more light on the patterns of productivity growth in aquaculture and its determinants based on productivity studies of several species in different countries. But first, we will take a look at price and production developments over time. The following figures provide examples from some aquaculture species. Figure 3 shows the development in Atlantic salmon production and prices over time. Until 2002, production growth was accompanied by declining prices, suggesting that innovations leading to declining production costs were a central driver [5–7]. In the following years, production has increased with increasing prices, indicating that positive shifts in demand have been the dominant underlying driver of market growth. For seabream and seabass, we find, as shown in Figs. 4 and 5, that growth in the early 2000s was accompanied by sharply declining prices, suggesting a substantial underlying productivity growth allowing for cost reductions. Thereafter, production has doubled, but prices have not experienced a significant decline, suggesting that positive shifts in demand have driven market expansion. For whiteleg shrimp, we find that production increased substantially from around 2000, accompanied by a significant decline in prices that started a few years earlier. Again, this suggests innovations allowed larger volumes to be supplied to the market

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Fig. 3 Atlantic salmon aquaculture production (million tons) and inflation adjusted prices (USD/kg) from 1990 to 2019. (Source: FAO [1])

gilthead seabream - prices

Fig. 4 Gilthead seabream aquaculture production (million tons) and its prices (USD/kg) from 1990 to 2019. (Source: FAO [1])

at lower costs [6]. From 2006, production volumes continued to increase and more than doubled, but then at price levels which are stable or higher, suggesting that increased market demand was the main growth driver in the later period (Fig. 6). Delgado et al. [8] and Kobayashi et al. [9] show that economic growth is the most important factor in explaining the development of seafood consumption. Three of the four largest countries in the world by population are the three largest aquaculture producers, suggesting that domestic demand is a main driver of aquaculture production. Abate, Nilsen, and Tveterås [10] find that the stringency

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Fig. 5 European seabass aquaculture production (million tons) and its prices (USD/kg) from 1990 to 2019. (Source: FAO [1])

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Fig. 6 Whiteleg shrimp aquaculture production (million tons) and its prices (USD/kg) from 1990 to 2019. (Source: FAO [1])

of environmental regulation has also been a central factor in explaining growth in aquaculture production. Aquaculture production is found in most countries around the world [11]. However, Asia is the dominant region with 92% of global production, when we include all aquaculture production, not only fish, crustaceans, and mollusks. China is by far the world’s biggest producer, representing 57% of global aquaculture output. Over time, the production shares of different world regions have changed significantly, as shown in Fig. 7. Europe and North America have experienced rather dramatic declines in their global production shares and have become large net importers of farmed seafood. Asia has increased its production share most, while

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Fig. 7 Aquaculture production quantity share by region. (Source: FAO [1])

Africa and South America have also experienced an increase, although from a very low base. These production-share developments can probably be explained by a mix of not only population growth, growth in GDP per capita, and environmental and other regulation, but also productivity growth [8–11]. In this chapter, we survey studies of aquaculture productivity, and its determinants. The chapter is organized as follows: In section “Bioeconomic Modeling of Aquaculture Production,” we discuss bioeconomic modeling of aquaculture production, and the farmer’s optimization problem. Section “Empirical Analyses of Productivity and Efficiency in Aquaculture” provides an overview of empirical studies of productivity and efficiency in aquaculture. Next, in section “Analyses of Production Risk and Economic Risk,” we survey empirical studies of production risk. Section “Analyses of Environmental Externalities” provides a survey on empirical studies of relationships between productivity and environmental factors. In section “Analyses of Agglomeration Economies,” we survey studies of agglomeration economies in aquaculture. Section “Conclusion” provides concluding comments.

Bioeconomic Modeling of Aquaculture Production Understanding of productivity developments in aquaculture requires an understanding of the production processes in this sector, and further benefits from insights provided by the literature on bioeconomic modeling of aquaculture. Aquaculture has several inherent characteristics, most notably a long production period in

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combination with biological and environmental shocks, which represents departures from underlying assumptions of the standard model of the competitive firm, and has significant consequences for observed productivity across firms and over time. The aquaculture farmer’s economic optimization problem involves (1) choice of production technology and investment in fixed capital equipment, (2) timing release of fingerlings into the ponds or cages, (3) making decisions on feeding and other decisions that affect the survival and growth of the biomass of fish, and (4) harvesting the fish for sale [12]. Bioeconomic models can be applied in modeling of this optimization problem [13], and we will look more into these in the following.

Bioeconomic Models Several authors have examined the scientific production related to the use of bioeconomic modeling in aquaculture. Allen et al. [14] identified 22 specific examples of modeling aquaculture production processes during the period of 1974–1983. Leung [15] found 32 studies in which bioeconomic models were used in aquaculture during the period of 1984–1993. Cacho [16] provides a review of bioeconomic literature and presents 20 published papers from 1974 to 1995. Pomeroy et al. [17] identified and surveyed 28 papers that used integrated models for fish production during the period of 1994–2003, of which only seven were applied in the field of aquaculture. Llorente and Luna [13] present 40 published papers on bioeconomic modeling of aquaculture management during the period 2004–2015, which include both optimization and simulation models. Karp et al. [18] and Leung and Shang [19] considered the problem of determining optimal harvest and restocking times and levels for farmed shrimp. Bjørndal [20] developed the first optimal harvesting models for aquaculture based on the forestry literature. Several authors have extended Bjørndal’s model to emphasize specific aspects of the problem. Arnason [21] introduced dynamic behavior and presented a general comparative dynamic analysis. Furthermore, he introduced feeding as a decision variable. Heaps [22] modeled density-independent growth, while Heaps [23] allowed for density-dependent growth and looked at the culling of farmed fish. Mistiaen and Strand [24] demonstrated general solutions for optimal feeding schedules and harvesting times under conditions of piecewise-continuous, weight-dependent prices (harvest). Yu and Leung [25] develop a partial harvesting model that addresses discrete partial harvesting and other partial harvesting using impulsive control theory. They consider a general framework for identifying the optimal time to harvest an aquacultural crop in the case on heterogeneous production cycles. In bioeconomic models, there is usually a growth and survival function for the farmed fish, and it may also include other biological or environmental variables, functions, and constraints which influence the production process and its effects on the environment. The bioeconomic model can also include constraints on the production process from government regulation and physical capacities of the productions system. Choice variables may be the timing and volume of fingerlings

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released into the farm, feed volume, and timing and quantity of harvested fish. The bioeconomic optimization problem consists of finding the values of the choice variables that maximize the objective function, for example, profits. Following Asche and Bjørndal [12], the farmer’s maximization problem can be specified as follows:

max π(t) = V (t)e−rt −

{0≤t≤T }



t

Cf F (u)Re−(M+r)u du,

0

where π is profits, t is time, T is life expectancy of fish, V(t) is gross fish biomass value, Cf is feed costs per unit of feed, F is the feed quantity per fish, R is the number of recruits (i.e., fingerlings released into the water), M is fish mortality rate, and r is the interest rate. The gross biomass value is given by V(t) = p(w)B(t), where p is sales price as a function of fish weight w and B is biomass volume. The biomass is B(t) = N(t)*w(t), i.e., a product of the number of fish N and average weight of  the fish w. The feed quantity per fish at time t is F(t) = ft w (t), where ft is the feed  conversion ratio, i.e., the ratio of feed volume in kg to fish growth in kg, and w is the fish growth rate. The first order condition for profit maximization can be derived as: π  (t) = V  (t)e−rt − rV (t)e−rt − Cf F (t)Re−(M+r)t = 0 and can be rewritten as: Cf F (t ∗ ) p (w)   ∗  w  (t ∗ ) w t + =r +M + ∗ p(w) w (t ) p(w)w (t ∗ ) The marginal revenue per fish with respect to time is on the left-hand side, while marginal cost per fish is on the right-hand side. Cf F(t) are the feed costs per fish at time t, and p(w)w(t) is the value of the fish. Cf F(t)/p(w)w(t) is thus the relative feed cost, which combined with the interest rate r and the mortality rate M represents the cost of not harvesting the fish at time t. The fish farmer will continue to feed the fish as long as the marginal revenue is higher than marginal cost, and harvest when marginal revenue intercepts the marginal cost, as shown in Fig. 8. This figure also shows the effect of introducing feed costs on optimal harvesting time (t2 * ) as compared to a situation without feed costs (t1 * ).

The Rotation Problem Among the most important decision-making activities in production planning is that of determining the optimal rotation of live fish, i.e., finding the best sequence of release of fingerlings and harvesting of fish. This has impact on the farm’s cash flow

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marginal revenue

[r + M] CfF pw r+M

t*2

time

t*1

Fig. 8 The optimal harvesting time with and without feed costs. (Source: Asche and Bjørndal [12], Fig. A2)

$

0

t1

t2

t3

t4

time

Fig. 9 The rotation problem in aquaculture. (Source: Asche and Bjørndal [12], Fig. 9.6)

as well as the allocation of limited production resources, such as feed, fish, space, and environmental resources [16, 26]. The rotation problem in aquaculture, illustrated in Fig. 9, shares many features with rotation problems in forestry and traditional terrestrial livestock production. However, fish farming also exhibits specific features that demand a more flexible model than those constructed for other sectors. In Guttormsen [26], an extended version of the Faustmann model is presented, which is general enough to treat different species and technologies. Two particularly important aspects of the problem are emphasized: first, the possibilities for cycles in relative price relationships between fish of different weight, and second, restrictions in release time for certain species. An illustration of the model based on assumptions from salmon farming shows that the inclusion of these two features has major influences on rotation time, and hence harvest weight. Guttormsen [26] argues based on his extended Faustmann model that a well-developed production plan can mean the difference between loss and profit for a fish farm.

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Risk and Risk Aversion In modeling of the farmer’s economic optimization problem, one can assume a deterministic setting with certain prices and absence of production risk. This is often implicit in theoretical models of aquaculture [12] which maximize profits, or in a multiperiod setting net present value of profits. However, risk is an inherent characteristic of aquaculture, both in the production process and in markets, leading to both output risk and price risk. Fish diseases, algae blooms, and other biological and environmental shocks are sources of production risk. When risk is incorporated and the farmer is assumed to be risk neutral, then the problem is one of maximizing expected net present value. However, if the farmer is risk averse, then the optimization problem becomes an expected utility maximization problem [27–29, 101]. The EU model of the competitive firm is a member of a broad range of maximization problems that have been considered in the EU theory of choice under uncertainty. Many of these can be fitted into the general framework: Maxα E [U (ϕ(θ, α, W0 ))] where U(·) is a von Neumann-Morgenstern utility function, α is a control variable (assumed to take positive values), θ is an economically relevant random variable, W0 is initial wealth, and ϕ(·) is a function mapping actions α and realizations of θ into outcomes, normally taken to be wealth levels. In the theory of the firm, the control variable α might be the production level y or a vector of input levels x. The random variable θ might be the production level y or the output price p, or both. The argument of the utility function, ϕ(·), might be the profit function plus initial wealth W0 . A foundation for modeling of risky production technologies, and also the econometric study of heteroskedastic production technologies, which is fruitful also for the study of aquaculture has been provided by Just and Pope [30]. They suggested eight postulates for the stochastic production function which they claim to be reasonable on the basis of a priori theorizing and observed behavior. Furthermore, they specified a particular functional form which satisfies the eight postulates. This is known as the Just-Pope production function, which is given by [30] y = f (x; α) + h (x; β) ε,

(1)

where y is output level, x is a vector of input levels, ε is a stochastic term, and E[ε] = 0. The function f(·) is the mean production function, and h(·) is the variance production function. The parameter vectors α and β are the mean and variance function parameters, respectively. In Eq. (1), the effect of input changes has been separated into two effects: the effect on mean output and the effect on the variance of output. The Just-Pope production function is a heteroskedastic specification, because the variance of y is a function of the input vector x. An advantage of the

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Just-Pope model is that it allows us to analyze the effects of changing input levels on mean output and output risk separately. This can be seen by deriving from Eq. (1) the conditional variance of output, var.[y] = [h(x; β)]2 var[ε], and the conditional mean output, E[y] = f (x; α). There is a positive linear relationship between the moments of output and the moments of profit under JP production risk (Eq. 1), with the mean and variance of profit given by E(π) = p•f(x) − w x = p•E(y) − w x and Var(π) = p2 Var(y). In the extreme case of risk neutrality (i.e., dU/dVar(π) = 0), the producer is only concerned about mean profits (output) and ignores the output risk effects of input choices. However, the model implies that with increasing risk aversion among farmers, input choices reflect that reduction of profit (output) risk becomes more important at the expense of increasing mean profit (output). Later, we will see how this theoretical framework has been applied and extended in econometric studies of risk in aquaculture.

Biological Shocks and Price Dynamics Production risk does not only affect the individual farmer, but can also have influence on the entire market. The effects of production risk caused by biological shocks on aquaculture markets have been explored by Asche, Oglend, and Kleppe [31]. They specify a partial equilibrium model for a biological production industry (as fish farming) and derive the conditions for profit-maximizing harvest and harvest transitions in a competitive setting. In their example, the salmon farmer’s timing decision to harvest depends on a limited availability of each year class fish stock, and environmental shocks. These factors make the optimal transition between each year class stock to occurs when the difference between the stocks’ marginal value is at maximum, which is contrary to what is found in Faustman’s model. This harvest strategy affects the market by creating a temporary price spike until it is normalized by the marginal value of the new stock, contributing to a short-term salmon price volatility. Furthermore, this model implies that biological shocks have effects on productivity, also through changes in harvest decisions.

Productivity in Aquaculture The relationships between different measures of economic and productivity performance may not be straightforward in aquaculture. In a standard textbook model of the competitive firm – producing one homogeneous product, absence of uncertainty, or risk related to prices and output, and where firms only differ in the relationships between input use and output as represented by the production technology f(x) – the most profitable firms will also be the most productive as measured by a primal measure of total factor productivity (TFP). Fluctuations in output prices and timing of harvesting are one explanation for this. A firm may have high primal productivity relative to other firms, as measured by the ratio of physical output quantity to inputs

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(e.g., total factor productivity – TFP), but have a profitability performance that is relatively weaker due to timing of harvesting and the realized market prices at the time of harvesting. Presence of market and production risk and risk aversion may also influence productivity performance. Biological and environmental shocks may have adverse effects on productivity. These shocks are typically not of the same magnitude across farms and may consequently affect the relative productivity performance. Moreover, production shocks can affect the productivity development over time, for example, disease shocks leading to high loss of biomass and revenue tend to occur at infrequent time intervals. Risk-averse farmers may not only make input choices that reduce the level of risk, but also affect the ratio of expected and realized output level to input levels as measured by TFP. When farmers are heterogeneous with respect to risk preferences, as Kumbhakar [27, 28, 32] found, i.e., have different degrees of risk aversion, this can also affect the observed relative productivity performance across farms as measured by TFP.

Empirical Analyses of Productivity and Efficiency in Aquaculture Productivity and efficiency in the aquaculture industry have become the topic of an increasing number of studies over time, as shown by Table 1. The literature has been able to cover all the major farmed species – carp, tilapia, catfish, salmon, seabass, seabream, shrimp, and oysters. Studies cover aquaculture sectors worldwide. While most analyses have been performed of aquaculture production in Asia (e.g., Bangladesh, China, India, and Vietnam), there are still many studies of aquaculture sectors in Europe, North and South America, and Africa. Early during the industry development, works on assessing the industry’s performance applied production and cost structural models to estimate technical change and returns to scale. For instance, Salvanes [92, 93] and Bjørndal and Salvanes [94] found cost inefficiencies related to the overregulated salmon farming in Norway. In addition, a detailed analysis of production inefficiency and its sources has been essential to provide evidence to decision makers to increase productivity. The main parametric approach applied in the literature for efficiency has been stochastic frontier analysis (SFA). Its main advantage is the possibility of decomposing the deviation from the frontier into stochastic noise and technical inefficiency in production. However, it is also necessary to impose a particular parametric form for the underlying technology.1 An alternative has been to use nonparametric approaches such as Data Envelopment Analysis (DEA). This type of analysis does not require a parametric assumption of the underlying technology. However, the estimated frontier may be sensitive to stochastic noise in the data and can

1 Recently,

nonparametric SF models have been developed to avoid this problem. See Parmeter et al. [95], Zhou, Wang, and Kumbhakar (2020), and references in there.

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Table 1 Studies on productivity and efficiency of the aquaculture industry Authors Sharma and Leung [33] Iinuma, Sharma, and Leung [34] Sharma et al. [35] Dey et al. [36] Karagiannis et al. [37] Sharma and Leung [38]

Year 1998

Region Nepal

Species Carp

Approach SFA

1999

Malaysia

Carp polyculture

SFA

1999 2000 2000

China Philippines Greece

Carp polyculture Tilapia Seabream/seabass

DEA SFA SFA

2000

Carp

SFA

Awoyemi et al. [39] Irz and Mckenzie [40] Ara et al. [41] Chiang et al. [42] Martinez-Cordero and Leung [43] Dey et al. [44]

2003

South Asia (India, Bangladesh, Pakistan, and Nepal) Nigeria

Fish

SFA

2003

Philippines

Fish

SFA

2004 2004 2004

Bangladesh Taiwan Mexico

Fish Milkfish Shrimp

SFA SFA DEA

2005

China/India/Thailand/Vietnam

SFA

Cinemre et al. [45] Kaliba and Engle [46] Kaliba, Engle, and Dorman [47] Alam and Murshed-e-Jahan [48] Kareem, Aromolaran, and Dipeolu [49]

2006

Black Sea region, Turkey

Fresh water pond polyculture system Trout

DEA

2006

USA

Catfish

DEA

2007

USA

Catfish

DEA

2008

Bangladesh

Prawn/carp

DEA

2009

Nigeria

SPF

Singh et al. [50] Chang et al. [51] Nilsen [52] Ogundari and Akinbogun [53] Onumah et al. [54] Alam [55] Nielsen [56] Pantzios et al. [57]

2009 2010 2010 2010

India Taiwan Norway Nigeria

Polyculture (Clarias gariepinus/Tilapia guinensis mainly) Fish Shellfish Salmon Fish

SFA DEA SFA SFA

2010

Ghana

Fish

SFA

2011 2011 2011

Bangladesh Denmark Greece

Pangas Trout Seabream/seabass

DEA DEA SFA (continued)

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Table 1 (continued) Authors Alam et al. [58] Asche and Roll [59] Asche, Guttormsen, and Nielsen [60] Begum et al. [61] Bukenya et al. [62] Arita and Leung [63]

Year 2012 2013

Region Bangladesh Norway

Species Tilapia Salmon

Approach SFA SFA

2013

Norway

Salmon

DEA

2013 2013

Bangladesh Uganda

SFA SFA

2014

Hawaii

Nguyen and Fisher [64] Schrobback, Pascoe, and Coglan [65] Iliyasu and Mohamed [66]

2014

Vietnam

Shrimp Fish (catfish/Tilapia) Several (catfish, crustacean, foodfish, ornamental, mollusk, and others) Shrimp

2014

Australia

Oyster

DEA

2015

Malaysia

DEA

Iliyasu, Mohamed, and Hashim [67]

2015

Malaysia

Begum et al. [68] Iliyasu and Mohamed [69]

2016 2016

Bangladesh Malaysia

Sandvold [70] Anh Ngoc et al. [71] Nguyen et al. [72] Ton Nu Hai et al. [73] Bayazid et al. [74] Forleo et al. [75] Mitra et al. [76] Rahman et al. [97]

2016 2018

Norway Vietnam

Fresh water aquaculture (shrimp, prawn, Tilapia, catfish, and carp) Fresh water aquaculture (shrimp, prawn, Tilapia, catfish, and carp) Shrimp Fresh water aquaculture (shrimp, prawn, Tilapia, catfish, and carp) Juveniles/salmon Pangas

SFA DEA

2018 2018

Vietnam Vietnam

Catfish Lobster

DEA DEA

2019 2019 2019 2019

Bangladesh Italy Bangladesh Bangladesh

Floodplain

DEA DEA DEA DEA

Tilapia/catfish Pangas/Tilapia

DEA

DEA

DEA

SFA DEA

(continued)

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Table 1 (continued) Authors Rahman, Nielsen, and Khan [138] Rodrigues et al. [77] Scuderi and Chen [78] Aponte [79] Aripin et al. [80] Long et al. [[81], [82]] Mitra et al. [83] Fernández Sánchez et al. [84] Gutiérrez, Lozano, and Guillén [85] Ton Nu Hai and Speelman [86] Ton Nu Hai, Meensel, and Speelman [87] Long [88] Nielsen et al. [89] Khan, Roll, and Guttormsen [90] Hukom, Nielsen, and Nielsen [91]

Year 2019

Region Bangladesh

Species Pangas/Tilapia

Approach SFA

2019

Brazil

Fingerlings

DEA

2019

USA

Oyster

SFA

2020 Norway 2020 Malaysia 2020a;b Vietnam

Salmon Seabass Shrimp

DEA DEA DEA

2020 2020

Bangladesh Mediterranean Sea

Tilapia/catfish Seabream/seabass

DEA SFA

2020

EU members

Several (fish and shellfish)

DEA

2020

Vietnam

Lobster

DEA

2020

Vietnam

Lobster

DEA

2021 2021 2021

Vietnam Mediterranean Bangladesh

Shrimp Seabream/seabass Pangas

DEA DEA SFA

2021

Indonesia

Shrimp polyculture

DEA

SFA Stochastic Frontier Analysis, DEA Data Envelopment Analysis

be overestimated. Thus, the noise term should be introduced in nonparametric frontier models and needs additional procedures (e.g., bootstrapping) for statistical inferences [96]. Both SFA and DEA approaches have been widely applied for estimating productivity and efficiency of aquaculture sectors (Table 1). In general, parametric studies apply a stochastic production frontier model using a Cobb-Douglas specification, or a more flexible form, such as the translog. Most nonparametric studies employ a total factor productivity index with input-oriented specification [85]. Independent of the approach, the technical efficiency (TE) and cost efficiency (CE) measurements provide evidence that the industry has potential to increase its efficiency levels. Several drivers of efficiency have been identified in the literature. For the parametric approach, it has been possible to estimate simultaneously the effect of several factors affecting efficiency together with the frontier. For the nonparametric approach, a second stage has been incorporated by running different regression

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models (e.g., linear, truncated, tobit, and logit, probit, and fractional) on the potential explanatory variables of inefficiency. Human capital or skill is among the factors which have been studied. Education level, training, and experience are common variables representing human capital used to explain efficiency levels [36, 42, 45, 50, 51, 53, 54, 58, 61, 64–66, 68, 71– 73, 76, 78, 80–82, 87, 146]. It is expected that higher levels of education, training, and experience will contribute to better informed decisions with respect to use and allocation of production resources and inputs. Other factors related to farmers and farm management include whether it is an owner-operated farm (i.e., company type [45, 46, 51, 59, 74]), or if aquaculture is the farmer’s primary activity [38], their household size, if there is a hired manager, and if a record of activities is maintained [62]. Sharma et al. [35] and Hukom Nielsen and Nielsen [91] show that comanaged farms perform better than without comanagement. Basically, comanaged farms require more communication and collaboration among the farmers, which builds adaptative capacity and resilience leading to a better performance. However, Forleo et al. [75] found the opposite for Italian aquaculture firms, where the most inefficient farms are cooperatives. Credit availability is also a key driver for efficiency in aquaculture farms. Several studies of different species (trout, tilapia, and catfish) in different regions (Uganda and Bangladesh) found that credit constrains are associated with lower productivity [45, 62, 76, 97]. Given the diverse set of technologies used in aquatic farming (e.g., intensive vs. extensive systems, ponds vs. cages, and small vs. large), farms’ technical characteristics have been also considered as potential drivers of efficiency. For instance, intensive carp farms in Nepal [33] and in Malaysia [34] and shrimp farms in Vietnam [81] performed better than extensive ones. However, other studies show that small extensive farmers can be more technically and economically efficient [35, 48, 64]. Aponte [79] identified some benefits with respect to smaller salmon firms presenting higher revenue productivity than the larger firms. In general, the farm operation scale (e.g., cage, pond, and farm area), specialization (mono or polyculture), water management, quality and source, and other inputs used in production (from temperature, pH, and nitrogen to smolt quantity and price, feed and fertilizer management) have been considered aspects that influence production efficiency. Furthermore, environmental aspects influencing efficiency have received particular attention in the last decades. Diseases leading to slower growth and mortality together with pollution and climate change negatively affect the farms’ performance [43, 59, 64, 91]. However, farmers are able to innovate and learn to deal with these issues. For instance, catfish farmers in Vietnam which experienced some climate change impacts in the past, such as flooding or salinity intrusion, had a better productive performance [72]. Other exogenous variables that seem to influence how well farms perform are related to access to markets and prices, and other complementary industries. Ogundari and Akinbogun [53] found that access to markets was associated with significant lower technical inefficiency of farmers. Gutiérrez, Lozano, and Guillén

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[85] suggested countries with a significant and established capture, and fisheries sectors were also more efficient. This might be related to a better infrastructure and logistics already established to support the aquaculture industry.

Analyses of Production Risk and Economic Risk Production and economic risk are present for various farmed species and countries. In fact, these risks have contributed to boom and bust cycles in several aquaculture sectors [98]. Often, feed and capital are found as risk-enhancing inputs factors and labor as risk reducing [29, 53, 99, 100]. Tveterås [29] estimated primal panel data models for risky production on an unbalanced panel of Norwegian salmon farms using the Just-Pope model framework, presented in section “Bioeconomic Modeling of Aquaculture Production,” and analyzed how different specifications of functional forms and different estimators influenced the empirical results. Feed and fish input were found to have risk-increasing effects on output, while labor and materials input had risk-reducing effects on input. In other words, a marginal increase in feed input increased the variance of output while a marginal increase in labor reduced the output variance. The finding that feed is risk increasing and labor risk reducing has been found in several later studies, e.g., Tveterås [147], Kumbhakar [27, 28], Kumbhakar and Tveterås [101], and Kumbhakar and Tsionas [102]. Labor plays a particularly important role in production risk management. Farm workers’ main tasks are monitoring of the live fish in the pens, biophysical variables (sea temperature, salinity, oxygen concentration, algae concentrations, etc.), and the condition of the physical production equipment (pens, nets, feeding equipment, anchoring equipment, etc.). Thus, workers’ ability to detect and diagnose abnormal fish behavior, detect changes in biophysical variables, and make prognoses on future development is crucial to mitigate adverse production conditions and reduce production risk. There are several explanations why feed may increase the level of output risk. The feed is not all digested by the farmed fish, and residue is released into the environment as feed waste or feces. Salmon competes with this released organic waste for the limited oxygen available in the cages (organic waste consumes oxygen in the decomposition process). In addition, feed waste can also lead to production of toxic by-products, such as ammonia. The models of Tveterås [29] accounted for technical change in both the mean and variance production function. When farmers are risk averse, the change in the variance function should also be accounted for in an analysis of technical change. Tveterås finds that from 1985 to 1993, technical change led to not only higher mean output for average input levels, but also to higher output risk conditional on average input levels, as shown in Fig. 10. However, when technical change is analyzed using the first-order stochastic dominance (FSD) criterion, it is found that the improvement in mean output dominates the increase in output risk, as shown in Fig. 11. First-order stochastic dominance is present if the cumulative density function (cdf) of the new technology lies strictly to the right of the cdf of the

Fig. 10 Rate of technical change of the mean (TC) and variance production functions (TCV). (Source: Tveterås [29])

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0.9

1985

F(y)

0.8

1986

0.7

1987

0.6

1988

0.5

1989

0.4

1990

0.3

1991

0.2

1992 1993

1.7

1.6

1.5

1.4

1.3

1.2

1.1

0.9

0.8

0.7

0.6

0.5

0.4

0

1

0.1

y

Fig. 11 Cumulative density function (cdf) of salmon output evaluated in sample mean input levels. (Source: Tveterås [29])

old technology for all values of output y. Then both risk averse and risk neutral producers will prefer the new technology. FSD of the new technology relative to the old technology also implies that risk-neutral and risk-averse farmers can agree that there has been technical progress during the data period. Unlike Tveterås, who only estimated the structure of production risk, Kumbhakar [27] extended the production model to include not only risk, but also producer’s attitude toward risk, and technical inefficiency. Kumbhakar’s specification did not assume a parametric form of the utility function, or a specific distribution for the error term representing production risk, and it allowed for a generalized technical efficiency model. In Kumbhakar [27], the model is also estimated on a sample of Norwegian salmon farms. According to the econometric estimates, all farmers are risk averse. Production risk is found to be increasing with feed and decreasing with labor and capital. Furthermore, risk preference associated with production uncertainty is found to be stronger than that of technical inefficiency. Technical inefficiency is found to be positively related to feed and negatively related to labor and capital. The mean technical inefficiency for the sample farms is found to be 7.9%. Kumbhakar and Tveterås [101] use a similar model framework and same data as Kumbhakar [27]. Farm age is introduced as an additional variable, and the relative risk premium is estimated. Empirical evidence of production risk and farm heterogeneity is found based on the econometric estimates. Capital and labor are found to be risk reducing, while feed and fish input are risk increasing. They also find that farm age has a negative effect on mean output, with a sample mean elasticity of −7.2%. The accumulation of organic sediments below the cages over time may change the environmental conditions around and within the farm, resulting

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in less oxygen available to the animal, along with higher risk of diseases. Hence, the farm-age variable may capture both the firm’s positive learning effect and the negative effect on fish health and welfare from the changes in the environmental conditions. The results imply that the negative fish welfare effects dominate. Moreover, Kumbhakar, and Tveterås [101], not surprisingly, find salmon farmers to be risk averse. However, they also find that farmers are downside risk averse. This means that farmers generally avoid situations which offer the potential for substantial gains, but which also leave them even slightly vulnerable to losses below some critical level. To gain a better understanding of the importance of risk-averse behavior as well as differences in the degree of risk aversion among farms, the authors estimated the risk premium (RP) defined as the sure amount of money satisfying E(U(π)) = U(π − RP) for each farm in every year. Since the farms are heterogeneous, it was considered more appropriate to focus on relative risk premium (RRP) values, i.e., the RP as a percentage of mean profits, for the salmon farmers in the sample. The estimated values of relative risk premiums were found to be positive, but vary across farms, and over time. The overall mean RRP is 17.9%, while the mean (by farm) RRP ranges from 11.5% to 31.5%. Overall, this empirical evidence suggests substantial welfare loss associated with private risk bearing in this industry. Kumbhakar and Tsionas [102] deal with nonparametric estimation of risk and the risk preference function when producers face both production risk and output price risk. Models are specified to estimate risk preference of individual producers under (i) only production risk, (ii) only price risk, (iii) both production and price risks, (iv) production risk with technical inefficiency, (v) price risk with technical inefficiency, and (vi) both production and price risks with technical inefficiency. Norwegian salmon farming data is used for an empirical application of some of the proposed models. Based on their estimates, Kumbhakar and Tsionas find that salmon farmers are, in general, risk averse. Labor is found to be output-risk decreasing while capital and feed are found to be risk increasing. The nonparametric estimation approach of Kumbhakar and Tsionas [102] does not restrict the distribution of elasticity estimates to be symmetric. In fact, they find that none of the distributions are symmetric but are all skewed to the right. Thus, the median values of these elasticities are less than their mean values. For technical change (TC), they find mean technical progress at the rate of 4.6% per year, with a median value of TC 5.3%. A notable feature of the TC distribution is that it is bimodal. The two modal values of TC are 2.5% and 7.5% per annum, respectively. Although the mean TC is around 6% per year, some farms experienced technical progress at the rate of 2.5% while other “leading” farms experienced a much higher rate. Kumbhakar and Tsionas report the frequency distribution of elasticities of the risk function with respect to labor, capital, feed, age, and time. The mean (median) values of these elasticities for labor, capital, feed, age, and time are −0.049 (−0.043), 0.016 (0.011), 0.085 (0.016), −0.001 (−0.001), and 0.002 (0.002), respectively. The risk part of the production technology seems to be quite insensitive to changes in the age (experience) of farmers. Similarly, no significant change in

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production risk has taken place over time. The mean (median) values of relative risk premium (RRP) in the model with both production risk and price risk are 0.087 (0.0522). In other words, the median farm is willing to pay 5.22% of the mean profit as an insurance against possible profit loss due to both production risk and output price uncertainty. Production risk has also been studied for other countries and species. Ogundari and Akinbogun [53] model technical efficiency with production risk in inputs based on a sample of fish farms from Oyo State, Nigeria. The species cultured by the sampled farms include tilapia, catfish, and carp. Clarias (catfish) is the most frequently cultured fish, grown by over 80% of the sampled fish farms. The authors applied a stochastic frontier model using a similar flexible risk specification as Kumbhakar [27]. Their results suggest not only that labor, fertilizer, and feed influence the mean fish output, but also that these inputs influence production risk. As found in other studies (Tveterås 2000) [27, 28, 101, 102], fertilizer and feed seem to increase risk, while labor is risk reducing. Thus, an average risk-averse farmer in Oyo State in Nigeria is expected to use less of fertilizer and feed and more labor compared to a risk-neutral farmer. They also found that without accounting for the flexible risk component in the production technology specification, the efficiency score is overstated. Sarker et al. [103] also provide empirical estimates of production risk and technical efficiency, using an extended stochastic frontier model, based on Just and Pope framework and Kumbhakar (2002). They focus on Thai Koi (Anabas testudineus) in the northern part of Bangladesh, a high-valued species in the market, which has experienced highly volatile output across different years and farms. Results show that the main input for production is feed, which is also a risk-increasing input together with fingerlings, labor, and salt. On the other hand, zeolite, a waterpurifying product, and pesticide appear to be risk reducing. As in Ogundari and Akinbogun [53], the mean TE scores are overestimated for the conventional model (0.96) in comparison to when accounting for the risk in the production model (0.73). Another paper by Khan, Guttormsen, and Roll [100] also looks at Bangladesh aquaculture industry but focuses on estimating the structure of production risk of the species Pangas (Pangasius hypophthalmus), which has in the last decades become an important export-oriented farmed fish sector. They estimated meanand risk-stochastic production functions based on the Just–Pope framework. Their results show a significant different production risk between small and large farms. Moreover, in contrast to other studies (e.g., salmon farming), feed and capital seem to reduce risk, while fingerlings and farm size (pond area) increase risk. In their analysis, they find that investments in training and extension services could reduce production risk. More efficient use of feed and other resources could also increase production, in particular for small farms with capital constraints and facing high feed prices. The authors report that smaller farms tend to use lower-quality or homemade alternative feed due to barriers to access credit (e.g., needs for collateral). This leads to productivity loss and increased production risk. The study highlights the importance of studies in production risk, in particular in developing countries, where farmers are more vulnerable to risk.

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Analyses of Environmental Externalities The environmental performance and sustainability of aquaculture has been subject to increasing scrutiny over the last decades. There are concerns about both global and local environmental issues. The use of feed ingredients which are scarce or finite, e.g., fishmeal and fish oil, or have significant environmental footprints, has been one type of concern [104–106]. Local pollution of water in terms of excessive nutrient loads [64, 86, 107, 108], and genetic pollution of wild fish stocks through escape of genetically modified farmed fish, is another concern [109]. In addition, there are increasing concerns with respect to the animal welfare due to excessive mortality [86, 104]. All these environmental externalities can be regarded as a bad output from the production process in aquaculture. Based on that, several studies have specified a model that quantifies the relationship between the good and bad outputs as well as inputs use [108–110]. In fact, efficient regulations to address these environmental issues require knowledge about these relationships. For instance, for outputs with a joint production process, a reduction in the bad output due to regulations will also mean a reduction in the good output. So, these relationships provide information on producers’ incentives to address the negative externalities in the production process. In order to investigate the incentives that Norwegian salmon farmers have to address escapees, Pincinato, Asche and Roll [109] applied a multiproduct cost function with escaped salmon as the bad output and farmed salmon as the good output. They found that escapees have not affected the salmon production cost directly or indirectly. This means there are relatively little private incentives to address this issue. However, given the nonjoint production technology and input-output separability, it is possible to elaborate regulations that do not reduce the farm’s economic efficiency. Nevertheless, several studies on efficiency in aquaculture point out environmentrelated factors, such as water pollution, nutrient emissions, and diseases, as key drivers of farms performance [59, 64, 86–88, 107, 111]. Long [88] found that lower input levels of chemicals and drugs, and higher feed use per area, seem to increase the cost efficiency of shrimp farming in Vietnam. Similar results are found for lobster farms in Vietnam, for which the use of inputs more efficiently would improve environmental performance and lower production costs [86]. In addition, Hukom, Nielsen, and Nielsen [91] examined how environmental stressors such as industrial pollution, disease, land use change, and domestic waste were perceived by smallscale shrimp farmers in Indonesia. Their results suggest that farmers operating in comanagement are more efficient and less frequently exposed to industrial pollution. Inefficiency can also make environmental impacts worse, in particular by the overuse of inputs such as feed. Asche, Roll, and Tveterås (2009) found that both technical and allocative inefficiencies explain not only the level and variation of Norwegian salmon farm costs, but also that major environmental impacts in the production process are due to technical inefficiency. Thus, the degree of inefficiency in the industry matters for designing environmental regulations and the industry’s overall sustainability.

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In general, improvements related to the industry’s environmental performance have been reached mainly through technological innovation [112–115]. Feed containing ingredients considered more sustainable (e.g., algae), better feed formulation to avoid feed waste, vaccines, more environment-friendly treatments against parasites (e.g., laser), and biofloc technology are just some examples of improvements due to innovation that contributes to reduce the pressure on the environment. Sandvold and Tveterås [116] examined productivity development using a cost function model for Norwegian juvenile salmon farms, which is a key input in salmon production. The significant decline in production costs found in the study is attributed to innovations in breeding, feed, equipment, fish health, and water technology. These improvements have spilled over to salmon grow-out production farms by increasing survival rates and reducing disease outbreaks after juveniles were released to the sea. Diseases and parasites have over the years caused several supply socks in the seafood supply [117, 118], and several studies have tried to estimate the cost related to these externalities [86, 111, 117, 119, 120]. Abolofia, Asche, and Wilen [119] found significant private cost related to sea lice, a parasite that is currently considered one of the major environmental issues in salmon farming. Another interesting example is the global shrimp industry, which has been hit by disease outbreaks in several periods and places. Asche et al. [117] specified an economic model to estimate the disease outbreaks’ impact on economic risk. When disease outbreaks hit one region, it opens up a market opportunity for other regions to exploit, by starting up and/or expanding their farming industry. This situation creates incentives for relatively unsustainable production practices given the high short-run profitability of the industry. The literature so far has shown that, in general, addressing the environmental externalities can lead to improved economic performance. This is essential given the importance of farmed aquatic species as one of the most resource-efficient sources of protein for the global food system [121, 122].

Analyses of Agglomeration Economies One strand of studies of aquaculture productivity has focused on economic effects of geographic colocation of aquaculture farms, and also colocation of sectors which are related to aquaculture. These studies are concerned with the possibility that there may be positive externalities between firms which can be realized when there is a geographic agglomeration of related firms. Localization in a cluster of related firms and institutions can yield economic benefits, so-called agglomeration economies, or external returns to scale. A distinction is often made in the literature between two types of external agglomeration economies – localization and urbanization economies. The former increase returns within a single or more narrowly defined industry (industry clusters) and draws on seminal insights from Marshall [123], who argues that firms that colocate could enjoy external economies because of exchange of inputs, expertise,

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and division of labor [124]. The latter increase returns to a diversity of industries in a regional or urban economy [125] and emphasizes the positive externalities associated with new ideas across different sectors, as suggested by Jacobs [126]. These agglomeration economies have also been referred to as intra- (localization) and inter- (urbanization) clustering [127]. Agglomeration economies also have linkages to the concept of a regional innovation system. A technological innovation system can be defined as “a dynamic network of agents interacting in a specific economic/industrial area under a particular institutional infrastructure and involved in the generation, diffusion, and utilization of technology” [128, 129]. Private companies in an aquaculture innovation system include salmon farming companies and their suppliers, seafood processing industry, etc. Universities and research institutes are not only important institutions through the R&D they undertake, but also as suppliers of highly trained labor and researchers. Innovation systems related to aquaculture have been studied by Doloreux et al. [130] and Bergesen and Tveterås [131]. The latter study estimates models which aim to explain the determinants of collaboration and innovation. Bergesen and Tveterås’ find, based on their estimated models, that firms’ internal R&D resources are key to ensure collaboration with external organizations, in particular, research institutions. However, collaboration with R&D institutions has a limited effect on firms’ probability of innovating. Innovation rates are positively influenced by firms’ internal skills (i.e., R&D employees), and firm’s collaboration with other firms in the value chain. In particular, innovation seems to be concentrated at the beginning and end of the supply chain. For the aquaculture industry, input suppliers were found to be highly innovative, while farms incorporate innovations from their suppliers. Tveteras [132], Tveteras and Battese [133], and Asche, Roll, and Tveteras [134] find evidence of agglomeration economies in Norwegian salmon farming using different econometric modeling approaches. Tveteras [132] estimates long- and short-run flexible cost functions to test the structure of agglomeration economies discussed in Porter [135] in the context of the salmon farming industry in Norway. More specifically, these cost models make it possible to test effects of agglomeration on firms’ costs, scale economies, and input demands. According to the results, which seem to be robust to different econometric model specifications, there is evidence of agglomeration economies that lead to cost savings in salmon farm production. These savings are associated with increasing regional farm density and increasing regional industry size. Moreover, the external economies are also significant compared to the estimated internal scale economies. Under some circumstances, the estimated models predict that smaller firms in regions with a large aquaculture industry have lower production costs than larger firms located in regions with a small industry. Sources of agglomeration economies may be thicker input markets, localized knowledge spillovers, and complementarities due to better alignment of activities [132]. There are benefits to the industry as a role from sharing inputs such as the industry’s physical infrastructure capital, research and development, and specialized human capital. For instance, certain investments in capital will not be fully utilized by one single firm, but by sharing the cost and use of certain specialized, highly

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productive external inputs, firms can contribute to savings on materials and labor inputs. The aquaculture industry demands specialized expertise in management, export marketing, production monitoring, veterinary services, fish and marine biology, feed technology, etc. However, there is a minimum market size for certain specialized services to be provided. Long geographic distances and high transportation costs make regional input markets important. A bigger regional market can lead to more productive and specialized inputs. This is particularly the case for many producer services in the Norwegian salmon industry, which is spread over its long coastline. Much of the productivity increase in salmon farming can be associated to the process of learning by doing. This process is in general localized given the uncertainty and context specificity of the knowledge. In this case, knowledge transmission depends on physical proximity, so that the level of knowledge spillovers increases with the industry’s spatial density. This leads also to agglomeration economies in salmon farming. Moreover, the larger the regional industry size, the larger number of workers, and the greater the diversity of human capital. So, one can expect that the increase in the probability of new ideas and knowledge created by these factors, and exchanged in many more places, will also influence the level of knowledge spillovers [136, 137]. There is indirect evidence from Tveterås’ [132] results of increasing negative externalities due to fish diseases being associated to higher farm density. This might dominate the positive agglomeration economies associated with physical proximity when farm density becomes high. Continuing with salmon farming, Tveterås and Battese [133] estimate a stochastic frontier production function which accounts also for technical inefficiency, and two external economy indexes – regional industry size and regional farm density. The argument for this separation of effects is that localized markets for highly productive, specialized inputs and localized knowledge spillovers can lead to different regional production frontiers. Furthermore, increased levels of localized knowledge spillovers and substitution of internal inputs with external inputs can also lead to fewer errors in decision-making and execution of production tasks, thus causing firms to move closer to the production frontier. The estimated econometric models predict that an increase in regional industry size is associated with increases in both best-practice output and the level of technical efficiency for farms in that region. This implies increasing positive externalities, possibly due to knowledge spillovers and increased supply of specialized external inputs, when the regional industry grows. An increase in regional farm density has a negative effect on frontier output but has a positive effect on the level of technical efficiency. Overall, the effect of increasing regional farm density on output is negative, implying that negative congestion externalities associated with fish diseases dominate the positive externalities associated with knowledge spillovers and the sharing of specialized inputs. Asche, Roll, and Tveteras [134] conducted an empirical study of Norwegian salmon aquaculture where a translog profit function was estimated. This allows one to account for revenue effects in addition to productivity and cost advantages, which

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can be important particularly in high-cost clusters because it also enables firms to cover higher costs. Profits are a measure that accounts for potentially increased economic returns to firms’ equity capital owners from agglomeration and thus also provide an indication of the economic incentives of capital owners to contribute to the geographic concentration of production activities. Asche, Roll, and Tveteras [134] also distinguish between agglomeration externalities within salmon aquaculture (Marshallian externalities) and agglomeration externalities across sectors (Jacobs externalities). By using firm-level panel data, they distinguish between inter- and intraindustry agglomeration effects on firm profitability. The results support the presence of inter-industry effects related to the size of other seafood sectors and the food-processing sector. A larger manufacturing sector, measured by regional employment, is not found to have any effect on profit in salmon production. This indicates that the agglomeration effect on aquaculture from related industries is limited to firms in the food sector. Regional agglomeration externalities have also been present in the aquaculture industry in Bangladesh. Rahman, Nielsen, and Khan [138] examine the influence of these externalities on productivity and efficiency applying a stochastic frontier and inefficiency model using several indices: regional industry size, regional farm density, distances between farms and the point of sale of fish, and the point of purchase of fingerlings and feed. They find that an increase in regional industry size is associated with an increase in frontier output. However, an increase in regional industry size is also associated with a decrease in the level of technical efficiency of the farms in that region. The authors explain this increase by the access to specialized input and services, found also in other studies [132, 139], while the decrease in the level of technical efficiency can be explained by the specialized products’ affordability. In some places, and in particular for small-scale farmers [140], the specialized inputs prices preclude them to produce at-the-best practices frontier. An alternative explanation given by the authors is that larger industries can supply more to local markets, promoting competition and resulting in lower prices received by farmers in areas with higher industry concentrations. As in the salmon farming [133], there might be some negative externalities associated with biophysical congestion in the aquaculture industry in Bangladesh. However, the positive effects on the level of technical efficiency and the elasticity of mean output associated to learning-by-doing and knowledge spillovers outweigh the negative effects. The extensive nature of pond farming in Bangladesh may be the reason for this argument. This contrasts to what is found in the intensive salmon farming. Spatial and biosecurity planning may be key to avoid negative externalities associated with intensification and agglomeration. The results from Rahman, Nielsen, and Khan [138] also indicate distance-related externalities in aquaculture production. For instance, points of sale are important for farmers to sell their product in higher prices markets, e.g., larger urban markets [141], and for farmers to have access to inputs with quality, e.g., fingerling with quality avoid losses and overstocking. In order to exploit these opportunities, farmers must organize themselves (e.g., cooperatives), and increase their bargaining

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power, which can lead to higher selling prices and purchases [142]. Without this organization, these opportunities are in same places taken by middlemen [143].

Conclusion A growing world population will need more healthy seafood. Productivity growth is a main determinant of the expansion of seafood supply from aquaculture. This chapter has provided evidence that many factors influencing productivity and efficiency in aquaculture have been studied in the literature, using data from a wide range of species and countries. Studies may not provide a clear roadmap for improvement of productive performance, but they certainly give a menu of factors which deserve attention for industry and policy makers aiming to increase economic welfare. Both internal and external influences on productive performance have been estimated in the literature. Internal factors include not only conventional inputs such as feed, fingerlings, labor and capital, and scale of operation, but also farms’ human capital, ownership and management structure, and access to credit. External influences can be both negative and positive; they include fish diseases, environmental emissions, and agglomeration externalities. One strand of the literature also provides evidence of production risk, which is partly influenced by farmers’ decision-making, and that fish farmers are generally risk averse. Overall, the empirical results provide evidence that to achieve the productive potential of aquaculture sectors, several sources of market failure – such as externalities, risk, and risk aversion – have to be taken into account, clearly suggesting a pivotal role for government policies and regulation. A “tragedy of the commons” in the form of biological and environmental externalities that is not sufficiently mitigated by government policies can affect both aquaculture sectors’ own productivity and producer welfare, as well as other sectors of the economy. Achieving internalization of the costs of external effects among aquaculture firms is a necessary but far from trivial task for governments, which will require different public regulations and incentives depending on species and production country. Sustainable growth of aquaculture seafood supply depends on intensification of production, i.e., producing more using the same or less area, and at the same time maintaining high-fish welfare and sufficiently mitigating biological and environmental externalities. The literature finds not only productivity growth over time in many sectors, but also that it can be uneven and negative in periods, partly due to biological and environmental shocks such as fish diseases. The presence of significant technical inefficiency is estimated in many aquaculture sectors, as well as factors explaining it. In the infant or early stages of aquaculture sectors, there were many “low-hanging fruits” to be picked that could contribute to increasing productivity, both innovations that shifted the production frontier and other measures that could reduce technical inefficiency [144]. Examples are innovations in feed and its raw materials. As sectors have matured, fewer low-hanging fruits are available, but still there several avenues to increasing productivity significantly.

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One avenue is selective breeding, which is a central form of innovation to increase productivity in aquaculture, as pointed out by Gjedrem et al. [145]. For salmon and shrimp species, selective breeding has been essential in increasing growth rates, reducing mortality, and increasing quality. Still, most species are at an early stage with respect to selective breeding programs, with less than 10% of aquaculture production based on genetically improved stocks in 2012. There is scope for increasing productivity significantly through genetic innovation, as the gains in biological growth rates can be over 10% for each new generation. Aquaculture generally trails far behind plant and farm animal industries in utilizing selective breeding as a tool to improve biological productivity, according to Gjedrem et al. [145]. A systemic challenge to maintain sufficient productivity growth is to further develop aquaculture innovation systems, consisting of aquaculture value chain companies and supporting private and public institutions, including government agencies, education, and R&D institutions [131]. In successful innovation systems, the private aquaculture value chain itself may be financing and investing in innovations in some technology areas, for example, feed and production equipment. In some other knowledge and technology areas, funding is by government or organized by government because innovation is more prone to market failure in the form of private underinvestment. This may, for example, be related to some biological and environmental externalities where firms are not able to appropriate sufficient profits from investing in innovation processes. Global aquaculture can provide examples of innovation systems with high research and innovation investments and high innovation output. But there is no global blueprint for development of aquaculture innovation systems that can provide more sustainable production and higher productivity because of the huge variation in biological, environmental, technological, and institutional characteristics across species and countries.

Cross-References  The Economics of Production in Marine Fisheries

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Benchmarking in the European Water Sector

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Applications of Production Economics Alan Horncastle, Joseph Duffy, Chien Xen Ng, and Peter Krupa

Contents Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Why Benchmarking Is Important in the Water Sector? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Benchmarking Techniques in Regulation: An Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . England and Wales: Cost Benchmarking Prior to PR14 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The Structure of the Water Sector and Regulation in England and Wales . . . . . . . . . . . . . Ofwat’s Approach Prior to PR14 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The Use of Ofwat’s Approach Elsewhere and a Change in Approach . . . . . . . . . . . . . . . . England and Wales: Cost Benchmarking from PR14 Onwards . . . . . . . . . . . . . . . . . . . . . . . Ofwat’s Approach in PR14: A Change in Direction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Ofwat’s Approach to Cost Benchmarking in PR19 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Definition of the Inputs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Identifying the Outputs and Other Drivers of Costs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Data Collection, Validation, and Consultation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Benchmarking . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Forecasting Future Efficient Cost Levels . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Wholesale Enhancement Expenditure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Northern Ireland . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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The authors would like to thank Julie Skovgaard Hansen and Emil Heesche (DCCA) and Laura Brien (CRU) for helpful comments on respective sections; Subal Kumbhakar, and two anonymous referees, for general comments and suggestions; Pierpaolo Perna for support on the approach in Italy; Oxera’s efficiency team (including Charles Blake, Simona Castellini, Srini Parthasarathy and Hannes Seidel), particularly on all the work undertaken as part of PR19, many insights from which we have incorporated in the relevant sections; and Patricia Taylor for supporting us with all the research. Any views expressed in this chapter are solely those of the authors and not of Oxera. A. Horncastle () · J. Duffy · C. X. Ng · P. Krupa Oxera Consulting, Oxford, UK e-mail: [email protected]; [email protected]; [email protected]; [email protected] © Springer Nature Singapore Pte Ltd. 2022 S. C. Ray et al. (eds.), Handbook of Production Economics, https://doi.org/10.1007/978-981-10-3455-8_42

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Historical and Industry Context . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Efficiency Benchmarking in Northern Ireland . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Future Potential Changes in Approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Scotland . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Historical and Industry Context . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Efficiency Benchmarking in Scotland . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A Change in Direction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Ireland . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Historical and Industry Context . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Efficiency Benchmarking in Ireland . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Denmark . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Historical and Industry Context . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Recent Regulatory Framework Changes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Efficiency Benchmarking in Denmark . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Italy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Historical and Industry Context . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Cost Benchmarking . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Areas for Further Development . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Input Definition: Modelled Expenditure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Input Definition: Accounting for the Investment Cycle . . . . . . . . . . . . . . . . . . . . . . . . . . . Output Definition: Multiple Outputs and Cost–Service Trade-Offs . . . . . . . . . . . . . . . . . . Benchmarking: Input Requirement Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Benchmarking: Functional Form . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Forecasting Efficient Costs: Identifying “Efficient” Cost Levels, While Accounting for Error and Heterogeneity, and Alternative Estimation Approaches . . . . . . Forecasting Efficient Costs: The Consistency of Catch-Up, Frontier Shift and Input Price Inflation Assumptions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . International Comparisons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Concluding Comments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Cross-References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Abstract

This chapter reviews the use of cost (and output) benchmarking within the economic regulation of the water sector in the EU. Unlike other utility sectors, such as energy, economic regulation and benchmarking is not yet widespread across the water sector. However, where cost benchmarking is used, we review the approaches taken by water regulators, setting out the key steps in the process and the issues that have been raised with the approaches taken. Given the longer history of such benchmarking in England and Wales and the fact that its framework has been used by other water regulators, we initially focus on England and Wales before covering some other examples of cost benchmarking across Europe. We finish by considering potential areas for future development. Keywords

Regulation · Efficiency · Stochastic frontier models · Data envelopment analysis · Production · Cost

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Introduction The water sector is a natural monopoly. In the absence of competition, water companies may incur higher than efficient costs or provide a poorer quality of service. To protect customers, these companies can be regulated with regulators assessing what cost reductions and/or service improvements are possible by comparing the relative performance of companies. This process is called benchmarking. In this chapter, we discuss the approaches that economic regulators have used to benchmark water companies across Europe. The chapter is organized as follows. In section “Why Benchmarking Is Important in the Water Sector?,” we set out the importance of benchmarking in the water sector. In particular, the water sector provides an essential service that has important impacts on both public health and the environment, yet water is generally provided by natural monopolies, which, left unchecked, may not produce optimal outcomes. As such, economic regulation is often applied to the sector and, within such regulation, benchmarking is usually applied. In section “Benchmarking Techniques in Regulation: An Introduction,” we provide a brief overview of the different benchmarking techniques used by economic regulators. In the next seven sections, we examine the benchmarking approach undertaken by those European regulators that implement benchmarking using econometric modelling, such as ordinary least squares (OLS), random effects (RE) or stochastic frontier analysis (SFA), or data envelopment analysis (DEA). We first examine Ofwat’s, the regulator in England and Wales, approach from privatization in 1989 up to PR09, the price review in 2009, in section “England and Wales: Cost Benchmarking Prior to PR14,” before moving on to examine its approach in PR14 and PR19 in section “England and Wales: Cost Benchmarking from PR14 Onwards.” In all price control reviews, Ofwat has used econometric modelling (either OLS or RE) of costs to benchmark the efficiency of the companies it regulates. This review over a number of price controls is of interest as a number of analytical and practical issues have been discussed and examined over time, including in-depth reviews during appeals of the regulator’s final determinations. As a result, Ofwat’s approach has evolved over time, although it has not used either SFA or DEA as its primary analytical tool in any price control review. Quite closely related are the approaches of the regulators in Northern Ireland, Scotland, and Ireland, which we cover more briefly in sections “Northern Ireland,” “Scotland,” and “Ireland,” respectively. Although we note that WICS, the regulator in Scotland, has moved away from econometric benchmarking approaches entirely in recent years. In section “Denmark,” we set out the approach of the DCCA, the competition authority in Denmark with the remit to regulate water companies, which is based on SFA and DEA. The setting for these models is a cross-sectional data set. Given the relatively large number of varied water companies that are benchmarked, the regulator pays particular attention to identifying and dealing with outliers. Finally, in section “Italy,” we review ARERA’s, the regulator in Italy, use of panel SFA models for setting efficient cost allowances for the period 2020–2023.

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We conclude, in section “Areas for Further Development,” by drawing together some of the key themes across these jurisdictions and examining potential areas for future development for those jurisdictions that have already implemented cost benchmarking and those that are considering introducing such benchmarking going forward.

Why Benchmarking Is Important in the Water Sector? As stated above, the water sector, in particular its network of pipes, is a natural monopoly providing an essential service for customers. In areas of the value chain outside of these network services, it has been possible to introduce competition. In particular, some retail services have been separated from wholesale services and opened up to competition in Scotland, England, and Wales. In England, a subset of the activities required to connect new developments to the water network is also open to competition, although the number of market participants varies regionally (see Ofwat [89]). In England, the Department for Environment Food and Rural Affairs [33] set out the UK Government’s strategic policy statement for the water regulator, Ofwat, to promote further competitive markets in water resources and bioresources. However, these contestable parts of the value chain represent only a small proportion of costs in the sector and in other European countries even these small parts of the value chain have not been opened up to competition. As such, monopoly provision of water services remains, by far, the main form of supply across Europe. In contrast to the outcomes from competitive markets, monopolies may not produce outputs at a price or service level that is optimal for consumers. Prices can be too high and service performance too low. In addition to the issues that would typically arise from a natural monopoly, the water sector provides an essential service and impacts the lives of citizens in a number of ways. The OECD [111] sets out the additional challenges faced by the water and wastewater sector as: • The responsibility of the sector to balance a range of economic, social, and environmental interests in its activities • The important externalities generated by the sector, in particular with regard to public health, the economy, and the environment • The generally fragmented nature of actors in the sector, both horizontally and vertically As such, some form of government control or economic regulation is often applied in the sector. Such control can be in the form of legislation, direct public ownership, or both. In either case, governments or local authorities directly control outcomes and prices. While each member state has its own national legislation governing the water sector, there are also European wide directives. A key directive for members of

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the European Union is the EU Water Framework Directive (WFD), see European Parliament and the Council [41]. This sets out a number of important objectives for the sector, including environmental objectives as well as the principle of cost recovery for water services in Article 9 and is applicable regardless of ownership [41]. According to the European Commission [40], “a number of Member States have upgraded their water pricing” but progress is still needed particularly as increased investments are essential to meet the objectives of the WFD. In some countries, prices do not cover all costs and general government funds are used to subsidize services (e.g., Ireland and Greece). It is not necessary for water networks to be publicly owned for government or local authorities to exert control. Indeed, given that water networks require significant investment to renew and maintain infrastructure (parts of which are old and, in some cases, have been historically under-maintained) and comply with environmental and other obligations, it is often desirable to attract investment from private capital in some form. The participation of the private sector in the provision of water services may also assist in cost reflectivity. Where private participation is involved, different approaches can be taken to try and deliver better outcomes. Concessionary contracts can be tendered – that is, competition for the market – delegating management responsibility and (potentially) financing, while maintaining public ownership and control over outcomes and prices. Alternatively, large discrete capital schemes, such as Thames Tideway in the UK, can be tendered out (see Ofwat [79]). That is, competition for the market can be introduced. The competitive tendering process aims to achieve efficient delivery of services as companies compete with each other for the opportunity to serve the market over a specified period. As such, the objective is to secure bidders offering a superior combination of outcomes and prices than other bidders. An alternative approach is to have private ownership, but to impose controls either on returns to private capital only (i.e., “cost-plus” or rate of return regulation) or on prices (including costs and returns) and outcomes (i.e., price or revenue caps). In this chapter, we consider regulatory models which impose controls on prices and outcomes, as a more relevant context for the application of cost benchmarking. It should be noted that private ownership is not a prerequisite for such economic regulation – indeed, many publicly owned companies are also subject to controls on prices or outcomes – but economic regulation is considered a necessary condition for the viability of private ownership, in the absence of a prescriptive concession agreement. In addition to environmental, safety, and water quality regulation, the aim of this type of economic regulation is to achieve the optimal levels of price and outcomes for customers, the environment and stakeholders as a whole. The “optimal” level might, for example, reflect the priorities of society such as affordable billing for customers in financial difficulty. Figure 1 sets out when economic regulation (and implicitly benchmarking) may be necessary in the context of private ownership, from the perspective of ensuring that prices are fair and that service quality does not suffer. However, given the critical importance of the water sector, even where competitive forces in the market

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Can a competitive market be introduced, and will it deliver the best outcomes for the sector?

Yes

Then market forces may remove the need for regulation

No

Can competition for the market be introduced, and will it deliver the best outcomes for the sector?

Yes

Then market forces may remove the need for regulation

No Services can be delivered by a monopoly, subject to economic regulation.

Then

Regulation is essential to ensure prices are fair

Fig. 1 Is direct economic regulation necessary in the context of private ownership?

or for the market can be established and work well, government intervention to control outcomes may still be necessary. Independent economic regulators aim to achieve efficient delivery through a series of incentives and cost and output benchmarking. Often this takes the form of RPI-X regulation, whereby prices are allowed to increase in line with general inflation less some factor reflecting, among other things, expected efficiency improvements. Regulators often set prices for a fixed period of time, incentivizing companies to outperform their allowance by improving their efficiency, while still delivering the required outcomes. In order to determine these allowed prices or revenues, regulators must determine the efficient cost level required to deliver a number of outcomes over the period. The outcomes the firm is expected to achieve can be determined by the government, environmental regulators, consumers, and/or other stakeholders. Although it is less straightforward to represent the views of consumers, consumer priorities can be determined through willingness to pay surveys and/or representation by a representative body. To inform all of these interactions, economic regulators often undertake output or service performance benchmarking. This generally takes the form of simple comparisons of metrics or ratios, such as the amount of leakage per km length of mains, and the identification of some best practice benchmark (for other approaches, including that of ARERA, see the discussion below). The approach taken to cost benchmarking is the most likely to involve the application of theories and techniques from production economics. In order to determine

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an efficient cost level, economic regulators often undertake cost benchmarking using approaches such as econometric modelling or DEA. Here, regulators compare costs between service providers, taking into account the outputs to be delivered, such as water delivered or number of properties served, in order to establish, for each company, what the efficient cost level should be. This chapter reviews the use of cost (and output) benchmarking within the economic regulation of the water sector. While the alternative models discussed – direct government control, control through legislation, competition for the market or competition in the market – all benefit from cost and output benchmarking, it is most relevant for independent sector regulators to set prices. As such, applications of benchmarking are most advanced where an independent sector regulator has been established, and this chapter draws from such jurisdictions. Which approach to control and form of benchmarking is most appropriate depends on the structure of the sector in the particular country, which is often fragmented (as noted above). Of the 27 EU member states, many are unregulated, either based on local municipalities with thousands of local authority operators (e.g., Germany) or run on a concessions basis whereby private participation is secured through a tendering process (e.g., France). A relatively small number of regulators have implemented formal cost (or output) benchmarking using either an econometric approach or DEA. In the remaining sections of this chapter, we provide more details on the form of benchmarking undertaken in some of these. • England and Wales: A key focus for this chapter, Ofwat, the water regulator in England and Wales, has used formal cost and output benchmarking from the first price review in 1994 after the initial setting of price allowances following privatization in 1989 [67, 151]. • Northern Ireland: Largely follows the framework developed by Ofwat, taking into account the relevant characteristics of Northern Ireland Water, relative to companies operating in England and Wales. • Scotland: Historically adopted a similar benchmarking approach to Ofwat, in the context of the unique operating characteristics faced by Scottish Water relative to comparator companies in England and Wales. In recent years, WICS (the regulator of Scottish Water) has moved away from using cost benchmarking to determine allowed costs, with an alternative regulatory model to reduce information asymmetry. • Ireland: Largely follows the framework developed by Ofwat, taking into account the relevant characteristics of Irish Water, relative to companies operating in England and Wales. • Denmark: The Danish Competition and Consumer Authority (DCCA) currently undertakes cost benchmarking every 2 years, though this will move to every 4 years from 2022. DCCA uses both DEA and SFA to identify the potential for efficiency improvements. • Italy: In 2019, ARERA, the Italian water regulator, used a number of SFA models for setting efficient cost allowances for the period 2020–2023.

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Benchmarking Techniques in Regulation: An Introduction In this section, we provide an overview of how cost (and output) benchmarking methods may be applied in regulatory settings. The two main benchmarking techniques used to estimate the cost frontier are econometric approaches (including SFA) and DEA. A stylized SFA model can be written as: c = f (w, y) + v + u where the dependent variable c is costs, f (w, y) is a cost function with input prices w and outputs y, v is inefficiency, and u is noise. In practice, input requirement functions are also estimated, where cost drivers are used as explanatory variables instead of outputs and input prices. For a detailed discussion of SFA models and how they may be estimated, see Kumbhakar et al. [57]. Regulators have used various cost measures, including operating expenditure (OPEX), capital expenditure (CAPEX), and total expenditure (TOTEX). The cost drivers used by regulators include such factors as scale (such as number of properties served), the density of area served by the water company, the topography of the region, and the quality of the water abstracted or the required complexity of the treatment process. We refer the reader to sections “Denmark” and “Italy” for examples of SFA models used by economic regulators, and, while the regulators discussed in sections “England and Wales: Cost Benchmarking Prior to PR14,” “England and Wales: Cost Benchmarking from PR14 Onwards,” “Northern Ireland,” “Scotland,” and “Ireland” do not use SFA, the cost drivers discussed are also relevant. Once the SFA model has been estimated, it can be used to estimate the efficient cost levels for each company. However, in practice, each regulator uses the results from estimated efficiency models in different ways. For example, ARERA, the Italian water regulator, uses efficiency scores to place companies into clusters as an input to determine cost sharing rates (see section “Italy”). Regulators, such as Ofwat, that use econometric modelling (but not SFA) need to identify a benchmark in order to estimate efficient cost levels. This is achieved by making ad hoc assumptions such as using the upper quartile as the benchmark. For further details, see sections “England and Wales: Cost Benchmarking Prior to PR14” and “England and Wales: Cost Benchmarking from PR14 Onwards.” Cost frontiers may also be estimated using DEA. In a regulatory setting, a DEA model consists of specifying a set of inputs and outputs, with the model typically input oriented. For a detailed discussion of DEA, see Thanassoulis [144]. The inputs and outputs can be similar to those used in the econometric models discussed above. For example, the DCCA, the Danish water regulator, uses TOTEX as the input and composite “grid” variables formed by weighting assets as the outputs (see section “Denmark”). Once the DEA model has been estimated, it can be used to estimate the efficient cost levels for each company. The DCCA, in deriving a final efficiency estimate for

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each company, uses the most favorable estimated efficiency score for each company from its SFA and DEA modelling with the resultant cost reduction capped at 2% p.a. (see section “Denmark”).

England and Wales: Cost Benchmarking Prior to PR14 In this section, we review the cost benchmarking approaches used in the England and Wales water sector used in price controls up to PR09 (i.e., from privatization, in 1989, up to 2015). The regulation of this water sector is perhaps more developed/long standing compared to many other European countries. In addition, other regulators in Great Britain and Ireland have often used Ofwat’s approach to benchmark their regulated water company with those in England and Wales. As such, we set out a relatively detailed history of the approach used in England Wales, and focus on differences from this in the sections that follow, particularly for regulatory regimes in Great Britain and Ireland.

The Structure of the Water Sector and Regulation in England and Wales Water services are provided by privately owned companies in England. Glas Cymru, the owner of the majority provider of water services in Wales (Dwr Cymru or Welsh Water), is a company limited by guarantee, with no shareholders, and, as such any financial surpluses are retained for the benefit of its customers. Most water and sewerage companies are regional monopolies. Competition was introduced for the water supply of large non-domestic customers (above 50 Ml/day) in England in 2003 (see Ofwat and the Department for Food and Rural Affairs [109]) although domestic customers and small non-domestic customers could still not choose or switch their supplier. Over the time, the size threshold for non-domestic customers able to choose or switch their supply has fallen, with the entire market for non-domestic customers opened in 2017 [84]. As such, the industry is regulated by an independent economic regulator, Ofwat, to ensure better outcomes for customers. Of key relevance to production economics – and cost and quality of service benchmarking – is that one of Ofwat’s duties is to “promote economy and efficiency by water companies” (see Water Industry Act [152] as amended, Sect. 2). As such, as part of its price control reviews, Ofwat has always undertaken cost and outcomes benchmarking in some form. Ofwat’s approach to quality of service benchmarking has taken the form of comparative metrics or key performance indicators (KPIs), separate from cost benchmarking. Therefore, in this section, we focus on Ofwat’s approach to cost benchmarking.

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Ofwat’s Approach Prior to PR14 We start with a review of past price control reviews in this section and then examine the Ofwat’s approach to cost benchmarking in PR14 and PR19, the price control reviews for the periods 2015–2019 and 2020–2024, in section “England and Wales: Cost Benchmarking from PR14 Onwards.”

Modelled Costs Ofwat considered expenditure within three categories: • Operating expenditure – regular day-to-day expenditure required to deliver water and wastewater services (including labor, energy, and chemicals) • Capital maintenance – expenditure incurred to maintain the long-term capability of the existing asset base, such as restoring a damaged water main • Capital enhancement – expenditure required to make additions to the asset and enhance service, such as construction of a new water main or modification to a treatment works to improve water quality Up to the price control review in 2009, PR09, Ofwat modelled operating expenditure and capital expenditure separately (for example, see Ofwat [75]). Operating expenditure was modelled using econometric models, capital maintenance was assessed using both econometric modelling and unit cost comparisons; and capital enhancement was assessed using unit cost comparisons only. Unit cost comparisons for capital maintenance and enhancement were set out in the “cost base report” [25, 72, 74]. In the research commissioned by Ofwat for the 1994 price control review, PR94, Stewart ([136], p. 1) stated that “in principle economic efficiency requires companies to minimise costs in total and hence this variable [total expenditure] should be the focus of attention. However, there are problems with the definition of total costs in the water industry. Current cost operating profit reflects the return actually earned, rather than the cost of capital [ . . . ] profits in the water industry reflect to a considerable extent both historical factors and future investment needs [ . . . ] An alternative approach would attempt to derive capital costs by applying the cost of capital to (current cost, depreciated) asset values. However, there are difficulties of both principle and practice in doing this” [primarily measurement and consistency issues]. Given the now much longer time series of data available, the cost definition has been reexamined by Ofwat in more recent price control reviews (see section “England and Wales: Cost Benchmarking from PR14 Onwards”).

Model Specifications In order to benchmark water companies’ costs on a like-for-like basis, Ofwat developed a series of input requirement functions, whereby costs are explained using a number of cost drivers. For further discussion on input requirement functions, see

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Diewert [34], Kumbhakar and Heshmati [54], and Kumbhakar et al. ([57], Sect. 2.7). This was applied instead of a cost function approach, whereby costs are a function of outputs and input prices, see Kumbhakar et al. ([57], Sect. 4.2). As set out in Ofwat [68], the functional forms of its models were either linear or log-log, and Ofwat modelled costs at a functional level and an overall service level (for water services). For water services, prior to subsequent mergers, Ofwat had 32 observations (10 WASCs and 22 WOCs) and modelled expenditure on overall water services, business activities, resources and treatment, and distribution. Ofwat subsequently also introduced separate models of power costs, stating that “many companies have made savings on power expenditure and it was considered that it might be useful to look at this cost separately. Another reason is that the allocation of power expenditure between distribution and resources and treatment can be problematic” – see Ofwat ([69], Sect. 2.1.3). For wastewater, Ofwat had fewer observations (only ten WASCs) and thus used a combination of econometric models (at a works or area level) and unit cost models. Ofwat used the following models for wastewater: business activities (unit cost model), small treatment works (unit cost model), large treatment works (econometric model at works level), sewerage area (econometric model at an area level), and sludge treatment and disposal. Ofwat estimated the models using cross-sectional data, focusing on econometric (OLS) modelling to determine efficient expenditure. However, Ofwat also undertook or commissioned studies using DEA and SFA during various price control reviews. In PR94, for example, Ofwat used DEA results (see section “Forecasting Efficient Costs: Identifying “Efficient” Cost Levels, While Accounting for Error and Heterogeneity, and Alternative Estimation Approaches”) as a form of crosscheck on its econometric results, improving a company’s outcome if the DEA result was significantly better than the OLS result – see the Monopolies and Mergers Commission (MMC [59], p. 415).

Establishing Efficient Costs In Stewart [137], the accuracy of the estimated residuals was also examined (using both SFA and estimating the confidence intervals around the OLS residuals), identifying some companies whose costs were significantly above, at or below average. Stewart ([136], p. 26) stated, “as a final word of caution, it should be reiterated that the inefficiency of a company is an inherently ‘residual’ concept. The accuracy of the measures derived depends on the extent to which we are able to control for all relevant cost drivers.” From PR94 until PR09, to establish future efficient costs, Ofwat first identified a benchmark using a number of criteria (which themselves changed slightly over time). For example, the criteria used in PR09 were summarized in a subsequent report by the Competition Commission ([28], Appendix F, para 6) and included: Ofwat must have no concern about the company’s data or the independence of the company’s data; the company must have no unusual exogenous characteristics which significantly reduce its costs; and the company’s turnover must represent a

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reasonable proportion of the industry. A “reasonable proportion” was defined as the size of the smallest WASC, around 2.5–3% of water services turnover. Having established the cost efficiency benchmark, Ofwat categorized companies into various efficiency bands. For example, Ofwat [67] explains that, in PR94, three bands were used – “more efficient,” “average,” and “less efficient.” This seems to have been based on the analysis by Stewart [137–138], discussed above. In later reviews, the differentiation was extended and bands A–E were used, starting with Band A representing companies estimated to be 0–5% from the benchmark, and these were subsequently further subdivided into upper and lower bands (see Competition Commission [27], Appendix K). For setting cost allowances, Ofwat then took the mid-point of each half band as representative of all companies in that half band. Ofwat then applied a percentage catch-up assumption rather than requiring companies to fully catch-up to its estimated benchmark. That is, for the next price control period, companies were allowed their current cost level less a percentage reduction based on catching up by a certain percentage to the benchmark. In PR94, for example, Ofwat [67] set this at around half of the OPEX efficiency gap, while in PR09 Ofwat set this at 60% (see Ofwat ([74], p. 107) and Competition Commission [27]). In addition, a “glide path” was used, whereby companies had to achieve this catch up over a number of years (generally 5 years for OPEX, while for CAPEX, the timeframe altered between price control reviews). Ofwat’s reasoning, as reported in Competition Commission ([26], Appendix 4.1, para. 19), was that initially that “this cautious approach reflects the possibility of errors in the DGWS’s work, the difficulty of identifying the efficiency frontier, and the need to offer an incentive to outperform” [emphasis added]. Subsequently, in PR04, Ofwat [70] emphasized the incentive properties of the approach and, in particular, its “carrot and stick,” whereby the catch-up target was a “stick” for companies to improve efficiency, while the remaining gap provided a “carrot,” as any efficiency gains achieved over and above this were kept by the company for the duration of the price control period. Separately, modelling or data errors were explicitly accounted for through an explicit adjustment to the estimated residuals – a 10% adjustment to the residuals in water and a 20% adjustment in sewerage, see Ofwat ([70], p. 154). In addition to catching up to the estimated benchmark, Ofwat would also set companies targets to reduce their costs further through technological progress. As with the catch-up element, this was based on a proportion of what Ofwat considered could be achievable through frontier shift. The frontier shift assumption itself was determined through an exercise separate from the relative efficiency assessment, examining productivity in the UK economy. For example, see Ofwat ([67], p. 31). As well as periodically setting future efficient cost allowances at price control reviews, Ofwat also published annual reports on companies’ cost efficiency. This provided reputational efficiency incentives on companies to improve their efficiency. Having worked with companies and discussed this issue, the authors are aware that such reporting of efficiency rankings did indeed have strong incentive properties.

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The Use of Ofwat’s Approach Elsewhere and a Change in Approach Following PR09, Ofwat would subsequently change its overall regulatory approach significantly, moving to a TOTEX regime in line with regulatory developments in other network regulators in Great Britain, such as the energy regulator, Ofgem, see Ofgem [63]. The implications of this change in approach to cost benchmarking is set out in section “England and Wales: Cost Benchmarking from PR14 Onwards.” However, other water regulators in Great Britain and Ireland have used or still use an approach similar to Ofwat’s PR09 approach (see sections “England and Wales: Cost Benchmarking from PR14 Onwards,” “Northern Ireland,” and “Scotland”), although some have also indicated a need to change (for example, see UR [147], p. 22), or no longer use cost benchmarking to set allowed expenditure (see WICS ([156], p. 8).

England and Wales: Cost Benchmarking from PR14 Onwards In PR14, Ofwat amended its approach to quality of service. Introducing socalled “performance commitments” (or targets for quality of service performance) and associated outcome delivery incentives. These were developed following extensive customer engagement and willingness to pay surveys and evidence by companies. At PR19, this approach was broadened to include a number of “common performance commitments” and many more company-specific performance commitments for each company. For details, see Ofwat [90]. Any comparative analysis is based on simple metrics, as such, we do not cover this benchmarking further. In this section, we review the cost benchmarking approach used by Ofwat during PR14 and PR19, the price control review for the periods 2015–2019 [77] and 2020–2024 [104], respectively. From the perspective of benchmarking, these price controls can be characterized by Ofwat moving to a TOTEX approach, whereby the regulator seeks to remove “any undesirable incentives for companies to seek capital expenditure-intensive solutions where there may be better alternatives,” Ofwat ([77], p. 5). This is achieved by modelling operating expenditure and capital expenditure together, as well as changing other parts of the regulatory regime outside the scope of this chapter.

Ofwat’s Approach in PR14: A Change in Direction In recent price reviews, starting with PR14, Ofwat changed its cost assessment framework, which, until that point, had been relatively unchanged since 1994. Ofwat [77, 78] and CEPA [13] set out the key changes, including: • Modelling total expenditure (TOTEX), given a perception that, under the previous regulatory approach of modelling costs separately, companies were incen-

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tivized towards a capital expenditure bias – see Ofwat [76] and Ofwat ([77], Sect. 5.2) for a discussion on this possible bias and Ofwat’s response. In particular, Ofwat considered that the structure of incentives within its regulatory framework were such that underperformance (i.e., spending more than expected) was less costly for solutions weighted towards capital expenditure than for those with a greater proportion of operating expenditure. See section “Input Definition: Modelled Expenditure” for a discussion on TOTEX modelling. The introduction of more flexible functional forms, namely partial translog models (see sections Functional form, economies of Scale, Size, and Scope and “Benchmarking: Functional Form”). Use of panel data (i.e., using data across the various regulated companies and over time). As shown in Kumbhakar and Horncastle [55], moving to a panel data approach for modelling costs in the England and Wales water sector can result in “a considerable increase in precision” of the modelling compared to Ofwat’s previous cross-sectional modelling approach. Use of both pooled OLS and RE estimation approaches (see sections Model development and model selection, Wholesale Enhancement expenditure, and Forecasting efficient costs: identifying ‘efficient’ cost levels, while accounting for error and heterogeneity, and alternative estimation approaches). In PR14, modelling water services at the water service level and, in PR19, modelling across the value chain (a slightly different activity split compared to the previously used functions) and at the aggregate level (see section “Definition of the Inputs”). Using the upper quartile (UQ) as the benchmark (see sections “Forecasting Future Efficient Cost Levels,” 4.8, and “Forecasting Efficient Costs: Identifying “Efficient” Cost Levels, While Accounting for Error and Heterogeneity, and Alternative Estimation Approaches”). Using forecast of cost drivers (from companies’ or Ofwat’s own assumptions) to forecast efficient costs, rather than setting percentage cost reductions from companies’ own current cost levels (see section “Forecasting Future Efficient Cost Levels”). No longer publishing annual efficiency reports.

However, Ofwat’s cost assessment modelling in PR14 was criticized by Bristol Water and the Competition and Markets Authority, CMA, in Bristol Water’s appeal of PR14 – (see Bristol Water [10], Sect. 11) and CMA [23]. Areas of criticism included: • Level of aggregation – In PR14, Ofwat did not undertake cost modelling below the aggregate water service level (though Ofwat did separately models network and treatment costs for wastewater services). The CMA consider that it was “ambitious to seek to model the entire wholesale water business through this type of high-level econometric model, which may fail to take proper account of the wide range of factors that affect companies’ expenditure requirements.” • Investment timing – The CMA noted that companies’ investment requirements vary over time and thus differences between companies in total cash expenditure

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may be reflective of differences in their investment requirements and not efficiency. TOTEX models – Ofwat’s models included enhancement expenditure. The CMA considered that there were likely to be substantial differences between water companies, and over time, in enhancement expenditure requirements, which did not seem to be sufficiently taken account of in Ofwat’s models. The CMA considered that the estimated coefficients were counterintuitive in some cases and some specified relationships did not make sense (e.g., taking logarithms of variables expressed as proportions). The CMA considered that some models used a relatively large number of explanatory variables compared to the sample size (e.g., one model had 27 explanatory variables and a sample size of 90 observations). Translog models – Given the relatively small sample size, the CMA considered that the translog structure seemed “overly ambitious” and had, in practice, “compromised the results” as, for example, one of Ofwat’s models implied a form of diseconomies of scale which the CMA found counterintuitive. Upper quartile benchmark – The CMA report considered that using an upper quartile benchmark could be overly demanding and instead used an average benchmark. This, it stated, was a judgment in light of the issues it had identified both in its review of Ofwat’s econometric models and from its development of alternative models. Future cost prediction – Ofwat’s econometric models included a time trend. Its future expenditure allowances took account of these time trends, implying an annual change in costs of around RPI + 0.4% p.a. The CMA considered this “overly generous.” Instead, the CMA applied a cost trend of RPI–1% p.a., to capture the impact of input price inflation and productivity improvements.

This backdrop formed the basis for Ofwat’s development of its approach for PR19, which we cover in the next section. Ofwat [85] stated “our approach has taken into account learnings from PR14, industry feedback, and the Competition and Markets Authority (CMA) reference on Bristol Water’s PR14 price controls.”

Ofwat’s Approach to Cost Benchmarking in PR19 Ofwat’s cost modelling in PR19 can be split into the following key steps (Fig. 2): 1. Definition the inputs, or cost base, to be modelled, i.e.: (a) The activities across the value chain to be modelled (b) The cost elements to be modelled 2. Identifying the outputs and other drivers of cost 3. Data collection, validation, and consultation 4. Benchmarking using econometric models 5. Forecasting efficient cost levels for each company for the next control period These steps are examined below in turn.

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Definition of the inputs

Identifying the outputs and other drivers of costs

Data collection, validation and consultation

Costs, outputs, input prices, other cost drivers including environmental factors

Benchmarking

Specification of the model(s), model development/selection (including checking against economic and operational insight), model estimation

Forecasting efficient cost levels Fig. 2 Ofwat’s key benchmarking steps in PR19

Definition of the Inputs In its initial assessment of plans (IAP), Ofwat [91–94], Ofwat benchmarked water companies through the use of econometric models of base expenditure (or “BOTEX”), which consists of operating expenditure and capital maintenance expenditure. In its subsequent slow-track draft (i.e., companies who submitted business plans that Ofwat deemed to be cost inefficient and required a further challenge) and final determinations, Ofwat [95, 96], Ofwat modelled BOTEX plus (consisting of BOTEX as defined above plus some elements of enhancement expenditure, primarily relating to growth). For water services, these enhancement costs were new developments, new connections, and addressing low pressure; for wastewater services, these enhancement costs were new developments and growth, growth at sewage treatment works, and reduced flooding risk for properties. For water, the enhancement expenditure in each year was added to BOTEX, while for wastewater, the average enhancement expenditure over the sample period (2011/2012–2018/2019) was used.

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Aside from growth, there are many other types of enhancement expenditure. At the PR19 final determinations, there were 40 separate enhancement categories across both water and wastewater. Examples include meeting legislatively mandated environmental obligations, improving the quality of water supply and building a more resilient water supply. These remaining elements of enhancement expenditure in the slow-track draft and final determinations were considered separately from BOTEX plus, as Ofwat ([93], p. 7) considered that “enhancement costs tend to be non-routine and company specific.” The reasoning for moving to modelling BOTEX plus in the slow-track draft determinations was set out in Ofwat [95] and can be summarized as follows: • Ofwat considered the expenditure to be “routine” – companies have incurred it in the past and will incur it in the future. • Growth-related enhancement can be explained with similar cost drivers to operational and capital maintenance (e.g., company scale). • Ofwat did not expect to see a significant step change in drivers of growth enhancement expenditure during PR19. Some companies criticized Ofwat’s inclusion of growth expenditure along with base expenditure, but Ofwat maintained its approach in the final determinations (see section “Input Definition: Modelled Expenditure”). In response to DEFRA [33] and the resultant need to promote upstream markets for water resources and bioresources, Ofwat [81] set out its methodological framework for PR19, including setting separate price controls for wholesale water, water resources, wholesale wastewater and bioresources, and retail. This included the need to model at a more disaggregated level than that undertaken PR14. This change in approach was also consistent with issues raised in CMA [23] around the level of aggregation of the modelling in PR14. Though, as noted in Oxera ([114], p. 1) about Ofwat’s previous functional modelling approach, “Ofwat’s modelling is undertaken using functional models for different cost areas. This requires costs are separable across the different water activities and that cost allocation across the water companies is consistent.” Disaggregation also makes it difficult to capture economies of scale and of scope, although Saal and Parker [127] did not find any evidence of economies of scope between water and wastewater. Similarly, more recently, Saal ([125], p. 3) stated that “such an approach assumes that complex multiple output systems can be fully separated.” At the most granular level, Ofwat subdivided wholesale water services into four separate parts of the value chain – water resources (abstraction), raw water distribution (delivery to a water treatment works), water treatment (treatment of raw water), and treated water distribution (delivery of treated water to consumers). Wastewater was subdivided into five separate parts of the value chain – sewage collection, sewage treatment, sludge transport, sludge treatment, and sludge disposal. However, Ofwat ([94], p. 11) acknowledge that there may be “interaction[s] between services of the value chain” and “inherent choices and trade-offs across the

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Water resources

Raw water distribution Water treatment

Water resources plus

Aggregate

Treated water distribution

Fig. 3 Value chain in wholesale water. (Source: Ofwat [94]. Checked boxes indicate that Ofwat modelled the part of the value chain)

value chain.” Therefore, Ofwat aggregated water resources, raw water distribution, and water treatment into “water resources plus.” Ofwat “triangulated,” or averaged, across the outcomes from: • the sum of allowances from “bottom up” models at the water resources plus and treated water distribution level; with • models at the aggregate water service level. This is set out in Fig. 3. Ofwat ([94], p. 11) stated that it found that these models had more reasonable ranges of estimated company efficiencies and coefficients were more aligned with economic intuition than when modelling at more disaggregate levels, such as separate water resources models. At its industry consultation on models to be used at the price review, Ofwat ([85], p. 13), it also considered that models at this higher level of aggregation were less susceptible to misallocations of costs across services. Ofwat’s approach to wholesale wastewater is similar to wholesale water. Ofwat aggregated sludge transport, sludge treatment, and sludge disposal into a “bioresources,” adding sewage treatment to create “bioresources plus.” This is set out below (Fig. 4). Unlike water services, no aggregate wastewater models were developed (see Ofwat [94], p. 25). This modelling decision assumes separability (see Oxera [114]). Another challenge is that population density has different effects on expenditure in different parts of the value chain, such that costs are interrelated, as set out by South West Water ([134], p. 9): [population density] affects the wastewater network differently to the water network because the sewer network is expensive to construct, so sparse networks are generally built to serve small catchment areas. Densely populated urbanised areas have increased network costs associated with operating in congested areas but have reduced treatment and sludge costs as a result of having larger treatment works serving densely populated areas. Sparsely populated rural areas have increased treatments costs due to requiring many small works scattered across a sparsely populated region

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Sewage collection

Sewage treatment

Aggregate Bio-resources plus

Sludge transport

Sludge treatment

Bio-resources

Sludge disposal

Fig. 4 Value chain in wholesale wastewater. (Source: Ofwat [94]. Checked boxes indicate that Ofwat modelled the part of the value chain)

By aggregating the results of these models before establishing the benchmark, Ofwat reduced the risk of setting unachievable benchmarks. However, the lack of an aggregate wastewater model removes the ability to cross-check such disaggregated modelling.

Identifying the Outputs and Other Drivers of Costs Having defined the inputs or costs to be modelled, the next key step is to decide on what else should be included in the model. As with previous price controls, Ofwat did not define formal cost models. Instead, Ofwat focused on choosing an appropriate set of “cost drivers.” Input prices were not included in Ofwat’s models, although Ofwat did examine the impact of regional wages. As such, Ofwat’s cost benchmarking models are not cost functions in the formal sense. These model formulations are sometimes referred to as input requirement functions (as mentioned above). For wholesale water and wastewater, Ofwat [94] found that the four key categories of cost drivers to be consistently important in explaining variations in costs across companies were: • Scale variables, to measure the size of the network and/or level of output • Complexity variables, to capture the complexity of required treatment or the complexity of the network • Topography variables, to capture energy requirements for transporting or pumping water or wastewater • Density variables, to capture economies of scale at the treatment level and costs resulting from operating in highly dense (or sparse) areas It appears that the consultation process (see section “Data Collection, Validation, and Consultation”) was successful in ensuring that some industry insight was

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embedded within the modelling framework, as these categories are similar to those suggested by some of the water companies as part of the cost assessment consultation – for example, see South West Water ([134], Sect. 3).

Data Collection, Validation, and Consultation The critical next steps in any regulatory cost assessment exercise are: collating relevant data (including ensuring all the key cost drivers are collated), ensuring the data is consistent across companies, and ensuring that the models are aligned with industry and economic insight. During 2016 and 2017, Ofwat ran a series of cost assessment working groups with the industry to develop the data and cost assessment tools for PR19. In July 2017, companies submitted data on costs and cost drivers for wholesale water and wastewater services over the 6-year period, from 2011–2012 to 2016–2017. The data was subject to extensive quality assurance and was shared with the industry. In March 2018, Ofwat issued a cost assessment consultation, Ofwat [85]. Thirteen water companies and Ofwat submitted a number of cost models across the value chain. In total, 382 models were submitted. Each company then commented on the models that had been submitted. In February 2019, Ofwat published its approach and decisions regarding econometric modelling for PR19, including its model specifications, in Ofwat [94].

Benchmarking The next step in Ofwat’s framework was to undertaking the benchmarking exercise itself, which first involved the development of a number of econometric models for each part of the value chain in order to compare costs across companies. In wholesale water services, Ofwat specified five econometric models (two for water resources plus; one for treated water distribution; and two for wholesale water). In wholesale wastewater, Ofwat specified eight econometric models (two each for sewage collection, sewage treatment, bioresources, and bioresources plus).

Model Development and Model Selection As set out in Ofwat [85], Ofwat’s approach to model development and assessment was as follows: • Engineering, operational, and economic understanding was used to specify an econometric model and form expectations about the relationship between cost and cost drivers in the model. • The resultant estimated coefficients were assessed as to whether they were of the right sign and plausible magnitude. • The estimated coefficients were examined for robustness, including whether they were stable and consistent across different specifications and statistically

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significant. On this issue, Ofwat stated that they “do not consider that the common thresholds of statistical significance (e.g., 95% significance) need to be strictly followed for our model selection. The size of the sample has a large effect on statistical significance. With a relatively small sample we are careful not to dismiss mechanistically variables that are not strictly statistically significant, so long as the significance is still reasonable and the estimation seems robust,” Ofwat ([85], p. 9). • Ofwat checked the consequences/risk of perverse incentives of including endogenous cost drivers. • Ofwat examined the statistical validity of the model. With regards to its estimation approach, for its initial assessment of plans, Ofwat ([94], p. 7) decided upon using a RE specification. Ofwat justified this choice as it considered it reflected the panel structure of the data, and statistical significance of the coefficients and Breusch-Pagan tests supported its use over OLS.

Estimated Wholesale BOTEX Plus Cost Models The resultant models typically included one scale driver and up to four other cost drivers. Saal and Nieswand [126] criticized this as being too restrictive. By using models with different scale drivers in different parts of the value chain, it could be argued that this may partly capture multiple output production process of water and waste water service provision. However, the differences are quite limited in practice and no quality of service measures were included in Ofwat’s core suite of models. As noted above, quality of service and outcomes more generally are regulated separately by Ofwat. This separation was criticized by some water companies and is examined in more detail in section “Input Definition: Accounting for the Investment Cycle.” In water, Ofwat separately modelled three parts of the value chain – two at the disaggregate level (water resources plus, and treated water distribution) and one aggregate (wholesale water). Ofwat [96], sets out the estimated models for water service at final determinations. These are generally of the form,   ln BOT EX plus it = a + b1 .scaleit + b2 .treatment complexity it   + b3 .topography it + b4 .f density it + uit (1) and are set out in Table 1 (we indicate the correspondence between the specific variable in the table and the general cost driver area in the formula with the relevant coefficient). In wastewater, Ofwat separately modelled four parts of the value chain, all at the disaggregate level (sewage collection, sewage treatment, bioresources, and bioresources plus). Ofwat [96] sets out the estimated models for wastewater service at final determinations. These are generally of the form,

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Table 1 Water BOTEX econometric models Model name

Dependent variable (log) Connected properties (log), b1 Lengths of main (log), b1 Water treated at works of complexity levels 3–6 (%), b2 Weighted average treatment complexity (log), b2 Number of booster pumping stations per lengths of main (log), b3 Weighted average density (log), b4 Squared term of log of weighted average density, b4 Constant term, a Overall R-squared Number of observations

WRP1 WRP2 Water resources + Raw water distribution + Water treatment 1.007*** 1.007***

TWD1 WW1 WW2 Treated water distribution Wholesale water total 1.034*** 1.020*** 1.049***

0.008***

0.005***

0.486***

0.568***

0.455***

0.231**

0.256***

−1.647*** −0.981**

−3.120*** −2.220*** −1.789***

0.103***

0.056 (0.120)

0.248***

0.156***

0.125***

−4.274**

−6.607***

5.686***

−2.725**

0.93 141

0.92 141

0.97 141

−1.106 (0.483) 0.98 141

0.98 141

Source: Ofwat ([96], p. 162) Note: The dependent variable is modelled base costs in 2017/2018 prices, using the CPIH adjustment. P values expressed in parentheses are based on clustered standard errors at the company level. *, **, and *** denote significance at 10%, 5%, and 1%, respectively

  ln BOT EX plus it = a + b1 .scaleit + b2 .size prof ile of works it + b3 .topography i + b4 .treatment complexity it

(2)

+ b5 .density it + uit and are set out below (we indicate the correspondence between the specific variable in the table and the general cost driver area in the formula with the relevant coefficient) (Table 2). Ofwat [96] sets out the approach taken to arrive at an overall cost prediction from all these models. • First, where alternative models were used, Ofwat took the average of the model predictions within each part of the value chain (e.g., WRP1 and WRP2, and WW1 and WW2) to arrive at a prediction for that part of the value chain (e.g., WRP

0.93 80

Overall R-squared Number of observations

0.88 80

−6.416***

0.178 (0.146)

0.606***

SWT2

0.88 80

−5.228***

0.004***

0.045***

0.779***

0.87 80

−3.988***

0.004***

−0.013**

0.773***

Sewage treatment

SWT1

−0.389 (0.648) 0.82 80

−0.295**

1.274*** 0.057**

Bioresources

BR1

0.79 80

0.994*

0.397*

1.265**

BR2

0.92 80

−4.753***

0.005***

0.038*

0.765***

0.92 80

−3.709***

0.005***

−0.011**

0.762***

BRP1 BRP2 Bioresources + Sewage treatment

Source: Ofwat ([96], p. 163) Note: The dependent variable is modelled base costs in 2017/2018 prices, using the CPIH adjustment. P values expressed in parentheses are based on clustered standard errors at the company level. *, **, and *** denote significance at 10%, 5%, and 1%, respectively

−8.124***

Sewage treatment works per number of properties, b4 /b5 Constant term, a

0.998**

0.317*

Sewage collection 0.839*** 0.896***

Dependent variable (log) Sewer length (log), b1 Load (log), b1 Sludge produced (log), b1 Load treated in size bands 1–3 (%), b2 Load treated in size band 6 (%), b2 Pumping capacity per sewer length (log), b3 Load with ammonia consent below 3 mg/l (%), b4 Number of properties per sewer length (log), b5 Weighted average density (log), b5

SWC2

SWC1

Model name

Table 2 Wastewater BOTEX econometric models

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overall, and WW overall, respectively). Since there was only one TWD model, there was no averaging undertaken for this part of the value chain. • Ofwat then constructed a “bottom-up” and “top-down” view of the company’s efficient costs. In water, the bottom-up view was arrived at by summing together water resources plus (WRP overall) and treated water distribution (TWD), while the top-down view was provided by WW overall. • To arrive at an overall cost prediction for water, an average of the “bottom-up” and “top-down” views is taken. Similarly, for wastewater, a cost prediction for each part of the value chain was arrived at by averaging its constituent models. For example, the SWC cost prediction was calculated by average the cost predictions from the two models “SWC1” and “SWC2.” Then, the bottom-up view was the sum of sewage collection (SWC), sewage treatment (SWT), and bioresources (BR), while the top-down view was the sum of sewage collection (SWC) and bioresources plus (BRP). These top-down and bottom-up views are then also averaged to arrive at the final overall cost prediction.

Estimated Retail Costs Models For retail services, Ofwat modelled total retail costs, bad debt, and other retail costs. The models are set out below. These are generally of the form, ln (total retail costs per household it ) = a + b1 .average bill sizeit + b2 .propensity to def ault it + b3 .net migrationi + b4 .dual service customers it + b5 .metered customers it + b5 .number of households it + uit (3)

and are set out below (we indicate the correspondence between the specific variable in the table and the general cost driver area in the formula with the relevant coefficient) (Tables 3 and 4). In order to arrive at an overall cost prediction from all these models, Ofwat [96] followed a similar approach as that set out above for wholesale services, placing 25% weight on the bottom-up models and 75% weight on the top-down retail cost models. A greater weight was placed on top-down models to reflect Ofwat’s view of relative model quality, with the wider range of efficiency scores in the bad debt models given as specific evidence, Ofwat ([96], p. 119).

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Table 3 Bottom-up retail cost models Model name

Dependent variable (log) Average bill size (log), b1 Proportion of households with default (%), b2 Proportion of households income deprived (%), b2 Total migration (% of population), b3 Proportion of dual service households (%), b4 Proportion of metered households (%), b5 Number of connected households (log), b1 Constant term, a Overall R-squared Number of observations

RDC1 RDC2 Bad debt and bad debt management costs per household 1.190*** 1.158*** 0.067***

ROC1

ROC2

Other retail costs per household

0.076*** 0.035**

−6.032*** 0.77 105

−5.680*** 0.78 105

0.002*

0.002**

0.007***

0.007***

2.400*** 0.13 105

−0.039 (0.394) 2.909*** 0.15 105

Source: Ofwat ([96], p. 164) Table 4 Top-down retail cost models Model name Dependent variable (log) Average bill size (log), b1 Proportion of households with default (%), b2 Proportion of households income deprived (%), b2 Total migration (% of population), b3 Proportion of metered customers (%), b5 Number of connected households (log), b1 Constant term, a Overall R-squared Number of observations

RTC1 RTC2 Total retail costs per household 0.458*** 0.526*** 0.024 0.030** (0.106)

RTC3 0.603***

0.059***

0.004 (0.321)

0.004 (0.206) −0.059*

0.037** 0.002 (0.436) −0.116*

−0.014 (0.980) 0.67 105

0.226 (0.653) 0.70 105

0.200 (0.564) 0.71 105

Source: Ofwat ([96], p. 164)

Functional Form, Economies of Scale, Size, and Scope In PR19, Ofwat simplified its models relative to those it used PR14, and no longer used semi-translog specifications. Generally, the models used a log–log functional form, though Ofwat included squared terms on density/sparsity related measured to

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pick up a U-shape impact for wholesale water services (costs are expected to be higher in both dense and sparse regions, relative to regions of average density) – see Table 1. On this issue, Ofwat ([93], p. 10) stated: While the translog has appealing properties in that estimated elasticities vary with company size, in practice we find individual company elasticities can have a counter-intuitive sign, that some translog terms were highly insignificant and (individually) unstable, and that the specification takes up degrees of freedom that could be dispensed with more relevant cost drivers. Instead, for PR19 we built our models ‘bottom up’ by considering the main cost drivers in each service. We include non-linear terms where their inclusion aligns with economic or engineering rationale but not for the purpose of fitting with a preconceived functional form.

For water services, Ofwat modelled the WASCs and WOCs together, the data set consisting of 17 companies, which vary significantly in size. The largest company, Thames Water, is more than 50 times the size of some of the smallest companies (on the basis of costs). For wastewater services, there are 11 WASCs, which are more similar in size with the exception of the recently created Hafren Dyfrdwy. As discussed above, in PR14, Ofwat used semi-translog specifications – see CEPA [13]. This model estimated varying economies of scale across the sector. However, this modelling was criticized in the subsequent appeal – Bristol Water ([11], p. 50) stated “it is important that the model is consistent with economic theory. Oxera has shown that this is not the case with respect to Ofwat’s translog model.” The CMA [23] similarly stated that “Ofwat’s refined base expenditure models implied a form of diseconomies of scale with respect to the size of a company’s customer base, which we found to be counter-intuitive.” For PR19, the models used were of a log-log functional form and economies of scale or constant returns to scale were estimated. For example, the value of the scale coefficient in Ofwat’s model for water distribution (TWD1) is 1.0, i.e., for a 10% increase in length of mains, costs increase by 10%. Similarly, the value on the scale coefficients in Ofwat’s sewage collection model (SC1) is 0.84, i.e., for a 10% increase in scale, costs increase by 8.4%. For retail, if a WASC provides both waste and water, then clearly there will only be one bill. Where a WOC operates in a WASC area, there may be two separate bills, or companies may agree to send a combined bill. As such, there can be an issue when comparing costs between companies. On this issue, Ofwat ([94], p. 32) states, “Dual service customers receive both water and wastewater services from the same company. Dual customers may generate more contact and enquiries relative to single service customers, which in turn drives customer service costs.” To address this issue, for other retail costs, Ofwat modelled costs per household, with dual customers per household as one of the drivers – i.e., the models is used to estimate the economies of scale in serving dual customers through use of proportion of dual customers and combined bills. The coefficient is positive with a value of 0.002. In contrast, for bad debt and total retail cost models are unit cost models. As such, an increase in the number of total customers, independent of whether they are dual or single, by 1% increases the predicted costs by 1%.

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Forecasting Future Efficient Cost Levels Ofwat used the above econometric models to arrive at a view of each company’s efficient costs for the next control period, AMP7 (2020/2021–2024/2025). This was achieved in four steps. First, Ofwat accounted for company-specific factors that were not accounted for in its modelling, as “statistical models are not perfect and cannot take into account all relevant factors that affect costs. There may be instances where an adjustment is required to correct these imperfections,” Ofwat ([81], p. 148). As such, Ofwat’s framework allowed companies to submit so-called “cost adjustment claims” in their business plans, whereby companies presented evidence of unique operating circumstances, legal requirements, or atypical expenditure which drive higher efficient costs for a company relative to its peers. Ofwat would then assess these claims and make adjustments where, for example, it considered that the claim was not captured by its modelling, was material, was outside management control, had been mitigated to the extent possible, and the evidence on its impact on efficient costs was robust. For details, see Ofwat ([81, 82, 86, 87, 93], Sect. 7, [96], Sect. 9). Second, Ofwat estimated a historical benchmark and estimated an efficiency challenge to this. At the IAP and draft determinations, this historical benchmark was the “upper quartile.” That is, a corrected OLS (COLS) style approach is used but with the benchmark given by the upper quartile (i.e., the fifth company for water services, and between the third and fourth company for wastewater services). Ofwat ([93], p. 11) state that “the upper quartile level recognises imperfections of statistical analysis.” In its final determinations, Ofwat ([96], Sect. 3.1.3) moved the benchmark to be the fourth ranked company for water services and the third ranked company for wastewater services. Ofwat considered that: (i) “following changes to our data and modelling approach . . . , the stringency of the historical upper quartile as a catch up efficiency challenge has reduced” and (ii) “the cost adjustment claims Ofwat allowed for were one-sided in most cases, increasing allowances for companies,” Ofwat ([96], pp. 31–33). As such, Ofwat considered it appropriate to strengthen the challenge for final determinations. Third, Ofwat generated cost predictions for each company by using the model coefficients over the historical period and applying these to forecast of company cost drivers over AMP7. Typically, these forecasts were either: those developed by the companies as part of their business plan submissions; derived by Ofwat using third party sources (such as the ONS, the UK government statistics department), a linear time trend or average of the past values for the cost driver; or a combination of the companies’ and Ofwat’s views, Ofwat [96]. Ofwat ([96], p. 23), stated that “it is important to protect customers from potentially inflated forecasts that feed into cost estimates . . . It is therefore an important part of our incentive based regulation to develop an independent view of cost drivers over the forecast period . . . .” It considered that its mixed approach “better reflects what companies are expected to deliver during 2020–25 consistent with our final determinations, while maintaining the properties of our incentive-based regulation.” The historical benchmark challenge was applied to these forecasts.

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Finally, Ofwat also applied a frontier-shift challenge of 1.1% p.a. over the period 2020/2021–2024/2025. This was based primarily on separate analysis using total factor productivity (TFP) growth rates using the EU KLEMS database, Stehrer et al. [135]. For details, see Europe Economics [37–39]. Ofwat also allowed for real input price inflation. For retail expenditure, Ofwat used a different approach. In its initial assessment of plans and fast-track draft determinations (i.e., determinations for three companies that submitted high quality business plans, Ofwat [106], p. 2), instead of calculating a historical upper quartile benchmark and overlaying a frontier shift assumption, Ofwat [93] used a forward-looking upper quartile benchmark. For the slow-track draft determinations and final determinations, Ofwat used the average of the historical UQ and forward-looking UQ-based results. Ofwat [93] stated that this choice of a forward-looking UQ was driven by the decline in companies’ projected costs over AMP7 relative to the current level of expenditure and in the final determinations. Ofwat ([96], p. 121) expanded on this reasoning, stating that “the retail control has started as recently as 2015 and retail services can transform more quickly than wholesale services . . . The fact that the majority of companies submitted forecasts that are significantly more efficient than historical expenditure is evidence of the pace at which this service is transforming. It is important that customers share the benefits.”

Wholesale Enhancement Expenditure Ofwat used a slightly different approach for enhancement expenditure. In contrast to BOTEX plus, where all constituent costs were aggregated together and modelled, each type of enhancement activity was assessed separately and efficiency challenges were generally set at the individual activity level. Exceptions to this were made for some related types of expenditure such as supply-demand balance expenditure and the water industry national environmental program (WINEP) expenditure. The efficiency challenge depended on “the quality of the model and the spread of company cost projections around [Ofwat’s] benchmarks” (see Ofwat [93], p. 16). The final view of efficient cost was the minimum of the efficient cost estimated by modelling and the company’s requested enhancement spend. In total, Ofwat examined 16 separate capital enhancement activities for water services and 24 separate capital enhancement activities for wastewater services, for a total of 40 activities across both water and wastewater. Of these 40 enhancement activities, 12 were examined using econometric benchmarking analysis, while the others were based on an examination of companies’ business plans. Ofwat’s preferred assessment method was econometric benchmarking analysis, stating “Our preferred method of assessment is benchmarking analysis of forecast costs. Where the investment area does not lend itself to statistical modelling we rely more on the evidence provided by companies in their business plans,” Ofwat ([96], p. 49).

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Where Ofwat did use an econometric approach, the estimation approach was relatively similar to the BOTEX approach described above. The benchmarking models were typically limited to only one to two cost drivers. However, there were two important dimensions in which its approach to enhancement differed to that for BOTEX: • Use of forecast data – BOTEX plus models were estimated on outturn, or historical data, while enhancement models were sometimes estimated on forecast data only, or a mixture of both historical and forecast data. Generally, Ofwat modelled enhancement cost data over the business plan forecast period (2020/2021– 2024/2025), although in some areas, it also took into account historical data (2011/2012–2017/2018) to inform the allowance. However, this was typically modelled in a separate model to the forecast data, rather than combining all the data into one model, for an example, see Ofwat [100]. • Choice of benchmark – The efficiency challenge applied varied across different activities, including an average benchmark, a company-specific challenge based on its historical base expenditure performance or an upper quartile benchmark. • Modelling was undertaken on both cross-sectional and panel data – While data is available for each company over a number of years, in some areas, Ofwat collapsed the data to one observation per company over the assessed period by summing across expenditure and cost driver. In other areas, the panel data structure (i.e., data across companies over time) was retained. This was typically modelled econometrically using the RE estimator, rather than the OLS estimator. However, Ofwat occasionally used the OLS estimator instead, for example, see Ofwat [100]. The models were used to generate cost predictions, efficiency scores, and in turn, the efficiency challenge. This efficiency challenge varied across models. As an example of how Ofwat’s benchmarking approach works for one area of enhancement expenditure, we set out the process for the “phosphorus removal” enhancement activity, which is associated with removing phosphorus from waste load to prevent eutrophication. This activity forms a part of a wider WINEP program. To assess phosphorus removal costs, Ofwat specifies a simple econometric model. This generates predicted costs for each company. However, rather than calculating the catch-up challenge for just phosphorus removals, Ofwat does so across all enhancement activities within WINEP. Ofwat ([96], p. 62) considers that this is appropriate because it accounts for potential cost allocation issues and the accuracy of the individual models used for each enhancement activity. Given the number of activities examined by Ofwat, we do not examine the other 11 individual modelling areas. For details on these models, the interested reader should review Ofwat [98–99]. One key difference between BOTEX and enhancement modelling is that the modelled benchmark did not necessarily form the efficiency challenge presented to companies. Citing the need to “[protect] customers from paying for inefficient, unrequired or undelivered investment in the control

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Gate 1

Need for adjustment?

Need for investment?

£

Gate 3 Management control?

Reallocated from appropriate line

Gate 4

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Gate 7 & 8

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Robust evidence?

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Best option for customers?

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Fig. 5 Shallow/deep-dive gates. (Source: Oxera ([118], p. 12), based on Ofwat ([93], p. 17))

period,” Ofwat capped the efficient expenditure level in an enhancement area to the minimum of the model prediction and the company’s submitted cost level, Ofwat ([93], p. 15). Companies have raised a number of challenges in response to Ofwat’s approach to enhancement modelling. For example, Ofwat collapses the panel data structure of phosphorus removals costs and cost drivers into a cross-sectional dataset, meaning that there are only 10 observations in total (one for each of the WASCs). As a result, Ofwat only accounted for a limited number of cost drivers. For instance, Yorkshire Water ([163], p. 63) argues that Ofwat’s approach does not fully account for the cost differences between different phosphorus removal solutions that are driven by differences in legislative obligations. Furthermore, Yorkshire Water [163] argues that the limited sample and reliance on forecast data mean that the use of an upper quartile benchmark is too challenging and risks classifying noise as inefficiency. In enhancement activities where Ofwat did not adopt an econometric benchmarking approach, Ofwat relied more heavily on the written evidence provided in company business plans. In these areas, either a so-called “shallow” or a “deep-dive” approach was taken, depending on whether the expenditure for a company was less or greater than 0.5% of water/wastewater total expenditure (TOTEX). Ofwat’s shallow/deep-dives were a quantitative and qualitative review of the companies’ enhancement submission in this area. Expenditure was assessed relative to a number of “gates,” as set out in Fig. 5. Ofwat applied a 20% challenge to expenditure that passed the first three gates and where it judged there to be insufficient evidence of optioneering (i.e., considering all relevant solutions) and/or robust evidence of efficient costs. In some instances, Ofwat applied an additional company-specific efficiency challenge. An assessment of affordability and board assurance was necessary only for the largest schemes.

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Northern Ireland In this section, we examine the cost benchmarking approach used in the Northern Irish water sector, which draws heavily on the approach used by Ofwat in PR09.

Historical and Industry Context Water and wastewater services in Northern Ireland are provided by NI Water, which has dual status as a government-owned company and a nondepartmental public body. The Department for Infrastructure is the sponsor and sole shareholder of NI Water. It monitors financial and performance reporting against nominated outputs as determined by the Utility Regulator (see below), has a challenge and advocacy role, and is responsible for paying the customer subsidy to NI Water. Water and sewerage services are regulated by the Northern Ireland Authority for Utility Regulation, Utility Regulator, or UR (previously Ofreg).

Efficiency Benchmarking in Northern Ireland PC15 was the third price control for NI Water covering a 6-year period from 2015–2016 to 2020–2021. The final determination for PC15 included separate cost benchmarking of operating expenditure and capital expenditure, following approaches previously used by Ofwat (in PR09 and before). The UR benchmarks NI Water’s costs against England and Wales water companies. Given the similarity to Ofwat’s approach, we only provide a high-level summary here. The interested reader should review the references provided. For operating expenditure, UR’s approach at PC15 included the following [146]: • Establish NI Water’s baseline OPEX. • Adjust for additions (and reductions) to base costs (e.g., additional OPEX due to new legal standards, improved drinking water, or treatment standards). • Assess transformation costs (in recognition that significant change was required to improve efficiency). • Assess OPEX from capex requirements (i.e., new expenditure arising from the capital program). • Determine allowances for special factors (rural/dispersed population, regional wages, electricity prices, legacy specialist wastewater treatment technology) and atypical expenditure. • Undertake cost benchmarking using unit cost and econometric modelling of costs for different functions (such as distribution, or resources and treatment). That is, a similar approach to that used by Ofwat in PR09 (see Ofwat [75]).

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• Estimate the relative efficiency gap between NI Water and the upper quartile. • Overlay assumptions on the frontier shift. • Consider how public private partnerships/private finance initiative (PPP/PFI) costs should be treated. • Review NI Water proposals. For capital expenditure, the UR followed a similar approach to that used by Ofwat in PR09. Two approaches were used – capital maintenance expenditure models (unit cost models or econometric models) and the “cost base.” The cost base was used to assess the relative efficiency of water and sewerage companies in procuring and delivering capital projects. These compare unit costs across a wide range of standard water and sewerage capital schemes (e.g., mains laying, mains rehabilitation, meter installation). For PC15, the UR primarily used upper quartile as its benchmark with cost base derived efficiencies triangulated against capital procurement efficiencies. The latter was derived though a panel of reporters drawn SMEs who examined and made recommendations on how NI Water might, over the course of the control period, improve procurement processes and practice to achieve enhanced efficiencies going forward.

Future Potential Changes in Approach For PC21, which covers 2021–2027, the UR does not envisage following the same approach to operational efficiency it took at PC10 and PC15 for assessing efficiencies, as this “no longer remains appropriate,” UR ([147], p. 22). Its approach to efficiencies in PC21 may include the following elements [148–149]: • Engaging with NI Water at a Cost Assessment Working Group (CAWG) • Setting a challenging efficiency target for NI Water, while recognizing NI Water’s progress in delivering efficiencies • Using a pooled dataset, including comparable data from the England and Wales companies • Benchmarking capital maintenance expenditure and/or BOTEX modelling • Accounting for special factors and atypical expenditure • Using COLS models • Reserving judgment on the specific rate of catch-up • Adopting a “triangulated” approach of combining econometric analysis, examination of frontier shift, and experience of rapid reductions in expenditure in similar regulated industries At the time of writing, the UR has not yet published its draft or final decisions. Therefore, its methodology may be subject to further change.

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Scotland In this section, we examine the cost benchmarking approach used in the Scottish water sector, which used to be based on the approach used by Ofwat in PR09, but has subsequently moved away from econometric cost benchmarking.

Historical and Industry Context Scottish Water is responsible for providing water and wastewater services to household customers and wholesale licensed providers. It is a public corporation accountable to Scottish Ministers and the Scottish Parliament. The Water Industry Commission for Scotland (WICS) is the economic regulator for Scottish Water, setting charges and reporting on costs and performance. The regulatory model in Scotland is in the process of changing from a traditional “adversarial” model to a more collaborative approach, characterized as Ethical Based Regulation (EBR), WICS [158]. This change has had implications for the role of benchmarking in the regulatory process, in particular reducing the need for benchmarking as a tool to reduce information asymmetry between regulator and company [158].

Efficiency Benchmarking in Scotland Under the previous regulatory model, the price setting process took place (typically) every 6 years. The process is set out in Fig. 6.

Regulator uses benchmarking to determine a sufficient price cap for the company to finance its plan Scottish government1 sets draft objectives for Scottish Water to achieve over the 6 year period

Scottish Water sets out a detailed business plan to meet these objectives to submit to WICS

WICS reviews the business plan and determines draft price caps

Scottish government1 and Scottish Water respond to WICS draft determination

WICS finalises price cap for the 6 year period in its final determination

Potential for company to determine the cost to deliver outcomes in alignment with government objectives

Fig. 6 Benchmarking under the previous regulatory model. (Source: WICS ([159], p. 9). Note: Quality regulators – such as the Drinking Water Quality Regulator and Scottish Environment Protection Agency – also contributed towards the objectives set for the water sector)

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The role for and nature of benchmarking has changed over time. At the first strategic review of charges for Scottish Water – Strategic Review of Charges 2002– 2006 – econometric benchmarking, in combination with bottom-up engineering evidence, was used to determine the potential for efficiency savings at Scottish Water. In addition, an estimate of the benefits of merging the three regional authorities into a single company was also overlaid, WICS [153]. The resulting efficiency challenge was substantial, with expected OPEX efficiencies of 37% by 2006 and a CAPEX efficiency target of 34%, WICS ([153], p. 15). As such, WICS provided for additional costs up front in order to facilitate the efficiency improvements, so-called spend to save. At this early stage in Scottish water regulation, the benchmarking techniques drew substantially from the models used in England and Wales in PR09 and before. As such, the key issues discussed in the section above on benchmarking in England and Wales benchmarking are also relevant for Scotland. WICS ([153], p. 14) stated: The Office of Water Services (Ofwat), in conjunction with Professor Mark Stewart at the University of Warwick, developed these econometric models [the models used to benchmark Scottish Water]. The models were used in the 1994 and 1999 price reviews in England and Wales. They have been held out as an example of good practice by the Cabinet Office and were reviewed by the Competition Commission last year. I have made only marginal adjustments to these models to ensure that they take fully into account the Scottish operating environment.

Although adjustments may have been “only marginal,” a persistent issue with benchmarking Scottish Water to companies in England and Wales has been the magnitude of unique or special factors related to operating in Scotland. WICS ([155], p. 95) sets out that these include, but are not limited to: • Scotland’s geography (size, remote islands, long coastline, topography) • Its population settlement patterns (remote communities, concentrated dense urban areas) • The extent of the assets required to serve customers in Scotland (long mains, small isolated treatment works) • The quality of the assets inherited by Scottish Water (condition and performance of the mains, sewers, treatment works, pumps) • The nature of the customer base • The fact that Scottish Water is in public ownership (political interest, Scottish Water’s duty to Scotland, remit and freedom of management) • The short time that Scottish Water has had to mature and improve Another complicating factor for the benchmarking has been Scottish Water’s inheritance of nine long-term Public Private Partnership contracts to operate wastewater treatment. These cover around 50% of Scotland’s wastewater treatment and 80% of its sludge treatment, WICS (ca. [157]). As the fees that Scottish Water pays for these contracts (around 10% of its annual spending) were set before the current company came into being, they are not controllable by current management.

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Thus, the expenditure cannot be considered as controllable or comparable to Scottish Water’s internal wastewater treatment expenditure and cannot be compared to companies in England and Wales without adjustment. Many of these contracts are not due to expire until the 2030s, WICS (ca. [157]). Over the course of subsequent reviews (2006–2010, 2010–2015) benchmarking with England and Wales has been applied in a number of ways by WICS and Scottish Water. Scottish Water [130] identified the main areas in which benchmarking has been used as: • • • •

Service levels using the Overall Performance Assessment (OPA). Operating expenditure using an econometric modelling approach. Capital maintenance using an econometric modelling approach. Capital enhancement using a cost base approach.

A Change in Direction In more recent reviews (SRC15 and SRC21), WICS has moved away from econometric benchmarking techniques. This has been driven by changes to the way Scottish Water is regulated and the position of the regulator on the efficacy of econometric benchmarking in the current context of the Scottish water sector. The move in regulatory reviews from 2015 onwards has been to empower Scottish Water to have full responsibility of its business plan. The combination of greater transparency (removing information asymmetries) and transferring ownership of determining what outputs and outcomes are required removes some of the need for benchmarking to increase the information available to the regulator and set hard budget constraints for companies. However, benchmarking can still play a role in informing the decision-making process even outside of a traditional “adversarial” regulatory framework, WICS [156]. WICS and Scottish Water have used higher level metrics, such as average unit OPEX, to inform the potential for efficiency improvements going forwards. Moving away from econometric benchmarks has been justified in WICS ([156], p. 8) on the following basis: Comparative benchmarking with the companies in England and Wales using econometric models – which has proved so useful in the past to drive improvements in Scotland – is no longer entirely adequate. The scale of Scottish Water’s improvement means that we would have to use more intrusive approaches to identify and measure gaps in performance. Ofwat, the economic regulator of the water sector in England and Wales, also appears to be moving away from its historical approach to econometric modelling.

We note that, while Ofwat’s approach has changed from price reviews prior to PR14, it has continued to place benchmarking methods at the center of its approach to cost assessment. In considering a comparison of the two approaches, it is also instructive to consider the different contexts the regulators operate within: WICS regulates a single public corporation owned by and accountable to government,

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while Ofwat regulates 17 companies – 16 of which are either publicly traded or owned by private equity.

Ireland In this section, we examine the cost benchmarking approach used in the Irish water sector , which draws heavily on the approach used by Ofwat in PR09.

Historical and Industry Context In Ireland, water services are provided by state-owned Irish Water, which was established as a single national public water utility in 2013. Local authorities currently continue to act as agents for Irish Water, providing services under Service Level Agreements. The Commission for Regulation of Utilities (CRU), previously the Commission for Energy Regulation (CER), is Irish Water’s economic regulator. The last revenue control covers the period from 2017 until 2019, having been extended by 1 year, CER [16] and CRU [18], while the current revenue control, RC3, covers the 5-year period from 2020 until 2024, CRU [20]. The Water Services Act [161] sets out that Irish Water’s revenue will be recovered through a mixture of Government subvention and customer charges. The CRU determines Irish Water’s revenue allowance (i.e., the level of funding Irish Water can collect from its customers). This involves the CRU reviewing Irish Water’s submissions, benchmarking Irish Water’s proposed costs against comparator companies, undertaking a public consultation process, and then setting revenue allowances. The rest of this section is based upon the consultation paper published by CRU, as, at the time of writing, the final decision had not been made.

Efficiency Benchmarking in Ireland With regards to benchmarking, the Commission for Regulation of Utilities in Ireland (CRU) assessed operating expenditure and capital expenditure separately. For operating costs, CRU [20] reviewed Irish Water’s costs and benchmarked them against water and wastewater utilities in other jurisdictions (including England and Wales water and wastewater companies, Scottish Water, and Northern Ireland Water). CRU reviewed Irish Water’s expenditure proposals and used unit cost comparisons and econometric benchmarking. For water services, OLS was used to model operating costs with one cost driver and time dummies. The cost driver was a composite scale variable (a weighted combination of distribution input, population and mains length), NERA [60]. CSV =(W ater Delivered)w1 x (Connected P roperties)w2 x (Mains Length)w3

(4)

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where w1, w2, and w3 are weightings on each of the scale variables. For wastewater services, operating costs were regressed against a composite scale variable (consisting of properties connected and sewer length), the number of WWTW and time dummies. CRU [20] stated that the impact of Irish Water’s specific characteristics on its comparative efficiency was considered, including considering Irish Water’s higher wages costs and its greater length of water network per connection. On the former, a wage adjustment was made, scaling costs up or down in order to improve comparability across companies before conducting cost benchmarking, as adopted by Ofgem [65] (the GB energy regulator) in its electricity distribution price control, RIIO-ED1. On the latter, CRU [20] noted that models developed by Ofwat, CMA (the UK competition authority) and UREGNI (the Northern Ireland regulator) tended to show that the number of connections (rather than network length) is the main cost driver. CRU [20] considered that the average level of operating costs was an appropriate target for Irish Water to move to over time. CRU [20] allowed Irish Water to reduce its costs over the RC3 period towards an efficient level of costs, as CRU considered that an immediate reduction would likely have a negative impact on the level of service. CRU also examined rates of improvement achieved elsewhere, in order to assess expected rates of improvement that Irish Water could achieve. For capital expenditure, CRU used a bottom-up/engineering approach, examining costs at a project level. As set out in Jacobs [45] and CRU [20], the review considered Irish Water’s maturity and approach to planning, prioritizing, and optimizing the work identified, including the need and timing of investments, to meet its obligations. Cost estimating processes were reviewed and a sample of projects examined in more detail to confirm whether the scope of work was appropriate, and whether the costing processes were applied as anticipated. In its final decision, for operating costs, CRU [22] required Irish Water to meet a 4% per annum efficiency gain on operating costs, starting with 2% in 2020 and rising to 6% in 2024. For capital expenditure, CRU [22] imposed a 3% per annum efficiency challenge. However, for spend that was already committed, comprising approximately one-third of capital investment, CRU [22] did not impose an efficiency challenge. In addition to the revenue control review, the CRU has developed a performance assessment framework for Irish Water, CER [17] and CRU [21]. This Framework provides a structured way for the CRU to assess Irish Water’s performance over time. Irish Water’s Performance Assessment Reports are published approximately every year, CRU [19]. CRU also notes that the publication of reports under the Framework incentivizes Irish water to improve its performance and service delivery and allows stakeholders to monitor that performance. This is in contrast to Ofwat, the UK water regulator, which discontinued publication of its annual efficiency reports. Key performance indicators or metrics used in Irish Water’s Performance Assessment which are categorized under the following headings:

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Customer service Environmental performance Water supply – quality of service Security of water supply Sewerage service

Denmark In this section, we examine the cost benchmarking approach used in the Danish water sector – a combination of DEA and SFA.

Historical and Industry Context The Danish drinking water sector is highly decentralized. It consists of approximately 2600 public waterworks, with around 87 municipally owned drinking water companies, which in total comprise approximately 330 waterworks. The remainder are privately owned, either as independent individual waterworks or collected together into small utility companies with additional facilities, usually owned by the consumer, Danish Water and Wastewater Association ([30], p. 8). All companies, with more than 200,000 m3 water p.a. are regulated, while around 300 of the largest companies (providing more than 800,000 m3 water p.a.) are subject to a benchmarking exercise covered by the Danish Water Sector Act [31], Konkurrence- og Forbrugerstyrelsen ([47], p. 4). The Act provides rules for the water supply companies to keep their revenues within a set limit (a revenue cap) and stipulates rules for efficiency requirements. The current timetable is that this will change from 2022 onwards, after which privately owned companies, which provide less than 800,000 m3 water p.a., will be able to choose to withdraw from regulation entirely. All companies who remain regulated will be part of the benchmarking exercise (so the benchmarking will not just include those with more than 800,000 m3 water p.a.). The Secretariat for Water Supply (Forsyningssekretariatet) is the Danish economic regulator for water supply companies. The Secretariat is a part of the Danish Competition and Consumer Authority (DCCA) under the Ministry of Industry, Business and Financial Affairs. Cost benchmarking was first used by the regulator to set individual requirements for efficiency improvements in 2012. The Secretariat conducted yearly operating expenditure (OPEX) benchmarking of Danish water companies above a size threshold to uncover possible efficiency improvements. Companies smaller than this threshold (providing below 800,000 m3 of water per year) are exempted, but can voluntarily participate and need to meet a general efficiency requirement of 1.7% of total costs.

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Recent Regulatory Framework Changes In 2015, there was cross-party agreement for a new regulatory framework for the water sector, Danish cross-party water sector control [29]. On 10 October 2016, this agreement was passed into law. It was amended on 28 June 2018 [35]. One of the key aims of this new regime was to promote more efficiency in the water sector. The overall goal was to achieve 1.3 billion DKK efficiency improvements in the Danish water sector from 2015 to 2020 ([35], p. 2). To allow companies to budget over a longer time horizon and reduce the administrative burden associated with regulation, multi-year binding price ceilings have been introduced. After a phasing-in period of one- and multi-year frameworks from 2017 onwards, 4-year revenue caps will be introduced from 2022 onwards for wastewater companies and from 2023 onwards for water companies ([29], p. 3). Currently, the benchmarking is undertaken and the revenue caps adjusted every 2 years, with water utilities benchmarked in even-numbered years and the wastewater utilities benchmarked in odd-numbered years. Although, companies who were identified to be fully efficient at the previous review have already received 4-year revenue caps. The required efficiency improvements are based on estimates of the potential for frontier shift and catch-up. For all water companies which process more than 800,000 m3 water per year: • The annual frontier shift for all companies is calculated in a separate exercise from the benchmarking exercise. It is based on the productivity performance in the construction sector and in the market economy, using information published by Statistics Denmark, Energi-, Forsynings- og Klimaministeriet ([35], Chaps. 5 and 6). • The individual catch-up efficiency requirement is based on cost benchmarking using DEA and SFA. The outcome is capped at 2% p.a., Danish cross-party water sector control ([29], p. 2). The regulatory framework has also been designed to encourage greater consolidation in the water sector and, thus, improve scale efficiency. This is achieved, in part, by excluding water companies’ merger related counselling expenditure from cost benchmarking, Danish cross-party water sector control ([29], p. 6).

Efficiency Benchmarking in Denmark To establish the catch-up efficiency requirements, a regular benchmarking exercise is conducted by the Forsyningssekretariatet. The benchmarking exercise currently takes place every 2 years, but will occur every 4 years from 2022 onwards, Danish cross-party water sector control ([29], p. 3). First, the water companies report information on their underlying cost drivers, investments, and operating costs. These reports are quality assured. For water

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companies that fail to submit financial information by the deadline, the Forsyningssekretariatet will issue an efficiency target based on estimated numbers, up to a maximum efficiency target of 2% p.a., Danish cross-party water sector control ([29], Chap. 5, paragraph 9, Sect. 6). Second, the models are then defined in terms of inputs and outputs: • In 2017, DCCA introduced a total expenditure (TOTEX) benchmarking framework. Modelled costs consist of the sum of operating expenditure, depreciation, and financial expenses. The costs are the actual costs in 2016 less depreciation for certain investments that are in the category of “other assets.” Investment is categorized as “other” if it is relatively unique and there is no corresponding output driver for these costs. However, even if excluding from the modelling, if a company efficiency challenge is identified in the benchmarking, it will still be applied to such costs, Konkurrence- og Forbrugerstyrelsen ([47], pp. 15–16). TOTEX, as assessed by the DCCA, also excludes noncontrollable costs (such as taxes) which do not have an efficiency challenge applied. • Two output variables (cost drivers) are used to explain the differences in efficient expenditure across water companies (Konkurrence- og Forbrugerstyrelsen ([47], p. 10). Based on the information provided by the water companies, two network volume measures are calculated, which are intended to describe the totality of the companies’ activities. The cost drivers are both based on the cost of asset replacement: – OPEX grid volume is intended to capture OPEX. – CAPEX grid volume, intended to CAPEX. Both cost drivers are adjusted in some years based on the density of the population that the utility serves and/or the age of the network if there is statistical evidence to support such an adjustment, Konkurrence- og Forbrugerstyrelsen (2019a, p. 14). The water companies are then benchmarked using these grid volume measures and water companies’ actual costs using DEA and SFA. For DEA, an inputoriented DEA model with constant returns to scale is used, Konkurrence- og Forbrugerstyrelsen [48]. For SFA, a Cobb–Douglas functional form is used (i.e., the inputs and outputs are modelled in logs) and a half-normal distribution is assumed for the inefficiency term, Konkurrence- og Forbrugerstyrelsen ([46], p. 6). As part of this benchmarking, the DCCA identifies outliers to be excluded in both its DEA and SFA analysis, taking a qualitative approach supported by the following procedures (Konkurrence- og Forbrugerstyrelsen ([49], p. 16): • In DEA, the so-called “super-efficiency” criterion is used in support of DCCA’s qualitative assessment of whether a company should be considered an outlier. This involves estimating each company’s efficiency against a frontier that excludes the company itself. If the super efficiency score exceeds a particular limit, then the company is considered a potential outlier. The limit used is q(75) + 1.5 × (q(75) − q(25)), where q(75) and q(25) are the 75th and 25th percentiles, respectively. Furthermore, companies forming the frontier in DEA

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are contacted to assess whether or not they are representative, Konkurrence- og Forbrugerstyrelsen ([47], p. 16). • In SFA, Cook’s Distance is used in support of DCCA’s qualitative outlier identification. If the maximum Cook’s Distance exceeds a certain limit, given 4 by N −k−1 (where N is the number of companies in the regression and k is the number of outputs), then the company associated with the maximum Cook’s Distance is considered a potential outlier. If removed, this procedure is then repeated. The qualitative assessment involves contacting all companies which constitute the frontier to assess whether they have some exceptional favorable conditions which are not comparable with the remaining companies. In addition, if a company only participates in one of the two possible wastewater activities (collection or treatment), then it is not allowed to constitute the general frontier for companies conducting both activities. When estimating the specialized collection (transportation) companies’ efficiency, all collection (transportation) companies are included in the benchmarking. Table 5 presents the parameter estimates from the estimated SFA models for water for the year 2019–2020. This establishes the efficiency frontier, which is used to calculate efficiency scores for each company. DCCA also calculates efficiency scores for each outlier company for both DEA and SFA. To do this, the outlier is added back into the model and the model is reestimated to obtain the efficiency score for the outlier. This procedure is then repeated, where each outlier is added back in to the model one at a time. The efficiency scores are corrected for special conditions. This correction is intended to ensure that account is taken of companies’ individual circumstances. This efficiency score is then used to calculate the companies’ efficient cost level. In deriving a final efficiency estimate for each company, the highest estimated efficiency score (and thus most favorable result) for each company is used, a

Table 5 Parameter estimates for SFA models Dependent variable

Constant OPEX grid volume CAPEX grid volume Ratio of variance of inefficiency to variance of noise Number of outliers

Unadjusted TOTEX excluding noncontrollable costs 0.928* 0.758*** 0.329*** 3.22**

Age-adjusted TOTEX excluding noncontrollable costs 0.0887* 0.823*** 0.264*** 2.67**

Density-adjusted TOTEX excluding noncontrollable costs 0.613* 0.556*** 0.517*** 2.90**

4

4

4

Source: Konkurrence- og Forbrugerstyrelsen ([50], Tables 3.1–3.3)

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so-called “best of two” approach, see Konkurrence- og Forbrugerstyrelsen ([51], p. 22) and Konkurrence- og Forbrugerstyrelsen [52]. As stated above, the cost reduction is also capped at 2% p.a., see Danish cross-party water sector control ([29], Chap. 5, paragraph 9, Sect. 6).

Italy In this section, we examine the cost benchmarking approach developed by the Italian water regulator, ARERA, in 2019.

Historical and Industry Context The Italian drinking water sector is highly fragmented. Around 2000 entities provide water services in Italy. The vast majority of these are municipalities or other public entities serving around 17% of the Italian population. Only in a limited number of instances are water utilities part of listed mixed ownership multi-utility groups or owned by private investors. Both national and local authorities in Italy are responsible for the water sector in Italy. The national water regulatory authority (Autorità di regolazione per Energia, Reti e Ambiente, ARERA) establishes tariff rules on the basis of a common methodology, in place since 2012. In 2017, ARERA introduced quality of service regulation, including rules to determine financial rewards and penalties as well as reputational incentives related to greater transparency in quality standards. The third regulatory period for water distribution started in 2020, with three main objectives: fostering investments, sector consolidation, and cost efficiency. With regards to cost efficiency, proposals were first published in 2019 to use a formal cost benchmarking approach for the next price control period – albeit not to determine allowed expenditure but to determine the proportion of any historical underperformance or overspending that should be passed on to customers through higher tariffs. It is expected that the scope of the cost benchmarking approach will be further strengthened in subsequent regulatory reviews.

Cost Benchmarking ARERA [5] set out the proposals for the regulatory framework for the period 2020– 2023, which considered different approaches to OPEX efficiency benchmarking. The proposed approach included: • Using a panel data set of 98 companies over 4 years (2014–2017), covering a population of around 42 million, or around 70% of the population • Using cost models with a Cobb–Douglas functional form and the following variables: – Inputs: operating costs

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– Input prices: cost of electricity supply (PE), labor costs (PL), wholesale water purchase cost (WS) – Outputs: volume of water invoiced (V), length of network (L), resident population (Pa), population equivalent (AE), availability and reliability of measurement data (PREQ1_4), compliance with the legislation on urban wastewater management (PREQ3), water losses (M1a) • Using SFA models of Battese and Coelli [9], Battese and Coelli [8], and Pitt and Lee [122] Table 6 provides an overview of model coefficients. The outcome of this modelling is estimated inefficiency scores ranging between 1% and 82%, with average inefficiency scores of 22–29%. Given the lower estimated inefficiency scores from the Pit and Lee [122] model, ARERA [5] proposed, and confirmed in [6], using that model. The Pitt and Lee model is also the only model where the factor M1a (variable relative to technical quality in terms of linear water losses) is statistically significant, which constituted an additional reason to be considered as a potential candidate to be used to assess cost efficiency. Rather than using the efficiency scores to estimate efficiency targets, ARERA instead groups companies based on two metrics: the companies’ historical unit OPEX and the companies’ “efficient” unit OPEX derived from the model prediction. ARERA sorts companies into one of six classes (based on the level of historical outturn unit OPEX) and one of three clusters (based on its econometric model’s prediction of historical efficient unit OPEX), and determines the underperformance sharing rate on this basis. In other words, the model is used to establish which proportion of any underperformance or overspending in 2016 should be passed on to customers through higher tariffs in “controllable” OPEX over the next regulatory period (2020–2023). Table 6 SFA model coefficients Variable PE PL WS V L Pa AE PREQ1_4 PREQ3 M1a Constant

Battese and Coelli [9] model 0.907** 0.261*** 0.664*** 0.210*** 0.142*** 0.510*** 0.118*** −0.099 −0.037 0.023 3.381***

Source: ARERA [5] Note: *p < 0.05; **p < 0.01; ***p < 0.001

Battese and Coelli [8] model 0.932*** 0.260*** 0.661*** 0.209*** 0.141*** 0.512*** 0.118*** −0.101 −0.037 0.023 3.378***

Pitt and Lee [122] model 1.032*** 0.282*** 0.784*** 0.226*** 0.146*** 0.469*** 0.142*** −0.075 −0.061 0.028* 3.277***

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The sharing rate is determined by the G parameter in the following formula: Opex aend = Opex 2018 end ∗

a t=2019

     OP ∗ max {0; Opex} 1 + l t − 1 + γi,j (5)

For each year, a = {2020, 2021, 2022, 2023}, “controllable” OPEX Opex aend is determined as follows: • Opex 2018 end is the cost component defined by MTI-2, calculated to determine 2018 tariff. • lt is the inflation rate. • Opex represents the difference between allowed controllable, OPEX in 2016, and outturn OPEX, COeff , in the same year. OP is the operator coefficient. It is determined by the operator class i (relating • γi,j to the operators’ historical outturn unit costs) and the operator cluster j (relating to the prediction of efficient historical unit costs derived from ARERA’s SFA model). Table 7 shows the operator clustering matrix, which determines the gamma parameter. To illustrate how this application of benchmarking works, it is instructive to consider a few examples. If a company is in class A (the first row) and cluster C (the rightmost column), implying that it has incurred low unit costs and the econometric model predicted that it should have high unit costs, then the underperformance sharing rate is 100% – i.e., the company’s customers bear the full costs of any historical cost overrun that the company incurred through higher bills over the

Table 7 OPEX clustering approach and sharing rate (G) setting Sharing rate (G) Outturn unit OPEX, i=A Outturn unit OPEX, i = B1 Outturn unit OPEX, i = B2 Outturn unit OPEX, i = C1 Outturn unit OPEX, i = C2 Outturn unit OPEX, i = Cover

SFA-predicted unit OPEX, cluster j = A −90%

SFA-predicted unit OPEX, cluster j = B −100%

SFA-predicted unit OPEX, cluster j = C −100%

−88%

−90%

−100%

−83%

−90%

−100%

−75%

−83%

−90%

−50%

−75%

−90%

0%

−50%

−88%

Source: ARERA [5] Note: Unit OPEX value ranges are ranked in increasing order (i.e., cluster A is characterized by the lowest unit OPEX)

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regulatory period. By contrast if a company that has the highest level of outturn unit costs (class Cover , the bottom row), but ARERA’s econometric model predicts an average unit cost (cluster B, the middle column), then it has an underperformance sharing rate of 50% – i.e., any cost overrun is split equally between company and customer bills. In its final decision, ARERA [6] stated that they will consider reviewing the methodology set out above for the second half of the price control period (2022– 2023).

Areas for Further Development In this section, we draw together some of the potential issues with the current application of cost benchmarking within the European water sector. We restrict the scope of this section to specific technical issues with cost or output benchmarking in regulatory jurisdictions where it is formally applied. Even then, the issues set out here are not exhaustive. This section does not address justification for/arguments against the decision to not apply formal benchmarking in countries outside the jurisdictions covered in this chapter.

Input Definition: Modelled Expenditure For an efficiency benchmarking assessment to provide a true reflection of a company’s relative efficiency, the modelled costs should be defined as broad as possible as there are trade-offs between cost categories – companies may focus on OPEX solutions or CAPEX solutions, and benchmarking on one cost category may result in unachievable targets. This creates an issue in that CAPEX is can be lumpy and thus difficult to compare across companies. Two approaches have been adopted with regards to this issue. • Many water regulators have used a “cash cost” approach to a total cost measurement, with annual or average enhancement expenditure and capital maintenance expenditure added to annual operating expenditure. For example, in PR19, Ofwat models operating expenditure combined with capital maintenance expenditure on an annual basis, while mostly assessing enhancement on an average basis. This is similar in style to that adopted by other water regulators that have followed Ofwat’s approach – i.e., CRU, UR, and WICS (historically) – and ARERA, whose cost benchmarking models include only OPEX. As ARERA’s cost benchmarking models include only OPEX, there may be some issues around not accounting for trade-offs with capital costs.

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• An alternative cost measures, such as a capital stock approach or an economic cost approach, could be also undertaken (see section “Input Definition: Accounting for the Investment Cycle”). For example, the DCCA models TOTEX as a combination of OPEX, depreciation, and financial expenses. Portela [123] provides further discussion on why, in a cost assessment exercise where prices are not taken into account (which is often the case in regulatory settings), an overall cost figure should be used. Another key issue when determining the cost base is to ensure that there is an appropriate correspondence between the inputs and the outputs used in the model. A case study on these issues, based on Ofwat’s approach at PR19, is provided below.

Ofwat’s Approach to Input Definition at PR19 In its draft determinations for PR19, Ofwat extended its cost definitions from BOTEX (OPEX and capital maintenance) to include some enhancement expenditure areas, including growth expenditure, calling the new definition BOTEX plus. Ofwat argued that growth expenditure was that more “routine” than other enhancement expenditure. In their responses to the draft determinations, a number of companies considered that Ofwat’s BOTEX plus modelling approach failed to properly account for the added enhancement expenditure areas, see, for example, Anglian Water [2]. In its final determinations, Ofwat ([96], p. 20) “accept[ed] that the integrated models may suffer from missing growth variables and that may lead to the base econometric models only funding the average historical growth rate across the industry.” As such, Ofwat slightly amended its approach, providing additional cost allowances for those companies in high growth regions and lower cost allowances for those companies in low growth regions. Nevertheless, Ofwat’s BOTEX plus approach might still be susceptible to some of the issues identified by companies, as the additional allowance is based on unit growth costs of the upper quartile BOTEX plus companies. Two alternative approaches may be worth pursuing going forward. • Ofwat’s BOTEX plus models could have potentially been improved by considering growth related cost drivers and reformulating the general model specification prior to testing down. That is, a renewed model development exercise could be undertaken. Indeed, such models were submitted by some companies as part of Ofwat’s cost assessment consultation [85] – for example, see some of the models submitted by South West Water, Ofwat ([88], pp. 21–22; pp. 31–32; pp. 62–64; pp. 73–75) and South West Water ([134], p. 1). However, the authors understand that such (continued)

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extensions were harder to develop using the data set available for the final determinations than at the modelling consultation stage. • Alternatively, growth expenditure could be modelled separately, as per Ofwat’s approach with other enhancement cost areas. That is, growth expenditure could be modelled in a similar way to Ofwat’s growth models in IAP (see Ofwat [97]), but using more sophisticated models as its growth models at IAP included only one cost driver (number of new connections). For example, see Anglian Water [4] and Anglian Water [3]. This approach does, however, assume that enhancement costs are separable from base expenditure. This issue is yet to be resolved as the time of writing. Following the PR19 final determinations, four companies appealed Ofwat’s decision. In their statements of case to the CMA, which set out their key arguments against Ofwat’s final determinations, several of these companies described issues with Ofwat’s approach that potentially understated a higher growth cost, see Oxera [120], Anglian Water [4], and Bristol Water [12]. In its provisional findings (CMA [24]), the CMA agreed that Ofwat’s approach was “imperfect” and “has some limitations,” but provisionally concluded that Ofwat’s approach was a “sensible and pragmatic approach.”

Regulators must manage the trade-off between including more of the cost base in a single benchmarking model (rather than separate models) – making it more challenging to account for all the relevant drivers and to appropriately measure capital expenditure – and including a smaller proportion of the cost base (such as OPEX only benchmarking) – potentially failing to account for cost trade-offs and cost allocation issues. An issue that regulators face in applying empirical methods from production economics to regulated companies is the scope for regulatory approaches to bias company behavior, for instance only benchmarking OPEX can lead to companies seeking CAPEX solutions even where this is not necessarily the most cost-effective solution. This has resulted in several regulators using broader cost definitions.

Input Definition: Accounting for the Investment Cycle A related input definition concern in the water sector is that a significant proportion of the cost base funds the construction and maintenance of very long-lived infrastructure assets. Indepen [43] identified that water and wastewater services are among the most capital-intensive sectors in the UK economy. This is likely to also be the case outside the UK. Incidentally, often the justification for regulating such companies (and, implicitly, the use of benchmarking) is that it would be uneconomic

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for another company to construct similar assets to serve a particular area, making it uneconomic for competitors to enter the market. This creates the following problems for benchmarking such companies: • The drivers of capital expenditure are often difficult to measure or at least measure consistently (for example, the amount of remaining capacity in the water network, or the condition of the underlying asset base). • The average asset life of the infrastructure typically significantly exceeds the period of data available to conduct the analysis (stretching to over 100 years for certain types of pipe) – implying that all companies do not need to undertake the same level of renewal activity over a typical analysis period. • A consequential outcome of underinvestment is an increase in the probability of low likelihood catastrophic events (due to a reduction in the underlying asset condition), which are challenging to measure directly as any impact may not occur for many years and is unlikely to have been observed in the analysis period. In the context of assessing Ofwat’s PR14 models as part of the Bristol Water appeal, CMA [23] noted that companies’ investment requirements vary over time – and that these requirements are not captured by any explanatory variables to control for these differences. Therefore, differences between companies in total cash expenditure may be reflective of differences in their investment requirements and not efficiency. This criticism remains valid for Ofwat’s models at PR19. In a recent publication on asset health, Ofwat [83], no mention is made of the relationship between asset health and cost benchmarking assessments, although a key focus of the document is how to ensure companies measure asset health on a consistent basis. Indeed, Ofwat has more recently initiated an industry project on asset resilience (see Ofwat [107]). In Scotland, the regulator has cited a key reason to move away from setting hard budget constraints is the restrictions this places on the ability of the regulated water company to take a long-term approach to managing asset replacement, WICS [160]. In particular, the concern of the regulator is that, by setting a fixed price cap, the company may be incentivized to take forward the interventions that require the smallest cash outlay in the short run, rather than taking a long-term approach, WICS [160]. This was a key reason for WICS to stop using benchmarking to determine an efficient cost level, in contrast to other regulatory jurisdictions set out in this chapter. This concern relating to the incentives that price cap regulatory regimes create for investment is also raised in Ofwat ([83], p. 68), which mentions potential problems around short term thinking imposed by the 5-year regulatory cycle, in particular, “ . . . we were a little concerned to hear that some considered the 5 year planning cycle to be a potential barrier to innovation, in the belief that it meant that investment returns need to be recouped in the 5 year period and this does not encourage taking a risk on new technology. This viewpoint requires further understanding, consideration and resolution.” Potential benchmarking solutions to address this issue are set out below.

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• Using a capital stock or economic cost definition for modelled costs rather than cash costs (as per DCCA’s approach, see Konkurrence- og Forbrugerstyrelsen [47]), or cross-checking the efficiency assessment of cash costs with assessments derived from asset values and the implied cost to replace the existing asset base. Saal et al. ([129], p. 49, pp. 78–79, p. 98, p. 114) criticized Ofwat’s cash cost approach, stating, “all OPEX is a cost while CAPEX, regardless of whether it is for enhancements or maintenance, is investment which contributes to a capital stock. This capital stock then has associated depreciation and capital financing costs.” • Using a cash cost approach but extending the timeframe of the dataset used for analysis or carefully considering what benchmarking period is appropriate (i.e., avoiding potential troughs in expenditure for companies that define the frontier). On this, Ofwat [105] state “We agree that capital maintenance is ‘lumpy’ and in some periods companies may need to spend more than in others. To address the issue of lumpy expenditure and ensure that we are setting an efficient allowance for the long term, we use eight years of data, which is the longest historical data set we have ever used in models, to ensure that our input data includes a wide range of company peaks, troughs and atypical lumps.” • Including measures of asset health or risk and/or measures of company activity – such as pipes replaced as in Ofwat ([88], pp. 52–53) – to address asset health as cost drivers and to ensure that companies are incentivized to maintain a high level of asset quality, while being careful to avoid perverse incentives of gold plating. Variables previously considered in capital maintenance modelling in England and Wales can be found in Ofwat [71] – these included measures of: the size of the asset base (such as modern equivalent asset value); asset type (such as large bore water mains); and asset condition (such as the proportion of assets by condition grade). The incentive structure around such drivers needs to be considered carefully to avoid “gold plating” – i.e., companies delivering a program of asset replacement beyond that desired by stakeholders. Indeed, the CMA, in its redetermination of PR19, was “concerned that [such] measures are within the control of a company” (CMA [24], p. 136). Age of the asset base has, however, been used, for example, the DCCA adjust their CAPEX grid volume measure based on the age of the network and Anglian Water [1] included age in some of their cost models. With regards to asset age, Ofwat has further argued that “asset age does not directly correlate to asset performance or service to customers in the water sector” and provide an example of companies claiming higher costs for relatively new assets due to “plastic pipes installed in the 1960s and 1970s” (see Ofwat [105], p. 6). Ultimately, in the case of the redetermination of PR19, the CMA provisionally concluded that “there should be no systematic underfunding in the long run” and “provisionally decide[d] not to adjust [its] approach to setting capital maintenance allowances” (see CMA [24], p. 136 and 139). Key to any approach designed to ensure that companies have enough money to fund costs associated with asset replacement will be to ensure that customers do fund the consequences of historical underinvestment.

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Cost benchmarking approaches to set capital expenditure allowances remain more controversial than in other areas, such as operating expenditure. Given the difficulties in capturing all drivers of capital maintenance expenditure within a DEA or econometric model, regulators have often considered other approaches to either set a capital maintenance allowance or to cross-check the outcome from econometric models, examples of these include the Common Framework (UKWIR [145] and Ofwat [73]), the Asset Management Assessment or AMA [27], and broad equivalence [154]. One potential development that could potentially increase the robustness of benchmarking capital maintenance is the development of more sophisticated indices of risk, such as that being developed by Ofgem and the industry for assessment of gas distribution costs (see Ofgem [66]). Ofwat’s recently initiated project on asset resilience (see Ofwat [107]) may also result in similar measures being developed in the England and Wales water industry. The use of such drivers in cost benchmarking models could address several of the challenges set out in this section.

Output Definition: Multiple Outputs and Cost–Service Trade-Offs Once appropriate inputs have been defined, there are two issues that regulators have faced in including outputs (including quality) in their models, particularly given constraints imposed by small datasets and limited data availability: • Controlling for multiple outputs, such as the number of customers served, the number of connected properties, the amount of water delivered, the sewage load treated, the size of network (capturing the distance water or sewage needs to be pumped or transported)). • Controlling for quality of service. With regard to controlling for multiple outputs, regulators have historically employed a range of approaches, including: • Controlling for multiple outputs in the same model, such as ARERA’s inclusion of the volume of water invoiced, the length of network, resident population, and population equivalent in its models; DCCA’s use of aggregate measures such as OPEX or CAPEX grid volumes; or the CRU’s and UK energy regulators’ use of composite scale variables (see NERA [60], Ofgem [64] and Ofgem [65]). • Using a single key output as a scale driver, and using other normalized cost drivers to pick up the impact of delivering other outputs (such the PR19 approach in Ofwat [96] of controlling for sewer length as the main scale driver and the number of properties per sewer length to capture the higher costs associated with more properties to collect sewage from for a given sewer length). • Ofwat’s PR19 approach of modelling across the value chain and controlling for the most relevant driver at each level of the value chain, for instance using sewer

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length as the most relevant variable for sewer collection and load as the most relevant variable for sewer treatment (see Ofwat [96]). As companies generally need to increase inputs in order to improve service quality and given that quality of service is a key outcome for consumers, the environment, and other stakeholders, it is important that regulators account for any differences in quality of service and outputs delivered in determining the regulatory settlement. This has been considered within regulatory frameworks either within the cost benchmarking assessment itself or separately, for example, through financial incentives for companies to deliver a relatively high level of service quality. The latter has tended to be the approach adopted by regulators historically. • In PR19, while some quality related measures were included, Ofwat’s cost models primarily excluded quality of service measures, which were determined and incentivized separately (see below). • Ireland and Northern Ireland adopt largely the same framework, with quality of service considered outside of benchmarking. • In Scotland, when WICS used benchmarking to set price control limits, it used a broadly similar framework to Ofwat. As set out above, quality of service performance was benchmarked against that of companies in England and Wales, but this was separate from cost assessment. • In Denmark, quality of service measures are currently not directly controlled for in the cost benchmarking framework, although the DCAA is considered accounting for quality in future modelling (see Konkurrence- og Forbrugerstyrelsen [53]). • In Italy, the regulator has identified compliance with the legislation on urban wastewater management and water losses as two quality of service related output variable it will aim to control for (see ARERA [5]). ARERA [5] also used a driver to capture the availability and reliability of measurement data as an additional control. A case study on these issues, based on Ofwat’s approach at PR19, is provided below.

Ofwat’s Approach to Accounting for Quality of Service at PR19 In the England and Wales water sector, a number of water companies considered that Ofwat’s separate approaches to cost assessment and service outcome (or performance commitments) in PR19 ignored the trade-offs between costs and performance – see, for example, Yorkshire Water [162]. Companies argued that more output and service performance measures, and the drivers of these (e.g., legislative changes), should be included within the cost models, (continued)

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as the exclusion of such measures can result in biased efficiency estimates. While some output measures were included by Ofwat, such as consent levels (i.e., the quality of discharges legally allowed following treatment), other important quality measures, required or targeted to improve substantially over AMP7 (Asset Management Period 7, over 2020/2021–2024/2025), were not included. Some modelling including additional service performance measures and the drivers of these was submitted by a number of companies – for example, see Oxera [119], NERA [61], and Yorkshire Water [163]. Given this, Ofwat ([101], pp. 36–42) further examined this issue for its final determinations, but generally maintained its approach of separately accounting for service performance and cost performance. However, Ofwat [96] did make some adjustments to its modelling and tried to account for the additional costs of: phosphorus removal, by including legislative changes in some of its modelling, and adjusted its allowance for Yorkshire Water [102]; and leakage, by including the distance from the upper quartile 2024–2025 leakage target in the model, and adjusted its allowance for Anglian Water (Ofwat ([96], p. 35, Ofwat [103]); PwC [124] and Ofwat [105]). Nevertheless, the issue remains important going forward and has continued to be debated as part of the appeals of PR19 – see Oxera [120], Anglian Water [4], Bristol Water [12], Northumbrian Water [62], and Yorkshire Water [163]. In its redetermination of PR19, the CMA provisionally concluded that quality of service measures were “substantially under management control” and their inclusion in the cost models “is likely to lead to endogeneity problems and thus biased coefficient estimates” (see CMA [24], pp. 125–131). However, this ignores the omitted variable bias problem caused by not including such variables.

Clearly, accounting for quality of service is a key consideration for cost benchmarking exercises and, while endogeneity issues require careful consideration, ignoring quality of service can bias the results. It terms of further development, the large number of potential outputs and service performance measures for the water industry suggests that it may not be possible to estimate a model that captures all of these measures separately. It may, therefore, be necessary to: • Focus on capturing only the key service performance measures, see ARERA [5] • Create composite output measures (similar in concept to DCCA’s aggregate asset-based output measures, see Konkurrence- og Forbrugerstyrelsen [47]), CRU’s use of a composite scale variable, see NERA [60], or Ofwat’s previous use of service incentive mechanism (SIM) scores, see Ofwat [80] • Use such models to quantify company-specific adjustments outside of the main industry-wide models (for example, Ofwat [96]) • Adjust costs prior to cost benchmarking based on willingness to pay evidence (see, for example, Energiavirasto [36] for such an approach in the energy sector)

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Benchmarking: Input Requirement Functions In developing cost benchmarking models, regulators have used both input requirement functions and cost functions. In Denmark, the regulator’s model consists of one input (total expenditure) and two “outputs.” These two outputs measures are both asset-based measures of network volume, with some adjustment for the density of the population and the age of the network. The two output measures are not outputs of the production process. As such, the model captures the costs expected for a particular size of network. In England and Wales, Ofwat’s PR19 models, while including output measures, did not include any input prices, as such their models were not cost functions but input requirement functions. Ofwat did, however, examine the impact of regional wages but found these not to be significant cost drivers. Ofwat ([94], p. 15) stated: “We have consistently found that the regional wage level is not a robust cost driver. In many specification the variable has very low predictive power, and sometimes it showed a counterintuitive negative sign (albeit statistically insignificant) [ . . . ] We recognise that variation in labour cost can have an impact on costs although companies can exercise control to mitigate this impact. We consider also that the inclusion of a density variable, and a square of density, in our models, capture the effect of regional wage as the two are correlated.” Ofwat’s model’s up to PR09 also did not include input prices. (Though differences in regional wages were accounted for separately). As such, other water regulators that have historically followed a similar approach to Ofwat – for example, CRU, UR, WICS (historically) – have also used input requirement functions. As has the DCCA. In contrast, in PR14, Ofwat included regional wages in its cost models. Although only one input price was captured and its application was subsequently criticized in CMA [23] as the estimated coefficient for regional wages varied substantially across Ofwat’s PR14 models, and, in some cases, a coefficient greater than one had been estimated. The CMA’s own models developed as part of the appeal of PR14 included regional wages, but any specifications that resulted in negative coefficients or coefficients greater than 1 were dropped. In Italy, ARERA’s model has three input prices (cost of electricity supply, labor costs, wholesale water purchase cost) and four outputs, plus a number of compliance dummies. The models are thus closer, conceptually, to cost functions. However, the unit costs that are included as proxies for input prices are endogenously (rather than exogenously) determined variables. For example, the labor price variable is based on each companies’ own employment costs. As such, any inefficiency in these variables (e.g., paying above the market wage rate) is not captured in the final estimated inefficiency. Instead, regional market prices, i.e., the regional wage rate using government statistics, should be used. In addition, it appears that the provision of reliable data is included in the modelling by way of an incentive effect to improve the provision of such data. However, it is unclear if the impact of such variables can be appropriately picked up in an econometric cost model. Such factors may be better incentivized outside the cost benchmarking.

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This area warrants further examination. While a possible extension of the input requirement function is relatively straightforward in principle, experience so far, such as that of Ofwat at PR19, has demonstrated that this may be challenging in practice. If there is minimal variation in input prices across regions, then this is possibly a less significant issue in practice. It may be that some input prices, such as raw material prices or capital input prices, do not greatly vary across regions within a country. Alternatively, where there is some variation in input prices, such as regional wages, the variation may still be limited. For example, London may have relatively high wages compared to the rest of England and Wales, but there may not be much variation across the remaining regions. It is also possible that the variation in regional wages may be correlated with other factors such as density, and so already partly captured through existing cost drivers.

Benchmarking: Functional Form Ofwat’s models in PR19 were simpler than those used in PR14, with log linear functional forms for all cost drivers other than density, which was also included in squared form to capture a “U-shape” effect of density/sparsity. Other water regulators have similar used relatively simple log-linear functional forms – with CRU, UR, and WICS (historically) following Ofwat’s approach pre PR14 and ARERA and DCCA (for its SFA models) using a Cobb–Douglas functional form. While Ofwat’s approach might overcome some of the criticisms in CMA [23], it may be that the relationships are more complex and that it is most likely that the relationships are not the same for all companies. Criticisms of such possible over-simplicity were made in Saal and Nieswand [126]. While this might not make a significant difference for the industry as a whole, it could result in an excessive challenge or a windfall gain for individual companies. For example, Portsmouth Water was estimated by Ofwat to be 16% more efficient than its benchmark in PR19 (though Ofwat capped this at 10% – see Ofwat [96], p. 13). This could be the result of an inappropriate model specification for Portsmouth Water and/or the impact of Ofwat’s choice of an ad hoc blanket benchmark (see below). Similarly, the models in ARERA [5] appear to estimate close to constant returns to scale, but with such a wide variation in company scale, a more flexible functional form may be more appropriate to better capture the relationship between company size and cost. A potential solution to this problem is to estimate more flexible functional forms, such as the translog model, and then test whether the estimated company-specific elasticities are aligned with economic and engineering insight for each company. As discussed above, the semi-translog models used in PR14 did not always align with economic and engineering insight. This was shown to be the case in the appeal and subsequently amended by the CMA. Non-parametric or semi-parametric econometric approaches or DEA could also be investigated. Indeed, as well as SFA, the DCCA has used DEA to estimate cost

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efficiency. While potentially more flexible, the DCCA used an input-oriented DEA model with constant returns to scale (see Konkurrence- og Forbrugerstyrelsen [50], p. 8).

Forecasting Efficient Costs: Identifying “Efficient” Cost Levels, While Accounting for Error and Heterogeneity, and Alternative Estimation Approaches Having developed the cost models, a critical step is then to use the models to estimate each regulated company’s efficient cost level. With regards to this step, approaches that do not use SFA or DEA (including those used by Ofwat, CRU, UR, and WICS (historically)) have a number of related issues: • Ad hoc blanket benchmark identification – these regulators have focused on defining the frontier through the use of an upper quartile adjustment (or, in Ofwat’s final determinations at PR19, the third or fourth ranked company), applied to either OLS or RE models. However, this adjustment is a subjective judgment, the benchmark could have equally been drawn, for example, at the average. Indeed, CRU [20] considered that the average level of operating costs was an appropriate target for Irish Water. Similarly, in Bristol Water’s appeal of Ofwat’s PR14 final determination, the CMA ([23], p. 117) chose an average benchmark, in part, because it was “concerned that an efficiency benchmark based on an upper quartile efficiency concept would be overly demanding if applied to the results of the econometric models that we used. This was a judgment in the light of the issues we had identified both from our review of Ofwat’s econometric models and from our development of alternative models.” Ad hoc blanket adjustments, like the upper quartile, also assume that the same degree of noise is present for each company (although this point is most pertinent to Ofwat as the other examples relate to the identification of the efficient cost level for only one company). In contrast, it is well known in econometrics that the accuracy of the model prediction decreases as you move farther out from the central data (i.e., noise is company specific) and the accuracy of the model prediction decreases as the sample size decreases. • limited estimation approaches – these regulators relied on either OLS or RE outcomes and did not always use alternative estimation approaches (such as DEA or SFA) that might have resulted in different outcomes for individual companies. An interesting contrast can be made with the German energy sector, in which the legislation underpinning the regulatory framework states that both DEA and SFA should be used to estimate efficiency (see [110], Anreizregulierungsverordnung, AregV, Sect. 12). The best outcome from both the DEA and SFA models is then used to set companies’ efficiency scores, subject to a minimum 60%. This benefit of doubt approach is similar to that also used by DCCA. Similarly, in PR94, Ofwat used DEA as a cross-check on the outcomes from its econometric

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modelling. The MMC ([59], p. 415) noted that “in most cases the [DEA – based] results were similar to those of the regressions. If they were significantly better, [Ofwat] moved the company up one band” (see also Thanassoulis [141] and Thanassoulis [142]). In addition, some water sector regulators have mitigated against the reliance on one top-down estimation approach by using bottom-up assessments to cross-check the outcomes from the econometric analysis. • Heterogeneity – when estimating inefficiency using the results from the RE models, Ofwat ([93], p. 11) implicitly assumed that, with the exception of the subsequent upper quartile adjustment, all of the estimated company-specific effect is due to inefficiency. However, given the use of relatively parsimonious models, it is likely that the company-specific effect is partly due to company heterogeneity/specific drivers not included in the model of the expenditure, and not inefficiency. With regards to heterogeneity, these company-specific effects could be controlled for by removing the company-specific effect from the estimated inefficiency (as per a true RE model approach as in Greene [42], pp. 7–32) or certain panel SFA models could be used to control for these effects. An alternative approach, deployed by both DCCA and Ofwat, is to adjust for special characteristics outside of the modelling to try to account for companies’ individual circumstances. For example, heterogeneity is particularly pronounced in Italy – the 98 companies in ARERA’s SFA modelling are very diverse – for example, their operating costs vary from A C1.6 m to A C340 m in 2017, and their length of network varies from 151 km to 19,783 km. As such, accounting for this heterogeneity is particularly important in this instance. However, other than the included cost drivers, ARERA’s models do not allow for company heterogeneity and so can conflate inefficiency with company specific factors and thus may overestimate inefficiency. This could be mitigated by using models that capture company heterogeneity – such models include Kumbhakar and Heshmati [54], Greene [42], Wang and Ho [150], Colombi et al. [15], and Kumbhakar et al. [56]. The DCCA’s SFA model uses cross-sectional data, so the SFA models that can control for heterogeneity are not applicable. Instead, the DCCA controls for heterogeneity by examining whether each company is an outlier (see Konkurrenceog Forbrugerstyrelsen [47]). As discussed above, in addition to its qualitative assessment, the DCCA undertakes a quantitative assessment. For the DEA modelling, potential outliers are identified using “super-efficiency,” while for SFA, Cook’s Distance is used to identify potential outliers. As part of the qualitative assessment, companies are contacted to assess whether or not they are representative. Adjustments are also made for special characteristics outside of the modelling, Konkurrence- og Forbrugerstyrelsen ([47], p. 16). In addition to accounting for heterogeneity, SFA could be used to account for modelling noise, providing company-specific separation of noise and inefficiency, and confidence intervals around the inefficiency estimates. While distributional assumptions are required, SFA avoids the need for the identification of an ad hoc blanket benchmark.

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While not used by Ofwat in PR19, the potential use of SFA was discussed, but the approach was dismissed based on factual errors. For example, in CEPA ([14], p. 38), some of the reasons for dismissing its use included “since SFA is not a statistical technique, it is not possible to implement tests to evaluate the accuracy of the results”; and “although [SFA models have] been considered by regulators, they are rarely pursued.” Clearly, SFA is a statistical technique, enabling testing to be undertaken, and has widely been used in regulatory contexts across sectors and jurisdictions by both regulators and companies – see, for example, ORR [112], Deloitte [32], Swiss Economics, Sumicsid and IAEW [140], and Oxera [115]. Evidence using SFA has also been used in the England and Wales water sector – see, for example, Stewart [137–138], Oxera [114], Oxera [116], and Saal et al. [128]. DEA can also potentially help with heterogeneity as companies’ efficiency will be estimated through comparisons with peer companies that have similar characteristics (to the extent that those characteristics are included in the model). In terms of specific applications of DEA in water regulatory assessments: • DCCA’s DEA model consists of one input and two output measures, under the assumption of constant returns to scale (which assumes companies can be scaled up or down to form virtual peers). See Konkurrence- og Forbrugerstyrelsen [47]. As such, the model’s ability to account for heterogeneity is quite limited although, DCCA does correct the estimated efficiency scores for special conditions. (Clearly, alternative assumptions for returns to scale could also be assumed). • In PR94, while econometric modelling was the focus of Ofwat’s assessment, separate DEA models were carried out for water distribution, water treatment, and sewerage services, with the models based on the econometric models. As the models included multiple output measures (e.g., properties, length of mains, and water delivered for the water distribution model; population, length of sewers, area served, and pumping capacity for sewerage network) slightly greater heterogeneity was accounted for (although constant returns to scale was also assumed to be consistent with the estimated econometric models at the time). See Thanassoulis [141], Thanassoulis [142], and Thanassoulis [144]. With regards to separating noise and inefficiency, although not a direct solution, DEA is a more rigorous approach to establishing an efficiency frontier than using an upper quartile adjustment to OLS or RE, and can be extended to account for noise (see Simar and Wilson [131], and Simar and Wilson [132]).

Forecasting Efficient Costs: The Consistency of Catch-Up, Frontier Shift and Input Price Inflation Assumptions The potential to catch-up to the frontier (technical efficiency), the potential for further frontier shift (technical change) improvements, and the impact of input price inflation are interrelated. Clearly catch-up and frontier shift are closely related to

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each other as the former is measured relative to the position of the frontier. Equally, the input price inflation that a sector experiences is affected by the productivity performance of the sectors related to those inputs. Productivity growth typically drives increases in real wages such that, in the long run, economy-wide real wage growth is broadly in line with economy-wide labor productivity growth (see, for example, the International Labour Organisation [44], p. 10). The expectations for changes in all these elements are clearly specific to the industry in question. For example, the digital sector clearly has greater rates of technological progress than the water sector. As such, it is important that regulators ensure consistency between the catch-up, frontier shift, and input price inflation assumptions. As discussed above, DCCA’s frontier-shift challenge was based on separate analysis to its relative efficiency analysis, Ofwat’s frontier-shift challenge was also based on separate analysis to its relative efficiency analysis and separate to its analysis of input price inflation. These approaches may, therefore, result in inconsistencies. Indeed, with regards to Ofwat’s approach, some companies argued that an inconsistency had occurred with Ofwat’s initial assumptions of a frontier shift of 1.5% p.a. and no input price inflation allowance. Ofwat ([95], p. 139) stated “Economic Insight and Oxera argue that it is inconsistent to assume that there will be no real wage growth yet still apply a frontier shift which implies significant growth in productivity, as wage rates and labour productivity will be linked.” In its draft determinations, Ofwat ([95], p. 129) made some allowance of input price inflation, stating “In contrast to the initial assessment, we are including a real price effect adjustment for real wages to reflect improvements in labour productivity.” This is one solution to improving consistency – namely, to reassess the outcomes from separate analyses and try to ensure consistency in those assumptions by making some adjustments. Though such an approach may not guarantee consistency. Another approach would be to model all components (technical efficiency, technical change, and input price inflation) simultaneously. This could be achieved using SFA and a subsequent decomposition of the components – see, for example, Kumbhakar et al. ([57], Chap. 11), Ashton [7], and Saal et al. [128]. (Clearly, to also be able to decompose the impact of input price inflation, input prices would need to be included in the models, i.e., a cost function would first need to be modelled). The same decomposition can be achieved using DEA and Malmquist Indices – see Maniadakis and Thanassoulis [58]. In this case, the critical assumption is that past performance in each of these elements is a good predictor for future performance. As such, further adjustments may still be required. Such a decomposition for SFA requires careful consideration of the pattern of efficiency over time. There are quite a few approaches that can be taken. On this point, we note that the models used by ARERA are quite restrictive in how they allow inefficiency to change over time. For example, Pitt and Lee [122] and Battese and Coelli [8] have time invariant inefficiency, i.e., these models assume inefficiency does not change over time, while Battese and Coelli [9] assume a fixed (exponential)

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shape to inefficiency over time. Less restrictive models, such as Kumbhakar and Heshmati [54], could instead be used.

International Comparisons An additional complexity for UR and CRU is the necessity to compare NI Water and Irish Water to operators outside the country. This makes achieving like-for-like comparisons more complicated. The first key issue is data comparability. In the case of NI Water, the company was required to submit similar data as was previously the case in England and Wales via Ofwat’s use of, the now disused, June Returns [108]. On this basis, most of NI Water’s dataset was comparable to the comparator set used by the UR, save for minor differences in pollution categorization regime, for example. There is also a need to convert to the same currency, if necessary. In CRU’s case, cost for the England and Wales companies were converted into euros using the OECD’s Purchasing Power Parities (PPP) for private consumption. Clearly, there are implementation issues and uncertainties around conversion rates so some sensitivity testing may be required. The second key issue is the ability to control for differences between companies. This is more complicated in an international setting. This can be accounted for in a number of ways, including: • Accounting for differences through the variables chosen in the model, ensuring, in particular, that the key differentiating characteristics between countries are accounted for. CRU [20] stated that it considered Irish Water’s greater length of water network per connection but noted that models developed by other regulators tended to show that the number of connections, rather than network length, was the main cost driver. • Accounting for potential differences in technology (that is, differences in the engineering process that companies use, or could use, as well as the physical and policy environments in which the companies operate). This implies that it is important to test whether the estimated (DEA or SFA) frontier differs by country. • Making pre- or post-modelling adjustments to account for atypical characteristics. For example, the UR made some allowances through the introduction of negative special cost factors after conducting cost benchmarking, such as a regional wage adjustment for wage differentials between Ireland and England and Wales. Similarly, CRU [20] accounted for Irish Water’s higher wages costs by scaling costs up or down before conducting cost benchmarking. With any international benchmarking exercise, a number of issues need to be accounted for. For the interested reader, Oxera [113] summarizes a number of issues with international benchmarking and how they can be mitigated in relation to ORR’s benchmarking of National Rail with other European network operators [133]. Similarly, Oxera [121] summarizes a number of issues with international

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benchmarking and how they can be mitigated in relation to CEER/Sumicsid’s benchmarking of European TSOs [139]. Many of the issues discussed in these papers are also relevant to international benchmarking in the water sector.

Concluding Comments Although economic regulation and benchmarking in the water sector is not widespread across Europe, some cost benchmarking is undertaken by a number of European regulators. With the exception of those regulators that have benchmarked the regulated water company with companies in another jurisdiction (namely, England and Wales), the approaches taken have varied considerably across regulators. This variation has been in, among other things: the form of the benchmarking approach taken (e.g., OLS, RE, SFA, or DEA); the costs benchmarked (e.g., OPEX, BOTEX, TOTEX); the outputs and other cost drivers used in the models; and how future efficient cost are ultimately established. That is, there has been little consensus in a number of fundamental aspects of the cost benchmarking exercise. While regulators need to be pragmatic and take into account local issues, including differences in the regulatory framework, the variation in approaches is perhaps somewhat surprising. As such, despite cost benchmarking having been undertaken in the water sector for a number of decades, there still remains a number of areas for future research and development.

Cross-References  Application of Production Economics in the Electricity Distribution Sector  Empirical Analysis of Production Economics: Applications to Banking

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124. Saal D (2018) Comments on CEPA’s methodological approach in its PR19 econometric benchmarking models for Ofwat, May 125. Saal D, Nieswand M (2019) A review of Ofwat’s January 2019 wholesale water and wastewater Botex cost assessment modelling for PR19, March 126. Saal DS, Parker D (2000) The impact of privatization and regulation on the water and sewerage industry in England and Wales: a translog cost function model. Manag Decis Econ 21(6):253–268 127. Saal D, Parker D, Weyman-Jones T (2007) Determining the contribution of technical, efficiency and scale change to productivity growth in the privatized English and welsh water and sewerage industry: 1985–2000. J Prod Anal 28:127–139 128. Saal D, Ferrari A, Nieswand M (2017) Independent review of Anglian Water’s preliminary regulatory cost modelling for PR2019, September 129. Scottish Water (2012) Submission to Scottish parliament, September 130. Simar L, Wilson PW (2000) A general methodology for bootstrapping in non-parametric frontier models. J Appl Stat 27(6):779–802 131. Simar L, Wilson PW (2011) Inference by the m out of n bootstrap in nonparametric frontier models. J Prod Anal 36:33–53 132. Smith A (2008) International benchmarking of Network Rail’s maintenance and renewal costs: an econometric study based on the LICB dataset (1996–2006): report for the Office of Rail Regulator, October 133. South West Water Limited (2018) Cost model consultation response, May 134. Stehrer R, Bykova A, Jäger K, Reiter O, Schwarzhappel M (2019) Industry level growth and productivity data with special focus on intangible assets, October. https://euklems.eu/ 135. Stewart M (1993a) Ofwat research paper number 2: modelling water costs 1992–93: further research into the impact of operating conditions on company costs: main report, December 136. Stewart M (1993b) Ofwat research paper number 4: modelling sewage treatment costs 1992– 93: research into the impact of operating conditions on the costs of sewage treatment: main report, December 137. Stewart M (1994) Ofwat research paper number 3: modelling sewerage costs 1992–93: research into the impact of operating conditions on the costs of the sewerage network: main report, January 138. Sumicsid (2019) Pan-European cost-efficiency benchmark for electricity transmission system operators main report, July 139. Swiss Economics, Sumicsid and IAEW (2019) Efficiency comparison of Electricity Distribution System Operators for the third regulatory period (EVS3). Produced for Bundesnetzagentu, April 140. Thanassoulis E (2000a) DEA and its use in the regulation of water companies. Eur J Oper Res 127(1):1–13 141. Thanassoulis E (2000b) The use of data envelopment analysis in the regulation of UK water utilities: water distribution. Eur J Oper Res 126(2):436–453 142. Thanassoulis E (2001) Introduction to the theory and application of data envelopment analysis. Kluwer, Dordrecht 143. Thanassoulis E (2002) Comparative performance measurement in regulation: the case of English and Welsh sewerage services. J Oper Res Soc 53:292–302 144. UK Water Industry Research Limited (UKWIR) (2002) Capital maintenance planning: a common framework 145. Utility Regulator (UR) (2014) Water & sewerage services, price control 2015–21. Final determination – main report, December 146. Utility Regulator (UR) (2018) Price control for water and sewerage services 2021–2027: our overall approach, June 147. Utility Regulator (UR) (2019a) PC21 Opex ‘minded to’ methodology, March 148. Utility Regulator (UR) (2019b) PC21 Capex ‘minded to’ methodology, March 149. Wang H-J, Ho C-W (2010) Estimating fixed-effect panel stochastic frontier models by model transformation. J Econ 157:286–296

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150. Water Act (1989) (c.15) UK government 151. Water Industry Act (1991) (c.15) UK government 152. Water Industry Commission for Scotland (WICS) (2001) Strategic review of Charges 2002– 06, November 153. Water Industry Commission for Scotland (WICS) (2004a) Our work in regulating the Scottish water industry: the calculation of prices, September 154. Water Industry Commission for Scotland (WICS) (2004b) Our work in regulating the Scottish water industry: the scope for operating cost efficiency, October 155. Water Industry Commission for Scotland (WICS) (2013) Strategic review of charges 2015– 21: innovation and choice, May 156. Water Industry Commission for Scotland (WICS) (ca. 2014) Staff paper 5: public private partnership costs. https://www.watercommission.co.uk/UserFiles/Documents/Staff%20paper %205.pdf 157. Water Industry Commission for Scotland (WICS) (2018) Strategic review of charges 2021– 27: methodology refinements and clarifications, November 158. Water Industry Commission for Scotland (WICS) (2019) What can regulation achieve, August 159. Water Industry Commission for Scotland (WICS) (2020) Prospects for prices. Strategic review of charges 2021–27. Final decision paper, February 160. Water Services Act (2017) Government of Ireland 161. Yorkshire Water (2019) Cost efficiency – Yorkshire water draft determination representation REDACTED 162. Yorkshire Water (2020) PR19 Redetermination Yorkshire water services: statement of case, April

The Economics of Sports

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Joshua Congdon-Hohman and Victor Matheson

Contents Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Team Sports Versus Individualistic Sports . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . What Is a Sports Team’s Objective Function? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Sports Economics and the Production Function for Attendance and Revenue Success (Off-Field Success) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Stadiums and Sports Infrastructure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Producing Attendance and Revenue: Teams . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Producing Attendance and Revenue: Leagues . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Sports Economics and the Production Function for On-Field Success . . . . . . . . . . . . . . . . . Measuring Monopsony Power . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Management and Strategic Efficiency . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Optimal Levels and Distribution of Inputs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Worker Effort and Compensation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Abstract

Spectator sports present a unique area of study for economists both because of their global popularity and because they require a high level of cooperation among rival firms to produce the product. This chapter explores how leagues allocate labor among teams to maximize member revenues. It also examines whether individual teams try to maximize on-field success (in terms of wins) or on-field success (in terms of revenues or profits) and how expenditures on talent and sports infrastructure contribute to these goals.

J. Congdon-Hohman · V. Matheson () College of the Holy Cross, Worcester, MA, USA e-mail: [email protected]; [email protected] © Springer Nature Singapore Pte Ltd. 2022 S. C. Ray et al. (eds.), Handbook of Production Economics, https://doi.org/10.1007/978-981-10-3455-8_43

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Keywords

Sports · Marginal revenue product · Labor economics · Attendance · Stadiums

Introduction Spectator sports represent a fairly unique area of study for economists for a wide variety of reasons. First, sports command a degree of attention within popular culture that far exceeds its financial contributions. Even before COVID-19 wreaked havoc on the bottom lines of sports promoters across the world, the total revenue generated by the five largest professional sports leagues in the United States (the National Football League (NFL), Major League Baseball (MLB), the National Basketball Association (NBA), the National Hockey League (NHL), and Major League Soccer (MLS) totaled only around $40 billion dollars (US) in 2018. Adding in the “Big 5” European soccer leagues (Germany’s Bundesliga, the UK’s Premier League, Italy’s Serie A, Spain’s La Liga, and France’s Ligue 1) raises the total by another roughly $17 billion. Including in Formula One auto racing, Nippon Professional Baseball, the Australian Football League (AFL), the Indian Premier League (IPL – Cricket), the Professional Golfers Association (PGA), the International Olympic Committee (IOC), and FIFA (soccer) brings the worldwide total average annual revenue to around $65 billion, a combined figure that would not even crack the top 100 largest firms in the world. However, these sports dominate media coverage around the world and can capture the attention of entire countries. Indeed, one of the most visible and dramatic signs that the coronavirus had moved from “potential threat” to “critical global pandemic” was the empty stadiums and arenas across the globe. Next, while sports remain a relatively small segment of the economy, they have experienced rapid growth over the past century. Technology in the form of television and other forms of media distribution has aided the growth of sports. In most highlevel leagues the amount of revenue generated by media rights exceeds the revenue generated by live, in-person attendance. In addition, it has been generally observed, especially in more recent studies, that attendance at games is a luxury good. As incomes grow, consumers have more disposable income and more free time to spend on entertainment [102]. Third, in order to produce its product, sports require a level of cooperation among erstwhile competitors that is uncommon among other industries. Thus, while firms in traditional industries such as consumer electronics, air travel, or automobiles would normally try to drive their competitors out of business in order to capture the entire market for themselves, in sports, a team that successfully drove each of its important rivals from the market would find itself unable to produce its product at all. Indeed, this distinction was at the heart of Walter Neale’s [85] seminal work on sports economics. Neale dubbed this phenomenon the “Louis-Schmeling Paradox” after the prominent heavy-weight boxers Joe Louis and Max Schmeling. Neale notes

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that without a prominent opponent like Max Schmeling, Louis would have “no one to fight and therefore no income.” In few other sectors of the economy is a firm’s production function so closely dependent on the existence of a competing firm in the same industry. Finally, sports provide researchers with a vast trove of publicly available data that is often not found in other industries. For example, perhaps more than any other industry, scholars can examine the contributions of individual workers to the success of the firm as a whole since the sports pages routinely publish statistics on specific players. In addition, sports provide data at a level of frequency that is rare in other industries. While national accounting practices may require firms to release financial information only on an annual or, perhaps, quarterly basis, sports performance data is typically available on a game-by-game basis that may be as frequent as daily matches or even individual events within a competition. One should keep in mind, however, that while sports benefit from a wealth of easily accessible productivity data, analysis of the industry has generally been hampered by a lack of profit or revenue data. Many sports teams, especially in North America, are privately held businesses and are therefore not required to release audited financial data unlike publicly held firms that dominate many other industries. Given the unique nature of sports, beginning in the 1950s, academic economists began to devote serious attention to the industry. In the late 1990s, industry practitioners began to adopt the quantitative methodology first proposed by economists, leading to a statistics-based revolution that has swept through essentially every sport and league. Today nearly every major sports team employs experts in quantitative analysis in order to optimize team performance. This chapter examines this on-going revolution and how economists measure production and productivity in spectator sports.

Team Sports Versus Individualistic Sports While individualistic sports such as tennis, various combatant sports such as wrestling, mixed martial arts, or boxing, golf, athletics, and horse and auto racing do play an important role in the world of sports, the general interest in as well as the revenue generated by these sports is small relative to that generated by team sports. Furthermore, the economic analysis of individualistic sports is generally less complex, or at least less unique, than that of team sports. In addition, there are fewer cross-country differences in the organization of individual sports as opposed to team sports. For example, marathons and golf tournaments are organized similarly in most parts of the world while sports leagues have vastly different organizational principles in North America as opposed to the major leagues in Europe [112]. Therefore, the primary focus of this chapter will be on production in team sports; however, it is useful to at least include a brief overview of individual sports here. The majority of the academic work on individual sports has typically focused on the incentive structures that lead to the maximum effort on the part of the participants. The literature includes theoretical models of effort [114] which

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generally suggest a payoff function for individual contestants that is increasing in prize money, decreasing in effort, and affected by the number and distribution of the participants. More commonly, given the fact that sporting contests provide easy access to data, the results of individual sports contests can also be used to test the hypotheses of sports specific tournament theory or more general labor economics tournament models such as that of Lazear and Rosen [70]. Empirical analyses of pay and performance can be found for tennis [93], golf [11, 29], distance running [40, 77], weight lifting [87], and horse racing [32, 49].

What Is a Sports Team’s Objective Function? In most industries, the objective function of a firm’s manager is assumed to simply be the maximization of profit. This is not necessarily the assumption made in sports. In sports, it is common to assume that sports teams could have one of two possible objective functions. The first possibility is that sports teams are just like any other firm and operate like profit-maximizers. The other option is that teams are simply trying to win as many games as possible subject to being able to stay in business. This option is typically referred to as “win-maximization” or “utilitymaximization.” The concept of teams as win-maximizers was first developed, apparently independently, by both Sloane [105] and El-Hodiri and Quirk [31]. It is plausible to believe that sports teams may choose to operate as utilitymaximizers rather than profit-maximizers for at least two reasons. First, many professional sports teams, including many of the biggest soccer teams in the world, such as Barcelona and Real Madrid, operate as clubs where the owners of the team are the fans themselves. Fans purchase memberships in these clubs not to make money on their investment but to qualify for the right to purchase tickets to games and to have a say in electing the team’s management. Indeed, it would be very unusual to believe that a fan-based ownership group would act in order to earn money rather than win games. Second, ownership of a sports team may bring nonmonetary benefits to the owner in terms of fame or popularity or can be seen as a form of entertainment or consumption akin to a hobby. For example, on-again, off-again Italian Prime Minister Silvio Berlusconi initially came to prominence at least in part as the owner of AC Milan, the most popular and successful soccer team in Italy. Similarly, Russian oligarch Roman Abramovich, the owner of the Chelsea soccer club of the English Premier League, has happily acknowledged that he is in the sports team ownership business not for the profits, but instead for the glory. The concept of win-maximization typically only applies to the labor or talent acquisition side of a sports team. A win-maximizing team will attempt to acquire players beyond the point at which the marginal revenue derived from the additional talent will cover the marginal cost of paying for that talent. However, both winmaximizing and profit-maximizing teams are likely to behave similarly on the revenue side of the ledger. Because the vast majority of costs in operating sports team are fixed in nature, at least over a given season, once a specific level of

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talent is chosen, profit-maximizers will largely want to engage in pricing strategies that maximize revenues. But win-maximizers will also want to engage in pricing strategies for tickets, media, concessions, and sponsorships (as well as appropriate marketing and promotions) that maximize revenue as they want to generate funds in order to be able to fund the highest possible level of talent acquisition while still remaining financially solvent. Finally, it is important to note that there are important differences between the objective function of a league and the objective functions of the individual teams that make up a league. Because competitive sports are a zero-sum game, for every team that generates a win there must be a corresponding team that takes a loss, it is impossible for a league to operate in a win-maximizing fashion. It is also the case that the behavior that maximizes profit for the league as whole may not maximize profit for specific individual teams within that league. Of course, even teams that are profit-maximizers are also concerned with the production of on-field success as it is an empirical fact that team quality is strongly correlated with demand and therefore revenue generated. The remainder of this chapter examines how teams and leagues generate revenue and attendance and then examines how teams generate on-field success.

Sports Economics and the Production Function for Attendance and Revenue Success (Off-Field Success) Stadiums and Sports Infrastructure Most of the literature on both on-field and off-field sports team and league success focuses on the labor side of the production function, or the athletes themselves. However, capital, specifically stadium and arena infrastructure, also plays an important role in generating revenue and potentially generating on-field success as well (primarily through the generation of additional revenue that can be used to purchase additional labor talent). The bulk of the research in stadium economics focuses on the public finance aspects of stadiums and arena in an attempt to answer the question as to whether these facilities represent a wise investment of taxpayer dollars. Overwhelmingly, independent research on sports facilities from the very earliest studies in the late 1980s and early 1990s [5, 6] throughout the US stadium boom in the 1990s and 2000s [17], and up to the current day [52, 78] finds that new stadiums have little or no net economic benefits to the cities where they are constructed, although occasionally significant neighborhood effects can be uncovered. However, even if stadiums are not winners from a public finance standpoint, it is quite clear that new stadiums can result in substantial benefits to the teams playing in them. These benefits come in three forms. First, new stadiums increase attendance and total ticket sales. Second, new stadiums allow teams to sell tickets at a higher

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price. Finally, new stadiums increase the ability of teams to sell complementary goods and services like concessions, premium seating, merchandise, and parking. The higher attendances experienced by sports teams following a stadium construction project are not generally the result of increasing stadium capacity. In fact, most Major League Baseball and Major League Soccer stadiums built during the most recent stadium boom in the United States since 1990 were smaller than the stadiums they replaced, and most stadiums in the National Football League, National Basketball League, and National Hockey League constructed during the same time periods were roughly equal in size to their predecessors [72]. Instead, most researchers chalk up the increase in attendance to a “novelty effect” or “honeymoon effect,” and the interest among fans in seeing a game in the new facility. This effect appears to persist for between five and ten seasons in most sports including baseball, basketball, football [18], hockey [106], and minor league baseball [2]. In addition to the novelty effect of new stadiums, sports facilities in the past three decades have tended to substitute quantity for quality, improving the game experience for fans and thereby also increasing fans’ willingness to pay for the experience. Because the “best seats in the house” are by their very nature limited in supply, courtside seats also serve as a positional good or an example of Veblen’s [115] “conspicuous consumption.” Newer sports facilities have generally increased luxury amenities in order to cater to the very highest spending spectators by emphasizing luxury boxes, club seats, and exclusive lounge areas [72]. Finally, it is clear that teams produce a multifaceted good to live audiences including both on-field entertainment as well as other in-stadium offerings such as concessions, merchandise, and parking [19]. In earlier days, food and beverage options at stadiums were quite limited. In fact, in the late 1800s as baseball began to emerge as “America’s pastime,” one major difference between the National League and its rival league the American Association was the sale of alcohol on the grounds. The practice was prohibited at National League stadiums while allowed at American Association parks earning the latter league the moniker “the Beer and Whiskey League” [86]. Modern stadiums dedicate significantly more space and attention to the sale of ancillary products which, along with the expansion of luxury amenities, explains why the amount of capital dedicated to the constructions of stadiums has increased so rapidly in recent years.

Producing Attendance and Revenue: Teams The standard production model for a sports team is that a team chooses a level of spending on talent in order to either maximize profit or maximize wins subject to a budget constraint. It is assumed, and also verified empirically (see Coates and Humphreys [19] and García and Rodríguez [44] among many others), that all other things equal, revenues rise with on-field success. On-field success rises with spending on talent. The return on investment in talent, however, eventually experiences diminishing marginal returns for two reasons. First, if a team becomes

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too good relative to its competition, fan interest may wane. The “uncertainty of outcome hypothesis,” a concept dating back to Rottenberg’s [94] seminal paper on sports economics, states that fan interest, and therefore team revenue, is dependent on having a relatively close contest. Thus, at a certain point the purchase of additional talent by a team does not generate additional revenue as the benefits from increased on-field performance are balanced out by the harm associated with lower uncertainty of outcome. In addition, as also noted by Rottenberg [94], because every sport limits the number of players that can participate in a contest, purchasing additional talent eventually results in lower marginal product with respect to on-field performance. There is only so much playing time, so eventually additional talent will just end up sitting on the bench. Thus, a team that is a profit maximizer will purchase talent up to the point where the marginal revenue they receive from the additional on-field success that that talent produces, or the marginal revenue product (MRP), is equal to the marginal cost of that talent. A win-maximizer will purchase talent up to the point where profit is zero. Obviously, many demand-side factors also go into determining attendance including the number of buyers (i.e., the size of the market), weather, rivalries, promotional events, the availability of other forms of entertainment, etc., but ultimately, the primary production decision by the team is the level of talent to use as an input to the production process. It is important to note that because fans in different markets will not react in the same way to additional purchases of talent, either due to differences in market size or other differences in the local market, there is no reason to believe that teams will all decide to purchase the same level of talent. However, due to the uncertainty of outcome hypothesis, it is likely that teams will wish to form leagues with other teams of similar levels of overall talent [28].

Producing Attendance and Revenue: Leagues The sporting industry is quite unique in that each individual firm requires the cooperation of an erstwhile competitor in order to produce its product. Thus, individual teams from the very beginning of organized spectator sports have joined together to form leagues. There are at least three reasons why the formation of leagues provides a production advantage to their member teams. The first is that a league provides a uniform set of playing rules for competition. After all, the game of football looks much different if you play by the Rugby rules where extreme physical contact and the use of hands is allowed as opposed to playing with Association rules where handling the ball is not allowed and the level of physical contact is more strictly controlled. (Indeed, it is from the term As-“soc”-iation rules that the term “soccer” derives its name.) In many ways this is no different than industry groups that set uniform technical standards for things like electrical voltages or standardized sizing. Leagues also form to allow the members to act as a cartel in order to exert monopoly (or monopsony) power on the sale of their product or the purchase of

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inputs. In terms of sale of the product, league cartels in closed leagues like those in the United States, league typically attempt to carve out geographical monopolies for each franchise in the league leading to only relatively rare cases where more than one team serves a particular metropolitan area. In so-called open leagues as are seen in Europe (also known as promotion-relegation leagues), the membership of the leagues changes year to year as the worst performing teams are relegated to lower level leagues and the best performing teams from lower divisions are promoted to higher level leagues. Since open leagues have far less control over which teams belong to the league at any given time and since individual teams in open market leagues do not have the ability to exclude competitors from large markets, it is common to see multiple teams in large European metro areas like London or Madrid [72]. Of course, leagues have long acted as cartels in terms of the purchase of labor. Essentially every league has had at some point in their history some version of the reserve clause, a rule that binds a particular player to a specific team and prevents other teams in the league from bidding for that player’s services. Beginning in the 1970s, players’ unions in various sports began to successfully fight back against league labor cartels winning the fight to at least some form of free agency, that is, the right to freely contract with any team in the league. That being said, many sports leagues still exert significant monopsony power on their players. The most notable example is the National Collegiate Athletic Association (NCAA) which has imposed amateurism on the athletes of its member institutions despite these institutions generating combined revenues from athletics in the billions of dollars [59]. The final reason for league formation is the most unique to sports. In order to provide an interesting product on the field or court that will be in high demand, the two opponents must be of relatively similar quality [34], a factor known as competitive balance. Competitive balance can mean many different things. It can describe whether a particular season or contest is close in nature, known as “intraseason or within season competitive balance.” And it can also refer to whether the same teams tend to win year after year even if the competition within any given season is fairly close. This is known as “inter-season or between season competitive balance.” Significant attention has been paid in the academic literature as to how one measures different types of competitive balance [51]. The next step in determining an optimal league production function is to estimate what the optimal level of competitive balance should be in the league in order to maximize the joint profits of the league members. While this is a basic element of the many theoretical models of sports league formation such as El-Hodiri and Quirk [31], Fort and Quirk [34], or Dietl, Grossmann, and Lang [27], empirically measuring the effect competitive balance on attendance or league revenues has been notoriously difficult (see, e.g., Schmidt and Berri [95] or Forrest and Simmons [33]). Indeed, it is even difficult (or at least it is rare in the literature) to develop a model for consumer preferences that leads to fans actually wanting their team to lose on occasion Humphreys and Miceli [53]. Most studies suggest, however, that extremely unbalanced competition, at least, is clearly harmful to overall league economic performance [51].

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The final step in designing an optimal league production function is to implement league policies that result in the desired level of competitive balance that produces the maximum profit or attendance. Potential league policies might include roster limits, restrictions on player allocation through actions like reverse order drafts or various forms of the reserve clause, salary caps or floors, and revenue sharing. It is important to note that both theoretical models and the observed reality suggests that revenue sharing alone will be insufficient to promote the desired level of competitive balance unless policies are also put into place that provide incentives for weaker teams to actually spend any revenue transfers [61]. Furthermore, Rottenberg himself stated in his seminal work that the reserve clause that restricted the freedom of players’ movements in the name of competitive balance would also be an ineffective tool to promote this goal as the best players would ultimately end up with the teams that valued their services the most. Thus, reserve clause contributed little to competitive balance and only served to reduce players’ bargaining power and hence their earnings. This concept has been dubbed “Rottenberg’s Invariance Principle.”

Sports Economics and the Production Function for On-Field Success The detailed production data available in sports allows for a number of applications examining the production process, optimization, and the value of various types of inputs. By using the detailed performance measures for athletes in various sports and examining team success as the desired output of production, researchers have been able to test economic theories in ways that the production process in other industries are not as well suited. Specifically, academic studies have tested theories regarding the value of monopsony power in labor markets, the impact of management efficiency, input optimization in regard to equality of labor skill and compensation within an organization, and testing for evidence of shirking based on various forms of compensation. In most cases, the first step in the analysis deals with estimating the production process itself, either through a production function or a production frontier. The initial attempts to identify a team’s production process came in professional baseball. In addition to being the most popular team sport in America for almost a century, baseball has unique characteristics that allow individual worker contributions to the production process to be identified. Specifically, a baseball game is a combination of predominantly individual activities. A single defensive player (the pitcher) is responsible for initiating a play by throwing a baseball at a specific speed, location and with an intended amount of movement as it travels through the air. A single offensive player (the batter) attempts to hit the ball in a way that will allow him to gain as many bases as possible. Teammates are part of each play as either offensive players who have reached bases in previous at bats and will attempt to advance on a particular hit ball, or as defensive players who will attempt to stop the batter and other offensive players from progressing after a ball has been hit.

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Measuring Monopsony Power Economists began to empirically examine the wage structure in baseball in the mid1970s, but the assertion that sports leagues had monopsony power in the labor market for athletes was identified much earlier. Rottenberg [94] first identified and codified the idea that Major League Baseball could be characterized as the combination of a number of different inputs (players) and that the owners as a collective group had market power as the only employer of high skilled baseball players. Specifically, US antitrust law carves a formal legal exemption for the professional baseball league. Davenport [23] also cited monopsonistic power when discussing the faster growth in baseball team revenues than player wages. Due to this monopsonistic power, he asserted that players must be paid a wage between their marginal revenue product and the wage they would earn in a non-baseball job. Gerald Scully [99] was the first to attempt to statistically measure the degree to which Major League Baseball teams used their monopsony power to pay players wages that were below their marginal revenue product (MRP). Scully first set out to predict revenue as it related to a team’s on-field success, controlling for market and stadium characteristics. Once Scully had an estimate of the value of winning baseball games, he turned to estimating the contribution of each team’s statistical output to the success of the team, which then were used to value the contribution of each player’s individual statistical accomplishments. With these values, Scully estimated each individual player’s contribution to a team’s predicted revenue, which then could be compared to the wages that the players were paid by the owners. Ultimately, Scully found that the most skilled baseball players were only paid 10– 20% of their net MRP. This high degree of monopsonistic power reflects the labor market structure under the reserve clause. Individual players were claimed by teams when they entered the league and were not allowed to negotiate a contract with any other team at any point in their career without the permission of the team that owned the exclusive rights to that player. After Scully’s seminal work, others applied variations on his model to evaluate monopsonist power as the structure of the labor market in baseball transformed beginning in the 1970s. In that decade, players in Major League Baseball gained the right to appeal team salary offers to an arbitrator after a few years in the league (“arbitration”) and the right to freely negotiate with all league teams after a longer period (“free agency”). Sommers and Quinton [109] examined the salary outcomes of the first group of players to negotiate contracts through free agency. As a competitive labor market model would suggest, they found that players were able to negotiate a wage near their estimated MRP. Other players who were still restricted were found to continue to be underpaid. Scully [100] revisited his model to examine the outcomes of free agent contracts in the mid-1980s. He found that free agents were paid only 28% of their MRP. Such a low value was likely due to collusion among owners to not compete for free agent players in this period. Zimbalist [120] found that the average free agent was actually slightly overpaid relative to their estimated MRP using a modified approach to Scully’s analysis, but players earn significantly less than their estimated MRP prior to meeting the

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tenure requirements for free agency. Rather than looking across class of players based on tenure, Rockerbie [92] focused on different outcomes within the class of free agent players. He found that those players who earn the highest contracts in free agency tend to be paid more than their estimated MRPs while all other free agents earn at or below their estimated MRPs. Humphreys and Pyun [54] used a modified-Scully approach to examine the evolution of monopsony power by examining the monopsony exploitation ratios (MER) across changes to the collective bargaining agreement between baseball owners and the player’s union. They found free agents were getting paid salaries closer to their predicted MRP with each progressive agreement, but other classes of players with less tenure did not see similar gains. The Scully approach to measuring a player’s MRP and evaluating the level of monopsonistic power in baseball has not been without its detractors. As exemplified by Bradbury [10] and Krautmann [64], the disagreement is focused primarily on the estimation of the value of players, not on the production function estimates. The Scully approach estimates the values of team success based on approximations of revenue streams for teams. The critique leveled is that rather than using a poor approximation of revenue and the value of a win, why not rely on the wage outcomes of free agents as the MRP since the market for these players should be competitive and therefore the wage offer reveals an owner’s estimation of the player’s MRP. The production function estimate can then be used to identify the value of individual production. A number of papers have been written comparing the traditional estimates of MRP to various alternative techniques. Fort and Quirk [35] found that estimates using the Scully approach overstate the size of the monopsonistic salary suppression. Krautmann [63] compared traditional MRP estimates and those based on the free market revelations of MRP through free agent contracts. He concluded that the traditional methods overvalued players’ MRPs. Based on these alternative methods, Krautmann concludes that only those players who are not eligible for arbitration are underpaid (at only 25% of their estimated MRP) while those players who are eligible for arbitration are rewarded in the same way as free agents, which is assumed to equal their MRP. Krautmann, Gustafson, and Hadley [67] find similar results, though they find slight differences based on a player’s race and identify that the underpayment of players only recoups a portion of the authors’ estimated player development costs. Fort, Lee, and Oh [37] found that the accuracy of traditional model estimates of MRP and therefore monopsonistic power are related to the markets in which teams play. Specifically, they found that MRP is overestimated for teams in small markets (those in cities with smaller populations), while it is underestimated for teams in larger markets. Monopsonistic power in the labor market has also been tested for other North American sports leagues that share a similar labor market structure to Major League Baseball. Scott, Long, and Somppi [98] found that players in the National Basketball Association (NBA) are paid at levels below their MRP when they first enter the league and they are restricted to playing for only one team. Once professional basketball players reach the requisite tenure, they are free to negotiate

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with all teams and the authors find that their compensation rises to the level of their MRP. A few years later, Atkinson, Stanley, and Tschirhart [4] extended this type of analysis to football and the National Football League (NFL). The authors identified an overpayment of players beyond what simple profit maximization might suggest, which they believe suggests that owners’ maximization problem must also include personal, nonmonetary-based utility. Brown [12] also examined the NFL to identify the degree to which monopsonistic exploitation in college may be offset by professional earnings. Though college football players receive limited, education-based compensation, the author found sizeable values for their MRP after controlling for different levels of revenue-generating abilities through quantile regression methods. Brown estimated that only 33–38% of college players earn NFL incomes that offset their non-realized MRP during college play. Krautmann, von Allmen, and Berri [68] directly compared the monopsonistic power across the three largest North American sports leagues: the NFL, MLB, and the National Hockey League (NHL, ice hockey). They found that owners in all three leagues exercised monopsony power during the period when player movement between teams is limited. When players are most restricted immediately after entering each league, the authors find that baseball players make the lowest percentage of their MRP, but basketball players provide the largest value of surplus for owners.

Management and Strategic Efficiency Often missing in the models described above is the role of team management and strategy in the estimation of the production function. The contribution of management and production methods is an important question in all firms’ production as owners look to get the most output possible from given levels of labor and capital. In sports, the quantifiable nature of both inputs and outputs makes the calculation of production efficiency more testable than it is in most industries. Though baseball has the most easily discernible measures of individual labor inputs, the examination of efficiency is much more reliant on aggregate inputs and therefore is more easily examined in the context of many different sports. Managerial efficiency was first explicitly quantified in Zak, Huang, and Siegfried [118] when estimating a production function for a small number of teams in the NBA. The authors used the measurable activities in basketball including shooting percentages, rebounds, fouls, steals, and blocks to estimate the production function. Teams were then assigned a measure of production efficiency based on how close the teams’ production is to its frontier. Later, Zech [119] included a measure of management that was insignificant in his estimation of the baseball production function. Porter and Scully [89] examined managerial efficiency and a manager’s MRP, finding that managers contribute significantly to the production process and that the MRP of the best managers was similar to the MRP of star players. Kahn [58] found similar evidence that good baseball managers increase win production, and

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also that good management can improve individual player performance. Scully [101] identified a strong link between estimated managerial efficiency and the likelihood of continuing in their position, supporting the earlier work of Chapman and Southwick [16] that suggested variation in manager productivity is based on the quality of the job match, with highly productive and mutually beneficial matches likely to continue. More recently, Volz [116] used data envelopment analysis techniques to show that a baseball team’s production efficiency was more important than winning percentage in predicting the retention of managers. In addition to the work of Zak, Huang, and Siegfried [118] and the NBA, other researchers looked to examine managerial efficiency in sports other than baseball relatively early. Carmichael and Thomas [13] examined the production inefficiency in English rugby by estimating the production function for Rugby Football League teams and identifying teams’ potential given their available resources. Dawson and Dobson [24] extended this type of analysis to measure managerial efficiency in English soccer. They examined which traits and experiences were associated with managers who were able to have success closer to a team’s production frontier given the team’s quality. They found that prior experience and familiarity with the specific club through prior affiliation club lead to the highest efficiency gains. Frick and Simmons [41] examined managerial quality in the German Bundesliga. They found that quality coaching improved team performance by limiting technical inefficiency and that coaches were paid below their estimated MRP. Using ice hockey, Kahane [56] estimated the stochastic production frontier to identify inefficiencies in the NHL. He found that inefficiencies were associated with coaching ability, as well as team ownership, management experience, and the share of players from specific areas. Fort, Lee, and Berri [36] used a similar approach to estimate the technical inefficiency of basketball coaches in the NBA. Like Scully [101], they found that job retention was strongly linked to a coach’s technical efficiency. In addition to manager efficiency, some have used production function analysis to evaluate various sports strategies. In cricket, Schofield [96] examined data from English country cricket to evaluate on-field strategies in addition to player selection, lineups, and development of particular skills. Carmichael, Thomas, and Ward [14] estimated the production function using the Opta Index as a quantifiable measure of player contributions in English Premiership football. They identified the specific activities on the pitch that most closely determined wins and identified the positive impact of “aggressive play.” In a follow-up using the Italian Serie A football league, Carmichael, Rossi, and Thomas [15] constructed composite measures of performance using factor loading and identified offensive performance as more critical than defensive performance. In baseball, Lee [71] evaluated a managerial strategy referred to as “small ball.” He found that aggressive baserunning and sacrificial activities to move runners closer to scoring were detrimental to the ultimate goal of scoring as many runs as possible and a source of production inefficiency.

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Optimal Levels and Distribution of Inputs In addition to evaluating the production efficiency with a given set of inputs, sports data has also been used to test theories regarding the level of various inputs and the optimal mix. Primarily, this literature examines whether the level and distribution of spending is directly related to team success, both through winning and through revenue earned. These examinations are testing the efficiency of personnel management and the nature of demand for the product. The optimizing the level of inputs is more obviously generalized to non-sports industries as it is essentially asking whether shear expenditures on skilled workers leads to positive outcomes. The question of the optimal mix of production inputs is also generalizable and the question is often posed to address questions of worker motivation based on earnings differentials between coworkers. When investigating the relationship between payroll and on-field success, researchers have found evidence that the structure of the labor market is a key determinant. Early examinations of the relationship between performance and payroll took a relatively naive approach. Szymanski and Smith [113] found that higher payrolls in English football (soccer) were positively related to a team’s on-field success and profits, but this relationship turned negative when controlling for a team’s endowments such as market and stadium sizes. Quirk and Fort [90] examined correlations between winning percentages and payroll rank across the four major North American sports leagues and found a small, but positive relationship in only the NHL and NFL. Hall, Szymanski, and Zimbalist [47] more formally tested the relationship between payroll and team success in both MLB and English soccer using Granger causality tests. In baseball, they found only a weak causal relationship running from performance to payroll which strengthened as league revenue became more disparate in the late 1990s. In soccer, a Granger causality test could not reject the hypothesis that payroll improved performance. Hall et al. believe that freer player movement in soccer is likely responsible for the difference in these results. Similarly, Simmons and Forrest [104] found that the relationship between payroll and performance is weaker in North American sports leagues compared to European football leagues. They also believe that labor market interventions in North American sports leagues are likely responsible for this difference. Payroll efficiency has also been tested using data envelopment analysis, with Einolf [30] and Lewis, Sexton, and Lock [76] finding lower levels of efficiency for largemarket baseball teams, though Einolf did not find a similar relationship for NFL teams. Data envelopment analysis has also been used to examine the efficiency of player contracts [50] and hall of fame voting in baseball [82]. In an attempt to address expenditure efficiency, sports management has renewed their focus on analytics to find undervalued player attributes. This relatively new trend was popularized in the book Moneyball by Michael Lewis [75] which focused on player contributions in MLB. Lewis examined the approach of a general manager in baseball who took advantage of undervalued statistical contributions to the production process to optimize team output despite a relatively small player payroll. These advantages tend to be short lived, as the labor market responds to successful

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roster management strategies for baseball, as documented in Hakes and Sauer [46] and Congdon-Hohman and Lanning [21]. A similar analytics sea change has occurred across many sports. Rather than examining total payroll, many researchers have used sports data to examine the question of how to distribute that payroll amongst the employees. There is a debate in the broader economic literature as to how workers respond to a large variance in wages between workers. Tournament theory, as established by Lazear and Rosen [70] and Milgrom and Roberts [83], suggests that disparity in compensation motivates workers to put forth more effort due to the clear rewards established in the wage structure, which in turn increases overall worker productivity. The cohesion theory (as established by Akerlof and Yellen [3] and Levine [74]) suggests that a compressed wage distribution promotes harmony in the workplace and results in higher worker productivity. Sports data is particularly appropriate to test these competing hypotheses since the nature of the work is cooperative, production is meticulously measured, and salaries are known to coworkers and the researcher. Additionally, since team production is often the output, most sports are ripe for examination. In most sports, the results when estimating wage inequality (as measured in various ways, including Gini coefficients, Herfindahl-Hirschman Indexes [HHI], and other measures of salary variance) as part of the team’s production functions suggest a negative relationship between payroll disparity and team success. Using baseball data, Richards and Guell (using a measure of variance to represent wage inequality, [91]), Bloom (Gini, [9]), Depken (HHI, [26]), Frick, Prinz, and Winkelmann (Gini, [43]), Jewell and Molina (Gini, [55]), and DeBrock, Hendrick, and Koenker (HHI, [25]) all find a strong, negative relationship between wage inequality and a team’s on-field success. That said, DeBrock, Hendrick, and Koenker [25] found that wage differences driven by differences in player quality did not lead to the same negative effect. Research examining the impact of wage inequality in professional ice hockey, soccer, and American football have found a similar relationship. Sommers [108] found weak evidence that inequality (as measured by a Gini coefficient) had a negative impact on team success in the NHL, while Stefanec [110] found a stronger negative result when looking at wage disparity within NHL position groups on a team. Similarly, Franck and Nüesch (Gini and measure of variance, [38]) and Coates, Frick, and Jewell [20] found negative relationships between wage disparity and team outcomes in the German Bundesliga and American Major League Soccer, with Franck and Nüesch [38] also finding that teams with higher inequality in pay played more individualistically. In American football, Frick, Prinz, and Winkelmann [43] found a negative but insignificant relationship between wage inequality and team success using data from the NFL, while Mondello and Maxcy [84] found similar results but also that wage dispersion improves team revenue production. Research examining the NBA produced one of the few results supporting the tournament theory regarding wage inequality and worker motivation. Frick, Prinz, and Winkelmann [43] found that the NBA was the only major sports league in North America where teams did better with more wage disparity (measured using a Gini

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coefficient). Using different methodology and an HHI to measure wage inequality, Berri and Jewell [7] find no evidence of a link between wage inequality and firm productivity. Simmons and Berri [103] used changes to the salary structure in the NBA due to changes to the league’s collective bargaining agreement with the players’ union to show that salary dispersion based on variation in individual’s talent levels showed a positive relationship to winning but “unjustified” inequality had no significant impact. Katayama and Nuch [60] found no relationship between salary dispersion and team performance using the general method of moments (GMM) to control for the effect of prior team success on current salary distribution. Disparity in worker characteristics other than salary have also been examined using sports data. Using data from MLB, Papps, Bryson, and Gomez [88] found that baseball teams with a more even distribution of talent perform better than teams with a more disparate distribution, though also that there is an optimal mix of talent that outperforms teams with a very low dispersion. Gelade [45] also found that a large spread in abilities in European soccer led to negative team performance, though with more goals scored for the team and its opponents. Kahane, Longley, and Simmons [57] examined the impact of cultural heterogeneity in the workplace through nationality in the NHL. They found that more European players from a particular country improves team performance, but foreign workers from various countries override the gains from player diversity.

Worker Effort and Compensation Sports data has been used extensively to examine the value of players as discussed above, but also to examine whether players modify their behavior and output based on the inherent incentives of the labor market structure in their sport and from their individual contract. Sports performance data is particularly apt to test agency theory, which examines the relationship of unaligning incentives between the principle (owners) and the agents (players). Though the principle-agent problem has been investigated in other contexts (CEO pay in Abowd [1] and transitions from hourly pay to piece rates in Lazear [69], for example), the nature of professional sports contracts gives researchers a clear case where incentives for players change at clearly distinguishable points. Again, professional baseball is the focus of much of the research on opportunistic behavior by labor because of the individual nature of the sport. Krautmann [62] and Scoggins [97] examined the period immediately following the introduction of free agency among players and found no evidence of improved performance before a new contract was signed nor evidence of a decrease in performance after (often referred to as “shirking”). Using later data, Sommers [107], Maxcy [80], Maxcy, Fort, and Krautmann [81], and Krautmann and Donley [65] also found no evidence of strategic behavior among baseball players following a new, longterm contract. Alternatively, Sommers [107] found evidence of shirking following arbitration contacts. Woolway [117] found that players’ MRP declined in the year following a new long-term contract. Krautmann and Solow [66] found evidence of

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shirking, but only among players identified as unlikely to play after their current contract expired. Many studies did find evidence that players spend more time unavailable to play due to injury after signing a contract (Lehn [73], Scoggins [97], and Maxcy, Fort, and Krautmann [81]), but this cannot necessarily be attributed to shirking as teams may instead be protecting the player from more severe injury given their long-term investment in the player’s labor. Outside of baseball, basketball has been the most fertile ground for a similar examination of player performance following long-term contracts and have found mixed results. Harder [48] found that over-rewarded players tend to play in a more team-oriented manner than under-rewarded players in the NBA, suggesting free agents respond positively following a new contract. Berri and Krautmann [8] find that a player’s measured output declines following a long-term contract, but their MRP does not. Stiroh [111] found evidence of opportunistic behavior with players’ performances improving prior to signing a multi-year contract but declining following the contract. He found this difference extended to team production, as teams with many players with expiring contracts improving play while teams with many players recently signed to multi-year contracts show a decline in performance. Opportunistic behavior has also been tested in other sports. Fernie and Metcalf [32] examine the horse racing industry and found that jockeys performed worse when they were paid on a non-contingent basis through guaranteed salaries. In the NFL, Conlin and Emerson [22] found that players in the last year of their contract started more games than predicted (suggesting more effort) and Frick, Dilger, and Prinz [42] found that teams who devoted a larger portion of their payroll to guaranteed bonuses performed worse than teams with a lower portion. Frick [39] used quantile regression to show that players in the German Bundesliga performed significantly better in the final year of their contracts.

Conclusion Sports teams and athletes bring an interesting theoretical twist to standard economic models of competition and production by requiring cooperation among competitors to produce their product. In addition, professional sports’ widespread popular appeal, growing economic importance, as well as the industry’s appealing availability of highly specialized data has made sports economics a rapidly growing field within economics. One study showed that frequency of published papers covering sports economics topics in a general interest journal had risen by a factor of nearly 20 between the 1970s and the 2010s [79], and the number of colleges and universities offering sports economics courses had risen from just a handful nationwide in the 1990s to well over 100 by 2020. In fact, sports economics even recently earned its own JEL code in 2015. At the same time, the high stakes world of professional sports labor markets, where the contract for a single athlete can now reach into the hundreds of millions of dollars, has increasingly turned to the sports analytics models first developed by economists like Gerald Scully ushering in an era of quantitative analysis in sports

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that was unheard of even 25 years ago. A modern general manager or vice president of player acquisition is more likely to be Ivy League trained economist than a former star player, and there is every indication that sports metrics will be even more integral to the game in the future. Indeed, it seems clear that sports economics has only just kicked off and that there is a long game ahead for the discipline.

References 1. Abowd JM (1990) Does performance-based managerial compensation affect corporate performance. Ind Labor Relat Rev 43(3):52S–73S 2. Agha N (2013) The economic impact of stadiums and teams: the case of minor league baseball. J Sports Econ 14(3):227–252 3. Akerlof GA, Yellen JL (1990) The fair wage-effort hypothesis and unemployment. Q J Econ 105(2):255–283 4. Atkinson SE, Stanley LR, Tschirhart J (1988) Revenue sharing as an incentive in an agency problem: an example from the National Football League.