Productivity: Concepts, Measurement, Aggregation, and Decomposition (Contributions to Economics) 3030754472, 9783030754471

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Contributions to Economics

Bert M. Balk

Productivity Concepts, Measurement, Aggregation, and Decomposition

Contributions to Economics

The series Contributions to Economics provides an outlet for innovative research in all areas of economics. Books published in the series are primarily monographs and multiple author works that present new research results on a clearly defined topic, but contributed volumes and conference proceedings are also considered. All books are published in print and ebook and disseminated and promoted globally. The series and the volumes published in it are indexed by Scopus and ISI (selected volumes).

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Bert M. Balk

Productivity Concepts, Measurement, Aggregation, and Decomposition

Bert M. Balk Rotterdam School of Management Erasmus University Rotterdam, The Netherlands

ISSN 1431-1933 ISSN 2197-7178 (electronic) Contributions to Economics ISBN 978-3-030-75447-1 ISBN 978-3-030-75448-8 (eBook) https://doi.org/10.1007/978-3-030-75448-8 © The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2021 This work is subject to copyright. All rights are solely and exclusively licensed 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 Switzerland AG The registered company address is: Gewerbestrasse 11, 6330 Cham, Switzerland

Preface

The main purpose of this book is to develop the theory of (intertemporal) productivity measurement without relying on the usual neo-classical assumptions, such as the existence of a production function characterized by constant returns to scale, optimizing behaviour of the economic agents, and perfect foresight. The theory can be applied to all the usual levels of aggregation (micro, meso, and macro), and half of the book is devoted to accounting for the links existing between the various levels. Basic insights from National Accounts are thereby used. The book emphasizes the fundamental equivalence of multiplicative and additive models. The main instruments used are (multiplicative) indices and (additive) indicators. Throughout the book, the fundamental equivalence of these measurement tools is made explicit. The final chapter is devoted to the decomposition of productivity change into the contributions of efficiency change, technological change, scale effects, and inputor output-mix effects. An application on a real-life data set shows the empirical feasibility of the theory. The book is directed to a variety of overlapping audiences: statisticians involved in measuring productivity change (e.g., in national or international statistical agencies); economists interested in growth accounting; researchers relating macroeconomic productivity change to its industrial sources; micro-data researchers; and business analysts interested in performance measurement (through time or space). The book tries to build a bridge between theory and practice. On the one hand, the textbook-orientated theorist is guided through a toolbox full of instruments, each with specific limitations and instructions for application. On the other hand, the practice-orientated policy developer is shown that the question “What is the (current) percentage of productivity change?” is easy to pose, but not so easy to answer. The book then provides the necessary background information. The book can be seen as a sort of exposition of the programme sketched in “Chapter 2: Empirical productivity indices and indicators,” in The Oxford Handbook of Productivity Analysis, edited by E. Grifell-Tatjé, C. A. K. Lovell, and R. C. Sickles, Oxford University Press, 2018. As the level of mathematics and economic theory never surpasses that of basic undergraduate courses, the book can be used v

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for more advanced courses on measurement in economics, in academic settings, or elsewhere.

Provenance of the Various Chapters Nine of the ten chapters are based on previous publications. Most of these have been revised thoroughly, and consolidated to avoid overlap. The mathematical notation has been harmonized. References have been brought up-to-date, and extensions—some of which based on newer insights—have found an appropriate place. Nevertheless, readers familiar with the subject will recognize the sources of the various chapters. In principle, the chapters can be read independently. Here follows an overview. • Section 1.1 is based on the corresponding section of “The residual: On monitoring and benchmarking firms, industries, and economies with respect to productivity”, Journal of Productivity Analysis 20 (2003), 5–47. • Chapter 2 is based on “An assumption-free framework for measuring productivity change”, The Review of Income and Wealth 56 (2010), Special Issue 1, S224– S256, reprinted in National Accounting and Economic Growth, edited by John M. Hartwick, The International Library of Critical Writings in Economics No. 313 (Edward Elgar, Cheltenham UK, Northampton MA, 2016). Enhanced with parts of “Empirical Productivity Indices and Indicators,” written in 2016, available at SSRN: http://ssrn.com/abstract=2776956. In particular Sect. 2.2.5, on the equivalence of multiplicative and additive models, is new. Added is a second Sect. 2.3.2 on growth accounting, and a Sect. 2.6 containing an empirical illustration of the approach developed in this chapter. Next, to make the book as self-contained as possible, Appendices A (on indices and indicators) and B (on decompositions of the value-added ratio) have been expanded. Appendix C, on the famous but frequently misunderstood Domar factor, is new. • Chapter 3 is based on “Measuring and decomposing capital input cost”, The Review of Income and Wealth 57 (2011), 490–512. Sections 3.3 (on the classic formulas) and 3.5 (on rates of return) have been expanded. Appendices A (Decompositions of time-series depreciation) and B (Geometric profiles) are new. • Chapter 4 updates “Consistency issues in the construction of annual and quarterly productivity measures”, International Productivity Monitor 37 (Fall 2019), 144– 155. • Chapter 5 expands “The dynamics of productivity change: A review of the bottom-up approach”, in Productivity and Efficiency Analysis, edited by W. H. Greene, L. Khalaf, R. C. Sickles, M. Veall, and M.-C. Voia, Springer Proceedings in Business and Economics, Springer International Publishing Switzerland, 2016. In particular, Sect. 5.4.4, discussing cases where not all the data were accessible, is new. Section 5.5.2 (on the TRAD, CSLS, and GEAD decompositions) has been expanded. Sections 5.6 (on the logmean approach) and 5.7 (on the geometric and

Preface

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harmonic approach) are new. Section 5.8 (on the monotonicity paradox) has been expanded. Finally, Appendices A (Reinsdorf’s expansion of the GR method), B (Exercises on the Netherlands’ manufacturing industry, 1984–1999), and C (Generalization of the OP decomposition) are new. Chapter 6 combines “Dissecting aggregate output and labour productivity change”, Journal of Productivity Analysis 42 (2014), 35–43, and “Dissecting aggregate output and labour productivity change: A postscript on the role of relative prices” (with Jesus C. Dumagan), Journal of Productivity Analysis 45 (2015), 117–119. Appendix A (The Tang and Wang method) has been expanded. Chapter 7 is based on “Measuring and relating aggregate and subaggregate productivity change without neoclassical assumptions”, Statistica Neerlandica 69 (2015), 21–48. Section 7.10, on growth accounting, is new. Chapter 8 is based on “A novel decomposition of aggregate total factor productivity change”, Journal of Productivity Analysis 53 (2020), 95–105. Appendix A, analysing Baumol’s “growth disease”, is new. Chapter 9 updates “Aggregate productivity and productivity of the aggregate: Connecting the bottom-up and top-down approaches”, in Productivity and Inequality, edited by W. H. Greene, L. Khalaf, P. Makdissi, R. C. Sickles, M. Veall, and M.-C. Voia, Springer Proceedings in Business and Economics, Springer International Publishing AG, 2018. Chapter 10 is entirely new, based on “The many decompositions of total factor productivity change” (with J. L. Zofío), Report No. ERS-2018-003-LIS, Series Research in Management, Erasmus Research Institute of Management, Erasmus University, Rotterdam, available at SSRN: http://ssrn.com/abstract=3167686.

As almost all the chapters are based on contributions to peer-reviewed journals and book collections, I want to thank all those colleagues who, in an entirely disinterested way, have spent time on reviewing initial versions of these chapters. The same applies to papers that did not make it to journals. I thank them for questions, criticism, and suggestions. In particular, I thank José Luis Zofío for the most pleasant collaboration in the research underlying Chap. 10 and Paul de Boer for going through the almost-final version of the manuscript. This book is dedicated to the Rotterdam School of Management, Erasmus University, especially its Department of Technology & Operations Management that continued to provide an intellectually stimulating environment after my formal retirement in 2011. Receive this book as a sort of farewell gift! Amersfoort, The Netherlands

Bert M. Balk

Contents

1

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1 The Origins of a Concept . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2 The Neo-Classical Approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3 The Contents of This Book . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1 1 3 5

2

A Framework Without Assumptions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 The Basic Input-Output Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.1 Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.2 Total Factor Productivity Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.3 Growth Accounting (1) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.4 Total Factor Productivity Indicator . . . . . . . . . . . . . . . . . . . . . . . . 2.2.5 The Equivalence of Multiplicative and Additive Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.6 Partial Productivity Measures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3 Different Models, Similar Measures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3.1 The KL-VA Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3.2 Growth Accounting (2) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3.3 The K-CF Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4 More Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.5 Capital Utilization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.6 An Empirical Illustration. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.7 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Appendix A: Indices and Indicators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Indices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Indicators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Appendix B: Decompositions of the Value-Added Ratio . . . . . . . . . . . . . . . . . . Appendix C: The Domar Factor. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . And the TFP Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . And the Labour Productivity Index. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

9 9 11 15 17 20 23 24 28 32 32 36 38 40 43 44 46 48 48 54 57 61 61 63

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Contents

Capital Input Cost . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 Defining Capital Input Cost . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3 The Relation with Capital Stock Measures and the Classic Formulas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4 The Relation Between Asset Price and Unit User Cost . . . . . . . . . . . . 3.5 Rates of Return . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.6 Aggregation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.7 Some Implementation Issues . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.8 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Appendix A: Decompositions of Time-Series Depreciation . . . . . . . . . . . . . . . Appendix B: Geometric Profiles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

65 65 66 71 75 79 83 85 88 89 91

4

Annual and Quarterly Measures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2 A Simple Case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2.1 Productivity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2.2 Productivity Change . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2.3 Relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3 A More Realistic Case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4 The System View . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.5 Subperiod Productivity Indices as Approximations . . . . . . . . . . . . . . . . 4.6 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

93 93 95 95 96 97 100 103 105 106

5

Dynamics: The Bottom-Up Approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2 Accounting Identities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3 Continuing, Entering, and Exiting Production Units . . . . . . . . . . . . . . . 5.4 Productivity Indices and Levels . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.4.1 Indices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.4.2 Levels . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.4.3 Linking Levels and Indices. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.4.4 When Not All the Data Are Accessible . . . . . . . . . . . . . . . . . . . 5.5 Decompositions: Arithmetic Approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.5.1 The First Three Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.5.2 Interlude: The TRAD, CSLS, and GEAD Decompositions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.5.3 The Fourth and Fifth Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.5.4 Another Five Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.5.5 Provisional Evaluation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.6 Decompositions: Logmean Approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.7 Decompositions: Geometric and Harmonic Approach . . . . . . . . . . . . . 5.8 Monotonicity Paradox? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.9 The Olley-Pakes Decomposition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.10 The Choice of Weights . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.11 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

109 109 112 114 115 116 118 120 121 128 129 132 136 138 141 142 145 148 150 153 154

Contents

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Appendix A: Reinsdorf’s Extension of the GR Method . . . . . . . . . . . . . . . . . . . 155 Appendix B: Exercises on the Netherlands Manufacturing Industry, 1984–1999 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 158 Appendix C: Generalization of the OP Decomposition . . . . . . . . . . . . . . . . . . . . 162 6

7

8

The Top-Down Approach 1: Aggregate Output and Simple Labour Productivity Indices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2 Conventional Decompositions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.3 Symmetric Decompositions of Aggregate Output Change . . . . . . . . . 6.4 Symmetric Decompositions of Simple Labour Productivity Change. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Appendix A: The Tang and Wang Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Appendix B: Dumagan’s Decomposition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Appendix C: Proof of Expression (6.13). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The Top-Down Approach 2: Aggregate Total Factor Productivity Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.2 First Decomposition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.3 Second Decomposition. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.4 Third Decomposition. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.5 Asymmetric Decompositions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.6 The Differential Price Change Term . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.7 The Case of a Dynamic Ensemble. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.8 Dynamic Ensemble: More Decompositions . . . . . . . . . . . . . . . . . . . . . . . . 7.9 Linking Value-Added Based to Gross-Output Based Productivity Change. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.10 Growth Accounting (3) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.11 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The Top-Down Approach 3: Aggregate Total Factor Productivity Level . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.2 Prerequisites . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.3 Decomposing Value-Added Based Total Factor Productivity Change. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.4 Decomposing the Reallocation Factor into Contributions of Separate Primary Inputs. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.5 Introducing Gross-Output Based Total Factor Productivity Change. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.6 The Zero Profit Case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.7 Going Beyond Total Factor Productivity Change . . . . . . . . . . . . . . . . . . . 8.8 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Appendix: Analyzing Baumol’s ‘Growth Disease’ . . . . . . . . . . . . . . . . . . . . . . . .

165 165 166 167 171 177 177 183 184 187 187 188 192 195 196 201 203 205 209 211 212 215 215 216 217 222 224 228 230 232 233

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Contents

9

Connecting the Two Approaches. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.2 The Connection Defined . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.3 Value-Added Based Total Factor Productivity . . . . . . . . . . . . . . . . . . . . . . 9.3.1 General Case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.3.2 Additivity Imposed . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.4 Value-Added Based Labour Productivity . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.4.1 General Case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.4.2 Simple Labour Productivity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.4.3 Additivity Imposed . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.5 Gross-Output Based Productivity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.5.1 Simple Labour Productivity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.5.2 Total Factor Productivity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.5.3 Some Empirical Evidence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.6 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

235 235 236 238 238 241 244 244 244 246 247 248 250 253 253

10

The Components of Total Factor Productivity Change . . . . . . . . . . . . . . . . 10.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.2 Basic Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.2.1 Technologies, Distance Functions, and Related Measures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.2.2 Measuring Productivity Change and Level . . . . . . . . . . . . . . . . 10.3 Decomposing a Malmquist Productivity Index by Output Distance Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.3.1 The Base Period Viewpoint . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.3.2 The Comparison Period Viewpoint . . . . . . . . . . . . . . . . . . . . . . . . 10.3.3 The ‘Geometric Mean’ Viewpoint . . . . . . . . . . . . . . . . . . . . . . . . . 10.4 Decomposing a Malmquist Productivity Index by Input Distance Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.4.1 The Base Period Viewpoint . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.4.2 The Comparison Period Viewpoint . . . . . . . . . . . . . . . . . . . . . . . . 10.4.3 The ‘Geometric Mean’ Viewpoint . . . . . . . . . . . . . . . . . . . . . . . . . 10.5 Decomposing a Moorsteen-Bjurek Productivity Index . . . . . . . . . . . . . 10.5.1 Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.5.2 Decompositions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.6 Decomposing a Lowe Productivity Index . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.6.1 Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.6.2 Decompositions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.7 Decomposing a Cobb-Douglas Productivity Index . . . . . . . . . . . . . . . . . 10.7.1 Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.7.2 Decompositions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.8 An Empirical Application. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.8.1 Data and DEA Approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.8.2 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

255 255 257 257 260 261 262 269 272 275 275 278 279 280 280 283 289 289 290 295 295 296 297 297 298

Contents

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10.9 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 302 Appendix A: Components of Mˇ o0 (x 1 , y 1 , x 0 , y 0 ) Along Path A . . . . . . . . . . 310 Appendix B: Data Envelopment Analysis (DEA) . . . . . . . . . . . . . . . . . . . . . . . . . . 312 Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 313 Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 325

List of Figures

Fig. 2.1

Measuring productivity change . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

47

Fig. 10.1 Decomposing productivity change (1) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 263 Fig. 10.2 Decomposing productivity change (2) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 276

xv

List of Tables

Table 2.1

Table 2.2

Table 5.1

Netherlands Manufacturing industry, 1995–2008. Geometric mean annual TFP change (percentage). Exogenous rate of return . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Netherlands Manufacturing industry, 1995–2008. Geometric mean annual TFP change (percentage). Endogenous rate of return . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

45

46

Netherlands Manufacturing industry, 1984–1999. Mean annual value-added based TFP change (percentage) and decomposition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161

Table 10.1 Descriptive statistics of the Taiwanese banking example, 2006 and 2010 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Table 10.2 Geometric mean Malmquist productivity index: output orientated decomposition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Table 10.3 Geometric mean Malmquist productivity index: input orientated decomposition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Table 10.4 Geometric mean Moorsteen-Bjurek productivity index: output orientated decomposition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Table 10.5 Geometric mean Moorsteen-Bjurek productivity index: input orientated decomposition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Table 10.6 Fisher productivity index: output orientated decomposition . . . . . . Table 10.7 Fisher productivity index: input orientated decomposition . . . . . . . Table 10.8 Törnqvist productivity index: output orientated decomposition . . Table 10.9 Törnqvist productivity index: input orientated decomposition . . . .

299 301 303 304 305 306 307 308 309

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Chapter 1

Introduction

Productivity . . . is a measure of the efficiency with which resources are converted into the commodities and services that men want. Solomon Fabricant, “Introduction”, in Kendrick (1961). . . . productivity measurement is all about comparing outputs with inputs, . . . Ross Gittins, “Productivity should be a spin-free zone”, The Sydney Morning Herald, June 23–24, 2007. Gains in productivity are the primary driver of wages and living standards. Sylvia Nasar, 2011, Grand Pursuit: The Story of Economic Genius (Fourth Estate, HarperCollinsPublishers, London) The traditional measure of the pace of innovation and technological change is total factor productivity (TFP) — output divided by a weighted average of labor and capital input. Robert J. Gordon (2016, 537)

1.1 The Origins of a Concept As several distinguished scholars have written about the history of productivity measurement and analysis this introduction can be kept brief. The reader needs only sufficient information to place the contents of this book in a proper frame. For details the reader is referred to Griliches (2001), the first section of which is a reworked version of Griliches (1996) on “the discovery of the residual”, Hulten (2001), Grifell-Tatjé and Lovell (2015, Chapter 1), and Grifell-Tatjé et al. (2018). The first mention of total factor productivity (TFP) change as the ratio of an output quantity index and an input quantity index occurs in a contribution by Copeland (1937) to what, with hindsight, could be called the national income accounting approach. Stimulated by institutions such as the National Bureau of Economic Research, in the post-war period several studies were published, a typical one being Stigler (1947). These studies were mainly dealing with industry- or economy-wide aggregates. Although the TFP index was sometimes referred to as © The Author(s), under exclusive license to Springer Nature Switzerland AG 2021 B. M. Balk, Productivity, Contributions to Economics, https://doi.org/10.1007/978-3-030-75448-8_1

1

2

1 Introduction

a measure of the efficiency of the economic process, the common opinion was best voiced by Abramowitz (1956), who called it a “measure of our ignorance.”1 The other, production-theoretic approach appears to go back to Tinbergen (1942). He extended the Cobb-Douglas production function with a time trend variable. The difference between the growth rate of real output and a weighted average of the growth rates of real capital and labour input was interpreted variably as efficiency change, technical development, or “Rationalisierungsgeschwindigkeit”. The basic and very influential contribution of Solow (1957) can be conceived as a sort of linkage of both traditions. He showed that under certain conditions the parameters of the Cobb-Douglas production function could be equated to observable statistical magnitudes, and the residual interpreted in terms of a ratio of output and input quantity index numbers. This is why the TFP index came to be known as the “Solow residual”, although the name “residual” appears to have been used by Domar (1961) for the first time. Solow interpreted the residual as a measure of technical change.2 Since the inception of the concept of TFP change there have been two main lines of research. The first was directed at explanation. The second was directed at better measurement, primarily of the input factors capital and labour. In the beginning, the second style was more prominent than the first. For example, Jorgenson and Griliches (1967) claimed that using the “correct” index number framework and the “right” measurement of inputs would largely eliminate the role of the residual. The residual disappeared indeed, but not at all due to better measurement techniques. The economy-wide disappearance of productivity growth in the seventies, its reappearance later on, and the search for the factors behind this world-wide phenomenon came to be known as the “productivity slowdown discussion”. The emphasis shifted from measurement problems to explanation, and Griliches’ work provides a clear demonstration of this shift. The main explanatory factors he considered were education, R&D expenditures, and patents. The measurement problems, however, remained important. Looking back at a life-long of research in this area, Griliches (2001) said: It is my hunch that at least part of what happened [namely, the productivity slowdown] is that the economy and its various technological thrusts moved into sectors and areas in which our measurement of output are especially poor: services, information activities, health, and also the underground economy.

but at the end of the day he concluded that There have been many reasonable attempts to explain the productivity slowdown (. . . ), but no smoking gun has been found, and no single explanation appears to be able to account for all the facts, leaving the field in an unsettled state until this day.

1 This

has become a frequently repeated quote; its last occurrence in Gordon (2016, 543). Lipsey and Carlaw (2000) once concluded that “TFP is as much a measure of our ignorance as it is a measure of anything positive.”, but what was meant by this “positive” remained unexplained! 2 On Solow and the NBER approach, exemplified by Kendrick, see Kleiman et al. (1966).

1.2 The Neo-Classical Approach

3

Twenty years later, Banerjee and Duflo (2019) extensively reviewed the state of the art and drew basically the same conclusion. It is instructive to consult case studies, for instance those concerning the Indian economy. Madsen et al. (2010) based their study on aggregate data over the period 1950–2005 and manufacturing firms data over the period 1993–2005. They concluded that there was “no robust long-run relationship between TFP and research activity” as well as that “TFP growth cannot be explained by growth in research activity.” Ghosh and Parab (2021) continued by applying various endogenous growth models to aggregate TFP growth data over the period 1970–2017. Their results were mixed and, as far as positive, weak. There was a modest role for human capital and R&D, and a bit larger role for foreign direct investment (FDI). Until the nineties, the research on productivity change typically made use of the concept of the ‘representative firm’ in combination with aggregate empirical data material provided by statistical agencies. The increased availability of longitudinal enterprise microdata sets has opened up many new, exciting research avenues. Researchers are now able to track large numbers of individual firms over time. This has led to a completely new area of research, with its own conferences and research centers.

1.2 The Neo-Classical Approach A cornerstone of the neo-classical approach is the assumption that the technology governing the production unit or units under consideration (be it enterprises, industries, or economies) can be represented by a sufficiently neat primal or dual function of input and output quantities and/or prices and time. Thus, for example, let the technology that governs a certain unit’s production process at time period t be characterized by the cost function C(w, y, t). This function provides the minimum cost for producing the output quantities y (an M-dimensional vector) when the input prices are given by w (an N -dimensional vector). Let the actual cost incurred by the production unit producing y(t) be w(t) · x(t), where x(t) denotes the vector of input quantities at period t and · denotes the inner product. One of the neo-classical key assumptions is that the production unit is a cost minimizer; that is, actual cost is assumed to be equal to minimum cost, or, in terms of the formulas just introduced: w(t) · x(t) = C(w(t), y(t), t).

(1.1)

Assuming that all functions are continuous, one can differentiate with respect to time t to obtain growth rates. The left-hand side of expression (1.1) yields d ln w(t) · x(t)  d ln wn (t)  d ln xn (t) = + , sn (t) sn (t) dt dt dt n n

(1.2)

4

1 Introduction

where the sn (t)’s denote the actual cost shares of the inputs. The right-hand side of expression (1.1) yields d ln C(w(t), y(t), t) = dt  ∂ ln C(.) d ln wn (t)  ∂ ln C(.) d ln ym (t) ∂ ln C(.) + + . ∂ ln wn dt ∂ ln ym dt ∂t n m

(1.3)

The assumption of cost minimization combined with Shephard’s Lemma leads to the well-known conclusion that ∂ ln C(.) = sn (t) (n = 1, . . . , N ); ∂ ln wn

(1.4)

that is, the partial derivatives occurring in the first term on the right-hand side of expression (1.3) are equal to the actual cost shares of the inputs. The second assumption is that the output prices are proportional to marginal cost; that is pm (t) ∝

∂C(.) (m = 1, . . . , M). ∂ym

(1.5)

Simple manipulations with this relation lead to the following expression:  ∂ ln C(.) ∂ ln C(.) = um (t) (m = 1, . . . , M), ∂ ln ym ∂ ln ym m

(1.6)

where the um (t)’s are the actual revenue shares of the outputs. The third assumption is that the technology exhibits constant returns to scale. This implies that  ∂ ln C(.) m

∂ ln ym

= 1.

(1.7)

Thus, second and third assumption together imply that the partial derivatives occurring in the second term on the right-hand side of expression (1.3) are equal to the actual revenue shares of the outputs. We can now put the various pieces together. Due to the cost minimization assumption the right-hand sides of expressions (1.2) and (1.3) are identically equal. Thus, substituting the two results just obtained, the following expression is obtained:  m

um (t)

∂ ln C(.) d ln ym (t)  d ln xn (t) − =− . sn (t) dt dt ∂t n

(1.8)

1.3 The Contents of This Book

5

The left-hand side of this expression measures TFP change (in growth rate form) and is known as the Solow residual. The right-hand side is the minimum cost decrease associated with the mere passage of time. This is conventionally interpreted as a measure of technological change. Thus, under the stated assumptions, TFP change is equal to technological change. The assumptions can be summarized as follows. First, the production unit is seen as cost efficient; that is, technically efficient—acting at the frontier of the current technology—, and allocatively efficient—the input quantities have the optimal mix. Second, the production unit is assumed to act in a competitive environment. Third, the technology is assumed to exhibit constant returns to scale. In the next chapter we will argue that as fourth assumption perfect foresight must be added.

1.3 The Contents of This Book The measurement of productivity change (or difference) is usually based on models relying on strong assumptions such as competitive behaviour and constant returns to scale. Chapter 2 discusses the basics of (intertemporal) productivity measurement and shows that one can dispense with most if not all of the usual, neo-classical assumptions. Various models are reviewed and their relationships discussed. Throughout the chapter the equivalence of multiplicative and additive models, as well as the equivalence of productivity measurement and growth accounting, is highlighted. By virtue of their structural features, the various measurement models are applicable to individual establishments and aggregates such as industries, sectors, or economies. Unlike labour productivity change, the measurement of total factor productivity change (or difference) crucially depends on the measurement and decomposition of capital input cost. Chapter 3 discusses the various measurement issues and shows that one can dispense with the usual neo-classical assumptions. There are several models and several alternatives for the ‘rate of return’. However, the particular rate that must be used remains at the discretion of the researcher or the statistical agency. Productivity change is generally measured in index form as ratio of an output quantity index over an input quantity index. Several statistical agencies publish quarterly as well as annual productivity index numbers or growth percentages, constructed from what appear to be basically the same sources. This raises the question whether, apart from measurement errors, consistency between quarterly and annual indices can be expected. Chapter 4 explores, from a theoretical perspective, the options for obtaining consistency between annual and quarterly (or more general: between period and subperiod) measures of productivity change. An industry is usually an ensemble of individual firms (decision making units) which may or may not interact with each other. Similarly, an economy is an ensemble of industries. In National Accounts terms this is symbolized by the fact that the nominal value added produced by an industry or an economy is the simple sum of firm-, or industry-specific nominal value added. From this viewpoint it is

6

1 Introduction

natural to expect that there is a relation between (aggregate) industry or economy productivity and the (disaggregate) firm- or industry-specific productivities. This is the topic of Chaps. 5–9. Chapter 5 considers the relation between (total factor) productivity measures for lower-level production units and aggregates thereof such as industries, sectors, or entire economies. In particular, this chapter contains a review of the so-called bottom-up approach, which takes an ensemble of individual production units, be it industries or enterprises, as the fundamental frame of reference. At the level of industries the various forms of shift-share analysis are reviewed. At the level of enterprises the additional features that must be taken into account are entry (birth) and exit (death) of production units. The top-down approach is pursued in Chaps. 6–8. Chapter 6 is concerned with the relation between output and labour productivity measures for individual production units and for aggregates such as industries, sectors, or economies. In the framework of discrete time periods several useful, symmetric expressions are derived and confronted with results from the literature. Chapter 7 proceeds by considering the relation between total factor productivity measures for individual production units and for aggregates such as industries, sectors, or economies. Though this topic has been treated in a number of influential publications, this chapter’s distinctive feature is that all kinds of (neo-classical) structural and behavioural assumptions are avoided, such as assumptions about the existence of production frontiers with certain properties, or optimizing behaviour of the production units. In addition, the chapter treats dynamic ensembles of production units, characterized by entry and exit. Thus, a greater level of generality is achieved from which the earlier results follow as special cases. In Chap. 7 three time-symmetric decompositions of aggregate value-added-based total factor productivity change were developed. In Chap. 8 a fourth decomposition will be developed. A notable difference with the previous chapter is that the development is cast in terms of levels rather than indices. Various aspects of this new decomposition will be discussed and links with decompositions found in the literature unveiled. It turns out that also here one can dispense with the usual neoclassical assumptions. As said, productivity analysis is carried out at various levels of aggregation. In microdata studies the emphasis is on individual firms (or plants), whereas in sectoral studies it is on (groupings of) industries. Microdata researchers do not care too much about the interpretation of the weighted means of firm-specific productivities employed in their analyses. In Chap. 9 the consequences of this attitude are explored, based on a review of the literature. However, a structurally similar phenomenon happens in sectoral studies, where the productivity change of industries is compared to each other and to the productivity change of some next-higher aggregate, which is usually the (measurable part of the) economy. Though there must be a relation between sectoral and economy-level measures, in most publications by statistical agencies and academic researchers this aspect is more or less neglected.

1.3 The Contents of This Book

7

The point of departure of Chap. 9 is that aggregate productivity must be interpreted as productivity of the aggregate. It is shown that this implies restrictive relations between the productivity measure, the set of weights, and the type of mean employed. The final chapter, Chap. 10, delves into the components of (total factor) productivity change. By making minimal assumptions about underlying technologies it appears that productivity change, here defined as output quantity change divided by input quantity change, can be seen as the combined result of (technical) efficiency change, technological change, a scale effect, and input- and output-mix effects. Given a certain functional form for the productivity index, the problem is then how to decompose such an index into factors corresponding to these five components. A basic insight offered in this chapter is that meaningful decompositions of productivity indices can only be obtained for indices that are transitive in the main variables. Using a unified approach, decompositions for Malmquist, MoorsteenBjurek, Lowe, and Cobb-Douglas productivity indices are obtained. A unique feature of this chapter is that all the decompositions are applied to the same dataset, a real-life panel of decision-making units, so that the extent of the differences between the various decompositions can be judged.

Chapter 2

A Framework Without Assumptions

2.1 Introduction The methodological backing of productivity measurement and growth accounting usually goes like this. The (aggregate) production unit considered has an input side and an output side, and there is a production function that links output quantities to input quantities. This production function includes a time variable, and the partial derivative of the production function with respect to the time variable is called technological change (or, in some traditions, multi- or total factor productivity change). Further, it is assumed that the production unit acts in a competitive environment; that is, input and output prices are assumed as given. Next, it is assumed that the production unit acts in a profit maximizing manner (or, it is said to be ‘in equilibrium’), and that the production function exhibits constant returns to scale. Under these assumptions it then appears that output quantity growth (defined as the output-share-weighted mean of the individual output quantity growth rates) is equal to input quantity growth (defined as the input-share-weighted mean of the individual input quantity growth rates) plus the rate of technological change (or, multi- or total factor productivity growth). For the empirical implementation one then turns to National Accounts, census and/or survey data, in the form of nominal values and deflators (price indices). Of course, one cannot avoid dirty hands by making various imputations where direct observations failed or were impossible (as in the case of labour input of self-employed workers). In the case of capital inputs the prices, necessary for the computation of input shares, cannot be observed, but must be computed as unit user costs. The single degree of freedom that is here available, namely the rate of return, is used to ensure that the restriction implied by the assumption of constant returns to scale, namely that profit equals zero, is satisfied. This procedure is usually rationalized by the assumption of perfect foresight, which in this case means that the ex post calculated capital input prices can be assumed as ex ante given to the

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2021 B. M. Balk, Productivity, Contributions to Economics, https://doi.org/10.1007/978-3-030-75448-8_2

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2 A Framework Without Assumptions

production unit, so that they can be considered as exogenous data for the unit’s profit maximization problem. This account is, of course, somewhat stylized, since there occur many, smaller or larger, variations on this theme in the literature. An example was provided in Sect. 1.2. Recurring, however, are a number of so-called neo-classical assumptions: (1) a technology that exhibits constant returns to scale, (2) competitive input and output markets, (3) optimizing behaviour, and (4) perfect foresight. A fine example from academia is provided by Jorgenson et al. (2005, 23, 37), while the Sources and Methods publication of Statistics New Zealand (2006) shows that the neo-classical model has also deeply invaded official statistical agencies.1 Another interesting example where neo-classical assumptions have invaded the measurement system is the World Productivity Database of the United Nations Industrial Development Organisation (UNIDO); see Isaksson (2009). An interesting position is taken by the EU KLEMS Growth and Productivity Accounts project. Though in their main text Timmer et al. (2007) adhere to the Jorgenson, Ho and Stiroh framework, there is a curious footnote saying Under strict neo-classical assumptions, MFP [multifactor productivity] growth measures disembodied technological change. In practice [my emphasis], MFP is derived as a residual and includes a host of effects such as improvements in allocative and technical efficiency, changes in returns to scale and mark-ups as well as technological change proper. All these effects can be broadly summarized as “improvements in efficiency”, as they improve the productivity with which inputs are being used in the production process. In addition, being a residual measure MFP growth also includes measurement errors and the effects from unmeasured output and inputs.

There are more examples of authors who exhibit similar concerns, without, however, feeling the need to adapt their conceptual framework. For an official statistical agency, whose main task it is to provide statistics to many different users for many different purposes, it is discomforting to have such, strong and often empirically refuted, assumptions built into the methodological foundations of productivity and growth accounting statistics. This especially applies to the behavioural assumptions numbered 2, 3 and 4. There is ample evidence that, on average, markets are not precisely competitive; that producers’ decisions frequently turn out to be less than optimal; and that managers almost invariably lack the magical feature of perfect foresight. Moreover, the environment in which production units operate is not so stable as the assumption of a fixed production function seems to claim. Fortunately, it is possible to avoid making such assumptions. In a sense this book proposes to start where the usual story ends, namely at the empirical side.2 For

1 The

neo-classical model figured already prominently in the 1979 report of the U. S. National Research Council’s Panel to Review Productivity Statistics (Rees 1979). An overview of national and international practice is provided by the regularly updated OECD Compendium of Productivity Indicators (OECD 2019). 2 There is another, minor, difference between the approach defended here and the usual story. The usual story runs in the framework of continuous time in which periods are of infinitesimal short

2.2 The Basic Input-Output Model

11

any production unit, the total factor productivity index is then defined as an output quantity index divided by an input quantity index. There are various options here, depending on what one sees as input and output, but the basic feature is that, given price and quantity (or value) data, this is simply a matter of index construction. There appear to be no behavioural assumptions involved, and this even applies—as will be demonstrated—to the construction of capital input prices. Surely, a number of imputations must be made (as in the case of the self-employed workers) and there is fairly large number of more or less defendable assumptions involved (for instance on the depreciation rates of capital assets), but this belongs to the daily bread and butter of economic statisticians. Structural as well as behavioural assumptions enter the picture as soon as it comes to the explanation of productivity change. Then there are, depending on the initial level of aggregation, two main directions: (1) to explain productivity change at an aggregate level by productivity change and other factors operating at lower levels of aggregation; (2) to decompose productivity change into factors such as technological change, technical efficiency change, scale effects, input- and output-mix effects, and chance. In this case, to proceed with the analysis one cannot sidestep a technology model with certain specifications. The contents of this chapter unfold as follows. Section 2.2 outlines the architecture of the basic, KLEMS-Y, input-output model, with its total and partial measures of productivity change. This section also links productivity measurement and growth accounting. Section 2.3 proceeds with the KL-VA and K-CF models. Four additional input-output models are briefly introduced in Sect. 2.4. This section also contains a comparison of all the models. Section 2.5 introduces the capital utilization rate. Section 2.6 reviews an empirical illustration. Section 2.7 concludes by discussing the main decomposition methods.

2.2 The Basic Input-Output Model Let us consider a single production unit. This could be an establishment or plant, a firm, an industry, a sector, or even an entire economy. We will simply speak of a ‘unit’. For the purpose of productivity measurement, such a unit is considered as a (consolidated) input-output system. What does this mean? For the output side as well as for the input side there is some list of commodities (according to some classification scheme). A commodity is thereby defined as a set of closely related items (goods or services) which, for the purpose of analysis, can be considered as “equivalent”, either in the static sense of their quantities being additive or in the dynamic sense of displaying equal relative price or quantity changes.

duration. When it then comes to implementation several approximations must be assumed. The approach in this book does not need this kind of assumptions either, because entirely based on accounting periods of finite duration, such as years.

12

2 A Framework Without Assumptions

Ideally, then, for any accounting period considered (ex post), say a year, each commodity comes with a value (in monetary terms) and a price and/or a quantity. If value and price are available, then the quantity is obtained by dividing the value by the price. If value and quantity are available, then the price is obtained by dividing the value by the quantity. If both price and quantity are available, then value is defined as price times quantity. In any case, for every commodity it must be so that value equals price times quantity, the magnitudes of which of course must pertain to the same accounting period. Technically speaking, the price concept used here is the unit value. At the output side, the prices must be those received by the unit, whereas at the input side, the prices must be those paid. Consolidation (also called net-sector approach) means that the unit does not deliver to itself. Put otherwise, all the intra-unit deliveries are netted out. The inputs are customarily classified according to the KLEMS format. The letter K denotes the class of owned, reproducible capital assets. The commodities here are the asset-types, sub-classified by age category. Cohorts of assets are assumed to be available at the beginning of the accounting period and, in deteriorated form (due to ageing, wear and tear), still available at the end of the period. Investment during the period adds entities to these cohorts, while desinvestment, breakdown, or retirement remove entities. Examples include buildings and other structures, land, machinery, transport and ICT equipment, tools. Theory implies that quantities sought are just the quantities of all these cohorts of assets (together representing the productive capital stock), whereas the relevant prices are their unit user costs (per type-age combination), constructed from imputed interest rates, depreciation profiles, (anticipated) revaluations, and tax rates. The sum of quantities times prices then provides the capital input cost of a production unit. The productive capital stock may be underutilized, which implies that not all the capital costs are incurred in actual production. See Schreyer (2001, section 5.6) for a general discussion of this issue. We return to this issue in Section 2.5.3 The letter L denotes the class of labour inputs; that is, all the types of work that are important to distinguish, cross-classified for instance according to educational attainment, gender, and experience (which is usually proxied by age categories). Quantities are measured as hours worked (or paid), and prices are the corresponding wage rates per hour. Where applicable, imputations must be made for the work carried out by self-employed persons. The sum of quantities times prices provides the labour input cost (or the labour bill, or labour compensation, as it is sometimes called).4

3A

classic treatment in the neo-classical framework, based on the distinction between shortrun and longrun output or cost, is provided by Morrison Paul (1999). For a modern treatment (with keywords: representative firm, dynamic cost minimization, Cobb-Douglas production function with constant returns to scale, Hicks-neutral technological change) the reader is referred to Comin et al. (2020). 4 The utilization rate of the labour input factors is assumed to be 1. Over- or underutilization from the point of view of jobs or persons is reflected in the wage rates. At the economy level, unutilized labour is called ‘unemployment’.

2.2 The Basic Input-Output Model

13

The classes K and L concern so-called primary inputs. The letters E, M, and S denote three, disjunct classes of so-called intermediate inputs. First, E is the class of energy commodities consumed by a production unit: oil, gas, electricity, and water. Second, M is the class of all the (physical) materials consumed in the production process, which could be sub-classified into raw materials, semi-fabricates, and auxiliary products. Third, S is the class of all the business services which are consumed for maintaining the production process. This includes the services of leased capital assets and outsourced activities. Though it is not at all a trivial task to define precisely all the intermediate inputs and to classify them, it can safely be assumed that at the end of each accounting period there is a quantity and a price associated with each of those inputs. Then, for each accounting period, production cost is defined as the sum of primary and intermediate input cost. Though this is usually not implemented at every aggregation level, there are good reasons to exclude R&D expenditure from production cost, the reason being that such expenditure is not related to the current production process but to a future one. Put otherwise, by performing R&D, production units try to shift the technology frontier. When it then comes to explaining productivity change, the non-exclusion of R&D expenditure might easily lead to a sort of double-counting error.5 At the output side, the letter Y denotes the class of commodities, goods and/or services, which are produced by the unit. Though in some industries, such as services industries or industries producing mainly unique goods, definitional problems are formidable, it can safely be assumed that for each accounting period there are data on quantities produced. For units operating on the market there are also prices. The sum of quantities times prices then provides the production revenue, and, apart from taxes on production, revenue minus cost yields profit.6 There is, of course, discussion possible about what to include or exclude at the input- and output-sides. We are here more or less tacitly assuming a broad production viewpoint, where for instance marketing services are included in the set S. A broader viewpoint would take into account sales and uses from inventories.7 Profit is an important financial performance measure. A somewhat less obvious, but equally useful, measure is ‘profitability’, defined as revenue divided by cost. Profitability gives, in monetary terms, the quantity of output per unit of input, and is thus a measure of return to aggregate input (and in some older literature called ‘return to the dollar’).

5 There

are a number of issues here, such as the separation of the R&D part of labour input, the precise definition of knowledge assets, and the distribution of R&D expenditures over the parts of multinational enterprises. See Diewert and Huang (2011) and De Haan and Haynes (2018). 6 Sometimes zero profit is imposed by considering profit as the remuneration (price) for entrepreneurial activity (of which the quantity is set equal to one), and adding this to business services S. In terms of National Accounts profit equals gross operating surplus (GOS) minus the imputed income of self-employed persons and capital input cost. 7 The inclusion of natural capital (including subsoil assets) in K is only meaningful if the production units are economies; see Brandt et al. (2017).

14

2 A Framework Without Assumptions

Monitoring the unit’s performance over time is here understood to mean monitoring the development of its profit or its profitability. Both measures are, by nature, dependent on price and quantity changes, at the two sides of the unit. If there is (price) inflation and the unit’s profit has increased then that mere fact does not necessarily mean that the unit has been performing better. Also, though general inflation does not influence the development of profitability, differential inflation does. If output prices have increased more than input prices then any increase of profitability does not necessarily imply that the unit has been performing better. Thus, for measuring the economic performance of the unit one wants to remove the effect of price changes, irrespective of whether those prices are within or beyond the unit’s control. Profit and profitability are different concepts. The first is a difference measure, the second is a ratio measure. Change of a variable through time, which will be our main focus, can also be measured by a difference or a ratio. It is important to realize that, apart from technical details—such as, that a ratio does not make sense if the variable changes sign or becomes equal to zero—, these two ways of measuring change are equivalent. Thus there appear to be a number of ways of mapping the same reality in numbers, but differing numbers do not necessarily imply differing realities.8 Profit change stripped of its price component will be called real profit change, and profitability change stripped of its price component will be called real profitability change.9 Another name for real profit (-ability) change is (total factor) productivity change. Thus, productivity change can be measured as a ratio (namely as real profitability change) or as a difference (namely as real profit change). At the economy level, productivity change can be related to some measure of overall welfare change. A down-to-earth approach would use the National Accounts to establish a link between labour productivity change and real-income-per-capita change. A more sophisticated approach, using economic models and assumptions, was provided by Basu and Fernald (2002). For a non-market unit the story must be told somewhat differently. For such a unit there are no output prices; hence, there is no revenue. Though there is cost, like for market units, there is no profit or profitability. National accountants usually resolve the problem here by defining the revenue of a non-market unit to be equal to its cost, thereby setting profit equal to 0 or profitability equal to 1.10 But this leaves the problem that there is no natural way of splitting revenue change through time in real

8 It

is easy to see, for example, that increasing profit can occur simultaneously with decreasing profitability. 9 Note that real change means nominal change deflated by some price index, not necessarily being a (headline) CPI. ‘Stripping’ is of course a vague term, and a more precise definition will be given later. 10 This approach goes back to Hicks (1940).

2.2 The Basic Input-Output Model

15

and monetary components. This can only be carried out satisfactorily when there is some output quantity index that is independent from the input quantity index.11 It is useful to remind the reader that the notions of profit and profitability, though conceptually rather clear, are difficult to operationalize. One of the reasons is that cost includes the cost of owned capital assets, the measurement of which exhibits a substantial number of degrees of freedom, as we will see in the remainder of this chapter. Also, labour cost includes the cost of self-employed persons, for which wage rates and hours of work usually must be imputed. It will be clear that all these, and many other, uncertainties spill over to operational definitions of the profit and profitability concepts.

2.2.1 Notation Let us now introduce some notation to define the various concepts we are going to use. As stated, at the output side we have M items, each with their price (received) t and quantity y t , where m = 1, . . . , M, and t denotes an accounting period (say, pm m a year). Similarly, at the input side we have N items, each with their price (paid) wnt and quantity xnt , where n = 1, . . . , N . To avoid notational clutter, simple vector notation will be used throughout. All the prices and quantities are assumed to be positive, unless stated otherwise. The ex post accounting point-of-view will be used; that is, quantities and monetary values of the so-called flow variables (output and labour, energy, materials, services inputs) are realized values, complete knowledge of which becomes available after the accounting period has expired. Similarly, the cost of capital input is calculated ex post. This is consistent with official statistical practice. The unit’s revenue, that is, the value of its (gross) output, during the accounting period t is defined as M 

R t ≡ pt · y t ≡

t t pm ym ,

(2.1)

wnt xnt .

(2.2)

m=1

whereas its (total) production cost is defined as C t ≡ wt · x t ≡

N  n=1

The unit’s profit (including taxes on production) is then given by its revenue minus its cost; that is,

11 More

on this in Diewert (2018).

16

2 A Framework Without Assumptions

t ≡ R t − C t = p t · y t − w t · x t .

(2.3)

The unit’s (gross-output based) profitability (also including taxes on production) is defined as its revenue divided by its cost; that is, ϒ t ≡ R t /C t = pt · y t /w t · x t .

(2.4)

Given positive prices and quantities, it will always be the case that R t > 0 and C t > 0. Thus, profitability ϒ t is always positive, but profit t can be positive, negative, or zero. Profit depends on the size of the production unit, but profitability not. Thus, for comparing the performance of production units having different sizes profitability is the measure to use. The next two performance concepts are margins. The first is the profit-cost margin of the production unit, defined as profit over cost, μt ≡

t . Ct

(2.5)

The relation between profit-cost margin and profitability is then given by μt = ϒ t − 1;

(2.6)

that is, the profit-cost margin is the profitability expressed as a percentage (which is usually published as μt × 100%). But what precisely does this mean? To get a clue, consider first the single-output case; that is M = 1. Then the production unit’s profitability reduces to   ϒ t = pt y t /C t = pt / C t /y t ;

(2.7)

that is, price over cost per unit. Put otherwise, the profit-cost margin μt is then the markup of price over unit cost. For the general, multi-output case, suppose that the cost can be allocated to the M t t t various outputs; that is, C t = m=1 Cm , where Cm is the cost of producing ym units of output m (m = 1, . . . , M). Then the unit’s profitability can be decomposed as M M t t t t  Cm pm m=1 pm ym ϒt =  = . M t t t t C Cm /ym m=1 Cm m=1

(2.8)

Thus profitability is a cost-share weighted mean of output-specific price over unitcost relatives. Put otherwise, the profit-cost margin is a cost-share weighted mean of output-specific markups,

2.2 The Basic Input-Output Model

17

μt =

M t  Cm μt , Ct m

(2.9)

m=1

t /(C t /y t ) − 1 (m = 1, . . . , M). where μtm ≡ pm m m The second margin is the profit-revenue margin, defined as profit over revenue,

νt ≡

t . Rt

(2.10)

The relation between profit-revenue margin and profitability is given by ν t = 1 − 1/ϒ t .

(2.11)

The profit-revenue margin is the percentage of revenue that is considered as profit (and usually published as ν t × 100%). Therefore this margin is also known as the return on sales. The two margin concepts are connected by the profitability concept. Connecting the three definitions yields ϒ t = μt /ν t ;

(2.12)

that is, profitability equals profit-cost margin divided by profit-revenue margin. Notice that the two margin concepts share with profit the property of being positive, zero, or negative. As stated, we are concerned with intertemporal comparisons. Moreover, in this chapter only bilateral comparisons will be considered, say comparing a certain period t to another, adjacent or non-adjacent, period t  . Without loss of generality it may be assumed that period t  precedes period t. We also may assume that periods t  and t are adjacent, and that the commodity sets at input- and output-side of the production unit are invariant. If not, and data of intermediate periods are available, then the technique of chaining indices may be used. To further simplify notation, the two periods will be labeled by t = 1 (which will be called the comparison period) and t  = 0 (which will be called the base period).

2.2.2 Total Factor Productivity Index The development over time of profitability is, rather naturally, measured by the ratio (R 1 /C 1 )/(R 0 /C 0 ). How to decompose this into a price and a quantity component? By noticing that ϒ1 R 1 /C 1 R 1 /R 0 = = ϒ0 R 0 /C 0 C 1 /C 0

(2.13)

18

2 A Framework Without Assumptions

we see that the question reduces to the question how to decompose the revenue ratio R 1 /R 0 and the cost ratio C 1 /C 0 into two parts. The natural answer is to grab from the economic statistician’s toolkit a price index P (.) and a quantity index Q(.) that satisfy the Product Test. Appendix A provides a summary of the contents of the toolkit. It appears that there are several possibilities, each with advantages and disadvantages. A good choice is the pair of Fisher price and quantity indices, since these satisfy not only the basic axioms of price and quantity measurement, but also a number of other relatively important requirements (such as the Time Reversal Test). Thus we are using here the ‘instrumental’ or ‘axiomatic’ approach for selecting measures for aggregate price and quantity change, an approach that goes back to Fisher (1922).12 When the temporal distance between the periods 0 and 1 is not too large, then any index that is a second-order differential approximation of the Fisher index may instead be used.13 Throughout this book, when it comes to solving such decomposition problems the reader is free to assume that Fisher indices are used, though nothing in the argument depends on this particular choice. Anyway, the revenue ratio is decomposed as R1 = P (p1 , y 1 , p0 , y 0 )Q(p1 , y 1 , p0 , y 0 ) R0 ≡ PR (1, 0)QR (1, 0),

(2.14)

where the second line serves to define our shorthand notation. In the same way the cost ratio is decomposed as C1 = P (w 1 , x 1 , w 0 , x 0 )Q(w 1 , x 1 , w 0 , x 0 ) C0 ≡ PC (1, 0)QC (1, 0).

(2.15)

Of course, the dimensionality of the indices in expressions (2.14) and (2.15) will usually be different. The functional forms need not be the same. The subscripts R and C are used because, as will appear later, there are more output and input concepts. The number of items distinguished at the output side (M) and the input side (N ) of a production unit can be very high. To accommodate this, (detailed) classifications are used, by which all the items are allocated to hierarchically organized (sub-)aggregates. The calculation of output and input indices then proceeds in

12 Balk

(2008) provides an up-to-date treatment, including its history and all the historical references. 13 Note, however, that this is not unproblematic. For instance, when the Törnqvist price index P T (.) is used, then the implicit quantity index (p 1 · y 1 /p 0 · y 0 )/P T (.) does not necessarily satisfy the Identity Test. The Identity Test for a quantity index prescribes that such an index equals unity whenever quantities have not changed.

2.2 The Basic Input-Output Model

19

stages. Theoretically, it suffices to distinguish only two stages. At the first stage one calculates indices for the subaggregates at some level, and at the second stage these subaggregate indices are combined to aggregate indices. Consequentially, in expressions (2.14) and (2.15) instead of one-stage also two-stage indices may be used; that is, indices of indices for subaggregates (see Appendix A for precise definitions). Since most indices are not Consistent-inAggregation, a decomposition by two-stage indices of whatever functional form will in general numerically differ from a decomposition by one-stage indices of the same form. Fortunately, one-stage and two-stage Fisher indices are second-order differential approximations of each other (as shown by Diewert 1978). Using the two relations (2.14) and (2.15), the profitability ratio can be decomposed as ϒ1 PR (1, 0) QR (1, 0) . = PC (1, 0) QC (1, 0) ϒ0

(2.16)

The gross-output based total factor productivity (TFP) index, for period 1 relative to period 0, is then defined by ITFPRODY (1, 0) ≡

QR (1, 0) . QC (1, 0)

(2.17)

Thus ITFPRODY (1, 0) is the real or quantity component of the profitability ratio. Put otherwise, it is the ratio of an output quantity index to an input quantity index; ITFPRODY (1, 0) is the factor with which the output quantities on average have changed relative to the factor with which the input quantities on average have changed. If the ratio of these factors is larger (smaller) than 1, there is said to be productivity increase (decrease).14 Notice that ITFPRODY (1, 0) is a function of all the output and input prices and quantities; that is ITFPRODY (1, 0) = ITFPRODY (p1 , y 1 , w 1 , x 1 , p0 , y 0 , w 0 , x 0 ). Its properties follow from those of the output and input quantity indices Q(p1 , y 1 , p0 , y 0 ) and Q(w 1 , x 1 , w 0 , x 0 ). In particular, • In the single-input-single-output case (N = M = 1) the TFP index reduces to 1 0 1 1 ITFPRODY (1, 0) = yx 1 /y = yy 0 /x ; that is, the ratio of the productivities in the /x 0 /x 0 two periods compared. • ITFPRODY (1, 0) exhibits the desired monotonicity properties: nondecreasing in y 1 , nonincreasing in x 1 , nonincreasing in y 0 , nondecreasing in x 0 . • ITFPRODY (1, 0) exhibits proportionality in input and output quantities; that is, ITFPRODY (p1 , μy 0 , w 1 , λx 0 , p0 , y 0 , w 0 , x 0 ) = μ/λ (λ, μ > 0). Expression (2.17) is the generic definition of the gross-output based TFP index. Using expressions (2.14) and (2.15), there appear to be three other, equivalent representations of the TFP index, namely 14 This

approach follows Diewert (1992), Diewert and Nakamura (2003), and Balk (2003b).

20

2 A Framework Without Assumptions

ITFPRODY (1, 0) =

(R 1 /R 0 )/PR (1, 0) (C 1 /C 0 )/PC (1, 0)

(2.18)

=

(R 1 /R 0 )/PR (1, 0) QC (1, 0)

(2.19)

=

QR (1, 0) . (C 1 /C 0 )/PC (1, 0)

(2.20)

Put in words, we are seeing here respectively a deflated revenue index divided by a deflated cost index,15 a deflated revenue index divided by an input quantity index, and an output quantity index divided by a deflated cost index. We will return to these expressions shortly. The other part of the profitability ratio decomposition (2.16) is called grossoutput based total price recovery (TPR) index: ITPRY (1, 0) ≡

PR (1, 0) . PC (1, 0)

(2.21)

This index measures the extent to which input price change is recovered by output price change. If revenue change equals cost change, R 1 /R 0 = C 1 /C 0 (for which zero profit in the two periods is a sufficient condition), then it follows that ITFPRODY (1, 0) =

1 ; ITPRY (1, 0)

(2.22)

that is, the TFP index is equal to the inverse of the TPR index. Though the inverse TPR index is sometimes used as a dual TFP index it is good to be aware of the fact that dual and primal index are generally different. For a non-market unit expression (2.17) cannot be used because there are no output prices available for use in the output quantity index. But if there is some prices-free output quantity index Q(y 1 , y 0 ), then the TFP index, for period 1 relative to period 0, is naturally defined by Q(y 1 , y 0 )/QC (1, 0). An alternative expression is obtained by replacing the input quantity index by the deflated cost index, Q(y 1 , y 0 )/[(C 1 /C 0 )/PC (1, 0)].

2.2.3 Growth Accounting (1) The foregoing definitions are already sufficient to provide examples of simple but useful analysis. Consider relation (2.19), and rewrite this as

15 See

Sheng et al. (2017) for an application to Australian agriculture over the period 1949–2012.

2.2 The Basic Input-Output Model

21

R 1 /R 0 = ITFPRODY (1, 0) × QC (1, 0) × PR (1, 0).

(2.23)

Taking logarithms, one obtains ln(R 1 /R 0 ) = ln ITFPRODY (1, 0) + ln QC (1, 0) + ln PR (1, 0).

(2.24)

This relation, implemented with Fisher and Törnqvist indices, was used by Dumagan and Ball (2009) for an analysis of the U. S. agricultural sector. Recall that revenue change through time is only interesting in so far it differs from general inflation. Hence, it makes sense to deflate the revenue ratio, R 1 /R 0 , by a general inflation measure such as the (headline) Consumer Price Index (CPI). Doing this, the last equation can be written as  ln

R 1 /R 0 CP I 1 /CP I 0



 = ln ITFPRODY (1, 0) + ln QC (1, 0) + ln

PR (1, 0) CP I 1 /CP I 0

 .

(2.25) Lawrence et al. (2006) basically used this relation to decompose ‘real’ revenue change into three factors: TFP change, input quantity change (which can be interpreted as measuring change of the unit’s size), and ‘real’ output price change respectively. Our second example follows from rearranging expression (2.20) and taking logarithms. This delivers the following relation: ln(C 1 /C 0 ) = ln PC (1, 0) + ln QR (1, 0) − ln ITFPRODY (1, 0).

(2.26)

This relation was also used by Dumagan and Ball (2009).16 A further rearrangement gives 

C 1 /C 0 ln QR (1, 0)

 = ln PC (1, 0) − ln ITFPRODY (1, 0).

(2.27)

We see here that the growth rate of average cost can be decomposed into two factors, namely the growth rate of input prices and a residual which is the negative of TFP growth. Put otherwise, in the case of stable input prices the growth rate of average cost is equal to minus the productivity growth rate. Our third example follows from rearranging expression (2.18) as ITFPRODY (1, 0) =

16 Note

1 1 + μ1 PC (1, 0), 1 + μ0 PR (1, 0)

that it is not necessary to assume that R t = C t (t = 0, 1).

(2.28)

22

2 A Framework Without Assumptions

where the profit-cost margin μt was defined by expression (2.5). Expression (2.28) can be read as picturing the distribution of the fruits of productivity increase: to the owner(s) of the production unit as the profit-cost margin increase, or to the customers as output prices decrease, or to the suppliers (including employees) as input prices (including wages) increase. Taking logarithms and rearranging a bit delivers the following relation: 

1 + μ1 ln PR (1, 0) = ln 1 + μ0

 + ln PC (1, 0) − ln ITFPRODY (1, 0),

(2.29)

This relation analyses output price change as resulting from three factors: change of the profit-cost margin, input price change, and, with a negative sign, TFP change. This relation may be used for regulatory purposes. For example, if the profit-cost margin is not allowed to grow then output prices may on average rise as input prices do, but adjusted by the percentage of TFP growth. All these are examples of what is called growth accounting. The relation between index number techniques and growth accounting techniques can, more generally, be seen as follows. Recall the generic definition of the TFP index in expression (2.17), and rewrite this as QR (1, 0) = ITFPRODY (1, 0) × QC (1, 0).

(2.30)

Taking logarithms, this multiplicative expression can be rewritten as ln QR (1, 0) = ln ITFPRODY (1, 0) + ln QC (1, 0).

(2.31)

For index numbers in the neighbourhood of 1 the logarithms thereof reduce to percentages, and the last expression can then be interpreted as saying that the percentage change of output volume equals the percentage change of input volume plus the percentage change of TFP. Growth accounting economists like to work with equations expressing output volume growth in terms of input volume growth plus a residual that is interpreted as TFP growth, thereby suggesting that the last two factors cause the first. However, TFP change cannot be considered as an independent factor since it is defined as output quantity change minus input quantity change. Put otherwise, a growth accounting table is nothing but an alternative way of presenting TFP growth and its contributing factors. And decomposition does not imply anything about causality.17 The relation between the last two expressions and production functions is explored in Balk (2018).

17 Thus,

saying that output growth outpaced input growth because TFP increased is “like saying that the sun rose because it was morning”, to paraphrase Friedman (1988, p. 58). Of course, when TFP change is decomposed into factors such as technological change or efficiency change, and one is able to measure such factors independently, more can be said.

2.2 The Basic Input-Output Model

23

2.2.4 Total Factor Productivity Indicator Let us now turn to profit and its development through time. This is naturally measured by the difference 1 − 0 . Of course, such a difference only makes sense when the two money amounts involved, profit from period 0 and profit from period 1, are deflated by some general inflation measure (such as the headline CPI). In the remainder of this paper, when discussing difference measures, such a deflation is tacitly presupposed. How to decompose the profit difference into a price and a quantity component? By noticing that 1 − 0 = (R 1 − R 0 ) − (C 1 − C 0 ),

(2.32)

we see that the question reduces to the question how to decompose revenue change R 1 − R 0 and cost change C 1 − C 0 into two parts. We now grab from the economic statistician’s toolkit—see Appendix A—a price indicator P(.) and a quantity indicator Q(.) that satisfy the analogue of the Product Test. A good choice is the pair of Bennet (1920) price and quantity indicators, since these indicators satisfy not only the basic axioms, but also a number of other relatively important requirements (such as the Time Reversal Test). But any indicator that is a secondorder differential approximation to the Bennet indicator may instead be used. The Bennet indicators are difference analogues to Fisher indices. Being indicators, however, their aggregation properties are much simpler. Thus, let R 1 − R 0 = P(p1 , y 1 , p0 , y 0 ) + Q(p1 , y 1 , p0 , y 0 ) ≡ PR (1, 0) + QR (1, 0),

(2.33)

and similarly, C 1 − C 0 = P(w 1 , x 1 , w 0 , x 0 ) + Q(w 1 , x 1 , w 0 , x 0 ) ≡ PC (1, 0) + QC (1, 0).

(2.34)

Notice that the dimensionality of the indicators in these two decompositions is in general different. Also the functional forms may or may not be the same. The profit difference can then be written as 1 − 0 = PR (1, 0) + QR (1, 0) − [PC (1, 0) + QC (1, 0)] = PR (1, 0) − PC (1, 0) + QR (1, 0) − QC (1, 0).

(2.35)

24

2 A Framework Without Assumptions

The first two terms at the right-hand side of the last equality sign provide the price component, whereas the last two terms provide the quantity component of the profit difference. Thus, based on this decomposition, the gross-output based TFP indicator is defined by DTFPRODY (1, 0) ≡ QR (1, 0) − QC (1, 0);

(2.36)

that is, an output quantity indicator minus an input quantity indicator. Notice that TFP change is now measured as an amount of money. An amount larger (smaller) than 0 indicates TFP increase (decrease).18 The equivalent expressions for difference-type TFP change are DTFPRODY (1, 0) = [R 1 − R 0 − PR (1, 0)] − [C 1 − C 0 − PC (1, 0)]

(2.37)

= [R 1 − R 0 − PR (1, 0)] − QC (1, 0)

(2.38)

= QR (1, 0) − [C − C − PC (1, 0)],

(2.39)

1

0

which can be useful in different situations. The gross-output based total price recovery (TPR) indicator is defined as DTPRY (1, 0) ≡ PR (1, 0) − PC (1, 0).

(2.40)

This is also an amount of money. It is clear that 1 − 0 = DTFPRODY (1, 0) +DTPRY (1, 0), and that, if 1 = 0 then DTFPRODY (1, 0) = −DTPRY (1, 0).

(2.41)

Put otherwise, if profit does not change then the negative of the total price recovery indicator measures TFP change. For a non-market production unit, a TFP indicator is difficult to define. Though one might be able to construe an output quantity indicator, it is hard to see how, in the absence of output prices, such an indicator could be given a money dimension.

2.2.5 The Equivalence of Multiplicative and Additive Models The fundamental equivalence of profit and profitability is expressed by the following relation:

18 This

approach follows Balk (2003b).

2.2 The Basic Input-Output Model

25

t = LM(R t , C t ) (t = 0, ϒ t = 1), ln ϒ t

(2.42)

where LM(.) denotes the logarithmic mean.19 Put in words, this expression says that profit divided by the logarithm of profitability is equal to the logarithmic mean of revenue and cost. Using this expression, the profit difference can be written as 1 − 0 = LM(R 1 , C 1 ) ln ϒ 1 − LM(R 0 , C 0 ) ln ϒ 0 ,

(2.43)

which can be decomposed symmetrically as20 1 LM(R 0 , C 0 ) + LM(R 1 , C 1 ) ln(ϒ 1 /ϒ 0 ) 2  + ln(ϒ 0 ϒ 1 )1/2 LM(R 1 , C 1 ) − LM(R 0 , C 0 ) .

1 − 0 =

(2.44)

Hence, given that the profitability ratio can be decomposed as ϒ 1 /ϒ 0 = ITFPRODY (1, 0) × ITPR  Y (1, 0), the profit difference  decomposes as the sum of a productivity effect, 12 LM(R 0 , C 0 ) + LM(R 1 , C 1 ) ln ITFPRODY (1, 0), a price   recovery effect 12 LM(R 0 , C 0 ) + LM(R 1 , C 1 ) ln ITPRY (1, 0), and a level effect   ln(ϒ 0 ϒ 1 )1/2 LM(R 1 , C 1 ) − LM(R 0 , C 0 ) . Reversely, the logaritm of the profitability ratio can be written as ln(ϒ 1 /ϒ 0 ) = ln ϒ 1 − ln ϒ 0 =

0 1 − , LM(R 1 , C 1 ) LM(R 0 , C 0 )

(2.45)

which can be decomposed symmetrically as 0



 1 1 + (1 − 0 ) LM(R 0 , C 0 ) LM(R 1 , C 1 )   1 1 1 − . + (0 + 1 ) 2 LM(R 1 , C 1 ) LM(R 0 , C 0 )

1 ln(ϒ /ϒ ) = 2 1

(2.46)

Hence, given that the profit difference can be decomposed as 1 − 0 = DTFPRODY (1, 0) + DTPRY (1, 0), the logarithm of the profitability ratio decomposes as the sum of a productivity effect 12 (1/LM(R 0 , C 0 ) + 1/LM(R 1 , C 1 ))DTFPRODY (1, 0), a price recovery effect 12 (1/LM(R 0 , C 0 ) + 1/LM(R 1 , 19 For

any two strictly positive real numbers a and b their logarithmic mean is defined by LM(a, b) ≡ (a − b)/ ln(a/b) when a = b, and LM(a, a) ≡ a. It has the following properties: (1) min(a, b) ≤ LM(a, b) ≤ max(a, b); (2) LM(a, b) is continuous; (3) LM(λa, λb) = λLM(a, b) (λ > 0); (4) LM(a, b) = LM(b, a); (5) (ab)1/2 ≤ LM(a, b) ≤ (a + b)/2; (6) LM(a, 1) is concave. See Balk (2008, 134–136) for details. 20 This is based on the identity a 1 b1 − a 0 b0 = (1/2)(a 0 + a 1 )(b1 − b0 ) + (1/2)(b0 + b1 )(a 1 − a 0 ).

26

2 A Framework Without Assumptions

C 1 ))DTPRY (1, 0), and an inverse level effect 12 (0 + 1 )(1/LM(R 1 , C 1 ) − 1/LM(R 0 , C 0 )). An alternative set of decompositions sheds more light on the relation between the two models. We now depart from expressions (2.14) and (2.15), where the revenue ratio and the cost ratio are decomposed into price and quantity indices. Taking the logarithm and applying the definition of the logarithmic mean to the left-hand side, these expressions can be written as R 1 − R 0 = LM(R 0 , R 1 ) ln PR (1, 0) + LM(R 0 , R 1 ) ln QR (1, 0),

(2.47)

C 1 − C 0 = LM(C 0 , C 1 ) ln PC (1, 0) + LM(C 0 , C 1 ) ln QC (1, 0),

(2.48)

and

respectively. According to expression (2.32) the profit difference then decomposes as 1 − 0 = LM(R 0 , R 1 ) ln PR (1, 0) − LM(C 0 , C 1 ) ln PC (1, 0) + LM(R 0 , R 1 ) ln QR (1, 0) − LM(C 0 , C 1 ) ln QC (1, 0).

(2.49)

The first line gives the price effect, and the second line gives the quantity effect. The price effect in turn can be decomposed symmetrically as LM(R 0 , R 1 ) ln PR (1, 0) − LM(C 0 , C 1 ) ln PC (1, 0) = 1 LM(R 0 , R 1 ) + LM(C 0 , C 1 ) ln ITPRY (1, 0) 2  1 + ln (PR (1, 0)PC (1, 0)) LM(R 0 , R 1 ) − LM(C 0 , C 1 ) . 2

(2.50)

The first factor on the right-hand side of this expression is the price recovery effect, which vanishes if and only if output price change equals input price change, PR (1, 0) = PC (1, 0). Following Grifell-Tatjé and Lovell (2015, 208–224), the second factor can be called a margin effect. A sufficient condition for becoming equal to 0 is that mean revenue equals mean cost, LM(R 0 , R 1 ) = LM(C 0 , C 1 ) (a sufficient condition for which is that in both periods profit equals 0), or output price change is reciprocal to input price change, PR (1, 0) = 1/PC (1, 0). Similarly, the quantity effect can be decomposed as LM(R 0 , R 1 ) ln QR (1, 0) − LM(C 0 , C 1 ) ln QC (1, 0) = 1 LM(R 0 , R 1 ) + LM(C 0 , C 1 ) ln ITFPRODY (1, 0) 2  1 + ln (QR (1, 0)QC (1, 0)) LM(R 0 , R 1 ) − LM(C 0 , C 1 ) . 2

(2.51)

2.2 The Basic Input-Output Model

27

The first factor on the right-hand side of this expression is the productivity effect, which vanishes if and only if output quantity change equals input quantity change, QR (1, 0) = QC (1, 0). The second factor can again be called a margin effect. A sufficient condition for becoming equal to 0 is that mean revenue equals mean cost, LM(R 0 , R 1 ) = LM(C 0 , C 1 ) (a sufficient condition for which is that in both periods profit equals 0), or output quantity change is reciprocal to input quantity change, QR (1, 0) = 1/QC (1, 0). Summarizing, given that the profitability ratio can be decomposed into two factors, ϒ 1 /ϒ 0 = ITFPRODY (1, 0)ITPRY (1, 0), the profit difference 1 − 0 decomposes into four factors, namely a productivity effect, a price recovery effect, and two margin effects. The existence of these margin effects “makes the differencebased model of profit change richer than the ratio-based model of profitability change”, according to Grifell-Tatjé and Lovell (2015, 201). This conclusion, however, is a bit superficial as will be demonstrated. We now depart from expressions (2.33) and (2.34), where the revenue difference and the cost difference are decomposed into price and quantity indicators. Applying the definition of the logarithmic mean to the left-hand side, these expressions can be written as ln(R 1 /R 0 ) =

1 1 PR (1, 0) + QR (1, 0), LM(R 0 , R 1 ) LM(R 0 , R 1 )

(2.52)

ln(C 1 /C 0 ) =

1 1 PC (1, 0) + QC (1, 0), LM(C 0 , C 1 ) LM(C 0 , C 1 )

(2.53)

and

respectively. According to expression (2.13) the logarithm of the profitability ratio then decomposes as ln(ϒ 1 /ϒ 0 ) = +

1 1 PR (1, 0) − PC (1, 0) LM(R 0 , R 1 ) LM(C 0 , C 1 ) 1 1 QR (1, 0) − QC (1, 0). 0 1 LM(R , R ) LM(C 0 , C 1 )

(2.54)

The first line gives the price effect, and the second line gives the quantity effect. The price effect in turn can be decomposed symmetrically as 1 1 PR (1, 0) − PC (1, 0) = LM(R 0 , R 1 ) LM(C 0 , C 1 )   1 1 1 + DTPRY (1, 0) 2 LM(R 0 , R 1 ) LM(C 0 , C 1 )   1 1 1 − . + (PR (1, 0) + PC (1, 0)) 2 LM(R 0 , R 1 ) LM(C 0 , C 1 )

(2.55)

28

2 A Framework Without Assumptions

The first factor on the right-hand side of this expression is the price recovery effect, which vanishes if and only if output price change equals input price change, PR (1, 0) = PC (1, 0). The second factor is a margin effect. A sufficient condition for vanishing is that mean revenue equals mean cost, LM(R 0 , R 1 ) = LM(C 0 , C 1 ) (a sufficient condition for which is that in both periods profit equals 0), or output price change is the negative of input price change, PR (1, 0) = −PC (1, 0). Similarly, the quantity effect can be decomposed as 1 1 QR (1, 0) − QC (1, 0) = 0 1 LM(R , R ) LM(C 0 , C 1 )   1 1 1 + DTFPRODY (1, 0) 2 LM(R 0 , R 1 ) LM(C 0 , C 1 )   1 1 1 − + (QR (1, 0) + QC (1, 0)) . 2 LM(R 0 , R 1 ) LM(C 0 , C 1 )

(2.56)

The first factor on the right-hand side of this expression is the productivity effect, which vanishes if and only if output quantity change equals input quantity change, QR (1, 0) = QC (1, 0). The second factor is again a margin effect. A sufficient condition for vanishing is that mean revenue equals mean cost, LM(R 0 , R 1 ) = LM(C 0 , C 1 ) (a sufficient condition for which is that in both periods profit equals 0), or output quantity change is the negative of input quantity change, QR (1, 0) = −QC (1, 0). Summarizing, given that the profit difference can be decomposed into two factors, 1 −0 = DTFPRODY (1, 0)+DTPRY (1, 0), the profitability ratio ϒ 1 /ϒ 0 decomposes into four factors, namely a productivity effect, a price recovery effect, and two margin effects. The overall conclusion of this section is that the two models, the multiplicative model based on profitability and the additive model based on profit, are completely equivalent.

2.2.6 Partial Productivity Measures The productivity index ITFPRODY (1, 0) and indicator DTFPRODY (1, 0) bear the adjective ‘total factor’ because all the inputs are taken into account. To define partial productivity measures, in ratio or difference form, additional notation is necessary.21 All the items at the input side of our production unit are assumed to be allocatable to the five, mutually disjunct, categories mentioned earlier, namely capital (K), labour (L), energy (E), materials (M), and services (S). The entire input price and

21 Though,

taken strictly, ‘multi factor’ differs from ‘total factor’, the two adjectives are generally used as synonymous.

2.2 The Basic Input-Output Model

29

t , w t , w t , w t , w t ) and quantity vectors can then be partitioned as wt = (wK L E M S t t t t t t x = (xK , xL , xE , xM , xS ) respectively. Energy, materials and services together form the category of intermediate inputs, that is, inputs which are acquired from other production units or imported. Capital and labour are called primary inputs. Consistent with this distinction the price and quantity vectors can also be partitioned t , wt t t t t t t t as w t = (wKL EMS ) and x = (xKL , xEMS ), or as w = (wK , wL , wEMS ) and t t t t x = (xK , xL , xEMS ). Since monetary values are additive, total production cost can be decomposed in a number of ways, such as

Ct =



wnt xnt +

n∈K

≡ ≡ ≡



wnt xnt +

n∈L

t t CK + CLt + CEt + CM t t CK + CLt + CEMS t t CKL + CEMS .



wnt xnt +

n∈E



wnt xnt +

n∈M



wnt xnt

n∈S

+ CSt

(2.57)

Based on the last line total production cost change can be decomposed as 0 1 0 C1 CEMS CEMS CKL C1 KL = + , 0 0 C0 C 0 CKL C 0 CEMS

(2.58)

but also as ⎛

−1

−1 ⎞−1 1 1 1 1 CEMS CEMS CKL CKL C1 ⎠ , =⎝ 1 + 0 0 C0 C C1 CKL CEMS

(2.59)

and then each ratio can be decomposed further. For instance, the labour-cost ratio can be decomposed as CL1 CL0

= P (wL1 , xL1 , wL0 , xL0 )Q(wL1 , xL1 , wL0 , xL0 ) ≡ PL (1, 0)QL (1, 0),

(2.60)

for a suitable pair of price and quantity indices. The gross-output based labour productivity index for period 1 relative to period 0 is defined by ILPRODY (1, 0) ≡

QR (1, 0) ; QL (1, 0)

(2.61)

that is, the ratio of an output quantity index to a labour input quantity index. Notice that labour is here not considered as an homogeneous commodity. The class L

30

2 A Framework Without Assumptions

consists of several types of labour, each with their quantities xnt (say, hours worked) and prices wnt (say, wage per hour). Conventionally however, labour is considered as homogeneous, so that the  t denote the total labour input quantities xnt can be added up. Let Lt ≡ x n∈L n of our production unit (say, measured in hours worked of whatever type) during period t (t = 0, 1).22 Then the gross-output based simple labour productivity index for period 1 relative to period 0 is defined by ISLPRODY (1, 0) ≡

QR (1, 0) . L1 /L0

(2.62)

Formally, L1 /L0 is a simple sum or Dutot quantity index. The ratio of the two labour productivity indices, LQUAL(1, 0) ≡

QL (1, 0) ISLPRODY (1, 0) = , ILPRODY (1, 0) L1 /L0

(2.63)

is said to measure the shift in ‘labour quality’; actually this ratio measures the shift in the composition of labour input. In precisely the same way one can define the gross-output based capital productivity index23 IKPRODY (1, 0) ≡

QR (1, 0) QK (1, 0)

(2.64)

and the other partial productivity indices: IEPRODY (.), IMPRODY (.), and ISPRODY (.) for energy, materials, and services, respectively. The ratio ILPRODY (1, 0) QK (1, 0) = IKPRODY (1, 0) QL (1, 0)

(2.65)

is called the index of ‘capital deepening’. Loosely speaking, this index measures the change of the quantity of capital input per unit of labour input. The relation between total factor and partial productivity indices can be derived as follows. For instance, let QC (1, 0) be a two-stage index of the form QC (1, 0) ≡ QF (Qk (1, 0), Ck1 , Ck0 ; k = K, L, E, M, S)

22 The

(2.66)

notation used might be slightly confusing, yet has been chosen so as to stay in line with conventional practice. Capital L without super- or subscripts denotes the set of labour types. Capital L with a time supercript denotes the total number of labour units. 23 Capital productivity is closely related to the business accounting concept ‘(total) asset turnover ratio’, defined as net sales divided by (the value of) total assets of a company. In the literature this is considered as a measure of efficiency.

2.2 The Basic Input-Output Model

31

where all the Qk (1, 0) are arbitrary indices and the second-stage index is a Fisher index. It is straightforward to check that then ITFPRODY (1, 0) = = 

QR (1, 0) QC (1, 0)

(2.67)

QR (1, 0) 1/2   0 0 −1 1 1 −1/2 k Qk (1, 0)Ck /C k Qk (1, 0) Ck /C



−1/2

1/2  Qk (1, 0) C 0  QR (1, 0) C 1 k k = QR (1, 0) C 0 Qk (1, 0) C 1 k

k

 =

k

Ck0 (IkPRODY (1, 0))−1 C0

−1/2  k

Ck1 IkPRODY (1, 0) C1

1/2 ,

where k = K, L, E, M, S. This is not a particularly simple relation; the first factor at the right-hand side is an harmonic mean, but the second factor is an arithmetic mean. If instead of the Fisher as second-stage quantity index the Cobb-Douglas functional form was chosen, that is,   Qk (1, 0)αk where αk = 1 (αk > 0), (2.68) QC (1, 0) ≡ k

k

then it appears that ln ITFPRODY (1, 0) =



αk ln IkPRODY (1, 0).

(2.69)

k

This is a very simple relation between total factor productivity change and partial productivity change. Notice, however, that this simplicity comes at a cost. Definition (2.68) implies for the relation between aggregate and subaggregate input price indices that PC (1, 0) =

 k

Pk (1, 0)αk 

C 1 /C 0 0 αk 1 k (Ck /Ck )

.

(2.70)

Such an index does not necessarily satisfy the fundamental Identity Test; that is, if all the prices in period 1 are the same as in period 0 then PC (1, 0) does not necessarily deliver as outcome 1. Can we also define partial productivity indicators? Given a set of indicators, the labour-cost difference between periods 0 and 1 is decomposed as CL1 − CL0 = P(wL1 , xL1 , wL0 , xL0 ) + Q(wL1 , xL1 , wL0 , xL0 )

32

2 A Framework Without Assumptions

≡ PL (1, 0) + QL (1, 0).

(2.71)

In the same way one can decompose the capital, energy, materials, and services cost difference. However, since costs are additive, it turns out that the TFP indicator can be written as  Qk (1, 0). (2.72) DTFPRODY (1, 0) = QR (1, 0) − k=K,L,E,M,S

By definition, the left-hand side measures real profit change. The right-hand side provides the contributing factors. The contribution of category k to real profit change is simply measured by the amount Qk (1, 0). A positive amount, which means that the aggregate quantity of input category k has increased, means a negative contribution to real profit change. Ball et al. (2015) called QR (1, 0) − QE (1, 0) a partial energy productivity indicator. However, notice that summing all those partial indicators,  k=K,L,E,M,S (QR (1, 0) − Qk (1, 0)), does not lead to the overall indicator, DTFPRODY (1, 0), which makes the interpretation of such partial indicators difficult.

2.3 Different Models, Similar Measures The previous section laid out the basic features of what is known as the KLEMS model of production. This framework is currently used by a number of statistical agencies for productivity measures at the industry or economy level of aggregation; for instance, the U. S. Bureau of Labor Statistics (Dean and Harper 2001; Eldridge et al. 2018), Statistics Canada (Baldwin et al. 2007), and Statistics Netherlands (Van den Bergen et al. 2008, De Haan et al. 2014; the last publication is specifically devoted to the effects of extending the scope of the capital input component). The KLEMS model, or, as it will be denoted in this book, the KLEMS-Y model delivers gross-output based total or partial productivity measures. However, there are more models in use, differing from the KLEMS-Y model by their input and output concepts. Since these models presuppose revenue as measured independently from cost, they are not applicable to non-market units.

2.3.1 The KL-VA Model The first of these models uses value added (VA) as its output concept. The production unit’s value added is defined as its revenue minus the costs of energy, materials, and services; that is,

2.3 Different Models, Similar Measures

33

t VAt ≡ R t − CEMS t t = pt · y t − wEMS · xEMS .

(2.73)

The value-added concept subtracts the total cost of intermediate inputs from the revenue obtained, and in doing so essentially conceives the unit as producing value added (that is, money) from the two primary input categories capital and labour. It is assumed that VAt > 0.24 The fundamental accounting relation of our production unit was given by expression (2.3) as C t + t = R t . Subtracting from the left- and the right-hand side t intermediate inputs cost CEMS , using expressions (2.57) and (2.73), this accounting t + t = VAt . Thus the same production unit can be relation transforms into CKL described by two different accounting relations, featuring different input and output concepts. Although gross output, represented by the quantity vector y t , is the natural output concept, the value-added concept is important when one wants to aggregate single units to larger entities. Gross output consists of deliveries to final demand and intermediate destinations. The split between these two output categories depends very much on the level of aggregation. Value added is immune to this problem. It enables one to compare (units belonging to) different industries. From a welfaretheoretic point of view the value-added concept is important because value added can be conceived as the income (from production) that flows into society.25 In the KL-VA input-output model it is natural to define (value-added based) profitability as the ratio of value added to primary inputs cost, t t ≡ VAt /CKL , ϒVA

(2.74)

and the natural starting point for defining a productivity index is to consider the development of this ratio through time. Since 1 ϒVA 0 ϒVA

=

VA1 /VA0 1 /C 0 CKL KL

,

(2.75)

24 An early advocate of the value added output concept was Burns (1930). Specifically, he favoured

what later in this chapter will be defined as net value added. Burns was aware of the possibility that for very narrowly defined production units and small time periods value added may sometimes become non-positive. The Covid-19 pandemic caused in economies all over the world dramatic declines of (nominal or real) value added, at various levels of aggregation, but never to the extent of becoming negative. 25 In between the KLEMS-Y model and the KL-VA model figures the KLE”M”S-Margin model, applicable to distributive trade units. Here the set of material inputs M is split into two parts, M  denoting the goods for resale and M  the auxiliary materials. Likewise E, the set of energy inputs, is split into E  and E  . The Margin is then defined as R t −CEt  ∪M  . See Inklaar and Timmer (2008).

34

2 A Framework Without Assumptions

we need a decomposition of the value-added ratio and a decomposition of the primary inputs cost ratio. The question how to decompose a value-added ratio in a price and a quantity component cannot be answered unequivocally. There are several options here, the technical details of which are deferred to Appendix B. Suppose, thus, that a satisfactory decomposition is somehow available; that is, VA1 = PVA (1, 0)QVA (1, 0). VA0

(2.76)

And using one- or two-stage indices, let the primary inputs cost ratio be decomposed as 1 CKL 0 CKL

1 1 0 0 1 1 0 0 = P (wKL , xKL , wKL , xKL )Q(wKL , xKL , wKL , xKL )

≡ PKL (1, 0)QKL (1, 0).

(2.77)

The value-added based total factor productivity (TFP) index for period 1 relative to period 0 is then defined as ITFPRODVA (1, 0) ≡

QVA (1, 0) . QKL (1, 0)

(2.78)

This index measures the ‘quantity’ change of value added relative to the quantity change of primary input; or, ITFPRODVA (.) can be seen as an index of real value added relative to an index of real primary input. This is by far the most common model, used by many official statistical agencies in their productivity statistics. For an up-to-date overview the reader is referred to OECD (2019). As observed above, profit in the KL-VA model is the same as profit in the KLEMS-Y model, and the same applies to the price and quantity components of profit differences. Given the canonical form of price and quantity indicators, one easily checks that DTFPRODVA (1, 0) ≡ QVA (1, 0) − QKL (1, 0) = QR (1, 0) − QC (1, 0) = DTFPRODY (1, 0);

(2.79)

that is, the TFP indicators are the same in the two models. This, however, does not hold for the TFP indices. One usually finds that ITFPRODVA (1, 0) = ITFPRODY (1, 0). In Appendix C it is shown that if profit is zero in both periods, that is, R t = C t (t = 0, 1), then, for certain two-stage indices that are second-order differential approximations to Fisher indices,

2.3 Different Models, Similar Measures

35

ln ITFPRODVA (1, 0) = D(1, 0) ln ITFPRODY (1, 0),

(2.80)

where D(1, 0) ≥ 1 is the (mean) Domar factor (= ratio of revenue over value added). Usually expression (2.80) is, in a continuous-time setting, derived under a set of strong neo-classical assumptions (see, for instance, Gollop 1979, Jorgenson et al. 2005, 298, or Schreyer 2001, 143), so that it seems to be some deep economic-theoretical result. Appendix C shows, however, that the inequality of the value-added based TFP index and the gross-output based TFP index is only due to the mathematics of ratios and differences. There is no underlying economic phenomenon. Neither is one index more ‘natural’ than the other. The value-added based labour productivity index for period 1 relative to period 0 is defined as ILPRODVA (1, 0) ≡

QVA (1, 0) , QL (1, 0)

(2.81)

where QL (1, 0) was defined by expression (2.60). The index defined by expression (2.81) measures the ‘quantity’ change of value added relative to the quantity change of labour input; or, ILPRODVA (.) can be seen as an index of real value added relative to an index of real labour input. Recall that the labour quantity index QL (1, 0) is here defined as a genuine quantity index, acting on the prices and quantities of all the types of labour that are being distinguished. Suppose that the units of measurement of the various types are in some sense the same; that is, the quantities of all the types are measured in hours, or in full-time equivalent jobs, or in some other common unit. Then one frequently considers, instead of QL (1, 0), the Dutot or simple sum quantity index, QD L (1, 0) ≡

 n∈L

xn1 /



xn0 = L1 /L0 .

(2.82)

n∈L

The value-added based simple labour productivity index, defined as ISLPRODVA (1, 0) ≡

QVA (1, 0) , L1 /L0

(2.83)

has the alternative interpretation as an index of real value added per unit of labour. This measure frequently figures at the left-hand side—that is, as explanandum— in a growth accounting equation. However, for deriving such a relation nothing spectacular is needed, as will now be shown.

36

2 A Framework Without Assumptions

2.3.2 Growth Accounting (2) The simplest growth accounting equation is a rewritten version of expression (2.83), QVA (1, 0) = ISLPRODVA (1, 0) × L1 /L0 .

(2.84)

Applied to GDP, this equation gives rise to ‘explanations’ of economic growth such as “mainly due to an increase of hours worked rather than an increase of labour productivity.” Such sentences, however, exhibit the mistake to consider labour productivity as an independent factor that can cause economic growth. It is, however, nothing but the result of dividing value-added based growth by labour input growth. Thus, an effect rather than a cause. Consider instead the definition of the value-added based TFP index, given by expression (2.78), and rewrite this as QVA (1, 0) = ITFPRODVA (1, 0) × QKL (1, 0).

(2.85)

Dividing both sides of this identity by the Dutot labour quantity index, and applying the definition of the simple labour productivity index, given by expression (2.83), one obtains a relation between the two productivity indices, ISLPRODVA (1, 0) = ITFPRODVA (1, 0) ×

QKL (1, 0) QL (1, 0) × . QL (1, 0) L1 /L0

(2.86)

Taking logarithms and, on the assumption that all the index numbers are in the neighbourhood of 1, interpreting these as percentages, the last equation can be interpreted as: (simple) labour productivity growth equals TFP growth plus ‘capital deepening’ plus ‘labour quality’ growth. Again, TFP change is measured as a residual and, thus, the three factors on the right-hand side of the last equation can in no way be regarded as causal factors. If, continuing our previous example (in Section 2.2.6), the primary inputs quantity index was defined as a two-stage index of the form QKL (1, 0) ≡ QK (1, 0)α QL (1, 0)1−α (0 < α < 1),

(2.87)

where the reader recognizes the simple Cobb-Douglas functional form,26 then the index of ‘capital deepening’ reduces to the particularly simple form   QKL (1, 0) QK (1, 0) α = ≡ CAPDEEP(1, 0)α . QL (1, 0) QL (1, 0)

26 The

typical magnitude of α is 0.3, as in Gordon (2016, 546).

(2.88)

2.3 Different Models, Similar Measures

37

The ‘labour quality’ index, QL (1, 0)/(L1 /L0 ), basically measures compositional shift or structural change among the labour types in the class L.27 Indeed, by using the linear homogeneity of the quantity index QL (.) we see that LQUAL(1, 0) ≡

QL (1, 0) = Q(wL1 , xL1 /L1 , wL0 , xL0 /L0 ). L1 /L0

(2.89)

The right-hand side of this equation is an index comparing the comparison period vector of labour shares xL1 /L1 to the base period vector of labour shares xL0 /L0 . Expression (2.86) may then be rewritten as ISLPRODVA (1, 0) =

(2.90)

ITFPRODVA (1, 0) × CAPDEEP(1, 0)α × LQUAL(1, 0). This is an interesting relation, materializing time and again in the growth accounting literature. In terms of logarithms, the usual reading is that simple labour productivity growth comes from three sources: TFP growth—reflecting technological change—, capital deepening, and labour quality growth. Notice that it can occur that labour productivity growth is positive (negative) while TFP growth is negative (positive). It all depends on the countervailing forces of capital deepening and labour quality. Expression (2.90) indeed appears as the centerpiece of Solow’s growth theory, as exposed in Chapter 5 of Banerjee and Duflo (2019). In the long run—whatever that may be—any economy is supposed to be on a “balanced growth” path. This means that there is no more capital deepening, and labour quality does not increase anymore. Then, obviously, simple labour productivity moves in tandem with TFP; or, economic growth is determined by TFP growth plus growth of, say, the labour force. Instead of capital deepening as defined in expression (2.88), one can opt for simple capital deepening, defined as SCAPDEEP(1, 0) ≡

QK (1, 0) . L1 /L0

(2.91)

Expression (2.86) then reduces to ISLPRODVA (1, 0) =

(2.92)

ITFPRODVA (1, 0) × SCAPDEEP(1, 0)α × LQUAL(1, 0)1−α . This is the growth accounting equation employed by Niebel and Saam (2016). Capital deepening as a ‘driving force’ can be replaced by (value-added based) capital intensity, defined as 27 An

example is provided by Niebel et al. (2017).

38

2 A Framework Without Assumptions

CAPINTENSVA (1, 0) ≡

QK (1, 0) . QVA (1, 0)

(2.93)

Notice that the capital intensity index is the reciprocal of the capital productivity index (both value-added based). Then ISLPRODVA (1, 0) =

(2.94)

ITFPRODVA (1, 0)1/(1−α) × CAPINTENSVA (1, 0)α/(1−α) × LQUAL(1, 0) is an alternative version of expression (2.86). Another alternative of this expression is obtained by substituting back the definition of ITFPRODVA (1, 0), and replacing the two occurences of QKL (1, 0) by QK (1, 0), ISLPRODVA (1, 0) =

QVA (1, 0) QK (1, 0) QL (1, 0) . × × QK (1, 0) QL (1, 0) L1 /L0

(2.95)

At the right-hand side we see, resectively, value-added based capital productivity change, capital deepening, and labour quality change. Thus, using the various definitions, expression (2.95) may be written as ISLPRODVA (1, 0) =

(2.96)

IKPRODVA (1, 0) × CAPDEEP(1, 0) × LQUAL(1, 0). It is tempting to consider the Eqs. (2.90), (2.92), (2.94), and (2.96) as a kind of production functions. One should be aware, however, that these equations are nothing but rewritten versions of the definition of value-added based TFP change, combined with the definition of labour productivity change, several other definitions, and/or a special choice of the functional form for the primary inputs quantity index.

2.3.3 The K-CF Model The next model features cash flow (CF) as output concept. A production unit’s cash flow is defined as its revenue minus the costs of labour and intermediate inputs; that is t CF t ≡ R t − CLEMS

= =

t p · y − wLEMS VAt − CLt . t

t

(2.97) t · xLEMS

2.3 Different Models, Similar Measures

39

This input-output model basically sees cash flow as the return to capital input. However, cash flow should not be confounded with the business-economical concept of ‘Return on Assets’ (ROA). This will be discussed in Sect. 2.5. It is assumed that CF t > 0. Of course, if there is no owned capital (that is, all capital assets are leased), t = 0, and this model does not make sense.28 then CK The counterpart to profitability is now the ratio of cash flow to capital input t , and the natural starting point for defining a productivity index is to cost, CF t /CK 1 )/CF 0 /C 0 ) = consider the development of this ratio through time. Since (CF 1 /CK K 0 1 0 1 (CF /CF )/(CK /CK ), we need a decomposition of the cash-flow ratio and a decomposition of the capital input cost ratio. Decomposing a cash-flow ratio in a price and a quantity component is structurally similar to decomposing a value-added ratio (see Appendix B). Thus, suppose that a satisfactory decomposition is somehow available; that is, there are price and quantity indices such that CF 1 = PCF (1, 0)QCF (1, 0). CF 0

(2.98)

Next it is assumed that the capital input cost ratio can be decomposed as 1 CK 0 CK

1 1 0 0 1 1 0 0 = P (wK , xK , wK , xK )Q(wK , xK , wK , xK )

≡ PK (1, 0)QK (1, 0),

(2.99)

where P (.) and Q(.) are certain price and quantity indices. The cash-flow based total factor productivity (TFP) index for period 1 relative to period 0 is then defined as ITFPRODCF (1, 0) ≡

QCF (1, 0) . QK (1, 0)

(2.100)

This index measures the change of the quantity component of cash flow relative to the quantity change of capital input; or, ITFPRODCF (.) can be seen as an index of real cash flow relative to an index of real capital input. In the K-CF model the counterpart to profit is the difference of cash flow and t , and the natural starting point for defining a TFP capital input cost, CF t − CK indicator is to consider the development of this difference through time. However, since costs are additive, we see that

28 Cash

flow is also called gross or variable profit. The National Accounts concept gross operating surplus (GOS) corresponds to cash flow plus the labour cost of self-employed persons. In some sectors it occasionally occurs that production units exhibit negative cash flows during certain periods. An example of such a sector is agriculture.

40

2 A Framework Without Assumptions t t t CF t − CK = R t − CLEMS − CK

= Rt − C t .

(2.101)

Thus, profit in the K-CF model is identical to profit in the KLEMS-Y model, and the same applies to the price and quantity components of profit differences. Again, one easily checks that DTFPRODCF (1, 0) ≡ QCF (1, 0) − QK (1, 0) = QR (1, 0) − QC (1, 0) = DTFPRODY (1, 0);

(2.102)

that is, the TFP indicators are the same in the two models. This, however, does not hold for the TFP indices. In general it will be the case that ITFPRODCF (1, 0) = ITFPRODY (1, 0). Following the reasoning of Appendix C it is possible to show that, if profit is zero in both periods, that is, R t = C t (t = 0, 1), then, for certain twostage indices which are second-order differential approximations of Fisher indices, ln ITFPRODCF (1, 0) = E(1, 0) ln ITFPRODY (1, 0),

(2.103)

where E(1, 0) ≥ 1 is the ratio of mean revenue over mean cash flow. Since CF t ≤ VAt , it follows that E(1, 0) ≥ D(1, 0).

2.4 More Models The K-CF model provides a good point of departure for a discussion of the measurement of capital input cost. Cash flow, as defined in the foregoing, is the (ex post measured) monetary balance of all the flow variables. Capital input cost is different, since capital is a stock variable. Basically, capital input cost is measured as the difference between the book values of the production unit’s owned capital stock at beginning and end of the accounting period considered. The theory, for which no behavioural or other far-reaching assumptions appear to be required, is developed in Chap. 2. Anticipating on this development it appears that capital input cost can rather naturally be split into four meaningful components, t t t t t CK = CK,w + CK,e + CK,u + CK,tax ,

(2.104)

respectively denoting the aggregate cost of waiting, anticipated time-series depreciation, unanticipated revaluation, and tax. This leads to four additional input-output models. The first two models are variants of the KL-VA model. The idea here is that the (ex post) cost of time-series depreciation plus tax should be treated in the same

2.4 More Models

41

way as the cost of intermediate inputs, and subtracted from value added. Hence, the output concept is called net value added, and defined by   t t t . + CK,u + CK,tax NVAt ≡ VAt − CK,e

(2.105)

The remaining input cost is the sum of labour cost, CLt , and waiting cost of capital, t . CK,w Some argue that this model is to be preferred from a welfare-theoretic point of view. If the objective is to hold owned capital (including investments during the accounting period) in terms of money intact, then depreciation—whether expected or not—and tax, which can be considered as payment for government services, should be treated like intermediate inputs (Spant 2003). This model was also strongly defended by Rymes (1983). Apart from land, Rymes considered labour and waiting as the only primary inputs, and connected this with a Harrodian model of technological change. Diewert et al. (2005) and Diewert and Lawrence (2006) suggested to consider unanticipated revaluation, which is the unanticipated part of time-series depreciation, as a monetary component that must be added to profit. The result could be called ‘profit from normal operations of the production unit’. Following this suggestion, the output concept becomes   t t NNVAt ≡ VAt − CK,e , + CK,tax

(2.106)

which could be called normal net value added. As inputs are considered labour, t CLt , and waiting cost of capital, CK,w . And profit from normal operations is t + t 29 CK,u . The final two models are variants of the K-CF model. Here also the idea is that the (ex post) cost of time-series depreciation plus tax should be treated in the same way as the cost of intermediate inputs, and subtracted from cash flow. Hence, the output concept is called net cash flow, and defined by   t t t . NCF t ≡ CF t − CK,e + CK,u + CK,tax

(2.107)

t The remaining input cost is the waiting cost of capital, CK,w . A variant of the K-NCF model is obtained by considering unanticipated revaluation, which is the unanticipated part of time-series depreciation, as a component that must be added to profit. Hence, the output concept becomes

  t t NNCF t ≡ CF t − CK,e , + CK,tax

29 This

(2.108)

also seems to be the model favoured by Hulten and Schreyer (2010). Anyway they consider what is here called NNVA as “entry point for measuring welfare change.”

42

2 A Framework Without Assumptions

which could be called normal net cash flow. The only input category is the waiting t t . cost of capital, CK,w , and profit from normal operations is t + CK,u A number of observations can now be made. First, all the input-output models—KLEMS-Y, KL-VA, KL-NVA, KL-NNVA, K-CF, K-NCF, and K-NNCF, respectively—lead to different TFP indices. However, most of these differences are artefacts, caused by a different mixing of subtraction and division.30 Thus, it depends on purpose and context of a study which particular model is chosen for the presentation of results. When TFP indicators are compared, the real difference turns up, namely between the KL-NNVA and K-NNCF models on the one hand and the rest on the other hand. The reason is that the KL-NNVA and K-NNCF are based t . on a different profit concept, namely ∗t ≡ t + CK,u Second, as will be shown in the next chapter, the waiting cost of capital is determined by an interest rate r t . Setting in the accounting relation of the K-NCF model, t + t = NCF t , CK,w

(2.109)

profit t equal to zero, and solving this equation then for r t delivers the so-called ‘endogenous’, or ‘internal’, or ‘balancing’ rate of return. However, one could do the same with the accounting relation of the K-NNCF model, t CK,w + ∗t = NNCF t .

(2.110)

This would lead to a concept one could call the ‘normal endogenous’ rate of return. The important point to stress here is that there appears to be no single concept of the endogenous rate of return. There is rather a continuum of possibilities, depending on the way one wants to deal with unanticipated revaluations. Third, an endogenous rate of return, of whatever variety, can only be calculated ex post. Net cash flow as well as normal net cash flow require for their computation that the accounting period has expired. Fourth, as the name suggests, a TFP index or indicator suggests that all the relevant inputs and outputs are taken into account. Uncovered inputs and outputs have impact on the definition of profit, and hence on the interpretation of TFP change. For example, restricting capital input K to reproducible assets implies that the cost of nonreproducible assets, such as intangible or subsoil assets, is included in profit t . Since an endogenous rate of return can be said to absorb profit, the extent of undercoverage has also implications for the interpretation of the rate of return. Put otherwise, since an endogenous rate of return closes the gap between the input side and the output side of the production unit, it is influenced by all in- or exclusion decisions. Moreover, there are various kinds of measurement error.31

30 Rymes

(1983) would single out the KL-NVA model as the “best” one, but this is clearly not backed by the argument presented here. 31 For more details see Schreyer (2010) and the sensitivity analysis carried out by Inklaar (2010).

2.5 Capital Utilization

43

The question whether to use, for a certain production unit, an endogenous or an exogenous rate of return belongs, according to Diewert (2008), to the list of still unresolved issues. The practice of official statistical agencies is mixed.32 An interesting empirical comparison on Australian agriculture over the period 1949– 2012 was carried out by Sheng et al. (2017).

2.5 Capital Utilization Until now it was tacitly assumed that the productive capital stock was fully used in actual production. We want to make this assumption explicit. For introducing the capital utilization rate, let us return to the K-CF model, which is governed by the equation t CK + t = CF t .

(2.111)

Cash flow, if positive, is seen as the return to the productive capital stock. But what if this stock is only partly used in productive operations? Here the simplest approach will be outlined. Let θKt (0 < θKt < 1) denote the fraction of the productive capital stock (averaged over all the type-age classes) actually used during period t. Such information could be obtained by surveying managers of production units. Then the production unit can be split into two virtual subunits with accounting relations, respectively,

32 Here

t θKt CK + ∗∗t = CF t

(2.112)

t t − (1 − θKt )CK = 0, (1 − θKt )CK

(2.113)

are some examples. The U. S. Bureau of Labor Statistics (see Eldridge et al. 2018) and Statistics Canada (see Baldwin et al. 2014) use mainly endogenous rates, except that for outcomes that are deemed unrealistic endogenous rates are replaced by more appropriate exogenous rates. Moreover, Statistics Canada replaces annual rates by 3-year moving averages. The Australian Bureau of Statistics uses, per production unit considered, the maximum of the endogenous rate and a certain exogenous rate (set equal to the annual percentage change of the CPI plus 4%) (see Roberts 2008). Statistics New Zealand (2014) uses throughout an exogenous rate, namely the annual percentage change of the CPI plus 4%. The Swiss Federal Statistical office employs an intricate system: per production unit the simple mean of the endogenous rate and a certain exogenous rate is used as the final exogenous rate (see Rais and Sollberger 2008). Concerning the endogenous rates, however, these sources are not clear as to which concept is used precisely. Statistics Netherlands sets the interest rate equal to the so-called Internal Reference Rate, which is the interest rate that banks charge to each other, plus 1.5% (see Van den Bergen et al. 2008). It seems that Jorgenson (2009) is also proposing endogenous rates of return for the four sectors considered.

44

2 A Framework Without Assumptions

t + t . The first relation describes the process whereby the with ∗∗t ≡ (1 − θKt )CK t fraction θK of the capital stock produces cash flow CF t . The second relation says that the cost of the remaining fraction of the capital stock, that is sitting idle, must be offset by a negative profit of the same magnitude.33 Adding these two accounting relations delivers expression (2.111). It is now obvious that the measurement of productivity change should be based on the accounting relation of the first subunit, since the second subunit does not produce anything. Thus the TFP index for period 1 relative to period 0 is defined as

ITFPRODCFU (1, 0) ≡

QCF (1, 0) (θK1 /θK0 )QK (1, 0)

.

(2.114)

This is the index of real cash flow divided by the index of real capital input multiplied by the change of the capital utilization rate. Using expression (2.100) we obtain the following relation between the utilization-rate-adjusted TFP index and the unadjusted TFP index: ITFPRODCFU (1, 0) =

ITFPRODCF (1, 0) θK1 /θK0

.

(2.115)

Thus, increasing capital utilization implies a downward adjustment, and decreasing capital utilization implies an upward adjustment of TFP change. The tables in the next section provide an idea of the magnitude of such an adjustment. Another example was provided by Coremberg (2008). It is straightforward to check that the utilization rate can be introduced in any of the models discussed in this chapter. This exercise, however, is left to the reader.

2.6 An Empirical Illustration In the foregoing sections a number of input-output models were introduced, as well as a number of profit concepts. Potentially each choice leads to a different magnitude of TFP change. How important are such differences in practice? Unfortunately there are not many studies delivering comparative results on real-life data.34 The most comprehensive study was carried out by Vancauteren et al. (2012) on data of the Netherlands. In a growth accounting framework the sensitivity analysis of

33 It

is assumed here that the unit user costs of used and unused assets are the same. See Hulten (2010) for a brief discussion of this issue. 34 There is some scattered evidence, such as Lovell and Lovell (2013), who compared ITFPROD VA and ITFPRODY , both with t = 0, on the Australian Coal Mining industry, 1991–2017. They found differences in line with the Domar factor.

2.6 An Empirical Illustration

45

Table 2.1 Netherlands Manufacturing industry, 1995–2008. Geometric mean annual TFP change (percentage). Exogenous rate of return Profit t + t (1 − θKt )CK

Model

t

t t + CK,u

t + t (1 − θKt )CK

t + θKt CK,u

KLEMS-Y KL-VA KL-NVA K-CF K-NCF

0.74 2.43 2.82 5.87 9.12

0.77 2.56 2.89 6.20 8.84

0.77 2.57 2.89 6.10 9.36

0.79 2.69 2.93 6.44 9.93

Niebel and Saam (2016) is worth considering. As an illustration certain key results from the Vancauteren et al. study will now be reviewed. For the years 1995–2008 TFP changes were calculated from detailed data of supply and use tables (containing values in current year prices and in prices of the previous year) according to the KLEMS-Y, KL-VA, KL-NVA, K-CF and K-NCF models, with exogenous and endogenous interest rates, for nine industrial sectors and their aggregate. The TFP indices are based on Laspeyres quantity indices. Capital input excluded subsoil assets, land, and inventories. The exogenous interest rate is the inter-bank interest rate plus 1.5%. As capital utilization rates, obtained from a business survey conducted by Statistics Netherlands, were only available for the Manufacturing industry, the results for this industry were more detailed. Over the period 1995–2008 the profit-cost margin of the Manufacturing industry varied between 4 and 8%. The first column of Table 2.1 contains the baseline suite of results. Going down the column, as expected, the percentages of TFP change increase, which also happens in the other columns. The difference between the KLEMS-Y row and the K-NCF row is remarkable. The last row shows percentages more than 10 times larger than those in the first row. The second column of this table shows the effect of removing unanticipated revaluation of the industrial assets from the denominator of profitability. Notice that the effect is modest, but also that there is no systematic relation between the percentages in the first two columns. The third column shows the effect of restricting the cost component of the TFP index to the assets actually used. Overall, compared to the baseline column, the percentages are higher. The final column shows the combined effect of removing unanticipated revaluations and the cost of unutilized assets from the capital cost component. Table 2.2 presents the TFP percentages under endogenous rates of return. The differences between corresponding cells of the two tables are relatively small. Notice, however, that there is no systematic relation between corresponding rows. Replacing the exogenous interest rate by an endogenous rate increases TFP change for the KLEMS-Y and the K-CF models, but decreases TFP change for the KL-VA, KL-NVA, and K-NCF models.

46

2 A Framework Without Assumptions

Table 2.2 Netherlands Manufacturing industry, 1995–2008. Geometric mean annual TFP change (percentage). Endogenous rate of return

Model

Profit t =0

t t + CK,u =0

t + t (1 − θKt )CK =0

t + t (1 − θKt )CK t t + θK CK,u = 0

KLEMS-Y KL-VA KL-NVA K-CF K-NCF

0.76 2.27 2.56 6.12 8.89

0.79 2.38 2.69 6.34 9.27

0.78 2.44 2.68 6.35 9.12

0.81 2.53 2.80 6.57 9.50

All in all, it is clear that there is no unique answer to the question: what was mean annual TFP change in the Netherlands Manufacturing industry over the period 1995–2008? The answer ranges between 0.74 and 9.93 percent. Thus, before being satisfied with a particular answer one should decide on the suitability of both measurement model and technical assumptions for the context of the question.35

2.7 Conclusion After measurement comes explanation. Conditional on the initial level of aggregation, there appear to be two main directions. The first is disaggregation: the explanation of productivity change at some aggregate level (economy, sector, or industry) by productivity change at a lower level (firm, or plant) and other factors, collectively subsumed under the heading of re-allocation (expansion, contraction, entry, and exit of production units). This topic will be covered in Chaps. 4–8. As will appear, this type of research is of economic-statistical nature, and there is no need to involve neo-classical assumptions. The second direction is concerned with the decomposition of productivity change in factors such as technological change, technical efficiency change, scale effects, input- and output-mix effects, and chance. The basic idea can be explained, with help of Fig. 2.1, as follows. To start with, for each time period t the technology to which the production unit under consideration has access is defined as the set S t of all the input-output quantity combinations which are feasible during t. Such a set is assumed to have

35 For

an interesting corroboration of this statement the reader is referred to Gandhi et al. (2017). Staying in a strictly neo-classical framework, these authors employed plant-level Manufacturing data from Colombia (1981–1991) and Chili (1979–1996) to estimate gross-output based as well as value-added based production functions in order to obtain productivity figures. They found out that the “dispersion in productivity appears far more stable both across industries and across countries when measured via gross output as opposed to value added”, and thus that “insights derived under value added, compared to gross output, could lead to significantly different policy conclusions.”

2.7 Conclusion

47

y 6

•c • (x1 , y 1 ) •d

•b

S1 S0

•a

• (x0 , y 0 ) -

x

Fig. 2.1 Measuring productivity change

nice properties like being closed, bounded, and convex. Of particular interest is the subset of S t , called its frontier, consisting of all the efficient input-output combinations. An input-output quantity combination is called efficient when output cannot be increased without increasing some input and input cannot be decreased without decreasing some output. From base period to comparison period our production unit moves from inputoutput combination (x 0 , y 0 ) ∈ S 0 to (x 1 , y 1 ) ∈ S 1 , but these two combinations are not necessarily efficient. Decomposition of productivity change means that between these two points some hypothetical path must be constructed, the segments of which have a distinct interpretation. For example, consider the projection, in the output direction, of (x 0 , y 0 ) on the frontier of S 0 (that is, point a), and (x 1 , y 1 ) on the frontier of S 1 (that is, point c). Comparing the base period and comparison period distances between the original points and their projections provides a measure of efficiency change. Two more points are given by projecting (x 0 , y 0 ) also on the frontier of S 1 (that is, point d), and (x 1 , y 1 ) also on the frontier of S 0 (that is, point b). The distances between the two frontiers at the base and comparison period projection points (that is, between points a and d, and b and c) provide (local) measures of technological change. And, finally, moving along each frontier (which is a surface in N + M-

48

2 A Framework Without Assumptions

dimensional space) from a base period to a comparison period projection point provides measures of the scale and input-output mix effects. The construction of all those measures is discussed in more detail in Chap. 9. Since there is no unique path connecting the two input-output combinations, there is no unique decomposition either. And here come the neo-classical assumptions, at the end of the day rather than at its beginning. Suppose that the production unit always stays on the frontier, that its input- and output-mix is optimal at the, supposedly given, input and output prices, and that the two technology sets exhibit constant returns to scale, then total factor productivity change reduces to technological change.36 The technology sets are thereby supposed to reflect the true state of nature, which rules out chance as a factor also contributing to productivity change.37

Appendix A: Indices and Indicators The basic measurement tools used are price and quantity indices and indicators. Indices are ratio-type measures, and indicators are difference-type measures. What, precisely, are the requirements for good tools? The following just serves to introduce and illustrate some concepts used in the main text of this book. For a complete treatment and historical information the reader is referred to Balk (2008). We consider two time periods and an aggregate consisting of M items (goods and 0 to p 1 and quantities going from y 0 to services) with (unit) prices going from pm m m 1 ym (m = 1, . . . , M). The first subsection reviews indices, and the second subsection reviews indicators. It is important to note that, though stated in terms of output, the theory surveyed below is equally applicable to other situations after appropriate modification of the number of items and the interpretation of the variables. When there are more than two time periods involved one can choose between direct (bilateral) indices/indicators or chained (multilateral) indices/indicators. The following survey is restricted to bilateral functions.

Indices A bilateral price index is a positive, continuously differentiable function P (p1 , y 1 , p0 , y 0 ) : 4M ++ → ++ that, except in extreme situations, correctly indicates any increase or decrease of the elements of the price vectors p1 or p0 , conditional on the quantity vectors y 1 and y 0 . A bilateral quantity index is a positive, continuously differentiable function of the same variables

36 See 37 On

Balk (1998, Section 3.7) for a formal proof. stochastic productivity measurement see Chambers (2008).

Appendix A: Indices and Indicators

49

Q(p1 , y 1 , p0 , y 0 ) : 4M ++ → ++ that, except in extreme situations, correctly indicates any increase or decrease of the elements of the quantity vectors y 1 or y 0 , conditional on the price vectors p1 and p0 . The number M is called the dimension of the price or quantity index. The basic requirements (aka axioms) on price and quantity indices comprise (1) that they exhibit the correct monotonicity properties; (2) that they are linearly homogeneous in comparison period prices (quantities, respectively); (3) that they satisfy the Identity Test; (4) that they are homogeneous of degree 0 in prices (quantities, respectively); (5) that they are invariant to changes in the units of measurement of the commodities. Interchanging the variables pt and y t (t = 0, 1) transforms any price index into a quantity index and vice versa. The Product Test requires that price index times quantity index equals the value ratio; that is, P (p1 , y 1 , p0 , y 0 ) × Q(p1 , y 1 , p0 , y 0 ) = p1 · y 1 /p0 · y 0 . This requirement is especially important in the context of deflation. Deflation of a nominal value means dividing it by a price index in order to obtain a real value; that is, something that behaves as a scalar quantity. In our two-period situation, by deflating the nominal comparison period value p1 · y 1 by the price index P (p1 , y 1 , p0 , y 0 ) we get as real comparison period value p1 ·y 1 /P (p1 , y 1 , p0 , y 0 ). The Product Test then implies that this can be written as p0 ·y 0 ×Q(p1 , y 1 , p0 , y 0 ); that is, base period nominal value inflated by a quantity index. This behaves as a scalar quantity if and only if the quantity index satisfies the basic requirements. However, that the price index satisfies the basic requirements is not at all a sufficient condition for the quantity index to do the same. As defined, price and quantity indices are functions of prices and quantities. In practice it is usually easier to observe (or estimate) price or quantity relatives and (nominal) values or value shares. It turns out, however, that any function P (p1 , y 1 , p0 , y 0 ) or Q(p1 , y 1 , p0 , y 0 ), that is invariant to changes in the units of measurement, can be written as a function of only 3M variables, namely the price 1 /p 0 or the quantity relatives y 1 /y 0 , respectively, and the values v t ≡ relatives pm m m m M m t t t t ≡ vt / pm ym (m = 1, . . . , M; t = 0, 1). Value shares are defined as sm m m=1 vm (m = 1, . . . , M; t = 0, 1).

The Main Formulas The indices materializing in this book will now be listed. The reader is invited to check their properties. We start by considering the Dutot price index, defined as P D (p1 , y 1 , p0 , y 0 ) =

M  m=1

1 pm /

M 

0 pm .

(2.116)

m=1

Notice that this index does not depend on quantities. Likewise, the Dutot quantity index is defined as

50

2 A Framework Without Assumptions

QD (p1 , y 1 , p0 , y 0 ) =

M 

1 ym /

m=1

M 

0 ym .

(2.117)

m=1

This index does not depend on prices. Dutot indices are not invariant to changes in the units of measurement of the items. The implication of this is that these indices can only be used when the underlying items (goods and services) can reasonably be considered as homogeneous (or interchangeable). Notice that P D (p1 , y 1 , p0 , y 0 ) × QD (p1 , y 1 , p0 , y 0 ) = p1 · y 1 /p0 · y 0 , except in special cases. It is also interesting to notice that the implicit Dutot price index, that is, the value index divided by the Dutot quantity index, is known as the unit-value index, M M 1 1 1 p1 · y 1  D 1 1 0 0 m=1 pm ym / m=1 ym Q (p , y , p , y ) = .   M M 0 0 0 p0 · y 0 m=1 pm ym / m=1 ym

(2.118)

This is a ratio of mean prices (aka unit values). The main disadvantage of the unitvalue index is that in general the Identity Test for price indices is violated. We continue by considering the Laspeyres price index. As function of prices and quantities this index is defined as P L (p1 , y 1 , p0 , y 0 ) ≡ p1 · y 0 /p0 · y 0 ,

(2.119)

which can be written as a function (arithmetic mean) of price relatives and (base period) value shares, P L (p1 , y 1 , p0 , y 0 ) =

M 

1 0 0 (pm /pm )vm /

m=1

M  m=1

0 vm =

M 

0 1 0 sm (pm /pm ),

(2.120)

m=1

The Laspeyres quantity index is defined as QL (p1 , y 1 , p0 , y 0 ) ≡ p0 · y 1 /p0 · y 0 .

(2.121)

Notice that P L (p1 , y 1 , p0 , y 0 ) × QL (p1 , y 1 , p0 , y 0 ) = p1 · y 1 /p0 · y 0 , except in special cases. The Paasche price index is defined as P P (p1 , y 1 , p0 , y 0 ) ≡ p1 · y 1 /p0 · y 1 ,

(2.122)

which can be written as a function (harmonic mean) of price relatives and (comparison period) value shares,

Appendix A: Indices and Indicators

P (p , y , p , y ) = P

1

1

0

0

M 

51

0 1 1 (pm /pm )vm /

m=1

M 



−1 =

1 vm

m=1

M 

−1 1 1 0 −1 sm (pm /pm )

.

m=1

(2.123) The Paasche quantity index is defined as QP (p1 , y 1 , p0 , y 0 ) ≡ p1 · y 1 /p1 · y 0 .

(2.124)

Notice that P P (p1 , y 1 , p0 , y 0 ) × QP (p1 , y 1 , p0 , y 0 ) = p1 · y 1 /p0 · y 0 , except in special cases. However, P L (p1 , y 1 , p0 , y 0 ) × QP (p1 , y 1 , p0 , y 0 ) = P P (p1 , y 1 , p0 , y 0 ) × QL (p1 , y 1 , p0 , y 0 ) = p1 · y 1 /p0 · y 0 . The Fisher price index is defined as the geometric mean of the Laspeyres and Paasche indices,  1/2 P F (p1 , y 1 , p0 , y 0 ) ≡ P L (p1 , y 1 , p0 , y 0 ) × P P (p1 , y 1 , p0 , y 0 )   M

0 1 0 m=1 sm (pm /pm ) M 1 1 0 −1 m=1 sm (pm /pm )

=

(2.125)

1/2 .

The Fisher quantity index is similarly defined as  1/2 QF (p1 , y 1 , p0 , y 0 ) ≡ QL (p1 , y 1 , p0 , y 0 ) × QP (p1 , y 1 , p0 , y 0 ) . (2.126) Notice that P F (p1 , y 1 , p0 , y 0 ) × QF (p1 , y 1 , p0 , y 0 ) = p1 · y 1 /p0 · y 0 . The geometric alternative to the Laspeyres price index, called GeoLaspeyres price index, is defined as P

GL

(p , y , p , y ) ≡ 1

1

0

0

M 

0

1 0 sm (pm /pm ) .

(2.127)

m=1

The GeoLaspeyres quantity index is defined as QGL (p1 , y 1 , p0 , y 0 ) ≡

M 

0

1 0 sm (ym /ym ) .

(2.128)

m=1

Notice that P GL (p1 , y 1 , p0 , y 0 ) × QGL (p1 , y 1 , p0 , y 0 ) = p1 · y 1 /p0 · y 0 , except in special cases. The GeoPaasche price index is defined as P GP (p1 , y 1 , p0 , y 0 ) ≡

M 

1

1 0 sm (pm /pm ) ,

m=1

(2.129)

52

2 A Framework Without Assumptions

and the GeoPaasche quantity index is defined as QGP (p1 , y 1 , p0 , y 0 ) ≡

M 

1

1 0 sm (ym /ym ) .

(2.130)

m=1

Notice that P GP (p1 , y 1 , p0 , y 0 ) × QGP (p1 , y 1 , p0 , y 0 ) = p1 · y 1 /p0 · y 0 , except in special cases. The geometric mean of the GeoLaspeyres and GeoPaasche price indices is called the Törnqvist price index, P (p , y , p , y ) ≡ T

1

1

0

0

M 

1 0 (sm +sm )/2 (pm /pm ) . 0

1

(2.131)

m=1

The Törnqvist quantity index is defined as P T (p1 , y 1 , p0 , y 0 ) ≡

M 

1 0 (sm +sm )/2 (ym /ym ) . 0

1

(2.132)

m=1

Notice that P T (p1 , y 1 , p0 , y 0 ) × QT (p1 , y 1 , p0 , y 0 ) = p1 · y 1 /p0 · y 0 , except in special cases. An alternative to the Törnqvist indices are the Sato-Vartia indices. Here the price or quantity relatives are weighed with logarithmic mean value shares, normalized to ensure that these add up to 1. Thus, the Sato-Vartia price index is P

SV

(p , y , p , y ) ≡ 1

1

0

0

M 

0

1

1 0 LM(sm ,sm )/ (pm /pm )

M

0 1 m=1 LM(sm ,sm )

,

(2.133)

m=1

where LM(.) denotes the logarithmic mean.38 The Sato-Vartia quantity index is defined as QSV (p1 , y 1 , p0 , y 0 ) ≡

M 

0

1

1 0 LM(sm ,sm )/ (ym /ym )

M

0 1 m=1 LM(sm ,sm )

.

(2.134)

m=1

Notice that P SV (p1 , y 1 , p0 , y 0 ) × QSV (p1 , y 1 , p0 , y 0 ) = p1 · y 1 /p0 · y 0 . A second alternative are the Montgomery-Vartia indices. The price and quantity indices are defined as, respectively,

38 Recall

that for any two strictly positive real numbers a and b their logarithmic mean is defined by LM(a, b) ≡ (a − b)/ ln(a/b) when a = b, and LM(a, a) ≡ a.

Appendix A: Indices and Indicators

P MV (p1 , y 1 , p0 , y 0 ) ≡

53 M 

0

1

0 ,V 1 )

0

1

0 ,V 1 )

1 0 LM(vm ,vm )/LM(V (pm /pm )

,

(2.135)

,

(2.136)

m=1

QMV (p1 , y 1 , p0 , y 0 ) ≡

M 

1 0 LM(vm ,vm )/LM(V (ym /ym )

m=1

M t where V t ≡ m=1 vm (t = 0, 1). Now the price and quantity relatives are weighed with ratios of means instead of means of ratios. These weights do not add up to 1, though generally the discrepancy appears to be negligible. Notice that P MV (p1 , y 1 , p0 , y 0 ) × QMV (p1 , y 1 , p0 , y 0 ) = p1 · y 1 /p0 · y 0 .

Two-Stage Indices It is important to consider the concept of two-stage indices. Let the aggregate under consideration be denoted by A, and let A be partitioned arbitrarily into K disjunct subaggregates Ak ,  A = ∪K k=1 Ak , Ak ∩ Ak  = ∅ (k = k ).

Each subaggregate consists of a number of items. Let Mk ≥ 1 denote K the number of items contained in Ak (k = 1, . . . , K). Obviously M = k=1 Mk . Let (pk1 , yk1 , pk0 , yk0 ) be the subvector of (p1 , y 1 , p0 , y 0 ) corresponding to the t ≡ p t y t is the value of item m at period t. Then subaggregate Ak . Recall that vm m m  t t Vk ≡ m∈Ak vm (k = 1, . . . , K) is the value of subaggregate Ak at period t, and  K t t = V t ≡ m∈A vm k=1 Vk is the value of aggregate A at period t. (1) (K) Let P (.), P (.), . . . , P (.) be price indices of dimension K, M1 , . . . , MK respectively that satisfy the basic requirements. Recall that any such price index can be written as a function of price relatives and base and comparison period item values. Replace in P (.) the price relatives by subaggregate price indices and the item values by subaggregate values. Then the price index defined by P ∗ (p1 , y 1 , p0 , y 0 ) ≡ P (P (k) (pk1 , yk1 , pk0 , yk0 ), Vk1 , Vk0 ; k = 1, . . . , K)

(2.137)

is of dimension M and also satisfies the basic requirements. The index P ∗ (.) is called a two-stage index. The first stage refers to the indices P (k) (.) for the subaggregates Ak (k = 1, . . . , K). The second stage refers to the index P (.) that is applied to the subindices P (k) (.) (k = 1, . . . , K). A two-stage index such as defined by expression (2.137) closely corresponds to the calculation practice at statistical agencies. All the subindices are then usually of the same functional form, for instance Laspeyres or Paasche indices. The aggregate, second-stage index may or may not be of the same functional form. This could be, for instance, a Fisher index.

54

2 A Framework Without Assumptions

If the functional forms of the subindices P (k) (.) (k = 1, . . . , K) and the aggregate index P (.) are the same, then P ∗ (.) is called a two-stage P (.)-index. Continuing the first example, the two-stage Laspeyres price index reads P ∗L (p1 , y 1 , p0 , y 0 ) ≡

K 

P L (pk1 , yk1 , pk0 , yk0 )Vk0 /

k=1

K 

Vk0 ,

(2.138)

k=1

and one simply checks that the two-stage and the single-stage Laspeyres price indices coincide; that is, P ∗L (p1 , y 1 , p0 , y 0 ) = P L (p1 , y 1 , p0 , y 0 ).

(2.139)

However, this is the exception rather than the rule. For most indices, two-stage and single-stage variants do not coincide. Put otherwise, most indices are not Consistentin-Aggregation (CIA). Two-stage quantity indices can be defined analogously.

Indicators 1 1 0 0 The continuous functions P(p1 , y 1 , p0 , y 0 ) : 4M ++ →  and Q(p , y , p , y ) : 4M ++ →  will be called price indicator and quantity indicator, respectively, if they satisfy the following basic requirements (aka axioms): (1) the functions exhibit the correct monotonicity properties; (2) they satisfy the Identity Test; (3) they are homogeneous of degree 1 in prices or quantities, respectively; (4) they are invariant to changes in the units of measurement of the commodities. Interchanging the variables pt and y t (t = 0, 1) transforms any price indicator into a quantity indicator and vice versa. The analogue of the Product Test requires that price indicator plus quantity indicator equals the value difference; that is, P(p1 , y 1 , p0 , y 0 )+Q(p1 , y 1 , p0 , y 0 ) = p1 ·y 1 −p0 ·y 0 . Notice that these functions may take on negative or zero values. As values are additive, we have  1 0 V1 −V0 = (vm − vm ). (2.140) m∈A

Given a pair of price and quantity indicators satisfying the Product Test, it must then be the case that P(p1 , y 1 , p0 , y 0 ) + Q(p1 , y 1 , p0 , y 0 ) =  1 1 0 0 1 1 0 0 P(pm , ym , pm , ym ) + Q(pm , ym , pm , ym ) , m∈A

(2.141)

Appendix A: Indices and Indicators

55

where the dimension of the indicators at the right-hand side equals 1. of this equation 1 , y 1 , p0 , y 0 ) = 1 , y 1 , p0 , y 0 ) A natural requirement is then that P(p P(p m m m m m∈A  1 , y 1 , p 0 , y 0 ). Any such indicator is CIA, as and Q(p1 , y 1 , p0 , y 0 ) = m∈A Q(pm m m m P(p1 , y 1 , p0 , y 0 ) =

K 

P(pk1 , yk1 , pk0 , yk0 ) =

K  

1 1 0 0 P(pm , ym , pm , ym ),

k=1 m∈Ak

k=1

(2.142) and Q(p1 , y 1 , p0 , y 0 ) =

K 

Q(pk1 , yk1 , pk0 , yk0 ) =

K  

1 1 0 0 Q(pm , ym , pm , ym ).

k=1 m∈Ak

k=1

(2.143) Recall that any function P(p1 , y 1 , p0 , y 0 ) or Q(p1 , y 1 , p0 , y 0 ), that is invariant to changes in the units of measurement, can be written as a function of only 1 /p 0 or the quantity relatives y 1 /y 0 , 3M variables, namely the price relatives pm m m m t respectively, and the values vm (m = 1, . . . , M; t = 0, 1). This leads to further specifications of the indicators. It is also useful to know that any indicator can be transformed into an index, and vice versa (Balk 2008, 128–129). In the process of transformation, however, certain properties may get lost.

The Main Formulas The indicators materializing in this book will now be reviewed. The Laspeyres price indicator as function of prices and quantities is defined as P L (p1 , y 1 , p0 , y 0 ) ≡ (p1 − p0 ) · y 0 .

(2.144)

This indicator can be written as a function of price relatives and (base period) values, P L (p1 , y 1 , p0 , y 0 ) =

M 

1 0 0 (pm /pm − 1)vm .

(2.145)

m=1

The Laspeyres quantity indicator is similarly defined as QL (p1 , y 1 , p0 , y 0 ) ≡ (y 1 − y 0 ) · p0 .

(2.146)

Notice that P L (p1 , y 1 , p0 , y 0 ) + QL (p1 , y 1 , p0 , y 0 ) = p1 · y 1 − p0 · y 0 , except in special cases. The Paasche price indicator, defined as P P (p1 , y 1 , p0 , y 0 ) ≡ (p1 − p0 ) · y 1

(2.147)

56

2 A Framework Without Assumptions

can be written as a function of price relatives and (comparison period) values, P P (p1 , y 1 , p0 , y 0 ) =

M 

0 1 1 (1 − pm /pm )vm .

(2.148)

m=1

The Paasche quantity indicator is defined as QP (p1 , y 1 , p0 , y 0 ) ≡ (y 1 − y 0 ) · p1 .

(2.149)

Notice that P P (p1 , y 1 , p0 , y 0 ) + QP (p1 , y 1 , p0 , y 0 ) = p1 · y 1 − p0 · y 0 , except in special cases. However, P L (p1 , y 1 , p0 , y 0 ) + QP (p1 , y 1 , p0 , y 0 ) = P P (p1 , y 1 , p0 , y 0 ) + QL (p1 , y 1 , p0 , y 0 ) = p1 · y 1 − p0 · y 0 . The Bennet price indicator is usually defined by P B (p1 , y 1 , p0 , y 0 ) ≡ (1/2)(p1 − p0 ) · (y 0 + y 1 ).

(2.150)

It is the arithmetic mean of the Laspeyres and Paasche indicators; that is,   P B (p1 , y 1 , p0 , y 0 ) = (1/2) P L (p1 , y 1 , p0 , y 0 ) + P P (p1 , y 1 , p0 , y 0 ) , (2.151) which can be written as  M  M   1 0 0 0 1 1 P B (p1 , y 1 , p0 , y 0 ) = (1/2) (pm /pm − 1)vm + (1 − pm /pm )vm . m=1

m=1

(2.152) The Bennet quantity indicator is defined by QB (p1 , y 1 , p0 , y 0 ) ≡ (1/2)(y 1 − y 0 ) · (p0 + p1 ).

(2.153)

One easily verifies that P B (p1 , y 1 , p0 , y 0 )+QB (p1 , y 1 , p0 , y 0 ) = p1 ·y 1 −p0 ·y 0 . 1 − p 0 are multiplied In the Bennet price indicator the price differences pm m 0 + y 1 )/2. by arithmetic means of base and comparison period quantities (ym m The Montgomery price indicator uses instead logarithmic mean values divided by logarithmic mean prices; that is, P M (p1 , y 1 , p0 , y 0 ) ≡

M 0 , v1 )  LM(vm m 0 (p1 − pm ). 0 , p1 ) m LM(pm m

(2.154)

m=1

The Montgomery quantity indicator is defined as QM (p1 , y 1 , p0 , y 0 ) ≡

M 0 , v1 )  LM(vm m 0 (y 1 − ym ). 0 , y1 ) m LM(ym m

m=1

(2.155)

Appendix B: Decompositions of the Value-Added Ratio

57

Using the definition of the logarithmic mean one easily verifies that P M (p1 , y 1 , p0 , y 0 ) + QM (p1 , y 1 , p0 , y 0 ) = p1 · y 1 − p0 · y 0 .

Appendix B: Decompositions of the Value-Added Ratio Value added is defined as revenue minus the cost of intermediate inputs. Let the revenue ratio R 1 /R 0 as in expression (2.14) be decomposed as R1 = P (p1 , y 1 , p0 , y 0 )Q(p1 , y 1 , p0 , y 0 ) R0 ≡ PR (1, 0)QR (1, 0),

(2.156)

for certain price and quantity indices, and let the intermediate inputs cost ratio 1 /C 0 CEMS EMS be decomposed by, not neccessarily the same, price and quantity indices as 1 CEMS 0 CEMS

1 1 0 0 1 1 0 0 = P (wEMS , xEMS , wEMS , xEMS )Q(wEMS , xEMS , wEMS , xEMS )

≡ PEMS (1, 0)QEMS (1, 0).

(2.157)

How can the (nominal) value added ratio 1 R 1 − CEMS VA1 = 0 VA0 R 0 − CEMS

then be decomposed into meaningful price and quantity components? It is clear that any of these components potentially requires a lot of data, namely 1 , x 1 , w 0 , x 0 , p 1 , y 1 , p 0 , y 0 ), data that are not always immediately (wEMS EMS EMS EMS available. Therefore, traditionally, one had recourse to so-called single deflation, where, for instance, a revenue-based price index is used to deflate nominal value sd (1, 0) ≡ P (1, 0), and let the quantity index be the added. That is, one sets PVA R remainder, Qsd VA (1, 0) ≡

VA1  PR (1, 0). VA0

(2.158)

Using the various definitions, this can be written as Qsd VA (1, 0) = =

0 1 /C 0 CEMS CEMS R 0 R 1 /R 0 EMS − VA0 PR (1, 0) VA0 PR (1, 0) 0 CEMS R0 PEMS (1, 0) , Q (1, 0) − QEMS (1, 0) R 0 0 PR (1, 0) VA VA

(2.159)

58

2 A Framework Without Assumptions

but also as Qsd VA (1, 0) =   −1 −1 1 C R1 (1, 0) P EMS QEMS (1, 0) QR (1, 0)−1 − EMS . PR (1, 0) VA1 VA1

(2.160)

Two important observations must be made. First, if VAt > 0 (t = 0, 1) then Qsd VA (1, 0) > 0. Second, it appears that in this setup the quantity component of the value added ratio depends on the price index of intermediate inputs relative to the price index of gross output, PEMS (1, 0)/PR (1, 0). Since it is not at all certain that this ratio dwells in the neighbourhood of 1, this can create undesirable outcomes.  A variant is mentioned by SNA (2008, par. 15.135), namely Qsd VA (1, 0) ≡ QR (1, 0); that is, the growth rate of real value added is set equal to the growth rate of gross output. It appears that 

Qsd VA (1, 0) =

R 1 /VA1 sd QVA (1, 0). R 0 /VA0

(2.161)

Thus, if the ratio of revenue to value added does not change then the two singledeflation variants coincide. In the literature this ratio is known as the Domar factor. When the necessary data are available, SNA (2008, par. 15.133) prefers double deflation, which means that the value-added based quantity index is defined as39 Qdd1 VA (1, 0) ≡

0 CEMS R0 Q (1, 0) − QEMS (1, 0), R VA0 VA0

(2.162)

or alternatively as  Qdd2 VA (1, 0) ≡

1 CEMS R1 −1 Q (1, 0) − QEMS (1, 0)−1 R VA1 VA1

−1 .

(2.163)

Both quantity indices have a two-stage structure. The first stage is given by the quantity indices for gross output and intermediate inputs, QR (1, 0) and QEMS (1, 0), respectively.40 The second stage is in the case of expression (2.162) a Laspeyres index (with weights from the base period), and in the case of expression (2.163) a Paasche index (with weights from the comparison period). Notice the minus signs for the parts concerning intermediate inputs. It is then rather natural to propose the geometric mean of the two quantity indices as the desired quantity index. Put otherwise, the quantity component of the 

sd interesting comparison of Qdd1 VA (.) and QVA (.) on UK data was carried out by Oulton et al. (2018). 40 Hence the name ‘double deflation’. This should be the standard, rather than an option. 39 An

Appendix B: Decompositions of the Value-Added Ratio

59

value added ratio is defined as the Fisher index of the subindices QR (1, 0) and QEMS (1, 0); that is, ⎤1/2 0 CEMS R0 Q (1, 0) − Q (1, 0) R EMS ⎥ ⎢ VA0 VA0 ⎥ . QFVA (1, 0) ≡ ⎢ ⎦ ⎣ R1 1 CEMS −1 −1 (Q (1, 0)) − (Q (1, 0)) R EMS VA1 VA1 ⎡

(2.164)

Similarly, the price component of the value added ratio is defined as the Fisher index of the subindices PR (1, 0) and PEMS (1, 0); that is, ⎤1/2 0 CEMS R0 P (1, 0) − P (1, 0) R EMS ⎥ ⎢ VA0 VA0 F ⎥ . (1, 0) ≡ ⎢ PVA ⎦ ⎣ R1 1 CEMS −1 −1 (P (1, 0)) − (P (1, 0)) R EMS VA1 VA1 ⎡

(2.165)

F (1, 0)QF (1, 0) = VA1 /VA0 . This pair of indices was One easily checks that PVA VA proposed by Geary (1944), though Karmel (1954) discloses some earlier sources. For studying the behaviour of the quantity index QFVA (1, 0) the following relation is useful:

QFVA (1, 0) = QR (1, 0) ⎡ ⎤1/2 0 1 CEMS QEMS (1, 0) CEMS ⎢1 − ⎥ 1− R 0 QR (1, 0) ⎢ ⎥ R1 ⎢ ⎥ .   0 −1 1 ⎣ ⎦ C CEMS QEMS (1, 0) 1 − EMS 1 − R0 QR (1, 0) R1

(2.166)

t When VAt > 0 then 1 − CEMS /R t > 0 (t = 0, 1). However, the function 0 /R 0 )Q QFV A (1, 0) is undefined when QR (1, 0) ≤ (CEMS EMS (1, 0) or QR (1, 0) ≥ 1 1 (R /CEMS )QEMS (1, 0). Moreover, Eq. (2.166) implies the following relations:

QR (1, 0) > QEMS (1, 0) ⇒ QFVA (1, 0) > QR (1, 0) QR (1, 0) = QEMS (1, 0) ⇒ QFVA (1, 0) = QR (1, 0)

(2.167)

QR (1, 0) < QEMS (1, 0) ⇒ QFVA (1, 0) < QR (1, 0). Karmel (1954), however, showed that in the case of chained indices these relations might be violated. Thus there may occur situations where Fisher indices are undefined. An alternative decomposition, which does not exhibit this defect, can be developed as follows. For the logarithm of the value added ratio we get by repeated application of the logarithmic mean LM(a, b),

60

2 A Framework Without Assumptions



VA1 ln VA0

=

VA1 − VA0 = LM(VA1 , VA0 )

(2.168)

C1 − C0 R1 − R0 − EMS 1 EMS = 1 0 LM(VA , VA ) LM(VA , VA0 ) 1 , C 0 ) ln(C 1 /C 0 ) L(R 1 , R 0 ) ln(R 1 /R 0 ) LM(CEMS EMS EMS EMS − . LM(VA1 , VA0 ) LM(VA1 , VA0 )

Using the decompositions of the revenue ratio and the intermediate inputs cost ratio, the logarithm of the value added ratio can be expressed as

VA1 ln VA0

=

LM(R 1 , R 0 ) ln(PR (1, 0)QR (1, 0)) − LM(VA1 , VA0 ) 1 , C 0 ) ln(P LM(CEMS EMS (1, 0)QEMS (1, 0)) EMS

LM(VA1 , VA0 )

.

(2.169)

This can be rearranged as VA1 QR (1, 0)φ(1,0) PR (1, 0)φ(1,0) = , PEMS (1, 0))ψ(1,0) QEMS (1, 0))ψ(1,0) VA0

(2.170)

where φ(1, 0) ≡ LM(R 1 , R 0 )/LM(V A1 , V A0 ), that is, mean revenue over mean 1 , C 0 )/LM(V A1 , V A0 ), that is, mean value added, and ψ(1, 0) ≡ LM(CEMS EMS intermediate inputs cost over mean value added. Thus, value-added based price and quantity indices can rather naturally be defined by MV PVA (1, 0) ≡

PR (1, 0)φ(1,0) PEMS (1, 0))ψ(1,0)

(2.171)

QMV VA (1, 0) ≡

QR (1, 0)φ(1,0) . QEMS (1, 0))ψ(1,0)

(2.172)

These indices generalize the conventional Montgomery-Vartia indices (Appendix A). Their disadvantage is that PR (1, 0) = PEMS (1, 0) (or QR (1, 0) = QEMS (1, 0)) MV (1, 0) = P (1, 0) (or QMV (1, 0) = Q (1, 0)). not necessarily implies that PVA R R VA Formally stated, these indices fail the Equality Test. The reason behind this failure is that generally φR (1, 0) − φEMS (1, 0) =

1 , C0 ) LM(R 1 , R 0 ) − LM(CEMS EMS

LM(VA1 , VA0 )

(2.173)

Appendix C: The Domar Factor

61

is close but not equal to 1 because LM(a, 1) is a concave function. Fortunately, in ‘normal’ situations the discrepancy appears to be unimportant. Summarizing the foregoing, we see that • single deflation is simple, leads to a quantity index that is always positive, but this index may be biased; • double deflation by a Fisher index leads to a quantity index that is unbiased but may not always be well-defined; • double deflation by a Montgomery-Vartia index leads to a quantity index that is always well-defined but fails the Equality Test. Thus, there is no theoretically entirely satisfactory solution to the problem of decomposing a value-added ratio into a price index and a quantity index.

Appendix C: The Domar Factor And the TFP Index According to the definition of the gross-output based TFP index41 ln ITFPRODY (1, 0) = ln QR (1, 0) − ln QC (1, 0).

(2.174)

t + Ct We know that C t = CKL EMS (t = 0, 1). It is assumed that the primary 1 /C 0 = inputs cost ratio is decomposed into a price and a quantity index, CKL KL PKL (1, 0)QKL (1, 0). It is also assumed that the intermediate inputs cost ratio is 1 /C 0 decomposed into a price and a quantity index, CEMS EMS = PEMS (1, 0) × QEMS (1, 0). Then QC (1, 0) can be defined as the Montgomery-Vartia index of QKL (1, 0) and QEMS (1, 0);42 that is,

ln QC (1, 0) = 1 , C0 ) LM(CKL KL LM(C 1 , C 0 )

(2.175) ln QKL (1, 0) +

1 , C0 ) LM(CEMS EMS LM(C 1 , C 0 )

ln QEMS (1, 0),

where LM(a, b) is the logarithmic mean of a and b. Then, substituting this expression into expression (2.174), we obtain ln ITFPRODY (1, 0) = ln QR (1, 0)

41 The

(2.176)

first part of this Appendix is based on Balk (2009). (1978) showed that any superlative index is a second-order approximation of the Montgomery-Vartia index at the point where price and quantity relatives are equal to 1.

42 Diewert

62

2 A Framework Without Assumptions

−

1 , C0 ) 1 , C0 ) LM(CEMS LM(CKL EMS KL ln Q ln QEMS (1, 0). (1, 0) − KL LM(C 1 , C 0 ) LM(C 1 , C 0 )

t (t = 0, 1). Let Next, the definition of value added is rewritten as R t = VAt + CEMS QR (1, 0) be defined as the Montgomery-Vartia index of QVA (1, 0) and QEMS (1, 0). Then the logarithm of the value-added based quantity index can be backed out as

ln QV A (1, 0) = LM(R 1 , R 0 ) LM(VA1 , VA0 )

(2.177) ln QR (1, 0) −

1 , C0 ) LM(CEMS EMS LM(VA1 , VA0 )

ln QEMS (1, 0).

Substituting this into the definition of the value-added based TFP index delivers ln ITFPRODVA (1, 0) = −

LM(R 1 , R 0 )  ln QR (1, 0) LM(VA1 , VA0 )

(2.178)

1 , C0 )  LM(CEMS LM(VA1 , VA0 ) EMS ln Q ln Q (1, 0) − (1, 0) . KL EMS LM(R 1 , R 0 ) LM(R 1 , R 0 )

Combining the two expressions (2.176) and (2.178) by eliminating ln QR (1, 0) delivers LM(R 1 , R 0 )  ln ITFPRODY (1, 0) LM(VA1 , VA0 )

1 , C0 ) LM(CKL LM(VA1 , VA0 ) KL − (2.179) + ln QKL (1, 0) LM(C 1 , C 0 ) LM(R 1 , R 0 )

1 , C0 ) 1 , C0 )  LM(CEMS LM(CEMS EMS EMS + − (1, 0) . ln Q EMS LM(C 1 , C 0 ) LM(R 1 , R 0 )

ln ITFPRODVA (1, 0) =

The factor in front of the square brackets, D(1, 0) ≡ LM(R 1 , R 0 )/LM(VA1 , VA0 ),

(2.180)

is known as the Domar factor: the ratio of (mean) revenue over (mean) value added. One easily checks that if revenue R t is equal to cost C t (or value added V At is equal t , or profit t is equal to 0) (t = 0, 1), then expression to primary inputs cost CKL (2.179) reduces to ln ITFPRODVA (1, 0) = D(1, 0) ln ITFPRODY (1, 0).

(2.181)

This corresponds to Proposition 1 of Gollop (1979). See Jorgenson et al. (2005, 298) for a modern derivation under the usual neo-classical assumptions.

Appendix C: The Domar Factor

63

And the Labour Productivity Index According to the definition of the value-added based labour productivity index, expression (2.81), ln ILPRODVA (1, 0) = ln QVA (1, 0) − ln QL (1, 0).

(2.182)

Substituting expressions (2.177) and (2.180) delivers ln ILPRODVA (1, 0) = D(1, 0) ln QR (1, 0) −

1 , C0 ) LM(CEMS EMS

LM(VA1 , VA0 )

(2.183)

ln QEMS (1, 0) − ln QL (1, 0).

According to the definition of the gross-output based labour productivity index, expression (2.61), ln ILPRODY (1, 0) = ln QR (1, 0) − ln QL (1, 0).

(2.184)

Substituting this into expression (2.183) delivers ln ILPRODVA (1, 0) = D(1, 0) ln ILPRODY (1, 0) −

1 , C0 ) LM(CEMS EMS

LM(VA1 , VA0 )

(2.185)

ln QEMS (1, 0) + (D(1, 0) − 1) ln QL (1, 0).

The assumption that QR (1, 0) is the Montgomery-Vartia index of QVA (1, 0) and QEMS (1, 0) implies that the sum of its weights is approximately equal to 1; that is, 1 , C0 ) LM(VA1 , VA0 ) LM(CEMS EMS + ≈ 1. LM(R 1 , R 0 ) LM(R 1 , R 0 )

(2.186)

Employing the definition of the Domar factor, this implies that 1 , C0 ) LM(CEMS EMS

LM(VA1 , VA0 )

≈ D(1, 0) − 1.

(2.187)

Substituting this into expression (2.185) delivers ln ILPRODVA (1, 0) ≈ D(1, 0) ln ILPRODY (1, 0) 

 QL (1, 0) + (D(1, 0) − 1) ln , QEMS (1, 0)

(2.188)

64

2 A Framework Without Assumptions

which is our final result. Notice that the second term at the right-hand side of this approximate equality does not vanish if revenue R t equals cost C t . The term vanishes if labour quantity change QL (1, 0) equals the aggregate quantity change of intermediate inputs QEMS (1, 0).

Chapter 3

Capital Input Cost

3.1 Introduction Measuring the value of capital as stock or service flow and the decomposition in price and quantity components is a thorny issue, with deep historical roots and a large literature, summarized by Diewert and Schreyer (2008). A classic paper is Hulten (1990). SNA (2008, Chapter 20) provides a non-technical introduction, building on the important OECD (2009) manual. The focus of this chapter is on capital as input of a production process.1 The background is measurement of total (or multi-) factor productivity change, for which the measurement and decomposition of capital input cost is crucial. The theory is developed with a view to practical implementation. Along the way it is shown that there is no need for the usual neo-classical assumptions. The novel elements of this chapter can be summarized in the following points: 1. Though the literature acknowledges the fact that for the treatment of capital input it is necessary to distinguish between time periods as points of time (that is, instances of a real variable) and time periods as intervals with a definite length, in the subsequent mathematics this distinction often is not maintained. We are here suggesting a notation to deal with this problem. 2. Investments are made at irregular time intervals. The literature by and large seems to assume that investments made during a certain time period already had become operational at the beginning of the same period. However, it is here explicitly assumed that investments become operational at each period’s midpoint. 3. The literature grosso modo neglects the fact that investments not only concern new assets but also used assets, and that there is a substantial trade in used assets. However, it is here explicitly assumed that investments can be of various ages.

1 Parts of this chapter are based on joint work with Dirk A. van den Bergen at Statistics Netherlands

in the years 2000–2010. © The Author(s), under exclusive license to Springer Nature Switzerland AG 2021 B. M. Balk, Productivity, Contributions to Economics, https://doi.org/10.1007/978-3-030-75448-8_3

65

66

3 Capital Input Cost

4. The detailed discussion of implementation issues highlights the role of models, assumptions and approximations, and is instrumental for the design of various types of sensitivity analysis. In this chapter the ex post accounting point of view is used. This is consistent with the statistician’s point of view, which is by and large accepted for the components of value added and labour input cost. For revenue, intermediate inputs cost, and labour input cost one uses simply the observed (money) values, whereby at the lowest level of aggregation prices are computed as ratios of observed values and observed quantities; that is, prices are unit values, ex post measured. Of course, a distinctive feature of capital input cost is that this cost as such cannot be observed. Imputations must be made, not only to arrive at (an estimate of) the cost, but also to enable one to split the cost in price and quantity components. Imputations are always more or less arbitrary, and depend on the purpose of the accounting exercise. One has to be clear about this. The layout of this chapter is as follows. The fundamental KLEMS-Y input-output model of a production unit was introduced in Sect. 2.2 of Chap. 2. This provides the framework for what follows in the present chapter. Section 3.2 introduces the notation, derives the unit user cost formulas for assets available at the beginning of an accounting period as well as for assets invested during this period, and discusses the decomposition of capital input cost change in price and quantity indices. Section 3.3 discusses the relation with capital stock measures. Unit user cost depends on prices (or valuations) of the assets, but these prices are not observable. Hence, Sect. 3.4 discusses the definition of such prices from expected values of the variables involved. This gives rise to a decomposition of total user cost into four components, namely the cost of waiting, the cost of anticipated time-series depreciation, the cost of unanticipated revaluation, and the cost of tax. An important role in the cost of waiting is played by the interest rate, which is also called the ‘rate of return’. There appear to be several concepts of this rate; they are discussed in Sect. 3.5. In Sect. 3.6 the issue of aggregation is considered. Section 3.7 discusses some issues of implementation, and Sect. 3.8 concludes.

3.2 Defining Capital Input Cost Cash flow is defined as revenue minus intermediate inputs cost and labour input cost. Cash flow minus capital input cost equals profit. When positive, cash flow can be seen as the (gross) return to capital input. In Sect. 2.3.3 this was called the K-CF model. Of course, this model makes only sense when the production unit actually owns capital assets. But let us assume that this is indeed the case. The K-CF model provides a good point of departure for a discussion of the measurement of capital input cost. Cash flow, as defined in the foregoing, is the (ex post measured) monetary balance of all the flow variables. Capital input cost is different, since capital is a stock variable. Basically, capital input cost is measured

3.2 Defining Capital Input Cost

67

as the difference between the book values of the production unit’s owned capital stock at beginning and end of the accounting period considered. The notation must reflect this. The beginning of period t is denoted by t − , and its end by t + . Thus a period is an interval of time t = [t − , t + ], where t − = (t − 1)+ and t + = (t + 1)− . Occasionally, the variable t will also be used to denote the midpoint of the period. All the assets are supposed to be economically born at midpoints of periods, whether this has occurred inside or outside the production unit under consideration. Thus the age of an asset of type i at (the midpoint of) period t is a non-negative integer number j = 0, . . . , Ji . The age of this asset at the beginning of the period is j − 0.5, and at the end j + 0.5. The economically maximal service life of asset type i is denoted by Ji .2 The opening stock of capital assets is the inheritance of past investments and desinvestments; hence, the opening stock consists of cohorts of assets of various types, each cohort comprising a number of assets of the same age. By convention, assets that are discarded (normally retired or prematurely scrapped) or sold during a certain period t are supposed to be discarded or sold at the end of that period; that is, at t + . Second-hand assets that are acquired during period t from other production units are supposed to be acquired at the beginning of the next period, (t + 1)− . However, all the other acquisitions of second-hand assets and those of new assets are supposed to happen at the midpoint of the period, and such assets are supposed to be immediately operational. Hence, all the assets that are part of the opening stock remain active through the entire period [t − , t + ]. The period t investments are supposed to be active through the second half of period t, that is, [t, t + ]. Put otherwise, the stock of capital assets at t, the midpoint of the period, is the same as the stock at t − , the beginning of the period, but 0.5 period older. At the midpoint of the period the investments, of various age, are added to the stock. Notice, however, that the closing stock at t + , the end of the period, is not necessarily identical to the opening stock at (t + 1)− , because of the convention on sale, acquisition, and discard of assets. Let Kijt denote the quantity (number) of asset type i (i = 1, . . . , I ) and age j (j = 1, . . . , Ji ) at the midpoint of period t. These quantities are non-negative; most of them might be equal to 0. Further, let Iijt denote the (non-negative) quantity (number) of asset type i (i = 1, . . . , I ) and age j (j = 0, . . . , Ji ) that is added to the stock at the midpoint of period t. The following expressions capture our assumptions: −

t t Ki,j −0.5 = Kij (j = 1, . . . , Ji ) +

t t = Ki,0.5 Ii0

2 It

(3.1) (3.2)

is of course a simplification to assume that the economically maximal service life of an asset type is some given constant. Actually a time superscript should be added. See Diewert (2009) for some theoretical considerations.

68

3 Capital Input Cost +

t Iijt + Kijt = Ki,j +0.5 (j = 1, . . . , Ji ) (t+1)−

+

+

t t Ki,(j +1)−0.5 = Ki,j +0.5 + Bi,j +0.5 (j = 0, . . . , Ji − 1) (t+1)−

Ki,(Ji +1)−0.5 = 0,

(3.3) (3.4) (3.5)

+

t + where Bi,j +0.5 denotes the balance of sale, acquisition, and discard at t . The first expression states that all the assets available at the beginning of period [t − , t + ] are still available at the midpoint of this period, but 0.5 period older. The second expression states that new investments, of age 0 at the midpoint of the period, are still available at the end of the period, but then of age 0.5. The third expression states that acquisitions of used assets (not from fellow production units) are added to the available assets, and all remain available till the end of the period [t − , t + ]. The fourth expression states that all these assets, minus those discarded and sold or acquired from fellow production units, go over to the next period [(t +1)− , (t +1)+ ]. The final expression states that all the assets which have reached their maximal service life at the midpoint of period [t − , t + ] simply vanish at the end of this period. In general, estimates of the K, I , and B variables are generated from detailed investment and desinvestment surveys, combined with the Perpetual Inventory Method. The level of detail can vary between production units. For operational details the reader is referred to the OECD (2009) manual. We are now ready to define the concept of user cost for assets that are owned by the production unit. The first distinction that must be made is between assets which are part of the opening stock of a period, and investments which are made during this period. Consider an asset of type i that has age j at the midpoint of period t. Its price t− (or valuation) at the beginning of the period is denoted by Pi,j −0.5 , and its price +

t (or valuation) at the end of the period by Pi,j +0.5 . For the time being, we consider such prices as being given, and postpone their precise definition to a next section. The prices are assumed to be non-negative; many might be equal to 0. In any case, t+ Pi,J = 0; that is, an asset that has reached its economically maximal age in i +0.5 period t is valued with a zero price at the end of this period. The (ex post) unit user cost over period t of an opening stock asset of type i that has age j at the midpoint of the period is then defined as

 − t− t t+ t utij ≡ r t Pi,j −0.5 + Pi,j −0.5 − Pi,j +0.5 + τij (j = 1, . . . , Ji ).

(3.6)

There are three components here. Let us start with the second, most important, one. t− t+ This part of expression (3.6), Pi,j −0.5 − Pi,j +0.5 , is the value change of the asset between beginning and end of the accounting period. Among National accountants this value change is called (nominal) time-series depreciation. It combines the effect of the progress of time, from t − to t + , with the effect of ageing, from j − 0.5 to j + 0.5. In general, the difference between the two prices (valuations) comprises the

3.2 Defining Capital Input Cost

69

effect of exhaustion, deterioration, and obsolescence. Decompositions are discussed in Appendix A. The third component, τijt , denotes the specific tax(es) that is (are) levied on the use of an asset of type i and age j during period t.3 t− Finally, the first component, r t Pi,j −0.5 , is the price (or valuation) of this asset at the beginning of the period, when its age is j − 0.5, times an interest rate r t . This component reflects the premium that must be paid by the user of the asset to its owner to prevent that it be sold, right at the beginning of the period, and the revenue used for immediate consumption; it is therefore also called the price of ‘waiting’.4 Another interpretation is to see this component as the actual or imputed interest cost to finance the monetary capital that is tied up in the asset; it is then called ‘opportunity cost’. Anyway, it is a sort of remuneration which, because there might be a risk component involved, is specific for the production unit and the asset, though the last complication is usually not taken into account.5 Unit user cost as defined by expression (3.6) is also called ‘rental price’, because it can be considered as the rental price that the owner of the asset as owner would charge to the owner as user. Put otherwise, unit user cost is like a lease price.6 Though expression (3.6) has been justified from various viewpoints, the expression basically goes back to Walras (1874, translated 1954, 269).7 One particular justification is worth recalling. Rearranging expression (3.6) delivers  t− t t t+ = u − τ (3.7) (1 + r t )Pi,j ij ij + Pi,j +0.5 (j = 1, . . . , Ji ). −0.5 Selling the asset at the beginning of the period and generating a return from the proceeds of the sale should cover the rental price minus tax plus the money necessary for buying back the asset at the end of the period when it is a full period older. Thus Eq. (3.7) could be seen as the outcome of an arbitrage process. There is, however, no behavioural assumption involved, because the rental price is not an independent variable, but defined by the same equation. The rental price is precisely equal to the cost of using this asset during one period, which is the sum of value change, tax, and opportunity cost.8

3 Fatica

(2017) multiplies the first and second component by a factor accounting for corporate taxation. 4 According to Rymes (1983) this naming goes back to Pigou. 5 SNA (2008, par. 6.130) implicitly prescribes that for non-market units the interest rate r t must be set equal to 0. 6 It should be noted that all the operational costs associated with the use of a particular asset are accounted for as intermediate or labour inputs cost. For investment decisions expected values of such cost components must of course be considered together with expected user cost. 7 The slightly different concept of ex ante user cost goes back to Hicks (1946, 193–194). 8 In the case of irreversible investments, such as electricity networks, basically a slight modification of expression (3.6) must be used, as argued by Diewert et al. (2009, Chapter 10). The first part

70

3 Capital Input Cost

Let us now turn to the unit user cost of an asset of type i and age j that is acquired at the midpoint of period t. To keep things simple, this user cost is, analogous to expression (3.6), defined as  t t t+ t + Pi,j − Pi,j vijt ≡ (1/2)r t Pi,j +0.5 + (1/2)τij (j = 0, . . . , Ji ).

(3.8)

The difference from the previous formula is that here the second half of the period instead of the entire period is taken into account.9 Total user cost over all asset types and ages, for period t, is then naturally defined by ≡

t CK

Ji I  

utij Kijt

+

i=1 j =1

Ji I  

vijt Iijt .

(3.9)

i=1 j =0

We see that the set of commodities K consists of two subsets, corresponding respectively to the type-age classes of assets that are part of the opening stock and thetype-age classes of assets that are acquired later. The dimension of the first set is Ii=1 Ji , and the dimension of the second set is Ii=1 (1 + Ji ). The input prices are given by expressions (3.6) and (3.8) respectively, while the quantities are given by Kijt and Iijt respectively. 1 /C 0 The next task is to split, for any two periods t = 0, 1, the cost ratio CK K 0 1 into a price index and a quantity index, or the cost difference CK − CK into a price indicator and a quantity indicator. There is some choice here and guidance can be obtained from Appendix A of Chap. 1. For example, the Laspeyres quantity index reads Ji 0 1 i=1 j =1 uij Kij I Ji 0 0 i=1 j =1 uij Kij I

QL K (1, 0)

≡

Ji 0 1 i=1 j =0 vij Iij   + Ii=1 Jj i=0 vij0 Iij0 +

I

.

(3.10)

When all or a number of the maximal life times Ji are time-dependent, a detour might be necessary. First one computes a price index PK (1, 0) on the common 1 /C 0 )/P (1, 0). vintages, and next an implicit quantity index as (CK K K If all the variables occurring in expression (3.9) were observable, then our story would almost end here. However, this is not the case. Though the quantity variables are in principle observable, the price variables are not. To start with, the expressions (3.6) and (3.8) contain prices (valuations) for all asset types and ages, but, except

remains as it is; in the second part, now called amortisation amount, instead of the actual end-ofperiod price the expected end-of-period price must be used; and the tax component disappears. 9 The factor (1/2)r t is meant as an approximation to (1 + r t )1/2 − 1, and the factor (1/2)τ t as an ij −

t 1/2 − 1)P t . approximation to ((1 + τijt /Pi,j i,j −0.5 )

3.3 The Relation with Capital Stock Measures and the Classic Formulas

71

for new assets and where markets for second-hand assets exist, such prices are not observable. Thus, we need models. But first we turn to the relation between measures of capital input cost and capital stock.

3.3 The Relation with Capital Stock Measures and the Classic Formulas The set of quantities {Kijt , Iijt ; i = 1, . . . , I ; j = 0, . . . , Ji } represents the so-called productive capital stock of the production unit. This is an enumeration of the assets that make production during period t possible. The total value of these assets at the midpoint of period t can be calculated as NCSt ≡

Ji I  

Pijt Kijt +

i=1 j =1

Ji I  

Pijt Iijt .

(3.11)

i=1 j =0

This value is called the net (or wealth) capital stock. Notice that, though the quantities in expression (3.11) are the same as those occurring in expression (3.9), the prices are different. In expression (3.9) the productive capital stock is valued at unit user costs, whereas in expression (3.11) market prices are used. For any two periods t = 0, 1, using well-known index number tools, the ratio NCS1 /NCS0 can also be split into a price index and a quantity index. For example, the Laspeyres quantity index reads Ji 0 1 i=1 j =1 Pij Kij I Ji 0 0 i=1 j =1 Pij Kij I

QL NCS (1, 0)

≡

Ji 0 1 i=1 j =0 Pij Iij  I Ji + i=1 j =0 Pij0 Iij0 +

I

.

(3.12)

L Both quantity indices, QL NCS (1, 0) and QK (1, 0), measure the volume change of the productive capital stock. The prices, used to weigh the quantities, are different, however. The quantity index QL NCS (1, 0) is called a volume index of the (1, 0) is called a volume index of capital services. Their capital stock, whereas QL K numerical difference can be appreciable (see, for example, Coremberg 2008). If there are no transactions in second-hand assets, then the number of assets Kijt t−j

is equal to the number of new investments of j periods earlier, Ii0 , adjusted for the t−j t ;i = probability of survival. The productive capital stock then reduces to {Ii0 , Ii0 1, . . . , I ; j = 1, . . . , Ji }. If, in addition, new investments are supposed to start their economic life at the beginning of the first-next period, then the productive capital t−j stock becomes {Ii0 ; i = 1, . . . , I ; j = 1, . . . , Ji }, and expression (3.9) reduces to

72

3 Capital Input Cost

⎛ ⎞ Ji I   t−j t ⎝ CK = utij Ii0 ⎠ , i=1

(3.13)

j =1

which is the classic formula for the value of capital services. This formula can be rewritten as ⎛ ⎞ Ji I   t−j t = uti1 ⎝ (utij /uti1 )Ii0 ⎠ , (3.14) CK j =1

i=1

provided that uti1 = 0. If it could be assumed that, for each asset type, the unit user cost ratios are independent of time10 , that is, utij /uti1 = φij (i = 1, . . . , I ; j = 1, . . . , Ji ),

(3.15)

then expression (3.14) reduces to t CK =

I 

⎛ ⎞ Ji  t−j uti1 ⎝ φij Ii0 ⎠ ,

i=1

(3.16)

j =1

Notice that φi1 = 1 (i = 1, . . . , I ). Moreover, common sense suggests that unit user costs decline with age so that 0 < φij ≤ 1 (i = 1, . . . , I ; j = 1, . . . , Ji ). For each asset i = 1, . . . , I , the coefficients φij transform (linearly) assets of age 2 to Ji into assets of age 1. The set {φi1 , . . . , φiJi } is called the age-efficiency profile of asset  t−j type i. The productive capital stock can now be depicted as { Jj i=1 φij Ii0 ; i = 1, . . . , I }. All the assets are measured in efficiency units, that is, in units of age 1, and the only unit user costs needed for the computation of the value of capital services are the user costs of 1 year old assets. Under the same two assumptions—no transactions in second-hand assets, and new investments start their economic life at the beginning of the first-next period— expression (3.11) reduces to ⎛ ⎞ Ji I   t−j ⎝ NCSt = Pijt Ii0 ⎠ . i=1

(3.17)

j =1

which is the classic formula for the net capital stock value. This formula can be rewritten as

10 Authors

such as Hulten (1990) suggest that “in equilibrium” this will be the case. Hulten (2010) calls it a “strong assumption.”

3.3 The Relation with Capital Stock Measures and the Classic Formulas

NCSt =

I 

⎛ ⎞ Ji  t−j t ⎝ t Pi0 (Pijt /Pi0 )Ii0 ⎠ ,

73

(3.18)

j =1

i=1

t = 0. If it could be assumed that, for each asset type, the price provided that Pi0 (valuation) ratios are independent of time, that is, t = ϕij (i = 1, . . . , I ; j = 1, . . . , Ji ), Pijt /Pi0

(3.19)

then expression (3.18) reduces to NCSt =

I 

⎛ ⎞ Ji  t−j t ⎝ Pi0 ϕij Ii0 ⎠ ,

i=1

(3.20)

j =1

Since ageing in general reduces the value of an asset, it is to be expected that the coefficients ϕij are smaller than 1 and their successive magnitudes declining. The set {ϕi1 , . . . , ϕiJi } is called the age-price profile of asset type i. The productive capital  t−j stock can now be depicted as { Jj i=1 ϕij Ii0 ; i = 1, . . . , I }. Per asset type all the vintages are transformed by means of relative prices into new investments, of age 0 in period t, and then added together. For the computation of the value of the capital stock one then only needs the prices of new investments.11 Though the two profiles, age-efficiency and age-price, are in general different, in Appendix B it is shown that under mild regularity conditions a geometric ageefficiency profile, φij = (1 − δi )j −1 , implies a geometric age-price profile, ϕij = (1 − δi )j , and vice versa.12 If geometric profiles can be assumed to hold for all asset types it is interesting to compare capital services and capital stock under a simple steady growth model. In such a model new investments are assumed to grow each t−1 t−2 year by constant percentages θi ; that is, Ii0 = (1 + θi )Ii0 (i = 1, . . . , I ). This assumption, equivalently, means that t−j

Ii0

t−1 = Ii0 /(1 + θi )j −1 (i = 1, . . . , I ; j = 1, . . . , Ji ).

(3.21)

The two assumptions, geometric depreciation and steady growth, together imply that expression (3.16) reduces to t CK =

I  i=1

⎛ t−1 uti1 ⎝Ii0

 Ji   1 − δi j −1 j =1

11 In

1 + θi

⎞ ⎠.

(3.22)

the literature one sometimes finds the productive capital stock at current prices defined as  t Ji φ I t−j . P i=1 i0 j =1 ij i0

I

12 See

SNA (2008, par. 20.22–24) for a simple proof.

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3 Capital Input Cost

Generally the depreciation rates δi appear to be somewhere in the range 0.03–0.16, and the typical growth rates θi lie below 0.10. Thus (1 − δi )/(1 + θi ) < 1. If Ji is sufficiently large, then  Ji   1 − δi j −1 j =1

1 + θi

≈

1 1−

1−δi 1+θi

=

1 + θi , δi + θi

and expression (3.22) reduces to t CK ≈

I  i=1

uti1

t Ii0 . δi + θi

(3.23)

Likewise it can be checked that expression (3.20) reduces to NCSt ≈

I  i=1

t Pi0 (1 − δi )

t Ii0 . δi + θi

(3.24)

Thus for each asset type i = 1, . . . , I the volume of the capital stock available for t /(δ + θ ), the volume of new production purposes during period t is given by Ii0 i i investment during period t divided by the sum of the depreciation and growth rates. For the value of capital services the capital stock volume must be valued at the unit user cost of a one period old asset uti1 . For the net capital stock value the valuation t (1 − δ ). The adjustment reflects the fact that is at the adjusted new asset price Pi0 i the youngest elements of the capital stock are of age 1; thus, valueing them at new asset prices would be inadequate. Blades (2015), building on OECD (2009, Annex C), suggested the method outlined here for estimating the capital stock at the economy level, as a way to circumvent the labourious route via (des-) investment surveys and applications of the Perpetual Inventory Method. Though geometric depreciation is both used and accepted widely, the validity of the steady growth assumption, especially at lower levels of aggregation, is doubtful. It is well known that investment is by and large a lumpy process. Moreover, for the estimation of growth curves we need data obtained from direct sources providing information about investment, desinvestment, service lives, and survival. It is hard to see how one can make progress without such information. All in all, the primitive, classic expressions for the value of capital services and capital stock remain expressions (3.13) and (3.17), respectively, as suggested by Diewert and Lawrence (2000). Recall, however, that these expressions are special cases of expressions (3.9) and (3.11), respectively. We now return to where we left off at the end of the previous section.

3.4 The Relation Between Asset Price and Unit User Cost

75

3.4 The Relation Between Asset Price and Unit User Cost Consider expression (3.6) and rewrite it in the form −

+

t t utij − τijt = (1 + r t )Pi,j −0.5 − Pi,j +0.5 (j = 1, . . . , Ji ).

(3.25)

For any asset that is not prematurely discarded it will be the case that its value at the end of period t is equal to its value at the beginning of period t + 1; formally, (t+1)− t+ Pi,j +0.5 = Pi,(j +1)−0.5 . Substituting this into expression (3.25), and rewriting again, one obtains −

t Pi,j −0.5 =

1  (t+1)− t t P (j = 1, . . . , Ji ). + u − τ ij ij i,(j +1)−0.5 1 + rt

(3.26)

This expression links the price of an asset at the beginning of period t with its price at the beginning of period t + 1, being then 1 period older. But a similar relation links its price at the beginning of period t + 1 with its price at the beginning of period t + 2, being then again 1 period older,  1 (t+2)− t+1 t+1 P + u − τ i,j +1 i,j +1 (j = 1, . . . , Ji ). i,(j +2)−0.5 1 + r t+1 (3.27) This can be continued until (t+1)−

Pi,(j +1)−0.5 =

(t+J −j )−

i Pi,Ji −0.5

=



1 1 + r t+Ji −j

(t+J −j +1)−

i Pi,Ji +0.5

(t+Ji −j +1)− since we know that Pi,Ji +0.5

t+J −j

+ ui,Ji i

(t+Ji −j )+ Pi,Ji +0.5

= into (3.26), etcetera, one finally obtains

t+Ji −j

− τi,Ji



(j = 1, . . . , Ji ),

(3.28) = 0. Substituting expression (3.27)

−

t Pi,j −0.5 =

utij − τijt 1 + rt

(3.29) +

t+1 ut+1 i,j +1 − τi,j +1

(1 + r t )(1 + r t+1 )

t+J −j

+ ... +

ui,Ji i

t+Ji −j

− τi,Ji

(1 + r t ) . . . (1 + r t+Ji −j )

.

Here materializes what is known as the fundamental asset price equilibrium equation. Notice, however, that there was no equilibrium—whatever that may mean—assumed here, and that there are no other economic behavioural assumptions involved; it is just a mathematical result. Expressions (3.25) and (3.29) are dual. The first derives the (ex tax) unit user cost from discounted asset prices, while the second derives the asset price as the sum of discounted future (ex tax) unit user costs; the discounting is executed by means of future interest rates. A mathematical truth like expression (3.29), however, is not immediately helpful in the real world. At the beginning, or even at the end of period t most if not all of the t− t+ data that are needed for the computation of the asset prices Pi,j −0.5 and Pi,j +0.5 are

76

3 Capital Input Cost

not available. Thus, in practice, expression (3.29) must be filled in with expectations, and these depend on the point of time from which one looks at the future. A rather − natural vantage point is the beginning of period t; thus, the operator E t placed before a variable means that the expected value of the variable at t − is taken.13 Modifying expression (3.29), the price at the beginning of period t of an asset of type i and age j − 0.5 is given by −

t Pi,j −0.5 ≡

(3.30) −

−

E t (utij − τijt ) −

1 + Et rt

+

−

t+1 E t (ut+1 i,j +1 − τi,j +1 ) −

t+E t Ji −j − i,E t Ji

−

E t (u

−

(1 + E t r t )(1 + E t r t+1 ) −

t+E t Ji −j ) − i,E t Ji

−τ

−

+ ... +

−

(1 + E t r t ) . . . (1 + E t r t+E

t− J

i −j

. )

Notice in particular that in this expression the economically maximal age, as − expected at the beginning of period t, E t Ji , occurs. Put otherwise, at the beginning − of period t the remaining economic lifetime of the asset is expected to be E t Ji − 14 j +0.5 periods. For each of the coming periods there is an expected (ex tax) rental, and the (with expected interest rates) discounted rentals are summed. This sum then constitutes the price (value) of the asset. Similarly, the price at the end of period t of an asset of type i and age j + 0.5 is given by (t+1)−

+

t Pi,j +0.5 = Pi,(j +1)−0.5 ≡

(3.31)

−

t+1 E (t+1) (ut+1 i,j +1 − τi,j +1 ) −

1 + E (t+1) r t+1 −

+

t+E (t+1) Ji −j − i,E (t+1) Ji

E (t+1) (u −

−

−

t+2 E (t+1) (ut+2 i,j +2 − τi,j +2 ) −

−

(1 + E (t+1) r t+1 )(1 + E (t+1) r t+2 )

+ ... +

−

t+E (t+1) Ji −j ) − i,E (t+1) Ji

−τ

−

(1 + E (t+1) r t+1 ) . . . (1 + E (t+1) r t+E

(t+1)− J

i −j

. )

Notice that this price depends on the economically maximal age, as expected at the − beginning of period t + 1 (which is the end of period t), E (t+1) Ji , which may or may not differ from the economically maximal age, as expected one period earlier, − E t Ji . The last mentioned expected age plays a role in the price at the end of period t of an asset of type i and age j + 0.5, as expected at the beginning of this period,

13 Notice

that there is no underlying probability distribution assumed.

14 See Erumban (2008) on the estimation of expected lifetimes for three types of assets in a number

of industries.

3.4 The Relation Between Asset Price and Unit User Cost −

77

+

t E t Pi,j +0.5 ≡

(3.32)

t−

t+1 E (ut+1 i,j +1 − τi,j +1 ) −

1 + E t r t+1 −

t+E t Ji −j − i,E t Ji

−

E t (u

t−

t+2 E (ut+2 i,j +2 − τi,j +2 )

+

−

+ ... +

−

t+E t Ji −j ) − i,E t Ji

−τ

−

−

(1 + E t r t+1 )(1 + E t r t+2 )

−

(1 + E t r t+1 ) . . . (1 + E t r t+E

t− J

i −j

. )

Expression (3.32) was obtained from expression (3.30) by deleting its first term as − well as the first-period discount factor 1 + E t r t . This reflects the fact that at the end of period t the asset’s remaining lifetime has become shorter by one period. − t+ t− Generally one may expect that E t Pi,j +0.5 ≤ Pi,j −0.5 . Expression (3.31) differs from expression (3.32) in that expectations are at (t + 1)− instead of t − . Since one may expect that, due to technological progress, − − the remaining economic lifetime of any asset shortens, that is, E (t+1) Ji ≤ E t Ji , expression (3.31) contains fewer terms than expression (3.32). Generally one may t+ t− t+ expect that Pi,j +0.5 ≤ E Pi,j +0.5 ; that is, the actual price of an asset at the end of a period is less than or equal to the price as expected at the beginning. Armed with these insights we return to the unit user cost expressions (3.6) and (3.8). Natural decompositions of these two expressions are utij =

(3.33)

 − t− t t− t+ + P − E P r t Pi,j −0.5 i,j −0.5 i,j +0.5 +  − + t t+ t E t Pi,j +0.5 − Pi,j +0.5 + τij (j = 1, . . . , Ji ),

and vijt =

(3.34) 

+



t t t + Pi,j − E t Pi,j (1/2)r t Pi,j +0.5 +  t+ t+ t E t Pi,j +0.5 − Pi,j +0.5 + (1/2)τij (j = 0, . . . , Ji ).

As before, the first term at either right-hand side represents the price of waiting. The second term, between brackets, is called anticipated time-series depreciation, and could be decomposed further into the anticipated effect of time (or, anticipated revaluation) and the anticipated effect of ageing (or, anticipated cross-sectional depreciation). See Appendix A for such decompositions. The third term, also between brackets, is called unanticipated revaluation. Anticipated time-series depreciation plus unanticipated revaluation equals (nominal) time-series depreciation. We will return to these terms later.

78

3 Capital Input Cost

The underlying idea is that, at the beginning of each period or, in the case of investment, at the midpoint, economic decisions are based on anticipated rather than realized prices. When anticipated and realized prices coincide then the third term vanishes and the second term becomes (nominal) time-series depreciation. The fourth term in the two decompositions is again the tax term. It is here assumed that with respect to waiting and tax anticipated and realized prices coincide. Substituting expressions (3.33) and (3.34) into expression (3.9), one obtains the following aggregate decomposition, t CK = Ji I   i=1 j =1

(3.35) −

t t r t Pi,j −0.5 Kij +

Ji I   t t (1/2)r t Pi,j Iij + i=1 j =0

Ji  Ji  I  I    t− t− t+ t t t+ t Pi,j K Pi,j − E P + − E t Pi,j ij −0.5 i,j +0.5 +0.5 Iij + i=1 j =1

i=1 j =0

Ji  Ji  I  I    t− t+ t+ t t+ t+ t E Pi,j +0.5 − Pi,j +0.5 Kij + E t Pi,j +0.5 − Pi,j +0.5 Iij + i=1 j =1

Ji I  

i=1 j =0

τijt Kijt +

i=1 j =1

Ji I   (1/2)τijt Iijt . i=1 j =0

On the first line after the equality sign we have the aggregate cost of waiting, t CK,w ≡ r t UCSt ,

(3.36)

where UCS ≡ t

Ji I   i=1 j =1

t− t Pi,j −0.5 Kij

Ji I   t t + (1/2)Pi,j Iij .

(3.37)

i=1 j =0

Notice that this differs slightly from the net capital stock NCSt as defined by expression (3.11). Midperiod prices Pijt are replaced by beginning-of-period prices −

t Pi,j −0.5 , and for investments the factor 1/2 represents the fact that these are used only during the second half of the period [t − , t + ]. Thus, UCSt can be interpreted as the value of the production unit’s productive capital stock as used during period t. On the second line after the equality sign in expression (3.35) we have the aggregate cost of anticipated time-series depreciation,

3.5 Rates of Return

t CK,e ≡

79

Ji  I   i=1 j =1

Ji  I   t− t− t+ t t t t+ t Pi,j K P − E P + − E P ij i,j −0.5 i,j +0.5 i,j +0.5 Iij . i=1 j =0

(3.38) On the third line we have the aggregate cost of unanticipated revaluation, t CK,u ≡

Ji  Ji  I  I    − t+ t+ t t t+ t+ t E t Pi,j K E − P + P − P ij +0.5 i,j +0.5 i,j +0.5 i,j +0.5 Iij . i=1 j =1

i=1 j =0

(3.39) Finally, on the fourth line we have the aggregate cost of tax, t CK,tax ≡

Ji I   i=1 j =1

τijt Kijt +

Ji I   (1/2)τijt Iijt .

(3.40)

i=1 j =0

Using all these definitions, expression (3.35) reduces to t t t t t CK = CK,w + CK,e + CK,u + CK,tax .

(3.41)

Thus, capital input cost can rather naturally be split into four meaningful components.

3.5 Rates of Return The K-CF model, see Sect. 2.3.3, is governed by the following accounting identity, where input categories are placed left and output categories are placed right of the equality sign: t t t t CK,w + CK,e + CK,u + CK,tax + t = CF t ,

(3.42)

where CF t denotes ex post cash flow generated by the operations of the production unit during period t, and profit t is defined by this identity. The next model is based on the idea that the (ex post) cost of time-series depreciation plus tax should be treated in the same way as the cost of intermediate inputs, and thus subtracted from cash flow. Hence, the output concept is called net cash flow, and the remaining input cost is the waiting cost of capital. This is called the K-NCF model, which is governed by the accounting relation t t t t CK,w + t = CF t − (CK,e + CK,u + CK,tax ) ≡ NCF t .

(3.43)

This relation provides an excellent point of departure for a discussion of the interest t rate r t , which determines the aggregate cost of waiting or opportunity cost CK,w

80

3 Capital Input Cost

according to expression (3.36). Thus Eq. (3.43) can be rewritten as r t UCSt + t = NCF t .

(3.44)

Provided that NCF t ≥ t ≥ 0, Eq. (3.44) then says that, apart from profit, net cash flow provides the return to the (owner of the) capital stock. This is the reason why r t is also called the ‘rate of return’. In principle, the value of the capital stock as well as the net cash flow are empirically determined. That leaves an equation with two unknowns, namely the rate of return r t and profit t . Setting t = 0 and solving Eq. (3.44) for r t delivers the so-called ‘endogenous’, or ‘internal’, or ‘balancing’, rate of return. This solution is, of course, specific for the production unit. Net cash flow is calculated ex post, since it contains total time-series depreciation. Thus, the endogenous rate of return as calculated from expression (3.44) is also an ex post concept. The alternative is to specify some reasonable, exogenous value for the rate of return, say the annual percentage of headline CPI change plus something. Then, of course, profit follows from Eq. (3.44) and will in general be unequal to 0. The endogenous rate of return is defined by the equation t rendo UCSt = NCF t .

(3.45)

Combining this with expression (3.44) delivers the following relation between the endogenous and some exogenous rate of return: t = r t + t /UCSt . rendo

(3.46)

t > r t . Then relation (3.46) can be Hence, profit t is positive if and only if rendo t interpreted as saying that the endogenous rate of return, rendo , absorbs profit. A variant of the K-NCF model is obtained by considering unanticipated revaluation, that is, the unanticipated part of time-series depreciation, as a component that must be added to profit:15 t t t CK,w + ∗t = CF t − (CK,e + CK,tax ) ≡ NNCF t .

(3.47)

This identity at the same time serves as the definition of profit from normal operations ∗t . The relation between the two profit concepts is t . ∗t = t + CK,u

(3.48)

The accounting identity of the K-NNCF model, given by expression (3.47), can be rewritten as r t UCSt + ∗t = NNCF t . 15 Adding

(3.49)

total (= unanticipated plus anticipated) revaluation to profit would be consistent with SNA (2008)’s prescription for non-market units.

3.5 Rates of Return

81

Now, provided that NNCF t ≥ ∗t ≥ 0, normal net cash flow is seen as the return to the (owner of the) capital stock. Setting ∗t = 0 and solving Eq. (3.49) for r t delivers what can be called the ‘normal endogenous’ rate of return, defined by ∗t rendo UCSt = NNCF t .

(3.50)

t , and using expresssion (3.45) it appears that Recalling that NNCF t = NCF t + CK,u ∗t t t rendo = rendo + CK,u /UCSt .

(3.51)

Thus, the normal endogenous rate of return absorbs not only profit, but also the monetary value of all the unanticipated asset revaluations. Alternatively, as in the previous model, one can specify some reasonable, exogenous value for the rate of return. Then, of course, ∗t follows from Eq. (3.49), and by subtracting the value t , one obtains profit t . of all the unanticipated asset revaluations, CK,u The two expressions (3.44) and (3.49) and their underlying models must be considered as polar cases. In the first all the unanticipated revaluations (that is, the t ) are considered as intermediate cost, whereas in the second they are whole of CK,u considered as belonging to profit. Clearly, positions in between these two extremes are thinkable. For some asset types unanticipated revaluations might be considered as intermediate cost and for other types these revaluations might be considered as belonging to profit. For the next rate of return we go back to the accounting identity of the K-CF model in expression (3.42), now written as t t t r t UCSt + CK,e + CK,u + CK,tax + t = CF t .

(3.52)

t t t + CK,u + CK,tax + t = 0. The Solow-Kuznets (S-K) It is now assumed that CK,e rate of return is then defined by the equation t UCSt = CF t . rSK

(3.53)

The interpretation is simple: the S-K rate of return is the unit price of the productive capital stock, obtained by dividing the cash flow by the value of the productive capital stock. Another name is ‘realized rate of return’, and the concept closely corresponds to what business accountants call ‘Return on Assets’ (ROA). This connection can be seen more clearly when the two assumptions made in Sect. 3.3—no transactions in used assets, and new investments start their economic life at the beginning of the first-next time period—are reinstated. Then expression (3.37) reduces to UCSt =

Ji I   i=1 j =1

−

t−j

t Pi,j −0.5 Ii0 .

(3.54)

82

3 Capital Input Cost −

t t If the valuations Pi,j −0.5 may be replaced by Pij , then the productive capital stock t UCS turns into the net capital stock, defined according to expression (3.17). t t t Back to expression (3.53). Recalling that CF t = CK,e + CK,u + CK,tax + NCF t and using the definition of the endogenous rate of return in expression (3.45), we obtain the following relationship: t t t t t = rendo + (CK,e + CK,u + CK,tax )/UCSt rSK

(3.55)

As only the unanticipated revaluation cost can be negative, generally the S-K rate of return will exceed the endogenous rate. If profit t is positive, then the S-K rate will also exceed the chosen exogenous rate.16 A number of conclusions can be drawn. First, there is no single concept of the endogenous rate of return. There is rather a continuum of possibilities, depending on the way one wants to deal with unanticipated revaluations. Second, an endogenous rate of return, of whatever variety, can only be calculated ex post. Net cash flow as well as normal net cash flow require for their computation that the accounting period has expired. Third, we are assuming that all the inputs and outputs are correctly observed. Unobserved inputs and outputs and measurement errors lead to a distorted profit figure. Since an endogenous rate of return can be said to absorb profit—see expressions (3.46) and (3.51) –, the extent of undercoverage has also implications for the interpretation of the rate of return.17 Put otherwise, since an endogenous rate of return closes the gap between the input and the output side of the production unit, it is influenced by all sorts of measurement errors. Finally, the concept of an endogenous rate of return does not make sense for nonmarket units, since there is no accounting identity based on independent measures at the input and the output side. Yet, non-market units use capital like market units do. The accounting point of view as used in this chapter implies an ex post user cost concept. However, economic decision processes are usually based on anticipated magnitudes of certain variables. In particular, investment and other production decisions are based on ex ante user costs. How do the two concepts relate? This is one of the questions considered in an intriguing article by Oulton (2007). He proposed a “hybrid approach”, which in our setup can be summarized as follows. Ex ante capital input cost is calculated as t t t t Cˆ K ≡ CK,w + CK,e + CK,tax ,

16 Lovell

(3.56)

and Lovell (2013) compared the value-added based TFP indices with exogenous and S-K rates of return for the Australian Coal Mining industry over the period 1991–2007 and found that the differences were almost negligible. 17 See Schreyer (2010) or Görzig and Gornig (2013). The last authors specifically consider the effect on the S-K rate of return of including intangible capital at the input and output side of the production units.

3.6 Aggregation

83

t where CK,w is based on some required rate of return r t (perhaps derived from past t is based on expected end-of-period prices. endogenous rates of return) and CK,e 1 /C ˆ 0 , can The ex ante capital input cost ratio for period 1 relative to period 0, Cˆ K K ˆ K (1, 0), and it is be decomposed into a price index PˆK (1, 0) and a quantity index Q this quantity index that is supposed to act as “the” capital input quantity index. Ex post capital input cost plus profit appears to be t t t t CK,w + CK,e + CK,u + CK,tax + t ,

(3.57)

which is by definition equal to CF t . The cash-flow based total factor productivity (TFP) index for period 1 relative to period 0 is generically defined as QCF (1, 0)/QK (1, 0), where QCF (1, 0) is a cash-flow based output quantity index and QK (1, 0) is the quantity index component of the ex post capital input cost ratio (see Section 2.3.3). It seems that Oulton (2007) suggests to calculate the TFP index instead by the following formula: QCF (1, 0) . 0 ˆ ˆ K (1, 0) (CK /CF 0 )Q

(3.58)

We see here that in the denominator the ex ante capital input quantity index is multiplied by the share of ex ante capital input cost in ex post cash flow (which is equal to ex post capital input cost under an endogenous rate of return). Using the familiar product relations linking price index, quantity index, and value ratio, Oulton’s TFP index can be written as QCF (1, 0) CF 1 /PCF (1, 0) = . 0 /CF 0 )Q 1 /Pˆ (1, 0) ˆ K (1, 0) (Cˆ K Cˆ K K

(3.59)

This is deflated ex post cash flow divided by deflated ex ante capital input cost. It is not completely clear what this ratio is supposed to measure.

3.6 Aggregation Let us now consider an ensemble of production units K. Let there be a common classification of asset types and ages. All the price, quantity, and value variables discussed in the foregoing should be adorned with the superscript k. Then, for each production unit the K-NCF accounting relation reads r kt UCSkt + kt = NCF kt (k ∈ K).

(3.60)

84

3 Capital Input Cost

If there are no tax wedges, so that the values of outgoing and incoming trade flows between the production units cancel, then aggregation reduces to simple addition, and the K-NCF accounting relation for the aggregate, considered as a big production unit, reads   r Kt UCSkt + Kt = NCF kt . (3.61) k∈K

k∈K

After having empirically filled in the capital stock and cash flow variables we are left with a large number of interrelated unknowns. For reaching consistency, there are two approaches, a bottom-up approach and a top-down approach, respectively. The bottom-up approach starts, not unexpected, with the relations (3.60). Aggregate profit is then set equal to the sum of individual profits, Kt =



kt ,

(3.62)

k∈K

and the aggregate rate of return is set such that Eq. (3.61) holds. But this means that r Kt



UCSkt =

k∈K



r kt UCSkt ,

(3.63)

k∈K

or r

Kt



k∈K = 

r kt UCSkt

k∈K UCS

kt

.

(3.64)

Thus the rate of return for the aggregate is a weighted mean of the rates of return for the individual units, the weights being shares of the value of the aggregate productive capital stock as used during period t. Notice that if all the individual rates of return are endogenous, then also the aggregate rate of return. This follows immediately from expression (3.62) in combination with the assumption that individual profits are zero. The top-down approach starts with relation (3.61). For each individual production unit one then sets the individual rates of return equal to the rate of return for the aggregate production unit, r kt = r Kt (k ∈ K).

(3.65)

One checks immediately that this implies that Eq. (3.62) holds; that is, aggregate profit isthe sum of all the individual profits. If r Kt is endogenous, then aggregate profit, k∈K kt , is equal to zero, but notice that the individual profits, kt (k ∈ K), need not be equal to zero. Put otherwise, the endogenous rate of return for the aggregate production unit does not necessarily coincide with the endogenous rates of return for the individual production units.

3.7 Some Implementation Issues

85

3.7 Some Implementation Issues There remain a number of implementation issues to discuss. For this, the reader is invited to return to expression (3.35). To ease the presentation, a period is now assumed to be a year. The quantities {Kijt ; i = 1, . . . , I ; j = 1, . . . , Ji } and {Iijt ; i = 1, . . . , I ; j = 0, . . . , Ji }, making up the productive capital stock, are not measured as such. Instead, as is usually the case, the Perpetual Inventory Method generates estit−1 t mates of the opening stock of assets at period t − 1 prices {Pi,j −0.5 Kij = −

t−1 t Pi,j −0.5 Ki,j −0.5 ; i = 1, . . . , I ; j = 1, . . . , Ji }, and an Investment Survey generates estimates of mid-period values {Pijt Iijt ; i = 1, . . . , I ; j = 0, . . . , Ji }. To accomodate this situation, expression (3.35) is modified to

t CK

=

Ji I  

−

t Pi,j −0.5

t

r

t−1 Pi,j −0.5

i=1 j =1 −

t−1 t Pi,j −0.5 Kij +

+

−

−

t−1 Pi,j −0.5

t Pi,j −0.5 −

+

+

−

i=1 j =1

t−1 Pi,j −0.5

t Pi,j −0.5 Ji I   τijt i=1 j =1

t Pi,j

t t−1 Pi,j Pi,j −0.5

(3.66)

+

t−1 t Pi,j −0.5 Kij

−

Ji t t t I   E t Pi,j +0.5 − Pi,j +0.5 Pi,j −0.5

t t (1/2)r t Pi,j Iij +

i=1 j =0

−

Ji t t t I  t  Pi,j −0.5 − E Pi,j +0.5 Pi,j −0.5 i=1 j =1

Ji I  

+

Ji t − Et P t I   Pi,j i,j +0.5 t Pi,j

i=1 j =0 +

t−1 t Pi,j −0.5 Kij +

t−1 t Pi,j −0.5 Kij +

t t Pi,j Iij +

+

Ji t t I   E t Pi,j +0.5 − Pi,j +0.5 t Pi,j

i=1 j =0

Ji I  

(1/2)

i=1 j =0

τijt t Pi,j

t t Pi,j Iij +

t t Pi,j Iij .

−

t−1 t In “normal” times the ratios Pi,j −0.5 /Pi,j −0.5 will be approximately equal to 1 as they measure asset price change between the middle of a period and the end of it. Models for time-series depreciation are briefly discussed in Appendix A. The time-series depreciation of an asset of type i and age j that is available at the beginning of period t is in practice frequently modeled as +

t Pi,j +0.5 −

t Pi,j −0.5

=

PPI ti PPI ti

+ −

(1 − δij ),

(3.67)

where PPI ti denotes the Producer Price Index (or a kindred price index) that is applicable to new assets of type i, and δij is the annual cross-sectional depreciation rate that is applicable to an asset of type i and age j . This depreciation rate ideally comes from an empirically estimated age-price profile. Thus, time-series depreciation is modeled as a simple, multiplicative function of + − two, independent factors. The first, PPI ti /PPI ti , which is 1 plus the annual rate of

86

3 Capital Input Cost

price change of new assets of type i, concerns the effect of the progress of time on the value of an asset of type i and age j . The second, 1 − δij > 0, concerns the effect of ageing by 1 year on the value of an asset of type i and age j . Ageing by 1 year causes the value to decline by δij × 100 percent. Similarly, anticipated time-series depreciation is modeled as

+

−

t E t Pi,j +0.5 −

t Pi,j −0.5

=E

t−

PPI ti PPI ti

+

−

(1 − δij ).

(3.68)

In this expression, instead of the annual rate of price change of new assets, as observed ex post, the annual rate as expected at the beginning of period t is taken. But what to expect? There are, of course, several options here. The first that comes to mind is to use some past, observed rate of change of PPI i or a more general PPI. Second, one could assume that expectedly the rate of price change of new assets is equal to the rate of change of a (headline) Consumer Price Index (CPI), and use the ‘realized expectation’:18 E

t−

PPI ti PPI ti

+

=

−

CPI t CPI t

+ −

(3.69)

.

Under the last assumption anticipated time-series depreciation is measured as −

+

−

t t t Pi,j −0.5 − E Pi,j +0.5 −

t Pi,j −0.5

=1−

CPI t CPI t

+ −

(1 − δij ),

(3.70)

and, combining expressions (3.67) and (3.70), unanticipated revaluation is measured by −

+



+

t t E t Pi,j +0.5 − Pi,j +0.5 −

t Pi,j −0.5

=

CPI t CPI t

+ −

−

PPI ti PPI ti

+ −

(1 − δij ).

(3.71)

Similar expressions hold for assets that are acquired at the midpoint of period t, except that we make a distinction between new and used assets. The time-series depreciation for an asset of type i and age j is modeled as +

t Pi,0.5 t Pi,0

18 This

+

PPI ti = (1 − δi0 ) PPI ti

corresponds to the SNA (2008, par. 12.87) advice on calculating “neutral holding gains.”

3.7 Some Implementation Issues +

87 +

t Pi,j +0.5

PPI ti = (1 − δij /2) (j = 1, . . . , Ji ). PPI ti

t Pi,j

(3.72)

The anticipated time-series depreciation is, again, measured by +

t − Et P t Pi,0 i,0.5 t Pi,0

+

= 1−

CPI t (1 − δi0 ) CPI t

= 1−

CPI t (1 − δij /2) (j = 1, . . . , Ji ), CPI t

+

t − Et P t Pi,0 i,j +0.5 t Pi,j

+

(3.73)

and unanticipated revaluation is measured by +

t Pi,0 +



+

t t E t Pi,0.5 − Pi,0.5

=

+

t t E t Pi,j +0.5 − Pi,j +0.5 t Pi,j

=

+

+

PPI ti CPI t − CPI t PPI ti

+

+

PPI ti CPI t − CPI t PPI ti

(1 − δi0 )

(3.74)

(1 − δij /2) (j = 1, . . . , Ji ).

With respect to the tax terms on the last line of expression (3.66) one should notice t measures tax as a fraction of the price of an asset. In the first of these that τijt /Pi,j t /P t−1 , which might be two terms this fraction must be adjusted by the ratio Pi,j i,j −0.5

approximated by (PPI ti /PPI t−1 i )(1 − δij /2). An important question is in which circumstances the unit user costs utij and vijt become non-positive? Consider, for instance, expression (3.33), and substitute expressions (3.70) and (3.71). This yields utij −

t Pi,j −0.5

= r +1− t

= rt + 1 −

CPI t CPI

t−

PPI ti



+

(1 − δij ) +

+

(1 − δij ) + t−

PPI i

CPI t CPI

+

τijt −

−

t−

t Pi,j −0.5

PPI ti

+

t−

PPI i

(1 − δij ) +

τijt −

t Pi,j −0.5

(3.75)

.

Hence, utij ≤ 0 if and only if PPI ti PPI ti

−

+ −

≥

t 1 + r t + τijt /Pi,j −0.5

1 − δij

.

(3.76)

88

3 Capital Input Cost

In certain, extreme cases this could indeed happen. Consider assets with a very low cross-sectional depreciation rate (such as certain buildings or land) and a very high revaluation rate (or rate of price increase). A low interest-plus-tax rate then could lead to negative unit user costs. Put otherwise, when the ex post revaluation (as measured by a PPI) more than offsets interest plus tax plus depreciation then the unit user cost of such an asset becomes negative. If the unanticipated revaluation is excluded from the user cost, that is, unit user cost is measured by utij t−

Pi,j −0.5

=r +1− t

CPI t CPI

+

t−

(1 − δij ) +

τijt −

t Pi,j −0.5

,

(3.77)

then utij ≤ 0 if and only if CPI t CPI t

−

+ −

≥

t 1 + r t + τijt /Pi,j −0.5

1 − δij

.

(3.78)

The likelihood that such a situation will occur is small. For this to happen, expected revaluation (as measured by a CPI) must more than offset interest plus tax plus depreciation.

3.8 Conclusion The approach followed in this chapter is that capital input cost is the cost of using capital assets, much like labour input cost is the cost of employing workers. However, unlike the building blocks for labour input cost, quantities of assets and unit user costs cannot simply be observed. The usual, neo-classically inspired approach to measuring and decomposing capital input cost seems to rest on a number of daring, behavioural assumptions. We are following here the accounting approach which on the one hand avoids such assumptions and on the other hand shows the degrees of freedom one has in implementing measurement. Some would say that freedom means arbitrariness, or even anarchy. This, however, is not the case. Surely, there is a lot of freedom, but that forces us to think about the choices that must be made. If choices are made to accommodate neo-classical behavioural assumptions, fine. But within the framework explicated in this chapter alternative choices can be argued for. Put otherwise, the accounting framework is so general that the consequences of variant assumptions can be explored and the assumptions themselves be tested at their relevance. To wrap up it is useful to revisit the main steps. As said, the point of departure is that capital input cost is the cost of using (owned) assets. The quantities of those assets, according to type-age classes at some level of detail, at beginning and end of each accounting period are estimated from investment surveys and

Appendix A: Decompositions of Time-Series Depreciation

89

other data sources, combined with some variant of the Perpetual Inventory Method. These quantities constitute the productive capital stock of the production unit under consideration. There appears to be no need for the concept of capital service, or the assumption that the service rendered by a certain asset is proportional to its quantity. This concept is as it were replaced by the user cost concept. The unit user cost of an asset of certain type and age is basically determined by the difference between its beginning- and end-of-period price, plus an opportunity cost determined by a certain interest rate, and an amount of tax. There is no behavioural assumption underlying this specification, but there are a number of as yet undetermined variables. Prices and unit user costs appear to be mutually determined by the fundamental asset price equation, which here materializes as a purely mathematical result. Prices appear to depend on future unit user costs, but the future is unknown; thus, this is where expectations enter the picture. It appears useful to distinguish between actual and expected prices, and to decompose the difference between beginning- and end-of-period actual price into an expected and an unexpected part, where it is assumed that expectations are formed at the beginning of the period. The difference between beginning- and (expected) end-ofperiod prices is then conventionally modeled by a simple multiplicative relation, in which Producer and Consumer Price Indices figure, as well as empirically determined asset-specific depreciation rates. The important remainder is the determination of the opportunity cost component, which turns out to be equal to the value of the productive capital stock times an interest rate. The key assumption of the neo-classical setup is to determine this interest rate, called rate of return, endogenously by setting capital input cost equal to cash flow (or gross operating surplus); put otherwise, by setting profit, as the difference between cash flow and capital input cost, equal to zero. This of course accommodates the assumption of competitive profit maximization under a constant-returns-to-scale technology, but there is nothing in the accounting system that necessitates such an assumption. And the assumption itself is at variance with much empirical evidence. Moreover, as demonstrated in Sect. 3.5, there is not a unique endogenous rate of return, but a continuum of possibilities, depending on the way one wants to deal with unexpected revaluations. The bottom line could be an advice to statistical agencies to use the degrees of freedom available in the measurement system to accommodate different sorts of users.

Appendix A: Decompositions of Time-Series Depreciation Time-series depreciation of an asset of type i and age j over period t is, according to t− t+ expression (3.6), defined as Pi,j −0.5 − Pi,j +0.5 , which is the (nominal) value change of the asset between beginning and end of the period. This value change combines the effect of the progress of time, from t − to t + , with the effect of ageing, from

90

3 Capital Input Cost

j − 0.5 to j + 0.5. Since value change is here measured as a difference, a natural decomposition of time-series depreciation into these two effects is −

+

t t Pi,j −0.5 − Pi,j +0.5 =   − t t+ t− t+ − P ) + (P − P ) (1/2) (Pi,j −0.5 i,j −0.5 i,j +0.5 i,j +0.5 +  −  t t− t+ t+ (1/2) (Pi,j − P ) + (P − P ) −0.5 i,j +0.5 i,j −0.5 i,j +0.5 .

(3.79)

This decomposition is symmetric; it has the same structure as the decomposition of a value difference into Bennet indicators. The first term at the right-hand side of the equality sign measures the effect of the progress of time on an asset of unchanged age; this is called revaluation. The revaluation, as measured here, is the arithmetic mean of the revaluation of a j − 0.5 periods old asset and a j + 0.5 periods old asset, and may be said to hold for a j periods old asset. The second term c oncerns the effect of ageing, which is measured by the price difference of two, otherwise identical, assets that differ precisely one period in age. This is called Hicksian or cross-sectional depreciation. The arithmetic mean is taken of cross-sectional depreciation at beginning and end of the period, and, hence, may be said to hold at the midpoint of period t. The ratio-type counterpart of expression (3.79) is a decomposition into two Fisher-type indices, 

+

t Pi,j +0.5 −

t Pi,j −0.5

=

+

+

−

−

t t Pi,j −0.5 Pi,j +0.5 t t Pi,j −0.5 Pi,j +0.5

1/2 

+

−

+

−

t t Pi,j +0.5 Pi,j +0.5

1/2

t t Pi,j −0.5 Pi,j −0.5

.

(3.80)

The first term at the right-hand side of the equality sign measures revaluation. The second ter m measures cross-sectional depreciation. As one sees, revaluation depends on age, and cross-sectional depreciation depends on time. In the usual model, these two dependencies are assumed away. Revaluation is approximated by + − Pit /Pit , the price change of a new asset of type i from beginning to end of period t. Cross-sectional depreciation is approximated by 1 − δij , where δij is the (positive) percentage of annual depreciation applying to an asset of type i and age j . The specific formulation highlights the fact that ageing usually diminishes the value of an asset. Under these two assumptions, the basic time-series depreciation model for an asset of type i and age j , over period t, is given by +

t Pi,j +0.5 −

t Pi,j −0.5

=

Pit Pit

+ −

(1 − δij ) (j = 1, . . . , Ji ).

(3.81)

For assets that are acquired at the midpoint of period t it is useful to distinguish between new and used assets. Over the second half of period t, the model reads

Appendix B: Geometric Profiles +

t Pi,0.5 t Pi,0

91 +

Pt = i t (1 − δi0 ) Pi

+

t Pi,j +0.5 t Pi,j

+

=

Pit (1 − δij /2) (j = 1, . . . , Ji ), Pit

(3.82)

where (1 − δij /2) serves as an approximation to (1 − δij )1/2 . The percentage of annual depreciation, δij , ideally comes from an empirically estimated age-price profile for asset-type i. Under a geometric profile one specifies δi0 = δi /2 and δij = δi (j = 1, . . . , Ji ). t+ t+ t+ t− t+ t t+ t t+ Replacing Pi,j +0.5 by E Pi,j +0.5 , Pi,0.5 by E Pi,0.5 , and Pi,j +0.5 by E Pi,j +0.5 delivers expressions for anticipated time-series depreciation.

Appendix B: Geometric Profiles Consider the fundamental asset price (equilibrium) equation in expression (3.29). To avoid notational clutter it is assumed that there is no tax. It is further assumed that the future interest rate is constant; that is, r t+j = r (j = 0, . . . , Ji ). Then utij

−

t Pi,j −0.5 =

1+r

+

ut+1 i,j +1 (1 + r)2

t+J −j

+ ... +

ui,Ji i

(1 + r)Ji −j +1

(3.83)

.

A geometric age-efficiency profile19 means that utij /uti1 = φij = (1 − δi )j −1 (i = 1, . . . , I ; j = 1, . . . , Ji ). Equivalently, uti,j +1 /utij = 1 − δi . By substituting this into expression (3.83) one obtains t− Pi,j −0.5

=

utij 1+r

+

ut+1 ij (1 − δi ) (1 + r)2

t+Ji −j

+ ... +

uij

(1 − δi )Ji −j

(1 + r)Ji −j +1

(3.84)

.

For a one-period older asset of the same type one then has, similarly but with one term less, −

t Pi,j +0.5 =

uti,j +1 1+r

+

ut+1 i,j +1 (1 − δi ) (1 + r)2

t+J −j −1

+ ... +

i ui,j +1

(1 − δi )Ji −j −1

(1 + r)Ji −j

.

(3.85)

Using again the geometric-profile assumption, expression (3.85) can be written as

19 See

Diewert (2009) for a model justifying such a profile.

92

3 Capital Input Cost

t− Pi,j +0.5

=

utij (1 − δi ) 1+r

+

2 ut+1 ij (1 − δi )

(1 + r)2

t+Ji −j −1

+...+

uij

(1 − δi )Ji −j

(1 + r)Ji −j

,

(3.86)

or ⎛ t−

Pi,j +0.5 =(1 − δi ) ⎝

utij 1+r

+

ut+1 ij (1 − δi ) (1 + r)2

t+Ji −j −1

+ ... +

uij

(1 − δi )Ji −j −1

(1 + r)Ji −j

⎞ ⎠.

(3.87) −

t The part between brackets appears to be identical to Pi,j −0.5 minus its last term, so that

t− Pi,j +0.5

t− Pi,j −0.5

= (1 − δi )

t+J −j

−

ui,Ji i

(1 + r)Ji −j +1

.

(3.88)

t+J −j

Now ui,Ji i is the user cost of an asset, of type i that is j years old in period t, in the final year of its (economic) existence. If Ji → ∞ then this user cost will expectedly tend to 0. The denominator (1 + r)Ji −j +1 will then tend to ∞. Together it is reasonable to expect that −

−

t t Pi,j +0.5 ≈ (1 − δi )Pi,j −0.5 ,

(3.89)

which means that the age-price profile of the asset is approximately geometric. Reversely, consider expression (3.25) without the tax component: −

+

t t utij = (1 + r t )Pi,j −0.5 − Pi,j +0.5 (j = 1, . . . , Ji ).

(3.90)

For a one-period older asset of the same type then −

+

t t uti,j +1 = (1 + r t )Pi,j +0.5 − Pi,j +1.5 .

(3.91)

t = ϕ j Assume now a geometric age-price profile; that is, Pijt /Pi0 ij = (1 − δi ) t t (i = 1, . . . , I ; j = 1, . . . , Ji ). Equivalently, Pi,j +1 /Pij = 1 − δi . Substituting this into expression (3.91) delivers −

+

t t t uti,j +1 = (1 + r t )(1 − δi )Pi,j −0.5 − (1 − δi )Pi,j +0.5 = (1 − δi )uij ,

which means that the age-efficiency profile of the asset is geometric.

(3.92)

Chapter 4

Annual and Quarterly Measures

4.1 Introduction As we have seen in Chap. 2, productivity change is generally measured in index form as the ratio of an output quantity index over an input quantity index. The presentation is usually in the form of a percentage change (aka growth rate). Specific measures materialize by selecting the output concept to be used (such as gross output or value added) and the number of inputs to be considered (resulting in single, multiple, or total factor productivity indices). The frequency with which such indices are compiled varies. The 2018 edition of the OECD Compendium of Productivity Indicators lists annual data for 44 countries. However, a fair number of official statistical agencies, as well as some international organisations, publish also quarterly data (Haine and Kanutin 2008). A well-known example is the US Bureau of Labor Statistics where such data have been published since 1967 (Eldridge et al. 2008). In most cases it appears that annual and quarterly data are constructed independently from basically the same source materials. This raises the issue of consistency between annual and subannual index numbers. However, even when annual and quarterly index numbers are by construction not independent, there are issues for concern. An interesting example is provided by the quarterly labour productivity statistics published by the UK Office for National Statistics. The basic building block appears to be a quarter-of-current-year-relative-to-previous-quarter productivity index, and quarterly productivity change is this index turned into a percentage. The productivity index for a quarter of the current year relative to some reference year (which is currently 2016) is obtained by chaining the quarter-of-current-year-relative-toprevious-quarter productivity indices (and normalizing to get 2016 = 100). The productivity index for the current year is then defined as the unweighted arithmetic mean of the productivity indices for its four quarters, after seasonal adjustment of the time series. Annual productivity change is obtained by dividing the productivity

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2021 B. M. Balk, Productivity, Contributions to Economics, https://doi.org/10.1007/978-3-030-75448-8_4

93

94

4 Annual and Quarterly Measures

index for the current year by the same index for the previous year, and turning this ratio into a percentage. Though in this setup quarterly productivity change has a definite meaning, the situation is less clear for the concept of annual change. Anyway, the functional forms of the annual and quarterly productivity indices are grossly different, and likely to coincide only in exceptional circumstances. Consistency as discussed in this chapter is a step beyond the consistency concept that figures in the National Accounts literature. There one is concerned with the requirement that annual (real) GDP must be equal to the sum of quarterly (real) GDP, and annual hours worked must be equal to quarterly hours worked, the realisation of which usually invokes some smoothing algorithm such as developed by Denton (1971). However, as will be shown in more detail in what follows, this kind of consistency does not necessarily imply that annual productivity (change) is equal to a simple mean of quarterly productivity (change). The situation is much more complex. This is a sort of “open nerve”. Though users of productivity statistics are well aware of the fact that, due to the complexity of all the survey and compilation processes, one can hardly expect that an independently compiled annual measure of productivity change is equal to a simple or less simple mean of subannual measures, the question of what conceptually is at stake here seems to have been avoided. Though Diewert (2008), in his retrospective survey of the two OECD workshops held in 2005–2006, lists the lack of consistency between quarterly estimates of productivity growth and annual estimates as one of the 12 measurement problems where further research is required, as yet no one has taken up this challenge. This chapter explores, from a theoretical perspective, the options for obtaining consistency between annual and quarterly (or more general: between period and subperiod) measures of productivity change. It is thereby assumed that all the necessary data are given, without statistical error. We are thus not talking about approximation errors. Section 4.2 considers the simple case of a single-input single-output production unit. In this case one can talk about productivity (level) as output quantity divided by input quantity. Annual output or input quantity is the simple sum of subannual quantities. Annual productivity then appears to be a weighted mean of subannual productivities. Annual productivity change is simply defined as the ratio of two annual productivities. For the definition of subannual productivity change there are several options: (1) compare adjacent subperiods; (2) compare corresponding subperiods of adjacent years; (3) compare a subperiod to an earlier year. It will appear that, whatever choice is being made, the relation between annual and subannual indices is anything but simple. Section 4.3 generalizes this to a multiple-input multiple-output situation, where input and output prices are fixed. Section 4.4 considers the general case. The question is, if it is assumed that price, quantity, and productivity indices satisfy some very fundamental axioms, will it then be possible to obtain consistency between annual and subannual indices? The answer appears to be negative.

4.2 A Simple Case

95

Section 4.5 considers the use of subperiod productivity indices as approximations to or forecasts of period indices. Section 4.6 concludes.

4.2 A Simple Case Let us for a start consider a single-input single-output production unit through two adjacent periods, called 0 and 1 respectively, of equal length. Each period consists of Q subperiods, also of equal length. The quantity of output produced during subperiod q of period t will be denoted by y tq (t = 0, 1; q = 1, . . . , Q). Likewise, the quantity of input used during subperiod q of period t will be denoted by x tq (t = 0, 1; q = 1, . . . , Q). All these quantities are assumed to be strictly positive. The quantity of output produced during the entire period t is evidently measured as the sum of the subperiod quantities, yt ≡

Q 

y tq (t = 0, 1).

(4.1)

q=1

Likewise, the quantity of input used during the entire period t is evidently measured by x ≡ t

Q 

x tq (t = 0, 1).

(4.2)

q=1

These two relations are basic for what follows.

4.2.1 Productivity In the case of a single-input single-output unit one can unambiguously talk about productivity as the quantity of output per unit of input. Hence, the productivity in subperiod q of period t is measured by PROD(tq) ≡ y tq /x tq (t = 0, 1; q = 1, . . . , Q),

(4.3)

and the productivity in the entire period t by PROD(t) ≡ y t /x t (t = 0, 1).

(4.4)

It is straightforward to check, using expressions (4.1) and (4.3), that the productivity of any period can be expressed as a weighted arithmetic average of its subperiod

96

4 Annual and Quarterly Measures

productivities, PROD(t) =

Q  (x tq /x t )PROD(tq),

(4.5)

q=1

the weights being input quantity shares. Alternatively, by expressions (4.2) and (4.3) the productivity of any period can be expressed as a weighted harmonic average of its subperiod productivities, ⎛

⎞−1 Q  PROD(t) = ⎝ (y tq /y t )(PROD(tq))−1 ⎠ ,

(4.6)

q=1

the weights now being output quantity shares. It is tempting to ask whether PROD(t) can also be expressed as a geometric mean of the subperiod productivities PROD(tq) (q = 1, . . . , Q). The answer appears to be negative. Employing the logarithmic mean1 one obtains ln PROD(t) =

Q  LM(x tq , y tq ) q=1

LM(x t , y t )

ln PROD(tq),

(4.7)

or PROD(t) =

Q 

tq

PROD(tq)φ ,

(4.8)

q=1

where φ tq ≡ LM(x tq , y tq )/LM(x t , y t ) (q = 1, . . . , Q). Put otherwise, the temporal aggregate productivity PROD(t) is a weighted product of the subperiod productivities PROD(tq) (q = 1, . . . , Q), where the weights are symmetric in input and output quantities. Note however that, due to the concavity of the function LM(a, 1), the sum of these weights is less than or equal to 1, though the difference is usually small. Thus expression (4.8) is not a geometric mean.

4.2.2 Productivity Change Productivity change between two (sub-) periods, as measured in ratio form, is naturally defined as the ratio of the productivities of the two (sub-) periods

1 Recall that for any two strictly positive real numbers a and b their logarithmic mean is defined by LM(a, b) ≡ (a − b)/ ln(a/b) when a = b, and LM(a, a) ≡ a.

4.2 A Simple Case

97

considered. In this way the productivity change between periods 0 and 1 is measured by IPROD(1, 0) ≡

y 1 /x 1 PROD(1) = 0 0. PROD(0) y /x

(4.9)

When considering subperiods, there are a number of possibilities. In line with the previous definition one could consider the productivity change between two adjacent subperiods q − 1 and q of period t; that is, IPROD(tq, t q − 1) ≡ =

PROD(tq) PROD(t q − 1) y tq /x tq (t = 0, 1; q = 1, . . . , Q), y t q−1 /x t q−1

(4.10)

where we will use the convention that subperiod 0 of period t is the same as subperiod Q of period t − 1. A second possibility is to compare the productivity of a certain subperiod to the productivity of the corresponding previous subperiod; that is, IPROD(1q, 0q) ≡

y 1q /x 1q PROD(1q) = 0q 0q (q = 1, . . . , Q). PROD(0q) y /x

(4.11)

A third possibility is to compare the productivity of a certain subperiod to the productivity of the entire previous period; that is, IPROD(1q, 0) ≡

y 1q /x 1q PROD(1q) = 0 0 (q = 1, . . . , Q). PROD(0) y /x

(4.12)

These three are the most usual modes of comparison.

4.2.3 Relations The interesting question now is: which relations exist between subperiod productivity indices, of whatever type, and period indices? Let us first look at the subperiod-to-period type indices. By setting t = 1 in expression (4.5) and dividing both sides by PROD(0), we obtain IPROD(1, 0) =

Q  (x 1q /x 1 )IPROD(1q, 0); q=1

(4.13)

98

4 Annual and Quarterly Measures

that is, IPROD(1, 0) can be written as a weighted mean of IPROD(1q, 0) (q = 1, . . . , Q). The weights are the subperiod input quantity shares of period 1, x 1q /x 1 (q = 1, . . . , Q). What error do we make by replacing these weights by 1/Q? Consider the following modification of the last expression: IPROD(1, 0) =

Q  (1/Q)IPROD(1q, 0) q=1 Q  + (x 1q /x 1 − 1/Q)IPROD(1q, 0).

(4.14)

q=1

The second factor at the right-hand side of this expression can be conceived as the covariance of the input quantity shares x 1q /x 1 and the subperiod productivity indices IPROD(1q, 0). If this covariance happens to be equal to 0, then IPROD(1, 0) is equal to the unweighted arithmetic mean of IPROD(1q, 0) (q = 1, . . . , Q). This assumption, however, is rather strong and, moreover, concerns the comparison period 1, which is unfortunate from the viewpoint of computation in real time. Similarly, based on expression (4.6) we obtain ⎛ ⎞−1 Q  IPROD(1, 0) = ⎝ (y 1q /y 1 )IPROD(1q, 0)−1 ⎠

(4.15)

q=1

=

Q  (1/Q)IPROD(1q, 0)−1 q=1

+

Q −1  (y 1q /y 1 − 1/Q)IPROD(1q, 0)−1 . q=1

The second factor at the right-hand side of this expression can be conceived as the covariance of the output quantity shares y 1q /y 1 and the inverse subperiod productivity indices 1/IPROD(1q, 0). If this covariance happens to be equal to 0, then IPROD(1, 0) is equal to the unweighted harmonic mean of IPROD(1q, 0) (q = 1, . . . , Q). This is also a strong assumption. Finally, based on expression (4.8) we obtain IPROD(1, 0) =

Q  q=1

1/Q

IPROD(1q, 0)

Q 

PROD(1q)φ

1q −1/Q

.

(4.16)

q=1

The first factor at the right-hand side is an unweighted geometric mean. The second factor is not necessarily equal to 1.

4.2 A Simple Case

99

The relation between IPROD(1, 0) and the subperiod-to-corresponding-subperiod indices IPROD(1q, 0q) (q = 1, . . . , Q) is less simple. Again, from expression (4.5) it appears that IPROD(1, 0) =

Q  1q  x PROD(0q) q=1

x 1 PROD(0)

 IPROD(1q, 0q) ;

(4.17)

that is, IPROD(1, 0) can be written as a linear combination of the subperiod indices IPROD(1q, 0q) (q = 1, . . . , Q). One verifies immediately that the weights x 1q PROD(0q)/x 1 PROD(0) don’t add up to 1. Sufficient conditions for these weights to be equal to 1/Q are that the subperiod input quantity shares are invariant through time, x 1q /x 1 = x 0q /x 0 (q = 1, . . . , Q), and that all the output quantity shares of period 0 are the same, y 0q /y 0 = 1/Q (q = 1, . . . , Q). From a practical point of view, such conditions are difficult to justify. Alternatively, from expression (4.6), it appears that we can write ⎛

⎞−1 Q 1q PROD(0)  y −1 IPROD(1, 0) = ⎝ IPROD(1q, 0q) ⎠ . y 1 PROD(0q)

(4.18)

q=1

Thus, IPROD(1, 0) can be written as an harmonic combination of the subperiod indices IPROD(1q, 0q) (q = 1, . . . , Q). But note that the weights y 1q PROD(0)/y 1 PROD(0q) also don’t add up to 1. Finally, using expression (4.8), we obtain IPROD(1, 0) =

1q Q  PROD(1q)φ

q=1

PROD(0q)φ

0q

(4.19)

.

This expression can be decomposed in a number of ways. Using the period 0 viewpoint, we get IPROD(1, 0) =

Q 

IPROD(1q, 0q)φ

0q

q=1

Q 

PROD(1q)φ

1q −φ 0q

PROD(0q)φ

1q −φ 0q

.

(4.20)

.

(4.21)

q=1

Using the period 1 viewpoint, we get IPROD(1, 0) =

Q 

IPROD(1q, 0q)φ

q=1

Using the “mean” viewpoint, we get

1q

Q  q=1

100

4 Annual and Quarterly Measures

IPROD(1, 0) =

Q 

IPROD(1q, 0q)(φ

0q +φ 1q )/2

×

(4.22)

q=1 Q 

(PROD(0q)PROD(1q))(φ

1q −φ 0q )/2

.

q=1

It may be clear that the right-most factors of these three expressions are not necessarily equal to 1. The adjacent subperiod indices IPROD(tq, t q − 1) (t = 0, 1; q = 1, . . . , Q) can be related to the subperiod-to-corresponding-subperiod indices by chaining, IPROD(1q, 0q) = q 

(4.23)

IPROD(1μ, 1 μ − 1)

μ=1

12 

IPROD(0μ, 0 μ − 1) (q = 1, . . . , Q).

μ=q+1

The right-hand side of expression (4.23) can then be inserted into expression (4.17), (4.18), (4.20), (4.21), or (4.22) to obtain a relation between the period 0 to period 1 productivity index IPROD(1, 0) and the adjacent subperiod indices. But this relation does not have a simple form. The conclusion is that already in the extremely simple case of a single-input single-output unit temporal aggregation of productivity indices proves difficult. It is possible to relate subperiod and period productivity indices to each other, but the resulting expressions are not simple.

4.3 A More Realistic Case Let us now consider a production unit that produces M outputs and uses N inputs. The quantity of output m produced during subperiod q of period t will be denoted tq by ym (m = 1, . . . , M; t = 0, 1; q = 1, . . . , Q). Likewise, the quantity of input tq n used during subperiod m of period t will be denoted by xn (n = 1, . . . , N; t = 0, 1; q = 1, . . . , Q). It is assumed that in each subperiod at least one input quantity and one output quantity is strictly positive. The quantity of output m produced during the entire period t is evidently measured by t ym ≡

Q  q=1

tq

ym (m = 1, . . . , M; t = 0, 1).

(4.24)

4.3 A More Realistic Case

101

Likewise, the quantity of input n used during the entire period t is evidently measured by xnt ≡

Q 

tq

xn (n = 1, . . . , N; t = 0, 1).

(4.25)

q=1

When there are multiple inputs and multiple outputs the concept of productivity (level) is no longer unambiguous. Prices are necessary to aggregate quantities. Thus, suppose we have a set of fixed (strictly positive) output prices p ≡ (p1 , . . . , pM ) and (strictly positive) input prices w ≡ (w1 , . . . , wN ). The aggregate output quantity produced during subperiod q of period t is then given by M 

p · y tq =

tq

pm ym (t = 0, 1; q = 1, . . . , Q),

(4.26)

m=1

where vector notation is used to simplify notation and highlight the analogies to the expressions in the previous section. One could also say that p·y tq is the subperiod tq output value expressed in constant prices. The aggregate output quantity produced during the entire period t is naturally given by p · yt =

M 

t pm ym =

m=1

Q 

p · y tq (t = 0, 1).

(4.27)

q=1

Likewise, the aggregate input quantity used during subperiod m of period t is given by w·x

tq

=

N 

tq

wn xn (t = 0, 1; q = 1, . . . , Q).

(4.28)

n=1

This is the subperiod tq input value expressed in constant prices. The aggregate input quantity used during the entire period t is also naturally given by w · xt =

N  n=1

wn xnt =

Q 

w · x tq (t = 0, 1).

(4.29)

q=1

Recall that it is assumed that all these values are given, without statistical error. Conditional on input prices w and output prices p, the productivity (level) in subperiod m of period t is measured by PROD(tq) ≡ p · y tq /w · x tq (t = 0, 1; q = 1, . . . , Q),

(4.30)

102

4 Annual and Quarterly Measures

and the productivity (level) in the entire period t by PROD(t) ≡ p · y t /w · x t (t = 0, 1).

(4.31)

This can be expressed in terms of subperiod productivity levels in three ways, namely PROD(t) =

Q  (w · x tq /w · x t )PROD(tq),

(4.32)

q=1

⎛ ⎞−1 Q  PROD(t) = ⎝ (p · y tq /p · y t )(PROD(tq))−1 ⎠ ,

(4.33)

q=1

and ln PROD(t) =

Q   LM(w · x tq , p · y tq ) q=1

LM(w · x t , p · y t )

 ln PROD(tq) .

(4.34)

The definitions of productivity change between two periods, between two subperiods, and between a subperiod and a period are straightforward. For instance, productivity change between periods 0 and 1 is measured by IPROD(1, 0) ≡

p · y 1 /w · x 1 PROD(1) = . PROD(0) p · y 0 /w · x 0

(4.35)

It is simple to check that the following relations hold: IPROD(1, 0) =

Q  (w · x 1q /w · x 1 )IPROD(1q, 0),

(4.36)

q=1

which generalizes expression (4.13), and IPROD(1, 0) =

 Q   w · x 1q PROD(0q) IPROD(1q, 0q) , w · x 1 PROD(0)

(4.37)

q=1

which generalizes expression (4.17). Similarly, generalizations of expressions (4.15), (4.16), (4.18), (4.20), (4.21), and (4.22) can be obtained. Moreover, analogous to the way it was done in the previous section, any subperiod-to-corresponding-

4.4 The System View

103

subperiod productivity index IPROD(1q, 0q) can be written as a chain of adjacent subperiod indices. Summarizing, by using a set of fixed input and output prices, any multiple-input multiple-output situation can effectively be reduced to a single-input single-output situation.

4.4 The System View It is clear that the productivity index IPROD(1, 0), as defined by expression (4.35), can be re-expressed as IPROD(1, 0) =

p · y 1 /p · y 0 ; w · x 1 /w · x 0

(4.38)

that is, as the ratio of an output quantity index and an input quantity index. The same holds for the other productivity indices considered in the previous section. These quantity indices have a specific functional form; they are so-called Lowe indices (see Balk 2008, 68). An important disadvantage of a Lowe quantity index is that its dual price index violates a rather fundamental axiom. Consider for instance the output quantity index p · y 1 /p · y 0 . The dual price index is obtained by dividing the quantity index into the value ratio p1 · y 1 /p0 · y 0 , where pt (t = 0, 1) denotes the vector of period t output prices. The result is p1 · y 1 /p · y 1 . p0 · y 0 /p · y 0

(4.39)

It is clear that this price index violates the Identity Axiom, which requires a price index to deliver the outcome 1 whenever the price vectors of the two periods compared are equal, p1 = p0 . Such a violation is generally considered to be undesirable. An integrated system of price, quantity, and productivity statistics requires functional forms Po (.), Pi (.), Qo (.), Qi (.), such that p1 · y 1 /p0 · y 0 = Po (p1 , y 1 , p0 , y 0 )Qo (p1 , y 1 , p0 , y 0 )

(4.40)

w 1 · x 1 /w 0 · x 0 = Pi (w 1 , x 1 , w 0 , x 0 )Qi (w 1 , x 1 , w 0 , x 0 ),

(4.41)

and a reasonable number of fundamental axioms (or regularity conditions) for price and quantity indices are satisfied (see Appendix A of Chap. 2). Here pt , w t (t = 0, 1) denote the vectors of period t output and input prices respectively. Notice that the functional forms used at the output side may or may not be the same as those used at the input side (apart from the dimension of the price and quantity vectors involved).

104

4 Annual and Quarterly Measures

Given these functional forms the productivity index for period 1 relative to period 0 is defined as IPROD(1, 0) ≡

Qo (p1 , y 1 , p0 , y 0 ) ; Qi (w 1 , x 1 , w 0 , x 0 )

(4.42)

that is, output quantity index divided by input quantity index. Consider now the subperiods. The relation between period and subperiod quantities was presented in expressions (4.24) and (4.25). Let p tq and w tq (t = 0, 1; q = 1, . . . , Q) denote the vectors of subperiod output and input prices respectively. The relation between period and subperiod prices is, rather naturally, given by t pm

≡

Q 

tq tq

(4.43)

tq tq

(4.44)

t pm ym /ym (m = 1, . . . , M; t = 0, 1)

q=1

wnt ≡

Q 

wn xn /xnt (n = 1, . . . , N; t = 0, 1),

q=1

and the relation between period and subperiod output and input values is similarly given by pt · y t =

Q 

ptq · y tq (t = 0, 1)

(4.45)

w tq · x tq (t = 0, 1).

(4.46)

q=1

wt · x t =

Q  q=1

Thus, whereas period quantities are simple sums of subperiod quantities, and the same holds for values, period prices are defined as unit values (given subperiod prices). Corresponding to expression (4.42), the productivity index for subperiod 1q relative to period 0 is then defined as IPROD(1q, 0) ≡

Qo (p1q , y 1q , p0 , y 0 ) (q = 1, . . . , Q). Qi (w 1q , x 1q , w 0 , x 0 )

(4.47)

Can these subperiod indices be related to the period index? The answer is obtained by looking at the so-called profitability ratio for period 1 relative to period 0, where profitability is defined as the ratio of output value over input value. On the one hand the profitability ratio can be decomposed as

4.5 Subperiod Productivity Indices as Approximations

p1 · y 1 /p0 · y 0 Po (p1 , y 1 , p0 , y 0 ) Qo (p1 , y 1 , p0 , y 0 ) . = w 1 · x 1 /w 0 · x 0 Pi (w 1 , x 1 , w 0 , x 0 ) Qi (w 1 , x 1 , w 0 , x 0 )

105

(4.48)

But on the other hand, by temporal disaggregation, one obtains Q  p1 · y 1 /p0 · y 0 w 1q · x 1q p1q · y 1q /p0 · y 0 = w 1 · x 1 /w 0 · x 0 w 1 · x 1 w 1q · x 1q /w 0 · x 0

(4.49)

q=1

=

 Q  1q  w · x 1q Po (p1q , y 1q , p0 , y 0 ) Qo (p1q , y 1q , p0 , y 0 ) . w 1 · x 1 Pi (w 1q , x 1q , w 0 , x 0 ) Qi (w 1q , x 1q , w 0 , x 0 ) q=1

By combining the last two expressions, using the definitions in expressions (4.42) and (4.47), one obtains IPROD(1, 0) = (4.50)   Q  w 1q · x 1q Po (p1q , y 1q , p0 , y 0 )/Po (p1 , y 1 , p0 , y 0 ) IPROD(1q, 0) . w 1 · x 1 Pi (w 1q , x 1q , w 0 , x 0 )/Pi (w 1 , x 1 , w 0 , x 0 ) q=1

This relation between period-to-period and subperiod-to-period productivity indices is not particularly simple. More importantly, since the productivity index at the lefthand side, IPROD(1, 0), is based on the same functional form(s) for the quantity indices as the productivity indices at the right-hand side, IPROD(1q, 0), the relation generates restrictions on those functional forms. It turns out that these restrictions are impossible to satisfy, except when Qo (.) and Qi (.) exhibit the Lowe functional form; that is, Qo (.) = p · y 1 /p · y 0 and Qi (.) = w · x 1 /w · x 0 where p and w are fixed output and input prices, respectively. But then the dual Po (.) and Pi (.) violate the fundamental Identity Axiom.

4.5 Subperiod Productivity Indices as Approximations Given that a consistent system encompassing period and subperiod productivity indices is impossible, can the latter be used as approximations or forecasts of the former? Can, for instance, the subperiod indices IPROD(1q, 0q) for q = 1, . . . , Q be used as approximations or forecasts of the period index IPROD(1, 0)? Consider again the period-to-period profitability ratio; that is, the left-hand side of expression (4.48). Notice that this ratio can be expressed as (1/Q)p1 · y 1 /p0 · y 0 . (1/Q)w1 · x 1 /w 0 · x 0

(4.51)

106

4 Annual and Quarterly Measures

Let δ tq and  tq (t = 0, 1; q = 1, . . . , Q) be defined by δ tq ≡ w tq · x tq − w t · x t /Q 

tq

≡p

tq

·y

tq

− p · y /Q; t

t

(4.52) (4.53)

that is, δ tq and  tq are deviations of actual subperiod values from average subperiod values. A first-order Taylor series expansion then delivers p1q · y 1q /p0 · y 0 (1/Q)p1 · y 1 /p0 · y 0 = + R(δ 1q ,  1q ) (q = 1, . . . , Q), 1q 1q 0 0 w · x /w · x (1/Q)w1 · x 1 /w 0 · x 0 (4.54) where the remainder term R(.) tends to zero when its arguments tend to zero. Thus, if δ 1q and  1q are small random fluctuations around 0, then the subperiod-to-period profitability ratios can be seen as approximations to the period-to-period profitability ratio. By decomposing both sides of expression (4.54) and rearranging we obtain IPROD(1q, 0) = (w 1q , x 1q , w 0 , x 0 )

(4.55) (p1 , y 1 , p0 , y 0 )

Pi Po IPROD(1, 0) 1q 1q 0 0 Po (p , y , p , y ) Pi (w 1 , x 1 , w 0 , x 0 ) +

Pi (w 1q , x 1q , w 0 , x 0 ) R(δ 1q ,  1q ) (q = 1, . . . , Q). Po (p1q , y 1q , p0 , y 0 )

The factor in front of IPROD(1, 0) can be rewritten as Pi (w 1q , x 1q , w 0 , x 0 )/Pi (w 1 , x 1 , w 0 , x 0 ) . Po (p1q , y 1q , p0 , y 0 )/Po (p1 , y 1 , p0 , y 0 )

(4.56)

The numerator is an index comparing subperiod input prices w1q to period input prices w 1 , and the denominator is an index comparing subperiod output prices p1q to period output prices p1 . Thus, somewhat loosely stated, if the seasonality of input and output prices is the same, than any subperiod index IPROD(1q, 0) is an unbiased forecaster of IPROD(1, 0).

4.6 Conclusion It appears that the goal of full consistency between period and subperiod price, quantity and productivity indices is unattainable. Moreover, as argued in Sect. 4.4,

4.6 Conclusion

107

this conclusion is independent of the specific functional forms used for the various indices. This impossibility theorem implies that choices must be made. The first choice concerns what is to be seen as the most natural accounting period for the production unit considered. In most, if not all, cases this will be a year. Annual price, quantity, and productivity comparisons can be based on indices that satisfy the basic axioms (or regularity conditions) and together form a consistent system. Given the need for sub-annual productivity information, the second choice concerns the type of index to use. As shown, every choice entails at best an approximate relationship between sub-annual and annual indices. The nature of this approximation should be clearly communicated to the public.

Chapter 5

Dynamics: The Bottom-Up Approach

5.1 Introduction In Chap. 2 we considered the measurement of productivity change for a single, consolidated production unit. The present chapter continues by studying an ensemble of such units. The classical form is a so-called sectoral shift-share analysis. The starting point of such an analysis is an ensemble of industries, according to some industrial classification (such as ISIC or NAICE), at some level of detail. An industry is a set of enterprises1 engaged in the same or similar kind of activities. In the case of productivity analysis the ensemble is usually confined to industries for which independent measurement of input and output is available. Such an ensemble goes by different names: business sector, market sector, commercial sector, or simply measurable sector. Data are published and/or provided by official statistical agencies. Let us, by way of example, consider labour productivity, in particular valueadded based labour productivity. The output of industry k at period t is then measured as real value added RVAkt ; that is, nominal value added VAkt (= revenue minus intermediate inputs cost) deflated by a suitable, ideally industry-specific, price index. Real value added is treated as ‘quantity’ of a single commodity, that may or may not be added across the production units belonging to the ensemble studied, and over time. At the input side there is usually given some simple measure of labour input, such as total number of hours worked Lkt ; rougher measures being persons employed or full-time equivalents employed. Then labour productivity of industry k at period t is defined as RVAkt /Lkt .

1 There

is no unequivocal naming here. So, instead of enterprises one also speaks of firms, establishments, plants, or kind-of-activity units. The minimum requirement is that realistic annual profit/loss accounts can be compiled. © The Author(s), under exclusive license to Springer Nature Switzerland AG 2021 B. M. Balk, Productivity, Contributions to Economics, https://doi.org/10.1007/978-3-030-75448-8_5

109

110

5 Dynamics: The Bottom-Up Approach

In the ensemble the industries are of course not equally important, thus some kind of weights reflecting relative importance, θ kt , adding up to 1, are necessary. In the literature there is some discussion as to the precise nature  of these weights. kt / kt Should the weights reflect (nominal) value-added shares VA k VA   ? ktor real kt kt kt value-added shares RVA / k RVA ? or labour input shares L / k L ? We return to this discussion later on. Aggregate labour productivity at period t is then defined as a weighted mean,   kt either arithmetic k θ kt RVAkt /Lkt or geometric k (RVAkt /Lkt )θ , and the focus 2 of interest is the development of such a mean over time. There are clearly two main factors here, shifting importance and shifting productivity, and their interaction. The usual product of a shift-share analysis is a table which provides detailed decomposition results by industry and time periods compared. Special interest can be directed thereby to industries which are ICT-intensive, at the input and/or the output side; industries which are particularly open to external trade; industries which are (heavily) regulated; etcetera. Things become only slightly more complicated when value-added based total factor productivity is considered. At the input side one now needs per industry and time period nominal capital and labour cost as well as one or more suitable deflators. kt , which can be treated as ‘quantity’ of The outcome is real primary input, XKL another single commodity. Total factor productivity of industry k at period t is then kt . The issue of the precise nature of the weights gets some defined as RVAkt /XKL additional complexity, since we now also could contemplate the use of nominal or real cost shares to measure the importance of the various industries. More complications arise when one wants to base the analysis on gross-output based total factor productivity. For the output side of the industries one then needs nominal revenue as well as suitable, industry-specific deflators. For the input side one needs nominal primary and intermediate inputs cost together with suitable deflators. The question of which weights to use is aggravated by the fact that industries deliver to each other, so that part of one industry’s output becomes part of another industry’s input. Improper weighting can then easily lead to doublecounting of productivity effects. Since the early 1990s an increasing number of statistical agencies made (longitudinal) microdata of enterprises available for research. Economists could now focus their research at production units at the lowest level of aggregation and dispense with the age-old concept of the ‘representative firm’ that had guided so much theoretical development. At the firm or enterprise level one usually has access to nominal data about output revenue and input cost detailed to various categories, in addition to data about employment and some aspects of financial behaviour. Lowest level quantity data are usually not available, so that industrylevel deflators must be used. Also, at the enterprise level the information available is generally insufficient to construct firm-specific capital stock data. Notwithstanding 

θ kt (RVAkt /Lkt )−1 conditions under which this mean materializes as the natural one.

2 Curiously, the literature neglects the harmonic mean

k

−1

, though there are

5.1 Introduction

111

such practical restrictions, microdata research has spawned and is still spawning lots of interesting results. A landmark contribution, including a survey of older results, is Foster et al. (2001). Good surveys were provided by Bartelsman and Doms (2000) and, more recently, Syverson (2011). Interesting examples of research are collected in a special issue on firm dynamics of the journal Structural Change and Economic Dynamics 23 (2012), 325–402. Of course, dynamics at the enterprise level is much more impressive than at the industry level, no matter how fine-grained. Thought-provoking features are the growth, decline, birth, and death of production units. Split-ups as well as mergers and acquisitions occur all over the place. All this is exacerbated by the fact that the annual microdata sets are generally coming from (unbalanced, rotating) samples, which implies that any superficial analysis of given datasets is likely to draw inaccurate conclusions. This chapter contains a review and discussion of the so-called bottom-up approach, which takes an ensemble of individual production units as the fundamental frame of reference. The theory developed here can be applied to a variety of situations, such as (1) a large company consisting of a number of subsidiaries, (2) an industry consisting of a number of enterprises, or (3) an economy or, more precisely, the ‘measurable’ part of an economy consisting of a number of industries. The top-down approach is the subject of Chaps. 6–8. The connection between the two approaches, bottom-up and top-down, is discussed in Chap. 9. What may the reader expect in this chapter? Sections 5.2 and 5.3 describe the scenery: a set of production units with their accounting relations, undergoing temporal change. Section 5.4 defines the various measurement devices, in particular productivity indices, levels, and their links. The second half of this section is devoted to a discussion of the gap between theory and practice; that is, what to do when not all the data wanted are accessible? And what are the consequences of approximations? Aggregate productivity change can be measured in different ways. First, as the development through time of arithmetic means of production-unit-specific productivity levels. Section 5.5 reviews the various decompositions proposed in the extant literature, and concludes with a provisional evaluation. Section 5.6 employs the logarithmic mean to derive a new, symmetric, decomposition with nice properties. Next, Sect. 5.7 briefly discusses the alternatives which emerge when arithmetic means are replaced by geometric or harmonic means. Section 5.8 discusses the monotonicity “problem”, revolving around the so-called Fox “paradox”: an increase of all the individual productivities not necessarily leads to an increase of aggregate productivity. It is argued that this is not a paradox at all but an essential feature of aggregation. Section 5.9 delves into the foundations of the much-used Olley-Pakes decomposition and distinguishes between valid and fallacious use. In the bottom-up approach aggregate productivity is some weighted mean of individual, production-unit-specific productivities. There is clearly a lot of choice here: in the productivity measure, in the weights of the units, and in the type of mean. Section 5.10 formulates the problem; the actual connection, however, between the

112

5 Dynamics: The Bottom-Up Approach

bottom-up and top-down approaches is discussed in Chap. 9. Section 5.11 concludes with a summary of the main lessons.

5.2 Accounting Identities We consider an ensemble (or set) Kt of consolidated production units,3 operating during a certain time period t in a certain country or region. For each unit the KLEMS-Y ex post accounting identity in nominal values (or, in current prices) reads kt kt CKL + CEMS + kt = R kt (k ∈ Kt ),

(5.1)

kt denotes the primary input cost, C kt where CKL EMS the intermediate inputs cost, kt R the revenue, and kt the profit (defined as remainder). Intermediate inputs cost (on energy, materials, and business services) and revenue concern generally tradeable commodities. It is presupposed that there is some agreed-on commodity kt classification, such that CEMS and R kt can be written as sums of quantities times (unit) prices of these commodities. Of course, for any production unit most of these quantities will be zero. It is also presupposed that output prices are available from a market or else can be imputed. Taxes on production are supposed to be allocated to the K and L classes. The commodities in the capital class K concern owned tangible and intangible assets, organized according to industry, type, and age class. Each production unit uses certain quantities of those assets, and the configuration of assets used is in general unique for the unit. Thus, again, for any production unit most of the asset cells are empty. Prices are defined as unit user costs and, hence, capital input cost kt is a sum of prices times quantities. CK Finally, the commodities in the labour class L concern detailed types of labour. Though any production unit employs specific persons with certain capabilities, it is usually their hours of work that count. Corresponding prices are hourly wages. Like the capital assets, the persons employed by a certain production unit are unique for that unit. It is presupposed that, wherever necessary, imputations have been made for self-employed workers. Henceforth, labour input cost CLkt is a sum of prices times quantities. kt = Total primary input cost is the sum of capital and labour input cost, CKL kt kt kt CK + CL . Profit  is the balancing item and thus may be positive, negative, or zero. We are operating here outside the neoclassical framework where profit always equals zero due to the structural and behavioural assumptions involved.

3 “Consolidated”

means that intra-unit deliveries are netted out. At the industry level, in some parts of the literature this is called “sectoral”. At the economy level, “sectoral” output reduces to GDP plus imports, and “sectoral” intermediate input to imports. In terms of variables to be defined kkt = R kkt = 0. below, consolidation means that CEMS

5.2 Accounting Identities

113

The KL-VA accounting identity then reads kt kt + kt = R kt − CEMS ≡ VAkt (k ∈ Kt ), CKL

(5.2)

where VAkt denotes value added, defined as revenue minus intermediate inputs cost. In this chapter it will always be assumed that VAkt > 0.4 We now consider whether the ensemble of production units Kt can be considered as a consolidated production unit. Though aggregation basically is addition, addingup the KLEMS-Y relations over all the units would imply double-counting because of deliveries between units. To see this, it is useful to split intermediate input cost and revenue into two parts, respectively concerning units belonging to the ensemble Kt and units belonging to the rest of the world. Thus, kt = CEMS





k kt ekt CEMS + CEMS ,

(5.3)

k  ∈Kt 

k kt is the cost of the intermediate inputs purchased by unit k from unit k  , where CEMS ekt is the cost of the intermediate inputs purchased by unit k from the world and CEMS beyond the ensemble K. Similarly,

R kt =





R kk t + R ket ,

(5.4)

k  ∈Kt 

where R kk t is the revenue obtained by unit k from delivering to unit k  , and R ket is the revenue obtained by unit k from delivering to units outside of Kt . Adding up the KLEMS-Y relations (5.1) then delivers  k∈Kt

kt CKL +

 



k kt CEMS +

k∈Kt k  ∈K

  k∈Kt

k  ∈Kt



ekt CEMS +

k∈Kt 

R kk t +





kt =

k∈Kt

R ket .

(5.5)

k∈Kt

If for all the tradeable commodities output prices are identical to input prices (which is ensured by National Accounting conventions), then the two intra-Kt -trade terms cancel, and the foregoing expression reduces to5

4 This

is a necessary but innocuous assumption. Only in exceptional cases value added is nonpositive, for instance when the accounting period is so short that revenue and intermediate inputs cost are booked in different periods. Value added is an accounting concept, without normative connotations. After all, value added must be used to pay for capital and labour expenses. 5 As Aulin-Ahmavaara and Pakarinen (2007) show, if one does not want to make such as   k  kt − R kk  t ) assumption then at the left-hand side of the next equation the term k∈K k  ∈K (CEMS must be added to account for the differences between purchaser’s prices at the input side and

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kt CKL +

k∈Kt



ekt CEMS +

k∈Kt



kt =

k∈Kt



R ket .

(5.6)

k∈Kt

Recall that capital assets and hours worked are unique for each production unit, which implies that primary input cost may simply be added over the units, without any fear for double-counting. Thus expression (5.6) is the KLEMS-Y accounting relation for the ensemble Kt , considered as a consolidated production unit. The corresponding KL-VA relation is then 

kt CKL +

k∈Kt



kt =

k∈Kt



R ket −

k∈Kt



ekt CEMS ,

(5.7)

k∈Kt

which can be written as6 Kt Kt CKL + K t = R K t − CEMS ≡ VAK t . t

t

t

t

t

(5.8)

Kt t ≡  kt Kt t ≡  kt Kt t ≡  ket where CKL t CKL ,  t  , R k∈ K k∈ K k∈Kt R , and  tt K ekt CEMS ≡ k∈Kt CEMS . One verifies immediately that  t VAK t = VAkt . (5.9) k∈Kt

The structural similarity between expressions (5.2) and (5.8), together with the additive relation between all the elements, is the reason why the KL-VA production model is the natural starting point for studying the relation between individual and aggregate measures of productivity change.7 We will see however that the bottomup approach basically neglects this framework.

5.3 Continuing, Entering, and Exiting Production Units As indicated in the previous section the superscript t denotes a time period, the usual unit of measurement being a year. Though data may be available over a longer time span, any comparison is concerned with only two periods: an earlier period 0 (also called base period), and a later period 1 (also called comparison period).

basic prices at the output side. Put otherwise, this term measures net taxes on intermediates. In the ensuing analysis the term can be treated as another intermediate input cost component or be merged with profit. 6 If K is an economy and Kt = 0 then this expression reduces to the familiar identity of gross domestic income and gross domestic product. 7 For a slightly more complicated reasoning, highlighting the role of taxes, trade and transport margins, see Jorgenson and Schreyer (2013, Sections 2, 3).

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115

These periods may or may not be adjacent. When the production units are industries, then the ensemble K0 will usually be the same as K1 . But when the production units studied are enterprises, this will in general not hold, and we must distinguish between continuing, exiting, and entering production units. In particular, K0 = C 01 ∪ X 0

(5.10)

K1 = C 01 ∪ N 1 ,

(5.11)

where C 01 denotes the set of continuing units (that is, units active in both periods), X 0 the set of exiting units (active in the base period only), and N 1 the set of entering units (active in the comparison period only). The sets C 01 and X 0 are disjunct, as are C 01 and N 1 . It is important to observe that in any application the distinction between continuing, entering, and exiting production units depends on the length of the time periods being compared, and on the time span between these periods. Of course, when the production units studied form a balanced panel, then the sets X 0 and N 1 are empty. The same holds for the case where the production units are industries. These two situations will in the sequel be considered as specific cases. The theory developed in the remainder of this chapter is cast in the language of intertemporal comparisons. By redefining 0 and 1 as countries or regions, and conditioning on a certain time period, the following can also be applied to cross-sectional comparisons. There is one big difference, however. Apart from mergers, acquisitions and the like, enterprises have a certain perseverance and can be observed through time. But a certain enterprise cannot exist at the same time in two countries or regions. Hence, in cross-sectional comparisons the lowestlevel production units can only be industries, and ‘entering’ or ‘exiting’ units correspond to industries existing in only one of the two countries or regions which are compared.

5.4 Productivity Indices and Levels As explained in Sect. 5.2, the various components of the accounting identity (5.1) are nominal values, that is, sums of prices times quantities. We are primarily interested in their development through time, as measured by ratios. It is assumed that all the detailed price and quantity data, underlying the values, are accessible. This is, of course, the ideal situation, which in practice is not likely to occur. Nevertheless, for conceptual reasons it is good to use this as our starting point. More mundane situations, deviating to a higher or lesser degree from the ideal, will then be considered later.

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5.4.1 Indices Using index number theory, each nominal value ratio can be decomposed as a product of two components, one capturing the price effect and the other capturing the quantity effect. Thus, let there be price and quantity indices such that for any two periods t and t  the following relations hold: 

kt kt k CKL /CKL = PKL (t, t  )QkKL (t, t  ) 

kt kt k CEMS /CEMS = PEMS (t, t  )QkEMS (t, t  ) 

R kt /R kt = PRk (t, t  )QkR (t, t  ).

(5.12) (5.13) (5.14)

Capital cost and labour cost are components of primary input cost, thus it can also be assumed that there are functions such that 

kt kt /CK = PKk (t, t  )QkK (t, t  ) CK 

CLkt /CLkt = PLk (t, t  )QkL (t, t  ).

(5.15) (5.16)

We are using here the shorthand notation introduced in Chap. 1. All these price and quantity indices are supposed to be, appropriately dimensioned, functions of the prices and quantities at the two periods that play a role in the value ratios; e.g. PLk (t, t  ) is a labour price index for production unit k, based on all the types of labour distinguished, comparing hourly wages at the two periods t and t  , conditional on hours worked at these periods. These functions are supposed to satisfy some basic axioms ensuring proper behaviour, and, dependent on the time span between t and t  , may be direct or chained indices (see Chapter 2, Appendix A). There may or may k (t, t  ) and the subindices not exist functional relations between the overall index PKL k k   PK (t, t ) and PL (t, t ) (or, equivalently, between the overall index QkKL (t, t  ) and the subindices QkK (t, t  ) and QkL (t, t  )). The construction of price and quantity indices for value added was discussed in Appendix B of Chapter 2. Thus there are also functions such that 

k VAkt /VAkt = PVA (t, t  )QkVA (t, t  )

(5.17)

Formally, the relations (5.12)–(5.16), and (6.6) mean that the Product Test is satisfied. Notice that it is not required that all the functional forms of the price and quantity indices are the same. However, the Product Test in combination with the axioms rules out a number of possibilities. We recall some definitions. The value-added based TFP index for period 1 relative to period 0 was defined in Chapter 2 as

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117

ITFPRODkVA (1, 0) ≡

QkVA (1, 0) QkKL (1, 0)

.

(5.18)

This index measures the ‘quantity’ change component of value added relative to the quantity change of all the primary inputs. The two main primary input components are capital and labour; both deserve separate attention. The value-added based capital productivity index for period 1 relative to period 0 is defined as IKPRODkVA (1, 0) ≡

QkVA (1, 0) QkK (1, 0)

.

(5.19)

This index measures the ‘quantity’ change component of value added relative to the quantity change of capital input. Similarly, the value-added based labour productivity index for period 1 relative to period 0 is defined as ILPRODkVA (1, 0) ≡

QkVA (1, 0) QkL (1, 0)

.

(5.20)

This index measures the ‘quantity’ change component of value added relative to the quantity change of labour input. Recall that the labour quantity index QkL (t, t  ) is here defined as an index acting on the prices and quantities of all the types of labour that are being distinguished. Suppose now that the units of measurement of the various types of labour are in some sense the same; that is, the quantities of all the labour types are measured in hours, or in full-time equivalent jobs, or in some other common unit. Then it makes sense to define the total labour quantity of production unit k at period t as Lkt ≡



xnkt ,

(5.21)

n∈L 

and to use the ratio Lkt /Lkt as quantity index.8 Formally, this is a Dutot or simple sum quantity index. The ratio of a genuine labour quantity index, i.e. an index based  on types of labour, QkL (t, t  ), and the simple sum labour quantity index Lkt /Lkt is an index of labour quality (or composition). The value-added based simple labour productivity index for production unit k, for period 1 relative to period 0, is defined as

8 Recall our notation: L without super- or subscripts denotes the set of labour types, but Lkt

the total number of labour units worked in production unit k at period t.

denotes

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5 Dynamics: The Bottom-Up Approach

ISLPRODkVA (1, 0) ≡

QkVA (1, 0) , Lk1 /Lk0

(5.22)

which can then be interpreted as an index of real value added per unit of labour.

5.4.2 Levels As one sees, some ‘level’-language has crept in. The bottom-up approach freely talks about productivity (change) in terms of levels. But what precisely are levels, and what is the relation between levels and indices? Intuitively, indices are just ratios of levels, so that it seems that the difference is merely in the kind of language one prefers. It appears, however, that a closer look is warranted. For each production unit k ∈ Kt real value added is (ideally) defined as k (t, b); RVAk (t, b) ≡ VAkt /PVA

(5.23)

that is, nominal value added at period t divided by (or, as one says, deflated by) a production-unit-k-specific value-added based price index for period t relative to a certain reference period b, where period b may or may not precede period 0. Notice that this definition tacitly assumes that production unit k, existing in period t, also existed or still exists in period b; otherwise, deflation by a production-unit-k-specific index would be impossible. When production unit k does not exist in period b then for deflation a non-specific index must be used. On the complications thereby we will come back at a later stage. The foregoing definition implies that k RVAk (b, b) = VAkb /PVA (b, b) = VAkb ,

(5.24)

since any price index, whatever its functional form, returns the outcome 1 for the reference period. Thus, at the reference period b, real value added equals nominal value added. k (t, b) is a Paasche-type double For example, one easily checks that when PVA deflator, then real value added RVAkt is period t value added at prices of period b (recall Chapter 2, Appendix B). The rather intricate form at the left-hand side of expression (5.23) serves to make clear that unlike VAkt , which is an observable monetary magnitude, RVAk (t, b) is the outcome of a function. Though the outcome is also monetary, its magnitude depends on the reference period and the deflator chosen. The first kind of dependence becomes clear by considering RVAk (t, b ) for some b = b. One immediately checks that RVAk (t, b )/RVAk (t, b) = k (t, b)/P k (t, b ), which is a measure of the (k-specific value-added based) PVA VA price difference between periods b and b. Put otherwise, real value added depends

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119

critically on the price level of the reference period, which is the period for which nominal and real value added coincide. k (t, b) is a PaascheAs to the other dependence, it of course matters whether PVA type or a Laspeyres-type or a Fisher-type double deflator. Here the difference in general increases with increasing the time span between the periods t and b. Another way of looking at real value added is to realize that, by using expression (6.6), RVAk (t, b) = VAkb QkVA (t, b). Put otherwise, real value added is a (normalized) quantity index. Like real value added, real primary, or capital-and-labour, input, relative to reference period b, is (ideally) defined as deflated primary input cost, k kt k XKL (t, b) ≡ CKL /PKL (t, b);

(5.25)

real capital input, relative to reference period b, is (ideally) defined as deflated capital cost, k kt (t, b) ≡ CK /PKk (t, b); XK

(5.26)

and real labour input, relative to reference period b, is (ideally) defined as deflated labour cost, XLk (t, b) ≡ CLkt /PLk (t, b),

(5.27)

Of course, similar observations as above apply to these two definitions. In particular, it is important to note that at the reference period b real primary input equals nominal k (b, b) = C kb , real capital input equals nominal capital cost, input cost, XKL KL k (b, b) = C kb , and real labour input equals nominal labour cost, X k (b, b) = C kb . XK K L L It is important to observe that, whereas nominal values are additive, real values k (t, b) = X k (t, b) + X k (t, b) for t = b. It is easy to are generally not; that is, XKL K L see, by combining expressions (5.25)–(5.27), that requiring additivity means that k (t, b) must be a second-stage Paasche index of the the overall price index PKL k two subindices PK (t, b) and PLk (t, b). When we are dealing with chained indices it is impossible to satisfy this requirement. An operationally feasible solution was proposed by Balk and Reich (2008).9 Using the foregoing building blocks, the value-added based TFP level of production unit k at period t is defined as real value added divided by real primary input,

TFPRODkVA (t, b) ≡

9 Of

RVAk (t, b) k (t, b) XKL

.

(5.28)

course, a trivial solution would be to use the same deflator for all the nominal values. Such a strategy was proposed for the National Accounts by Durand (2004).

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5 Dynamics: The Bottom-Up Approach

Notice that numerator as well as denominator are expressed in the same price level, namely that of period b. Thus TFPRODkVA (t, b) is a dimensionless variable. The foregoing definition immediately implies that at the reference period b value-added based total factor productivity equals nominal value added divided by kb . Now recall the KL-VA nominal primary input cost, TFPRODkVA (b, b) = VAkb /CKL kt accounting identity (5.2) and assume that profit  is constrained to equal 0 for all production units at all time periods. Then reference period total factor productivity of all production units equals 1, TFPRODkVA (b, b) = 1 (k ∈ Kt ). Likewise, the value-added based labour productivity level of unit k at period t is defined as real value added divided by real labour input, LPRODkVA (t, b) ≡

RVAk (t, b) XLk (t, b)

(5.29)

.

This is also a dimensionless variable. For the reference period b we obtain LPRODkVA (b, b) =

VAkb CLkb

=

kb V Akb CKL kb C kb CKL L

(5.30)

.

Hence, when profit kt = 0 for all production units at all time periods then production unit k’s labour productivity at reference period b, LPRODkVA (b, b) equals kb /C kb . This is the reciprocal of k’s labour cost share at period b. CKL L In case the simple sum quantity index is used for labour, one obtains LPRODkVA (t, b) =

RVAk (t, b) CLkt /PLk (t, b)

=

RVAk (t, b) CLkb QkL (t, b)

=

RVAk (t, b) (CLkb /Lkb )Lkt

,

(5.31)

where subsequently expressions (5.27) and (5.16) were used. The constant in the denominator, CLkb /Lkb , is the mean price of a unit of labour at reference period b. The simple value-added based labour productivity level of unit k at period t is defined by SLPRODkVA (t, b) ≡

RVAk (t, b) . Lkt

(5.32)

It is not unimportant to notice that its dimension is money-of-period-b per unit of labour.

5.4.3 Linking Levels and Indices We now turn to the relation between levels and indices. One expects that taking the ratio of two levels would deliver an index, but let us have a look. Dividing unit k’s

5.4 Productivity Indices and Levels

121

total factor or labour productivity level at period 1 by the same at period 0 delivers, using the various definitions and relations (5.17), (5.12) and (5.16), TFPRODkVA (1, b) TFPRODkVA (0, b) LPRODkVA (1, b) LPRODkVA (0, b) SLPRODkVA (1, b) SLPRODkVA (0, b)

=

=

=

QkVA (1, b)/QkVA (0, b) QkKL (1, b)/QkKL (0, b) QkVA (1, b)/QkVA (0, b)

,

(5.33)

,

(5.34)

QkVA (1, b)/QkVA (0, b) , Lk1 /Lk0

(5.35)

QkL (1, b)/QkL (0, b)

respectively. Surely, if QkVA (t, t  ), QkKL (t, t  ) and QkL (t, t  ) are well-behaving functions then the right-hand sides of expressions (5.33)–(5.35) have the form of an output quantity index divided by an input quantity index, both for period 1 relative to period 0. When b = 0, 1 one easily checks that expression (5.33) reduces to ITFPRODkVA (1, 0), that expression (5.34) reduces to ILPRODkVA (1, 0), and that expression (5.35) reduces to ISLPRODkVA (1, 0). But, when b = 0, 1, then TFPRODkVA (1, b)/TFPRODkVA (0, b) = ITFPRODkVA (1, 0) if and only if the quantity indices QkVA (t, t  ) and QkKL (t, t  ) are transitive (that is, satisfy the Circularity Test). Similarly, LPRODkVA (1, b)/LPRODkVA (0, b) = ILPRODkVA (1, 0) if and only if the quantity indices QkVA (t, t  ) and QkL (t, t  ) are transitive, and SLPRODkVA (1, b)/SLPRODkVA (0, b) = ISLPRODkVA (1, 0) if and only if the quantity index QkVA (t, t  ) is transitive. Unfortunately, transitive quantity indices are in practice seldomly used. Moreover, they would lead to price indices which fail some basic axioms.

5.4.4 When Not All the Data Are Accessible The word ‘ideally’ was deliberately inserted in front of definitions (5.23), (5.25), (5.26) and (5.27). This word reflects the assumption that all the detailed price and quantity data, necessary to compile production-unit-specific price and quantity index numbers, are accessible. In practice, especially in the case of microdata, though the data are available at the enterprises, because revenue and cost are

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sums of quantities produced or used at certain unit prices, they are usually not accessible for researchers, due to the excessive cost of obtaining such data, their confidentiality, the response burden experienced by enterprises, or other reasons. In such cases researchers have to fall back at indices which are estimated for a higher aggregation level. This in turn means that real values are contaminated by differential price developments between the production units considered and the higher level aggregate. It is useful to review a number of situations.

5.4.4.1

Sectoral Studies

The typical sectoral study starts with estimates of nominal value added (that is, value added at current prices) and deflators per industry. These data come from, and are ideally consistent with, the overall National Accounts. Ideally, also the components of value added are given: revenue (or, sales) and cost of intermediate inputs (energy, materials, and services). Using product level Producer Price Indices (PPIs), nominal revenue and the various cost components can be converted into real values, the balance of which is real value added. The ratio of nominal to real value added is then the value-added deflator. Thus, ideally, a value-added deflator turns out to be a so-called “double deflator”, which is a technical term saying that a value-added deflator is some function of the revenue and intermediate inputs cost deflators. The quality of the industry-specific value-added deflators critically depends on the nature of the underlying sample data, their level of detail, and the various compilation, estimation and imputation procedures used; e.g. are hedonic methods used to account for quality change? do the procedures take sufficient care of new as well as obsolete products? is sample rotation adequately accounted for? Sometimes one discovers that there is not enough information to execute a proper deflation of some or all of the cost components, and auxiliary assumptions must be invoked; e.g. that the price development of energy is the same as that of (raw) materials, or that the price development of services can be approximated by a Consumer Price Index (CPI). Sometimes there is no deflator at all for intermediate inputs, so that it turns out that value added is in fact deflated by a revenue-based price index. In such a case one speaks of a “single deflator”. An interesting question then arises: what is the best output concept, value added deflated by a single deflator, which means that any differential price change of intermediate inputs relative to gross output contaminates the output measure, or revenue deflated by a specific deflator (that is, gross output is the output measure) and ignoring intermediate inputs entirely? There is no general answer here; it all depends on the purpose of the study.10

10 See

also Timmer et al. (2010, 219–221).

5.4 Productivity Indices and Levels

123

How much harm is caused by the use of an improper deflator for value added?11 Let our target be to measure simple labour productivity RVAk (t, b)/ Lkt for all k ∈ Kt , where b is some reference period. Suppose that, instead of a value-added k (t, b), a revenue-based price index P k (t, b) was used to based price index PVA R deflate nominal value added VAkt . As one easily checks, this means that, instead k (t, b)/P k (t, b) is used in the numerator of the labour of RVAk (t, b), RVAk (t, b)PVA R productivity measure. The bias of the labour productivities is RVAk (t, b) Lkt



k (t, b) PVA

PRk (t, b)

−1

(k ∈ Kt ),

(5.36)

but in general nothing can be said about this distribution. And how large the bias is for any particular industry can only be determined in a case-by-case analysis for which detailed industry data are indispensable. Is there an aggregate effect? Suppose we are interested in weighted mean  simple labour productivity, k θ k RVAk (t, b)/Lkt , with certain weights θ k summing to  1. The use of improper deflators then means that we are looking instead k k k k kt at k θ (PVA (t, b)/ PR (t, b))RVA (t, b)/L . Put otherwise, the simple labour k (t, b)/P k (t, b)) (k ∈ Kt ), k productivities are not weighted by θ but by θ k (PVA R and these weights do not sum to 1. Now it appears that, using the definition of covariance,  k θ k (PVA (t, b)/PRk (t, b))RVAk (t, b)/Lkt = k

 k

k θ k (PVA (t, b)/PRk (t, b))



θ k RVAk (t, b)/Lkt

k

 k + Covar PVA (t, b)/PRk (t, b), RVAk (t, b)/Lkt ; θ k ,

(5.37)

where Covar(a k , bk ; θ k ) denotes the (finite) covariance of scalar variables a k and bk , weighted by θ k , where the range of k is a finite set. Assume  kthatk this covariance equals 0. Then the actual mean labour productivity, k θ (PVA (t,  b)/PRk (t, b))RVAk (t, b)/Lkt , differs from the target mean, k θ k RVAk (t, b)/Lkt ,  k k by the factor θ (PVA (t, b)/PRk (t, b)). This factor is the mean ratio of the valueadded based deflators and the revenue based deflators. The magnitude of such a mean, and especially whether it is greater or less than 1, is an empirical issue. For

11 A

study casting empirical light on this question is Cassing (1996). In terms of our notation she compared a. single-deflated value added, that is VAkt /PRk (t, b), where the revenue-based price k (t, b), where the valueindex is a Törnqvist, b. double-deflated value added, that is VAkt /PVA added based price index is a Paasche, and c. double-deflated value added where the price index is a Törnqvist. Using Indonesian manufacturing data over the period 1975–1988 she found the differences in the development of deflated value added more striking the less aggregated the data.

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individual cases this is of little help, but the mean ratio can be used to adjust the actually measured labour productivities so that their mean becomes equal to the target mean. The message of the preceding two paragraphs is simply that the way value added is deflated influences the distributions of the ensueing productivity levels; and the same holds at the input side. The situation here can roughly be depicted as follows. Per industry the labour cost of its employees is usually available, together with an estimate of the number of employees or jobs. Such numbers must be converted to hours worked (or paid) by auxiliary survey data. In an interesting recent study, Burda et al. (2012) showed the effect of using different types of surveys for such conversion factors. Next, estimates of hours worked (or paid) must be generated for self-employed workers. Introducing types of labour, e.g. according to gender, skill, education, experience, is usually considered a bridge too far. An interesting application with two types of labour, nationals and immigrants, was recently provided by Kangasniemi et al. (2012). Consequentially, labour productivity indices, whether value-added or gross-output based, are usually of the simple type; and quality change of labour is neglected. For the computation of capital input cost one ideally needs detailed estimates of the capital stock, by industry, type, and age of the assets, together with sufficient information to generate unit user costs. A general recipe was outlined in Chap. 2. If this is followed, and all the necessary data are available, then one ends up with quantities and prices and, thus, industry-specific quantity and price indices can be compiled. Differing assumptions, with respect to the rate of return, the extent of asset revaluation, and the utilisation rates have impact on the outcomes. So does the scope of the capital concept: are only tangibles included, or also intangibles, land, and subsoil assets? The NBER Technical Working Paper by Bartelsman and Gray (1996), though rather old, is still a useful source of information on the many problems that must be solved in the course of constructing a dataset suitable for analyzing industrial productivity change. The paper also contains some interesting tables highlighting the effect of using this or that type of deflator. A more recent survey was provided by Timmer et al. (2010, Sections 3.3–7).

5.4.4.2

Microdata Studies

Let us now turn to microdata, and use Foster et al.’s (2001) study of the U. S. manufacturing sector over the 1977–1987 period as a first example. This study was based on plant-level data from the Census of Manufactures (CM). The following quote from the Appendix of their article pictures the data situation: The Census of Manufactures (CM) plant-level data includes value of shipments, inventories, book values of equipment and structures, employment of production and nonproduction workers, total hours of production workers, and cost of materials and energy usage. Real gross output is measured as shipments adjusted for inventories, deflated by the four-digit output deflator for the industry in which the plant is classified. All output and materials

5.4 Productivity Indices and Levels

125

deflators used are from the four-digit NBER Productivity Database (Bartelsman and Gray 1996, recently updated by Bartelsman, Becker and Gray). Labor input is measured by total hours for production workers plus an imputed value for the total hours for nonproduction workers. The latter imputation is obtained by multiplying the number of non-production workers at the plant (a collected data item) times the average annual hours per worker for a nonproduction worker from the Current Population Survey. We construct the latter at the 2digit industry level for each year and match this information to the CM by year and industry. [. . . ] Materials input is measured as the cost of materials deflated by the 4-digit materials deflator. Capital stocks for equipment and structures are measured from the book values deflated by capital stock deflators (where the latter is measured as the ratio of the current dollar book value to the constant dollar value for the two-digit industry from Bureau of Economic Analysis data). Energy input is measured as the cost of energy usage, deflated by the Gray-Bartelsman energy-price deflator. The factor elasticities are measured as the industry average cost shares, averaged over the beginning and ending year of the period of growth. Industry cost shares are generated by combining industry-level data from the NBER Productivity Database with the Bureau of Labor Statistics (BLS) capital rental prices. The CM does not include data on purchased services (other than that measured through contract work) on a systematic basis (there is increased information on purchased services over time).

Some features are worth highlighting. First, there are two types of labour (production and nonproduction workers), and two types of capital assets (equipment and structures) distinguished. Second, the imputation for the hours of nonproduction workers at production unit k is based on information from higher-level aggregates, obtained from sources outside k. Third, all the deflators used to obtain k-specific real values are not k-specific, but at best specific for the next-higher aggregate to which k belongs. This implies that all these real values, at the output as well as the input side, contain residual price differences (between k and the higher aggregate). Put otherwise, any productivity measure is contaminated by price differentials. Fourth, there is no information on services input, so only multi-factor productivity (MFP) measures can be compiled. Cast in our notation, it appears that the following formula for the MFP level was used by Foster et al. (2001, 318): MFPRODkFHK (t, b) ≡

k (t, b)αKe X k (t, b)αKs XKe Ks

(5.38)

Y k (t, b)  αL , k (t, b)αE X k (t, b)αM kt Lkt + L X p np E M

where Y k (t, b) is deflated revenue,12 e denotes equipment, s structures, p production worker, and np nonproduction worker. The denominator is easily recognized as a generalised Cobb-Douglas aggregator, the α’s being average industry cost shares, adding up to 1. Notice that the two types of labour are aggregated by a simple sum function. Labour productivity levels were compiled as 12 Formally,

according to expression (5.14), Y k (t, b) ≡ R kt /PRk (t, b).

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5 Dynamics: The Bottom-Up Approach

LPRODkFHK (t, b) ≡

Y k (t, b) . kt Lkt p + Lnp

(5.39)

Against the background of the previous sections all this is clearly not ideal, especially the fact that what is presented as productivity change is contaminated by all kinds of price effects. As Foster et al. (2001, 354) put it in their concluding remarks: Ideally, we would like to measure outputs, inputs, and associated prices of outputs and inputs at the establishment level in a way that permits the analysis of aggregate productivity growth in the manner discussed in this paper. Current practices at statisticsl agencies are far from this ideal with many of the components collected by different surveys with different units of observation (e.g., establishments vs. companies) and indeed by different statistical agencies.

Our second example is the data description provided by Dobbelaere and Mairesse (2013): We use an unbalanced panel of French manufacturing firms over the period 1978–2001, based mainly on firm accounting information from EAE (Enquête Annuelle d’Enterprise, Service des Etudes et Statistiques Industrielles (SESSI). We only keep firms for which we have at least 12 years of observations, ending up with an unbalanced panel of 10,646 firms with the number of observations for each firm varying between 12 and 24. We use real current production deflated by the two-digit producer price index of the French industrial classification as a proxy for output (Y ). Labor (L) refers to the average number of employees in each firm for each year and material input (M) refers to intermediate consumption deflated by the two-digit intermediate consumption price index. The capital stock (K) is measured by the gross book value of tangible assets as reported in the firm balance sheets at the beginning of the year (or at the end of the previous year), adjusted for inflation. The shares of labor (αL ) and material input (αM ) are constructed by dividing, respectively, the firm total labor cost and undeflated intermediate consumption by the firm undeflated production and by taking the average of these ratios over adjacent years.

It is unclear whether energy and services are included in material input or not, but that does not concern us here. In our notation their total factor productivity index (called Solow residual) for adjacent years t and t − 1 appears to be IMFPRODkDM (t, t − 1; b) ≡ 

(5.40)

Y k (t, b)/Y k (t − 1, b)      , 1−α L −αM k (t, b)/X k (t − 1, b) kt /Lk,t−1 αL X k (t, b)/X k (t − 1, b) αM XK L K M M

k,t−1 kt /R kt ) /R k,t−1 + CM where αL ≡ (CLk,t−1 /R k,t−1 + CLkt /R kt )/2 and αM ≡ (CM /2. Notice that the denominator is a Törnqvist quantity index, and that the indices for output, capital, and material input are dependent on the reference year of the deflators (whereby it is here assumed that all those deflators employ the same reference year). Notice also that the definition of the weights 1 − αL − αM , αL and αM implies that profit equals zero, kt = 0, for all production units at all time periods.

5.4 Productivity Indices and Levels

127

A pervasive feature of microdata studies, of which two examples were discussed, is the use of higher-level instead of production-unit specific deflators.13 Continuing the analysis of the previous subsection we can ask: how bad is this? Let us again assume that our target is to measure simple labour productivity RVAk (t, b)/Lkt for all k ∈ Kt , where b is some reference period.14 Suppose that, instead of the kk (t, b), a Kt -specific index P Kt (t, b) specific value-added based price indices PVA VA were used. The bias distribution would be

k (t, b) RVAk (t, b) PVA (5.41) − 1 (k ∈ Kt ), t K Lkt PVA (t, b) about which in general nothing can be said with any certainty. How large the bias is for particular production units can only be determined in a case-by-case analysis, using unit-specific data, the very data we are missing. Is there an aggregate effect?  Again, suppose we are interested in weighted mean simple labour productivity, k θ k RVAk (t, b)/Lkt , where θ k are certain weights adding up to 1. In this case it appears that 

k θ k (PVA (t, b)/PVKA (t, b))RVAk (t, b)/Lkt = t

k

 k

k K θ k (PVA (t, b)/PVA (t, b)) t



θ k RVAk (t, b)/Lkt

k

 k Kt + Covar PVA (t, b)/PVA (t, b), RVAk (t, b)/Lkt ; θ k .

(5.42)

Assume that the covariance equals 0. Then labour productivity  the kactual mean Kt (t, b)). This factor is differs from the target mean by the factor k θ k (PVA (t, b)/PVA the mean of the relative k-specific value-added based deflators, and this mean might be expected to be approximately equal to 1. Thus, in the zero-covariance case there is probably no aggregate effect. However, for determining whether this is indeed the case, one needs all the individual data. Without such data the situation seems a bit hopeless. There is some literature on the effect of using industry-level deflators instead of enterprise-level deflators on the estimation of production functions and the analysis of productivity change. See the early study of Abbott (1991) and, more recently, Mairesse and Jaumandreu (2005) and Foster et al. (2008). Of course, for such studies one needs enterprise-level price data, which severely limits the possibilities.15 In

13 An

exception is the study of Spanish manufacturing firms by Escribano and Stucchi (2014). For MFP (no services included) a simplified version of expression (5.38) was used. 14 It is assumed here that all units k ∈ Kt exist(-ed) in reference period b. 15 The alternative is to resort to enterprises with a single well-defined output. Famous are the US studies on the ready-mix concrete industry, a recent one being Backus (2020).

128

5 Dynamics: The Bottom-Up Approach

the literature, productivity based on revenue or value added deflated by an industrylevel price index is sometimes called ‘revenue productivity’, to distinguish it from our concept that is then called ‘(physical) output productivity’.16 Smeets and Warzynski (2013) found that physical productivity exhibited more dispersion than revenue productivity. A similar feature was unveiled by Eslava et al. (2013). In the last study it was also found that the correlation coefficient of the two measures was low. On the failure of revenue productivity measures to identify within-plant efficiency gains from exporting, see Marin and Voigtländer (2013). From the cross-sectional perspective this issue was studied by van Biesebroeck (2009).

5.5 Decompositions: Arithmetic Approach Let us now assume that productivity levels, real output divided by real input, are somehow available.17 We denote the productivity level of unit k at period t by PRODkt . Each production unit comes with some measure of relative size (importance) in the form of a weight θ kt . These weights add up to 1 for each period, that is   θ k0 = θ k1 = 1. (5.43) k∈K0

k∈K1

We concentrate here on the productivity levels as introduced in the previous section; that is, PRODkt has the form of real value added divided by real primary input or real labour input. Then, ideally, the relative size measure θ kt must be consistent with either of those measures. Though rather vague, this assumption is for the time being sufficient; we will return to this issue in Sect. 5.9. The aggregate (or mean) productivity level at period t is quite naturally defined as the weighted  arithmetic average of the unit-specific productivity levels, that is PRODt ≡ k∈Kt θ kt PRODkt , where the summation is taken over all production units existing at period t. The weighted geometric average, which is a natural alternative, as well as the weighted harmonic average, will be discussed in Sect. 5.7. Aggregate productivity change between periods 0 and 1 can be measured as a ratio or a difference. In line with much of the literature we start with the difference of productivity levels,

16 The

distinction between revenue productivity and physical productivity is a central issue in the microdata study of Hsieh and Klenow (2009), where Indian, Chinese, and U. S. manufacturing plants/firms were compared over the period 1977–2005. However, the authors did not have access to plant/firm-level deflators. Using some theoretical reasoning, real value added was estimated as RVAk (t, b) = (VAkt )3/2 , so that the ratio of physical productivity, calculated as k (t, b), and revenue productivity, calculated as VAkt /X k (t, b), becomes equal to RVAk (t, b)/XKL KL (VAkt )1/2 (k ∈ Kt ). It comes as no surprise then that physical productivity exhibits more dispersion than revenue productivity. 17 This section updates Balk (2003b, Section 6).

5.5 Decompositions: Arithmetic Approach



PROD1 − PROD0 =

129

θ k1 PRODk1 −

k∈K1



θ k0 PRODk0 .

(5.44)

k∈K0

Given the distinction between continuing, exiting, and entering production units, as defined by expressions (5.10) and (5.11), expression (5.44) can be decomposed as PROD1 − PROD0 =  θ k1 PRODk1 k∈N 1

+



θ k1 PRODk1 −

k∈C 01

−





θ k0 PRODk0

k∈C 01

θ k0 PRODk0 .

(5.45)

k∈X 0

The first term at the right-hand side of the equality sign shows the contribution of entering units, the second and third term together show the contribution of continuing units, whereas the last term shows the contribution of exiting units. The  contribution of continuing units, k∈C 01 θ k1 PRODk1 − k∈C 01 θ k0 PRODk0 , is the joint outcome of intra-unit productivity change, PRODk1 − PRODk0 , and inter-unit relative size change, θ k1 − θ k0 , for all k ∈ C 01 . The problem of decomposing this joint outcome into the contributions of the two factors happens to be structurally similar to the index number (or indicator) problem. Whereas in index number theory we talk about prices, quantities, and commodities, we are here talking about sizes, productivity levels, and (continuing) production units. It can thus be expected that in reviewing the various decomposition methods familiar names from index number theory, such as Laspeyres, Paasche, and Bennet, will turn up.

5.5.1 The First Three Methods The first method decomposes the contribution of the continuing units into a Laspeyres-type contribution of intra-unit productivity change and a Paasche-type contribution of relative size change: PROD1 − PROD0 =  θ k1 PRODk1 k∈N 1

+



θ k0 (PRODk1 − PRODk0 ) +

k∈C 01

−



k∈X 0



(θ k1 − θ k0 )PRODk1

k∈C 01

θ k0 PRODk0 .

(5.46)

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5 Dynamics: The Bottom-Up Approach

The second term at the right-hand side of the equality sign relates to intra-unit productivity change and uses base period weights. It is therefore, using the language of index number theory, called a Laspeyres-type measure. The third term relates to relative size change and is weighted by comparison period productivity levels. It is therefore called a Paasche-type measure. This decomposition was used in the early micro-data study of Baily et al. (BHC) (1992). One feature is important to notice. Disregard for a moment entering and exiting production units. Then aggregate productivity change is entirely due to continuing units, and is the sum of two terms. Suppose that all the units experience productivity increase, that is, PRODk1 > PRODk0 for all k ∈ C 01 . Counterintuitively, however, aggregate productivity change is not necessarily positive, because the relative-size change term k∈C 01 (θ k1 − θ k0 )PRODk1 can exert a negative influence. Though in various contexts noticed by various authors, this ‘paradox’ was extensively discussed by Fox (2012).18 We will return to this issue in Section 5.8. Since base period and comparison period weights add up to 1, we can insert an arbitrary scalar a, and obtain PROD1 − PROD0 =  θ k1 (PRODk1 − a) k∈N 1



+

θ k0 (PRODk1 − PRODk0 ) +

k∈C 01



−



(θ k1 − θ k0 )(PRODk1 − a)

k∈C 01

θ k0 (PRODk0 − a).

(5.47)

k∈X 0

At this point it is useful some additional notation. Let PRODX ≡   to introduce k0 PRODk0 / k0 be the mean productivity level of the exiting units, θ θ 0 k∈X 0 k∈X 1 k1  k1 k1 be the mean productivity and let PRODN ≡ k∈N 1 θ PROD / k∈N 1 θ level of the entering units. Then expression (5.47) can be written as 0

PROD1 − PROD0 =  1 ( θ k1 )(PRODN − a) k∈N 1



+

θ k0 (PRODk1 − PRODk0 ) +

k∈C 01

−(





(θ k1 − θ k0 )(PRODk1 − a)

k∈C 01

θ k0 )(PRODX − a). 0

(5.48)

k∈X 0

18 The

productivity ‘paradox’ is structurally similar to the unit-value ‘paradox’ in index number theory: all the individual prices may increase or decrease, yet the unit value (= quantity-weighted mean price) may decrease or increase, respectively. See Balk (2008, Section 3.3.4).

5.5 Decompositions: Arithmetic Approach

131

Thus, entering units contribute positively to aggregate productivity change when their mean productivity level exceeds a, and exiting units contribute positively when their mean productivity level falls short of a. The net effect of entrance and exit is given by the sum of the first and the fourth right-hand side term,   1 0 ( k∈N 1 θ k1 )(PRODN −a)−( k∈X 0 θ k0 )(PRODX −a). It is interesting to notice that this effect not only depends on relative importancies and mean productivities, but also on the value chosen for the arbitrary scalar a. However, as we will see, there are a number of reasonable options here. The second method uses a Paasche-type measure for intra-unit productivity change and a Laspeyres-type measure for relative size change. This leads to PROD1 − PROD0 =  1 ( θ k1 )(PRODN − a) k∈N 1



+

θ k1 (PRODk1 − PRODk0 ) +

k∈C 01

−(



(θ k1 − θ k0 )(PRODk0 − a)

k∈C 01



X0

θ k0 )(PROD

− a).

(5.49)

k∈X 0

There appears to be no application of this decomposition in the literature. It is possible to avoid the choice between the Laspeyres-Paasche-type and the Paasche-Laspeyres-type decomposition. The third method uses for the contribution of both intra-unit productivity change and relative size change Laspeyres-type measures. However, this simplicity is counterbalanced by the necessity to introduce a covariance-type term: PROD1 − PROD0 =  1 ( θ k1 )(PRODN − a) k∈N 1



+

θ k0 (PRODk1 − PRODk0 ) +

k∈C 01



+



(θ k1 − θ k0 )(PRODk0 − a)

k∈C 01

(θ k1 − θ k0 )(PRODk1 − PRODk0 )

k∈C 01

−(



θ k0 )(PRODX − a). 0

(5.50)

k∈X 0

As we see, the contribution of the continuing units consists of three components. The first, k∈C 01 θ k0 (PRODk1 − PRODk0 ) is usually called the ‘within effect’; the k0 k1 k0 second,  k∈C 01 (θ −θ )(PROD −a) is called the ‘static reallocation effect’; and k1 k0 the third, k∈C 01 (θ −θ )(PRODk1 −PRODk0 ) is called the ‘dynamic reallocation effect’.

132

5 Dynamics: The Bottom-Up Approach

In view of the overall Laspeyres-type perspective, a natural choice for the arbitrary scalar now seems to be a = PROD0 , the base period aggregate productivity level. This leads to the decomposition originally proposed by Haltiwanger (1997) and preferred by Foster et al. (FHK) (2001) (there called method 1). This method has been employed inter alia by Foster et al. (2006), Foster et al. (2008), CollardWexler and de Loecker (2015), and Erumban et al. (2019).19 The FHK method is also used in OECD’s MultiProd project (Berlingieri et al. 2017). 0 Baldwin and Gu (2006) suggested to set a = PRODX , the base period mean productivity level of the exiting units. It is clear that then the final right-hand side term in expression (5.50) vanishes, and that the net effect of entrance and exit  1 0 becomes equal to ( k∈N 1 θ k1 )(PRODN − PRODX ). It is as if the entering units have replaced the exiting units, and that the mean productivity surplus is all that matters. Choosing a = 0 brings us back to the BHC decomposition. Nishida et al. (2014) provided interesting comparisons of the BHC and FHK decompositions on Chilean, Colombian and Slovenian micro-level data.

5.5.2 Interlude: The TRAD, CSLS, and GEAD Decompositions Let us pause for a while at expression (5.50) and consider the case where there is neither exit nor entry; that is K0 = K1 = C 01 . Such a case materializes, for instance, when we are considering a certain economy with a fixed set of industries. Then this expression reduces to PROD1 − PROD0 =  θ k0 (PRODk1 − PRODk0 ) k∈C 01

+



(θ k1 − θ k0 )(PRODk0 − a)

k∈C 01

+



(θ k1 − θ k0 )(PRODk1 − PRODk0 ).

(5.51)

k∈C 01

In order to transform to (forward-looking) percentage changes (aka growth rates) both sides of this expression are divided by PROD0 , which delivers

19 Altomonte

and Nicolini (2012) applied the FHK method to aggregate price-cost margin change. For any individual production unit the price-cost margin was defined as nominal cash flow (= value kt kt added minus labour cost)  divided by nominal revenue, CF /R . These margins were weighted by market shares R kt / k∈Kt R kt (k ∈ Kt ).

5.5 Decompositions: Arithmetic Approach

133

PROD1 − PROD0 = PROD0

k0  PRODk1 − PRODk0 k0 PROD θ PROD0 PRODk0 01 k∈C



+

k∈C 01



+

k∈C 01

PRODk0 θ k0 PROD0 PRODk0 θ k0 PROD0



 θ k1 PRODk0 − a − 1 θ k0 PRODk0



 θ k1 PRODk1 − PRODk0 −1 . θ k0 PRODk0

(5.52)

Consider now simple labour productivity, that is, real value added per unit of labour; thus PRODkt = SLPRODkV A (t, b) ≡ RVAk (t, b)/Lkt (k ∈ C 01 ). Let the relative size of a production unit be given by its labour share; that is, θ kt ≡

kt  L

C 01 ).

k∈C 01

Lkt

(k ∈ It is straightforward to check that then the weights occurring in each k0 right-hand side term of expression (5.52), θ k0 PROD , reduce to real-value-added PROD0 shares,

k  RVA (0,b) k 01 k∈C RVA (0,b)

(k ∈ C 01 ), so that

PROD1 − PROD0 = PROD0  k∈C 01

+

RVAk (0, b)  k k∈C 01 RVA (0, b)

 k∈C 01

+

 k∈C 01

RVAk (0, b)  k k∈C 01 RVA (0, b) RVAk (0, b)  k k∈C 01 RVA (0, b)

In view of the fact that as



k∈C 01 (θ

PROD1 − PROD0 = PROD0  k∈C 01



RVAk (0, b)  k k∈C 01 RVA (0, b)

PRODk1 − PRODk0 PRODk0 



 θ k1 PRODk0 − a − 1 θ k0 PRODk0

(5.53)

 θ k1 PRODk1 − PRODk0 −1 . θ k0 PRODk0

k1 −θ k0 )



= 0, expression (5.53) can also be written

PRODk1 − PRODk0 PRODk0

134

5 Dynamics: The Bottom-Up Approach

+





k∈C 01

+

 k∈C 01

RVAk (0, b) k k∈C 01 RVA (0, b)

RVAk (0, b)  k k∈C 01 RVA (0, b)





 θ k1 PRODk0 − a − 1 θ k0 PRODk0

(5.54)

 PRODk1 − PRODk0 − a  θ k1 −1 . θ k0 PRODk0

for another arbitrary scalar a  . Now, choosing a = 0 and a  = 0 yields the TRAD(itional) way of decomposing aggregate labour productivity change into contributions of the various industries, according to three main sources: a withinsector effect, a reallocation level effect (aka Denison effect), and a reallocation growth effect (aka Baumol effect), respectively.20 Choosing a = PROD0 and a  = PROD1 − PROD0 yields the CSLS decomposition (which has been developed at the Centre for the Study of Living Standards; Sharpe 2010). Alternatively, let the relative size of a production unit be given by its combined labour and relative price share; that is, θ kt ≡

kt  L k∈C 01

Lkt

k (t,b) PVA K (t,b) PVA

(k ∈ C 01 ), where

K (t, b) is some general, non-k-specific deflator. Notice that these weights do not PVA add up to 1. It is straightforward to check that in this case the weights occurring in k0 the right-hand side terms of expression (5.52), θ k0 PROD , reduce to nominal-valuePROD0

added shares,

k0  VA k0 01 k∈C VA

(k ∈ C 01 ), so that

PROD1 − PROD0 = PROD0

 PRODk1 − PRODk0 VAk0  k0 PRODk0 k∈C 01 VA 01 k∈C

+

 k∈C 01

+

 k∈C 01

VAk0  k0 k∈C 01 VA VAk0  k0 k∈C 01 VA





 θ k1 PRODk0 − a − 1 θ k0 PRODk0

(5.55)

 θ k1 PRODk1 − PRODk0 − a  −1 , θ k0 PRODk0

where a and a  are arbitrary scalars. For a = 0 and a  = 0 this appears to be the Generalized Exactly Additive Decomposition (GEAD), going back to Tang and Wang (2004) and explored by Dumagan (2013a). Recalling the definitions given above, aggregate simple labour productivity in the GEAD is defined as

20 These

names seem to go back to Nordhaus (2002). See for various other names and the history of these concepts De Avillez (2012).

5.5 Decompositions: Arithmetic Approach



PRODt ≡

135

k (t, b)SLPRODkV A (t, b), L˜ kt P˜VA

(5.56)

k∈C 01

where L˜ kt ≡

kt  L k∈C 01

Lkt

k (t, b) ≡ and P˜VA

k (t,b) PVA K (t,b) PVA

(k ∈ C 01 ). As we see, there

are three factors here that determine the development of aggregate productivity: the labour shares, the relative output prices, and the individual labour productivities. The GEAD effectively considers only two factors, namely the combination of labour shares and relative output prices vis-a-vis the individual productivities. This raises the question whether it is possible to split aggregate productivity change into the three factors. Diewert (2015b, 2016) contributed to answering this question. His approach consisted in first noticing that the aggregate productivity ratio, PROD1 /PROD0 , can be written as a weighted arithmetic mean, k  ˜ k1 ˜ k PROD1 k0 L PVA (1, b) SLPRODV A (1, b) , = s VA k (0, b) SLPRODk (0, b) PROD0 L˜ k0 P˜VA VA 01

(5.57)

k∈C

k0 ≡ where sVA

k0  VA k0 k∈C 01 VA

(k ∈ C 01 ) are the nominal-value-added shares in period 0.

A first-order approximate decomposition of the aggregate productivity growth rate then appears to be  PROD1 k0 −1≈ sVA 0 PROD 01



k∈C



+

k∈C 01

˜k  L˜ k1 k0 PVA (1, b) −1 −1 + sVA L˜ k0 P˜ k (0, b) 01 k∈C

k0 sVA

SLPRODkV A (1, b) SLPRODkV A (0, b)

(5.58)

VA

−1 .

Unfortunately, the complete decomposition appears to contain higher-order terms that are mixtures of the three factors one wants to separate—labour shares, relative output prices, and individual labour productivities. Moreover, the right-hand side of expression (5.57) is asymmetric in the sense that period 0 shares are used as weights. The aggregate productivity ratio, PROD1 /PROD0 , can as well be written as a weighted harmonic mean, ⎛

−1 ⎞−1 k1 P˜ k (1, b) SLPRODk (1, b)  ˜ PROD1 L VA VA k1 ⎠ , =⎝ sVA k (0, b) SLPRODk (0, b) PROD0 L˜ k0 P˜VA VA 01

(5.59)

k∈C

k1 ≡ where sVA

k1  VA k1 k∈C 01 VA

(k ∈ C 01 ) are the nominal-value-added shares in period

1. This of course leads to a different approximate decomposition of the aggregate productivity growth rate.

136

5 Dynamics: The Bottom-Up Approach

Fortunately, symmetry can be attained, without having to use approximations, by employing the generalized, three-factor Sato-Vartia decomposition proposed by Balk (2003a). This will be discussed in Section 5.6. De Avillez (2012) provided an interesting empirical comparison of TRAD, CSLS, and GEAD. He found that “despite some similarities, all three decomposition formulas paint very different pictures of which sectors drove labour productivity growth in the Canadian business sector during the 2000–2010 period.” The difference between TRAD and CSLS not unexpectedly hinges on the role played by the scalars a and a  . Varying a and/or a  implies varying magnitudes of the two reallocation effects, not at the aggregate level—because the sums are invariant—but at the level of individual production units (in this case, industries). The difference between TRAD and CSLS on the one hand and GEAD on the other evidently hinges on the absence or presence of relative price levels in the sectoral measures of importance. De Avillez found it “impossible to say which set of estimates provides a more accurate picture of economic reality because the GEAD formula is, ultimately, measuring something very different from the TRAD and CSLS formulas.” Basically this conclusion means that the answer cannot be found within the bottom-up perspective. The top-down perspective is required to obtain a decision. We will return to this issue in Section 9.4. De Avillez’s (2012) analysis was supplemented by Reinsdorf (2015). He compared CSLS, GEAD, and the Diewert decomposition on the same data and found also starkly different sectoral contributions to aggregate labour productivity growth. Another piece of evidence was recently provided by Nishi (2019). He compared GEAD, Diewert’s decomposition, and CSLS on detailed Japanese industrial data over the years 1970–2010. In all the decompositions the effect of within-industry labour productivity change appeared to be dominant. The other effects were small and extremely varying through time and between sectors; hence, difficult to interpret. Voskoboynikov (2020) applied TRAD, CSLS, and GEAD in a study of the Russian economy over the period 1995–2012. This study was specifically devoted to the shift of labour from the formal to the informal parts of the economy.

5.5.3 The Fourth and Fifth Method Let us now return to expression (5.50). Instead of the Laspeyres perspective, one might as well use the Paasche perspective. The covariance-type term accordingly appears with a negative sign. Thus, the fourth decomposition is PROD1 − PROD0 =  1 ( θ k1 )(PRODN − a) k∈N 1

5.5 Decompositions: Arithmetic Approach



+

137

θ k1 (PRODk1 − PRODk0 ) +

k∈C 01



−



(θ k1 − θ k0 )(PRODk1 − a)

k∈C 01

(θ k1 − θ k0 )(PRODk1 − PRODk0 )

k∈C 01

−(



θ k0 )(PRODX − a). 0

(5.60)

k∈X 0

The natural choice for a would now be PROD1 , the comparison period aggregate 1 productivity level. Choosing a = PRODN would lead to disappearance of the entry effect. The net effect of entrance and exit then becomes equal to  1 0 ( k∈X 0 θ k0 )(PRODN − PRODX ). It is left to the reader to explore the analogs to expressions (5.54) and (5.55) by using backward-looking percentage changes. It seems that there is no empirical application of this decomposition. The fifth method avoids the Laspeyres-Paasche dichotomy altogether, by using the symmetric Bennet method. This amounts to taking the arithmetic average of the first and the second method. The covariance-type term then disappears. Thus, PROD1 − PROD0 =  1 ( θ k1 )(PRODN − a) k∈N 1

+

 θ k0 + θ k1  PRODk1 − PRODk0 2 01

k∈C

+





(θ

k∈C 01

−(



k1

PRODk0 + PRODk1 −a −θ ) 2

k0

θ k0 )(PRODX − a). 0

(5.61)

k∈X 0

With respect to the scalar a there are several options available in the literature. A rather natural choice is a = (PROD0 + PROD1 )/2, the overall two-period mean aggregate productivity level. Then, entering units contribute positively to aggregate productivity change if their mean productivity level is above this overall mean. Exiting units contribute positively if their mean productivity level is below the overall mean. Continuing units can contribute positively in two ways: if their productivity level increases, or if the units with productivity levels above (below) the overall mean increase (decrease) in relative size. This decomposition basically corresponds to that used in the early microdata study of Griliches and Regev (1995), and will therefore be called the GR method. Because of its symmetry it is widely preferred. Moreover, Foster et al. (2001) argue that the GR method (there called method 2) is presumably less sensitive to (random) measurement errors than the

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5 Dynamics: The Bottom-Up Approach

asymmetric FHK method. The GR method was employed inter alia by Baily et al. (2001), Foster et al. (2008), and Figal Garone et al. (2020).21 However, other choices for the scalar a are also plausible. Baldwin and Gu (BG) 0 (2006) suggested to set a = PRODX , the base period mean productivity level of the exiting production units. Then, as we have seen before, the last term of expression (5.61) disappears. Put otherwise, entering units are seen as displacing exiting units, contributing positively to aggregate productivity change insofar their mean productivity level exceeds that of the exiting units. Baldwin and Gu (2008) considered two alternatives, to be applied to different types of industries. The first is to set a equal to the base period mean productivity level of the continuing units that are contracting; that is, the units k ∈ C 01 for which θ k1 < θ k0 . The second is to set a equal to the base period mean productivity level of the continuing units that are expanding. A specific extension of the GR method was developed by Reinsdorf (2015). For details the reader is referred to Appendix A. Balk and Hoogenboom-Spijker (2003) compared the five methods, defined by expressions (5.48)–(5.50), (5.60), and (5.61) respectively, on micro-level data of the Netherlands Manufacturing industry over the period 1984–1999. Though there appeared to be appreciable differences between the various decompositions, the pervasive fact was the preponderance of the productivity change of the continuing units (or, the ‘within’ term). The core of this study is reproduced in Appendix B, not only because of its interesting results, but also to show that there is quite a distance between theory and application, even in an environment stuffed with microdata.

5.5.4 Another Five Methods A common feature of the five decomposition methods discussed hitherto is that the productivity levels of exiting and entering production units are compared to a single overall benchmark level a, for which a number of options is available. It seems more natural to compare the productivity levels of exiting units to the mean level of the continuing units at the base period—which is the period of exit, and to compare the productivity levels of entering units to the mean level of the continuing units at the comparison period—which is the period of entrance. Thus, let the aggregate productivity level of the continuing production units at  01 kt  kt kt (t = 0, 1). period t be defined as PRODC t ≡ k∈C 01 θ PROD / k∈C 01 θ kt Since the weights θ add up to 1 for both periods—see expression (5.43)—, expression (5.44) can be decomposed as

21 The

GR method, without entering or exiting units, was proposed by Maital and Vaninsky (2000) as a method for overcoming ‘productivity paradoxes’. However, the third component at the righthand side of expression (5.61) can still be negative, and dominate the second, while PRODk1 > PRODk0 for all k ∈ C 01 .

5.5 Decompositions: Arithmetic Approach

139

PROD1 − PROD0 =   1 01 ( θ k1 ) PRODN − PRODC 1 k∈N 1

+ PRODC 1 − PRODC 0   0 01 −( θ k0 ) PRODX − PRODC 0 . 01

01

(5.62)

k∈X 0

This expression tells us that entering units contribute positively to aggregate productivity change if their mean productivity level is above that of the continuing units at the entrance period. Similarly, exiting units contribute positively if their mean productivity level is below that of the continuing units at the period of exit.  Let the relative size of continuing units be defined by θ˜ kt ≡ θ kt / k∈C 01 θ kt (k ∈ C 01 ; t = 0, 1). The contribution of the continuing units to aggregate productivity change can then be written as PRODC

01 1

− PRODC

01 0

=



θ˜ k1 PRODk1 −

k∈C 01



θ˜ k0 PRODk0 ,

(5.63)

k∈C 01

which has the same structure as the second and third term of expression (5.45),  the difference being that the weights now add up to 1; that is, k∈C 01 θ˜ kt = 1 (t = 0, 1). Thus the five methods discussed earlier can simply be repeated on the right-hand side of expression (5.62). For example, the third method delivers PROD1 − PROD0 =   1 01 ( θ k1 ) PRODN − PRODC 1 k∈N 1

+



θ˜ k0 (PRODk1 − PRODk0 ) +

k∈C 01

+



(θ˜ k1 − θ˜ k0 )(PRODk0 − a)

k∈C 01

(θ˜ k1 − θ˜ k0 )(PRODk1 − PRODk0 )

k∈C 01

−(





 0 01 θ k0 ) PRODX − PRODC 0 .

(5.64)

k∈X 0

The first right-hand side term of this expression refers to the entering production units. As we see, its magnitude is determined by the period 1 share of the entrants and the productivity gap with the continuing units. The last right-hand side term refers to the exiting production units. The magnitude of this term depends on the share of the exiting units and the productivity gap with the continuing units. The second, third, and fourth term refer to the continuing production units. They may contribute positively in three ways: if their productivity levels on average

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5 Dynamics: The Bottom-Up Approach

increase, if the units with productivity levels above (or below) a certain scalar a increase (or decrease) in relative size, or if there is a positive covariance between relative size change and productivity change. In the third term an arbitrary scalar a can be inserted, since the relative weights of the continuing production units add up to 1 in both periods. Though the term itself is invariant, the unit-specific components (θ˜ k1 − θ˜ k0 )(PRODk0 − a) are not. Expression (5.64) found an application in Kim (2019). The covariance term is a mathematical artefact, however, due to the base period perspective in the other two components. This term vanishes if the symmetric Bennet decomposition is selected. Then, PROD1 − PROD0 =   1 01 ( θ k1 ) PRODN − PRODC 1 k∈N 1

+

 θ˜ k0 + θ˜ k1  PRODk1 − PRODk0 2 01

k∈C

+



(θ˜ k1 − θ˜ k0 )

k∈C 01

−(



PRODk0 + PRODk1 −a 2

 0 01 θ k0 ) PRODX − PRODC 0 .

(5.65)

k∈X 0

This decomposition was developed by Diewert and Fox (DF) (2010), the discussion paper version of which was published in 2005. Though in a different context—the development of shares of labour types in plant employment—the DF decomposition can be detected in Vainiomäki (1999). Surprisingly there exist not many empirical applications in the field of productivity measurement.22 If there are no exiting or entering units, that is, K0 = K1 = C 01 , then the DF method (5.65) as well as the GR method (5.61) reduce to the simple Bennet decomposition.

22 Though Kirwan et al. (2012) contend to use the DF decomposition, it appears that their analysis is

simply based on expression (5.44). The part relating to continuing units is replaced by a weighted sum of production-function based unit-specific productivity changes plus residuals. Böckerman and Maliranta (2007) used the DF decomposition for the analysis of value-added based simple labour productivity and total factor productivity. Kauhanen and Maliranta (2019) applied a twostage DF decomposition to mean wage change. Another example is Ilmakunnas and Maliranta (2016).

5.5 Decompositions: Arithmetic Approach

141

5.5.5 Provisional Evaluation The overview provided in the foregoing subsections hopefully demonstrates a number of things, the first and most important of which is that there is no unique decomposition of aggregate productivity change as defined by expression (5.44).23 Second, one should be careful with giving the various components a sort of agency status. This in particular applies to the covariance-type term, since this term can be considered as a mere artefact, arising from the specific (Laspeyresor Paasche-) perspective chosen. Third, the undetermined character of the scalar a lends additional arbitrariness to the first set of five decompositions. At the aggregate level it is easily seen that letting a tend to 0 will lead to a larger contribution of the entering units, the exiting units, and the size change of continuing units, at the expense of intra-unit productivity change. The advantage of the second set of five decompositions, among which the symmetric DF method, is that the distribution of these four parts is kept unchanged. The remaining arbitrariness in expression (5.65) is in the size-change term and materializes only at the level of individual continuing production units. Fourth, what counts as ‘entrant’ or ‘exiting unit’ depends not only on the length of the time span between the periods 0 and 1, but also on the length of the periods themselves and the observation thresholds employed in sampling. All in all it can be expected that the outcome of any decomposition exercise depends to some extent on the particular method favoured by the researcher. This is not a problem as long as he or she realizes this. However, it is always good to check results for robustness, and to let the favoured results be accompanied by some alternatives. Finally, as demonstrated in the previous section, the productivity levels PRODkt depend on the price reference period of the deflators used. In particular this holds for the simple labour productivity levels RVAk (t, b)/Lkt . This dependence obviously extends to aggregate productivity change PROD1 − PROD0 . To mitigate its effect, one considers instead the forward-looking growth rate of aggregate productivity, (PROD1 −PROD0 )/PROD0 , and its decomposition, obtained by dividing each term by PROD0 . It would of course be equally justified to consider the backward-looking growth rate, (PROD1 −PROD0 )/PROD1 . A symmetric growth rate is obtained when the difference PROD1 − PROD0 is divided by a mean of PROD0 and PROD1 . When the logarithmic mean24 is used, one obtains PROD1 − PROD0 = ln(PROD1 /PROD0 ), LM(PROD0 , PROD1 ) 23 This

(5.66)

non-uniqueness should not come as a surprise and finds its parallel in index number theory (see Balk 2008) and in so-called structural decomposition analysis (widely used in input-output analysis; see Dietzenbacher and Los 1998 and De Boer and Rodrigues 2020). 24 Recall that the logarithmic mean, for any two strictly positive real numbers a and b, is defined by LM(a, b) ≡ (a − b)/ ln(a/b) if a = b and LM(a, a) ≡ a.

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5 Dynamics: The Bottom-Up Approach

which can be interpreted as a percentage change.25 Its decomposition contains arithmetic differences of the form PRODk1 − PRODk0 . These can of course be transformed into logarithmic differences, but that leads to pretty complicated expressions. However, in the midst of all these complications there is a treasure hidden that calls for being discovered. This is the topic of the next section.

5.6 Decompositions: Logmean Approach Let us return to expression (5.62), basic for the DF decomposition, and substitute this in expression (5.66) to obtain ln(PROD1 /PROD0 ) = 1 LM(PROD0 , PROD1 )

PROD1 − PROD0 = LM(PROD0 , PROD1 )    1 01 ( θ k1 ) PRODN − PRODC 1 k∈N 1

+ PRODC 1 − PRODC 0    0 01 −( θ k0 ) PRODX − PRODC 0 . 01

01

(5.67)

k∈X 0

  0 Recall that PRODX ≡ k∈X 0 θ k0 PRODk0 / k∈X 0 θ k0 is the mean productivity  1 k1  k1 k1 level of the exiting units, that PRODN ≡ k∈N 1 θ PROD / k∈N 1 θ 01 t C is productivity level of  the entering units, and that PROD ≡  the mean  kt PRODkt / kt = kt PRODkt is the mean productivity ˜ θ θ θ 01 01 01 k∈C k∈C k∈C level of the continuing production units at period t (t = 0, 1). Instead of employing the Bennet decomposition, the logarithmic mean is employed to rewrite the continuing units part of expression (5.67) as PRODC

25 Since

01 1

− PRODC

LM(PROD

C 01 0

LM(PROD

C 01 0

01 0

=

, PROD

C 01 1

, PROD

C 01 1

) ln

PRODC

01 1

PRODC 

01 0

=

θ˜ k1 PRODk1 ) ln  k0 ˜ k0 k∈C 01 θ PROD

k∈C 01

ln(a/a  ) = ln(1 + (a − a  )/a  ) ≈ (a − a  )/a  when (a − a  )/a  is small.

.

(5.68)

5.6 Decompositions: Logmean Approach

143

Then, the Sato-Vartia decomposition26 tells us that 

θ˜ k1 PRODk1 ln  k0 ˜ k0 k∈C 01 θ PROD

k∈C 01

=



 k∈C

θ˜ k1 PRODk1  (1, 0) ln θ˜ k0 PRODk0 01

k

,

(5.69)

with weights defined as

  k (1, 0) ≡ 

LM k∈C 01

θ˜ k0 PRODk0 θ˜ k1 PRODk1 , 01 C 01 0 PROD PRODC 1  k0 k0 k1 ˜ θ˜ PRODk1 LM θ PROD , 01 01 PRODC 0 PRODC 1

(k ∈ C 01 ),

 where LM(.) is the logarithmic mean. Notice that k∈C 01  k (1, 0) = 1. The righthand side of expression (5.69) can then be split into two parts, so that



01 1

PROD1 ln PROD0

26 Usually



=



θ˜ k1  (1, 0) ln θ˜ k0 01

 k∈C



PRODk1 ln  (1, 0) ln = + . 01 PRODk0 PRODC 0 01 k∈C k∈C (5.70) The first part provides the aggregate effect of inter-unit relative size changes, and the second part provides the aggregate effect of intra-unit productivity changes. Basically, expression (5.70) can be considered as the geometric variant of the GR decomposition in expression (5.61). Together, expressions (5.67), (5.68) and (5.70) provide a complete, timesymmetric decomposition of the growth rate of aggregate productivity, ln (PROD1 /PROD0 ), into four independent components: the contribution of entering production units, the contribution of exiting units, the effect of the reallocation of continuing units, and the effect of intra-unit productivity growth of the continuing units. This decomposition will be called the Diewert-Fox-Balk (DFB) decomposition. Though its formula might look a bit intimidating, if one has appropriated that a logarithmic mean is a mean, the interpretation of the various components is rather straightforward. As in Section 5.5.2 we now consider a situation where there is neither exit nor entry. The identity K0 = K1 = C 01 implies that θ˜ kt = θ kt (k ∈ C 01 , t = 0, 1), and expression (5.70) reduces to PRODC

k



θ k1  (1, 0) ln θ k0 01 k

 +



 k∈C

k



PRODk1  (1, 0) ln PRODk0 01 k

. (5.71)

the Sato-Vartia decomposition is applied to a value ratio, leading to Sato-Vartia price and quantity indices. See Appendix A of Chap. 2 and Balk (2008, 85–86).

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5 Dynamics: The Bottom-Up Approach

If productivity is simple labour productivity and the weights are labour shares then, k (t,b) kt 01 as we have seen earlier, θ kt PROD =  RVA RVA k (t,b) (k ∈ C , t = 0, 1), and PRODt k∈C 01

consequently

RVAk (1, b) RVAk (0, b) , LM  k k 01 RVA (0, b) k∈ C k∈C 01 RVA (1, b)

(k ∈ C 01 ).  k (1, 0) ≡  RVAk (0, b) RVAk (1, b) , k∈C 01 LM  k k k∈C 01 RVA (0, b) k∈C 01 RVA (1, b) Thus the weights  k (1, 0) (k ∈ C 01 ) are (normalized) logarithmic mean real-valueadded shares of base and comparison period. With these weights expression (5.71) should be compared with the TRAD and CSLS decompositions, expression (5.54): the coefficients have become time-symmetric, and the Baumol effect has vanished. Alternatively, if the weights are labour shares times relative output prices, that is, θ kt

k (t,b) PVA K kt P K (t,b) , where PVA (t, b) is some general, non-k-specific L k∈C 01 VA kt kt kt (k ∈ C 01 , t = 0, 1). Then θ kt PROD =  VA VAkt = sVA PRODt k∈C 01

≡

then

kt  L

 k (1, 0) ≡ 

k0 , s k1 ) LM(sVA VA k0 k1 k∈C 01 LM(sVA , sVA )

deflator,

(k ∈ C 01 ).

Thus the weights  k (1, 0) (k ∈ C 01 ) are (normalized) logarithmic mean nominalvalue-added shares of base and comparison period. With these weights expression (5.71) should be compared with the GEAD decomposition, expression (5.55): the coefficients have become time-symmetric, and the Baumol effect has vanished. We can now go one step further and show that the DFB decomposition has k (t, b) (k ∈ a feature the DF decomposition does not have. Since θ kt = L˜ kt P˜VA 01 C , t = 0, 1), expression (5.71) specializes into a three-factor decomposition,

PROD1 ln PROD0

=

 k∈C

+



L˜ k1  (1, 0) ln L˜ k0 01

 k∈C 01

k

 k (1, 0) ln

+

k∈C



k (1, b) P˜VA  (1, 0) ln P˜ k (0, b) 01



k

SLPRODkV A (1, b) SLPRODkV A (0, b)

VA

(5.72)

.

We see here that the growth rate of aggregate simple labour productivity is decomposed into three components: a weighted mean of growth rates of labour shares, a weighted mean of growth rates of relative output prices, and a weighted mean of growth rates of individual simple labour productivities. The weights are (normalized) logarithmic means of nominal-value-added shares for the two periods

5.7 Decompositions: Geometric and Harmonic Approach

145

compared. Notice that expression (5.72) is an identity and not an approximation such as expression (5.58). Interestingly, the output-prices component becomes identically equal to 0 if and K (t, b) is chosen such that only if the general deflator PVA ln

K (1, b) PVA

K (0, b) PVA

=



k

 (1, 0) ln

k∈C 01

k (1, b) PVA

k (0, b) PVA

(5.73)

.

Or, if the growth rate of the general deflator is equal to a symmetrically weighted mean of the individual output-price growth rates. This identity throws light on the fact that in empirical studies, where the production units are industries, the outputprices component generally appears to be negligible. Usually the general deflator is defined as some reasonable mean of the production-unit-specific deflators. When the time-span between periods 0 and 1 is not too long and output price behaviour is not too dispersed then the specific type of mean that has been selected does not matter much.

5.7 Decompositions: Geometric and Harmonic Approach In the geometric approach the aggregate productivity level is defined as a weighted geometric average of the unit-specific productivity levels, that is t ≡  kt θ kt t ≡ PROD k∈Kt (PROD ) . This is equivalent to defining ln PROD  kt kt k∈Kt θ ln PROD , which implies that, by replacing PROD by ln PROD, the entire story of Section 5.5 can be repeated. We will not do this. It is sufficient to mention that the geometric counterpart of the DF decomposition (5.65) is ln PROD1 − ln PROD0 =   1 01 ( θ k1 ) ln PRODN − ln PRODC 1 k∈N 1

+

 θ˜ k0 + θ˜ k1  ln PRODk1 − ln PRODk0 2 01

k∈C

+





(θ

k∈C 01

−(

˜ k1



k∈X 0

ln PRODk0 + ln PRODk1 −a −θ ) 2

˜ k0

 0 01 θ k0 ) ln PRODX − ln PRODC 0 ,

(5.74)

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5 Dynamics: The Bottom-Up Approach

 1 k1  C 01 t ≡ k1 k1 where now ln PRODN ≡ k∈N 1 θ ln PROD / k∈N 1 θ , ln PROD    kt / kt kt (t = 0, 1), and ln PRODX 0 ≡ k0 k∈C 01 θ ln k∈C 01 θ k∈X 0 θ PROD k0 k0 ln PROD / k∈X 0 θ . In expression (5.74) we are having logarithmic differences all over the place, but with the same weights as in expression (5.65). The second right-hand side term, reflecting productivity change of continuing production units, is a meanshare weighted sum of logarithmic differences. This term functionally resembles a Törnqvist price or quantity index. The advantage of the Geometric DF decomposition, expression (5.74), over the DF decomposition, expression (5.65), is that logarithmic changes can be interpreted immediately as percentage changes. This holds for the left-hand side of expression (5.74) as well as for the first, second, and fourth term at its righthand side. The disadvantage is that, as an aggregate level measure, a geometric  kt θ kt is less easy to understand than an arithmetic mean mean k∈Kt (PROD )  kt kt k∈Kt θ PROD . We let the top-down approach, see Chapter 9, advise which mean should be preferred. Using the approximation mentioned in footnote 25 two times, the entrants component of the Geometric DF decomposition can be approximated as (

 1 01 θ k1 ) ln PRODN − ln PRODC 1

 k∈N 1

≈(



k1

θ )

k∈N 1

⎛

≈ ln ⎝1 + (

PRODN PRODC



k∈N 1

01 1

θ k1 )

1

−1

PRODN PRODC

1

01 1

⎞ − 1 ⎠,

(5.75)

and the exit component can similarly be approximated. With these approximations the Geometric DF becomes the decomposition used by Böckerman and Maliranta (2007) in their study of the dynamics of regional productivity change. But the above expression makes clear that the same ingredients are needed for computing the approximation and the approximand. The Geometric DF decomposition as such was applied by Hyytinen and Maliranta (2013). They extended the decomposition to deal with age groups of firms. As indicated above there are also geometric versions of the GR, FHK, and BG decompositions. Baldwin and Gu (2011) compared these on Canadian retail trade and manufacturing industry microdata over the period 1984–1998. As in the comparative study of Balk and Hoogenboom-Spijker (2003) they found that in manufacturing the ‘within’ term was dominant. However, in retail trade the net effect of entry and exit appeared more important. In the harmonic approach the aggregate productivity level is defined as a weighted harmonic average of the unit-specific productivity levels, that is PRODt ≡

5.7 Decompositions: Geometric and Harmonic Approach

147

 ( k∈Kt θ kt (PRODkt )−1 )−1 . Though the literature does hardly pay any attention to this option, there are situations in which this type of average rather naturally emerges. An example is provided by Böckerman and Maliranta (2012). Though these authors were primarily concerned with the evolution of the aggregate real labour share through time, it turns out that their analysis is equivalent to an Harmonic DF decomposition on aggregate labour productivity, defined as weighted harmonic mean of LPRODkVA (t, t − 1) with weights defined as real-value-added shares at period t. Though an harmonic mean at first sight looks a bit unwelcoming, it is evident that by concentrating on 1/PRODt the entire story of the previous section can be repeated. But again, we will not do this. It is sufficient to mention that the harmonic counterpart of the DF decomposition, expression (5.65), is 1/PROD1 − 1/PROD0 =   1 01 θ k1 ) 1/PRODN − 1/PRODC 1 ( k∈N 1

+

 θ˜ k0 + θ˜ k1  1/PRODk1 − 1/PRODk0 2 01

k∈C

+

 k∈C

−(



1/PRODk0 + 1/PRODk1 −a (θ˜ k1 − θ˜ k0 ) 2 01



 0 01 θ k0 ) 1/PRODX − 1/PRODC 0 ,

(5.76)

k∈X 0

 1 k1  C 01 t ≡ k1 k1 where 1/PRODN ≡ k∈N 1 θ (1/PROD )/ k∈N 1 θ , 1/PROD    0 kt )/ kt kt (t = 0, 1), and 1/PRODX ≡ k0 k∈C 01 θ (1/PROD k∈C 01 θ k∈X 0 θ  k0 k0 (1/PROD )/ k∈X 0 θ . This, however, is not what we want. We want a decomposition of PROD1 − PROD0 or ln(PROD1 /PROD0 ), which allows the interpretation as a percentage change. The best we can do is divide both sides of expression (5.76) by the logarithmic mean of 1/PROD0 and 1/PROD1 so that the left-hand side reduces to 1/PROD1 − 1/PROD0 = − ln(PROD1 /PROD0 ). LM(1/PROD0 , 1/PROD1 )

(5.77)

In this way we obtain a decomposition of ln(PROD1 /PROD0 ) in the by now familiar four components.

148

5 Dynamics: The Bottom-Up Approach

5.8 Monotonicity Paradox? As already alluded to, all the definitions of aggregate productivity change, whether arithmetic, geometric or harmonic, suffer from what Fox (2012) called the “monotonicity problem” or “monotonicity paradox”. Again, disregard for a moment entering and exiting production units. Then aggregate productivity change is entirely due to continuing units, and is the combination (sum, product, or harmonic sum, respectively) of two terms. Suppose that all the units experience productivity increase, that is, PRODk1 > PRODk0 for all k ∈ C 01 . Then the ‘within’ term in the DF decomposition, expression (5.65), and in the Geometric DF decomposition, expression (5.74), is positive. In the Harmonic DF decomposition, expression (5.76), the ‘within’ term is formulated in inverse productivities and thus negative. One then would expect aggregate productivity change to be positive, or inverse productivity change to be negative. This, however, does not necessarily happen, because the relative-size-change terms can exert a counterbalancing influence and dominate the ‘within’ terms. Is it possible to define aggregate productivity change in such a way that ‘paradoxes’ can be avoided?  Fox (2012) noticed that the term k∈C 01 (θ k0 + θ k1 )(PRODk1 − PRODk0 )/2 as such has the desired monotonicity property, and proposed to extend this measure to the set C 01 ∪ X 0 ∪ N 1 = K0 ∪ N 1 = X 0 ∪ K1 . Aggregate productivity change is then defined as PRODF ox (1, 0) ≡  k∈C 01 ∪X 0 ∪N 1

θ k0 + θ k1  PRODk1 − PRODk0 . 2

(5.78)

Now, for all the exiting production units, k ∈ X 0 , it is evidently the case that in the later period 1 those units have size zero; that is, θ k1 = 0. It is then rather natural to set their virtual productivity level also equal to zero; that is, PRODk1 = 0. Likewise, entering units, k ∈ N 1 , have size zero in the earlier period 0; that is, θ k0 = 0. Their virtual productivity level at that period is also set equal to zero; that is, PRODk0 = 0. Then expression (5.78) can be decomposed as PRODF ox (1, 0) =  θ k1 PRODk1 (1/2) k∈N 1

+

 θ k0 + θ k1  PRODk1 − PRODk0 2 01

k∈C

− (1/2)

 k∈X 0

θ k0 PRODk0 .

(5.79)

5.8 Monotonicity Paradox?

149

Unfortunately, there is no geometric or harmonic analogue to expressions (5.78) and (5.79), because the logarithm or reciprocal of a zero productivity level is infinite. By using the logarithmic mean, one obtains PRODF ox (1, 0) =  (1/2) θ k1 PRODk1 k∈N 1

+

  θ k0 + θ k1 LM(PRODk0 , PRODk1 ) ln PRODk1 /PRODk0 2 01

k∈C

− (1/2)



θ k0 PRODk0 ,

(5.80)

k∈X 0

which, however, does not provide any advantage vis-a-vis expression (5.79). It is interesting to compare Fox’s proposal to the GR decomposition in expression (5.61) with a = 0. It then turns out that PROD1 − PROD0 =  θ k0 + θ k1   PRODk1 − PRODk0 θ k1 PRODk1 + 2 1 01 k∈N

+



k∈C

(θ k1 − θ k0 )

k∈C 01

 PRODk0 + PRODk1 − θ k0 PRODk0 = 2 0

PRODF ox (1, 0) + (1/2)



k∈X

θ k1 PRODk1

(5.81)

k∈N 1

+

 k∈C 01

(θ k1 − θ k0 )

 PRODk0 + PRODk1 − (1/2) θ k0 PRODk0 . 2 0 k∈X

The expression between first and second equality sign is the GR decomposition. The expression after the second equality sign contains Fox’s measure plus the remainder. It is remarkable that of the entire contribution of entering and exiting production units to PROD1 − PROD0 , half is considered as non-productivity change. It is difficult to envisage a solid justification for this.

150

5 Dynamics: The Bottom-Up Approach

5.9 The Olley-Pakes Decomposition Though aggregate, or weighted mean, productivity levels are interesting, researchers are also interested in the distribution of the unit-specific levels PRODkt (k ∈ Kt ), and the change of such distributions over time. A good example is Bartelsman and Dhrymes (1998), and a survey of recent research is provided by Bartelsman and Wolf (2018). One of the main aims of the MultiProd project of the OECD is providing tools to understand productivity dispersion (Berlingieri et al. 2017).27 Given the relative size measures θ kt —which are adding up to 1—a natural question is whether high or low productivity of a unit goes together with high or low size. Are big firms more productive than small firms? Or are the most productive firms to be found among the smallest? Questions multiply when the time dimension is taken into account. Does the ranking of a particular production unit in the productivity distribution sustain through time? Are firms ranked somewhere in a particular period likely to rank higher or lower in the next period? Is there a relation with the age, however determined, of the production units? Do the productivity distributions, and the behaviour of the production units, differ over the industries? When it comes to size a natural measure to consider is the covariance of weights t and levels. Let #(Kt ) be the number of units in Kt , let PROD ≡  productivity kt t t PROD /#(K ) be the unweighted mean of the productivity levels, and let k∈K θ¯ t ≡ k∈Kt θ kt /#(Kt ) = 1/#(Kt ) be the unweighted mean of the weights. One then easily checks that 

t t (θ kt − θ¯ t )(PRODkt − PROD ) = PRODt − PROD .

(5.82)

k∈Kt

This is a particular instance of a general relation derived by the statistician Bortkiewicz in 1923/24. Bortkiewicz showed that the difference between two differently weighted means has the form of a covariance. Wellknown, interesting applications can be found in index number theory (see Balk 2008). Olley and Pakes (OP) (1996, 1290) rearranged this relation to the form t

PRODt = PROD +



t (θ kt − θ¯ t )(PRODkt − PROD ),

(5.83)

k∈Kt

27 von

Brasch et al. (2020) studied the role of measurement error as an explanatory factor. They used an unbalanced panel of Norwegian food manufacturing establishments over the period 2000– 2014. Productivity was thereby defined as the logarithm of undeflated gross-output based simple labour productivity; that is, in our notation, ln(R kt /Lkt ). Their findings indicated that about 1% of measured productivity dispersion was attributable to measurement error. This small percentage could be due to the fact that their data were administrative rather than survey-based.

5.9 The Olley-Pakes Decomposition

151

and provided an interpretation which has been repeated, in various forms, by many researchers.28 The interpretation usually goes like this: There is some event (say, a certain technological innovation or some other shock) that gives rise to t a productivity level PROD . Next, this productivity level is transformed into an t aggregate level PROD by means of a mechanism called reallocation, the extent of which is measured by the covariance term in expression (5.83). So it seems that the aggregate productivity level PRODt is ‘caused’ by two factors, a productivity shock and a reallocation.29 We might call this the Olley-Pakes fallacy, because there are not at all two factors. The Bortkiewicz expression (5.82) is a mathematical identity: reallocation, defined as a covariance, is identically equal to the difference of two means, a weighted and an unweighted one. All that expression (5.83) does is featuring the unweighted mean rather than the weighted mean as the baseline variable. We need not dispute the usefulness of studying time-series or cross-sections of covariances such as seen at the left-hand side of expression (5.82).30 Notice that by replacing PROD by ln PROD or 1/PROD one obtains a geometric or harmonic variant respectively.31 Additional insight can be obtained when one replaces productivity levels PRODkt by productivity changes, measured as differences PRODk1 − PRODk0 or percentage changes ln(PRODk1 /PRODk0 ). As a descriptive device this is wonderful, especially for comparing ensembles (industries, economies)—see for instance Lin and Huang (2012) where such covariances are regressed on several background variables. In the cross-country study of Bartelsman et al. (2013) withinindustry covariances between size and productivity play a key role.32 The OP decomposition, expression (5.83), can of course be used to decompose aggregate productivity change PROD1 − PROD0 into two terms, the first being 1 0 PROD − PROD , and the second being the difference of two covariance terms. But then we are unable to distinguish between the contributions of exiting, continuing, and entering production units. Thus, it is advisable to restrict the OP decomposition to the continuing units, and substitute into expression (5.62). Doing this results in the following expression,

is straightforward to generalize the OP decomposition to the case where the ensemble Kt consists of a number of disjunct groups. See Appendix C. Collard-Wexler and de Loecker (2015) considered a case of two groups. 29 In Foster et al. (2001)’s article the OP decomposition, expression (5.83), was called method 3. On the accuracy of estimating the two factors from sample data, see Hyytinen et al. (2016). 30 An instructive example was provided by Karagiannis and Paleologou (2018). 31 The geometric variant was used by Gu (2019). 32 See also the special issue on “Misallocation and Productivity” of the Review of Economic Dynamics 16(1)(2013). There appears to be no unequivocal definition of ‘misallocation’. In Berlingieri et al. (2017) at least three different concepts are discussed. 28 It

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5 Dynamics: The Bottom-Up Approach

PROD1 − PROD0 =   1 01 ( θ k1 ) PRODN − PRODC 1 k∈N 1

C 01 1

C 01 0

− PROD + PROD  C 01 1 + (θ˜ k1 − 1/#(C 01 ))(PRODk1 − PROD ) k∈C 01

−



(θ˜ k0 − 1/#(C 01 ))(PRODk0 − PROD

k∈C 01

−(



C 01 0

)

 0 01 θ k0 ) PRODX − PRODC 0 ,

(5.84)

k∈X 0

C 01 t

where PRODC t is the weighted mean productivity level and PROD ≡  kt /#(C 01 ) is the unweighted mean productivity level of the continuing PROD 01 k∈C units at period t (t = 0, 1); #(C 01 ) is the number of those units. This then is the decomposition proposed by Melitz and Polanec (2015). Their paper contains an interesting empirical comparison of the GR method in expression (5.61), the FHK method in expression (5.50), and the extended OP method in expression (5.84). Hansell and Nguyen (2012) compared the BG method in expression (5.61), the DF method in expression (5.65), and the extended OP method in expression (5.84). Again, their overall conclusion on Australian data concerning the 2002–2010 period was that the “dominant source of labour productivity growth in manufacturing and professional services is from within firms.” Expression (5.84) figures, under the name ‘dynamic OP method’ in OECD’s MultiProd project (Berlingieri et al. 2017). Wolf (2011, 21–25), see also Bartelsman and Wolf (2014), used the OP decomposition to enhance the GR decomposition. By substituting expression (5.83) into expression (5.61), with a = (PROD0 + PROD1 )/2, one obtains 01

PROD1 − PROD0 =

 PROD0 + PROD1 k1 N1 θ ) PROD − ( 2 1 k∈N

+

 θ k0 + θ k1  PRODk1 − PRODk0 2 01

k∈C

+

 k∈C



0

PRODk0 + PRODk1 PROD + PROD − (θ k1 − θ k0 ) 2 2 01

1

5.10 The Choice of Weights

−



(θ

−θ )

k1

k0

k∈C 01

(θ

k∈K1

−(



0 (θ k0 − θ¯ 0 )(PRODk0 − PROD )

k∈K0



+

153



k∈X 0

k0

k1

 1 1 k1 ¯ − θ )(PROD − PROD ) /2



θ ) PROD

X0

PROD0 + PROD1 − 2

.

(5.85)

As one sees, the original GR ‘between’ term, the third right-hand side term in expression (5.61), is split into two parts. The first part, which is the third right-hand side term in expression (5.85), is relatively easy to understand: it is still a covariance between size changes and mean productivity levels. The second part, which is the fourth right-hand side term in expression (5.85), is far more complex. This part can  be rewritten as ( k∈N 1 θ k1 − k∈X 0 θ k0 ) times a mean covariance (of size and productivity level). It is unclear how this could be interpreted.

5.10 The Choice of Weights The question which weights θ kt are appropriate when a choice has been made as to the productivity levels PRODkt (k ∈ Kt ) has received some attention in the literature. Given that somehow PRODkt is output divided by input, should θ kt be outputor input-based? And how is this related to the type of mean—arithmetic, geometric, or harmonic? The literature does not provide us with definitive answers.33 Indeed, as long as one stays in the bottom-up framework it is unlikely that a convincing answer can be obtained. We need the complementary top-down view. A bit formally, the problem can be posed as follows. Generalizing the three definitions used in Sections 5.5 and 5.7, aggregate productivity is a weighted ‘mean’ of individual productivities PRODt ≡ M(θ kt , PRODkt ; k ∈ Kt ),

33 As

(5.86)

Karagiannis (2013) showed, the issue is not unimportant. He considered the OP decomposition in expression (5.83) on Greek cotton farm data. Output and input shares were used to weigh total factor productivity and labour productivity levels. The covariances turned out to be significantly different. An earlier example was provided by van Beveren (2012), using firm-level data from the Belgian food and beverage industry. De Loecker and Konings (2006) noted that there is no clear consensus on the appropriate  weights (shares) that should be used. In their work they used employment based shares Lkt / k Lkt to weigh value-added based total factor productivity indices QkVA (t, b)/QkKL (t, b).

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5 Dynamics: The Bottom-Up Approach

where the ‘mean’ M(.) can be arithmetic, geometric, or harmonic; the weights θ kt may or may not add up to 1; and PRODkt can be value-added based TFPRODVA , LPRODVA , or SLPRODVA , as defined in Section 5.4.2, or the analogous gross-output based TFPRODY or SLPRODY (see Section 9.5). The task then is: find the set of weights such that PRODt = PRODK t ; t

(5.87)

that is, such that aggregate productivity can be interpreted as productivity of the aggregate. It is clear that there are a number of options here, which will be discussed in Chapter 9.

5.11 Conclusion The main lessons can be summarized as follows: 1. Generically, productivity is defined as output over input. Yet most, if not all, empirical studies are not about productivity as such, because there is contamination by price effects at the input and/or at the output side of the production units considered. In many sectoral studies the available deflators are more or less deficient; for instance, value added is single-deflated instead of double-deflated. In almost all microdata studies there are simply no firmor plant-specific deflators available and higher-level substitutes must be used instead. All this may or may not matter at the aggregate (industry or economy) level, but it does matter when it comes to judging the contribution of specific (sets of) production units to aggregate productivity (change). 2. Economists appear to have a preference for working with levels; for example, with concepts such as real value added. It is good to realize, as pointed out in Sect. 5.4, that a level actually is a long-term index. And this implies that there is always some, essentially arbitrary, normalization involved. For instance, there is a time period for which real value added equals nominal value added; or, there is a period for which by convention total factor productivity is identically equal to 1. 3. Essentially the bottom-up approach consists in aggregating micro-level productivities with help of some set of size-related weights and then decomposing aggregate productivity change into contributions of (specific sets of) continuing, entering, and exiting units. We have seen that there is a large number of such decompositions available. Because of its symmetry and its natural benchmarks for exiting and entering production units the Diewert-Fox decomposition, defined by expression (5.65), is a good choice. If the weights are composite, then the Diewert-Fox-Balk decomposition, defined by expressions (5.67), (5.68) and (5.70), may be used.

Appendix A: Reinsdorf’s Extension of the GR Method

155

4. Beware of the covariance, so-called “reallocation”, terms; for example, in expressions (5.50), (5.60), or (5.83). They are statistical artefacts and there is not necessarily some underlying economic process involved. 5. In the bottom-up approach not every combination of micro-level productivities, weights, and aggregator function leads to a nice interpretation of aggregate productivity as productivity of the aggregate. The complementary top-down approach should be our guide here. The connection between the two approaches is the subject of Chapter 9.

Appendix A: Reinsdorf’s Extension of the GR Method The extension, proposed by Reinsdorf (2015), concerns the case where there is neither exit nor entry; that is K0 = K1 = C 01 . Typically, this is the situation of an economy consisting of a fixed set of industries. Then the GR method, expression (5.61), reduces to PROD1 − PROD0 =  θ k0 + θ k1  PRODk1 − PRODk0 2 01 k∈C

+

 k∈C 01

PRODk0 + PRODk1 −a , −θ ) 2

(θ

k1

k0

(5.88)

where a is an arbitrary scalar. A rather natural choice is a = (PROD0 + PROD1 )/2, the overall two-period mean aggregate productivity. The growth rate of aggregate productivity is obtained by dividing both sides of Eq. (5.88) by base period aggregate productivity, PROD0 , PROD1 − PROD0 = PROD0  θ k0 + θ k1 PRODk0 PRODk1 − PRODk0 PROD0 PRODk0

 PRODk0 + PRODk1 PROD0 + PROD1 k1 k0 + (θ − θ ) − , 2PROD0 2PROD0 01 k∈C 01

2

k∈C

which can be written as PROD1 − PROD0 = PROD0

(5.89)

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5 Dynamics: The Bottom-Up Approach

 1 PRODk0 PRODk1 − PRODk0 (θ k1 /θ k0 + 1)θ k0 2 PROD0 PRODk0 01

k∈C

+

 PRODk0   1 (θ k1 /θ k0 − 1) θ k0 PRODk1 /PRODk0 + 1 0 2 PROD 01

k∈C

  − θ k0 PROD1 /PROD0 + 1 .

(5.90)

Like in Section 5.5.2 we consider the case of simple labour productivity; that is, kt ≡ for each production unit k ∈ C 01 its productivity (level) is defined  as PROD k kt kt kt SLPRODVA (t, b), and its weight as its labour share, θ ≡ L / k∈C 01 L . One immediately checks that θ k0

PRODk0 RVAk (0, b) (k ∈ C 01 ). =  k (0, b) RVA PROD0 01 k∈C

(5.91)

If the reference period of all the deflators is selected as being b = 0, then, by definition, RVAk (0, 0) = VAk0 , and the real-value-added shares reduce to nominalvalue-added shares, θ k0

PRODk0 VAk0 k0 = ≡ sVA (k ∈ C 01 ).  k0 PROD0 VA k∈K

(5.92)

Expression (5.90) then reduces to PROD1 − PROD0 = PROD0 k1 k0  1 k0 PROD − PROD (θ k1 /θ k0 + 1)sVA k0 2 PROD 01 k∈C

+

   1 k0 (θ k1 /θ k0 − 1) sVA PRODk1 /PRODk0 + 1 2 01

k∈C

  − θ k0 PROD1 /PROD0 + 1 ,

(5.93)

which corresponds to Reinsdorf’s (2015) equations (24)–(25). The first term at the right-hand side of expression (5.93) was called the ‘within-industry productivity effect’, and the second term was called the ‘reallocation effect’. Recall that the identity holds for any functional form of the value-added based, production-unit specific deflators. Let now the joint reference period be selected as b = 1. By definition, k (0, 1). For any well-defined price index P (.) there exists RVAk (0, 1) = VAk0 /PVA ∗ a price index P (.) such that 1/P (0, 1) = P ∗ (1, 0). And if P (.) satisfies the Time

Appendix A: Reinsdorf’s Extension of the GR Method

157

Reversal Test then P ∗ (.) is identically equal to P (.). Consequently, the real-valueadded shares appear to be price-updated nominal-value-added shares, θ k0

∗k (1, 0) VAk0 PVA PRODk0 k01 ≡ sVA = (k ∈ C 01 ).  k0 ∗k PROD0 k∈C 01 VA PVA (1, 0)

(5.94)

Notice that in general these price-updated shares differ from the comparison period’s k01 = s k1 ≡ VAk1 /  k1 01 nominal-value-added shares; that is sVA k∈C 01 VA (k ∈ C ). VA Additivity of real value added holds if and only if 

∗k VAk0 PVA (1, 0) = (

k∈C 01



∗C VAk0 )PVA (1, 0), 01

(5.95)

k∈C 01

∗C (.) is a value-added based price index appropriate for the aggregate where PVA production unit C 01 . Then the price-updated shares can be simplified to 01

k01 k0 = sVA sVA

∗k (1, 0) PVA ∗C (1, 0) PVA 01

(k ∈ C 01 ),

(5.96)

and expression (5.90) reduces to PROD1 − PROD0 = PROD0 ∗k k1 k0  1 k0 PVA (1, 0) PROD − PROD (θ k1 /θ k0 + 1)sVA 01 2 PRODk0 P ∗C (1, 0) 01 k∈C

+

VA

∗k    1 k0 PVA (1, 0) (θ k1 /θ k0 − 1) sVA PRODk1 /PRODk0 + 1 01 2 P ∗C (1, 0) 01

k∈C

VA

  − θ k0 PROD1 /PROD0 + 1 ,

(5.97)

which corresponds to Reinsdorf’s (2015) equation (26). Though in the derivation of expression (5.97) the assumption of additivity of real value added appears to be required, there is no restriction on the functional form of the value-added based, production-unit specific deflators. Superficially, expressions (5.93) and (5.97) might look like being two different decompositions of the same entity. This, however, is not the case. The decomposition in expression (5.93) is based on productivities with reference period b = 0, whereas the decomposition in expression (5.97) is based on productivities with reference period b = 1. Nevertheless, Reinsdorf (2015) proposes to take a convex combination of these two decompositions.

158

5 Dynamics: The Bottom-Up Approach

In the empirical illustration, on United States data concerning the years 1998– 2012, a simplification of Reinsdorf’s decomposition method was compared with the GEAD-Diewert method. As expected, the differences appear to be in the industrial details.

Appendix B: Exercises on the Netherlands Manufacturing Industry, 1984–1999 This Appendix summarizes Balk and Hoogenboom-Spijker (2003). The data for this study came from the production surveys. These annual surveys contain detailed information on revenue and cost components of private firms; in particular, gross output, value added, capital input cost, labour cost, intermediate input cost, and number of employees. The statistical unit in the production surveys is the firm (kind-of-activity unit, establishment), considered to be the actual agent in the production process, characterised by its autonomy with respect to that process and by the sale of its goods or services to the market. A firm can consist of one or more juridical units or can be part of a larger juridical unit. Firms are classified according to their main economic activity. For 1984–1986 more data were available than for the years thereafter, since the observation threshold was changed in 1987. Prior to this year all firms with 10 or more employees were surveyed, while from 1987 all firms with at least 20 employees were surveyed. Firms with less employees were sampled. In this study bilateral comparisons were restricted to firms with 20 or more employees. The focus is on firms of the Manufacturing industry, consisting of the 2-digit industries 15–37 of the Standard Industrial Classification (SBI; an extension of NACE Rev. 1, and corresponding with ISIC Rev. 3.1) used in the Netherlands. There were no data for SBI 36631, social job creation. The industrial classification has been changed in 1993. This caused a break in the data series and led to some difficulties in finding appropriate deflators for the years prior to 1993. Because of that change data for 1992 were classified in two ways. Nominal gross output and value added were deflated by producer output price index numbers (PPI) for total turnover. Where available, the indices at the threedigit level of the Netherlands’ SBI were used, otherwise those at the two-digit level. To assign these sectoral price indices to firms, one must know to which industry a firm belongs. This can change through time, however. The pragmatic solution was, that per firm the industry of the comparison period was taken, unless there was no observation in that period. Then the industry of the base period was taken. The cost of energy, materials, and services was deflated by producer input price index numbers for total expenditures at the two-digit level of the SBI classification. Since the production surveys do not contain data on the capital stock, depreciation

Appendix B: Exercises on the Netherlands Manufacturing Industry, 1984–1999

159

cost was used as input variable. The nominal values of depreciation cost have not been deflated. For each year, firms with an incomplete data record and/or zero or negative values were deleted from the database. In addition, firms with unreasonable high or low profitabilities were deleted. The information in the data records allowed, where possible, to link the data over the years. After linking two adjacent periods, the firms which appear to be outlier in either one of the periods and the firms with incomplete data in either one of the two periods were deleted. The entry or exit status of a firm was determined from the remaining data. If a firm occured with data in a base period but not in a comparison period, then this firm was defined as an exiting firm. If a firm did not occur with data in a base period but did in a comparison period, then this firm was called an entering firm. The disadvantage of this approach is that the entry and exit sets are polluted with firms which were sampled in only one of the two periods. It would be better to define exit and entry from a business register. Register-based data allow firms to be tracked through time, because addition or removal of firms from the register usually reflects the entry and exit of firms in the ‘real world’. The pairs of time periods studied ranged from 1984–1985 to 1998–1999. The numbers of continuing firms, entering, and exiting firms hovered about 5000, 500, and 400, respectively. The main suite of decompositions was concerned with change of aggregate valueadded based TFP levels, defined as PRODt ≡



θ kt TFPRODkVA (t, b) =

k∈Kt



θ kt

k∈Kt

RVAk (t, b) k (t, b) XKL

,

(5.98)

where real value added is defined by expression (5.23), and real primary input is defined by expression (5.25). Anytime, two adjacent periods were considered, t = 0, 1, and the reference period was chosen as b = 0. The various elements of this expression will now be discussed. Real value added is nominal value added divided by a suitable price index number. Real primary input is nominal primary input cost divided by another suitable price index number. However, not all the theoretically necessary values and index numbers were available. Also the fact that there are entering firms in the comparison period (t = 1) necessitates an adaptation of the definitions. Obviously, for entering firms base period (t = 0) information does not exist. Therefore, to obtain commensurability with the real values of the continuing firms, comparison period value added of an entering firm was deflated by a suitable low-level PPI, and the same was carried out with capital and labour cost. Thus, operationally, the following definitions were used: TFPRODkVA (0, 0) ≡ TFPRODkVA (1, 0) ≡

VAk0 k0 CKL

for k ∈ C 01 ∪ X 0

VAk1 /PPI k (1, 0) k0 Qk (1, 0) CKL KL

for k ∈ C 01

(5.99)

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5 Dynamics: The Bottom-Up Approach

≡

VAk1 /PPI k (1, 0) k1 /PPI k (1, 0) CKL

for k ∈ N 1 ,

where PPI k (1, 0) denotes a producer price index number for the lowest-level industry group to which firm k belongs. Notice that, for the continuing firms, real primary input of period 1 is calculated as nominal primary input cost of period 0 times an input quantity index. The firm-specific quantity index was defined as  QkKL (1, 0)

≡

Lk1 Lk0

α k01

k1 CK

1−α k01

k0 CK

(k ∈ C 01 ),

(5.100)

kt is the depreciation cost of firm k where Lkt is the number of employees and CK k01 in period t. The exponents α are defined as mean (over two periods) labour cost shares; that is,

CLk1 1 CLk0 k01 α ≡ + k1 (5.101) (k ∈ C 01 ). k0 2 CKL CKL

The weights θ kt were defined as real input shares; that is, θ k0 ≡  θ k0 ≡  ≡ 

k0 CKL k∈C 01 ∪X 0

k0 CKL

for k ∈ C 01 ∪ X 0

(5.102)

k0 Qk (1, 0) CKL KL for k ∈ C 01 k0 Qk (1, 0) +  k1 /PPI k (1, 0) C C 01 1 k∈C k∈N KL KL KL k1 /PPI k (1, 0) CKL for k ∈ N 1 . k0 Qk (1, 0) +  k1 /PPI k (1, 0) C C k∈C 01 KL KL k∈N 1 KL

For each pair of adjacent years PROD1 − PROD0 was decomposed according to the five methods discussed earlier in this chapter. Respectively, • • • • • •

the BHC method, defined by expression (5.48), with a = 0; the dual BHC method, defined by expression (5.49), with a = 0; the FHK method, defined by expression (5.50), with a = PROD0 ; the dual FHK method, defined by expression (5.60), with a = PROD1 ; the GR method, defined by expression (5.61), with a = 0; the GR method, defined by expression (5.61), with a = (PROD0 + PROD1 )/2.

The decompositions were then turned into contributions to the percentage change ((PROD1 − PROD0 )/PROD0 ) × 100%, and averaged over time. The results are summarized in Table 5.1. It is straightforward to check that the mean of the BHC and dual BHC decompositions equals the GR decomposition (with a = 0), and that the mean of the FHK and dual FHK decompositions equals the GR decomposition

Appendix B: Exercises on the Netherlands Manufacturing Industry, 1984–1999

161

Table 5.1 Netherlands Manufacturing industry, 1984–1999. Mean annual value-added based TFP change (percentage) and decomposition

Source Total Within Between Cross Entry Exit

BHC a=0 2.12 2.16 0.49

Dual BHC a=0 2.12 1.62 1.03

6.88 −7.41

6.88 −7.41

FHK a = PROD0 2.12 2.16 0.48 −0.54 0.06 −0.04

Dual FHK a = PROD1 2.12 1.62 −0.04 0.54 −0.11 0.11

GR a=0 2.12 1.89 0.76 6.88 −7.41

GR a=

PROD0 +PROD1 2

2.12 1.89 0.22 −0.03 0.04

(with a = PROD +PROD ). It is also clear that decompositions where the scalar a has 2 been set equal to 0 exhibit (in absolute value) larger contributions of the entry, exit, and between components at the expense of the within component. The study considered a number of alternatives, the results of which are here summarized: 0

1

• Instead of computing value-added based TFP levels as arithmetic means geometric means were used. The differences appeared to be negligible. • Variations in the definition of the exponents α k01 appeared to be inconsequential. • The weights θ kt can be based on numbers of employees Lkt or on real output VAkt /PPI k (t, 0). Though the actual outcomes changed, the overall picture did not change. • Instead of value-added based TFP levels, the analysis can be based on labour productivity levels RVAk (t, b)/XLk (t, b) and weights as real labour input. Again, the overall picture did not change much. • For a number of years it was possible to compare value-added based results to gross-output based results. The differences were as expected. • For a number of years it was also possible to supplement exit and entry as defined from the production survey data with information about entry and exit flowing from the business register. It turned out that, as result of more precise definitions, for those years the entry, exit, and between components of aggregate productivity change almost vanished. . The study of Balk and Hoogenboom-Spijker (2003) was continued and extended by Polder et al. (2018). The continuation concerned the years 2007–2012. The extension concerned a broadening of the scope from Manufacturing to all industries; a more precise determination of the status of production units by using business register information; and special attention for the ICT intensity of the production units. The productivity measure used was simple value-added based labour productivity; the weights were based on numbers of full-time equivalent employees; and the aggregation method was the GR method, defined by expression (5.61), with a = (PROD0 + PROD1 )/2.

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5 Dynamics: The Bottom-Up Approach

As far as Manufacturing was concerned the results of both studies were consistent: aggregate productivity change turned out to be mainly due to the ‘within’ component of continuing production units.

Appendix C: Generalization of the OP Decomposition We start by recalling some definitions. Let the aggregate (or mean) productivity level at period t be defined as the weighted arithmetic mean of the unit-specific t ≡ kt kt productivity levels; that is, PROD t θ PROD , where the weights add up k∈ K  kt to 1, k∈Kt θ = 1. t t the number of production units in the set Kt . Then PROD ≡  Let #(K ) denote kt t arithmetic mean of the unit-specific k∈Kt PROD /#(K ) is the unweighted  kt /#(Kt ) = 1/#(Kt ) is the unweighted productivity levels, and θ¯ t ≡ θ k∈Kt arithmetic mean of the weights. The Bortkiewicz relation is 

t t (θ kt − θ¯ t )(PRODkt − PROD ) = PRODt − PROD .

(5.103)

k∈Kt

Let now the set Kt consist of J disjunct subsets; that is, Kt =

J 

Kjt , Kjt ∩ Kjt  = ∅ (j = j  ).

j =1

Then the aggregate productivity level PRODt is a weighted mean of the subaggregate productivity levels PRODj t (j = 1, . . . , J ), PRODt =

J 

ϑ j t PRODj t ,

(5.104)

j =1

where ϑ j t ≡



θ kt and PRODj t ≡ J 1, . . . , J ). Notice that j =1 ϑ j t = 1. k∈Kjt



k∈Kjt

θ kt PRODkt /



k∈Kjt

θ kt (j =

t

In the same way the overall unweighted mean PROD appears to be a weighted jt mean of the subaggregate means PROD (j = 1, . . . , J ), t

PROD =

J  j =1

jt

#(Kjt )PROD /#(Kt ),

(5.105)

Appendix C: Generalization of the OP Decomposition

163

 jt t kt t where PROD ≡ k∈Kjt PROD /#(Kj ) (j = 1, . . . , J ). Notice that #(K ) = J t j =1 #(Kj ). By substituting expressions (5.104) and (5.105) in the Bortkiewicz relation, expression (5.103), we obtain covar(Kt ) ≡



t (θ kt − θ¯ t )(PRODkt − PROD ) =

k∈Kt J 

ϑ j t PRODj t −

j =1 J 

J 

jt

#(Kjt )PROD /#(Kt ) =

j =1

J    jt jt + ϑ j t − #(Kjt )/#(Kt ) PROD = ϑ j t PRODj t − PROD

j =1 J  j =1

j =1

ϑ j t covar(Kjt ) +

J   jt ϑ j t − #(Kjt )/#(Kt ) PROD ,

(5.106)

j =1

where in the final step the Bortkiewicz relation was applied at the subset level. Thus the overall covariance between productivity levels and weights, covar(Kt ), can be decomposed into two parts: a weighted mean of subset covariances (aka the within-groups effect), and a covariance of unweighted subset productivity levels and relative subset weights (aka the between-groups effect). The decomposition in expression (5.106) is, however, not unique. The alternative is  t covar(Kt ) ≡ (θ kt − θ¯ t )(PRODkt − PROD ) = k∈Kt J  J    #(Kjt )/#(Kt ) covar(Kjt ) + ϑ j t − #(Kjt )/#(Kt ) PRODj t , j =1

(5.107)

j =1

but also a mean of the two decompositions could be contemplated. t ), to Expression (5.106) can be used to compare the overall covariance, J covar(K t  j t any of the subset covariances covar(Kj  ) (j = 1, . . . , J ). As j =1 ϑ = 1, we immediately see that covar(Kt ) − covar(Kjt  ) = J  j =1

J    jt ϑ j t − #(Kjt )/#(Kt ) PROD . ϑ j t covar(Kjt ) − covar(Kjt  ) + j =1

(5.108)

164

And, as

5 Dynamics: The Bottom-Up Approach

 j t − #(Kt )/#(Kt ) ϑ = 0, in the between-groups component j =1 j

J

of expression (5.108) the unweighted subset productivity levels PROD j t

jt

jt

may be

replaced by relative levels PROD − PROD . The resulting expression plays a big role in the analysis of Maliranta and Määttänen (2015); there called “augmented OP decomposition method”. Finally, by substituting expression (5.106) in expression (5.83), and using again J  j t t t the fact that j =1 ϑ − #(Kj )/#(K ) = 0, we obtain t

PROD = PROD + t

J 

ϑ j t covar(Kjt )

(5.109)

j =1

+

J    jt t ϑ j t − #(Kjt )/#(Kt ) PROD − PROD . j =1

This could be seen as a generalization of the original OP decomposition. The reallocation term is split into two terms, measuring intra-subset and inter-subset reallocation, respectively.

Chapter 6

The Top-Down Approach 1: Aggregate Output and Simple Labour Productivity Indices

6.1 Introduction Important economy-wide welfare measures are real GDP, real GDP per capita, and the development thereof. Real GDP is just the real value added produced by an economy, and it turns out that real GDP per capita is directly linked, via a number of demographic and labour force characteristics, to real GDP per unit of labour. A stylized fact is that the development of GDP per capita is dominated by the development of GDP per unit of labour, that is, by simple labour productivity.1 Aggregate output or productivity growth/decline is the outcome of numerous actions at the level of individual production units. To guide industry policy it is insufficient to relay only on statistics of aggregate behaviour, but it is important to be able to dissect those measures into contributions of lower level entities. For instance, one wants to see which sectors are primarily responsible for growth/decline at the aggregate level. Should industry policy be directed to stimulate particular sectors and neglect the rest? Should stimulants be administered to a sector’s input or output side, or both? Dissecting is also important in an international perspective. Similar growth rate figures can be turned out by differently structured economies, or different growth rate figures by similarly structured economies. In all such cases one should be able to delve underneath the surface of aggregate statistics. This chapter is about ex post accounting, and not about the mechanisms responsible for particular macro-economic outcomes. Yet, ex post statistics and decompositions provide insightful information for a number of purposes. In the literature one finds a number of methods by which aggregate output and/or labour productivity change can be decomposed into, say, industry contributions.

1 See

Marattin and Salotti (2011). Gandy and Mulhearn (2021) developed a simple method to adjust, at the economy level, conventionally measured GDP per hour for unemployment. The result provides an answer to a ‘what if?’ question and requires some assumptions. © The Author(s), under exclusive license to Springer Nature Switzerland AG 2021 B. M. Balk, Productivity, Contributions to Economics, https://doi.org/10.1007/978-3-030-75448-8_6

165

166

6 The Top-Down Approach 1: Aggregate Output and Simple Labour Productivity. . .

Section 6.2 introduces to and reviews the basic features of such methods in a discrete time periods framework. By way of example, and using the notational framework of this book, a recent decomposition method is thereby reconstructed. This reconstruction reveals a number of things. First, it appears that the example method is asymmetric, in the sense that the Laspeyres, forward-looking perspective is used. This calls for the development of a symmetric alternative. Second, the example method expresses aggregate output change as a weighted product or sum of the components of subaggregate output change rather then subaggregate output change itself. This calls for the development of an alternative that links aggregate output change, subaggregate output change, and the components thereof. Section 6.3 provides two solutions, given by expressions (6.11) and (6.14). Next, we turn to the decomposition of aggregate simple labour productivity change and deliver a number of alternatives, the main ones being given by expressions (6.22) and (6.29). Though structurally similar, these alternatives differ from each other in details. It will also be pointed out that these alternatives correspond to formulas put forward in the literature where the language of continuous time was used, as in Nordhaus (2002), Stiroh (2002), and Karagiannis (2017). It is well known that the mathematics of continuous time and infinitesimal changes tends to obfuscate interesting features which emerge when one uses the framework of discrete time periods. All this is the topic of Sect. 6.4. Section 6.5 summarizes.

6.2 Conventional Decompositions For the basic accounting identities the reader is referred to Sect. 5.2. In this chapter it is assumed that the ensemble of production units, Kt , is static. Thus the superscript t can be deleted. In the National Accounts it is usual to measure (aggregate) output as (aggregate) real value added, which is calculated as nominal value added deflated by an appropriate price index relative to some reference period b. Thus, formally, K RVAK (t, b) ≡ VAKt /PVA (t, b)  K = VAkt /PVA (t, b) k∈K

=

 kt K R kt − CEMS /PVA (t, b),

(6.1)

k∈K

K (t, b) is the appropriate price index. Let there also be given price and where PVA quantity indices such that revenue ratios and intermediate input cost ratios can be split into price and quantity components; that is, for any two periods t and t  , we have

6.3 Symmetric Decompositions of Aggregate Output Change 

R kt /R kt = PRk (t, t  )QkR (t, t  ) 

kt kt k CEMS /CEMS = PEMS (t, t  )QkEMS (t, t  ).

167

(6.2) (6.3)

In this chapter we consider aggregate output change, going from an earlier period 0 (also called base period) to a later period 1 (also called comparison period). The forward-looking growth rate of aggregate output is naturally measured by (RVAK (1, b) − RVAK (0, b))/RVAK (0, b). Tang and Wang (2015) showed that this growth rate can be decomposed as the sum of six terms: the aggregate effect of output quantity change, intermediate input quantity change, relative output price change, relative intermediate input price change, the interaction effect of output quantity and price change, and the interaction effect of intermediate input quantity and price change. Thus, four main effects and two interaction effects. A simple reconstruction is provided in Appendix A. Let Lkt be a scalar measure of the quantity of labour used by production unit k during period t (such as hours of work, or full-time-equivalent jobs), and let  kt . Conventional the aggregate quantity of labour be defined as LKt ≡ L k∈K practice defines aggregate labour productivity as aggregate real value added divided by the total quantity of labour; that is, what we called simple labour productivity: K Kt SLPRODK VA (t, b) ≡ RVA (t, b)/L . Its forward-looking growth rate is measured K K by (SLPRODVA (1, b) − SLPRODVA (0, b))/ SLPRODK VA (0, b). Proceeding as in the previous case, Tang and Wang (2015) could decompose this growth rate into two sets of three terms: change of a relative-price-index-adjusted labour-input share, quantity-per-unit-of-labour change, and their interaction; one set for output, and another for intermediate input. A slightly different decomposition was proposed earlier by Tang and Wang (2004), discussed by Diewert (2010), and more recently by Dumagan (2013a). Appendix B briefly reviews this material. Diewert (2015b) could separate the relative-price effect from the labour-input effect, albeit approximately. The final decomposition, containing fifteen terms (five main effects and eight interaction effects), was applied by Tang and Wang (2020). The decompositions developed by Tang and Wang (2015) are inherently asymmetric, in the sense that everywhere the forward-looking perspective is used. Though the backward-looking perspective is equally warranted, numerically there will usually be differences between the two perspectives. However, combining the two perspectives by taking some average leads to unsatisfactory, complicated expressions. Thus, this chapter presents simple, symmetric alternatives, first for output change, then for labour productivity change.

6.3 Symmetric Decompositions of Aggregate Output Change As stated, we consider aggregate output change, going from an earlier period 0 to a later period 1. When this change is measured as ratio, then by using the powerful instrument of the logarithmic mean, the following symmetric decomposition

168

6 The Top-Down Approach 1: Aggregate Output and Simple Labour Productivity. . .

emerges (see Balk 2003a or 2008, 85–86): 

VAK1



VAk1 = exp ψ (1, 0) ln VAk0 VAK0 k∈K k

! ,

(6.4)

where LM

VAk1

VAk0

, VAK1 VAK0

(k ∈ K) ψ (1, 0) ≡  VAk1 VAk0 , k∈K LM VAK1 VAK0 k

and LM(.) denotes the logarithmic mean.2 Aggregate nominal value-added change, measured as ratio, is thus equal to a weighted geometric mean of individual nominal value-added changes. Each coefficient ψ k (1, 0) is the (normalized) mean share of production unit k in aggregate nominal value added. Notice that these coefficients add up to 1; that is, 

ψ k (1, 0) = 1.

(6.5)

k∈K kt For the decomposition of VAk1 /VAk0 (k ∈ K) recall that VAkt = R kt − CEMS for t = 0, 1. Recall also that there are price and quantity indices such that revenue ratios and intermediate input cost ratios can be split into price and quantity components; see expressions (6.2) and (6.3). Then each individual nominal value-added ratio can be decomposed as (see Chapter 2, Appendix B) k

k

PRk (1, 0)φR (1,0) QkR (1, 0)φR (1,0) VAk1 = (k ∈ K), k k k0 φEMS (1,0) k (1, 0)φEMS (1,0) Qk VA PEMS (1, 0) EMS

(6.6)

where the exponent φRk (1, 0) is defined as φRk (1, 0) ≡ LM(R k1 , R k0 )/LM(VAk1 , VAk0 ); that is, mean nominal revenue over mean nominal value added; and the expok (1, 0) is defined as φ k (1, 0) ≡ LM(C k1 , C k0 )/LM (VAk1 , VAk0 ); nent φEMS EMS EMS EMS that is, mean nominal intermediate input cost over mean nominal value added. It is important to observe that the only requirement put on the price and quantity indices for revenue and intermediate input cost is that they satisfy product relations

2 Recall that the logarithmic mean, for any two strictly positive real numbers a and b, is defined by LM(a, b) ≡ (a − b)/ ln(a/b) if a = b and LM(a, a) ≡ a. Recall also that (1, 0) is a shorthand notation for all the prices and quantities playing a role in value added of period 1 and 0, respectively.

6.3 Symmetric Decompositions of Aggregate Output Change

169

(6.2) and (6.3). Thus a particular functional form for a price index implies via these relations a definite form for the corresponding quantity index and vice versa. k1 , The concavity of the logarithmic mean implies that LM(R k1 , R k0 ) − LM(CEMS k0 k k k1 k0 CEMS ) ≤ LM(VA , VA ). Hence, φR (1, 0) − φEMS (1, 0) ≤ 1 (k ∈ K). When we define the unit k price index and quantity index of value added respectively as k

k PVA (1, 0) ≡

PRk (1, 0)φR (1,0) k

k (1, 0)φEMS (1,0) PEMS

(k ∈ K)

(6.7)

(k ∈ K),

(6.8)

k

QkVA (1, 0) ≡

QkR (1, 0)φR (1,0) k

QkEMS (1, 0)φEMS (1,0)

then expression (6.6) can be simplified to VAk1 k = PVA (1, 0)QkVA (1, 0) (k ∈ K). VAk0

(6.9)

Technically seen, these value-added price and quantity indices are (generalized) Montgomery-Vartia (MV) indices. Substituting expression (6.6) into expression (6.4) delivers an expression which is perfectly symmetric: VAK1 VAK0



=



k

PRk (1, 0)φR (1,0)ψ

k (1,0)

k

k∈K

k (1, 0)φEMS (1,0)ψ PEMS

k (1,0)





k

QkR (1, 0)φR (1,0)ψ k

k∈K

k (1,0)

QkEMS (1, 0)φEMS (1,0)ψ

k (1,0)

.

(6.10) Notice that the exponent φRk (1, 0)ψ k (1, 0) can be interpreted as the share of nominal revenue of production unit k in aggregate nominal value added; or, as a so-called Domar weight. These weights, however, do not add up to 1. Similarly, the exponent k (1, 0)ψ k (1, 0) can be interpreted as the share of nominal intermediate input φEMS cost of unit k in aggregate nominal value added. These weights also do not add up to 1. Aggregate output change can now be defined as the quantity component of VAK1 /VAK0 ; hence as QK VA (1, 0) ≡ 



k

QkR (1, 0)φR (1,0)ψ k

k∈K

k (1,0)

QkEMS (1, 0)φEMS (1,0)ψ

k (1,0)

(6.11)

 k k Qk (1, 0)φR (1,0)ψ (1,0) . =  k∈K R k φEMS (1,0)ψ k (1,0) k Q (1, 0) k∈K EMS

The right-hand side of this expression enables us to decompose aggregate output change according to various viewpoints: the contribution of gross output and

170

6 The Top-Down Approach 1: Aggregate Output and Simple Labour Productivity. . .

intermediate input, the contribution of the production units making up the ensemble, or a combination of these. Switching to relative changes (aka growth rates), expression (6.11) delivers ln QK VA (1, 0) = 

φRk (1, 0)ψ k (1, 0) ln QkR (1, 0) −

k∈K



(6.12)

k φEMS (1, 0)ψ k (1, 0) ln QkEMS (1, 0),

k∈K





k (1, 0)ψ k (1, 0) ≤ 1. where k∈K φRk (1, 0)ψ k (1, 0) − k∈K φEMS Definition (6.11) is, however, not unique. As an alternative to expression (6.10) consider



k k   PRk (1, 0)ξR (1,0) QkR (1, 0)ξR (1,0) VAK1 = , (6.13) k (1,0) k (1,0) ξEMS ξEMS k k VAK0 k∈K PEMS (1, 0) k∈K QEMS (1, 0)

where the exponent ξRk (1, 0) is defined as  LM

ξRk (1, 0) ≡

R k1

R k0



, VAK1 VAK0

(k ∈ K),   k1 k1 k0   CEMS CEMS R R k0 − k∈K LM , , k∈K LM VAK1 VAK0 VAK1 VAK0

k (1, 0) as and the exponent ξEMS



k1 k0 CEMS CEMS , LM K1 VAK0 VA k

(k ∈ K). ξEMS (1, 0) ≡   k1 k1 k0   CEMS CEMS R R k0 − k∈K LM , , k∈K LM VAK1 VAK0 VAK1 VAK0 For the derivation of expression (6.13) the reader is referred to Appendix C.3 The exponent ξRk (1, 0) can be interpreted as the (normalized) mean share of nominal revenue of production unit k in aggregate nominal value added; or, again as a Domar k (1, 0) can be interpreted as the (normalized) weight. Similarly, the exponent ξEMS mean share of nominal intermediate input cost of unit k in aggregate  nominal value added. The difference of these exponents adds up to 1; that is, k∈K ξRk (1, 0) −  k k∈K ξEMS (1, 0) = 1.

3 Reinsdorf

and Yuskavage (2010) developed a similar formula in a slightly different context.

6.4 Symmetric Decompositions of Simple Labour Productivity Change

171

Based on expression (6.13), the quantity component of VAK1 /VAK0 , defining aggregate output change, is given by QK VA (1, 0)

≡





k

QkR (1, 0)ξR (1,0) k

k∈K

QkEMS (1, 0)ξEMS (1,0)



=

k ξRk (1,0) k∈K QR (1, 0) .  k (1,0) ξEMS k k∈K QEMS (1, 0)

(6.14)

Again, the right-hand side of this expression enables us to decompose aggregate output change according to various viewpoints: the contribution of gross output and intermediate input, the contribution of the production units making up the ensemble,4 or a combination of these. Switching to relative changes (aka growth rates) expression (6.14) delivers ln QK VA (1, 0) =

 k∈K

ξRk (1, 0) ln QkR (1, 0) −



k ξEMS (1, 0) ln QkEMS (1, 0),

k∈K

(6.15) where one should recall that the signed weights add up to 1. Notice that, though expressions (6.11) and (6.14), or (6.12) and (6.15), have the same structure, the exponents are different: ξRk (1, 0) = φRk (1, 0)× ψ k (1, 0) and k (1, 0) = φ k (1, 0) × ψ k (1, 0). The influence of these differences on the ξEMS EMS outcome will in general be immaterial. The difference between expressions (6.11) and (6.14) is that between a one-step and a two-step procedure. Expression (6.14) is the result of a one-step procedure: nominal revenue (or intermediate input cost) of unit k is immediately related to aggregate nominal value added. Expression (6.11) is the result of a two-step procedure: nominal revenue (or intermediate input cost) of unit k is first related to nominal value added of unit k, and then nominal value added of unit k is related to aggregate nominal value added. In a continuous time setup this difference disappears. However, when it then comes to empirical implementation, the difference reappears—as it were, via the backdoor.

6.4 Symmetric Decompositions of Simple Labour Productivity Change Recall that Lkt is a scalar measure of the quantity of labour used by production unit k during period t (such as hours of work, or full-time-equivalent jobs), and kt that LKt ≡ k∈K L . The measurement of simple labour productivity change starts from considering nominal value added per unit of labour and the development thereof. Applying expression (6.4) delivers

4 Thus

aggregate output change can reasonably be approximated by using data of the largest production units; see Gabaix (2011) on the granular residual.

172

6 The Top-Down Approach 1: Aggregate Output and Simple Labour Productivity. . .

ln

VAK1 /LK1

VAK0 /LK0 

= ln

VAK1

VAK0



VAk1 ψ (1, 0) ln VAk0 k∈K

−

k



LK 1 − ln LK 0



= 

ξLk (1, 0) ln

k∈K

(6.16)

Lk1 Lk0

 ,

where ψ k (1, 0) was defined earlier and ξLk (1, 0) is defined analogously as  ξLk (1, 0) ≡ 

LM

Lk1 Lk0 , LK1 LK0

k∈K LM





Lk1 Lk0 , LK1 LK0

(k ∈ K).

This definition implies that 

ξLk (1, 0) = 1.

(6.17)

k∈K

Next, we use expression (6.9) to obtain ln  k∈K

k ψ k (1, 0) ln PVA (1, 0) +



VAK1 /LK1 VAK0 /LK0

=

ψ k (1, 0) ln QkVA (1, 0) −

k∈K

(6.18)  k∈K

 ξLk (1, 0) ln

Lk1 Lk0

 .

Recall that unit k’s (value-added based) simple labour productivity index is defined as ISLPRODkVA (1, 0) ≡

QkVA (1, 0) (k ∈ K). Lk1 /Lk0

(6.19)

As above, let there be given price and quantity indices such that aggregate nominalvalue-added change can be decomposed as VAK1 VAK0

K = PVA (1, 0)QK VA (1, 0),

(6.20)

and define, analogous to expression (6.19), the aggregate (value-added based) simple labour productivity index by ISLPRODK VA (1, 0) ≡

QK VA (1, 0) . K L 1 /LK0

(6.21)

6.4 Symmetric Decompositions of Simple Labour Productivity Change

173

Expression (6.18) can then be rearranged in the following, analytically useful, form: ln ISLPRODK VA (1, 0)

=



k

ψ (1, 0) ln

k∈K

 k∈K

ψ

k

(1, 0) ln ISLPRODkVA (1, 0) +

k (1, 0) PVA

K (1, 0) PVA

+

(6.22)

   Lk1   k k −a , ψ (1, 0) − ξL (1, 0) ln Lk0

k∈K

 k   k where a is an arbitrary scalar. Recall that = 0. k∈K ψ (1, 0) − ξL (1, 0) Expression (6.22) decomposes aggregate simple labour productivity change into three components. The first component measures the aggregate effect of relative price change. K (1, 0) is defined Notice that this effect vanishes when the aggregate price index PVA k as a Sato-Vartia index of the unit-specific price indices PVA (1, 0) (k ∈ K). Thus one can argue that the relative price change effect is nothing but the consequence of using an inappropriate deflator for aggregate value added. The second component is a weighted mean of all the unit-specific simple labour productivity changes. The weights are the shares of the units in aggregate value added.   Setting a equal to the unweighted mean of ln Lk1 /Lk0 a nice interpretation of the third component emerges, namely as the covariance of labour quantity change and the gap between a unit’s value-added share and its labour share. This covariance is positive if a higher-than-average (lower-than-average) increase of the quantity of labour corresponds with a relatively big (small) gap between the unit’s share in value added and its labour share, the average gap being equal to zero. Due to the fact that the coefficients ψ k (1, 0) and ξLk (1, 0) add up to 1, the third component can also be written as

    Lk1   Lk1 /LK1 k k k −a = ψ (1, 0) − ξL (1, 0) ln ψ (1, 0) ln . Lk0 Lk0 /LK0 k∈K k∈K (6.23) This is a weighted mean of labour share ratios. Thus the covariance term in expression (6.22) can be interpreted as the effect of labour reallocation on aggregate simple labour productivity change. Expression (6.22) is the discrete-time counterpart of equation (7) of Stiroh (2002) and corresponds to equation (1) of Nordhaus (2002).5 The following alternative rearrangement of expression (6.18) is also worthwhile to consider:

5 The

Stiroh decomposition was applied by Timmer et al. (2010, 153).

174

6 The Top-Down Approach 1: Aggregate Output and Simple Labour Productivity. . .

ln ISLPRODK VA (1, 0)

=



k

ψ (1, 0) ln

k∈K



ξLk (1, 0) ln ISLPRODkVA (1, 0)+

k∈K

k (1, 0) PVA

K (1, 0) PVA

+

(6.24)

  ψ k (1, 0) − ξLk (1, 0) ln QkVA (1, 0) − b , k∈K

where b is an arbitrary scalar. The second component at the right-hand side of the equality sign in expression (6.24) is again a weighted mean of unit-specific simple labour productivity change, the weights now being equal to the labour shares. Setting b equal to the unweighted mean of ln QkVA (1, 0) the remainder term can be interpreted as the covariance of value-added quantity change and the gap between a unit’s value added share and its labour share. This covariance is positive if a higher-than-average (lower-than-average) increase of the quantity component of value added corresponds with a relatively big (small) gap between the unit’s share in value added and its labour share. Analogous to expression (6.23) the third component in expression (6.24) can be written as

   QkVA (1, 0) k k k k ψ (1, 0)−ξL (1, 0) ln QVA (1, 0)−b =− ξL (1, 0) ln , QK VA (1, 0) k∈K k∈K (6.25)  k k where ln QK k∈K ψ (1, 0) ln QVA (1, 0) is defined as the (two-stage) VA (1, 0) ≡ Sato-Vartia quantity index of aggregate value added. Thus the covariance term in expression (6.24) can be interpreted as the effect of real-value-added reallocation. Note the minus sign! Expressions (6.22) and (6.24) correspond to expression (20’) of Reinsdorf and Yuskavage (2010).6 Combining the first and third right-hand side components in expression (6.22), and using expression (6.23), we obtain ln ISLPRODK VA (1, 0) =  k∈K

6 Instead

ψ

k

(1, 0) ln ISLPRODkVA (1, 0) +

 k∈K

(6.26)

k (1, 0) k1 PVA L /LK1 ψ (1, 0) ln K PVA (1, 0) Lk0 /LK0 k

,

of nominal value added, Reinsdorf and Yuskavage (2010) started by considering value added expressed in artificial prices, defined as a weighted mean of period 0 and period 1 prices. K1 K0 The left-hand side of expression (6.18) can then be written as ln(QK VA (1, 0)/(L /L )), where QK (1, 0) is a value-added based aggregate output quantity index. The first term at the right-hand VA side of the said expression, however, not necessarily disappears.

6.4 Symmetric Decompositions of Simple Labour Productivity Change

175

which was called GEA-BMB by Dumagan (2013b). The labour reallocation term now contains the effect of relative price change between the units and the aggregate. The empirical examples provided by Dumagan (2013b) make clear that in- or exclusion of the relative price change effect matters. It appears that the individual Lk1 /LK1 Lk0 /LK0

contributions,

and

k (1,0) k1 PVA L /LK1 K (1,0) Lk0 /LK0 PVA

(k ∈ K), differ not only in magnitude

but occasionally also in sign. A third alternative emerges by substituting the definition of ln QkV A (1, 0) back into expression (6.18). This delivers ln ISLPRODK VA (1, 0)

=



k

ψ (1, 0) ln

k∈K



ψ k (1, 0)φRk (1, 0) ln QkR (1, 0) −

k∈K

 ξLk (1, 0) ln

k∈K

ψ

k

(1, 0)φRk (1, 0) ln

k∈K

+

K (1, 0) PVA

+

(6.27)

k ψ k (1, 0)φEMS (1, 0) ln QkEMS (1, 0) −

k∈K







k (1, 0) PVA



(ψ

k

Lk1 Lk0



QkR (1, 0) Lk1 /Lk0

=



k

ψ (1, 0) ln

k∈K

−



k (1, 0) PVA

K (1, 0) PVA

+

ψ

k

k (1, 0)φEMS (1, 0) ln

k∈K

k (1, 0)φRk (1, 0) − ψ k (1, 0)φEMS (1, 0) − ξLk (1, 0))

k∈K

QkEMS (1, 0) Lk1 /Lk0

  k1  L . ln Lk0

The first term at the right-hand side is the same relative-price-change term as we met earlier. The second term is a weighted sum of unit-specific gross-output based simple labour productivity indices, the weights being ψ k (1, 0)φRk (1, 0), that is Domar weights. The third term is a weighted sum of unit-specific indices of intermediate input deepening, the weights being the counterparts of the Domar weights, that is k (1, 0). Since φ k (1, 0) − φ k (1, 0) ≤ 1, the coefficients in the fourth ψ k (1, 0)φEMS EMS R k (1, 0) − ξ k (1, 0) ≤ ψ k (1, 0) − ξ k (1, 0), term ψ k (1, 0)φRk (1, 0) − ψ k (1, 0) φEMS L L and the discrepancy should usually be small. Thus the last term of expression (6.27) has the same structure as the last term of expression (6.22). Expression (6.27) was favoured by Karagiannis (2017). Expression (6.22) is, however, not unique. Instead of expression (6.6) one could use expression (6.13) to obtain ln

VAK1 /LK1 VAK0 /LK0

=

(6.28)

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6 The Top-Down Approach 1: Aggregate Output and Simple Labour Productivity. . .



ln

k

PRk (1, 0)ξR (1,0) k

k (1, 0)ξEMS (1,0) PEMS  k1   L k . − ξL (1, 0) ln Lk0

+

k∈K



ln

k∈K

k

QkR (1, 0)ξR (1,0) k

QkEMS (1, 0)ξEMS (1,0)

k∈K

Using expression (6.20) and the definition of a simple labour productivity index, the foregoing expression can be rearranged as ln ISLPRODK VA (1, 0)

=



ln

ξRk (1, 0) ln QkR (1, 0) −

k∈K



K (1, 0) + − ln PVA

k (1,0) k (1, 0)ξEMS PEMS

k∈K



k

PRk (1, 0)ξR (1,0)

k ξEMS (1, 0) ln QkEMS (1, 0) −

k∈K



(6.29) 

ξLk (1, 0) ln

k∈K

Lk1 Lk0

 .

This expression exhibits the same structure as expression (6.27), the weights in some terms being slightly different. If K ln PVA (1, 0)

=

 k∈K

ln

k

PRk (1, 0)ξR (1,0) k

k (1, 0)ξEMS (1,0) PEMS

,

that is, if the aggregate-value-added price index is equal to the generalized SatoVartia index of the price indices of the unit-specific gross-output and intermediateinput components, then expression (6.29) reduces to ln ISLPRODK VA (1, 0) = 

k ξRk (1, 0) − ξEMS (1, 0)

k∈K

+





QkR (1, 0) ln Lk1 /Lk0

+

 k∈K

(6.30)

k ξEMS (1, 0) ln

QkR (1, 0)

QkEMS (1, 0)

   Lk1   k − a . ξRk (1, 0) − ξEMS (1, 0) − ξLk (1, 0) ln Lk0

k∈K

   k (1, 0) − ξ k (1, 0) = 0. The first component Notice that k∈K ξRk (1, 0) − ξEMS L at the right-hand side of expression (6.30) is a weighted sum of all the unitspecific gross-output based simple labour productivity changes, the weights being (approximately equal to) the value-added shares of the units. The second component is the aggregate excess of gross-output quantity change over intermediate-inputs quantity change, the weights being the intermediate-inputs cost shares in aggregate value added. The third component expresses again the labour reallocation effect. The entire expression corresponds to expression (20) of Reinsdorf and Yuskavage (2010)

Appendix A: The Tang and Wang Method

177

and is a discrete-time version of expression (6) of Stiroh (2002), also discussed by Karagiannis (2017). The said expression was also used by Bosworth and Triplett (2007).

6.5 Conclusion When dealing with an ensemble of production units it is important to relate aggregate output change to output change at the level of the production units. An important measure of output change is real-value-added change. In this chapter we considered symmetric decompositions of aggregate real-value-added change. Two, slightly different, solutions are given by expressions (6.11) and (6.14). The difference is the result of using a two-step or a one-step procedure respectively. We proceeded by considering simple labour productivity change. The solutions are here basically given by expressions (6.22) and (6.29). Additional insight into the “causes” of aggregate simple labour productivity change can be obtained from expressions such as (6.22), (6.24), (6.27) and (6.30). Three of these expressions contain a labour reallocation term, also called Denison effect. As Bosworth and Triplett (2007) observe, the interpretation of such a term is not immediately intuitive. Their comment runs as follows: Consider a technological shock in industry A that raises (. . . ) labour productivity, and for the sake of the illustration we specify that technologies in other industries are unchanged. Unless the demand elasticity for industry A’s output is high, industry A will use fewer resources. If the released resources go to industries with lower productivity growth rates, the reallocation reduces aggregate and sector productivity rates (the direct rates). Reallocations thus provide a partial offset to the direct impact on the sector rates from industry A’s productivity gain.

Mathematically seen, the labour reallocation effect occurs to the extent that a unit’s k (1, 0) differs from its labour share value-added share ψ k (1, 0) or ξRk (1, 0) − ξEMS ξLk (1, 0). If all those shares would stay equal, there would be no reallocation effect. Finally, in all the simple labour productivity expressions labour quantities figure as scalars. When labour is more complex, so that indices rather than scalars must be used, these expressions can easily be generalized. All one has to do is replace each ratio Lk1 /Lk0 by a genuine quantity index QkL (1, 0), and each weight ξLk (1, 0) by production unit k’s share in aggregate labour cost.

Appendix A: The Tang and Wang Method In the National Accounts it is usual to measure (aggregate) output as (aggregate) real value added, which is calculated as nominal value added deflated by an appropriate price index relative to some reference period b. The formal definition is given by expression (6.1). Recall that there are price and quantity indices such that revenue ratios and intermediate input cost ratios can be split into price

178

6 The Top-Down Approach 1: Aggregate Output and Simple Labour Productivity. . .

and quantity components; see expressions (6.2) and (6.3). Real gross output is then defined by Y k (t, b) ≡ R kb QkR (t, b), real intermediate input use by k kb Qk ˜k (t, b) ≡ CEMS XEMS EMS (t, b), the relative gross output price index by PR (t, b) ≡ K (t, b), and the relative intermediate input price index by P˜ k (t, b) ≡ PRk (t, b)/PVA EMS k K (t, b). After substituting all these definitions into expression (6.1) it PEMS (t, b)/PVA appears that aggregate output can be written as RVAK (t, b) =

 k k P˜Rk (t, b)Y k (t, b) − P˜EMS (t, b)XEMS (t, b) .

(6.31)

k∈K

This is an important building block for what follows. Aggregate output change, going from an earlier period 0 to a later period 1, is naturally measured by RVAK (1, b) − RVAK (0, b). This can be written as RVAK (1, b) − RVAK (0, b) =  P˜Rk (1, b)Y k (1, b) − P˜Rk (0, b)Y k (0, b) −

(6.32)

k∈K

 k k k k P˜EMS (1, b)XEMS (1, b) − P˜EMS (0, b)XEMS (0, b) .

k∈K

The two parts at the right-hand side of this expression can be decomposed according to the Laspeyres, forward-looking perspective, yielding RVAK (1, b) − RVAK (0, b) =   P˜Rk (0, b) Y k (1, b) − Y k (0, b) k∈K

+



 Y k (0, b) P˜Rk (1, b) − P˜Rk (0, b)

k∈K

+

  P˜Rk (1, b) − P˜Rk (0, b) Y k (1, b) − Y k (0, b)

k∈K

−



k∈K

−



 k k k (0, b) XEMS (1, b) − XEMS (0, b) P˜EMS  k k k XEMS (0, b) P˜EMS (1, b) − P˜EMS (0, b)

k∈K

−

  k k k k P˜EMS (1, b) − P˜EMS (0, b) XEMS (1, b) − XEMS (0, b) .

k∈K

Switching to relative changes (forward-looking growth rates) delivers

(6.33)

Appendix A: The Tang and Wang Method

RVAK (1, b) − RVAK (0, b) RVAK (0, b)

179

=

(6.34)

 P˜ k (0, b)Y k (0, b) Y k (1, b) − Y k (0, b) R Y k (0, b) RVAK (0, b)

k∈K

+

 P˜ k (0, b)Y k (0, b) P˜ k (1, b) − P˜ k (0, b) R R R RVAK (0, b) P˜ k (0, b)

k∈K

+

R

 P˜ k (0, b)Y k (0, b) P˜ k (1, b) − P˜ k (0, b) Y k (1, b) − Y k (0, b) R R R Y k (0, b) RVAK (0, b) P˜ k (0, b)

k∈K

−

 P˜ k (0, b)Xk (0, b) Xk (1, b) − Xk (0, b) EMS EMS EMS EMS k∈K

−

RVAK (0, b)

k XEMS (0, b)

 P˜ k (0, b)Xk (0, b) P˜ k (1, b) − P˜ k (0, b) EMS EMS EMS EMS RVAK (0, b) P˜ k (0, b)

k∈K

−

R

EMS

 P˜ k (0, b)Xk (0, b) P˜ k (1, b) − P˜ k (0, b) Xk (1, b) − Xk (0, b) EMS EMS EMS EMS EMS EMS . k K k ˜ XEMS (0, b) RVA (0, b) (0, b) P

k∈K

EMS

Notice that, by using the definitions of P˜Rk (0, b), Y k (0, b), RVAK (0, b), and expression (6.2), it appears that P˜Rk (0, b)Y k (0, b)/RVAK (0, b) = R k0 /VAK0 , which is the base period share of nominal revenue of unit k in aggregate nominal value k (0, b)X k k0 K K0 added. Similarly it appears that P˜EMS EMS (0, b)/RVA (0, b) = CEMS /VA , which is the base period share of nominal intermediate input cost of unit k in aggregate nominal value added. Hence, these shares are independent of the reference period b. This, however, does not hold for the growth rates, contrary to the assertion of TW. Notice that, for instance, QkR (1, b) − QkR (0, b) Y k (1, b) − Y k (0, b) = , Y k (0, b) QkR (0, b) which in general stays dependent on reference period b data. The right-hand side of expression (6.34) consists of six terms: the first and the fourth give the aggregate effect of quantity change, the second and the fifth give the aggregate effect of relative price change, and the third and the sixth give the aggregate effect of the interaction of quantity and price change. The entire expression corresponds to expression (2) of Tang and Wang (2015). As noticed in the main text, this method can be extended to aggregate (simple) labour productivity change, but also to aggregate TFP change. Instead of decomposing aggregate output change according to the Laspeyresperspective, as in expression (6.33), one can use the Paasche, backward-looking

180

6 The Top-Down Approach 1: Aggregate Output and Simple Labour Productivity. . .

perspective. Then RVAK (1, b) − RVAK (0, b) =   P˜Rk (1, b) Y k (1, b) − Y k (0, b) k∈K

+



(6.35)

 Y k (1, b) P˜Rk (1, b) − P˜Rk (0, b)

k∈K

−

  P˜ k (1, b) − P˜ k (0, b) Y k (1, b) − Y k (0, b) R

k∈K

−



k∈K

−



R

 k k k (1, b) XEMS (1, b) − XEMS (0, b) P˜EMS  k k k XEMS (1, b) P˜EMS (1, b) − P˜EMS (0, b)

k∈K

+

  k k k k P˜EMS (1, b) − P˜EMS (0, b) XEMS (1, b) − XEMS (0, b) .

k∈K

Notice that the signs on the interaction terms in expression (6.35) differ from those in expression (6.33). Switching to backward-looking relative changes delivers RVAK (1, b) − RVAK (0, b) RVAK (1, b)

=

(6.36)

 P˜ k (1, b)Y k (1, b) Y k (1, b) − Y k (0, b) R Y k (1, b) RVAK (1, b)

k∈K

+

 P˜ k (1, b)Y k (1, b) P˜ k (1, b) − P˜ k (0, b) R R R RVAK (1, b) P˜ k (1, b)

k∈K

−

R

 P˜ k (1, b)Y k (1, b) P˜ k (1, b) − P˜ k (0, b) Y k (1, b) − Y k (0, b) R R R Y k (1, b) RVAK (1, b) P˜ k (1, b)

k∈K

−



R

k (1, b)X k P˜EMS EMS (1, b)

RVAK (1, b)

k∈K

−

 P˜ k (1, b)Xk (1, b) P˜ k (1, b) − P˜ k (0, b) EMS EMS EMS EMS RVAK (1, b) P˜ k (1, b)

k∈K

+

k k XEMS (1, b) − XEMS (0, b) k XEMS (1, b)

EMS

 P˜ k

k∈K

k EMS (1, b)XEMS (1, b)

RVAK (1, b)

k (1, b)−P˜ k (0, b) X k k P˜EMS EMS EMS (1, b)−XEMS (0, b) . k XEMS (1, b) P˜ k (1, b) EMS

Appendix A: The Tang and Wang Method

181

Now P˜Rk (1, b)Y k (1, b)/RVAK (1, b) = R k1 /VAK1 , which is the comparison period share of nominal revenue of unit k in aggregate nominal value added, k (1, b)X k k1 K K1 and P˜EMS EMS (1, b)/RVA (1, b) = CEMS /VA , which is the comparison period share of nominal intermediate input cost of unit k in aggregate nominal value added. The right-hand side of expression (6.36) consists of six terms: the first and the fourth give the aggregate effect of quantity change, the second and the fifth give the aggregate effect of relative price change, and the third and the sixth give the aggregate effect of the interaction of quantity and price change. Forward- and backward-looking relative changes are related in a straightforward way. For instance,  −1 Y k (1, b) − Y k (0, b) Y k (1, b) − Y k (0, b) =1− 1+ . Y k (1, b) Y k (0, b) The decompositions in expressions (6.33) and (6.35), (6.34) and (6.36), respectively, have the same structure, and it is by and large a matter of taste and convenience which of the two is preferred. Therefore, as another alternative the simple arithmetic mean of the Laspeyres- and Paasche-perspective decompositions, expressions (6.33) and (6.35), could be chosen. The resulting decomposition is named after Bennet and has the additional feature that the two interaction terms cancel: RVAK (1, b) − RVAK (0, b) =    (1/2) P˜Rk (1, b) + P˜Rk (0, b) Y k (1, b) − Y k (0, b) k∈K

+



k∈K

−



k∈K

−



(6.37)

  (1/2) P˜Rk (1, b) − P˜Rk (0, b) Y k (1, b) + Y k (0, b)   k k k k (1/2) P˜EMS (1, b) + P˜EMS (0, b) XEMS (1, b) − XEMS (0, b)   k k k k (1/2) P˜EMS (1, b) − P˜EMS (0, b) XEMS (1, b) + XEMS (0, b) .

k∈K

Switching to forward- or backward-looking relative changes would destroy the symmetry, as one easily checks. The symmetry can be retained when differences are related to logarithmic means, as in RVAK (1, b) − RVAK (0, b) LM(RVAK (1, b), RVAK (0, b))

= ln(RVAK (1, b)/RVAK (0, b)).

182

6 The Top-Down Approach 1: Aggregate Output and Simple Labour Productivity. . .

This delivers ln(RVAK (1, b)/RVAK (0, b)) =   P˜Rk (1, b) + P˜Rk (0, b) LM(Y k (1, b), Y k (0, b))

Y k (1, b) Y k (0, b)



P˜Rk (1, b) ln P˜ k (0, b)

2LM(RVAK (1, b), RVAK (0, b)) R  k (1, b) + P˜ k (0, b) LM(X k k  P˜EMS EMS EMS (1, b), XEMS (0, b))

k∈K

−

ln

2LM(RVAK (1, b), RVAK (0, b))   LM(P˜Rk (1, b), P˜Rk (0, b)) Y k (1, b) + Y k (0, b)

k∈K

+

(6.38)

2LM(RVAK (1, b), RVAK (0, b))  k (1, b), P˜ k (0, b)) X k k  LM(P˜EMS EMS EMS (1, b)+XEMS (0, b)

ln

k∈K

−

k∈K

2LM(RVAK (1, b), RVAK (0, b))

k XEMS (1, b)

k XEMS (0, b)

k (1, b) P˜EMS . ln P˜ k (0, b)

EMS

What do we see here? At the left-hand side we have the logarithmic difference of aggregate real value added. If small then this approximates a forward-looking percentage change or the negative of a backward-looking percentage change. The right-hand side decomposes this into four terms. Each term is a weighted sum of subaggregate changes. The first term gives the total effect of real gross output quantity change, where each individual term is weighted by the share of mean nominal revenue in mean nominal aggregate value added. The second term gives the total effect of relative gross output price change, where each individual term is also weighted by the share of mean nominal revenue in mean nominal aggregate value added. The third term gives the total effect of real intermediate input quantity change, where each individual term is weighted by the share of mean nominal intermediate input cost in mean nominal aggregate value added. The fourth term gives the total effect of relative intermediate input price change, where each individual term is also weighted by the share of mean nominal intermediate input cost in mean nominal aggregate value added. Though symmetric with respect to time, the weights in the first and second term, and those in the third and fourth term, are not equal, due to different combinations of arithmetic and logarithmic means. This is not completely satisfactory. Another disadvantage of this decomposition is that aggregate real value added change is not written as a weighted sum or product of subaggregate real value added changes.

Appendix B: Dumagan’s Decomposition

183

Appendix B: Dumagan’s Decomposition Dumagan (2013a) considered another decomposition of aggregate simple labour productivity. As in expression (6.1), aggregate real value added is defined as K (t, b), RVAK (t, b) ≡ VAKt /PVA

(6.39)

K (t, b) is an appropriate price index relative to some base period b. where PVA Similarly, every production unit’s real value added is defined as k (t, b) (k ∈ K), RVAk (t, b) ≡ VAkt /PVA

(6.40)

k (t, b) are also appropriate price indices relative to some base period b; where PVA K (t, b) there is no assumption about any relation between the aggregate deflator PVA k and the unit-specific deflators PVA (t, b) (k ∈ K); they may or may not exhibit the same functional form. Define for any production unit k its relative value added price index as k (t, b) ≡ P k (t, b)/P K (t, b) and its labour share as L ˜ kt ≡ Lkt /LKt . Since P˜VA VA VA  aggregate value added is the sum of unit-specific value added, VAKt = k∈K VAkt , it readily appears, by substituting all the foregoing definitions, that aggregate simple labour productivity can be written as k  RVAK (t, b) k ˜ kt RVA (t, b) . ˜VA = (t, b) L P Lkt LK t

(6.41)

k∈K

We see here three factors: relative price index, labour share, and unit-specific simple labour productivity. Next, aggregate simple labour productivity growth, again defined as RVAK (1, b)/LK1 − RVAK (0, b)/LK0 RVAK (0, b)/LK0

,

was decomposed according to the Laspeyres, forward-looking perspective, into three factors, respectively called pure productivity effect, Denison effect, and Baumol effect. This final expression, called GEAD, goes back to Tang and Wang (2004). However, this decomposition is not unique and the distinction between the two effects appears to be rather artificial. This was discussed in Section 5.5.2. Next, Dumagan considered the special case where the relation between aggregate and unit-specific deflators is given by K PVA (t, b)

=

 VAkt 

k∈K

VAKt

k PVA (t, b)

−1

−1 .

(6.42)

184

6 The Top-Down Approach 1: Aggregate Output and Simple Labour Productivity. . .

Thus the aggregate deflator is a two-stage Paasche index of the unit-specific deflators (see 2, Appendix A). Rewriting this expression leads to RVAK (t, b) =  Chapter k k∈K RVA (t, b), and then to k  RVAK (t, b) ˜ kt RVA (t, b) , = L Lkt LK t

(6.43)

k∈K

which is a simplification of expression (6.41) in the sense that there are but two factors left: labour share and unit-specific simple labour productivity. Aggregate simple labour productivity growth can then again be decomposed in a number of ways, one of which was called TRAD. The paper contains interesting empirical comparisons of the GEAD and TRAD decompositions. Another empirical comparison was provided by De Avillez (2012). Finally, notice that expression (6.43) also materializes when for aggregate and K (t, b) = unit-specific value added the same deflators are used; that is when PVA k (t, b) (k ∈ K). PVA

Appendix C: Proof of Expression (6.13) Our point of departure is the identity  VAk1 k∈K

VAK1

−

 VAk0 k∈K

VAK0

= 0.

(6.44)

Using the definition of value added, this can be rewritten as  R k1 − C k1 EMS VAK1

k∈K

−

 R k0 − C k0 EMS VAK0

k∈K

= 0,

(6.45)

which can be rearranged to   R k1 k∈K

VAK1

−

R k0 VAK0

 −





k1 CEMS

VAK1

k∈K

−

k0 CEMS

VAK0

= 0,

(6.46)

As we know, the logarithmic mean transforms differences into ratios. Thus the last expression can be rewritten as  k∈K

 LM

R k1

R k0

, VAK1 VAK0



 ln

R k1 /VAK1 R k0 /VAK0

Appendix C: Proof of Expression (6.13)



−

LM

k∈K

185

k1 k0 CEMS CEMS , VAK1 VAK0

ln

k1 /VAK1 CEMS

= 0.

k0 /VAK0 CEMS

(6.47)

This, finally, can be rearranged as ln

VAK1

=

VAK0



 ξRk (1, 0) ln

k∈K

R k1 R k0

 −



k ξEMS (1, 0) ln

k∈K

k1 CEMS

k0 CEMS

,

(6.48)

where  ξRk (1, 0) ≡

LM 



k∈K LM

k0 R k1 , R VAK1 VAK0

k0 R k1 , R VAK1 VAK0



−





k∈K LM



k1 CEMS C k0 , EMS K1 VA VAK0

 (k ∈ K),

and

k1 k0 CEMS CEMS , LM K1 VAK0 VA k

(k ∈ K). ξEMS (1, 0) ≡   k1 k1 k0   CEMS CEMS R R k0 − k∈K LM , , k∈K LM VAK1 VAK0 VAK1 VAK0 Replacing revenue and intermediate inputs cost ratios by products of price and quantity indices, we obtain ln 

ξRk (1, 0) ln(PRk (1, 0)QkR (1, 0)) −

k∈K

VAK1 VAK0



=

(6.49)

k k ξEMS (1, 0) ln(PEMS (1, 0)QkEMS (1, 0)) =

k∈K

 k k ξRk (1, 0) ln PRk (1, 0) − ξEMS (1, 0) ln PEMS (1, 0) + k∈K



k ξRk (1, 0) ln QkR (1, 0) − ξEMS (1, 0) ln QkEMS (1, 0) ,

k∈K

from which expression (6.13) immediately follows. The price and quantity components after the second equality sign in expression (6.49) generalize the conventional Sato-Vartia indices (see Appendix A of Chap. 2).

Chapter 7

The Top-Down Approach 2: Aggregate Total Factor Productivity Index

7.1 Introduction In Chap. 2 we considered the measurement of productivity change for a single, consolidated production unit. The present chapter continues by considering an ensemble of such units, and studying the relation between aggregate and subaggregate (or individual) measures of productivity change. The theory developed here is applicable to a variety of situations, such as (1) an enterprise consisting of a number of plants, (2) an industry consisting of a number of firms, or (3) an economy or, more precisely, the commercial sector of an economy consisting of a number of industries. There are a number of measures of productivity change. This chapter concentrates on total factor productivity (TFP); that is, all the input factors are taken into account. In the foregoing chapter labour productivity (LP) was considered. On an intuitive level the relation between aggregate and subaggregate, or individual, measures of productivity change is not too difficult to understand. Productivity is output quantity divided by input quantity and, thus, productivity change is output quantity change divided by input quantity change. Any aggregate is somehow the sum of its parts, which in the present context implies that aggregate productivity is somehow a weighted mean of subaggregate (or individual) productivities, where the weights somehow express the “importance” of the subaggregates, or individual units, making out the aggregate. Hence, there are two, independent, factors responsible for aggregate productivity change: (1) productivity change at subaggregate, or individual, level, and (2) change of the “importance” of the subaggregate, or individual, units. Coming down to practice, things are rapidly becoming complicated. First, firms produce and use multiple commodities, which brings input and output prices into play. Quantities of different commodities cannot be added, but must be aggregated by means of prices. Through time, prices are also changing, which implies that we cannot simply talk about aggregate prices and quantities, but must talk about © The Author(s), under exclusive license to Springer Nature Switzerland AG 2021 B. M. Balk, Productivity, Contributions to Economics, https://doi.org/10.1007/978-3-030-75448-8_7

187

188

7 The Top-Down Approach 2: Aggregate Total Factor Productivity Index

price and quantity index numbers. Moreover, aggregation rules are not unique anymore. Second, firms and industries deliver to each other, which implies that “simple” addition of production units easily leads to a form of double-counting of outputs and inputs. Third, especially when we are dealing with firms or plants an important fact to take into account is the dynamics of growth, decline, appearance, and disappearance of production units. All this is discussed in the coming sections. Sections 7.2 to 7.6 discuss the case of a static ensemble of production units. The theory developed in these sections is immediately applicable to the situation of an economy consisting of a number of industries. Sections 7.7 and 7.8 proceed with the case of a dynamic ensemble. All the relations developed in Sects. 7.2 to 7.8 are in terms of value added as output measure. Section 7.9 discusses the link between value-added based and gross-output based productivity measures. Section 7.10 considers the conversion of equations in terms of indices to growth accounting equations. Section 7.11 concludes.

7.2 First Decomposition For the accounting framework the reader is referred to Sect. 5.2. In this chapter we first consider a static ensemble K. Though its members may rise or fall through time, the membership list as such does not change. We start by considering the KL-VA accounting relations. Aggregate profitability in the KL-VA model is defined as aggregate value added divided by aggregate primary input cost, VAKt Kt CKL



= k∈K

VAkt

kt k∈K CKL

(7.1)

.

Using the logarithmic mean repeatedly, it appears that the logarithm of aggregate profitability is a linear function of the logarithms of all the individual profitabilities, ln

VAKt Kt CKL

=

 k∈K

φ kt ln

VAkt kt CKL

,

(7.2)

kt )/LM(VAKt , C Kt ) and LM(.) being the logarithmic with φ kt ≡ LM(VAkt , CKL KL 1 mean. Notice that, since LM(a, 1) is concave, the coefficients φ kt do not necessarily add up to 1. Notice further that, for expression (7.2) to be meaningful, at least kt , or kt = 0. two production units must exist with VAkt = CKL

1 Recall that the logarithmic mean, for any two strictly positive real numbers a and b, is defined by LM(a, b) ≡ (a − b)/ ln(a/b) if a = b and LM(a, a) ≡ a.

7.2 First Decomposition

189

Aggregate profitability change, going from an earlier period 0 (also called base period) to a later period 1 (also called comparison period), is rather naturally measured by the ratio of period 1 profitability to period 0 profitability, K1 )/(VAK0 /C K0 ). Using expression (7.2), the logarithm of this ratio can (VAK1 /CKL KL be decomposed symmetrically as

K1 VAK1 /CKL

= K0 VAK0 /CKL  k1 k0 (φ k1 − φ k0 )(ln(VAk1 /CKL ) + ln(VAk0 /CKL )) (1/2)

ln

k∈K

+ (1/2)



k1 k0 (φ k1 + φ k0 )(ln(VAk1 /CKL ) − ln(VAk0 /CKL ))

k∈K

⎛

1/2 ⎞ k1 VAk0 VA ⎠ = (φ k1 − φ k0 ) ln ⎝ k1 C k0 C KL KL k∈K   k1 k0 + (1/2)(φ k1 + φ k0 ) ln(VAk1 /VAk0 ) − ln(CKL /CKL ) . 

(7.3)

k∈K

Let there be given price and quantity indices such that for each production unit the value-added ratio and the primary-input-cost ratio can be decomposed into two components, k VAk1 /VAk0 = PVA (1, 0)QkVA (1, 0)

(7.4)

k1 k0 k CKL /CKL = PKL (1, 0)QkKL (1, 0).

(7.5)

Notice that it is not required that all the functional forms of the price and quantity indices be the same.2 Substituting this into the last part of expression (7.3) one obtains

K1 VAK1 /CKL ln = K0 VAK0 /CKL ⎛

1/2 ⎞ k1 k0  VA VA ⎠ (φ k1 − φ k0 ) ln ⎝ k1 C k0 C KL KL k∈K

2 Actually the arguments of the functions P (.) and Q(.) are all the prices and quantities of the commodities involved in the scope of these indices. Thus we are using the shorthand notation introduced earlier.

190

7 The Top-Down Approach 2: Aggregate Total Factor Productivity Index

+



k k (1/2)(φ k1 + φ k0 ) ln(PVA (1, 0)/PKL (1, 0))

k∈K

+



(1/2)(φ k1 + φ k0 ) ln ITFPRODkVA (1, 0),

(7.6)

k∈K

where individual value-added based total factor productivity (TFP) change is defined as the ratio of the output quantity index number over the input quantity index number, ITFPRODkVA (1, 0) ≡ QkVA (1, 0)/QkKL (1, 0) (see Section 2.3.1). Thus, according to expression (7.6), aggregate profitability change can be decomposed into three parts. The first part can be interpreted as the effect of size change (or reallocation) of the individual production units. The size of any individual production unit is thereby measured by the coefficient φ kt , which is the relative mean of the unit’s value added and primary inputs cost. The weight of each individual size change is the logarithm of mean profitability. This part vanishes if there is no size change at all (that is, φ k1 = φ k0 for all k ∈ K) or if size increases of certain production units compensate size decreases of other units. The second part gives the aggregate effect of differential price change at the output and input side (which is sometimes called terms-of-trade change). This part vanishes if   k k (1/2)(φ k1 + φ k0 ) ln PVA (1, 0) = (1/2)(φ k1 + φ k0 ) ln PKL (1, 0); (7.7) k∈K

k∈K

that is, if (approximate) mean output-side price change equals (approximate) mean k (1, 0) = P k (1, 0) for all input-side price change. A special case occurs if PVA KL k ∈ K; that is, if, for any production unit, output and input price change is the same. The third part gives the aggregate effect of productivity change at the individual unit level. Let there also be given price and quantity indices such that the aggregate valueadded ratio and the aggregate primary-input-cost ratio can be decomposed into two parts, K VAK1 /VAK0 = PVA (1, 0)QK VA (1, 0)

(7.8)

K1 K0 K CKL /CKL = PKL (1, 0)QK KL (1, 0),

(7.9)

and let aggregate value-added based TFP change be defined as ITFPRODK VA (1, 0) ≡ K (1, 0). Substituting all this into expression (7.6) and moving the QK (1, 0)/Q VA KL aggregate price index numbers from the left-hand side to the right-hand side delivers an expression for aggregate TFP change: ln ITFPRODK VA (1, 0) =

7.2 First Decomposition

191

⎛

1/2 ⎞ k1 VAk0 VA ⎠ (φ k1 − φ k0 ) ln ⎝ k1 C k0 CKL KL k∈K 

+



k k (1/2)(φ k1 + φ k0 ) ln(PVA (1, 0)/PKL (1, 0))

k∈K

+



(1/2)(φ k1 + φ k0 ) ln ITFPRODkVA (1, 0)

k∈K

K K − ln(PVA (1, 0)/PKL (1, 0)).

(7.10)

This can also be written as ln ITFPRODK VA (1, 0) = ⎛

1/2 ⎞ k1 k0  VA VA ⎠ (φ k1 − φ k0 ) ln ⎝ k1 C k0 C KL KL k∈K



k (1, 0) k (1, 0)  PVA PKL k1 k0 + (1/2)(φ + φ ) ln − ln K (1, 0) K (1, 0) P PKL VA k∈K  + (1/2)(φ k1 + φ k0 ) ln ITFPRODkVA (1, 0) k∈K



+



(1/2)(φ

k1

K K + φ ) − 1 ln(PVA (1, 0)/PKL (1, 0)). k0

(7.11)

k∈K

There are four terms here. The first presents, as before, the aggregate effect of size change. The second presents the aggregate effect of differential price change at the output and input side of the production units. This part vanishes if K ln PVA (1, 0) =



φ k1 + φ k0 k ln PVA (1, 0) k1 k0 k∈K (φ + φ )

(7.12)

φ k1 + φ k0 k ln PKL (1, 0); k1 k0 k∈K (φ + φ )

(7.13)



k∈K

and K ln PKL (1, 0) =

 k∈K



that is, if aggregate output price change is equal to mean output price change, and aggregate input price change is equal to mean input price change, respectively. A k (1, 0) = P K (1, 0) and P k (1, 0) = P K (1, 0) for all special case occurs if PVA VA KL KL k ∈ K; that is, if at the output side and at the input side all the price changes are the same.

192

7 The Top-Down Approach 2: Aggregate Total Factor Productivity Index

The third term is an (approximate) mean of all the unit-specific productivity changes. The fourth term concerns a concavity discrepancy (which, generally, should be negligible). Recall, however, that the decomposition in expression (7.11) is only useful in case there exist at least a certain number of production units with nonzero profit. In the next two sections we turn to decompositions which have a wider applicability.

7.3 Second Decomposition The second decomposition departs from the output-side related identity we have met in expression (6.4), ln

VAK1

VAK0





VAk1 ψ (1, 0) ln = VAk0 k∈K k

(7.14)

,

where LM

VAk1

VAk0

, VAK1 VAK0

(k ∈ K). ψ (1, 0) ≡  VAk1 VAk0 , k∈K LM VAK1 VAK0 k

Aggregate value-added change, measured as a ratio, is thus equal to a weighted geometric mean of individual value-added changes. Notice that the coefficients ψ k (1, 0) add up to 1. Each coefficient is the (normalized) mean (over two periods) share of production unit k in aggregate value added. K (1, 0) from the Now, applying product relations (7.4) and (7.8), and moving PVA left-hand to the right-hand side, we obtain ln QK VA (1, 0)

=



k

ψ (1, 0) ln

k∈K

k (1, 0)Qk (1, 0) PVA VA

K (1, 0) PVA

(7.15)

.

Subtracting from both sides ln QK KL (1, 0) and applying the definition of aggregate value-added based TFP change delivers ln ITFPRODK VA (1, 0)

=

 k∈K

k

ψ (1, 0) ln

k (1, 0)Qk (1, 0) PVA VA

K (1, 0)QK (1, 0) PVA KL

7.3 Second Decomposition

=

193





k

ψ (1, 0) ln

k∈K

k (1, 0)Qk (1, 0)Qk (1, 0) PVA VA KL

K (1, 0)QK (1, 0)Qk (1, 0) PVA KL KL

(7.16)

.

Using the definition of individual value-added based TFP change, the last expression can be rewritten as  ln ITFPRODK (1, 0) = ψ k (1, 0) ln ITFPRODkVA (1, 0) + VA 

k

ψ (1, 0) ln

k∈K

k∈K k (1, 0) PVA

+

K (1, 0) PVA



k

ψ (1, 0) ln

QkKL (1, 0)

.

QK KL (1, 0)

k∈K

(7.17)

Notice that the second term atthe right-hand side of this expression vanishes k K (1, 0) = k if and only if ln PVA k∈K ψ (1, 0) ln PVA (1, 0); that is, the aggregate output price index is a Sato-Vartia index of the production-unit-specific output price indices. It is straightforward to check that the definition of the coefficients ψ k (1, 0) (k ∈ K) implies that 

k

ψ (1, 0) ln

k∈K

k (1, 0) PVA



=−

K (1, 0) PVA

k

ψ (1, 0) ln

k∈K

QkVA (1, 0)

QK VA (1, 0)

(7.18)

.

Substituting this into expression (7.17) one obtains 

ln ITFPRODK VA (1, 0) = 

k

ψ (1, 0) ln

k∈K

ψ k (1, 0) ln ITFPRODkVA (1, 0) −

k∈K

QkVA (1, 0)

QK VA (1, 0)

+



k

ψ (1, 0) ln

k∈K

QkKL (1, 0)

QK KL (1, 0)

.

(7.19)

Alternatively, using product relations (7.5) and (7.9), the last term of expression (7.17) can be rewritten, which results in ln ITFPRODK VA (1, 0) = +



ψ k (1, 0) ln

k∈K

+

 k∈K

ψ k (1, 0) ln



ψ k (1, 0) ln ITFPRODkVA (1, 0)

k∈K k (1, 0) PVA

K (1, 0) PVA

k1 /C k0 CKL KL K1 /C K0 CKL KL

− ln

.

k (1, 0) PKL



K (1, 0) PKL

(7.20)

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7 The Top-Down Approach 2: Aggregate Total Factor Productivity Index

There are three terms here. The first is a weighted mean of unit-specific productivity changes. The second is the aggregate effect of differential price change at the output and input side of the production units. The third can be interpreted as the aggregate effect of relative size change, where the size of a unit is measured by its primaryinput cost share. k K (1, 0) =  k The second term vanishes if ln PVA k∈K ψ (1, 0) ln PVA (1, 0) and  k K k ln PKL (1, 0) = k∈K ψ (1, 0) ln PKL (1, 0). In particular, if there is no differential k (1, 0) = P K (1, 0) and P k (1, 0) = P K (1, 0) for price change at all, that is, PVA VA KL KL all k ∈ K. The third term vanishes if for all units k ∈ K and time periods t = 0, 1 value kt , or profit equals zero, kt = 0. Then added equals primary input cost, VAkt = CKL kt K t primary-input cost shares CKL /CKL equal value-added shares VAkt /VAKt , and by using the definition of the logarithmic mean one immediately checks the result. Under these two conditions expression (7.20) reduces to ln ITFPRODK VA (1, 0) =



ψ k (1, 0) ln ITFPRODkVA (1, 0).

(7.21)

k∈K

The right-hand side of this expression starkly looks like the third term on the righthand side of expression (7.11). The weights are slightly different. Expression (7.21) is a discrete-time version of the relation stated in Proposition 3 of Gollop (1979), and there proved using a large number of neo-classical assumptions. The first term on the right-hand side of expression (7.20) and the third term on the right-hand side of expression (7.11) correspond to what is called ‘the bottomup approach’ to the measurement of aggregate (total- or multi-factor) productivity change. These terms could form the basis for approximating aggregate TFP change from TFP change of a sample of production units, as in Gabaix (2011). In line with this approach, for example, Van den Bergen et al. (2008, 400) rather casually remark that “. . . MFP change (expressed as a percentage) of the big unit can be expressed as a weighted arithmetic average of MFP change of the small units.”3 It now appears that in such an approach the remainder terms in expressions (7.20) and (7.11) are neglected. Whether this is or is not important is an empirical issue. The magnitude of the remainder terms depends for a great deal on the level of aggregation of the production units and the specificity of the deflators that have been employed. Moreover, there are several counterbalancing mechanisms: growth versus decline of production units, and increasing versus decreasing prices at input and output sides. Also the length of the time span between base and comparison period plays a role, although implicitly. Expression (7.20) exhibits the same structure as a decomposition derived by Diewert (2015b), here reproduced as expression (7.28). Interestingly, Diewert’s

3 Recall

that for index numbers in the neighbourhood of 1 their logarithms approximate percentage changes.

7.4 Third Decomposition

195

computations on the Australian economy over the period 1995–2012 revealed that the remainder terms, though not all their components, were close to zero.

7.4 Third Decomposition The third decomposition departs, rather naturally, from the input-side related identity ln

K1 CKL

=

K0 CKL





k

ω (1, 0) ln

k∈K

k1 CKL

(7.22)

,

k0 CKL

where  LM ωk (1, 0) ≡



k1 k0 CKL CKL K1 , C K0 CKL KL



k∈K LM



k1 k0 CKL CKL K1 , C K0 CKL KL

 (k ∈ K).

Aggregate primary-input cost change is thus equal to a weighted geometric mean of individual primary-input cost changes. Notice that the coefficients ωk (1, 0) add up to 1. Each coefficient is the (normalized) mean share of production unit k in aggregate primary-input cost. Applying product relations (7.5) and (7.9) and performing steps similar to those in the previous section delivers ln ITFPRODK VA (1, 0) = 

k

ω (1, 0) ln

k∈K



ωk (1, 0) ln ITFPRODkVA (1, 0) −

k∈K k (1, 0) PKL

K (1, 0) PKL

−



k

ω (1, 0) ln

k∈K

QkVA (1, 0)

QK VA (1, 0)

.

(7.23)

It is straightforward to check that the definition of ωk (1, 0) (k ∈ K) implies that  k∈K

k

ω (1, 0) ln

k (1, 0) PKL

K (1, 0) PKL

=−



k

ω (1, 0) ln

k∈K

QkKL (1, 0)

QK KL (1, 0)

Substituting this into expression (7.23) one obtains ln ITFPRODK VA (1, 0) =

 k∈K

ωk (1, 0) ln ITFPRODkVA (1, 0) +

.

(7.24)

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7 The Top-Down Approach 2: Aggregate Total Factor Productivity Index



k

ω (1, 0) ln

k∈K

QkKL (1, 0)

QK KL (1, 0)

−



k

ω (1, 0) ln

k∈K

QkVA (1, 0)

QK VA (1, 0)

(7.25)

.

Alternatively, using product relations (7.4) and (7.8), expression (7.23) can be rewritten as  ln ITFPRODK (1, 0) = ωk (1, 0) ln ITFPRODkVA (1, 0) VA 

+

k

ω (1, 0) ln

k∈K



−

k

ω (1, 0) ln

k∈K

k∈K k (1, 0) PVA

K (1, 0) PVA

VAk1 /VAk0

− ln

k (1, 0) PKL



K (1, 0) PKL

(7.26)

.

VAK1 /VAK0

This expression exhibits the same structure as expression (7.20). Notice, however, the negative sign in front of the term measuring relative size change. This makes expression (7.26) less attractive than expression (7.20) for practical purposes. It is left to the reader to check that, if there is no differential price change and if for each production unit value added equals primary input cost, then expression (7.26) reduces to expression (7.21).

7.5 Asymmetric Decompositions We now turn to a number of asymmetric decompositions. The first departs from the output-side related identity: VAK1 VAK0

=

 VAk0 VAk1 . VAK0 VAk0 k∈K

(7.27)

Aggregate value-added change is here written as a weighted arithmetic mean of individual value-added changes. K (1, 0) from the left-hand Applying product relations (7.4) and (7.8), moving PVA K to the right-hand side, dividing both sides by QKL (1, 0), and using product relations (7.5) and (7.9), respectively, leads to ITFPRODK VA (1, 0) =  VAk0 k∈K

VAK0

P k (1, 0) ITFPRODkVA (1, 0) VA K (1, 0) PVA

(7.28)

k (1, 0) PKL

K (1, 0) PKL

−1

k1 /C k0 CKL KL

K1 /C K0 CKL KL

.

7.5 Asymmetric Decompositions

197

This decomposition was also obtained by Diewert (2015b, ex. (29)). If there is no differential price change and if for each production unit value added equals primary input cost, then expression (7.28) reduces to  VAk1

ITFPRODK VA (1, 0) =

VAK1 k∈K

ITFPRODkVA (1, 0),

(7.29)

or, to compare with the earlier expression (7.21), ITFPRODK VA (1, 0) − 1 =

 VAk1 VAK1 k∈K

(ITFPRODkVA (1, 0) − 1).

(7.30)

Notice that the individual productivity changes are here weighed with comparison period value-added (= primary-input-cost) shares, not with base period shares. The second asymmetric decomposition departs from the natural counterpart to expression (7.27), namely VAK1 VAK0

⎛ =

 VAk1 ⎝



VAK1 k∈K

VAk1 VAk0

−1 ⎞−1 ⎠

(7.31)

.

Here aggregate value-added change is written as a weighted harmonic mean of individual value-added changes. The same steps as above then lead to −1  = ITFPRODK VA (1, 0)  VAk1  k∈K

VAK1

P k (1, 0) ITFPRODkVA (1, 0) VA K (1, 0) PVA

(7.32)

k (1, 0) PKL

−1

K (1, 0) PKL

k1 /C k0 CKL KL

K1 /C K0 CKL KL

−1 .

If there is no differential price change and if for each production unit value added equals primary input cost, then expression (7.32) reduces to

ITFPRODK VA (1, 0)

−1  VAk0  k ITFPROD = (1, 0) VA VAK0 k∈K

−1 .

(7.33)

Notice that the individual productivity changes are here weighed with base period value-added (= primary-input-cost) shares. Returning to the more general expressions (7.28) and (7.32), we notice that the first is an arithmetic mean using base period value-added shares whereas the second is a harmonic mean using comparison period value added shares. They resemble the conventional Laspeyres and Paasche indices respectively. A symmetric expression

198

7 The Top-Down Approach 2: Aggregate Total Factor Productivity Index

for aggregate value-added based productivity change may be obtained, in the spirit of Fisher, by taking the geometric mean of the two right-hand sides. We now consider the input-side related identity:  C k0 C k1 KL KL = . K0 K0 C k0 CKL C k∈K KL KL

K1 CKL

(7.34)

Aggregate primary-input cost change is here written as a weighted arithmetic mean of individual primary-input cost changes. Applying product relations (7.5) and (7.9), K (1, 0) from the left-hand to the right-hand side, dividing both sides by moving PKL QK VA (1, 0), and using product relations (7.4) and (7.8), respectively, leads to  −1 ITFPRODK (1, 0) = VA

(7.35)

−1  C k0  k KL ITFPROD (1, 0) VA C K0 k∈K KL



k (1, 0) PVA

−1

K (1, 0) PVA

k (1, 0) PKL VAk1 /VAk0

K (1, 0) VAK1 /VAK0 PKL

.

If there is no differential price change and if for each production unit value added equals primary input cost, then expression (7.35) reduces to

ITFPRODK VA (1, 0)

−1  VAk1  ITFPRODkVA (1, 0) = K 1 VA k∈K

−1 .

(7.36)

Notice that the individual productivity changes are here weighed with comparison period value-added (= primary-input-cost) shares. The final decomposition departs from the natural counterpart to expression (7.34), namely K1 CKL K0 CKL

⎛ =⎝

 C k1 KL k∈K



K1 CKL

k1 CKL k0 CKL

−1 ⎞−1 ⎠ .

(7.37)

Aggregate primary-input cost change is here expressed as a weighted harmonic mean of individual primary-input cost changes. The same steps as above then lead to ITFPRODK VA (1, 0) =  C k1 KL

k (1, 0) PVA k ITFPROD (1, 0) VA K (1, 0) C K1 PVA k∈K KL



k (1, 0) PKL

K (1, 0) PKL

−1

VAk1 /VAk0 VAK1 /VAK0

(7.38)

−1 .

7.5 Asymmetric Decompositions

199

If there is no differential price change and if for each production unit value added equals primary input cost, then expression (7.38) reduces to ITFPRODK VA (1, 0) =

 VAk0 VAK0 k∈K

ITFPRODkVA (1, 0).

(7.39)

Notice that the individual productivity changes are here weighed with base period value-added (= primary-input-cost) shares. The four decompositions presented here, in expressions (7.28), (7.32), (7.35), and (7.38), respectively, exhibit the same structure. They decompose aggregate productivity change into the contributions of the individual production units. Each of these contributions consists of four components: an input or output based weight, individual productivity change, differential price change at the output and input side of the unit, and an input or output based measure of relative size change, respectively. A second step is needed for aggregating the three unit-specific components, productivity change, differential price change, and size change, respectively, so that aggregate productivity change can be seen as coming from these three “sources”. There are several methods available for carrying out this second step, such as the method of Gini (1937), which is a generalization of the Fisher price and quantity indices, or the simpler method developed by Balk (2003a). The Gini method leads to rather complicated final expressions for aggregate productivity change. The method advised by Balk (2003a), however, leads to final expressions that at first sight look rather complicated but at second sight appear to reduce to expressions (7.20) and (7.26). One example is sufficient to demonstrate this; the remaining cases are left to the reader. Consider expression (7.28), and let k PVA,diff (1, 0) ≡

k (1, 0) PVA

K (1, 0) PVA



k (1, 0) PKL

K (1, 0) PKL

−1 (k ∈ K)

(7.40)

denote unit-specific, value-added based differential price change, and let k SKL (1, 0) ≡

k1 /C k0 CKL KL

K1 /C K0 CKL KL

(k ∈ K)

(7.41)

denote unit-specific primary-input-cost based relative size change. With these definitions expression (7.28) reads ITFPRODK VA (1, 0) =  VAk0 VAK0 k∈K

k k ITFPRODkVA (1, 0)PVA,diff (1, 0)SKL (1, 0) =

(7.42)

200

7 The Top-Down Approach 2: Aggregate Total Factor Productivity Index



k k k k0 k∈K VA ITFPRODVA (1, 0)PVA,diff (1, 0)SKL (1, 0) ,  k k k k0 k∈K VA ITFPRODVA (0, 0)PVA,diff (0, 0)SKL (0, 0)

 k0 where the last line is based on the identity VAK0 = k∈K VA . Applying then the method advised by Balk (2003a) delivers the following multiplicative decomposition of aggregate productivity change: ITFPRODK (7.43) VA (1, 0) = θ k (1,0)   θ k (1,0)   θ k (1,0)  k k ITFPRODkVA (1, 0) PVA,diff SKL (1, 0) (1, 0) , k∈K

k∈K

k∈K

where the exponents are defined as θ k (1, 0) ≡ 

LM(sk1 , sk0 ) 1 0 k∈K LM(sk , sk )

(k ∈ K)

(7.44)

with skt ≡ 

k k (t, 0) (t, 0)SKL VAk0 ITFPRODkVA (t, 0)PVA,diff k k k k0 k∈K VA ITFPRODVA (t, 0)PVA,diff (t, 0)SKL (t, 0)

(k ∈ K; t = 0, 1). (7.45)

Expression (7.43) can be written additively as ln ITFPRODK VA (1, 0) =   k θ k (1, 0) ln ITFPRODkVA (1, 0) + θ k (1, 0) ln PVA,diff (1, 0) k∈K

+



(7.46)

k∈K

θ

k

k (1, 0) ln SKL (1, 0).

k∈K

This is the same expression as (7.20), except that the coefficients θ k (1, 0) look far more complex than the ψ k (1, 0). The ψ k (1, 0) were defined as (normalized) logarithmic means of base period and comparison period value-added shares, whereas the θ k (1, 0) were defined as (normalized) logarithmic means of base period and updated-base period value-added shares, updated with productivity change, differential price change, and size change. Inserting the various definitions, and using product relations (7.4) and (7.5), however, confirms that θ k (1, 0) = ψ k (1, 0) (k ∈ K). Thus expression (7.46) is indeed identical to expression (7.20).

(7.47)

7.6 The Differential Price Change Term

201

7.6 The Differential Price Change Term In the case of a static ensemble of production units, the preferred decomposition of aggregate TFP change is given by expression (7.20). Aggregate productivity change is decomposed according to three “sources”: unit-specific productivity change, differential price change, and relative size change. Especially the second term appears to be hard to sell to economists. It is readily acknowledged that productivity change at the level of production units “causes” productivity change at the level of the aggregate, and one is also ready to admit that relative growth or decline of production units contributes to aggregate productivity change. But in which sense can productivity change, defined as output quantity change divided by input quantity change, be dependent on price change? This section is devoted to an exploration of this topic. k First, careful consideration of the expression defining PVA,diff (1, 0) reveals that input or output price development as such does not play a role. It is relative price development, that is, price development of the production unit relative to the aggregate, that matters. If, at the production unit level, there is no dispersion of k input and output price developments, then PVA,diff (1, 0) = 1, and the contribution to aggregate productivity change disappears. Second, economists usually argue in terms of “levels”, and let aggregate output (input) be the sum of unit-specific output (input). The hidden assumption thereby is that output (input) is homogeneous over the production units. However, even if at the level of individual commodities the price is the same for every buyer/seller, then the “price” of the composite input and output commodity will vary over the production units. Simple addition of “quantities” of input or output, thus, is not admissible. The corresponding “prices” play a role in the aggregation process, and compositional differences in the two periods compared in principle influence the measurement of aggregate quantity change. Thus price change must be taken into account as a separate factor.4 Third, it is interesting to consider what happens if for all the production units the same deflators are used, which is a situation very common in empirical work on firm-level data. Thus, instead of expressions (7.4) and (7.5), which reflect the true decompositions of the value-added and primary-input-cost ratios, we use, for all k ∈ K, k∗ VAk1 /VAk0 = PVA (1, 0)Qk∗ VA (1, 0)

(7.48)

k1 k0 k∗ CKL /CKL = PKL (1, 0)Qk∗ KL (1, 0),

(7.49)

with

4 See

also Tang and Wang (2015) on the importance of the contribution of differential price change to aggregate productivity change.

202

7 The Top-Down Approach 2: Aggregate Total Factor Productivity Index k∗ K PVA (1, 0) ≡ PVA (1, 0)

(7.50)

k K k Qk∗ VA (1, 0) ≡ (PVA (1, 0)/PVA (1, 0))QVA (1, 0)

(7.51)

k∗ K PKL (1, 0) ≡ PKL (1, 0)

(7.52)

k K k Qk∗ KL (1, 0) ≡ (PKL (1, 0)/PKL (1, 0))QKL (1, 0).

(7.53)

Then we obtain, instead of expression (7.20), ln ITFPRODK VA (1, 0) =   k ψ k (1, 0) ln ITFPRODk∗ ψ k (1, 0) ln SKL (1, 0), VA (1, 0) + k∈K

(7.54)

k∈K

k∗ k∗ where ITFPRODk∗ VA (1, 0) ≡ QVA (1, 0)/QKL (1, 0) (k ∈ K). There are two terms left, the first of which has the same structure as the first term of expression (7.20), whereas the second is identical to the third term of expression (7.20). The differential price change term, the second term of expression (7.20), has vanished, but the unit-specific TFP change term is now “contaminated” by differential price changes, since k k ITFPRODk∗ VA (1, 0) = ITFPRODVA (1, 0)PVA,diff (1, 0) (k ∈ K).

(7.55)

Substituting the definitions of true TFP change and differential price change, the last expression transforms into ITFPRODk∗ VA (1, 0) =

K (1, 0) VAk1 /VAk0 PKL k1 /C k0 K CKL KL PVA (1, 0)

(k ∈ K).

(7.56)

This is unit-specific profitability change, deflated by the ratio of aggregate output and input price index numbers. In the literature this goes by the name “revenue” TFP change. The unavailability of firm-specific prices or price index numbers means that one cannot calculate firm-specific productivity index numbers. The best one can do is calculate firm-specific profitability ratios deflated by aggregate price index numbers. A contribution by Foster et al. (2008) sheds light upon the consequences of this shortcoming. These authors worked with a sample of (almost) single-output establishments producing physically homogeneous goods, so that establishment-level output prices and quantities were available and comparable across establishments. There were no deliveries between such establishments. Thus, in a KLEMS-Y framework they could calculate approximations to “revenue” TFP index numbers, k ITFPRODk∗ Y (1, 0), and genuine TFP index numbers, ITFPRODY (1, 0).

7.7 The Case of a Dynamic Ensemble

203

Two general lessons of this exercise are important to keep in mind. The first is that there appeared to be appreciable differences between the distributions of ITFPRODkY (1, 0) and ITFPRODk∗ Y (1, 0). The second is that one should be very careful with interpreting the index numbers ITFPRODk∗ Y (1, 0) as measures of technological change. This was confirmed in more recent research of Eslava et al. (2013). The topic as such—“revenue” versus “physical” TFP—was classified as one of the “big questions” in the Syverson (2011) survey.

7.7 The Case of a Dynamic Ensemble We now turn to a dynamic ensemble; that is, an ensemble the membership of which changes through time. Thus, wherever necessary, we must add a superscript t to K. The accounting identities as discussed in Sect. 5.2 remain valid. For any two time periods compared, whether adjacent or not, a distinction must then be made between continuing, exiting, and entering production units. In particular, as discussed in Sect. 5.3, K0 = C 01 ∪ X 0

(7.57)

K1 = C 01 ∪ N 1 ,

(7.58)

where C 01 denotes the continuing units (active in both periods), X 0 the exiting units (active in the base period only), and N 1 the entering units (active in the comparison period only).5 As before, aggregate profitability in period t is measured as VAKt Kt CKL



t

VAkt

k∈Kt

kt CKL

= k∈K

,

(7.59)

and aggregate profitability change from period 0 to period 1 is measured as the ratio of the two profitabilities. Using expression (7.2) and the distinction between the three groups of units, aggregate profitability change can be written as ln

5 The

VAK1 K1 CKL

− ln

VAK0 K01 CKL

=

(7.60)

definition of ‘active’ should of course be made precise in any empirical application on microdata to ascertain that exit of a certain production unit is due to quitting business and not due to e.g. obtaining a new name, merging with an other unit, splitting into two or more units, or falling below the observation threshold; likewise for entry.

204

7 The Top-Down Approach 2: Aggregate Total Factor Productivity Index





VAk1





VAk1

φ ln + k1 k1 CKL CKL 1 k∈N

  VAk0 VAk0 k0 k0 − φ ln φ ln − . k0 k0 CKL CKL 01 0 φ

k1

ln

k1

k∈C 01

k∈C

k∈X

The first minus the third term at the right-hand side can be decomposed symmetrically like expression (7.3). Then, using product relations (7.4), (7.5), (7.8), and (7.9), we finally obtain ln ITFPRODK VA (1, 0) = ⎛

1/2 ⎞ k1 k0  VA VA ⎠ (φ k1 − φ k0 ) ln ⎝ k1 C k0 CKL KL 01 k∈C

+



(7.61)

k (1/2)(φ k1 + φ k0 ) ln PVA,diff (1, 0)

k∈C 01

+



(1/2)(φ k1 + φ k0 ) ln ITFPRODkVA (1, 0)

k∈C 01

⎛

+⎝



⎞ K K (1/2)(φ k1 + φ k0 ) − 1⎠ ln(PVA (1, 0)/PKL (1, 0))

k∈C 01

+



φ

k1

ln

k∈N 1

VAk1

k1 CKL

−



φ

k0

ln

k∈X 0

VAk0 k0 CKL

.

As might be expected, if K0 = K1 = C 01 , then expression (7.61) reduces to expression (7.11). The last three terms on the right-hand side of expression (7.61) can be rewritten, using the fact that 

φ k1 +

k∈C 01



k∈C 01



φ k1 =

k∈N 1

φ k0 +



k∈X 0



φ k1 ≡ φ K1

(7.62)

φ k0 ≡ φ K0 .

(7.63)

k∈K1

φ k0 =



k∈K0

One then obtains ln ITFPRODK VA (1, 0) =

(7.64)

7.8 Dynamic Ensemble: More Decompositions

⎛

 k∈C

+

205

VAk1 VAk0 (φ k1 − φ k0 ) ln ⎝ k1 C k0 CKL KL 01 

1/2 ⎞ ⎠

k (1/2)(φ k1 + φ k0 ) ln PVA,diff (1, 0)

k∈C 01

+



(1/2)(φ k1 + φ k0 ) ln ITFPRODkVA (1, 0)

k∈C 01

+



⎛ φ k1 ln

k∈N 1

−



k1 ⎝ VA

k0 ⎝ VA

K (1, 0) PVA

−1/2 ⎞ ⎠

K (1, 0) PVA

1/2 ⎞ ⎠

K (1, 0) PKL

k1 CKL

⎛ φ k0 ln





K (1, 0) PKL   K K + (1/2)(φ K1 + φ K0 ) − 1 ln PVA (1, 0)/PKL (1, 0) . k∈X 0

k0 CKL

There are six terms here, four of which are already more or less familiar. The first gives the aggregate effect of size change of the continuing production units. The second gives the aggregate effect of differential price change of the continuing units. The third gives the aggregate effect of TFP change of the continuing units. The fourth gives aggregate profitability of the entering units, where each unit’s profitability is deflated by the square root of the ratio of aggregate output and input price indices. The fifth gives aggregate profitability of the exiting units, where each unit’s profitability is inflated by the square root of the ratio of aggregate output and input price indices. Taken together, the fourth and fifth term give the net effect of entering and exiting units on aggregate productivity change. The final term concerns, again, a concavity discrepancy (which should be negligible, in general). Recall that expression (7.64) is only meaningful if there exist a sufficient number of production units exhibiting nonzero profit.

7.8 Dynamic Ensemble: More Decompositions In the case of continuing, exiting and entering production units, the value-added identity (5.9) can be detailed as VAK0 =



VAk0 + VAX 0

(7.65)

VAk1 + VAN 1 ,

(7.66)

k∈C 01

VAK1 =

 k∈C 01

206

7 The Top-Down Approach 2: Aggregate Total Factor Productivity Index

  where VAX 0 ≡ k∈X 0 VAk0 and VAN 1 ≡ k∈N 1 VAk1 . Thus, depending on the period we are looking at, aggregate value added is the sum of aggregate value added of the continuing units and aggregate value added of the exiting or entering units. Similarly, aggregate primary-input cost is the sum of primary-input cost of the continuing units and primary-input cost of the exiting or entering units: 

K0 CKL =

k0 X0 CKL + CKL

(7.67)

k1 N1 CKL + CKL ,

(7.68)

k∈C 01



K1 CKL =

k∈C 01

 k0 k1 X0 ≡  N1 where CKL k∈X 0 CKL and CKL ≡ k∈N 1 CKL . It is thereby assumed that 0 1 the sets X and N are both non-empty. We first consider the value-added identities (7.65)–(7.66). In the now familiar way aggregate value-added change can be decomposed as ln

VAK1 VAK0

=

 k∈C



VAk1 ψ k (1, 0) ln VAk0 01



+ ψ X N (1, 0) ln

VAN 1 VAX 0

,

(7.69)

where LM

VAk1

VAk0

, VAK1 VAK0



(k ∈ C 01 ) ψ (1, 0) ≡  VAk1 VAk0 VAN 1 VAX 0 , , + LM k∈C 01 LM VAK1 VAK0 VAK1 VAK0 k



VAN 1 VAX 0 , LM VAK1 VAK0 XN



. ψ (1, 0) ≡  VAk1 VAk0 VAN 1 VAX 0 , , + LM k∈C 01 LM VAK1 VAK0 VAK1 V AK0  Notice that these coefficients add up to 1; that is, k∈C 01 ψ k (1, 0)+ψ X N (1, 0) = 1. Each coefficient ψ k (1, 0) is the (normalized) mean share of continuing production unit k in aggregate value added, whereas ψ X N (1, 0) is the (normalized) mean share of the aggregate of exiting and entering units in aggregate value added. Put otherwise, to the panel of continuing units we have added an artificial continuing unit for which base period value added is given by VAX 0 and comparison period value added by VAN 1 . Aggregate value-added change can then again be written as a weighted geometric mean of individual value-added changes.

7.8 Dynamic Ensemble: More Decompositions

207

K (1, 0) from the leftApplying then product relations (7.4) and (7.8), moving PVA hand to the right-hand side, applying the definitions of aggregate and individual value-added based TFP change, and using product relations (7.5) and (7.9), we obtain  ln ITFPRODK ψ k (1, 0) ln ITFPRODkVA (1, 0) VA (1, 0) =

+



k∈C 01 k ψ k (1, 0) ln PVA,diff (1, 0)

k∈C 01

+



k

ψ (1, 0) ln

k∈C 01

+ψ

XN

(1, 0) ln

k1 /C k0 CKL KL

K1 /C K0 CKL KL

K (1, 0) (VAN 1 /VAX 0 )/PVA

N 1 /C X 0 )/P K (1, 0) (CKL KL KL

X 0 N 1 CKL /CKL + ψ X N (1, 0) ln . K1 /C K0 CKL KL

(7.70)

At the right-hand side of the equality sign there are five terms. The first is a weighted sum of unit-specific TFP changes of the continuing units. The second is the aggregate effect of differential price change at the output and input side of the continuing production units. The third can be interpreted as the aggregate effect of relative size change of the continuing units, where the size of each unit is measured by its primary-input cost share. The remaining two terms are related to the artificial continuing unit. The first of these two terms measures profitability change deflated by the ratio of aggregate price index numbers. The second measures relative size change, where the size of the artificial unit is measured by its primary-input cost share. Taken together, these two terms measure the net contribution of entering and exiting units to aggregate TFP change. If for all units k ∈ Kt and time periods t = 0, 1 value added equals primary kt , and if there is no differential price change, then expression input cost, VAkt = CKL (7.70) reduces to ln ITFPRODK VA (1, 0) = +ψ since

XN

(1, 0) ln



ψ k (1, 0) ln ITFPRODkVA (1, 0)

k∈C 01

K (1, 0) (VAN 1 /VAX 0 )/PVA N 1 /C X 0 )/P K (1, 0) (CKL KL KL

(7.71)

208

7 The Top-Down Approach 2: Aggregate Total Factor Productivity Index



k

ψ (1, 0) ln

+ψ

VAK1 /VAK0

k∈C 01



VAk1 /VAk0

XN

(1, 0) ln

VAN 1 /VAX 0

VAK1 /VAK0

= 0.

(7.72) The first part of expression (7.71) is the same as expression (7.21) except that the  sum of the weights k∈C 01 ψ k (1, 0) ≤ 1. It is also interesting to recall that ITFPRODkVA (1, 0) =

QkVA (1, 0) QkKL (1, 0)

=

k (1, 0) (VAk1 /VAk0 )/PVA k1 /C k0 )/P k (1, 0) (CKL KL KL

(k ∈ K),

(7.73)

which is production unit k’s deflated profitability change. Thus the two terms on the right-hand side of expression (7.71) exhibit the same structure. Put otherwise, the second term can be interpreted as TFP change of the artificial unit. Now turning to the primary-input cost identities (7.67)–(7.68), we obtain as decomposition of primary-input cost change, ln

K1 CKL K0 CKL

=





k

ω (1, 0) ln

k∈C 01

k1 CKL k0 CKL



+ω

XN

(1, 0) ln

N1 CKL X0 CKL

,

(7.74)

where

k1 k0 CKL CKL , LM K1 C K0 CKL k

KL

(k ∈ C 01 ) ω (1, 0) ≡ k1 k0 X 0 N 1  CKL CKL CKL CKL , K0 + LM , k∈C 01 LM K 1 K1 C K0 CKL CKL CKL KL

N 1 CX 0 CKL KL , LM K1 C K0 CKL XN

KL

. ω (1, 0) ≡ k1 k0 N 1 CX 0  CKL CKL CKL KL , , + LM k∈C 01 LM K1 C K0 K1 C K0 CKL CKL KL KL  Notice that these coefficients add up to 1; that is, k∈C 01 ωk (1, 0)+ωX N (1, 0) = 1. Each coefficient ωk is the (normalized) mean share of continuing production unit k in aggregate primary-input cost, whereas ψ X N (1, 0) is the (normalized) mean share of the aggregate of exiting and entering units in aggregate primary-input cost. Performing the same steps as above we obtain ln ITFPRODK VA (1, 0) = +

 k∈C 01



ωk (1, 0) ln ITFPRODkVA (1, 0)

k∈C 01 k ωk (1, 0) ln PVA,diff (1, 0)

7.9 Linking Value-Added Based to Gross-Output Based Productivity Change

−



k

ω (1, 0) ln

k∈C 01

+ω

XN

(1, 0) ln

VAk1 /VAk0

209

VAK1 /VAK0

K (1, 0) (VAN 1 /VAX 0 )/PVA

N 1 /C X 0 )/P K (1, 0) (CKL KL KL

VAN 1 /VAX 0 − ωX N (1, 0) ln . VAK1 /VAK0

(7.75)

At the right-hand side of the equality sign there are again five terms. The first is a weighted sum of unit-specific TFP changes of the continuing units. The second is the aggregate effect of differential price change at the output and input side of the continuing production units. The third can be interpreted as the aggregate effect of relative size change of the continuing units, where the size of each unit is measured by its value-added share. Notice the minus sign. The remaining two terms are related to the artificial continuing unit. The first of these two terms measures profitability change deflated by the ratio of aggregate price index numbers. The second measures relative size change, where the size of the artificial unit is measured by its value-added share. Notice here also the minus sign. Taken together, these two terms measure the net contribution of entering and exiting units to aggregate TFP change. If for all units k ∈ Kt and time periods t = 0, 1 value added equals primary input kt , and if there is no differential price change, then the same result cost, VAkt = CKL is obtained as in the previous case, namely expression (7.71).

7.9 Linking Value-Added Based to Gross-Output Based Productivity Change In the previous sections we considered the link between aggregate value-added based TFP change, ITFPRODK VA (1, 0), and individual value-added based TFP change, ITFPRODkVA (1, 0) (k ∈ K). In this section we consider the link between aggregate value-added based TFP change and individual gross-output based TFP change, ITFPRODkY (1, 0) (k ∈ K). Consider a static ensemble and let for all k ∈ K revenue R kt be equal to cost C kt kt , or profit kt be equal to (or value added VAkt be equal to primary input cost CKL 0) (t = 0, 1). Then Appendix C of Chapter 2 tells us that under mild conditions ln ITFPRODkVA (1, 0) =

LM(R k1 , R k0 ) ln ITFPRODkY (1, 0). LM(VAk1 , VAk0 )

(7.76)

210

7 The Top-Down Approach 2: Aggregate Total Factor Productivity Index

This identity relates value-added based to gross-output based TFP change. The factor LM(R k1 , R k0 )/LM(VAk1 , VAk0 ) is known as the Domar factor: the ratio of (mean) revenue over (mean) value added. By substituting expression (7.76) into expression (7.20) (or (7.26)) we obtain ln ITFPRODK VA (1, 0) = 

ψ k (1, 0)

k∈K

+



LM(R k1 , R k0 ) ln ITFPRODkY (1, 0) LM(VAk1 , VAk0 )

k ψ k ln PVA,diff (1, 0),

(7.77)

k∈K

 with ψ k (1, 0) = LM(φ k1 , φ k0 )/ k∈K LM(φ k1 , φ k0 ) and φ kt = VAkt /VAKt (k ∈ K; t = 0, 1). If, in addition, it is assumed that there is no differential price change, that is, k PVA,diff (1, 0) = 1 for all k ∈ K, then expression (7.77) reduces to 

LM(R k1 , R k0 ) ln ITFPRODkY (1, 0). k1 , VAk0 ) LM(VA k∈K (7.78) This relation connects aggregate value-added based TFP change to disaggregate (or individual) gross-output based TFP changes. Recall that each Domar factor is greater than or equal to 1, so that the sum of the combined factors preceding ln ITFPRODkY (1, 0) is also greater than or equal to 1. Each combined factor approximates (mean) revenue of production unit k over (mean) aggregate value added, and is called a Domar aggregation weight. The general interpretation of these weights is that the impact of unit k’s productivity change is greater than what corresponds to its share in aggregate production, as measured by value added, because of the intermediate deliveries to other units.6 The derivations in Appendix C of Chap. 2 and above, however, make clear that Domar factors and weights are mathematical artefacts. The second right-hand side term occurring in expression (7.77) measures the aggregate effect of differential price change, which can be interpreted in several ways. Anyway, the term reflects linkages between the individual production units. Following Ten Raa (2011) we could perhaps see it as an industrial organization effect. The last expression, (7.78), corresponds to the relation derived in a continuoustime framework, using a number of structural and behavioural assumptions, by ln ITFPRODK VA (1, 0) =

6A

ψ k (1, 0)

typical quote from Jorgenson (2018, 881) reads “A distinctive feature of Domar weights is that they sum to more than one, reflecting the fact that an increase in the growth of the industry’s productivity has two effects: the first is a direct effect on the industry’s output and the second an indirect effect via the output delivered to other industries as intermediate inputs.”

7.10 Growth Accounting (3)

211

Hulten (1978) and Gollop (1979, Proposition 2). See Jorgenson et al. (2005, 375) or Jorgenson and Schreyer (2013, 201) for a modern derivation under the customary neo-classical assumptions. The expression can also be seen as generalizing Proposition 3 of Ten Raa and Shestalova (2011), where in an input-output framework it appears that the Domar-weighted product of industry-specific gross-output based TFP indices (computed on optimal prices and quantities) is a measure of aggregate technological change. It is also interesting to note that expression (7.77) bears a likeness to a relation obtained in an input-output framework by Wolff (1994). This relation, however, is less simple to interpret and, moreover, depends on crucial Leontief-induced assumptions. If one does not want to make assumptions about the equality of revenue and cost and the absence of differential price change, then one must revert to the more intricate expressions which result from the substitution of expression (2.179) into expression (7.11) or (7.20) or (7.26). Interestingly, if expression (2.179) is substituted into expression (7.19) then one obtains the general, discrete-time version of the augmented Domar aggregation formula as derived originally by Jorgenson et al. (1987) under the customary neo-classical assumptions.7

7.10 Growth Accounting (3) As we have seen in Sections 2.2.3 and 2.3.2, any definition of a productivity index can be converted into a growth accounting relation. This also holds for the relations derived in this chapter, as will now be demonstrated on expression (7.17). By substituting back the definition of the aggregate value-added based TFP index, this expression can be rewritten as ln QK VA (1, 0) =



ψ k (1, 0) ln ITFPRODkVA (1, 0) +

k∈K

 k∈K



k

ψ (1, 0) ln

k (1, 0) PVA

K (1, 0) PVA

+



ψ k (1, 0) ln QkKL (1, 0).

(7.79)

k∈K

Recall that the second term atthe right-hand side of this expression vanishes k K (1, 0) = k if and only if ln PVA k∈K ψ (1, 0) ln PVA (1, 0); that is, the aggregate output price index is a Sato-Vartia index of the production-unit-specific output price indices. Let us also assume that kt = 0 for all k ∈ K, so that Eq. (7.76) holds. Let k D (1, 0) ≡ ψ k (1, 0)LM(R k1 , R k0 )/LM(VAk1 , VAk0 ) denote the Domar weight of the individual production unit k ∈ K. 7 See

Zheng (2005) for a reproduction of this derivation.

212

7 The Top-Down Approach 2: Aggregate Total Factor Productivity Index

Under both assumptions expression (7.79) reduces to ln QK VA (1, 0) =



D k (1, 0) ln ITFPRODkY (1, 0) +

k∈K



ψ k (1, 0) ln QkKL (1, 0).

(7.80)

k∈K

The final step consists in choosing functional forms for the primary input indices QkKL (.). In principle these are production-unit-specific. However, it is simpler to use the same form for all the units. A convenient choice is the Cobb-Douglas form QkKL (1, 0) ≡ (QkK (1, 0))α (QkL (1, 0))1−α (0 < α < 1).

(7.81)

Then expression (7.80) reduces to ln QK VA (1, 0) =



D k (1, 0) ln ITFPRODkY (1, 0) +

k∈K

α



ψ k (1, 0) ln QkK (1, 0) + (1 − α)

k∈K



ψ k (1, 0) ln QkL (1, 0).

(7.82)

k∈K

This is a typical growth accounting relation. Aggregate value-added based growth (for instance, GDP growth) is expressed as a weighted sum of production-unitspecific (for instance, industry-specific) components, gross-output based TFP growth, capital input growth, and labour input growth, respectively.

7.11 Conclusion In this chapter we discussed what Hulten (2001, 2010) called the “top-down view of sectoral productivity analysis, in which the aggregate TFP residual is the point of reference.” For linking aggregate value-added based TFP change to subaggregate (industrial or individual) TFP change and other sources it is recommended to use, in the case of a static ensemble, the decomposition given by expression (7.20), and, in the case of a dynamic ensemble, provided that the sets of exiting and entering units are not empty, the extended decomposition given by expression (7.70). Otherwise, the decomposition given by expression (7.64) must be used. If one wants to replace subaggregate value-added based TFP change by gross-output based TFP change, and one does not want to make additional assumptions, then expression (2.179) is the relation to employ. All these relations between aggregate and subaggregate TFP change are mathematical identities. Of course, these relations can be used only when detailed price and quantity data are available. In particular, one must be able to split

7.11 Conclusion

213

(nominal) value added and primary input cost change into price and quantity components. Though at the sectoral level this requirement is usually met, be it more or less satisfactorily, at the micro level adequate price deflators are almost without exception absent. In addition, at the micro level one usually is dealing with samples rather than universes of production units. The exact relations derived in this chapter then provide a starting point for assessing the influence of all the simplifications one has to invocate. The “bottom-up approach [. . . ] takes the universe of plants or firms as the fundamental frame of reference.” Microdata studies differ from sectoral studies in two main respects: 1. one unreservedly talks about productivity in terms of levels, and 2. there is not too much concern about the precise form of the aggregator function. The main methods employed in the bottom-up approach were reviewed in Chap. 5. That chapter contains also an extensive discussion of data-quality issues at the sectoral as well as the micro level. The link between the top-down and bottom-up approaches will be discussed in Chap. 9. First, however, we turn to a top-down decomposition in terms of levels.

Chapter 8

The Top-Down Approach 3: Aggregate Total Factor Productivity Level

8.1 Introduction This chapter basically continues the foregoing. In Chap. 7 three (time-) symmetric decompositions of aggregate value-added based total factor productivity (TFP) change were developed. In the present chapter a fourth decomposition will be developed. A notable difference with the earlier decompositions is that the development is cast in terms of levels rather than indices. This chapter unfolds as follows. As we have seen in Sect. 5.4.2, value-added based TFP is defined as real value added divided by real primary input. Section 8.2 refreshes these two concepts. Section 8.3 shows that aggregate value-added based TFP change essentially consists of three components: a weighted mean of individual value-added based TFP changes, a factor reflecting reallocation between the production units, and a factor reflecting relative price changes at the input and output side of the production units. Section 8.4 shows how the reallocation factor can be decomposed further into the contributions of the separate primary inputs. Section 8.5 shows how the decomposition derived in Sect. 8.3 is enhanced if valueadded based productivity change is replaced by gross-output based productivity change. Section 8.6 contains a key result: under mild restrictions on the relation between aggregate and individual deflators, if profit equals 0 then the reallocation factor vanishes, and aggregate value-added based TFP change equals the product of Domar-weighted individual gross-output based TFP changes. In Sect. 8.7 we take a further step by assuming that the production units share the same time-invariant production function. We then obtain a decomposition in terms of technical efficiency change, scale, and mix effects.

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2021 B. M. Balk, Productivity, Contributions to Economics, https://doi.org/10.1007/978-3-030-75448-8_8

215

216

8 The Top-Down Approach 3: Aggregate Total Factor Productivity Level

8.2 Prerequisites We consider a (static) ensemble (or set) K of consolidated production units, operating during a certain time period t in a certain country or region. The accounting framework was described in Sect. 5.2, to which the reader is referred. For any production unit, real value added of period t, RVAk (t, b), is nominal k (t, b), for period t relative value added, VAkt , divided by a suitable price index PVA to a certain reference period b. Recall expression (5.23). Rearranging this definition gives k VAkt = PVA (t, b)RVAk (t, b) (k ∈ K).

(8.1)

Nominal value added is here decomposed into a price component and a quantity component. Without loss of generality it may be assumed that period b lies somewhere in the past and that the ensemble K already existed in period b. The functional form of the price indices may vary over the production units; in particular, the price indices may be direct or chained or mixed. It is assumed that k (b, b) = 1, so that RVAk (b, b) = VAkb (k ∈ K); that is, for the reference period PVA real value added is identical to nominal value added. For the ensemble, considered as a higher-level production unit, we have a similar relation, K VAKt = PVA (t, b)RVAK (t, b),

(8.2)

K (t, b) is a value-added based price index for the ensemble K for period t where PVA relative to the reference period b. For the time being it is sufficient to assume that this index is estimated from (a sample of) the data underlying the individual price k (t, b) (k ∈ K). indices PVA The additivity of nominal value added implies a restriction on the functional form K (t, b), which can be seen as follows. Substituting expressions (8.1) and (8.2) of PVA into the fundamental adding-up relation (5.9) and dividing both sides by real value added of the ensemble, RVAK (t, b), delivers a relation between the price index for the ensemble and the individual price indices, K (t, b) = PVA

 RVAk (t, b) RVAK (t, b) k∈K

k (t, b). PVA

(8.3)

It is also important to observe that, unlike nominal value added—see again expression (5.9)—, real value added generally appears to be not additive. The dual to expression (8.3) is RVAK (t, b) =

 P k (t, b) VA

P K (t, b) k∈K VA

RVAk (t, b).

(8.4)

8.3 Decomposing Value-Added Based Total Factor Productivity Change

217

k (t, b), For any individual production unit, the real primary input of period t, XKL kt , divided by a suitable price index is defined as nominal primary input cost, CKL k PKL (t, b) for period t relative to the reference period b. Recall expression (5.25). Rearranging this definition gives kt k k = PKL (t, b)XKL (t, b) (k ∈ K). CKL

(8.5)

The corresponding relation for the ensemble reads Kt K K = PKL (t, b)XKL (t, b), CKL

(8.6)

Kt ≡  kt K where CKL k∈K CKL and PKL (t, b) is a suitable deflator for the primary input cost of the ensemble K. The additivity of nominal primary input cost then implies that K (t, b) = PKL

 Xk (t, b) KL

XK (t, b) k∈K KL

k PKL (t, b).

(8.7)

It is also important to observe that, unlike nominal primary input cost, real primary input generally appears to be not additive. The dual to expression (8.7) is K XKL (t, b) =

 P k (t, b) KL

P K (t, b) k∈K KL

k XKL (t, b).

(8.8)

8.3 Decomposing Value-Added Based Total Factor Productivity Change Value-added based TFP was defined in expression (5.28) as real value added divided by real primary input; that is, for each individual production unit, TFPRODkV A (t, b) ≡

RVAk (t, b) k (t, b) XKL

(k ∈ K)

(8.9)

and for the aggregate, TFPRODK V A (t, b) ≡

RVAK (t, b) K (t, b) XKL

.

(8.10)

An interesting interpretation of value-added based TFP is obtained by substituting expression (8.5) into expression (8.9). This yields

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8 The Top-Down Approach 3: Aggregate Total Factor Productivity Level

TFPRODkV A (t, b) =

k (t, b) PKL kt /RVAk (t, b) CKL

(k ∈ K);

(8.11)

that is, value-added based TFP is identical to primary input price divided by unit cost, both normalized to reference period b (see also Balk 2018, 92). If profit equals zero then unit cost equals value-added based price index, and primal TFP equals dual TFP (defined as input price index divided by output price index). Going from (an earlier) period 0 to (a later) period 1, individual TFP change is measured by the ratio TFPRODkV A (1, b)/TFPRODkV A (0, b) (k ∈ K), and aggregate K TFP change by TFPRODK V A (1, b)/TFPRODV A (0, b). Can the last ratio be written as a function of all the production-unit-specific ratios?1 In the foregoing chapter three (time-period-) symmetric decompositions of the aggregate TFP index were developed. It will now be shown that there is a fourth decomposition. To start with, as we have seen in Sect. 7.3, the aggregate nominal value-added ratio, for period 1 relative to period 0, can be decomposed as ln

VAK1

VAK0





VAk1 ψ k (1, 0) ln = VAk0 k∈K

,

(8.12)

where  ψ k (1, 0) ≡ 

LM

VAk1 VAk0 , VAK1 VAK0



k∈K LM



VAk1 VAk0 , VAK1 VAK0

(k ∈ K),

and the function LM(.) is the logarithmic mean.2 Aggregate value-added change, measured as a ratio, is thus equal to a weighted geometric mean of individual valueadded changes. Notice that the coefficients ψ k (1, 0) add up to 1. Each coefficient is the (normalized) mean share of production unit k in aggregate nominal value added. Similarly, as we have seen in Sect. 7.4, the aggregate primary input cost ratio, for period 1 relative to period 0, can be decomposed as ln

K1 CKL

K0 CKL

=

 k∈K

ωk (1, 0) ln

k1 CKL k0 CKL

,

(8.13)

where

1 Recall

that the logarithm of any such ratio, if in the neighbourhood of 1, can be interpreted as a growth rate. 2 Recall that the logarithmic mean, for any two strictly positive real numbers a and b, is defined by LM(a, b) ≡ (a − b)/ ln(a/b) if a = b and LM(a, a) ≡ a.

8.3 Decomposing Value-Added Based Total Factor Productivity Change

 LM ωk (1, 0) ≡



k1 k0 CKL CKL K1 , C K0 CKL KL



k∈K LM

219



k1 CKL C k0 , KL K1 K0 CKL CKL

 (k ∈ K).

Aggregate primary-input cost change is thus equal to a weighted geometric mean of individual primary-input cost changes. Notice that the coefficients ωk (1, 0) add up to 1. Each coefficient is the (normalized) mean share of production unit k in aggregate primary-input cost. Substituting the expressions (8.1) and (8.2) into (8.12), and substituting the expressions (8.5) and (8.6) into (8.13) delivers, respectively, ln

K (1, b)RVAK (1, b) PVA

=

K (0, b)RVAK (0, b) PVA



k

ψ (1, 0) ln

k (1, b)RVAk (1, b) PVA

,

k (0, b)RVAk (0, b) PVA

k∈K

(8.14)

and ln

K (1, b)X K (1, b) PKL KL

K (0, b)X K (0, b) PKL KL

=



k

ω (1, 0) ln

k∈K

k (1, b)X k (1, b) PKL KL k (0, b)X k (0, b) PKL KL

.

(8.15) Subtracting equation (8.15) from Eq. (8.14), moving the aggregate price indices from the left-hand side to the right-hand side, using the fact that the coefficients add up to 1, and applying definition (8.10), delivers ln 



RVAk (1, b) ψ (1, 0) ln RVAk (0, b) k∈K

TFPRODK VA (1, b)

=

TFPRODK VA (0, b)

k

−



k

ω (1, 0) ln

k∈K

(8.16)

k (1, b) XKL

k (0, b) XKL

+ ln Prel (1, 0),

where 

ln Prel (1, 0) ≡

k

ψ (1, 0) ln

k∈K

 k∈K

k

ω (1, 0) ln

k (1, b)/P K (1, b) PVA VA

−

k (0, b)/P K (0, b) PVA VA

k (1, b)/P K (1, b) PKL KL

k (0, b)/P K (0, b) PKL KL

(8.17)

.

This expression concerns mean relative price change at the output side minus mean relative price change at the input side of the production units. If there is no relative k (t, b) = P K (t, b) and P k (t, b) = P K (t, b) for price change at all, that is, PVA VA KL KL

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8 The Top-Down Approach 3: Aggregate Total Factor Productivity Level

all k ∈ K and all time periods considered, then ln Prel (1, 0) = 0. However, such a situation is unlikely to occur. The following observation is more interesting. If ln

K (1, b) PVA

=

K (0, b) PVA



k

ψ (1, 0) ln

k (1, b) PVA

(8.18)

k (0, b) PVA

k∈K

and ln

K (1, b) PKL

K (0, b) PKL

=



k

ω (1, 0) ln

k∈K

k (1, b) PKL

(8.19)

k (0, b) PKL

then ln Prel (1, 0) = 0. Technically, the assumptions expressed in the foregoing two expressions mean that the price indices for aggregate value added and primary input are (second-stage) Sato-Vartia (S-V) indices of the price indices for the individual production units. As such, these two expressions provide specifications of expressions (8.3) and (8.7), respectively. Expression (8.16) can be decomposed in several ways. Applying definition (8.9), the expression can be written either as ln

TFPRODK VA (1, b)

=

TFPRODK VA (0, b)



ψ k (1, 0) ln

TFPRODkVA (1, b)

+

TFPRODkVA (0, b)

k∈K

(8.20)

k (1, b)  X KL ψ k (1, 0) − ωk (1, 0) ln − a  + ln Prel (1, 0), k (0, b) X KL k∈K or as ln

TFPRODK VA (1, b)

TFPRODK VA (0, b)



=

k∈K

k

ω (1, 0) ln

TFPRODkVA (1, b) TFPRODkVA (0, b)

+

(8.21)

 RVAk (1, b) k k  ψ (1, 0) − ω (1, 0) ln − a + ln Prel (1, 0), RVAk (0, b) k∈K or as the arithmetic mean of the former two expressions, ln

TFPRODK VA (1, b)

TFPRODK VA (0, b)

=

1 TFPRODkVA (1, b) k k ψ (1, 0) + ω (1, 0) ln 2 TFPRODkVA (0, b) k∈K

(8.22)

8.3 Decomposing Value-Added Based Total Factor Productivity Change

221

⎛ ⎞

1/2 k (1, b) X k (1, b)  RVA KL ψ k (1, 0) − ωk (1, 0) ⎝ln + − a  ⎠ k (0, b) X k (0, b) RVA KL k∈K + ln Prel (1, 0), where a  , a  and a  are arbitrary scalars. Either of the expressions (8.20)–(8.22) constitutes the fourth decomposition. In each case aggregate TFP change consists of three main factors. The first factor is a (with respect to time) symmetrically weighted mean of the production-unit-specific TFP changes, where the weights in expression (8.20) are nominal-value-added shares, in expression (8.21) are nominal-primaryinput-cost shares, and in expression (8.22) are the means of those shares. The second factor measures reallocation,3 in expression (8.20) from the viewpoint of primary inputs, in expression (8.21) from the viewpoint of output (real value added), and in expression (8.22) from a combined viewpoint. The third factor, which is the same in the three expressions, measures net mean relative price change,4 and vanishes if there is no relative price change or if S-V indices are used, as in expressions (8.18) and (8.19). Let us, by way of example, have a closer look at the reallocation factor in expression (8.20),

k (1, b)  X KL ln RALKL (1, 0) ≡ ψ k (1, 0) − ωk (1, 0) ln − a . k (0, b) X KL k∈K (8.23) That indeed reallocation is being measured can be seen by selecting the arbitrary K (1, b)/X K (0, b)). Then the reallocation factor reduces to scalar as a  = ln(XKL KL ln RALKL (1, 0) =

 k∈K



ψ (1, 0) − ω (1, 0) ln k

k



k (1, b)/X K (1, b) XKL KL

k (0, b)/X K (0, b) XKL KL

, (8.24)

which measures the impact of the change of relative real primary input between periods 0 and 1. Notice that the weights add up to 0; that is,  the  k (1, 0) − ωk (1, 0) = 0. Thus the right-hand side of expression (8.24) is ψ k∈K

3 There

is a large literature on the topic of reallocation, but no universal definition of the concept. Though the word ‘reallocation’ seems to have a normative undertone, in the present context it can best be read as ‘dynamics’: the process of (relative) growth and decline of production units. 4 The occurrence of such a factor in a decomposition of aggregate productivity change was discussed in Sect. 7.6. The central argument was that “. . . even if at the level of individual commodities the price is the same for every buyer/seller then the ‘price’ of the composite input and output commodity will vary over the production units.”

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8 The Top-Down Approach 3: Aggregate Total Factor Productivity Level

a covariance. A positive value of the reallocation factor means that primary inputs have shifted to production units whose value-added share ψ k (1, 0) is greater than their primary-input cost share ωk (1, 0).5 k (t, b)/X K (t, b) (k ∈ K) As real primary input is not additive, the relatives XKL KL donot to 1. Shares can be obtained by selecting the arbitrary scalar as a  =   add up  k k ln k∈K XKL (1, b)/ k∈K XKL (0, b) . Then the reallocation factor reduces to ln RALKL (1, 0) =







ψ (1, 0) − ω (1, 0) ln k

k

k∈K

k (1, b)/ XKL



Xk (1, b) k∈K KL k k (0, b) XKL (0, b)/ k∈K XKL

.

(8.25) Another alternative is to select the arbitrary scalar as a  = k (1, b)/X k (0, b)). Then the reallocation factor reduces to ln(XKL KL ln RALKL (1, 0) =

 k∈K

ψ k (1, 0) ln

k (1, b)/ XKL k (0, b)/ XKL

 



k∈K ω

k ω k∈K (XKL (1, b))

k (1,0)

k ωk (1,0) k∈K (XKL (0, b))

k (1, 0)

. (8.26)

Technically, exp{a  } is now the S-V quantity index of the individual primary input k (1, b)/X k (0, b) (k ∈ K). quantity indices XKL KL It is important to keep in mind that the reallocation factor itself is invariant to the choice made for the scalar a  . Its components, however, will be influenced.

8.4 Decomposing the Reallocation Factor into Contributions of Separate Primary Inputs The reallocation factor ln RALKL (1, 0), as defined in the previous section, reads in terms of joint primary inputs capital (K) and labour (L). To see the contributions of these two input classes separately one needs some additional prerequisites. The first is that there are separate, production-unit-specific deflators for nominal capital input cost and nominal labour input cost; that is, we have, analogous to expression (8.5), kt k CK = PKk (t, b)XK (t, b) (k ∈ K)

5 An

(8.27)

alternative interpretation in terms of primary inputs shifting to production units whose output k (t, b), is higher than average, VAKt /X K (t, b), as suggested per unit of primary inputs, VAkt /XKL KL k (t, b) = P K (t, b) (k ∈ K); that is, if there is no by Bollard et al. (2013), holds only if PKL KL differential price change at the input side.

8.4 Decomposing the Reallocation Factor into Contributions of Separate. . .

223

and CLkt = PLk (t, b)XLk (t, b) (k ∈ K),

(8.28)

k (t, b) and X k (t, b) are real where PKk (t, b) and PLk (t, b) are price indices and XK L inputs, for capital and labour respectively. As nominal primary input cost is additive kt = C kt + C kt ), it is clear that there must exist a relation between the joint price (CKL K L k (t, b) and the separate price indices P k (t, b) and P k (t, b), or between index PKL K L k (t, b) and the separate real inputs X k (t, b) and X k (t, b). joint real input XKL K L The second assumption then concerns the way these relations are modeled. It will here be assumed that joint real primary input is a convex combination of real capital and labour input; that is,

 α k  1−α k k k XLk (t, b) XKL (t, b) ≡ XK (t, b) (0 < α k < 1; k ∈ K),

(8.29)

k k (t, b) ≡ α k ln XK (t, b) + (1 − α k ) ln XLk (t, b) (k ∈ K). ln XKL

(8.30)

or

Then 

k ωk (1, 0) ln XKL (t, b)

(8.31)

k∈K

=



k∈K

=α

K



k ωk (1, 0)α k ln XK (t, b) +

ωk (1, 0)(1 − α k ) ln XLk (t, b)

k∈K

K ln XK (t, b) + (1 − α K ) ln XLK (t, b),

where αK ≡



ωk (1, 0)α k

(8.32)

k ωk (1, 0)α k ln XK (t, b)/α K

(8.33)

ωk (1, 0)(1 − α k ) ln XLk (t, b)/(1 − α K ).

(8.34)

k∈K

K (t, b) ln XK

≡



k∈K

ln XLK (t, b) ≡



k∈K

The reallocation factor, as represented by expression (8.26), can then be written as ln RALKL (1, 0) =



 k (1, b) K (1, b)  XK XK k k K + − α ln ψ (1, 0) α ln k (0, b) K (0, b) XK XK k∈K

(8.35)

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8 The Top-Down Approach 3: Aggregate Total Factor Productivity Level







ψ (1, 0) (1 − α ) ln k

k

k∈K

XLk (1, b) XLk (0, b)

K

− (1 − α ) ln

XLK (1, b) XLK (0, b)

 ,

where the contributions of the two primary input classes are nicely separated. Expression (8.35) bears a stark resemblance to the reallocation term(s) figuring in the decompositions obtained by Baldwin et al. (2013, expression (10)) and Erumban et al. (2019, expression (9)). Notice that expression (8.29) represents a production-unit-specific CobbDouglas aggregator function. This choice is not completely arbitrary, but in line with previous examples. In conventional empirical work the α k ’s are estimated and not production-unit-specific.

8.5 Introducing Gross-Output Based Total Factor Productivity Change On the right-hand side of expressions (8.20)–(8.22) we see weighted means of production-unit-specific value-added based TFP change. As gross-output (or revenue) stays closer to the actual operations of a production unit, it is desirable to replace value-added by gross-output based TFP change. Gross-output based TFP is defined as real revenue divided by real KLEMS input; that is, TFPRODkY (t, b) ≡

Y k (t, b) k XKLEMS (t, b)

(k ∈ K),

(8.36)

where nominal revenue is supposed to be decomposable as R kt = PRk (t, b)Y k (t, b) (k ∈ K)

(8.37)

and nominal (total) cost as kt kt k k + CEMS = PKLEMS (t, b)XKLEMS (t, b) (k ∈ K). C kt = CKL

(8.38)

Also nominal intermediate input cost is supposed to be decomposable as kt k k = PEMS (t, b)XEMS (t, b) (k ∈ K). CEMS

(8.39)

k k (t, b) are suitable deflators for nomi(t, b), and PEMS In the above PRk (t, b), PKLEMS nal revenue, nominal (total) cost, and nominal intermediate inputs cost, respectively; k k (t, b), and XEMS (t, b) their real counterparts. Decompositions and Y k (t, b), XKLEMS kt of primary input cost, CKL , and nominal value added, VAkt , were already provided by expressions (8.5) and (8.1), respectively.

8.5 Introducing Gross-Output Based Total Factor Productivity Change

225

Based on the fact that nominal value added plus intermediate inputs cost equals kt (k ∈ K), it is assumed that revenue, R kt = VAkt + CEMS 

Y k (1, b) ln Y k (0, b)





RVAk (1, b) LM(VAk1 , VAk0 ) = ln LM(R k1 , R k0 ) RVAk (0, b)

(8.40)



k1 , C k0 ) k XEMS (1, b) LM(CEMS EMS ln , + k LM(R k1 , R k0 ) XEMS (0, b) where LM(.) is the logarithmic mean. Basically this means that the revenuebased output quantity index for period t relative to period 0 is defined as the Montgomery-Vartia (M-V) index of the value-added based output quantity index and the intermediate inputs quantity index. In particular one should notice that the weights do not precisely add up to 1, due to the concavity of the logarithmic mean. Expression (8.40) is equivalent to the dual relation between the corresponding price indices,

k (1, b) PVA PRk (1, b) LM(VAk1 , VAk0 ) ln ln = (8.41) k (0, b) LM(R k1 , R k0 ) PRk (0, b) PVA

k1 , C k0 ) k (1, b) P LM(CEMS EMS EMS ln . + k (0, b) LM(R k1 , R k0 ) PEMS Expression (8.40) can be rearranged as

RVAk (1, b) ln RVAk (0, b) −

  k Y (1, b) LM(R k1 , R k0 ) ln = Y k (0, b) LM(VAk1 , VAk0 )

k1 , C k0 ) LM(CEMS EMS

LM(VAk1 , VAk0 )

ln

k XEMS (1, b) k XEMS (0, b)

(8.42)

.

Value-added based TFP was defined by expression (8.9). Hence, for period 1 relative to period 0, ln

TFPRODkV A (1, b) TFPRODkV A (0, b)



RVAk (1, b) = ln RVAk (0, b)

− ln

k (1, b) XKL k (0, b) XKL

=

(8.43)



  k k1 , C k0 ) k LM(CEMS XEMS (1, b) LM(R k1 , R k0 ) Y (1, b) EMS − ln ln k Y k (0, b) LM(VAk1 , VAk0 ) LM(VAk1 , VAk0 ) XEMS (0, b)

226

8 The Top-Down Approach 3: Aggregate Total Factor Productivity Level

− ln

k (1, b) XKL

k (0, b) XKL

.

Next, it is assumed that

k k1 , C k0 ) k (1, b) XKLEMS (1, b) XKL LM(CKL KL ln ln = k k (0, b) LM(C k1 , C k0 ) XKLEMS (0, b) XKL

(8.44)



k1 , C k0 ) k XEMS LM(CEMS (1, b) EMS ln , + k LM(C k1 , C k0 ) XEMS (0, b) which means that the total (KLEMS) input quantity index for period 1 relative to period 0 is defined as the M-V index of the primary input quantity index and the intermediate inputs quantity index. Notice that expression (8.44) is equivalent to the dual relation between the corresponding price indices, ln

k PKLEMS (1, b)

k PKLEMS (0, b)



k1 , C k0 ) k (1, b) PKL LM(CKL KL ln = k (0, b) LM(C k1 , C k0 ) PKL

(8.45)



k1 , C k0 ) k (1, b) PEMS LM(CEMS EMS ln . + k (0, b) LM(C k1 , C k0 ) PEMS Gross-output based TFP was defined by expression (8.36). Hence, for period 1 relative to period 0, ln

TFPRODkY (1, b) TFPRODkY (0, b)



Y k (1, b) = ln Y k (0, b)



 − ln

k XKLEMS (1, b) k XKLEMS (0, b)

.

(8.46)

After substituting expression (8.44) this becomes ln

TFPRODkY (1, b)

TFPRODkY (0, b)



Y k (1, b) = ln Y k (0, b)





k1 , C k0 ) k (1, b) XKL LM(CKL KL − ln k (0, b) LM(C k1 , C k0 ) XKL



k1 , C k0 ) k LM(CEMS XEMS (1, b) EMS − ln , k LM(C k1 , C k0 ) XEMS (0, b)

(8.47)

or, rearranged, 

Y k (1, b) ln Y k (0, b)



 = ln

TFPRODkY (1, b) TFPRODkY (0, b)



k1 , C k0 ) k (1, b) XKL LM(CKL KL ln + k (0, b) LM(C k1 , C k0 ) XKL

8.5 Introducing Gross-Output Based Total Factor Productivity Change



k1 , C k0 ) k XEMS LM(CEMS (1, b) EMS ln . + k LM(C k1 , C k0 ) XEMS (0, b)

227

(8.48)

Substituting expression (8.48) into the right-hand side of expression (8.43) finally delivers

 TFPRODkVA (1, b) TFPRODkY (1, b) LM(R k1 , R k0 ) ln ln = LM(VAk1 , VAk0 ) TFPRODkVA (0, b) TFPRODkY (0, b)

k1 , C k0 ) k (1, b) LM(CKL XKL LM(VAk1 , VAk0 ) KL + − (8.49) ln k (0, b) LM(C k1 , C k0 ) LM(R k1 , R k0 ) XKL



k1 , C k0 ) k1 , C k0 ) k LM(CEMS LM(CEMS (1, b) XEMS EMS EMS , + − ln k LM(C k1 , C k0 ) LM(R k1 , R k0 ) XEMS (0, b) which is a specific case of expression (2.179) in Appendix C of Chap. 2. The factor in front of the square brackets, LM(R k1 , R k0 )/LM(VAk1 , VAk0 ), is known as the Domar factor: the ratio of (mean) nominal revenue over (mean) nominal value added. Another decomposition of value-added based TFP change in terms of grossoutput based TFP change plus some additional factors, but avoiding the Domar factor, was obtained by Basu and Fernald (2002). If presented in the same mathematical language as above this alternative would exhibit the same structure— featuring real primary input change and real intermediate inputs change—with more complicated weights. It is useful to recall the specific assumptions made in the course of the derivation of expression (8.49): • For each production unit, the revenue-based output quantity index is an M-V index of the value-added based output quantity index and the intermediate inputs quantity index. • For each production unit, the total input quantity index is an M-V index of the primary input quantity index and the intermediate inputs quantity index. The functional forms of the quantity indices for value added, primary input, and intermediate inputs are left unspecified. However, if these indices were themselves M-V indices of the underlying price and quantity data then, due to the Consistencyin-Aggregation of M-V indices,6 both the revenue-based output quantity index and the total input quantity index would be M-V indices of the underlying data. Further, as Diewert (1978) has shown, at any given data point an M-V index differentially approximates to the second order any other time-symmetric index, such as Fisher or Törnqvist. Thus, if for revenue-based output quantity and total

6 See

Balk (2008, 111).

228

8 The Top-Down Approach 3: Aggregate Total Factor Productivity Level

input quantity instead of M-V indices other time-symmetric indices were used, then the equality sign in expression (8.49) must be replaced by an approximation sign. In the limit, that is, if period 0 approaches period 1, then approximation tends to equality.7

8.6 The Zero Profit Case It is important to consider what happens if for all the production units at any time period profit equals zero; that is, kt = 0 (k ∈ K). Such a situation materializes if the unit user cost of all the capital assets is based on endogenous interest rates (which, then, are production-unit-specific), or if actual profit is considered as cost of an additional input called entrepreneurial activity (the price of which, then, is kt production-unit-specific). Zero profit is equivalent to R kt = C kt or VAkt = CKL (k ∈ K). The first consequence is that the coefficients ψ k (1, 0) and ωk (1, 0) (k ∈ K) are identical, so that expressions (8.20)–(8.22) reduce to

TFPRODK VA (1, b)

= TFPRODK VA (0, b)

 TFPRODkVA (1, b) k + ln Prel (1, 0). ψ (1, 0) ln TFPRODkVA (0, b) k∈K

ln

(8.50)

Quite surprisingly, we conclude that the entire reallocation factor has vanished. The second consequence, easily checked, is that expression (8.49) reduces to ln

TFPRODkVA (1, b) TFPRODkVA (0, b)



TFPRODkY (1, b) LM(R k1 , R k0 ) ln = (k ∈ K). LM(VAk1 , VAk0 ) TFPRODkY (0, b) (8.51)

7 Diewert

(2015a) replaced the M-V indices in the two expressions (8.40) and (8.44) by Laspeyres and Paasche indices, which are only first-order differential approximations, and found that, under the zero-profit condition discussed below, the ratio of value-added based and gross-output based TFP growth rates approximates the asymmetric Domar factors, R k0 /VAk0 and R k1 /VAk1 , respectively. Two further assumptions, namely that geometric means can be approximated by arithmetic means and that Laspeyres and Paasche revenue-based output quantity indices are equal, made it possible to obtain a similar result in the case of Fisher indices. It is left to the reader to judge whether Diewert’s derivation method is “much simpler” than the one presented above. Using Australian data, Calver (2015) presents evidence on the variability of the Domar factors over industries and through time and on the accuracy of the approximations.

8.6 The Zero Profit Case

229

Notice that under the zero profit condition the Domar factors may alternatively be k1 , C k0 ) (k ∈ K); that is, reciprocals of (mean) expressed as LM(C k1 , C k0 )/LM(CKL KL primary input cost shares. In words, expression (8.51) means that value-added based TFP growth equals gross-output based TFP growth times the Domar factor.8 By substituting expression (8.51) into expression (8.50), one obtains

TFPRODK VA (1, b)

= TFPRODK VA (0, b)

 TFPRODkY (1, b) k + ln Prel (1, 0), D (1, 0) ln TFPRODkY (0, b) k∈K

ln

(8.52)

where the coefficients D k (1, 0) ≡ ψ k (1, 0)(LM(R k1 , R k0 )/LM(VAk1 , VAk0 )) (k ∈ K) measure (mean) individual nominal revenue over (mean) aggregate nominal value added; they are known as Domar weights. Their sum is greater than or equal to 1. Conventional wisdom attaches some economic interpretation to this fact. The derivation given in Section 8.5 (as well as in Appendix C of Chap. 2), however, makes clear that it is nothing but a mathematical artefact, caused by moving intermediate inputs cost from the denominator of a gross-output based productivity index to the numerator with a minus sign to get a value-added based productivity index. Recall from Sect. 8.3 that the relative price change term, ln Prel (1, 0), vanishes if S-V indices are used to aggregate production-unit-specific input and output price indices. It is useful to summarize our findings in the form of a theorem. Theorem 8.1 Let for any production unit k ∈ K suitable deflators for value k (t, b), added (VA), primary input (KL), and intermediate inputs (EMS) be given: PVA k k k PKL (t, b), and PEMS (t, b), respectively. Let the deflator for revenue, PR (t, b), be k (t, b) and P k (t, b), and let the deflator for total input cost, an M-V index of PVA EMS k k (t, b) and P k (t, b). Let the deflator for PKLEMS (t, b), be an M-V index of PKL EMS K aggregate value added, PVA (t, b), and the deflator for aggregate primary input cost, K (t, b), be S-V indices of the corresponding production-unit-specific deflators PKL k (t, b) and P k (t, b) (k ∈ K), respectively. If for any production unit profit PVA KL equals zero, that is, kt = 0 (k ∈ K), then aggregate value-added based TFP growth can be expressed (a) as a value-added-share-weighted or primaryinput-cost-share-weighted mean of production-unit-specific value-added based TFP growth,

8A

consequence is that the covariance of value-added based TFP growth and some other variable equals the covariance of gross-output based TFP growth and this variable times the Domar factor. It is good to keep this in mind when meeting such covariances in the literature on firm dynamics.

230

8 The Top-Down Approach 3: Aggregate Total Factor Productivity Level

ln

TFPRODK VA (1, b)

=

TFPRODK VA (0, b)



k

ψ (1, 0) ln

k∈K

TFPRODkVA (1, b)

TFPRODkVA (0, b)

,

(8.53)

or (b) as a Domar-weighted sum of production-unit-specific gross-output based TFP growth, ln

TFPRODK VA (1, b) TFPRODK VA (0, b)

=

 k∈K

k

D (1, 0) ln

TFPRODkY (1, b)

TFPRODkY (0, b)

.

(8.54)

In official statistical practice the assumptions concerning the use of M-V and S-V indices are not fulfilled because simpler indices such as Laspeyres or Fisher are used as deflators. Then the expressions in the Theorem hold only approximately.9 The better the indices actually used approximate M-V and S-V indices the better the final approximation will be. As the accuracy of any approximation hinges on the variance, over time and over production units, of the underlying price and quantity data, closeness of the time periods compared and similarity of the production units involved are crucial for obtaining a good approximation. The continuous time analogue of expression (8.54) plays a key role in Oulton’s (2001), (2016) analysis of what is usually called Baumol’s ‘Growth Disease’. See Appendix for details.

8.7 Going Beyond Total Factor Productivity Change Recall that production-unit specific gross-output based TFP was defined by expression (8.36). Using the assumption incorporated in expression (8.44) we obtained expression (8.47), here repeated as

TFPRODkY (1, b)

= (8.55) TFPRODkY (0, b)

 k  k k (1, b) X (1, b) X Y (1, b) EMS k10 k10 KL ln − ϑKL − ϑEMS (k ∈ K), ln ln k (0, b) k Y k (0, b) XKL XEMS (0, b)

ln

k10 ≡ LM(C k1 , C k0 )/LM(C k1 , C k0 ) and ϑ k10 k1 in which ϑKL EMS ≡ LM(CEMS , KL KL k0 )/LM(C k1 , C k0 ) (k ∈ K). Expression (8.55) is an example of the Solow CEMS residual: the growth rate of aggregate output minus a weighted mean of the growth rates of aggregate primary and intermediate inputs. However, as we did

9 Likewise,

sion (8.53).

Diewert’s (2016) expression (1.20) can be considered as an approximation of expres-

8.7 Going Beyond Total Factor Productivity Change

231

not introduce the usual neo-classical assumptions we cannot consider the Solow residual as a measure of technological change, or the impact of innovation (as Jorgenson 2018 does). In the absence of such assumptions, the Solow residual is what it is. In order to make progress we need to decompose the residual into economically meaningful components representing technical efficiency change, technological change, scale effects, and input and output mix effects. For this we need to assume the existence of a time-period-specific technology to which the production units belonging to the ensemble K have access, with features so regular that analytical techniques can be used, and which can be estimated from available data. This topic will be explored further in Chap. 10. It might, however, at this point be useful to provide a simple illustration. It is assumed that the technology can be represented by a simple, time-invariant CobbDouglas function; that is, k k Y k (τ, b) = k (τ, b)(XKL (τ, b))αKL (XEMS (τ, b))αEMS (k ∈ K, τ = 0, 1), (8.56) where 0 < k (τ, b) ≤ 1 measures the technical efficiency of production unit k ∈ K. By substituting expression (8.56) into expression (8.55) we obtain

ln

TFPRODkY (1, b) TFPRODkY (0, b)



k (1, b) = ln k (0, b)

k10 + (αEMS − ϑEMS ) ln



 k10 + (αKL − ϑKL ) ln

k XEMS (1, b) k XEMS (0, b)

k (1, b) XKL

k (0, b) XKL (8.57)

(k ∈ K).

One immediately recognizes here the familiar components of an empirical measure of TFP change: the first factor on the right-hand side of expression (8.57) measures technical efficiency change, whereas the second and third factor measure scale-andk10 and input-mix effects. These two factors vanish if the empirical cost shares ϑKL k10 ϑEMS —which, as we know, approximately add up to 1—coincide with the elasticities αKL and αEMS —which add up to 1 if constant returns to scale is assumed—, respectively. There is no role for technological change, as the production function is assumed to be time-invariant. By substituting expression (8.57) into expression (8.52) we obtain for aggregate value-added based TFP growth the following decomposition:

TFPRODK V A (1, b)

= TFPRODK V A (0, b)   k   (1, b) D k (1, 0) ln k (0, b)

ln

k∈K

(8.58)

232

8 The Top-Down Approach 3: Aggregate Total Factor Productivity Level

+



D

k

k10 (1, 0)(αKL − ϑKL ) ln

k∈K

+



k10 D k (1, 0)(αEMS − ϑEMS ) ln

k∈K

k (1, b) XKL

k (0, b) XKL

k XEMS (1, b) k XEMS (0, b)

+ ln Prel (1, 0).

Apart from some details, such as the possible role of fixed costs and the relative price change factor, this expression corresponds with the decomposition advocated by Petrin and Levinsohn (2012). Petrin and Levinsohn called the second and third factor on the right-hand side reallocation. However, as we have seen already, reallocation has vanished as a result of the zero profit assumption. Hence, as indicated, it is more appropriate to consider the second and third factor as measuring the aggregate effect of scale and input mix change.10

8.8 Conclusion A key element in any system of productivity statistics comprising various levels of aggregation (economy, industry, firm) is a relation connecting a productivity index at a certain level to those at lower levels. In this chapter such a relation was derived, without invoking any of the usual neo-classical assumptions (a technology exhibiting constant returns to scale, competitive input and output markets, optimizing behaviour of the agents, and perfect foresight), just by mathematically manipulating the various accounting relations. In the process also the famous Domar factor could be demystified to being nothing but a mathematical artefact. Our key relations (8.20)–(8.22) link higher level value-added based productivity growth to a weighted mean of lower level productivity growth, a reallocation factor (reflecting the aggregate effect of lower level dynamics), and a relative price change factor. If zero profit is imposed, then the three relations become identical and the reallocation factor vanishes. Moreover, lower level value-added based productivity growth can then be replaced by Domar-weighted gross-output based productivity growth. Finally, if the ‘correct’ deflators are used, then the relative price change factor also vanishes. All this underscores the fact that by and large in empirical work, at various levels of aggregation, reallocation and relative price change tend to play a minor role vis-avis lower level productivity growth as such. The minor role of relative price change

10 An

important part of the Petrin and Levinsohn (2012) article was devoted to an empirical comparison of the decomposition in expression (8.58), minus the relative price change factor, with a concept called ‘BHC productivity change’. However, the two concepts appear to measure different things, which makes a comparison rather meaningless.

Appendix: Analyzing Baumol’s ‘Growth Disease’

233

is probably also due to the scarcity of genuine, production-unit-specific, deflators at low levels of aggregation.

Appendix: Analyzing Baumol’s ‘Growth Disease’ The situation considered is an economy, consisting of a fixed number of industrial sectors, over discrete time periods (years) . . . , t − 1, t, t + 1, . . .. Under the hypotheses of Theorem 8.1 aggregate value-added based productivity growth between periods t − 1 and t is a weighted sum of sectoral productivity growth, ln

TFPRODK VA (t, b)



=

TFPRODK VA (t − 1, b)

D (t, t − 1) ln

TFPRODkY (t, b)

k

k∈K

TFPRODkY (t − 1, b)

,

(8.59) where the coefficients D k (t, t − 1) measure (mean) sectoral nominal revenue over (mean) aggregate nominal value added; they are known as Domar weights. The continuous-time analogue was derived by Oulton (2001, expression (19)), (2016, expression (4)) under neo-classical assumptions. Hartwig and Krämer (2019), in an article commemorating Baumol’s ‘Growth Disease’, called it “Oulton’s theorem”. Baumol’s original diagnosis, now more than 50 years old, concerns the factors responsible for rise or fall of aggregate productivity growth over time. It is stated that aggregate productivity growth will decline as service industries with low productivity growth receive an increasing weight in the economy. Thus we must look at differences of growth rates; that is, differences of the form ln

TFPRODK VA (t + 1, b)

− ln

TFPRODK VA (t, b)

TFPRODK VA (t, b)

TFPRODK VA (t − 1, b)

(8.60)

.

As we see, there are two groups of factors: the Domar weights, and the sectoral productivity growth rates. The contribution of both can be assessed by using the well-known Bennet decomposition, ln

TFPRODK VA (t + 1, b)

− ln

TFPRODK VA (t, b)





TFPRODK VA (t − 1, b)

D (t + 1, t) − D (t, t − 1) k

k

TFPRODK VA (t, b)

 ln

+ ln

=

TFPRODkY (t + 1, b)

k∈K



TFPRODkY (t, b) TFPRODkY (t − 1, b)

TFPRODkY (t, b)

 +

(8.61)

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8 The Top-Down Approach 3: Aggregate Total Factor Productivity Level



D (t + 1, t) + D (t, t − 1) k

k



 ln

TFPRODkY (t + 1, b) TFPRODkY (t, b)

k∈K

− ln 





D (t + 1, t) − D (t, t − 1) ln k

D (t + 1, t) + D (t, t − 1) k

k

TFPRODkY (t + 1, b)



TFPRODkY (t − 1, b)

 ln

TFPRODkY (t + 1, b)

k∈K

− ln

=

TFPRODkY (t − 1, b)

k∈K





TFPRODkY (t, b)

k

+

TFPRODkY (t, b)

TFPRODkY (t, b) TFPRODkY (t − 1, b)

 .

The situation is more complex than in a continuous-time setting. Sufficient conditions for rising aggregate productivity growth are 1. increasing Domar weight combined with productivity growth, and/or decreasing Domar weight combined with productivity decline, and 2. rising productivity growth for all sectors. But these conditions are by no means necessary. Declining average productivity growth (as measured by the second factor) can be counterbalanced by increasing (or decreasing) Domar weights combined with productivity growth (or decline) (as measured by the first factor). Empirical data must decide. Curiously, only the first condition, called structural change, got attention from researchers. Oulton (2001) considered the UK over the years 1973–1995 and found a positive structural change effect. Next, Oulton (2016) considered 18 European countries over the years 1970–2007, based on the EU KLEMS release of 2009/2011. In this case he concluded that “The effect of structural change is seen to be predominantly negative.” Hartwig and Krämer (2019), first, replicated Oulton’s (2001) results for the UK. Second, they extended the analysis to all the G7 countries (except Canada), based on EU KLEMS releases of 2009/2011 and 2017/2018. The overall impact of structural change was found to be negative, due to the combination of increasing Domar weights but productivity decline.

Chapter 9

Connecting the Two Approaches

9.1 Introduction In Chap. 2 we considered productivity measurement for a single, consolidated production unit. In terms of levels, productivity was defined as real output divided by real input. Real output or input means nominal output or input deflated by some output- or input-specific price index, respectively. For the production unit considered, productivity change (through time) can then be measured as a difference or a ratio of productivities. In the latter case it appears that productivity change can also be defined directly as output quantity index divided by input quantity index. The choice of the output and input concepts appears to be critical. As discussed in Chap. 2, three main models can be distinguished: KLEMS-Y, KL-VA, and K-CF. Taking the composition of capital input cost into account, as set out in Chap. 3, two more models can be added, namely KL-NVA and K-NCF. Assuming profit (defined as revenue minus total cost) to be equal to zero, or, what amounts to the same, replacing an exogenous interest rate by an endogenous rate, multiplies the number of models by two. And the introduction of a capital utilization rate further complicates the picture. Thus, there is a lot of choice here, with not unimportant empirical consequences, as illustrated in Sect. 2.6. Production units exist at various levels of aggregation. There are plants, enterprises, industries, countries, to name just some types of production units materializing in analytic studies of productivity change. Usually such units appear, more or less naturally, arranged into higher-level aggregates: a number of plants belonging to the same enterprise; a certain type of enterprises defining an industry; a number of industries defining the ‘measurable’ part of a national economy; national economies making up the world economy. It is not difficult to perceive several sorts of hierarchy here. As in any of these situations the structure is the same—there is an ensemble of production units, and the ensemble itself may or may not be considered as a higher

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2021 B. M. Balk, Productivity, Contributions to Economics, https://doi.org/10.1007/978-3-030-75448-8_9

235

236

9 Connecting the Two Approaches

level production unit—, it is interesting to study the relation between aggregate productivity (change) and productivity (change) of the aggregate. As we have seen there are basically two approaches here. Chapter 5 reviewed the so-called bottom-up approach, the approach that takes an ensemble of individual production units as the fundamental frame of reference. The top-down approach was the subject of the next three chapters, namely Chap. 6 on labour productivity, and Chaps. 7 and 8 on total factor productivity. The present chapter investigates the connection between the two approaches, bottom-up and top-down. Characteristic of the approach taken in this chapter is that aggregate productivity (change) should be interpreted as productivity (change) of the aggregate. It will be shown that this implies restrictive relations between the productivity measures involved, including the weights of the individual production units, and the type of mean employed. For instance, it appears that, assuming additivity, value-added based total factor productivities and output based weights require an harmonic mean. The order of this chapter is as follows. Section 9.2 defines the problem. Sections 9.3 and 9.4 consider value-added based total factor productivity and labour productivity, respectively. Section 9.5 considers gross-output based productivity. Section 9.6 concludes.

9.2 The Connection Defined We consider an ensemble (or set) Kt of consolidated production units, operating during a certain time period t in a certain country or region. The accounting framework was discussed in Sect. 5.2, to which the reader is referred. Let the productivity level1 of unit k at period t be denoted by PRODkt . The generic definition here employed is: real output divided by real input. Output can be measured as revenue (also called ‘gross output’) (R kt ) or as value added (VAkt ). kt kt + C kt ), as primary Input can be measured as total cost (C kt ≡ CKLEMS = CKL EMS kt kt input cost (CKL ), as labour input cost (CL ), or as total labour quantity (Lkt , where a common measurement unit must be used for the various types of labour). In all these cases, ‘real’ means nominal deflated by some price index, which may or may not be specific for each production unit. It is supposed that the reference period b, that is, the period for which the price index equals 1 by definition, is the same for all the units. Each production unit comes with some measure of relative size (or, importance) in the form of a weight θ kt . For each period these weights usually but not necessarily add up to 1. The question which weights θ kt are appropriate when a choice has been made for the productivity levels PRODkt (k ∈ Kt ) has received some attention in the 1 The

relation between levels and indices was discussed in Sect. 5.4.

9.2 The Connection Defined

237

literature. Given that somehow PRODkt is output divided by input, should the weight θ kt be output- or input-based? And how is this related to the type of mean—arithmetic, geometric, or harmonic? The literature does not provide us with definitive answers. Indeed, as long as one stays in the bottom-up framework it is unlikely that a convincing answer can be obtained. We need the complementary top-down view. A bit formally, the problem can be posed as follows. Generalizing the definitions introduced in Chap. 5, aggregate productivity is a weighted ‘mean’ of the individual productivities PRODt ≡ M(θ kt , PRODkt ; k ∈ Kt ),

(9.1)

where the ‘mean’ M(.) can be arithmetic, geometric, or harmonic; the weights θ kt may or may not add up to 1; and PRODkt can be value-added based total factor productivity (TFP), labour productivity, or simple labour productivity, or grossoutput based TFP, or simple labour productivity. Microdata studies, where the production units considered are plants or enterprises, then concentrate on the distributional characteristics of the (large) set of individual productivities PRODkt , the development over time of aggregate productivity PRODt , and the decomposition of this development with respect to several types of firms. Sectoral studies, where the production units considered are industries (according to some national or international classification), are usually interested in industryspecific productivity change and its components, such as capital deepening and labour-composition change. The number of industries distinguished is generally so small that separate attention can be devoted to each specific case. In both situations the ensemble Kt itself can be considered as a (consolidated) t higher level production unit. Using the same definitions, its productivity PRODK t can be calculated. In general it will then turn out, explicitly or implicitly, that t the productivity of the aggregate, PRODK t , is unequal to aggregate productivity, t 2 PROD , as defined above. Microdata analysis is usually not interested in the productivity of the aggregate. As a consequence the problem of the choice of weights and type of mean arises. Sectoral analysis usually does show productivity change of the aggregate (e.g., the economy) alongside productivity change of the component industries, however without an explicit discussion of their relationship. If there is some comparison of

2 PRODt

can be considered as a 2-stage aggregation procedure: first PRODkt aggregates over basic inputs and outputs per production unit k, and then PRODt aggregates over all the units k ∈ Kt . t PRODK t can be considered as a 1-stage aggregate of the same basic inputs and outputs. See Diewert (1980, 495–498) for a similar discussion in terms of variable profit (or, value added) functions and technological change (assuming continuous time and differentiability), and the PPI Manual (2004, Chapter 18) for the cases of revenue, intermediate-input-cost, and value-added t based price indices. Notice the double role of the variable t in PRODK t .

238

9 Connecting the Two Approaches

aggregate productivity change and productivity change of the aggregate at all, then their difference is classified as an “unexplained residual”. In this chapter we will pursue whether it is possible to find a set of weights and a type of ‘mean’ such that PRODt = PRODK t ; t

(9.2)

that is, such that aggregate productivity can be interpreted as productivity of the aggregate. As we know, there are a number of options here. We start with the case where t PRODkt and PRODK t is value-added based TFP. Next we consider value-added based labour productivity. Finally we turn to gross-output based labour productivity and TFP respectively.

9.3 Value-Added Based Total Factor Productivity 9.3.1 General Case The top-down approach starts with the adding-up relation (5.9). This relation tells us that nominal value added of the ensemble Kt is the sum of nominal value added of the individual production units k making up this ensemble. It is also important to recall that the KL-VA accounting identities of the individual units, given by expression (5.2), are structurally identical to the KL-VA accounting identity of the ensemble (5.8). This means that we can treat the ensemble as a higher level production unit, and that all the definitions of indices and levels can be applied to the individual units and the ensemble in the same way. For production unit k, recall that real value added, RVAk (t, b), is nominal value added, VAkt , divided (or, as one says, deflated) by a suitable price index with k (t, b). Rewriting this definition gives reference period b, PVA k VAkt = PVA (t, b)RVAk (t, b) (k ∈ Kt ).

(9.3)

Nominal value added is here decomposed into a price component and a quantity component. For the ensemble we have similarly K (t, b)RVAK (t, b), VAK t = PVA t

t

t

(9.4)

K (t, b) is a value-added based price index for the ensemble Kt for period where PVA t relative to the reference period b. This index is supposed to be estimated from a sample of enterprises and products. Substituting expressions (9.3) and (9.4) into expression (5.9) and dividing both Kt (t, b) delivers a relation between real value added of sides by the price index PVA t

9.3 Value-Added Based Total Factor Productivity

239

the ensemble and real value added of the individual units, RVAK (t, b) =

 P k (t, b) VA

t

t P K (t, b) k∈Kt VA

RVAk (t, b).

(9.5)

Thus, unlike nominal value added, real value added is generally not additive. For any individual production unit, real primary input is defined by k kt k XKL (t, b) ≡ CKL /PKL (t, b) (k ∈ Kt )

(9.6)

k (t, b) is a suitable deflator for the primary input cost of production unit where PKL k. For the ensemble the corresponding definition reads

K Kt K (t, b) ≡ CKL /PKL (t, b), XKL t

t

t

(9.7)

K (t, b) is a suitable deflator for the primary input cost of the ensemble where PKL Kt (t, b) and inserting on the t K . Now, dividing both sides of expression (9.5) by XKL k (t, b)/X k (t, b) = 1 (k ∈ Kt ), one obtains right-hand side XKL KL t

RVAK (t, b) t

Kt

XKL (t, b)

=

 P k (t, b) Xk (t, b) RVAk (t, b) VA KL K (t, b) X K (t, b) X k (t, b) PVA KL KL t

k∈Kt

t

.

(9.8)

At both sides of this identity we see value-added based TFP, as introduced in Chap. 5, for the aggregate and the individual production units, respectively. Thus expression (9.8) can be written as TFPRODK VA (t, b) = t

 P k (t, b) Xk (t, b) VA KL

K (t, b) P K (t, b) XKL k∈Kt VA t

t

TFPRODkVA (t, b).

(9.9)

This is our desired result. It means that if PRODkt is defined as value-added based TFP, TFPRODkVA (t, b), then the appropriate weights are given by φ kt ≡

k (t, b) k PVA XKL (t, b)

K (t, b) X K (t, b) PVA KL t

t

(k ∈ Kt ).

(9.10)

 If these weights are used, then aggregate productivity k∈Kt φ kt PRODkt can be interpreted as the value-added based TFP of the ensemble, considered as a higherlevel production unit. Notice that the weights φ kt (k ∈ Kt ) do not necessarily add up

240

9 Connecting the Two Approaches

to 1. Thus, though expression (9.9) is a weighted sum of individual productivities, it is not a genuine mean.3 There is, however, another way of looking at expression (9.9). To see this, k (t, b)/P Kt (t, b))TFPRODk (t, b) is so-called revenue TFP; that is, notice that (PVA VA VA k (t, b) but by the the result of deflating VAkt not by its unit-k-specific deflator PVA t K (t, b). Weighing these revenue TFPs by real input ensemble-specific deflator PVA t k K shares XKL (t, b)/XKL (t, b) then delivers aggregate TFP. Notice that these real input shares also do not necessarily add up to 1. Another interesting viewpoint emerges when the weights φ kt are decomposed as φ

kt

≡



k (t, b)X k (t, b) PVA KL k∈Kt

k (t, b)X k (t, b) PVA KL

k∈Kt

K (t, b)X K (t, b) PVA KL t

k (t, b)X k (t, b) PVA KL

t

(k ∈ Kt ).

(9.11)

The first factor on the right-hand side is a share, adding up to 1 when summed over all k ∈ Kt , whereas the second factor can be considered as an adjustment factor common to all the individual productivities TFPRODkVA (t, b). Then expression (9.9) is a weighted (arithmetic) mean of adjusted individual productivities. Expression (9.9) as a relation between aggregate and individual productivities is, however, not unique. To see this, instead of the adding-up relation for value added (5.9), we consider the adding-up relation for primary input cost, Kt CKL = t



kt CKL .

(9.12)

k∈Kt

Employing definitions (9.6) and (9.7), expression (9.12) can be rewritten as K XKL (t, b) = t

 P k (t, b) KL

P K (t, b) k∈Kt KL t

k (t, b). XKL

(9.13)

Thus, like real value added, real primary input is generally non-additive. Individual and aggregate real value added were defined by expressions (9.3) and t (9.4) respectively. Now, dividing both sides of expression (9.13) by RVAK (t, b) and inserting on the right-hand side RVAk (t, b)/RVAk (t, b) = 1 (k ∈ Kt ), one obtains K (t, b) XKL t

RVAK (t, b) t

=

 P k (t, b) RVAk (t, b) Xk (t, b) KL KL . Kt (t, b) RVAKt (t, b) RVAk (t, b) P k∈Kt KL

(9.14)

Again employing the definition of value-added based TFP, expression (9.14) can be written as

3 Expression

(2016).

(9.9) is the model underlying GEAD-TFP as implemented by Calver and Murray

9.3 Value-Added Based Total Factor Productivity

241

 −1 −1  P k (t, b) RVAk (t, b)  t k KL TFPROD TFPRODK (t, b) = (t, b) , t t VA VA P K (t, b) RVAK (t, b) k∈Kt KL (9.15) or Kt

TFPRODVA (t, b) =

 P k (t, b) RVAk (t, b)  KL K (t, b) RVAK (t, b) PKL t

k∈Kt

t

TFPRODkVA (t, b)

−1

−1 . (9.16)

This is our alternative result. Thus, aggregate total factor productivity can also be obtained as a weighted harmonic sum of individual productivities, with weights ψ kt ≡

k (t, b) PKL RVAk (t, b)

Kt (t, b) PKL

t RVAK (t, b)

(k ∈ Kt ).

(9.17)

Notice that these weights do not necessarily add up to 1. However, like above, the weights ψ kt can be decomposed such that expression (9.16) can be considered as a weighted (harmonic) mean of adjusted individual productivities. It is interesting to compare the structure of the two sets of weights φ kt and ψ kt . Both are hybrid. The first are based on real primary input shares and relative valueadded price levels, whereas the second are based on real output (value added) shares and relative primary input price levels. As such, these weights are unobservable. Summarizing, there is no unique relation between the individual TFPs and the TFP of the aggregate. One must either multiply the individual productivities by weights φ kt and add up, or use weights ψ kt and take the harmonic sum.

9.3.2 Additivity Imposed We observed that both real value added and real primary input are generally nonadditive.4 A sufficient condition for additivity is that deflators for the ensemble, for value added as well as primary input, are Paasche-type indices. This can be seen as follows. Additivity of real value added, RVAK (t, b) = t



RVAk (t, b),

(9.18)

k∈Kt

is, by inserting the definitions of real value added, equivalent to

4 Notice

that we are considering here additivity of production units, which is different from additivity of commodities as considered in Sect. 5.2.

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9 Connecting the Two Approaches

1 Kt (t, b) PVA

=

 VAkt k∈Kt

t VAK t

1 k (t, b) PVA

(9.19)

.

This relation simply expresses that the value-added based deflator for the ensemble is a Paasche index of the deflators for the individual production units (recall that nominal value added is additive). Similarly, if the primary-input based deflator for the ensemble is a Paasche index of the unit-specific deflators, then K XKL (t, b) = t



k XKL (t, b).

(9.20)

k∈Kt

It is straightforward to check that if conditions (9.18) and (9.20) are satisfied, then instead of expression (9.9) we obtain the simpler expression5 TFPRODK VA (t, b) = t

 Xk (t, b) KL

K (t, b) XKL k∈Kt t

TFPRODkVA (t, b),

(9.21)

and instead of expression (9.16) we obtain the simpler expression

t TFPRODK VA (t, b)

−1  RVAk (t, b)  k TFPROD = (t, b) t VA RVAK (t, b) k∈Kt

−1 .

(9.22)

In both cases the weights now do add up to 1. The result is simple to summarize. If one weighs individual TFPs by real input shares then the arithmetic mean must be used, but if one weighs them by real output shares then the harmonic mean must be used to arrive at an interpretable result.6 Mixing this leads to unwanted effects. For example, combining the harmonic mean with real input shares leads to understating the productivity of the aggregate:

−1  Xk (t, b)  KL TFPRODkVA (t, b) t K XKL (t, b) k∈Kt

−1 ≤ TFPRODK VA (t, b), t

(9.23)

and combining the arithmetic mean with real output shares leads to overstating the productivity of the aggregate:

5 This

is the model underlying the CSLS decomposition as implemented by Calver and Murray (2016). 6 This reflects the denominator rule and the numerator rule of Fare and Karagiannis (2017), respectively.

9.3 Value-Added Based Total Factor Productivity

 RVAk (t, b) k∈Kt

t RVAK (t, b)

243

TFPRODkVA (t, b) ≥ TFPRODK VA (t, b). t

(9.24)

Both results rest on combining the mathematical fact that an harmonic mean is always less than or equal to an arithmetic mean with expressions (9.21) and (9.22). Equality in expressions (9.23) and (9.24) holds only when all the individual productivities TFPRODkVA (t, b) (k ∈ Kt ) are the same. Interestingly, the left-hand side of expression (9.24) is the target variable considered by Olley and Pakes (1996). Also the type of mean matters. A geometric mean is greater than or equal to an harmonic mean, which implies that, using expression (9.22), RVAk (t,b)/RVAKt (t,b)   t TFPRODkVA (t, b) ≥ TFPRODK VA (t, b).

(9.25)

k∈Kt

Such a geometric mean was a target variable considered by Melitz and Polanec (2015). It is thus seen to overstate productivity of the aggregate. Finally, the choice of the weights is important. De Loecker and Konings (2006), for instance, concluded that there was no clear consensus  among researchers.t In their own work they used employment based shares Lkt / k∈Kt Lkt = Lkt /LK t to weigh value-added based TFPs. The aggregate productivity then, however, appears to be a biased measure of the value-added based TFP of the aggregate, since  Lkt t TFPRODkVA (t, b) = TFPRODK VA (t, b) + Kt t L t

k∈K

 k∈Kt



k (t, b) XKL Lkt TFPRODkVA (t, b). t − Kt (t, b) LK t XKL

(9.26)

The bias, that is, the second term on the right-hand side, is the covariance between TFP and difference of labour share and real input share. Its magnitude, and its sign, is of course an empirical matter. In an earlier study, Basu and Fernald (2002) consideredvalue-added based kt Kt t ) TFPs weighted with nominal value-added shares, that is, k∈Kt (VA /VA TFPRODkVA (t, b). This is also a biased measure of value-added based TFP of the ensemble, unless specific conditions apply.

244

9 Connecting the Two Approaches

9.4 Value-Added Based Labour Productivity 9.4.1 General Case For value-added based labour productivity the setup of the previous section can simply be repeated. The only thing one needs to do is replacing real primary input by real labour input. Thus, for the individual production units real labour input is defined as XLk (t, b) ≡ CLkt /PLk (t, b) (k ∈ Kt ).

(9.27)

Likewise, for the ensemble XLK (t, b) ≡ CLK t /PLK (t, b), t

t

t

(9.28)

 t t where CLK t ≡ k∈Kt CLkt and PLk (t, b) and PLK (t, b) are suitable deflators for the labour cost of the individual production units and the ensemble, respectively. In Sect. 5.4.2 labour productivity was defined as real value added divided by real labour input. Starting from the numerator of the labour productivity of the ensemble the decomposition appears to be LPRODK VA (t, b) = t

 P k (t, b) Xk (t, b) VA L

K (t, b) X K (t, b) PVA L t

k∈Kt

t

LPRODkVA (t, b),

(9.29)

whereas starting from the denominator one obtains

t LPRODK VA (t, b)

−1  P k (t, b) RVAk (t, b)  k L LPROD = (t, b) t t VA P K (t, b) RVAK (t, b) k∈Kt L

−1 .

(9.30) Both expressions relate value-added based labour productivity of the ensemble, considered as a higher level production unit, to the labour productivities of the constituent production units. Notice that the weights do not necessarily add up to 1. However, like shown before, these weights can be decomposed such that expressions (9.29) and (9.30) can be considered as weighted means of adjusted individual labour productivities.

9.4.2 Simple Labour Productivity Two special cases deserve our attention. First, when for labour the simple sum quantity index is used then for the individual production units labour productivity is given by

9.4 Value-Added Based Labour Productivity

LPRODkVA (t, b) =

RVAk (t, b) CLkt /PLk (t, b)

=

RVAk (t, b) CLkb QkL (t, b)

245

=

RVAk (t, b) (CLkb /Lkb )Lkt

(k ∈ Kt ), (9.31)

and real labour input is given by XLk (t, b) = (CLkb /Lkb )Lkt , where Lkτ denotes production unit k’s total labour quantity in period τ (τ = b, t). Real labour input is labour quantity valued at reference period unit values. For the ensemble similar expressions hold. Substitution, for the individual production units as well as for the ensemble, into expression (9.29), and some simplification, delivers the following expression, t  P k (t, b) Lkt RV Ak (t, b) RV AK (t, b) VA = . t Kt (t, b) LKt t Lkt LK t P k∈Kt VA

(9.32)

This is an expression in terms of simple labour productivities. Put otherwise, expression (9.32) can be written as SLPRODK VA (t, b) = t

 P k (t, b) Lkt VA k t t t SLPRODVA (t, b). K K L t PVA (t, b)

(9.33)

k∈K

It is quite natural to assume that the labour input of the ensemble, considered as a higher level production of the constituent unit, is a simple sum of the labour inputs t t units; that is, LK t = k∈Kt Lkt . Then the fractions Lkt /LK t (k ∈ Kt ) are labour shares, adding up to 1. Notice, however, that these labour shares are premultiplied by relative price levels, so that the weights of the labour productivities themselves do not necessarily add up to 1. However, like shown before, these weights can be decomposed such that expression (9.33) can be considered as weighted mean of adjusted individual labour productivities. The relative price levels vanish when k (t, b) = P Kt (t, b) (k ∈ Kt ); that is, when there is no differential output price PVA VA change among the production units. There is, however, another way of looking at expression (9.33). To see this, recall k (t, b)/P Kt (t, b))(RV Ak (t, b)/Lkt ) is so-called revenue labour producthat (PVA VA tivity; that is, the result of deflating value added, VAkt , not by its unit-k-specific k (t, b) but by the ensemble-specific deflator P Kt (t, b). Weighing these deflator PVA VA t revenue labour productivities by labour shares Lkt /LK t then delivers aggregate labour productivity. Finally, we notice that expression (9.32) or (9.33) is the model underlying the Generalized Exactly Additive Decomposition (GEAD) (see Sect. 5.5.2). Thus the GEAD is consistent in the sense that aggregate productivity is equal to productivity of the aggregate. But it now turns out that an alternative decomposition can be developed. To see this, notice that, by substituting expression (9.31) into expression (9.30)  and using the product relation CLkt /CLkt = PLk (t, t  )QkL (t, t  ), expression (9.30) reduces to

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9 Connecting the Two Approaches



t SLPRODK VA (t, b)

−1 RVAk (t, b)  k SLPROD = (t, b) t t VA t C K t /LK t RVAK (t, b) k∈Kt L 

CLkt /Lkt

−1 .

(9.34) This expression can be used to develop an alternative to the GEAD. If the unit labour prices are the same across production units, that is, CLkt /Lkt = α t t (k ∈ Kt ) and CLK t /LK t = α, then expression (9.34) further reduces to

t SLPRODK VA (t, b)

−1  RVAk (t, b)  k SLPROD = (t, b) t VA RVAK (t, b) k∈Kt

−1 .

(9.35)

An alternative route to obtain this expression is the following. The assumpt tion unit labour prices across production units implies that LK t =  of equal kt k∈Kt L . Then, starting with this relation, dividing its left- and right-hand sides t by RVAK (t, b), and inserting at the right-hand side RVAk (t, b)/RVAk (t, b) = 1 (k ∈ Kt ) one obtains expression (9.35). Notice that the weights in expression (9.35) do not add up to 1, unless additivity holds. However, like shown before, these weights can be decomposed such that expression (9.35) can be considered as weighted (harmonic) mean of adjusted individual labour productivities.

9.4.3 Additivity Imposed  t k Second, let us assume that additivity holds; that is, RVAK (t, b) = k∈Kt RVA  t (t, b) and LK t = k∈Kt Lkt . Instead of expression (9.29) we then obtain SLPRODK VA (t, b) = t

 Lkt SLPRODkVA (t, b), Kt t L t

(9.36)

k∈K

and instead of expression (9.30) we obtain

t SLPRODK VA (t, b)

−1  RVAk (t, b)  k SLPROD = (t, b) t VA RVAK (t, b) k∈Kt

−1 .

(9.37)

Now in both cases the weights add up to 1. The labour-share weighted arithmetic mean of simple labour productivities appears to be equal to the real-value-addedshare weighted harmonic mean of simple labour productivities, and both are equal to the simple labour productivity of the aggregate. Mixing means and weights leads to undesirable results. Using the general relation between harmonic and arithmetic means, we conclude that

9.5 Gross-Output Based Productivity



247

−1  Lkt  k SLPROD (t, b) t VA LK t t

−1 ≤ SLPRODK VA (t, b) t

(9.38)

k∈K

 RVAk (t, b) k∈Kt

t RVAK (t, b)

SLPRODkVA (t, b) ≥ SLPRODK VA (t, b). t

(9.39)

Thus, a labour-share weighted harmonic mean of simple labour productivities understates labour productivity of the aggregate, while a real-value-added-share weighted arithmetic mean overstates this. The second inequality was also obtained by van Biesebroeck (2008), though in a less direct way. Also here the type of mean matters. As a geometric mean is less then or equal to an arithmetic mean, we conclude that a labour-share weighted geometric mean of simple labour productivities understates simple labour productivity of the aggregate; that is,  

SLPRODkVA (t, b)

Lkt /LKt t

≤ SLPRODK VA (t, b). t

(9.40)

k∈Kt

Such a geometric mean features prominently in Melitz and Polanec (2015). In the Appendix of their paper decompositions based on the left-hand side and the righthand side of expression (9.40) are empirically compared. The geometric mean was also considered as target variable for firms by Hyytinen and Maliranta (2013), and Maliranta and Määttänen (2015). Notice that the right-hand side of expression (9.36) is the target variable of the TRAD and CSLS decompositions considered in Sect. 5.5.2. Thus these decompositions are consistent; that is, aggregate productivity can be interpreted as productivity of the aggregate. However, underlying this result is the assumption of additivity, which is quite restrictive.

9.5 Gross-Output Based Productivity There are not so many microdata studies dealing with the concept of gross-output based productivity. For any individual production unit gross-output based TFP is defined as TFPRODkY (t, b) ≡

Y k (t, b) k XKLEMS (t, b)

=

R kt /PRk (t, b) kt k CKLEMS /PKLEMS (t, b)

(k ∈ Kt ).

(9.41)

In the numerator we have real revenue, Y k (t, b); that is, nominal revenue R kt deflated by a k-specific revenue based price index with reference period b, PRk (t, b).

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9 Connecting the Two Approaches

k In the denominator we have real KLEMS input XKLEMS (t, b); that is, nominal kt KLEMS input cost CKLEMS deflated by a k-specific KLEMS input based price index k with the same reference period, PKLEMS (t, b); so that the ratio TFPRODkY (t, b) is a dimensionless variable. Similarly, gross-output based simple labour productivity is defined as

SLPRODkY (t, b) ≡

Y k (t, b) (k ∈ Kt ); Lkt

(9.42)

that is, real revenue per unit of labour. The dimension of this variable is money of reference period b. Suppose that we have access to production-unit specific data such that either of these measures can be compiled. Which weights would be appropriate? We review a number of typical studies.

9.5.1 Simple Labour Productivity Let us start with the target variable considered by Baily et al. (2001). This is SLPRODkY (t, b), though instead of unit-specific deflators industry-level deflators were used. The labour unit was an hour worked. These simple labour  productivities t were weighed by labour shares; that is, by Lkt /LK t = Lkt / k∈Kt Lkt . Thus, aggregate productivity was compiled as7 t LPRODK BBH (t, b)

≡

 k∈Kt



Lkt

k∈Kt

 SLPRODkY (t, b) Lkt

=

k∈Kt

Y k (t, b)

LK t t

.

(9.43) But what precisely does this mean? To see this, we must return to the accounting identities discussed in Sect. 5.2 and notice that     t R kt = R kk t + R K t . (9.44) k∈Kt

k∈Kt k  ∈Kt

Thus, total revenue is the sum of revenue obtained by internal deliveries (recall that  R kk t is the revenue obtained by unit k from delivering to unit k  , and that R kkt = 0) t and aggregate revenue R K t , which is the revenue obtained by the ensemble Kt , when the ensemble is considered as a consolidated production unit. Now, imposing additivity, that is, defining the aggregate revenue based price index as a Paasche

7 This

measure was also considered by Foster et al. (2001). Actually, two variants were considered, one where the labour unit was an hour worked and one where it was a worker. The geometric alternative was employed by Hyytinen and Maliranta (2013) for plants; labour quantity was thereby measured in full-time equivalents.

9.5 Gross-Output Based Productivity

249

index of the k-specific revenue based price indices, 1 Kt

PR (t, b)



≡

R kt



k∈Kt

k∈Kt

1 , R kt PRk (t, b)

(9.45)

implies that expression (9.44) can be written as PRK (t, b) t



 

Y k (t, b) =

k∈Kt



R kk t + R K t , t

(9.46)

k  ∈Kt

k∈Kt

or 

 Y (t, b) = k

k∈Kt



k  ∈Kt



R kk t

PRK (t, b)

k∈Kt

t

RK t t

+

PRK (t, b) t

(9.47)

.

If we define real revenue of the ensemble Kt , considered as a consolidated t t t production unit, by Y K (t, b) ≡ R K t /PRK (t, b), then expression (9.47) can be simplified to 

 Y k (t, b) =

k∈Kt



k  ∈Kt



R kk t

PRK (t, b)

k∈Kt

t

+ Y K (t, b). t

(9.48)

Substituting expression (9.48) into expression (9.43) and applying definition (9.42) to the ensemble considered as a production unit delivers the following relation: t LPRODK BBH (t, b)

=

t SLPRODK Y (t, b)

 1+

k∈Kt



k  ∈Kt t K R t



R kk t

.

(9.49)

Since nominal revenue is non-negative, it appears that aggregate BBH productivity overstates simple labour productivity of the aggregate, and that the magnitude of the bias depends on the relative extent of the intra-ensemble deliveries. The bias vanishes only when there are no intra-ensemble deliveries. Foster et al. (2001) considered simple labour productivities weighed by real output shares; that is, LPRODK FHK (t, b) ≡ t





k∈Kt

Y k (t, b) SLPRODkY (t, b). k k∈Kt Y (t, b)

(9.50)

Applying the arithmetic-harmonic mean inequality, and definitions (9.42) and (9.43) subsequently, we obtain t LPRODK FHK (t, b)

 ≥

k∈Kt

Y k (t, b)

t LK t

= LPRODK BBH (t, b). t

(9.51)

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9 Connecting the Two Approaches

The right-hand side is familiar from the foregoing. Combining expressions (9.51) and (9.49) we may conclude that, even in the case of industries exhibiting no t intra-ensemble trade, LPRODK FHK (t, b) overstates simple labour productivity of the t aggregate, SLPRODK Y (t, b).

9.5.2 Total Factor Productivity We now turn to TFPRODkY (t, b), a key variable considered by Bartelsman and Dhrymes (1998). They had industry and time effects removed econometrically, but that does not need to concern us here. The individual gross-output based TFPs  k k were weighed by real KLEMS input shares XKLEMSRR (t, b)/ k∈Kt XKLEMS (t, b), so that aggregate total factor productivity was compiled as TFPRODK BD (t, b) ≡ t





k∈Kt



= 

k XKLEMS (t, b)

k∈Kt

k∈Kt

k XKLEMS (t, b)

Y k (t, b)

k k∈Kt XKLEMS (t, b)

TFPRODkY (t, b)

(9.52)

.

Notice that, assuming that additivity at the output side holds, the numerator is given by expression (9.48). For the denominator a similar expression can be derived. To see this, we again return to the accounting identities in Sect. 5.2 and notice that 

 

kt CEMS =

k∈Kt

k∈Kt



k kt Kt CEMS + CEMS , t

(9.53)

k  ∈Kt

Kt t =  kt Adding at both sides CKL k∈Kt CKL , we obtain the following accounting relation:    kt k  kt Kt t CKLEMS = CEMS + CKLEMS . (9.54) k∈Kt k  ∈Kt

k∈Kt

Thus, total cost is the sum of cost incurred by internal deliveries (recall that k  kt is the cost incurred by unit k for purchases from unit k  ) and aggregate CEMS Kt t cost CKLEMS , which is the KLEMS input cost of the ensemble Kt , considered as a consolidated production unit. Now, imposing additivity at the input side, that is, defining the aggregate KLEMS input based price index as a Paasche index of the k-specific KLEMS input based price indices, 1 Kt

PKLEMS (t, b)

≡

 k∈Kt



kt CKLEMS

1

kt k k∈Kt CKLEMS PKLEMS (t, b)

,

(9.55)

9.5 Gross-Output Based Productivity

251

implies that expression (9.54) can be written as K PKLEMS (t, b) t



k XKLEMS (t, b) =

 



k kt Kt CEMS + CKLEMS , t

(9.56)

k∈Kt k  ∈Kt

k∈Kt

or 

 k XKLEMS (t, b)

k∈Kt

=



k  ∈Kt



k kt CEMS

K PKLEMS (t, b) t

k∈Kt

Kt CKLEMS t

+

K PKLEMS (t, b) t

.

(9.57)

If we define real KLEMS input of the ensemble Kt , considered as a consolidated Kt t Kt Kt (t, b) ≡ CKLEMS /PKLEMS (t, b), then expression (9.57) production unit, as XKLEMS can be simplified to 

 k XKLEMS (t, b)

=

k∈Kt

k∈Kt



k  ∈Kt



k kt CEMS

Kt PKLEMS (t, b)

K (t, b). + XKLEMS t

(9.58)

Substituting expressions (9.48) and (9.58) into expression (9.52) and applying definition (9.41) to the ensemble considered as a production unit delivers the following relation: TFPRODK BD (t, b) t

=

t TFPRODK Y (t, b)

   t 1 + k∈Kt k  ∈Kt R kk t /R K t .   k  kt /C Kt t 1 + k∈Kt k  ∈Kt CEMS KLEMS

(9.59)

As observed in Sect. 5.2, National Accounting conventions imply that revenue and cost of the intra-ensemble transactions are equal; that is,   k∈Kt k  ∈Kt



R kk t =

 



k kt CEMS .

k∈Kt k  ∈Kt

Thus the magnitude of the bias of aggregate BD TFP depends on the magnitude t Kt t of aggregate revenue R K t relative to aggregate KLEMS input cost CKLEMS . Put t otherwise, the magnitude of the bias depends on aggregate profit K t . If aggregate profit is positive (negative), then aggregate BD TFP understates (overstates) TFP of the aggregate. If aggregate profit equals 0, then the bias vanishes. A sufficient condition for zero aggregate profit is that kt = 0 for each individual production unit k ∈ Kt . Of course, the bias also vanishes in the trivial case when there are no intra-ensemble deliveries.

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9 Connecting the Two Approaches

Foster et al. (2001) considered total factor productivities weighed by real output shares; that is,8 TFPRODK FHK (t, b) ≡ t





k∈Kt

Y k (t, b) TFPRODkY (t, b). k (t, b) Y t k∈K

(9.60)

Applying the arithmetic-harmonic mean inequality and using the definitions in expressions (9.41) and (9.52) subsequently, we find that K TFPRODK FHK (t, b) ≥ TFPRODBD (t, b). t

t

(9.61)

Now expression (9.59) above tells us that, under additivity at the input and the output t side, TFPRODK BD (t, b) is an unbiased measure of TFP of the aggregate if there are no intra-ensemble deliveries. Thus, we may conclude that in the cases studied by Foster, Haltiwanger and Krizan, which were four-digit level industries where intrat industry deliveries are unlikely, TFPRODK FHK (t, b) most likely overstates TFP of t the aggregate, TFPRODK Y (t, b). The target variable of Eslava et al. (2013) appears to be TFPRODK EHKK (t, b) ≡ t



(TFPRODkY (t, b))Y

k (t,b)/  k∈Kt

Y k (t,b)

;

(9.62)

k∈Kt

that is, the geometric variant of the FHK measure defined by expression (9.60). Using subsequently the geometric-harmonic mean inequality, definition (9.41), and expression (9.52), we obtain K TFPRODK EHKK (t, b) ≥ TFPRODBD (t, b). t

t

(9.63)

As we have seen, the right-hand side of this expression may or may not approximate t TFPRODK Y (t, b). Finally, it is interesting to consider a recent paper by Collard-Wexler and de Loecker (2015). These authors also dealt with TFPRODkY (t, b) (k ∈ Kt ), but to obtain aggregate  productivity the individual TFPs were weighed by nominal revenue shares R kt / k∈Kt R kt . Thus aggregate productivity was defined as9 TFPRODK CWL (t, b) ≡ t

 k∈Kt



R kt k∈Kt

R kt

TFPRODkY (t, b).

(9.64)

8 Actually, their multi-factor productivity index, discussed in Sect. 5.4.4.2, can be seen as a special case of TFPRODkY (t, b). 9 Essentially the same method was used by Figal Garone et al. (2020), except that TFPRODk (t, b) Y was replaced by revenue productivity.

9.6 Conclusion

253

To obtain an interpretation for this mean, we relate it to the alternative where real  shares Y k (t, b)/ k∈Kt Y k (t, b) are used as weights, K TFPRODK CWL (t, b) = TFPRODFHK (t, b) t

+

 k∈Kt



R kt k∈Kt

R kt

−

t

 Y k (t, b) TFPRODkY (t, b). k (t, b) Y t k∈K

(9.65)

The first term on the right-hand side of this equation is the FHK measure as defined by expression (9.60). The second term is a covariance, between the difference of nominal and real revenue shares and TFP. There is in general no compelling reason for this covariance to be positive or negative, large or small. Taken together, on the assumption that the covariance in Eq. (9.65) equals 0, it seems likely that aggregate CWL productivity overstates the productivity of the aggregate.

9.5.3 Some Empirical Evidence The primary purpose of the classic paper by Foster et al. (2001) was to compare decompositions of intertemporal change of the three aggregate measures t Kt Kt TFPRODK FHK (t, b), LPRODFHK (t, b), and LPRODBBH (t, b). They specifically examined the Foster-Haltiwanger-Krizan (FHK) and the Griliches-Regev (GR) decomposition methods (see expressions (5.50) and (5.61) respectively). It turned out that, though the levels were of course different, the FHK decompositions of t Kt TFPRODK FHK (t, b) and LPRODFHK (t, b) were strikingly similar. The levels t Kt as well as the FHK decompositions of LPRODK FHK (t, b) and LPRODBBH (t, b) differed, however, remarkably. Interestingly, for the three aggregate measures the GR decomposition delivered almost the same results. Overall, the ‘within’ term appeared dominant.

9.6 Conclusion The overall conclusion of this chapter is that not every combination of micro-, or meso-level productivities, weights, and aggregator function (‘mean’) leads to a nice interpretation of aggregate productivity as productivity of the aggregate. Specifically: • An arithmetic ‘mean’ of value-added based TFPs requires weights based on relative real primary input times relative value-added based price levels. • An harmonic ‘mean’ of value-added based TFPs requires weights based on relative real value added times relative primary input price levels.

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9 Connecting the Two Approaches

• Under additivity the relative price levels disappear from the expressions. • Similar results hold for value-added based (simple) labour productivities. • An arithmetic mean of gross-output (revenue) based simple labour productivities weighed with (physical) labour input shares is likely to overstate its aggregate counterpart. • An arithmetic mean of gross-output (revenue) based TFPs weighed with real input shares approximates gross-output (revenue) based TFP of the aggregate; the magnitude of the bias depends on aggregate profit.

Chapter 10

The Components of Total Factor Productivity Change

10.1 Introduction This chapter provides a detailed exposition of the ideas set out in the Conclusion of Chap. 2. We discuss the decomposition of productivity change for a single production unit. Thus we don’t consider longitudinal microdata and issues connected with the dynamics of firm behaviour. By now, the reader should have noticed that the concept of productivity change in this book is broader than the usual neo-classical concept of TFP change. We must, therefore, be aware of a terminological problem. From the earlier literature it is well known that productivity change, in the sense used in this book, is the combined result of (technical) efficiency change, technological change, a scale effect, and input and output mix effects. Neo-classical TFP change appears to be identical to technological change, which is but one of the components of productivity change. Our approach is non-neoclassical in the sense that optimizing behaviour of the production unit under consideration is not assumed. Input and output prices and quantities are taken as given. The only assumption maintained in this chapter is that the data enable us to approximate time-period specific technological frontiers with regular characteristics, so that basic instruments from duality theory can be employed. The neo-classical approach as such can be embedded by introducing additional assumptions. Given a certain functional form for the productivity1 index, the problem then is how to decompose such an index into factors corresponding to the five components mentioned. As we know, every mathematical expression a can, given any other expression b, be decomposed as a = (a/b) × b. However, not all such decompositions are meaningful. At the very least, the two factors a/b and b should be independent of each other and admit clear economic interpretations to

1 In

this chapter ‘productivity’ is to be understood as ‘total factor productivity’ (TFP).

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2021 B. M. Balk, Productivity, Contributions to Economics, https://doi.org/10.1007/978-3-030-75448-8_10

255

256

10 The Components of Total Factor Productivity Change

be meaningful. One of the basic insights offered in this chapter is that meaningful decompositions of productivity indices can only be obtained for indices that are transitive in the main variables, input and output quantities. Such decompositions can be obtained in a systematic way by considering the various hypothetical paths in input and output quantity space that connect a production unit’s base period position to its comparison period position. For example, the Malmquist index that conditions on the base period cone technology— precise definitions provided later—admits six different decompositions, as does the index that conditions on the comparison period cone technology. Their geometric mean even admits eighteen different decompositions. By merging either the inputor the output-mix effect with the scale effect, it is possible to obtain two different decompositions that are symmetric in all their variables. The Moorsteen-Bjurek (MB) productivity index is defined as a ratio of Malmquist output and input quantity indices and hence contains a number of conditioning variables. If and only if these are specified independently of the main variables, the MB index is transitive and can be decomposed. Finally, the decomposition of price-weighted and value-shareweighted productivity indices is discussed. It is important to make a clear distinction between components and drivers of productivity change. In this chapter we are concerned with components. Bartelsman and Doms (2000), reviewing the yield of (mainly longitudinal) microdata research up to 2000, were primarily interested in the drivers of productivity change, without much acknowledgement of the various components. Ten years later, Syverson (2011) stayed more explicitly in the neo-classical tradition, essentially identifying productivity change with technological change.2 Thus, in our language, Syverson considered internal and external drivers of technological change. The lay-out of this chapter is as follows. Section 10.2 reviews basic definitions from production theory. We start with productivity indices that are functions of only quantities. Such indices can be used for market as well as non-market production units. Sections 10.3 and 10.4 discuss the problem of decomposing Malmquist productivity indices, first using the output orientation and then using the input orientation. We then turn to productivity indices that essentially are functions of quantities and prices. Section 10.5 considers the class of Moorsteen-Bjurek indices. Sections 10.6 and 10.7 treat the decomposition problem for conventional productivity indices, price-weighted (for example, Fisher) or value-share-weighted (for example, Törnqvist), respectively. Section 10.8 contains the application of the foregoing to a real-life dataset coming from a panel of individual production units. Finally, in Sect. 10.9 some general conclusions are drawn.

2 “Conceptually,

TFP differences reflect shifts in the isoquants of a production function: variation in output produced from a fixed set of inputs.” (Syverson 2011, 330).

10.2 Basic Definitions

257

10.2 Basic Definitions We consider a single production unit, for simplicity called a firm, which is observed during time periods of equal length.3 Such a firm is considered here as an entity transforming inputs into outputs. The input quantities are represented by an Ndimensional vector of non-negative real values x ≡ (x1 , . . . , xN ) ∈ N + −{0N }. The output quantities are represented by an M-dimensional vector of non-negative real values y ≡ (y1 , . . . , yM ) ∈ M + − {0M }. Thus there is always at least one positive input and output quantity. Vectors without superscripts, with or without primes, are used as generic variables, whereas vectors with superscripts represent observations. Thus, for instance, (x t , y t ) denotes the input and output quantities of our firm at period t.

10.2.1 Technologies, Distance Functions, and Related Measures We assume that this firm has access to a certain technology. The technology at period M t is defined as the set S t ⊂ N + × + of all the feasible input-output quantity 4 combinations. Alternatively, the period t technology can be described by the input sets Lt (y) ≡ {x | (x, y) ∈ S t } for all y, or by the output sets P t (x) ≡ {y | (x, y) ∈ S t } for all x. Thus Lt (y) is the set of all the input quantity vectors x that can produce the output quantity vector y in period t, and P t (x) is the set of all the output quantity vectors y that can be produced by the input quantity vector x in period t. Following Färe and Primont (1995) and Balk (1998) it is assumed that the technology has the following properties:5 P.1 P.2 P.3 P.4 P.5 P.6 P.7 P.8 P.9

3A

0M ∈ P t (x) for all x (inactivity is possible). If y ∈ P t (x) and y  ≤ y then y  ∈ P t (x) (strong disposability of outputs). P t (x) is bounded for all x (scarcity). P t (x) is closed for all x. If y = 0M then y ∈ / P t (0N ) (no free lunch). t If y ∈ P (x) and x  ≥ x then y ∈ P t (x  ) (strong disposability of inputs). Lt (y) is closed for all y. Lt (y) is convex for all y. P t (x) is convex for all x.

translation of the theory to spatial comparisons is rather simple. Instead of a single firm in two time periods, two firms at different locations are considered. Those firms must use the same inputs and supply the same outputs. 4 According to Førsund (2015, 198), this is the micro-unit ex ante viewpoint. 5 Notation: 0 is a vector of M zero’s and y  ≤ y means that y  ≤ y (m = 1, . . . , M). Diewert M m m and Fox (2017) provide a story without convexity assumptions.

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10 The Components of Total Factor Productivity Change

The (direct, radial) output distance function is defined by Dot (x, y) ≡ inf{δ | δ > 0, (x, y/δ) ∈ S t }.

(10.1)

Thus, (x, y/Dot (x, y)) is the point on the frontier of the period t technology that is obtained by holding the input quantity vector x constant while radially expanding the output quantity vector y. Put otherwise, the point (x, y/Dot (x, y)) could be called the radial projection of (x, y) on the frontier in the direction of y. The output distance function is positive, nonincreasing in x, and nondecreasing and linearly homogeneous in y. When M = 1 (the case of a single output), F t (x) ≡ y/Dot (x, y) = 1/Dot (x, 1) is the familiar production function. The (direct, radial) input distance function is defined by Dit (x, y) ≡ sup{δ | δ > 0, (x/δ, y) ∈ S t }.

(10.2)

Thus, (x/Dit (x, y), y) is the point on the frontier of the period t technology that is obtained by holding the output quantity vector y constant while radially contracting the input quantity vector x. Put otherwise, the point (x/Dit (x, y), y) could be called the radial projection of (x, y) on the frontier in the direction of x. The input distance function is positive, nondecreasing and linearly homogeneous in x, and nonincreasing in y. Both functions serve as measures of technical efficiency. The output distance function, Dot (x, y), measures output orientated technical efficiency with values between 0 and 1, and the inverse of the input distance function, 1/Dit (x, y), measures input orientated technical efficiency with values between 0 and 1. Both belong to the class of path-based measures as defined by Russell and Schworm (2018). The period t technology is said to exhibit global constant returns to scale (global CRS) if for all θ > 0, (θ x, θy) ∈ S t whenever (x, y) ∈ S t . This property can also be expressed as CRS

S t = θ S t (θ > 0).

Two equivalent conditions for global CRS are CRS

Dot (x, y) is homogeneous of degree −1 in x

and CRS

Dit (x, y) is homogeneous of degree −1 in y.

Associated with the (actual) technology is the cone technology. This is a virtual technology, defined as the conical envelopment of the actual technology S t , Sˇ t ≡ {(λx, λy) | (x, y) ∈ S t , λ > 0}.

(10.3)

10.2 Basic Definitions

259

M It is thereby assumed that Sˇ t is a proper subset of N + × + , which means that globally increasing returns-to-scale of the actual t technology is excluded. The output distance function of the cone technology is denoted by Dˇ ot (x, y), the input distance function by Dˇ it (x, y), and (when M = 1) the production function by Fˇ t (x). Their definitions are the same as the foregoing, except that S t is replaced by Sˇ t . It is immediately clear that Sˇ t exhibits global CRS, and that S t exhibits global CRS if and only if S t = Sˇ t , that is, if the actual technology coincides with the associated cone technology. It is straightforward to show that Dˇ it (x, y) = 1/Dˇ ot (x, y). Since S t ⊂ Sˇ t , Dˇ ot (x, y) ≤ Dot (x, y). The ratio

OSEt (x, y) ≡

Dˇ ot (x, y) Dot (x, y)

(10.4)

is called output-orientated scale efficiency. Notice that OSEt (x, y) is homogeneous of degree 0 in y—thus depends only on the output mix—, is always less than or equal to 1, and attains the value 1 for all x and y if and only if the technology exhibits global CRS. Likewise, since S t ⊂ Sˇ t , Dˇ it (x, y) ≥ Dit (x, y). The ratio ISEt (x, y) ≡

Dit (x, y) Dˇ t (x, y)

(10.5)

i

is called input-orientated scale efficiency. Notice that ISEt (x, y) is homogeneous of degree 0 in x—thus depends only on the input mix—, is always less than or equal to 1, and attains the value 1 for all x and y if and only if the technology exhibits global CRS. These two measures of scale efficiency were extensively discussed in Balk (2001). Through time, the firm changes its input and output quantities. The technological environment changes simultaneously. Various aspects of this complex of changes can be measured by distance-function based instruments. We are using here the notation introduced in the same article. Going from period 0 to period 1, technical efficiency change is measured by comparing the distances to the contemporaneous frontiers. Thus, output orientated, by ECo (x 1 , y 1 , x 0 , y 0 ) ≡ Do1 (x 1 , y 1 )/Do0 (x 0 , y 0 ),

(10.6)

or, input orientated, by ECi (x 1 , y 1 , x 0 , y 0 ) ≡ Di0 (x 0 , y 0 )/Di1 (x 1 , y 1 ).

(10.7)

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10 The Components of Total Factor Productivity Change

Technological change is measured by comparing, for a certain data point, its distances to the two frontiers. Thus, output orientated, by 0 1 TC1,0 o (x, y) ≡ Do (x, y)/Do (x, y),

(10.8)

1 0 TC1,0 i (x, y) ≡ Di (x, y)/Di (x, y),

(10.9)

or, input orientated, by

where (x, y) is any input-output quantity combination for which radial distances can be computed. The output-orientated scale effect experienced by going from x 0 to x 1 , within the technology of period t and conditional on certain output quantities y¯ is measured by t 0 SECto,M (x 1 , x 0 , y) ¯ ≡ OSEt (x 1 , y)/OSE ¯ (x , y). ¯

(10.10)

The input-orientated scale effect experienced by going from y 0 to y 1 , within the technology of period t and conditional on certain input quantities x¯ is measured by ¯ y 1 , y 0 ) ≡ ISEt (x, ¯ y 1 )/ISEt (x, ¯ y 0 ). SECti,M (x,

(10.11)

Recall that the function OSEt (x, y) is homogeneous of degree 0 in y and that the function ISEt (x, y) is homogeneous of degree 0 in x. Thus OSEt (x, y) = OSEt (x, y/  y ), and ISEt (x, y) = ISEt (x/  x , y), where y/  y  is the output mix of y and x/  x  is the input mix of x. The output mix effect of the move from y 0 to y 1 , within the technology of period t and conditional on certain input quantities x¯ can therefore be measured by ¯ y 1 , y 0 ) ≡ OSEt (x, ¯ y 1 )/OSEt (x, ¯ y 0 ). OMEtM (x,

(10.12)

The input mix effect of the move from x 0 to x 1 , within the technology of period t and conditional on certain output quantities y¯ can, similarly, be measured by t 0 ¯ ≡ ISEt (x 1 , y)/ISE ¯ (x , y). ¯ IMEtM (x 1 , x 0 , y)

(10.13)

The use of the additional subscript M, denoting Malmquist, in the scale and mix measures will be discussed later on.

10.2.2 Measuring Productivity Change and Level Generally, productivity change between the input-output situation (x, y) and the input-output situation (x  , y  ) is measured by some positive, finite function F :

10.3 Decomposing a Malmquist Productivity Index by Output Distance Functions

261

M 2 6 ((N + − {0N }) × (+ − {0M })) → ++ − {∞}. This function, with values    F (x , y , x, y), should be nonincreasing in x , nondecreasing in y  , nondecreasing in x, and nonincreasing in y.7 Moreover, this function should exhibit proportionality in input and output quantities; that is,

F (λx, μy, x, y) = μ/λ (λ, μ > 0).

(10.14)

In particular, this property implies that F (x, y, x, y) = 1; that is, F (x  , y  , x, y) satisfies the Identity Test. Taken together, the function F (x  , y  , x, y) should be such that by fixing input quantities x = x  = x¯ the function F (x, ¯ y  , x, ¯ y) behaves as an output quantity index, and by fixing output quantities y = y  = y¯ the function F (x  , y, ¯ x, y) ¯ behaves as the reciprocal of an input quantity index. See Appendix A of Chap. 2 for requirements for quantity indices. A function F (x  , y  , x, y) is called transitive in (x, y) if it satisfies the equality F (x  , y  , x, y) = F (x  , y  , x  , y  )F (x  , y  , x, y)

(10.15)

for any (x, y), (x  , y  ) and (x  , y  ). Transitivity implies that F (x  , y  , x, y) = G(x  , y  )/G(x, y)

(10.16)

for a certain function G(x, y). Reversely, any function F (x  , y  , x, y) that has the form of expression (10.16) is transitive. Proportionality, expression (10.14), then implies that the function G(x, y) must be linearly homogeneous in y and homogeneous of degree −1 in x.8 Put otherwise, if F (x  , y  , x, y) is a transitive measure of productivity change, then G(x, y) measures the productivity level at the input-output situation (x, y), up to a certain scalar normalization.

10.3 Decomposing a Malmquist Productivity Index by Output Distance Functions Well-known candidates for measuring productivity change are those from the class of Malmquist indices. We start by selecting a certain benchmark (or reference)

stated, F (x  , y  , x, y) satisfies the Determinateness Test. Notice that F (.) may involve prices, however not as main variables. 7 These monotonicity properties were considered to be fundamental by Agrell and West (2001). 8 A further specification, G(x, y) = Y (y)/X(x), leads to functions considered by O’Donnell in various articles. Here X(x) and Y (y) are aggregator functions which are nonnegative, nondecreasing, and linearly homogeneous. 6 Formally

262

10 The Components of Total Factor Productivity Change

technology, which must be conical in view of the required properties.9 The output-orientated Malmquist productivity index, conditional on the period t cone technology, is defined by Dˇ t (x  , y  ) . Mˇ ot (x  , y  , x, y) ≡ o Dˇ ot (x, y)

(10.17)

Notice that numerator and denominator are always finite. This index has indeed the required monotonicity and proportionality properties, and is by construction transitive in (x, y). Thus, the output distance function Dˇ ot (x, y) measures the productivity level at the input-output situation (x, y). Consider now the movement of our firm from a base period situation (x 0 , y 0 ) to a (later) comparison period situation (x 1 , y 1 ). These periods may or may not be adjacent. Which cone technology should then be selected for the Malmquist productivity index defined by expression (10.17)? Although, in principle, no relation needs to exist between the benchmark technology time period t and the observation periods 0 and 1, it is quite natural to identify t with one of those periods.10 Selecting the base period technology then leads to Mˇ o0 (x 1 , y 1 , x 0 , y 0 ) and selecting the comparison period technology leads to Mˇ o1 (x 1 , y 1 , x 0 , y 0 ). We also consider their geometric mean. Let us start with the first option.

10.3.1 The Base Period Viewpoint How do we decompose the base-period-output-orientated Malmquist index, Dˇ 0 (x 1 , y 1 ) Mˇ o0 (x 1 , y 1 , x 0 , y 0 ) = o , Dˇ o0 (x 0 , y 0 )

(10.18)

into meaningful, independent factors?11 It appears that this problem can be solved by breaking up the movement of the firm into hypothetical, independent segments. Figure 10.1 shows a single-input/single-output situation. The base period technology set S 0 is pictured by the area between the horizontal axis and the lower curve, whereas the comparison period technology set S 1 is pictured by the area between the horizontal axis and the upper curve. It is assumed that S 0 = S 1 . The figure suggests 9 Notice that “using

a CRS frontier as a reference does not mean that we assume CRS, it just serves as a reference for TFP measures.” (Førsund 2015, 214). 10 Natural but not necessary. For instance, Førsund (2016) considers as benchmark the conical envelopment of the pooled technologies of the two periods, S 0 ∪ S 1 . This is a special case of the “global Malmquist productivity index” as defined by Pastor and Lovell (2005). 11 The type of decomposition considered here differs from that studied by Färe et al. (2001). These authors considered a decomposition into components corresponding to subvectors of x and y.

10.3 Decomposing a Malmquist Productivity Index by Output Distance Functions

263

y 6

•c • (x1 , y 1 ) •d

•b

S1 S0

a

• (x0 , y 0 ) -

x

Fig. 10.1 Decomposing productivity change (1)

uniform technological progress and inefficient firm behaviour in both periods. The figure also suggests that Do0 (x 1 , y 1 ) is finite. We break up the firm’s journey into six segments. The first segment stretches from the actual base period position to its radial projection in the output direction on the base period technology frontier. This point is represented by a. Thus the first segment is formally defined by (x 0 , y 0 ) −→ (x 0 , y 0 /Do0 (x 0 , y 0 )).

(10.19)

The second segment stretches along the base period frontier from a to the point represented by b, which is the radial projection of the firm’s comparison period position on the base period frontier. Thus, on the assumption that Do0 (x 1 , y 1 ) is finite, the second segment is (x 0 , y 0 /Do0 (x 0 , y 0 )) −→ (x 1 , y 1 /Do0 (x 1 , y 1 )).

(10.20)

Integrating proposals by Balk (2001) and Lovell (2003), the second segment can actually be split into three subsegments, namely a part corresponding to radial change in input quantity space,

264

10 The Components of Total Factor Productivity Change

(x 0 , y 0 /Do0 (x 0 , y 0 )) −→ (λx 0 , y 0 /Do0 (λx 0 , y 0 )),

(10.21)

a remainder part in input quantity space, (λx 0 , y 0 /Do0 (λx 0 , y 0 )) −→ (x 1 , y 0 /Do0 (x 1 , y 0 )),

(10.22)

and a part corresponding to the change in output quantity space, (x 1 , y 0 /Do0 (x 1 , y 0 )) −→ (x 1 , y 1 /Do0 (x 1 , y 1 )),

(10.23)

where λ is some positive scalar, to be discussed later. It is assumed that both Do0 (x 1 , y 0 ) and Do0 (λx 0 , y 0 ) are finite. Notice that by virtue of the positive linear homogeneity in y of the output distance function, (λx 0 , y 0 /Do0 (λx 0 , y 0 )) = (λx 0 , μy 0 / Do0 (λx 0 , μy 0 )) for any positive scalar μ. The third segment stretches from the base period frontier at b to the comparison period frontier at c. This is the radial projection of the firm’s comparison period position on the comparison period frontier, (x 1 , y 1 /Do0 (x 1 , y 1 )) −→ (x 1 , y 1 /Do1 (x 1 , y 1 )).

(10.24)

The fourth segment stretches from point c back to the firm’s comparison period position, (x 1 , y 1 /Do1 (x 1 , y 1 )) −→ (x 1 , y 1 ).

(10.25)

Thus, summarizing, the entire journey from (x 0 , y 0 ) to (x 1 , y 1 ) is broken up into six segments, respectively defined by expressions (10.19), (10.21), (10.22), (10.23), (10.24), and (10.25), as pictured in the following frame. Path A: (x 0 , y 0 ) −→ (x 0 , y 0 /Do0 (x 0 , y 0 )) −→ (λx 0 , y 0 /Do0 (λx 0 , y 0 )) −→ (x 1 , y 0 /Do0 (x 1 , y 0 )) −→ (x 1 , y 1 /Do0 (x 1 , y 1 )) −→ (x 1 , y 1 /Do1 (x 1 , y 1 )) −→ (x 1 , y 1 ) Along each segment the index Mˇ o0 (x  , y  , x, y) can be computed (see Appendix A for details). By virtue of transitivity, multiplying the left-hand sides of the resulting six equations delivers precisely Mˇ o0 (x 1 , y 1 , x 0 , y 0 ), whereas multiplying the righthand sides provides a decomposition which can be summarized as 1 1 Mˇ o0 (x 1 , y 1 , x 0 , y 0 ) = ECo (x 1 , y 1 , x 0 , y 0 ) × TC1,0 o (x , y )×

SEC0o,M (λx 0 , x 0 , y 0 ) × SEC0o,M (x 1 , λx 0 , y 0 ) × OME0M (x 1 , y 1 , y 0 ).

(10.26)

There are thus five factors, respectively corresponding to technical efficiency change, technological change, a radial scale effect—recall that SEC0o,M (λx 0 ,

10.3 Decomposing a Malmquist Productivity Index by Output Distance Functions

265

x 0 , y 0 ) = SEC0o,M (λx 0 , x 0 , μy 0 ) for any positive μ —, an input mix effect, and an output mix effect, according to the definitions provided in Sect. 10.2.1. These five factors are indeed independent, as can be verified easily. First, if there is no technological change, that is, S 1 = S 0 , then TC1,0 o (x, y) = 1 for all x, y. Second, if the firm is technically efficient in both periods, then Do0 (x 0 , y 0 ) = 1 = Do1 (x 1 , y 1 ), and thus ECo (x 1 , y 1 , x 0 , y 0 ) = 1. Third, if x 1 = λx 0 for some λ > 0, then the input mix effect vanishes. Fourth, if y 1 = μy 0 for some μ > 0, then the output mix effect vanishes. (Notice that in the single-output case the output mix effect always vanishes.) If all these conditions are fulfilled, the only remaining part at the right-hand side of expression (10.26) is the radial scale effect SEC0o,M (λx 0 , x 0 , y 0 ). Using the linear homogeneity of the distance functions a number of times, we see that SEC0o,M (λx 0 , x 0 , y 0 ) =

μ μ μ 1 = = = , λ λDo0 (λx 0 , y 0 ) λDo0 (λx 0 , μy 0 ) λDo1 (x 1 , y 1 ) (10.27)

as it should be. Two important observations must be made: • Although the left-hand side of expression (10.26) and the efficiency change factor on the right-hand side are always well-determined, this is not necessarily the case for the other four factors on the right-hand side. • If the base period technology exhibits global CRS (that is, S 0 = Sˇ 0 ), then the last three factors on the right-hand side of expression (10.26) (that is, radial scale, input mix, and output mix effect) vanish. This can easily be checked by employing the various definitions. The decomposition in expression (10.26) extends a number of alternative decompositions occurring in the literature. By merging the radial scale effect and the input mix effect, we obtain the four-factor decomposition proposed by Balk (2001): 1 1 Mˇ o0 (x 1 , y 1 , x 0 , y 0 ) = ECo (x 1 , y 1 , x 0 , y 0 ) × TC1,0 o (x , y )×

SEC0o,M (x 1 , x 0 , y 0 ) × OME0M (x 1 , y 1 , y 0 ).

(10.28)

By merging the radial scale effect, the input mix effect, and the output mix effect, we obtain the three-factor decomposition proposed by Ray and Desli (1997): OSE0 (x 1 , y 1 ) 1 1 . Mˇ o0 (x 1 , y 1 , x 0 , y 0 ) = ECo (x 1 , y 1 , x 0 , y 0 ) × TC1,0 (x , y ) × o OSE0 (x 0 , y 0 ) (10.29)

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10 The Components of Total Factor Productivity Change

The last factor on the right-hand side of expression (10.29) was called ‘returns-toscale effect’ by Lovell (2003).12 It proves convenient to merge the technical efficiency change effect and the technological change effect. The combined effect, 1 1 Mo0 (x 1 , y 1 , x 0 , y 0 ) ≡ ECo (x 1 , y 1 , x 0 , y 0 ) × TC1,0 o (x , y ) =

Do0 (x 1 , y 1 ) , Do0 (x 0 , y 0 ) (10.30)

will be called the base-period-output-orientated CCD index.13 Expression (10.26) can then be written as Mˇ o0 (x 1 , y 1 , x 0 , y 0 ) = Mo0 (x 1 , y 1 , x 0 , y 0 ) × SEC0o,M (λx 0 , x 0 , y 0 )× SEC0o,M (x 1 , λx 0 , y 0 ) × OME0M (x 1 , y 1 , y 0 ).

(10.31)

Recall that if the base period technology exhibits global CRS, then the other three factors on the right-hand side of expression (10.31) become equal to 1, and we find that Mˇ o0 (x 1 , y 1 , x 0 , y 0 ) = Mo0 (x 1 , y 1 , x 0 , y 0 ). Let us now return to expression (10.26). As we have seen that the choice of μ is immaterial, the remaining task is to choose a suitable value for λ. Our choice would be the solution λ(1) of Do0 (λx 0 , y 0 ) = Do0 (x 1 , y 0 ),

(10.32)

which means that λx 0 and x 1 are on the same output isoquant of the base period technology.14

12 Balk

and Zofío (2018) also discuss proposals by Färe et al. (1989, 1994), Färe et al. (1994), Färe et al. (1997), and Zofío (2007). 13 The output-orientated CCD index, generically defined as Mot (x  , y  , x, y) ≡ Dot (x  , y  )/Dot (x, y), was introduced by Caves et al. (1982), and then believed to be a productivity index. However, it does not possess the proportionality property expressed in (10.14) unless the benchmark technology exhibits global CRS. Nevertheless, following established practice, we refer to Mot (.) as an index. 14 The same expression materialized in Peyrache (2014) and in Diewert and Fox (2017). Lovell (2003) suggested μ = 1/Do0 (x 1 , y 0 ) and λ = 1/Di0 (x 0 , μy 0 ), or Di0 (λx 0 , y 0 /Do0 (x 1 , y 0 )) = 1. Provided that some mild regularity conditions are met (see Färe 1988, Lemma 2.3.10), Dit (x, y) = 1 if and only if Dot (x, y) = 1, and this equation appears to be equivalent to Do0 (λx 0 , y 0 /Do0 (x 1 , y 0 )) = 1,

10.3 Decomposing a Malmquist Productivity Index by Output Distance Functions

267

Recall that the segment from a to b was split into three parts, respectively given by expressions (10.21), (10.22), and (10.23). Reversing the order in which changes in input and output space take place, and assuming that Do0 (x 0 , y 1 ) and Do0 (λx 0 , y 1 ) are finite, we get an alternative decomposition of this segment. The entire journey from (x 0 , y 0 ) to (x 1 , y 1 ) is now pictured in the next frame. Path B: (x 0 , y 0 ) −→ (x 0 , y 0 /Do0 (x 0 , y 0 )) −→ (x 0 , y 1 /Do0 (x 0 , y 1 )) −→ (λx 0 , y 1 /Do0 (λx 0 , y 1 )) −→ (x 1 , y 1 /Do0 (x 1 , y 1 )) −→ (x 1 , y 1 /Do1 (x 1 , y 1 )) −→ (x 1 , y 1 )

In the same way as demonstrated earlier, Path B leads to the following decomposition of the base-period-output-orientated Malmquist index: Mˇ o0 (x 1 , y 1 , x 0 , y 0 ) = Mo0 (x 1 , y 1 , x 0 , y 0 ) × SEC0o,M (λx 0 , x 0 , y 1 )× SEC0o,M (x 1 , λx 0 , y 1 ) × OME0M (x 0 , y 1 , y 0 ).

(10.33)

The differences between this decomposition and the previous one, expression (10.31), are subtle but noteworthy. The CCD index, capturing efficiency change and technological change, is the same. In expression (10.31) the radial scale effect and the input mix effect are conditional on y 0 , but in expression (10.33) they are conditional on y 1 . In a certain sense, the reverse happens with the output mix effect; in expression (10.31) this effect is conditional on x 1 but in expression (10.33) it is conditional on x 0 . As in the previous case, if the base period technology exhibits global CRS (that is, S 0 = Sˇ 0 ), then the last three factors on the right-hand side of expression (10.33) (that is, radial scale, input mix, and output mix effect) vanish. By merging the radial scale effect and the input mix effect, we now obtain Mˇ o0 (x 1 , y 1 , x 0 , y 0 ) = Mo0 (x 1 , y 1 , x 0 , y 0 ) × SEC0o,M (x 1 , x 0 , y 1 )× OME0M (x 0 , y 1 , y 0 ),

(10.34)

which should be compared to expression (10.28) to see the differences in the conditioning variables. The obvious choice for λ is now the solution λ(2) of

which brings us back to expression (10.32). We could also take the solution of Dˇ o0 (x 1 , y 0 /Do0 (x 1 , y 0 )) = Dˇ o0 (λx 0 , y 0 /Do0 (λx 0 , y 0 )). it is easily verified that this implies that SEC0o,M (x 1 , λx 0 , y 0 ) = 1; that is, the input mix effect vanishes.

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10 The Components of Total Factor Productivity Change

Do0 (λx 0 , y 1 ) = Do0 (x 1 , y 1 ).

(10.35)

Notice that in general λ(2) = λ(1) . A sufficient condition for equality is that y 1 = μy 0 for some μ > 0. This, however, would mean that the output mix effect vanishes. We must introduce the concept of output homotheticity for the formulation of a necessary and sufficient condition The period t technology is said to exhibit output homotheticity if Dot (x, y) = Dot (1N , y)Gt (x), where Gt (x) is some nonincreasing function that is consistent with the axioms, and 1N is a vector of N ones. Essentially, output homotheticity means that all the output sets P t (x) are radial expansions of P t (1N ). Theorem 10.1 λ(1) = λ(2) if and only if the base period technology exhibits output homotheticity. Proof The sufficiency follows immediately. For the necessity part we notice that Eqs. (10.32) and (10.35) imply that Do0 (x, y 1 )/Do0 (x, y 0 ) is independent of x. Thus g 0 (y 1 ) Do0 (x, y 1 ) = 0 0 0 0 Do (x, y ) g (y ) for some function g 0 (y). Thus, Do0 (x, y 1 ) = Do0 (x, y 0 )g 0 (y 1 )/g 0 (y 0 ), and since the left-hand side is independent of y 0 , the right-hand side must also be independent of y 0 , which implies that Do0 (x, y 0 )/g 0 (y 0 ) = h0 (x) for some function h0 (x). Thus Do0 (x, y 1 ) = h0 (x)g 0 (y 1 ). In particular Do0 (1N , y 1 ) = h0 (1N )g 0 (y 1 ), which upon substitution in the foregoing expression leads to Do0 (x, y 1 ) = Do0 (1N , y 1 )h0 (x)/ h0 (1N ). This means that the base period technology exhibits output homotheticity.



At this point we may conclude that there are two, equally meaningful, decompositions of the base-period-output-orientated Malmquist index Mˇ o0 (x 1 , y 1 , x 0 , y 0 ). They differ with respect to the radial scale effect, the input mix effect and the output mix effect. By taking the geometric mean of expressions (10.31) and (10.33), we obtain the third decomposition, Mˇ o0 (x 1 , y 1 , x 0 , y 0 ) = Mo0 (x 1 , y 1 , x 0 , y 0 )× [SEC0o,M (λ(1) x 0 , x 0 , y 0 )SEC0o,M (λ(2) x 0 , x 0 , y 1 )]1/2 ×

10.3 Decomposing a Malmquist Productivity Index by Output Distance Functions

269

[SEC0o,M (x 1 , λ(1) x 0 , y 0 )SEC0o,M (x 1 , λ(2) x 0 , y 1 )]1/2 × [OME0M (x 0 , y 1 , y 0 )OME0M (x 1 , y 1 , y 0 )]1/2 .

(10.36)

The first factor, that is the base-period-output-orientated CCD index, captures technological change and efficiency change, the second factor captures the radial scale effect, the third factor captures the input mix effect, and the fourth factor captures the output mix effect. Calculating this decomposition for a certain firm requires solving Eqs. (10.32) and (10.35). A sufficient condition for the existence of unique solutions is that the distance function Do0 (x, y) is continuously differentiable. If the underlying technology is approximated by DEA (see Appendix B) then such solutions may not exist.

10.3.2 The Comparison Period Viewpoint The second candidate productivity index is quite naturally given by the outputorientated Malmquist productivity index that conditions on the comparison period cone technology: Dˇ 1 (x 1 , y 1 ) . Mˇ o1 (x 1 , y 1 , x 0 , y 0 ) = o Dˇ o1 (x 0 , y 0 )

(10.37)

To decompose this index into meaningful factors, we consider the following path from (x 0 , y 0 ) to (x 1 , y 1 ): Path C: (x 0 , y 0 ) −→ (x 0 , y 0 /Do0 (x 0 , y 0 )) −→ (x 0 , y 0 /Do1 (x 0 , y 0 )) −→ (λx 0 , y 0 /Do1 (λx 0 , y 0 )) −→ (x 1 , y 0 /Do1 (x 1 , y 0 )) −→ (x 1 , y 1 /Do1 (x 1 , y 1 )) −→ (x 1 , y 1 )

in which λ is an as yet undetermined positive scalar. Referring back to Fig. 10.1, we see that the first segment connects the firm’s base period position to its radial projection on the base period frontier (point a). The second segment connects this point to the projection of the firm’s base period position on the comparison period frontier, which is depicted as point d. Next, we travel from point d to point c, which depicts the radial projection of the firm’s comparison period position on the comparison period frontier. This segment is divided into three subsegments, respectively corresponding to a radial movement in input space, a remainder movement in input space, and a movement in output space. The final segment connects point c to the firm’s comparison period position. It is thereby assumed that Do1 (x 0 , y 0 ), Do1 (λx 0 , y 0 ) and Do1 (x 1 , y 0 ) are finite.

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10 The Components of Total Factor Productivity Change

This leads to the following decomposition: Mˇ o1 (x 1 , y 1 , x 0 , y 0 ) = Mo1 (x 1 , y 1 , x 0 , y 0 ) × SEC1o,M (λx 0 , x 0 , y 0 )× SEC1o,M (x 1 , λx 0 , y 0 ) × OME1M (x 1 , y 1 , y 0 ),

(10.38)

where Do1 (x 1 , y 1 ) Do1 (x 0 , y 0 ) (10.39) is the comparison-period-output-orientated CCD index. It is straightforward to check from the various definitions that if the comparison period technology exhibits global CRS (that is, S 1 = Sˇ 1 ), then the last three factors on the right-hand side of expression (10.38) (that is, radial scale, input mix, and output mix effect) vanish. The obvious choice for λ is now the solution λ(3) of the equation 0 0 Mo1 (x 1 , y 1 , x 0 , y 0 ) ≡ ECo (x 1 , y 1 , x 0 , y 0 ) × TC1,0 o (x , y ) =

Do1 (λx 0 , y 0 ) = Do1 (x 1 , y 0 ).

(10.40)

Grifell-Tatjé and Lovell (1999) considered the same path, though with λ = 1/Di1 (x 0 , y 0 /Do1 (x 1 , y 0 )). However, under the regularity conditions mentioned in footnote 14, this equality appears to be equivalent to expression (10.40). By merging the radial scale effect SEC1o,M (λx 0 , x 0 , y 0 ) with the input mix effect SEC1o,M (x 1 , λx 0 , y 0 ), we obtain Mˇ o1 (x 1 , y 1 , x 0 , y 0 ) = Mo1 (x 1 , y 1 , x 0 , y 0 ) × SEC1o,M (x 1 , x 0 , y 0 )× OME1M (x 1 , y 1 , y 0 ).

(10.41)

The alternative path, assuming now that distance function values Do1 (x 0 , y 0 ), Do1 (λx 0 , y 1 ) and Do1 (x 0 , y 1 ) are finite, is defined by the following sequence: Path D: (x 0 , y 0 ) −→ (x 0 , y 0 /Do0 (x 0 , y 0 )) −→ (x 0 , y 0 /Do1 (x 0 , y 0 )) −→ (x 0 , y 1 /Do1 (x 0 , y 1 )) −→ (λx 0 , y 1 /Do1 (λx 0 , y 1 )) −→ (x 1 , y 1 /Do1 (x 1 , y 1 )) −→ (x 1 , y 1 ) This leads to the second decomposition of the comparison-period-output-orientated Malmquist index, defined by expression (10.37), namely as Mˇ o1 (x 1 , y 1 , x 0 , y 0 ) = Mo1 (x 1 , y 1 , x 0 , y 0 ) × SEC1o,M (λx 0 , x 0 , y 1 )× SEC1o,M (x 1 , λx 0 , y 1 ) × OME1M (x 0 , y 1 , y 0 ),

(10.42)

10.3 Decomposing a Malmquist Productivity Index by Output Distance Functions

271

the obvious choice for λ now being the solution λ(4) of the equation Do1 (λx 0 , y 1 ) = Do1 (x 1 , y 1 ).

(10.43)

Notice the subtle differences between expressions (10.38) and (10.42). Again, if the comparison period technology exhibits global CRS (that is, S 1 = Sˇ 1 ), then the last three factors at the right-hand side of expression (10.42) vanish. By merging the radial scale effect SEC1o,M (λx 0 , x 0 , y 1 ) with the input mix effect SEC1o,M (x 1 , λx 0 , y 1 ), expression (10.42) reduces to Mˇ o1 (x 1 , y 1 , x 0 , y 0 ) = Mo1 (x 1 , y 1 , x 0 , y 0 ) × SEC1o,M (x 1 , x 0 , y 1 )× OME1M (x 0 , y 1 , y 0 ).

(10.44)

This decomposition was also obtained by Balk (2001); notice the subtle differences with expression (10.41). Notice that in general λ(4) = λ(3) unless y 1 = μy 0 for some μ > 0, which, however, would mean that the output mix effect vanishes. Similar to the earlier theorem, one can prove that Theorem 10.2 λ(3) = λ(4) if and only if the comparison period technology exhibits output homotheticity. As before, the third decomposition of the comparison-period-output-orientated Malmquist index is obtained by taking the geometric mean of expressions (10.38) and (10.42), resulting in Mˇ o1 (x 1 , y 1 , x 0 , y 0 ) = Mo1 (x 1 , y 1 , x 0 , y 0 )× [SEC1o,M (λ(3) x 0 , x 0 , y 0 )SEC1o,M (λ(4) x 0 , x 0 , y 1 )]1/2 × [SEC1o,M (x 1 , λ(3) x 0 , y 0 )SEC1o,M (x 1 , λ(4) x 0 , y 1 )]1/2 × [OME1M (x 0 , y 1 , y 0 )OME1M (x 1 , y 1 , y 0 )]1/2 .

(10.45)

The first factor captures technological change and efficiency change, the second factor captures the radial scale effect, the third factor captures the input mix effect, and the fourth factor captures the output mix effect. Calculating this decomposition for a certain firm requires solving Eqs. (10.40) and (10.43). A sufficient condition for the existence of unique solutions is that the distance function Do1 (x, y) is continuously differentiable. If the underlying technology is approximated by DEA (see Appendix B) then such solutions may not exist.

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10 The Components of Total Factor Productivity Change

10.3.3 The ‘Geometric Mean’ Viewpoint Our third candidate productivity index is defined as the geometric mean of the two one-sided indices; that is, Mˇ o (x 1 , y 1 , x 0 , y 0 ) ≡ [Mˇ o0 (x 1 , y 1 , x 0 , y 0 ) × Mˇ o1 (x 1 , y 1 , x 0 , y 0 )]1/2 = 

Dˇ o0 (x 1 , y 1 ) Dˇ o1 (x 1 , y 1 ) Dˇ o0 (x 0 , y 0 ) Dˇ o1 (x 0 , y 0 )

1/2 .

(10.46)

This will be called the geometric-mean-output-orientated Malmquist index. As can be verified easily, there are nine possible decompositions, which can be obtained by combining, respectively, expression (10.31) with (10.38), (10.31) with (10.42), (10.31) with (10.45); (10.33) with (10.38), (10.33) with (10.42), (10.33) with (10.45); (10.36) with (10.38), (10.36) with (10.42), and (10.36) with (10.45). The last combination, which is the most comprehensive, is given by Mˇ o (x 1 , y 1 , x 0 , y 0 ) = [Mo0 (x 1 , y 1 , x 0 , y 0 )Mo1 (x 1 , y 1 , x 0 , y 0 )]1/2 × [SEC0o,M (λ(1) x 0 , x 0 , y 0 )SEC0o,M (λ(2) x 0 , x 0 , y 1 )× SEC1o,M (λ(3) x 0 , x 0 , y 0 )SEC1o,M (λ(4) x 0 , x 0 , y 1 )]1/4 × [SEC0o,M (x 1 , λ(1) x 0 , y 0 )SEC0o,M (x 1 , λ(2) x 0 , y 1 )× SEC1o,M (x 1 , λ(3) x 0 , y 0 )SEC1o,M (x 1 , λ(4) x 0 , y 1 )]1/4 × [OME0M (x 0 , y 1 , y 0 )OME0M (x 1 , y 1 , y 0 )× OME1M (x 0 , y 1 , y 0 )OME1M (x 1 , y 1 , y 0 )]1/4 .

(10.47)

We can distinguish between four main components. The first, on the second line, is the geometric-mean-output-orientated CCD index, [Mo0 (x 1 , y 1 , x 0 , y 0 ) Mo1 (x 1 , y 1 , x 0 , y 0 )]1/2 . This index captures technological change and efficiency change. The second main component, on the third and fourth lines, measures the radial scale effect. The third main component, on the fifth and sixth lines, measures the input mix effect. The fourth main component, on the seventh and eighth lines, measures the output mix effect. There are four equations to solve in order to obtain the lambda’s.

10.3 Decomposing a Malmquist Productivity Index by Output Distance Functions

273

The right-hand side of expression (10.47) would be symmetric in all its variables if λ(1) = λ(2) = λ(3) = λ(4) . In general, however, this is unlikely to happen. For the next result, we introduce the concept of implicit Hicks input-neutral technological change. This type of technological change holds if Do1 (x, y) = Do0 (x, y)A(y) for some function A(y) that is consistent with the axioms. Theorem 10.3 λ(1) = λ(2) = λ(3) = λ(4) if and only if the technologies S 0 and S 1 exhibit output homotheticity and technological change exhibits implicit Hicks input neutrality. Proof The sufficiency part is obvious. For the necessity part we notice that the former two theorems imply the property of output homotheticity for both technologies. Then, using the definition of output homotheticity, we see that Eqs. (10.32) and (10.35) imply that G0 (λx 0 ) = G0 (x 1 ), and that Eqs. (10.40) and (10.43) imply that G1 (λx 0 ) = G1 (x 1 ). Since these equations are assumed to hold for all x 0 , x 1 , it must be the case that the ratio G1 (x)/G0 (x) is independent of x. Thus, G1 (x) = αG0 (x) for some positive scalar α. But then one infers by simple substitution that Do1 (x, y) = Do1 (1N , y)αG0 (x) = Do0 (x, y)α

Do1 (1N , y) . Do0 (1N , y)

This means that technological change exhibits implicit Hicks input neutrality.



Of course, we could select any of the solutions of Eqs. (10.32), (10.35), (10.40), or (10.43) and then set λ(1) = λ(2) = λ(3) = λ(4) . This, however, would introduce an essential element of arbitrariness into the decomposition in expression (10.47). In view of the last theorem, in addition to the possibility that unique solutions of the four equations might not exist, we must be content with four-factor decompositions. By merging the radial scale effect and the input mix effect, expression (10.47) reduces to the following symmetric decomposition: Mˇ o (x 1 , y 1 , x 0 , y 0 ) = [Mo0 (x 1 , y 1 , x 0 , y 0 )Mo1 (x 1 , y 1 , x 0 , y 0 )]1/2 × [SEC0o,M (x 1 , x 0 , y 0 )SEC0o,M (x 1 , x 0 , y 1 )SEC1o,M (x 1 , x 0 , y 0 )SEC1o,M (x 1 , x 0 , y 1 )]1/4 × [OME0M (x 0 , y 1 , y 0 )OME0M (x 1 , y 1 , y 0 )OME1M (x 0 , y 1 , y 0 )OME1M (x 1 , y 1 , y 0 )]1/4 . (10.48)

By merging all the three effects, expression (10.48) further reduces to a Ray and Desli (1997) type decomposition. This decomposition plays a fundamental role in the forecasting exercise of Daskovska et al. (2010), and was used by Chen and Yang (2011) in an extension to meta-frontiers. The geometric mean index proposed by Balk (2001) corresponds to the combination of expressions (10.31) and (10.42), whereby radial scale and input mix effects are merged. Notice that if the base and

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10 The Components of Total Factor Productivity Change

comparison period technologies exhibit global CRS (that is, S 0 = Sˇ 0 and S 1 = Sˇ 1 ), then the scale and mix effects vanish. In the extant literature on productivity measurement, the geometric-mean-outputorientated CCD index frequently figures under the name of ‘the (output orientated) Malmquist productivity index’.15 Notice, however, that a CCD index does not satisfy the proportionality requirement. Following Färe et al. (2008, 572–574), the technological change component of this CCD index, [T Co1,0 (x 1 , y 1 )T Co1,0 (x 0 , y 0 )]1/2 , can be decomposed multiplicatively into three components, respectively measuring the output bias, the input bias, and the magnitude. The magnitude is thereby defined as T Co1,0 (x 0 , y 0 ). Magnitude and bias, however, are badly defined concepts. It is unclear why a particular inputoutput combination is singled out to provide the magnitude of technological change, and the other combination only its input- or output bias. We could as well measure the magnitude by T Co1,0 (x 1 , y 1 ) and define the bias components accordingly. The point is simply that unless there is output neutrality (Balk 1998, 98), these two magnitudes differ. It then makes sense to use their (geometric) mean as the overall magnitude of technological change and their spread as a measure of the extent of non-neutrality.16 An alternative decomposition of the geometric-mean-output-orientated Malmquist productivity index figuring in the literature (see Färe et al. 1994, Färe et al. 2008, 576) is  1,0 1/2 0 0 ˇ o (x 1 , y 1 )TC ˇ 1,0 × Mˇ o (x 1 , y 1 , x 0 , y 0 ) = ECo (x 1 , y 1 , x 0 , y 0 ) × TC o (x , y ) OSE1 (x 1 , y 1 ) . OSE0 (x 0 , y 0 )

(10.49)

The first right-hand side factor measures efficiency change. The second factor measures technological change as exhibited by the (virtual) cone technologies. The third factor conflates scale efficiency effects and technological change. Apart from the fact that the last two factors are not independent of each other, it is not so clear why technological feasibilities should be determined by a virtual rather than an actual technology.

15 The

geometric mean of two CCD indices was introduced as a measurement tool by Färe et al. (1989, 1994). 16 Balk and Althin (1996) generalized this idea to a situation with multiple production units and multiple time periods.

10.4 Decomposing a Malmquist Productivity Index by Input Distance Functions

275

10.4 Decomposing a Malmquist Productivity Index by Input Distance Functions Since a cone technology exhibits global CRS, Dˇ ot (x, y) = 1/Dˇ it (x, y), the productivity index defined by expression (10.17) can also be written as Mˇ ot (x  , y  , x, y) =

Dˇ it (x, y) ≡ Mˇ it (x  , y  , x, y); Dˇ t (x  , y  )

(10.50)

i

that is, as an input-orientated Malmquist index conditional on the period t cone technology. The productivity level at the input-output situation (x, y) is now measured by 1/Dˇ it (x, y). The productivity index at the right-hand side of expression (10.50) has the desired monotonicity and proportionality properties, is by construction transitive in (x, y), and its numerator and denominator are always finite. As before, we consider a number of options. The discussion is kept brief as the theoretical development runs parallel to the previous section.

10.4.1 The Base Period Viewpoint Let us first consider the index that conditions on the base period cone technology; that is, Mˇ i0 (x 1 , y 1 , x 0 , y 0 ) =

Dˇ i0 (x 0 , y 0 ) . Dˇ 0 (x 1 , y 1 )

(10.51)

i

The imagination will now be guided by Fig. 10.2. To start with, we consider the following path from the firm’s base period position to its comparison period position: Path E: (x 0 , y 0 ) −→ (x 0 /Di0 (x 0 , y 0 ), y 0 ) −→ (x 1 /Di0 (x 1 , y 0 ), y 0 ) −→ (x 1 /Di0 (x 1 , y 1 ), y 1 ) −→ (x 1 /Di1 (x 1 , y 1 ), y 1 ) −→ (x 1 , y 1 ) The first segment connects the firm’s base period position to its radial projection, now in the input direction, on the base period technology frontier (point a). We next travel along this frontier to point b, which represents the projection of the firm’s comparison period position on the base period frontier. This movement is split into two parts, respectively corresponding to the change in input quantity space and the change in output quantity space. The fourth segment bridges the distance between the two technology frontiers at point b, to arrive at point c. The final segment

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10 The Components of Total Factor Productivity Change

y 6

S1 c •

d•

•

a

(x1 , y 1 ) • • b

S0

(x0 , y 0 ) -

x

Fig. 10.2 Decomposing productivity change (2)

connects this point, that is, the radial projection of the firm’s comparison period position on the comparison period frontier to the firm’s comparison period position itself. By using the same technique as before, and assuming that Di0 (x 1 , y 0 ) and 0 Di (x 1 , y 1 ) are finite, this leads to the following four-factor decomposition: Mˇ i0 (x 1 , y 1 , x 0 , y 0 ) = Mi0 (x 1 , y 1 , x 0 , y 0 )× IME0M (x 1 , x 0 , y 0 ) × SEC0i,M (x 1 , y 1 , y 0 ),

(10.52)

where 1 1 Mi0 (x 1 , y 1 , x 0 , y 0 ) ≡ ECi (x 1 , y 1 , x 0 , y 0 ) × TC1,0 i (x , y ) =

Di0 (x 0 , y 0 ) Di0 (x 1 , y 1 ) (10.53)

10.4 Decomposing a Malmquist Productivity Index by Input Distance Functions

277

is the base-period-input-orientated CCD index, like its output-orientated counterpart introduced by Caves et al. (1982).17 For the definitions of the various functions the reader is referred to Sect. 10.2.1. The scale effect, SEC0i,M (x 1 , y 1 , y 0 ), can of course be decomposed into a radial effect (in output quantity space) and an output mix effect by introducing a scalar μ that plays a similar role as λ in the previous section. This complication is left to the reader. The alternative path is obtained by reversing the order in which, along the segment from a to b, changes in input and output quantity space are accounted for. Thus, assuming that Di0 (x 0 , y 1 ) and Di0 (x 1 , y 1 ) are finite, Path F: (x 0 , y 0 ) −→ (x 0 /Di0 (x 0 , y 0 ), y 0 ) −→ (x 0 /Di0 (x 0 , y 1 ), y 1 ) −→ (x 1 /Di0 (x 1 , y 1 ), y 1 ) −→ (x 1 /Di1 (x 1 , y 1 ), y 1 ) −→ (x 1 , y 1 ) This leads to the second decomposition, which reads Mˇ i0 (x 1 , y 1 , x 0 , y 0 ) = Mi0 (x 1 , y 1 , x 0 , y 0 )× IME0M (x 1 , x 0 , y 1 ) × SEC0i,M (x 0 , y 1 , y 0 ).

(10.54)

This decomposition was obtained by Balk (2001), and used in the empirical work of Pantzios et al. (2011). The third decomposition is obtained by taking the geometric mean of the previous two decompositions, expressions (10.52) and (10.54), resulting in Mˇ i0 (x 1 , y 1 , x 0 , y 0 ) = Mi0 (x 1 , y 1 , x 0 , y 0 )× [IME0M (x 1 , x 0 , y 0 )IME0M (x 1 , x 0 , y 1 )]1/2 × [SEC0i,M (x 0 , y 1 , y 0 )SEC0i,M (x 1 , y 1 , y 0 )]1/2 .

(10.55)

The first factor on the right-hand side of the equality sign captures technological change and efficiency change, the second main factor (on the second line) captures the input mix effect, and the main third factor (on the third line) captures the scale (including output mix) effect. Using the various definitions, it is straightforward to check that if the base period technology exhibits global CRS (that is, S 0 = Sˇ 0 ), then the scale and mix effects vanish.

generic definition of the input-orientated CCD index is Mit (x  , y  , x, y) ≡ Dit (x, y)/Dit (x  , y  ). Notice that generally this function does not exhibit the proportionality property required for a genuine productivity index.

17 The

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10 The Components of Total Factor Productivity Change

10.4.2 The Comparison Period Viewpoint Next, we consider the input-orientated Malmquist index which conditions on the comparison period cone technology; that is, Mˇ i1 (x 1 , y 1 , x 0 , y 0 ) =

Dˇ i1 (x 0 , y 0 ) . Dˇ 1 (x 1 , y 1 )

(10.56)

i

The following path connects the firm’s base period position with its comparison period position: Path G: (x 0 , y 0 ) −→ (x 0 /Di0 (x 0 , y 0 ), y 0 ) −→ (x 0 /Di1 (x 0 , y 0 ), y 0 ) −→ (x 1 /Di1 (x 1 , y 0 ), y 0 ) −→ (x 1 /Di1 (x 1 , y 1 ), y 1 ) −→ (x 1 , y 1 ) Recall Fig. 10.2. The first part takes us from the firm’s base period position to its projection on the base period frontier (point a). At this point we cross over to point d, which is the radial projection of the firm’s base period position on the comparison period frontier. Then we move along this frontier to point c, which is the radial projection of the firm’s comparison period position on the comparison period frontier. This movement is split into two parts, respectively corresponding to a movement in input quantity space and a movement in output quantity space. Finally, we connect point c to the firm’s comparison period position. Assuming that Di1 (x 0 , y 0 ) and Di1 (x 1 , y 0 ) are finite, we obtain the following four-factor decomposition: Mˇ i1 (x 1 , y 1 , x 0 , y 0 ) = Mi1 (x 1 , y 1 , x 0 , y 0 )× IME1M (x 1 , x 0 , y 0 ) × SEC1i,M (x 1 , y 1 , y 0 ),

(10.57)

where 0 0 Mi1 (x 1 , y 1 , x 0 , y 0 ) ≡ ECi (x 1 , y 1 , x 0 , y 0 ) × TC1,0 i (x , y ) =

Di1 (x 0 , y 0 )

Di1 (x 1 , y 1 ) (10.58) is the comparison-period-input-orientated CCD index, comprising efficiency change and technological change. Further, IME1M (x 1 , x 0 , y 0 ) is the input mix effect, and SEC1i,M (x 1 , y 1 , y 0 ) is the scale (including output mix) effect. The decomposition in expression (10.57) was obtained by Balk (2001). The alternative path is obtained by reversing the order in which, along the segment from d to c, changes in input and output quantity space are accounted for. Thus,

10.4 Decomposing a Malmquist Productivity Index by Input Distance Functions

279

Path H: (x 0 , y 0 ) −→ (x 0 /Di0 (x 0 , y 0 ), y 0 ) −→ (x 0 /Di1 (x 0 , y 0 ), y 0 ) −→ (x 0 /Di1 (x 0 , y 1 ), y 1 ) −→ (x 1 /Di1 (x 1 , y 1 ), y 1 ) −→ (x 1 , y 1 ) Assuming that Di1 (x 0 , y 0 ) and Di1 (x 0 , y 1 ) are finite, we obtain the second decomposition, which reads Mˇ i1 (x 1 , y 1 , x 0 , y 0 ) = Mi1 (x 1 , y 1 , x 0 , y 0 )× IME1M (x 1 , x 0 , y 1 ) × SEC1i,M (x 0 , y 1 , y 0 ).

(10.59)

Like before, the third decomposition is obtained by taking the geometric mean of these two decompositions, expressions (10.57) and (10.59), resulting in Mˇ i1 (x 1 , y 1 , x 0 , y 0 ) = Mi1 (x 1 , y 1 , x 0 , y 0 )× [IME1M (x 1 , x 0 , y 0 )IME1M (x 1 , x 0 , y 1 )]1/2 × [SEC1i,M (x 0 , y 1 , y 0 )SEC1i,M (x 1 , y 1 , y 0 )]1/2 .

(10.60)

The first factor on the right-hand side of the equality sign is the comparison-periodinput-orientated CCD index, capturing technological change and efficiency change. The second main factor (on the second line) captures the input mix effect, and the third main factor (on the third line) captures the scale (including output mix) effect. As before, it is straightforward to check that if the comparison period technology exhibits global CRS (that is, S 1 = Sˇ 1 ), then the scale and mix effects vanish.

10.4.3 The ‘Geometric Mean’ Viewpoint Finally, we consider the geometric mean of the two input-distance-function-based Malmquist productivity indices Mˇ i0 (x 1 , y 1 , x 0 , y 0 ) and Mˇ i1 (x 1 , y 1 , x 0 , y 0 ). Again there are nine possible decompositions. The completely symmetric and comprehensive one is obtained by combining the decompositions (10.55) and (10.60), resulting in Mˇ i (x 1 , y 1 , x 0 , y 0 ) ≡ [Mi0 (x 1 , y 1 , x 0 , y 0 )Mi1 (x 1 , y 1 , x 0 , y 0 )]1/2 × [IME0M (x 1 , x 0 , y 0 )IME0M (x 1 , x 0 , y 1 )IME1M (x 1 , x 0 , y 0 )IME1M (x 1 , x 0 , y 1 )]1/4 ×

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10 The Components of Total Factor Productivity Change

[SEC0i,M (x 0 , y 1 , y 0 )SEC0i,M (x 1 , y 1 , y 0 )SEC1i,M (x 0 , y 1 , y 0 )SEC1i,M (x 1 , y 1 , y 0 )]1/4 . (10.61) In the light of the theoretical framework developed here, we see that the geometric mean index proposed by Balk (2001) corresponds to the combination of expressions (10.54) and (10.57). Notice that [Mi0 (x 1 , y 1 , x 0 , y 0 )Mi1 (x 1 , y 1 , x 0 , y 0 )]1/2 is the geometric-meaninput-orientated CCD index. In the extant literature on productivity measurement, this index frequently figures under the name ‘the (input orientated) Malmquist productivity index’. Notice, however, that a CCD index does not satisfy the proportionality requirement. If the base and comparion period technologies both exhibit global CRS (that is, S 0 = Sˇ 0 and S 1 = Sˇ 1 ), then the scale and mix effects in expression (10.61) vanish.

10.5 Decomposing a Moorsteen-Bjurek Productivity Index 10.5.1 Definitions As introduced in Sect. 10.2.2, in general a productivity index is a function ¯ and F (x  , y  , x, y) which behaves as an output quantity index for x = x  = x, as the reciprocal of an input quantity index for y = y  = y. ¯ Thus it seems rather natural to define a productivity index as an output quantity index divided by an input quantity index. This is the basic idea behind the family of Moorsteen-Bjurek (MB) productivity indices. For any period t technology, a Malmquist output quantity index, comparing output quantities y  to y, conditional on certain input quantities x, ¯ can be defined as Qto (y  , y, x) ¯ ≡ Dot (x, ¯ y  )/Dot (x, ¯ y). Similarly, a Malmquist input quantity index, ¯ can comparing input quantities x  to x, conditional on certain output quantities y, t (x, y). be defined as Qti (x  , x, y) ¯ ≡ Dit (x  , y)/D ¯ ¯ Both indices can be traced back i to suggestions by Moorsteen (1961). Their properties were extensively discussed in Balk (1998, Sections 3.4 and 4.3).18 The family of Moorsteen-Bjurek (MB) productivity indices is then defined by MBt (x  , y  , x, y; x, ¯ y) ¯ ≡

¯ Qto (y  , y, x) ; Qti (x  , x, y) ¯

(10.62)

that is, the ratio of a period t Malmquist output quantity index and a period t Malmquist input quantity index, conditional on a vector of input quantities x¯ and a vector of output quantities y, ¯ respectively. There does not need to be any relation between the benchmark technology period t and the timing of the inputoutput combinations (x  , y  ), (x, y), or (x, ¯ y). ¯ Typically, however, in an empirical 18 See

also Diewert and Fox (2017) where slightly weaker regularity conditions were used.

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281

application involving many production units, x¯ and y¯ would be chosen as vectors of sample means. Expression (10.62) defines a family of indices. Each specific selection of t and (x, ¯ y) ¯ generates a member of this family. It is easily checked that any specific MB index exhibits the monotonicity and proportionality properties essential for a productivity index, and that for x¯ = x, x  and y¯ = y, y  the index is transitive ¯ ∈ St , in (x, y). Determinateness requires that (x, ¯ y) ∈ S t , y ∈ M ++ , and (x, y) N 19 x ∈ ++ . Actually, by substituting the two quantity index definitions, we see that an MB productivity index can be written in the form ¯ y) ¯ = MBt (x  , y  , x, y; x,

Dot (x, ¯ y  )/Dit (x  , y) ¯ ; t t Do (x, ¯ y)/Di (x, y) ¯

(10.63)

that is, as a ratio of two productivity levels. Up to a scalar normalization, and conditional on certain x¯ and y, ¯ the productivity level at the input-output situation (x, y) is thereby measured as Dot (x, ¯ y)/Dit (x, y). ¯ Thus, the family of MB indices belongs to the class of “multiplicatively complete” TFP indices, as defined by O’Donnell (2012).20 It is interesting to relate the MB indices to the CCD productivity indices introduced in the previous chapters. Using the definitions, we easily verify that ¯ y) ¯ = Mot (x, ¯ y  , x, ¯ y) × Mit (x  , y, ¯ x, y). ¯ MBt (x  , y  , x, y; x,

(10.64)

The first factor on the right-hand side of the equality sign is an output-orientated ¯ y). The second CCD index comparing the input-output combination (x, ¯ y  ) to (x, factor is an input-orientated CCD index comparing the input-output combination (x  , y) ¯ to (x, y). ¯ It is clear that if x  = x, then the MB index reduces to an output-orientated CCD index, and if y  = y, then the MB index reduces to an input-orientated CCD index.21 Expression (10.64) also throws light on the fact that

 | y  > 0}. Then check that the numerator is always finite, consider λ ≡ minm {ym /ym m   t (x, ¯ λy ) ≤ (x, ¯ y) and thus, by free disposability of outputs, (x, ¯ λy ) ∈ S . Then Dot (x, ¯ λy  ) ≤ 1 t  and, by linear homogeneity of the output distance function, Do (x, ¯ y ) ≤ 1/λ < ∞. To check that the denominator is always greater than 0, consider λ ≡ maxn {xn /xn | xn > 0}. Then (λ x  , y) ¯ ≥ (x, y) ¯ and thus, by free disposability of inputs, (λ x  , y) ¯ ∈ S t . Then Dit (λ(x  , y) ¯ ≥ 1 and, by linear homogeneity of the input distance function, Dit (x  , y) ¯ ≥ 1/λ > 0. This proof generalizes the proof provided by Briec and Kerstens (2011). 20 O’Donnell (2014) called the indices defined by expression (10.62) after Färe and Primont because the component output and input quantity indices were discussed in their 1995 book. The indices were called Bjurek productivity indices by Diewert and Fox (2017). To continue footnote 8, X(x) ≡ Dit (x, y) ¯ and Y (y) ≡ Dot (x, ¯ y). O’Donnell (2016) generalised the indices by including environmental variables. 21 Moreover, if x ∈ 1 (single input) and constant and the benchmark technology S t is equal to + ¯ y  , x, ¯ y) = Mˇ ot (x, ¯ y  , x, ¯ y) = Mˇ it (x, ¯ y  , x, ¯ y), and the scale, its DEA approximation, then Mot (x, 19 To

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10 The Components of Total Factor Productivity Change

MB and CCD indices share the monotonicity properties but not the proportionality property. This should settle the “healthy debate on the relative merits of the orientated [that is, CCD] and non-orientated [that is, MB] Malmquist productivity indices”, called for by Lovell (2016). We obtain the indices as originally considered by Bjurek (1996) as special cases, namely by selecting in the generic definition of the MB index x¯ = x and y¯ = y or x¯ = x  and y¯ = y  . The first employs the ‘Laspeyres’ perspective, MBt (x  , y  , x, y; x, y) =

Qto (y  , y, x) , Qti (x  , x, y)

(10.65)

and the second employs the ‘Paasche’ perspective, MBt (x  , y  , x, y; x  , y  ) =

Qto (y  , y, x  ) . Qti (x  , x, y  )

(10.66)

The third index considered by Bjurek is the geometric mean, " t   #1/2 MB (x , y , x, y; x, y)MBt (x  , y  , x, y; x  , y  ) All these indices were called Malmquist Total Factor Productivity indices, although the fact that they take all the inputs into account is not a feature that distinguishes them from the Malmquist indices discussed in the previous sections.22 One easily checks that neither the MB indices defined by expression (10.65), nor those defined by expression (10.66), nor their geometric means can be written as a ratio of productivity levels, so that these indices are not transitive. This has implications for the way in which these and other indices belonging to the MB family are decomposed into meaningful components. For the details the reader is referred to Balk and Zofío (2018). A further specification concerns the benchmark technology figuring in expressions (10.65) and (10.66). The base-period-, or Laspeyres-perspective MB index for period 1 relative to period 0 is then given by MB0 (x 1 , y 1 , x 0 , y 0 ; x 0 , y 0 ), and the comparison-period-, or Paasche-perspective MB index by MB1 (x 1 , y 1 , x 0 , y 0 ; x 1 , y 1 ). Conditions under which their geometric mean,

input- and output-mix effects vanish, as noticed by Karagiannis and Lovell (2016). Of course, a similar result holds for the single-output case. 22 Bjurek’s name-giving was continued by Grifell-Tatjé and Lovell (2015). The geometric mean index was called Hicks-Moorsteen TFP index by Färe et al. (2008), O’Donnell (2012), and Kerstens and Van de Woestyne (2014). Sometimes a reference is made to footnote 4 in Hicks (1961). On closer scrutiny, however, there appears to be insufficient evidence for ascribing a partial fathership to Hicks. Though Hicks definitely discussed the concepts of Malmquist output and input quantity indices in a qualitative, thus not formal, way, there is no hint that he considered their ratio as a measure of productivity change. See Epure et al. (2011) on the use of the MB indices in benchmarking.

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283

 1/2 MB0 (x 1 , y 1 , x 0 , y 0 ; x 0 , y 0 )MB1 (x 1 , y 1 , x 0 , y 0 ; x 1 , y 1 ) , coincides with the geometric-mean CCD indices materializing in expressions (10.48) and (10.61), [Mo0 (x 1 , y 1 , x 0 , y 0 )Mo1 (x 1 , y 1 , x 0 , y 0 )]1/2 and [Mi0 (x 1 , y 1 , x 0 , y 0 )Mi1 (x 1 , y 1 , x 0 , y 0 )]1/2 , respectively, were analyzed in Balk (1998, Section 4.6). They appear to be (1) that both technologies exhibit global CRS and output homotheticity or input homotheticity, or (2) that both technologies exhibit global CRS and that technological change is implicit Hicks output-neutral or input-neutral. The second set of conditions generalizes a result obtained by Mizobuchi (2017). O’Donnell (2012, 258) assumed inverse homotheticity, which is a stronger condition than (1).

10.5.2 Decompositions We now consider the general MB productivity index MBt (x  , y  , x, y; x, ¯ y) ¯ for fixed values of the conditioning variables t, x¯ and y. ¯ Consider path A and let λ = 1. The six segments then reduce to five. The MB index is evaluated along each segment. Using its transitivity, the parts can be joined and the following decomposition is obtained: 1 1 MBt (x 1 , y 1 , x 0 , y 0 ; x, ¯ y) ¯ = ECo (x 1 , y 1 , x 0 , y 0 ) × TC1,0 o (x , y ) ×



¯ Do0 (x 0 , y 0 ) Dit (x 0 , y) t 0 1 0 1 Do (x , y ) Di (x , y) ¯

 ×

 ¯ y 1 ) Do0 (x 1 , y 0 ) Dot (x, . Dot (x, ¯ y 0 ) Do0 (x 1 , y 1 )

(10.67)

The factors on the right-hand side of the equality sign represent, respectively, technical efficiency change (corresponding to the first and last segment of the path), technological change (corresponding to the fourth segment), radial scale including input mix effect (corresponding to the second segment, and conditional on y 0 ), and output mix effect (corresponding to the third segment, and conditional on x 1 ). The first two factors are familiar. Together they constitute the base-period-outputorientated CCD index as defined by expression (10.30). To facilitate comparisons with the components of the Malmquist indices, we denote the scale (including input mix) effect by 1 0 0 SEC0t ¯ ≡ o,MB (x , x , y ; y)

and the output mix effect by

¯ Do0 (x 0 , y 0 ) Dit (x 0 , y) , t 0 1 0 1 Do (x , y ) Di (x , y) ¯

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10 The Components of Total Factor Productivity Change

1 1 0 OME0t ¯ ≡ MB (x , y , y ; x)

¯ y 1 ) Do0 (x 1 , y 0 ) Dot (x, . Dot (x, ¯ y 0 ) Do0 (x 1 , y 1 )

Expression (10.67) can then be written as MBt (x 1 , y 1 , x 0 , y 0 ; x, ¯ y) ¯ = 1 0 0 1 1 0 ¯ × OME0t ¯ Mo0 (x 1 , y 1 , x 0 , y 0 ) × SEC0t o,MB (x , x , y ; y) MB (x , y , y ; x). (10.68) The third factor in expression (10.68) indeed measures the output mix effect. Using the linear homogeneity of the output distance functions, it appears that 1 1 0 1 1 1 0 0 OME0t ¯ = OME0t ¯ MB (x , y , y ; x) MB (x , y /  y , y /  y ; x).

To interpret the second factor, we assume that there is no technological change, that the firm is technically efficient in both periods, that x 1 = λx 0 for some positive scalar λ, and that y 1 = μy 0 for some positive scalar μ. Then expression (10.68) reduces to MBt (x 1 , y 1 , x 0 , y 0 ; x, ¯ y) ¯ =

1 Do0 (x 0 , y 0 ) = , λDo0 (λx 0 , y 0 ) λDo0 (λx 0 , y 0 )

(10.69)

since the firm is assumed efficient in period 0. By multiplying numerator and denominator by μ, using the linear homogeneity of the output distance function, and assuming efficiency in period 1, respectively, we obtain MBt (x 1 , y 1 , x 0 , y 0 ; x, ¯ y) ¯ =

μ μ μ = = , λ λDo0 (λx 0 , μy 0 ) λDo1 (x 1 , y 1 )

(10.70)

which is the outcome expected. If the base period technology exhibits global CRS (that is, S 0 = Sˇ 0 ), then its output distance function is identically equal to the inverse of its input distance function. The second factor then reduces to 1 0 0 SEC0t ¯ = o,MB (x , x , y ; y)

Di0 (x 1 , y 0 ) Dit (x 0 , y) ¯ . t 0 1 ¯ Di (x 0 , y 0 ) Di (x , y)

By using the linear homogeneity of the two input distance functions we obtain 1 0 0 SEC0t ¯ = o,MB (x , x , y ; y)

Di0 (x 1 /x 1 , y 0 ) Dit (x 0 /x 0 , y) ¯ , t 0 1 1 0 0 0 ¯ Di (x /x , y ) Di (x /x , y)

which means that the scale effect has vanished and only the input mix effect remains. Let us now compare the scale (including input mix) effect and the output mix effect with the corresponding components of the Malmquist indices. The

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285

scale (including input mix) components of the output-orientated Malmquist indices are based on SECto,M (x 1 , x 0 , y). ¯ Using the definition of this function— see Sect. 10.2.1—and the fact that a cone technology’s output distance function is the inverse of its input distance function, it appears that SECto,M (x 1 , x 0 ; y) ¯ =

¯ ¯ Dˇ it (x 0 , y) Dot (x 0 , y) . t 1 t 1 ˇ Do (x , y) ¯ Di (x , y) ¯

1 0 0 ¯ The differences are subtle, This has the same structure as SEC0t o,MB (x , x , y ; y). but noteworthy. The output mix components of the output-orientated Malmquist indices are based on

OMEtM (x, ¯ y1, y0) =

¯ y 1 ) Dot (x, ¯ y0) Dˇ ot (x, , ¯ y1) ¯ y 0 ) Dot (x, Dˇ ot (x,

1 1 0 ¯ defined above. which has the same structure as OME0t MB (x , y , y ; x), It is interesting to consider under which conditions the scale (including input mix) and output mix effects vanish entirely from expression (10.67). It is easily checked that a sufficient set of conditions is that (1) the base period technology is chosen as benchmark, (2) the base period technology exhibits global CRS, and (3) the conditioning variables are chosen as (x, ¯ y) ¯ = (x 1 , y 0 ). Then 0 1 1 0 0 1 0 0 1 1 0 0 MB (x , y , x , y ; x , y ) = Mo (x , y , x , y ). We now turn to the other paths, but present the corresponding decompositions without discussion. Along path B (λ = 1) we obtain

MBt (x 1 , y 1 , x 0 , y 0 ; x, ¯ y) ¯ = Mo0 (x 1 , y 1 , x 0 , y 0 ) ×

  ¯ Do0 (x 0 , y 1 ) Dit (x 0 , y) ¯ y 1 ) Do0 (x 0 , y 0 ) Dot (x, . × Do0 (x 1 , y 1 ) Dit (x 1 , y) ¯ Dot (x, ¯ y 0 ) Do0 (x 0 , y 1 )

(10.71)

The scale (including input mix) and output mix effects vanish from expression (10.71) if (1) the base period technology is chosen as benchmark, (2) the base period technology exhibits global CRS, and (3) the conditioning variables are chosen as (x, ¯ y) ¯ = (x 0 , y 1 ). Then MB0 (x 1 , y 1 , x 0 , y 0 ; x 0 , y 1 ) = Mo0 (x 1 , y 1 , x 0 , y 0 ). Along path C (λ = 1) we obtain MBt (x 1 , y 1 , x 0 , y 0 ; x, ¯ y) ¯ = Mo1 (x 1 , y 1 , x 0 , y 0 ) ×

  ¯ Do1 (x 0 , y 0 ) Dit (x 0 , y) ¯ y 1 ) Do1 (x 1 , y 0 ) Dot (x, . × Do1 (x 1 , y 0 ) Dit (x 1 , y) ¯ Dot (x, ¯ y 0 ) Do1 (x 1 , y 1 )

(10.72)

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10 The Components of Total Factor Productivity Change

The scale (including input mix) and output mix effects vanish from expression (10.72) if (1) the comparison period technology is chosen as benchmark, (2) the comparison period technology exhibits global CRS, and (3) the conditioning variables are chosen as (x, ¯ y) ¯ = (x 1 , y 0 ). Then MB1 (x 1 , y 1 , x 0 , y 0 ; x 1 , y 0 ) = 1 1 1 0 0 Mo (x , y , x , y ). Along path D (λ = 1) we obtain MBt (x 1 , y 1 , x 0 , y 0 ; x, ¯ y) ¯ = Mo1 (x 1 , y 1 , x 0 , y 0 ) ×

  ¯ Do1 (x 0 , y 1 ) Dit (x 0 , y) ¯ y 1 ) Do1 (x 0 , y 0 ) Dot (x, . × Do1 (x 1 , y 1 ) Dit (x 1 , y) ¯ Dot (x, ¯ y 0 ) Do1 (x 0 , y 1 )

(10.73)

The scale (including input mix) and output mix effects vanish from expression (10.73) if (1) the comparison period technology is chosen as benchmark, (2) the comparison period technology exhibits global CRS, and (3) the conditioning variables are chosen as (x, ¯ y) ¯ = (x 0 , y 1 ). Then MB1 (x 1 , y 1 , x 0 , y 0 ; x 0 , y 1 ) = 1 1 1 0 0 Mo (x , y , x , y ). In the next four decompositions, output distance functions are replaced by input distance functions. Thus, along path E, MBt (x 1 , y 1 , x 0 , y 0 ; x, ¯ y) ¯ = Mi0 (x 1 , y 1 , x 0 , y 0 ) ×

Di0 (x 1 , y 0 ) Dit (x 0 , y) ¯ ¯ y 1 ) Di0 (x 1 , y 1 ) Dot (x, . × ¯ Dot (x, ¯ y 0 ) Di0 (x 1 , y 0 ) Di0 (x 0 , y 0 ) Dit (x 1 , y)

(10.74)

At the right-hand side of this expression we see, respectively, the base-periodinput-orientated CCD index as defined by expression (10.53), the input mix effect (conditional on y 0 ), and the radial scale including output mix effect (conditional on x 1 ). These effects are structurally similar to the corresponding components of the input-orientated Malmquist indices. The input mix and scale (including output mix) effects vanish from expression (10.74) if (1) the base period technology is chosen as benchmark, (2) the base period technology exhibits global CRS, and (3) the conditioning variables are chosen as (x, ¯ y) ¯ = (x 1 , y 0 ). Then MB0 (x 1 , y 1 , x 0 , y 0 ; x 1 , y 0 ) = Mi0 (x 1 , y 1 , x 0 , y 0 ). Path F delivers similarly MBt (x 1 , y 1 , x 0 , y 0 ; x, ¯ y) ¯ = Mi0 (x 1 , y 1 , x 0 , y 0 ) ×

Di0 (x 1 , y 1 ) Dit (x 0 , y) ¯ Dot (x, ¯ y 1 ) Di0 (x 0 , y 1 ) . × ¯ Dot (x, ¯ y 0 ) Di0 (x 0 , y 0 ) Di0 (x 0 , y 1 ) Dit (x 1 , y)

(10.75)

The input mix and scale (including output mix) effects vanish from expression (10.75) if (1) the base period technology is chosen as benchmark, (2) the base period

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technology exhibits global CRS, and (3) the conditioning variables are chosen as (x, ¯ y) ¯ = (x 0 , y 1 ). Then MB0 (x 1 , y 1 , x 0 , y 0 ; x 0 , y 1 ) = Mi0 (x 1 , y 1 , x 0 , y 0 ). Along path G we obtain MBt (x 1 , y 1 , x 0 , y 0 ; x, ¯ y) ¯ = Mi1 (x 1 , y 1 , x 0 , y 0 ) ×



Di1 (x 1 , y 0 ) Dit (x 0 , y) ¯ Dot (x, ¯ y 1 ) Di1 (x 1 , y 1 ) × . ¯ Dot (x, ¯ y 0 ) Di1 (x 1 , y 0 ) Di1 (x 0 , y 0 ) Dit (x 1 , y)

(10.76)

The input mix and scale (including output mix) effects vanish from expression (10.76) if (1) the comparison period technology is chosen as benchmark, (2) the comparison period technology exhibits global CRS, and (3) the conditioning variables are chosen as (x, ¯ y) ¯ = (x 1 , y 0 ). Then MB1 (x 1 , y 1 , x 0 , y 0 ; x 1 , y 0 ) = 1 1 1 0 0 Mi (x , y , x , y ). Along path H we obtain MBt (x 1 , y 1 , x 0 , y 0 ; x, ¯ y) ¯ = Mi1 (x 1 , y 1 , x 0 , y 0 ) ×

Di1 (x 1 , y 1 ) Dit (x 0 , y) ¯ Dot (x, ¯ y 1 ) Di1 (x 0 , y 1 ) . × ¯ Dot (x, ¯ y 0 ) Di1 (x 0 , y 0 ) Di1 (x 0 , y 1 ) Dit (x 1 , y)

(10.77)

The input mix and scale (including output mix) effects vanish from expression (10.77) if (1) the comparison period technology is chosen as benchmark, (2) the comparison period technology exhibits global CRS, and (3) the conditioning variables are chosen as (x, ¯ y) ¯ = (x 0 , y 1 ). Then MB1 (x 1 , y 1 , x 0 , y 0 ; x 0 , y 1 ) = 1 1 1 0 0 Mi (x , y , x , y ). Summarizing, we have eight decompositions of the productivity index MBt (x 1 , y 1 , x 0 , y 0 ; x, ¯ y). ¯ By specifying t = 0 in expressions (10.67) and (10.71), we obtain two decompositions of MB0 (x 1 , y 1 , x 0 , y 0 ; x, ¯ y). ¯ Similarly, by specifying t = 1 in expressions (10.72) and (10.73), we obtain two decompositions of MB1 (x 1 , y 1 , x 0 , y 0 ; x, ¯ y). ¯ By taking their geometric mean, we obtain the following decomposition of the geometric-mean MB index: 1 1 1 0 0 ¯ y)MB ¯ (x , y , x , y ; x, ¯ y)] ¯ 1/2 = [MB0 (x 1 , y 1 , x 0 , y 0 ; x,

[Mo0 (x 1 , y 1 , x 0 , y 0 )Mo1 (x 1 , y 1 , x 0 , y 0 )]1/2 × 



¯ Do0 (x 0 , y 0 ) Di0 (x 0 , y) 0 1 0 0 Do (x , y ) Di (x 1 , y) ¯

¯ Do1 (x 0 , y 0 ) Di1 (x 0 , y) 1 1 0 1 1 Do (x , y ) Di (x , y) ¯



×

×

¯ Do0 (x 0 , y 1 ) Di0 (x 0 , y) × Do0 (x 1 , y 1 ) Di0 (x 1 , y) ¯

¯ Do1 (x 0 , y 1 ) Di1 (x 0 , y) 1 1 1 1 1 Do (x , y ) Di (x , y) ¯

1/4 ×

288

10 The Components of Total Factor Productivity Change





Do0 (x, ¯ y 1 ) Do0 (x 1 , y 0 ) Do0 (x, ¯ y 0 ) Do0 (x 1 , y 1 )

Do1 (x, ¯ y 1 ) Do1 (x 1 , y 0 ) 1 Do (x, ¯ y 0 ) Do1 (x 1 , y 1 )



 ×



 ×

 Do0 (x, ¯ y 1 ) Do0 (x 0 , y 0 ) × Do0 (x, ¯ y 0 ) Do0 (x 0 , y 1 )

Do1 (x, ¯ y 1 ) Do1 (x 0 , y 0 ) 1 Do (x, ¯ y 0 ) Do1 (x 0 , y 1 )

 1/4 (10.78)

.

This decomposition is based on output distance functions. The first component, on the second line, is the geometric-mean-output-orientated CCD index, combining technical efficiency change and technological change. The second component, between big brackets on the third and fourth lines, measures the scale including input mix effect. The third component, between big brackets on the fifth and sixth lines, measures the output mix effect. When output distance functions are replaced by input distance functions alternative decompositions are obtained. Thus, by specifying t = 0 in expressions (10.74) and (10.75), we obtain two decompositions of MB0 (x 1 , y 1 , x 0 , y 0 ; x, ¯ y). ¯ Similarly, by specifying t = 1 in expressions (10.76) and (10.77), we obtain two decompositions of MB1 (x 1 , y 1 , x 0 , y 0 ; x, ¯ y). ¯ By taking their geometric mean, we obtain another decomposition of the geometric-mean MB index, now in terms of input distance functions: 1 1 1 0 0 [MB0 (x 1 , y 1 , x 0 , y 0 ; x, ¯ y)MB ¯ (x , y , x , y ; x, ¯ y)] ¯ 1/2 =

[Mi0 (x 1 , y 1 , x 0 , y 0 )Mi1 (x 1 , y 1 , x 0 , y 0 )]1/2 × 

¯ Di0 (x 1 , y 0 ) Di0 (x 0 , y)

×

Di0 (x 0 , y 0 ) Di0 (x 1 , y) ¯

Di1 (x 1 , y 0 ) Di1 (x 0 , y) ¯

Di1 (x 0 , y 0 ) Di1 (x 1 , y) ¯ 



×

¯ y 1 ) Di0 (x 1 , y 1 ) Do0 (x, Do0 (x, ¯ y 0 ) Di0 (x 1 , y 0 )

Do1 (x, ¯ y 1 ) Di1 (x 1 , y 1 ) Do1 (x, ¯ y 0 ) Di1 (x 1 , y 0 )

×



Di0 (x 1 , y 1 ) Di0 (x 0 , y) ¯

Di0 (x 0 , y 1 ) Di0 (x 1 , y) ¯

¯ Di1 (x 1 , y 1 ) Di1 (x 0 , y)

1/4 ×

Di1 (x 0 , y 1 ) Di1 (x 1 , y) ¯

×

×

¯ y 1 ) Di0 (x 0 , y 1 ) Do0 (x, × Do0 (x, ¯ y 0 ) Di0 (x 0 , y 0 )

Do1 (x, ¯ y 1 ) Di1 (x 0 , y 1 ) Do1 (x, ¯ y 0 ) Di1 (x 0 , y 0 )

1/4 .

(10.79)

This decomposition has a similar structure as the foregoing; the differences are subtle. The first component on the right-hand side of the equality sign, on the second line, is the geometric-mean-input-orientated CCD index, combining technical efficiency change and technological change. The second component, between big brackets on the third and fourth lines, measures the input mix effect. The third

10.6 Decomposing a Lowe Productivity Index

289

component, between big brackets on the fifth and sixth lines, measures the scale including output mix effect. By further specifying the conditioning variables x¯ and y, ¯ we can establish linkages with decompositions proposed in the literature. For instance, by setting t = 0 and (x, ¯ y) ¯ = (x 0 , y 0 ) in expression (10.67), we obtain the decomposition contemplated by Grifell-Tatjé and Lovell (2015, 136–137). Related decompositions proposed by Nemoto and Goto (2005), Peyrache (2014), and Diewert and Fox (2017) were reviewed in Balk and Zofío (2018).

10.6 Decomposing a Lowe Productivity Index 10.6.1 Definitions In the case of a single input and a single output, productivity is naturally measured by the ratio of output quantity over input quantity; that is, y/x. The productivity index, measuring the change through time, is then unequivocally given by (y  /x  )/(y/x) = (y  /y)/(x  /x). This simplicity is lost in the multi-input-multioutput case. The natural generalization is to weigh output quantities y by certain output prices p, and input quantities x by certain input prices w, and to define the productivity level by p · y/w · x.23 A price-weighted productivity index can then be defined as LWPROD(x  , y  , x, y; w, p) ≡

p · y  /p · y p · y  /w · x  = , p · y/w · x w · x  /w · x

(10.80)

where p ≡ (p1 , . . . , pM ) ∈ M ++ denotes a vector of output prices and w ≡ N (w1 , . . . , wN ) ∈ ++ denotes a vector of input prices.24 We can verify that LWPROD(x  , y  , x, y; w, p) satisfies the fundamental monotonicity and proportionality requirements and, for fixed w and p, exhibits transitivity in (x, y). As the last part of expression (10.80) shows, the productivity index can be written as the ratio of a Lowe output quantity index and a Lowe input quantity index; see Balk (2008) for their axiomatic properties. Thus LWPROD(x  , y  , x, y; w, p) will be called a Lowe productivity index. A simple transformation of a Lowe productivity index produces an instance of the well-known Solow residual. Consider ln LWPROD(x  , y  , x, y; w, p)

23 Thus, 24 These

to continue footnote 8, X(x) ≡ w · x and Y (y) ≡ p · y. prices might be imputed or shadow prices. See for example Coelli et al. (2003).

290

10 The Components of Total Factor Productivity Change

    = ln p · y  /p · y − ln w · x  /w · x     w · (x  − x) p · (y  − y) − ln 1 + . = ln 1 + p·y w·x

(10.81)

If  y  − y  and  x  − x  are small, then the following approximation holds: ln LWPROD(x  , y  , x, y; w, p) ≈

p · (y  − y) w · (x  − x) − . p·y w·x

(10.82)

If we define revenue shares by um ≡ pm ym /p · y (m = 1, . . . , M) and cost shares by sn ≡ wn xn /w · x (n = 1, . . . , N ), then expression (10.82) can be written as ln LWPROD(x  , y  , x, y; w, p) ≈

M 

 −y  x  − xn ym m − sn n , ym xn N

um

m=1

(10.83)

n=1

which in continuous time would be written as ln LWPROD(x  , y  , x, y; w, p) ≈

M 

um d ln ym −

m=1

N 

sn d ln xn .

(10.84)

n=1

Thus the logarithm of a Lowe productivity index is approximately equal to a weighted mean of output quantity growth rates minus a weighted mean of input quantity growth rates, where the weights are revenue shares and cost shares, respectively. The right-hand side of expression (10.84) is the familiar Solow residual.

10.6.2 Decompositions It is straightforward to verify that path A with λ = 1 leads to the following decomposition: LWPROD(x 1 , y 1 , x 0 , y 0 ; w, p) = 

Do1 (x 1 , y 1 ) Do0 (x 1 , y 1 ) × × Do0 (x 0 , y 0 ) Do1 (x 1 , y 1 )

  0 1 0  Do0 (x 0 , y 0 ) w · x 0 Do (x , y ) p · y 1 × . Do0 (x 1 , y 0 ) w · x 1 Do0 (x 1 , y 1 ) p · y 0

(10.85)

The first factor measures technical efficiency change, and the second factor measures technological change. The third factor corresponds to the scale (including input mix) effect, since, under CRS, this factor reduces to 1 if x 1 = λx 0 (λ > 0). The fourth factor is a Lowe output quantity index number divided by a Malmquist

10.6 Decomposing a Lowe Productivity Index

291

output quantity index number. Due to the fact that the various parts are linearly homogeneous in y, the factor is a function of y 1 /  y 1  and y 0 /  y 0 ; hence, can be interpreted as a measure of output mix change. The first two factors are familiar; together they constitute the base-period-outputorientated CCD productivity index as defined by expression (10.30). To facilitate comparisons with the components of the Malmquist productivity indices, we denote the scale (including input mix) effect by SEC0o,LW (x 1 , x 0 , y 0 ; w) ≡

Do0 (x 0 , y 0 ) w · x 0 , Do0 (x 1 , y 0 ) w · x 1

and the output mix effect by OME0LW (x 1 , y 1 , y 0 ; p) ≡

Do0 (x 1 , y 0 ) p · y 1 . Do0 (x 1 , y 1 ) p · y 0

Expression (10.85) can then be written as LWPROD(x 1 , y 1 , x 0 , y 0 ; w, p) = Mo0 (x 1 , y 1 , x 0 , y 0 ) × SEC0o,LW (x 1 , x 0 , y 0 ; w) × OME0LW (x 1 , y 1 , y 0 ; p).

(10.86)

As in the previous section, it is easy to verify that SEC0o,LW (x 1 , x 0 , y 0 ; w) ¯ as defined in Sect. 10.2.1. Likewise structurally resembles SECto,M (x 1 , x 0 , y), 0 1 1 0 OMELW (x , y , y ; p) structurally resembles OMEtM (x, ¯ y 1 , y 0 ). The alternative path B delivers  LWPROD(x 1 , y 1 , x 0 , y 0 ; w, p) = Mo0 (x 1 , y 1 , x 0 , y 0 ) × 

 Do0 (x 0 , y 0 ) p · y 1 . Do0 (x 0 , y 1 ) p · y 0

 Do0 (x 0 , y 1 ) w · x 0 × Do0 (x 1 , y 1 ) w · x 1 (10.87)

Notice that the difference with expression (10.85) is in the conditioning variables of the scale and output mix effects. Taking the geometric mean of expressions (10.86) and (10.87) yields a third decomposition of LWPROD(x 1 , y 1 , x 0 , y 0 ; w, p). If we repeat this sequence of steps for the paths C and D, both with λ = 1, two decompositions are obtained which can be averaged as LWPROD(x 1 , y 1 , x 0 , y 0 ; w, p) = Mo1 (x 1 , y 1 , x 0 , y 0 ) × ⎤ ⎡ ⎤ ⎡

1 (x 0 , y 0 ) D 1 (x 0 , y 1 ) 1/2 w · x 0 1 (x 1 , y 0 ) D 1 (x 0 , y 0 ) 1/2 p · y 1 D D o o o o ⎦×⎣ ⎦. ⎣ w · x1 p · y0 Do1 (x 1 , y 0 ) Do1 (x 1 , y 1 ) Do1 (x 1 , y 1 ) Do1 (x 0 , y 1 )

(10.88)

292

10 The Components of Total Factor Productivity Change

Here Mo1 (x 1 , y 1 , x 0 , y 0 ) is the comparison-period-output-orientated CCD index, which was defined by expression (10.39). In turn, expressions (10.86), (10.87), and (10.88) can be averaged, with weights 1/4, 1/4, and 1/2, to get as final outputorientated decomposition LWPROD(x 1 , y 1 , x 0 , y 0 ; w, p) = [Mo0 (x 1 , y 1 , x 0 , y 0 )Mo1 (x 1 , y 1 , x 0 , y 0 )]1/2 × 



Do0 (x 0 , y 0 ) Do0 (x 0 , y 1 ) Do1 (x 0 , y 0 ) Do1 (x 0 , y 1 ) Do0 (x 1 , y 0 ) Do0 (x 1 , y 1 ) Do1 (x 1 , y 0 ) Do1 (x 1 , y 1 )

Do0 (x 1 , y 0 ) Do0 (x 0 , y 0 ) Do1 (x 1 , y 0 ) Do1 (x 0 , y 0 ) Do0 (x 1 , y 1 ) Do0 (x 0 , y 1 ) Do1 (x 1 , y 1 ) Do1 (x 0 , y 1 )

1/4

1/4

 w · x0 × w · x1

 p · y1 , p · y0

(10.89)

which is symmetric in all variables. This is the counterpart of expression (10.48). The first major factor on the right-hand side of the equality sign is the geometricmean-output-orientated CCD index, the second factor concerns the scale (including input mix) effect, and the third factor concerns the output mix effect. The whole exercise can be repeated with the paths specified in Sect. 10.4. For example, along path E we obtain, by using the definition of the input-orientated CCD index conditional on the base period technology—see expression (10.53)—,  LWPROD(x , y , x , y ; w, p) = 1

1

0

0

Mi0 (x 1 , y 1 , x 0 , y 0 ) × 

 Di0 (x 1 , y 1 ) p · y 1 . Di0 (x 1 , y 0 ) p · y 0

Di0 (x 1 , y 0 ) w · x 0 Di0 (x 0 , y 0 ) w · x 1

 ×

(10.90)

The first factor captures technical efficiency change and technological change. The second factor is a Malmquist input quantity index number divided by a Lowe input quantity index number. This factor compares x 1 /  x 1  to x 0 /  x 0 , and thus measures the input mix effect. The third factor measures the scale (including output mix) effect: under CRS, the factor reduces to 1 if y 1 = μy 0 (μ > 0). The paths F, G, and H are left to the reader. The completely symmetric decomposition, averaging over the four paths, reads LWPROD(x 1 , y 1 , x 0 , y 0 ; w, p) = [Mi0 (x 1 , y 1 , x 0 , y 0 )Mi1 (x 1 , y 1 , x 0 , y 0 )]1/2 × ⎡ ⎣

Di0 (x 1 , y 0 ) Di0 (x 1 , y 1 ) Di1 (x 1 , y 0 ) Di1 (x 1 , y 1 ) Di0 (x 0 , y 0 ) Di0 (x 0 , y 1 ) Di1 (x 0 , y 0 ) Di1 (x 0 , y 1 )

1/4

⎤ w · x0 ⎦ × w · x1

10.6 Decomposing a Lowe Productivity Index

⎡ ⎣

293

Di0 (x 1 , y 1 ) Di0 (x 0 , y 1 ) Di1 (x 1 , y 1 ) Di1 (x 0 , y 1 )

1/4

Di0 (x 1 , y 0 ) Di0 (x 0 , y 0 ) Di1 (x 1 , y 0 ) Di1 (x 0 , y 0 )

⎤ p · y1 ⎦ . p · y0

(10.91)

This is the counterpart of expression (10.61). The first major factor on the righthand side of the equality sign is the geometric-mean-input-orientated CCD index, the second factor concerns the input mix effect, and the third factor concerns the scale (including output mix) effect. Summarizing, we again have a large number of decompositions of the Lowe productivity index. Conditioning on the base period technology, there are six decompositions, three of which use output distance functions and three use input distance functions. Conditioning on the comparison period technology, there are also six decompositions. Taking the ‘average’ viewpoint, there are nine decompositions that use output distance functions, and nine that use input distance functions. The two completely symmetric decompositions are given by expressions (10.89) and (10.91). They differ by being based on either output distance functions or input distance functions. Moreover, the decompositions as such differ: apart from technical efficiency change, technological change, and the scale effect, expression (10.89) includes the output mix effect, whereas expression (10.91) includes the input mix effect. The number of different decompositions is still larger because we conditioned on the price variables (w, p). Every choice gives rise to a different Lowe productivity index and a different decomposition. Notice, however, that only the scale and (output or input) mix components are affected, since the CCD indices are independent of prices. Natural choices in the case of our firm are the base and comparison period prices. The corresponding productivity index numbers are then given by LWPROD(x 1 , y 1 , x 0 , y 0 ; w 0 , p0 ) and LWPROD(x 1 , y 1 , x 0 , y 0 ; w 1 , p1 ). The first is a Laspeyres productivity index number, the ratio of a Laspeyres output quantity index number and a Laspeyres input quantity index number. The second is a Paasche productivity index number, the ratio of a Paasche output quantity index number and a Paasche input quantity index number. We can again take the ‘mean’ viewpoint, and compute the geometric mean of these two index numbers, which is a Fisher productivity index number. Using expression (10.89), this index number can be decomposed as  1/2 LWPROD(x 1 , y 1 , x 0 , y 0 ; w 0 , p0 )LWPROD(x 1 , y 1 , x 0 , y 0 ; w 1 , p1 ) =  1/2 QF (p1 , y 1 , p0 , y 0 ) 0 1 1 0 0 1 1 1 0 0 = M (x , y , x , y )M (x , y , x , y ) × o o QF (w 1 , x 1 , w 0 , x 0 ) 

Do0 (x 0 , y 0 ) Do0 (x 0 , y 1 ) Do1 (x 0 , y 0 ) Do1 (x 0 , y 1 ) Do0 (x 1 , y 0 ) Do0 (x 1 , y 1 ) Do1 (x 1 , y 0 ) Do1 (x 1 , y 1 )

1/4

 1 × QF (w 1 , x 1 , w 0 , x 0 )

294

10 The Components of Total Factor Productivity Change



Do0 (x 1 , y 0 ) Do0 (x 0 , y 0 ) Do1 (x 1 , y 0 ) Do1 (x 0 , y 0 ) Do0 (x 1 , y 1 ) Do0 (x 0 , y 1 ) Do1 (x 1 , y 1 ) Do1 (x 0 , y 1 )



1/4

QF (p1 , y 1 , p0 , y 0 ) ,

(10.92) where QF (p1 , y 1 , p0 , y 0 ) is the Fisher output quantity index number, and QF (w 1 , x 1 , w 0 , x 0 ) is the Fisher input quantity index number (see Appendix A of Chap. 2). The alternative, input distance function based, decomposition of the Fisher productivity index number is obtained by using expression (10.91); thus,  1/2 LWPROD(x 1 , y 1 , x 0 , y 0 ; w 0 , p0 )LWPROD(x 1 , y 1 , x 0 , y 0 ; w 1 , p1 ) =  1/2 QF (p1 , y 1 , p0 , y 0 ) 0 1 1 0 0 1 1 1 0 0 = M (x , y , x , y )M (x , y , x , y ) × i i QF (w 1 , x 1 , w 0 , x 0 ) ⎡ ⎣

Di0 (x 1 , y 0 ) Di0 (x 1 , y 1 ) Di1 (x 1 , y 0 ) Di1 (x 1 , y 1 ) Di0 (x 0 , y 0 ) Di0 (x 0 , y 1 ) Di1 (x 0 , y 0 ) Di1 (x 0 , y 1 )

⎡ ⎣

1/4

Di0 (x 1 , y 1 ) Di0 (x 0 , y 1 ) Di1 (x 1 , y 1 ) Di1 (x 0 , y 1 ) Di0 (x 1 , y 0 ) Di0 (x 0 , y 0 ) Di1 (x 1 , y 0 ) Di1 (x 0 , y 0 )

⎤ 1 ⎦× QF (w 1 , x 1 , w 0 , x 0 )

1/4

⎤ QF (p1 , y 1 , p0 , y 0 )⎦ .

(10.93) These two decompositions have a similar structure. Their first component is a geometric-mean CCD index, output orientated and input orientated, respectively, capturing technical efficiency change and technological change. Their second component measures the scale (including input mix) effect and input mix effect, respectively. Their third component measures the output mix effect and the scale (including output mix) effect, respectively. In expression (10.92) the Fisher productivity index number is related to the geometric-mean-output-orientated CCD productivity index number. Grifell-Tatjé and Lovell (2015, 140–144) and Lovell (2016) related the Fisher productivity index number to the geometric mean MB index number, as follows:   QF (p 1 , y 1 , p 0 , y 0 ) 0 (x 1 , y 1 , x 0 , y 0 ; x 0 , y 0 )MB1 (x 1 , y 1 , x 0 , y 0 ; x 1 , y 1 ) 1/2 = ! MB QF (w1 , x 1 , w0 , x 0 )

1/2 p 0 · y 1 /Do0 (x 0 , y 1 ) p 1 · y 1 /Do1 (x 1 , y 1 ) × p 0 · y 0 /Do0 (x 0 , y 0 ) p 1 · y 0 /Do1 (x 1 , y 0 )

−1/2 w0 · x 1 /Di0 (x 1 , y 0 ) w1 · x 1 /Di1 (x 1 , y 1 ) × . (10.94) w0 · x 0 /Di0 (x 0 , y 0 ) w1 · x 0 /Di1 (x 0 , y 1 )

Consider the factor on the second line of this expression. The output vector in the first numerator, y 1 /Do0 (x 0 , y 1 ), lies at the same output isoquant as the output

10.7 Decomposing a Cobb-Douglas Productivity Index

295

vector in the first denominator, y 0 /Do0 (x 0 , y 0 ). The vectors y 1 /Do1 (x 1 , y 1 ) and y 0 /Do1 (x 1 , y 0 ) in the second ratio also lie at the same, but different, output isoquant. The whole factor is interpreted as measuring the mean effect of output mix change, going from a ray through y 0 to a ray through y 1 in output space. If y 1 = μy 0 (μ > 0), then this factor reduces to 1. In the same way the factor on the third line of expression (10.94) is interpreted as measuring the mean effect of input mix change, going from a ray through x 0 to a ray through x 1 in input space. If x 1 = λx 0 (λ > 0), then this factor reduces to 1. This interpretation does not come as a surprise. From the respective definitions it immediately follows that if x 1 = λx 0 and y 1 = μy 0 , then the Fisher productivity index number as well as any MB productivity index number attains the value μ/λ. Thus in general the difference between the two productivity index numbers is due to the mix effects.

10.7 Decomposing a Cobb-Douglas Productivity Index 10.7.1 Definitions A second quite natural generalization of the single-input-single-output productivity index is given by the value-share-weighted geometric index, 

M



CDPROD(x , y , x, y; s, u) ≡ m=1 N

 /y )um (ym m

 sn n=1 (xn /xn )

,

(10.95)

N where sn > 0 (n = 1, . . . , N ), n=1 sn = 1, um > 0 (m = 1, . . . , M), M m=1 um = 1, and it is supposed that all the quantities are positive. The index defined by expression (10.95) satisfies the fundamental monotonicity and proportionality requirements and, for fixed s and u, exhibits transitivity in (x, y). The output and input quantity indices in numerator and denominator, respectively, are known as Cobb-Douglas indices; see Balk (2008) for their axiomatic properties.25 Thus CDPROD(x  , y  , x, y; s, u) will be called a Cobb-Douglas productivity index. The relation between this index and the Solow residual is immediate, as expression (10.95) is equivalent to ln CDPROD(x  , y  , x, y; s, u) =

M 

 um (ln ym − ln ym ) −

m=1

N 

sn (ln xn − ln xn ),

n=1

(10.96)

25 To

continue footnote 8, now X(x) ≡

N

sn n=1 xn

and Y (y) ≡

M

um m=1 ym .

296

10 The Components of Total Factor Productivity Change

which in continuous time would be written as ln CDPROD(x  , y  , x, y; s, u) =

M 

um d ln ym −

m=1

N 

sn d ln xn .

(10.97)

n=1

If the output weights um (m = 1, . . . , M) are revenue shares and the input weights sn (n = 1, . . . , N ) are cost shares, then the right-hand side of expression (10.97) is the Solow residual.

10.7.2 Decompositions 1 , y 1 , x 0 , y 0 ; w, p), derived By replacing in the decompositions of LWPROD(x N 1 0 in Sect. 10.6.2, the ratio w · x /w · x by n=1 (xn1 /xn0 )sn , and the ratio p ·  1 0 um 1 y 1 /p · y 0 by M m=1 (ym /ym ) , we obtain eight decompositions of CDPROD(x , y 1 , x 0 , y 0 ; s, u), corresponding to the eight paths A-H. Natural choices for the share variables (s, u) in the case of our firm are the base and comparison period cost and revenue shares, which are defined by snt ≡ wnt xnt /w t · x t (n = 1, . . . , N) and t y t /p t · y t (m = 1, . . . , M), respectively (t = 0, 1). The corresponding utm ≡ pm m productivity index numbers are then given by CDPROD(x 1 , y 1 , x 0 , y 0 ; s 0 , u0 ) and CDPROD(x 1 , y 1 , x 0 , y 0 ; s 1 , u1 ). The first is a Geometric Laspeyres productivity index number, the ratio of a Geometric Laspeyres output quantity index number and a Geometric Laspeyres input quantity index number. The second is a Geometric Paasche productivity index number, the ratio of a Geometric Paasche output quantity index number and a Geometric Paasche input quantity index number. We again take the ‘mean’ viewpoint, which is the Törnqvist productivity index number, defined as ratio of Törnqvist output and input quantity index numbers. The two symmetric decompositions are

CDPROD(x 1 , y 1 , x 0 , y 0 ; (s 0 + s 1 )/2, (u0 + u1 )/2) =  1/2 CDPROD(x 1 , y 1 , x 0 , y 0 ; s 0 , u0 )CDPROD(x 1 , y 1 , x 0 , y 0 ; s 1 , u1 ) =  1/2 QT (p1 , y 1 , p0 , y 0 ) = Mo0 (x 1 , y 1 , x 0 , y 0 )Mo1 (x 1 , y 1 , x 0 , y 0 ) × T 1 1 0 0 Q (w , x , w , x ) 

Do0 (x 0 , y 0 ) Do0 (x 0 , y 1 ) Do1 (x 0 , y 0 ) Do1 (x 0 , y 1 ) Do0 (x 1 , y 0 ) Do0 (x 1 , y 1 ) Do1 (x 1 , y 0 ) Do1 (x 1 , y 1 )

1/4

 1 × QT (w 1 , x 1 , w 0 , x 0 )

10.8 An Empirical Application



297

Do0 (x 1 , y 0 ) Do0 (x 0 , y 0 ) Do1 (x 1 , y 0 ) Do1 (x 0 , y 0 ) Do0 (x 1 , y 1 ) Do0 (x 0 , y 1 ) Do1 (x 1 , y 1 ) Do1 (x 0 , y 1 )



1/4

QT (p1 , y 1 , p0 , y 0 ) , (10.98)

and CDPROD(x 1 , y 1 , x 0 , y 0 ; (s 0 + s 1 )/2, (u0 + u1 )/2) =  1/2 CDPROD(x 1 , y 1 , x 0 , y 0 ; s 0 , u0 )CDPROD(x 1 , y 1 , x 0 , y 0 ; s 1 , u1 ) =  1/2 QT (p1 , y 1 , p0 , y 0 ) 0 1 1 0 0 1 1 1 0 0 = M (x , y , x , y )M (x , y , x , y ) × i i QT (w 1 , x 1 , w 0 , x 0 ) ⎡ ⎣

Di0 (x 1 , y 0 ) Di0 (x 1 , y 1 ) Di1 (x 1 , y 0 ) Di1 (x 1 , y 1 ) Di0 (x 0 , y 0 ) Di0 (x 0 , y 1 ) Di1 (x 0 , y 0 ) Di1 (x 0 , y 1 )

1/4

⎤ 1 ⎦× QT (w 1 , x 1 , w 0 , x 0 )

⎡ ⎤

1/4 Di0 (x 1 , y 1 ) Di0 (x 0 , y 1 ) Di1 (x 1 , y 1 ) Di1 (x 0 , y 1 ) ⎣ QT (p1 , y 1 , p0 , y 0 )⎦ , Di0 (x 1 , y 0 ) Di0 (x 0 , y 0 ) Di1 (x 1 , y 0 ) Di1 (x 0 , y 0 ) (10.99) where QT (p1 , y 1 , p0 , y 0 ) is the Törnqvist output quantity index number, and QT (w 1 , x 1 , w 0 , x 0 ) is the Törnqvist input quantity index number (see Appendix A of Chap. 2). These two decompositions have a similar structure. Their first component is a geometric-mean CCD index, output orientated and input orientated, respectively, capturing technical efficiency change and technological change. Their second component measures the scale (including input mix) effect and input mix effect, respectively. Their third component measures the output mix effect and the scale (including output mix) effect, respectively.

10.8 An Empirical Application 10.8.1 Data and DEA Approach This section26 provides an empirical illustration of the various productivity index decompositions discussed so far. The data come from a balanced panel of 31 Taiwanese banks over the period 2006–2010, previously studied by Juo et al.

26 This

and the next subsection draw upon joint work with José L. Zofío, as reported in Balk and Zofío (2018).

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10 The Components of Total Factor Productivity Change

(2015).27 A complete discussion of the statistical sources, variables specification, and summary statistics can be found there. Regarding the technology and interrelations between inputs and outputs, the variables reflect the intermediation approach suggested by Sealey and Lindley (1977), whereby financial institutions, through labour and capital, collect deposits from savers to produce loans and other earning assets for borrowers. Inputs are financial funds (x1 ), labour (x2 ), and physical capital (x3 ). The output vector includes financial investments (y1 ) and loans (y2 ). By way of example Table 10.1 presents the main descriptive statistics for quantities and prices in the beginning year 2006 and the end year 2010. Firmspecific prices are calculated as unit values, that is, costs divided by quantities. What immediately catches the eye is that all the variables exhibit in the 2 years huge dispersion, and that relative prices have changed fairly from 2006 to 2010. The technique of Data Envelopment Analysis (DEA) is used for approximating the annual technological frontiers; see Appendix B for the basic idea. The main features and computational aspects are more fully discussed in Balk and Zofío (2018). As explained in Sect. 10.3.3, we conveniently focus on four-factor decompositions in which radial scale and input or output mix effects are merged.

10.8.2 Results All the individual results, productivity index numbers and their decompositions, are contained in the 64 tables of Balk and Zofío (2018). The following eight tables are a summary, limited to (descriptive statistics of) the geometric mean variants of the indices discussed in the previous sections. All these tables exhibit the same layout. The first block of four columns in the upper panel contains the productivity index numbers for the four adjacent-year comparisons for which data are available. The second block contains the technical efficiency change (EC) components, and the third block the technological change (TC) components. The lower panel differs slightly for output-orientated or input-orientated decompositions. In the output-orientated case the first block of four columns displays the scale-includinginput-mix (SEC+IME) effects, whereas the second block displays the output mix (OME) effects. In the input-orientated case the first block displays the input mix (IME) effects, whereas the second block displays the scale-including-output-mix (SEC+OME) effects. As we are not pursueing a study of Taiwanese banking, we refrain from interpreting the outcomes but let the numbers speak for themselves. General conclusions are drawn in the next section. We start with the geometric mean Malmquist index numbers  1/2 Mˇ ot (x k,t+1 , y k,t+1 , x kt , y kt )Mˇ ot+1 (x k,t+1 , y k,t+1 , x kt , y kt ) =

27 I

am grateful to these authors for sharing the data.

x1 x2 x3 w1 w2 w3 y1 y2 p1 p2

2006 Arithmetic average 628,856 3781 14,623 0.0203 1.1538 0.3106 111,098 506,372 0.0612 0.0365

Maximum 2,133,665 8463 74,448 0.0776 2.6926 0.8325 352,976 1,734,526 0.2745 0.0885

Minimum 26,162 203 494 0.0107 0.6975 0.0631 2,354 49,780 0.0001 0.0210

Standard deviation 578,979 2463 15,971 0.0124 0.4394 0.1750 108,651 468,063 0.0587 0.0119

2010 Arithmetic average 795,536 3826 13,393 0.0064 1.2586 0.3171 196,808 609,489 0.0349 0.0233

Table 10.1 Descriptive statistics of the Taiwanese banking example, 2006 and 2010 Maximum 3,171,493 9538 76,576 0.0186 2.2963 0.7625 904,580 2,091,100 0.3044 0.0803

Minimum 25,019 202 505 0.0025 0.7170 0.0725 1681 66,947 0.0059 0.0162

Standard deviation 768,008 2729 15,185 0.0026 0.3963 0.1697 215,063 582,854 0.0668 0.0111

10.8 An Empirical Application 299

300

10 The Components of Total Factor Productivity Change

 1/2 Mˇ t (x k,t+1 , y k,t+1 , x kt , y kt )Mˇ t+1 (x k,t+1 , y k,t+1 , x kt , y kt ) , i

i

for k = 1, . . . , 31 and t = 2006, 2007, 2008, 2009. They are decomposed in a symmetric way, output orientated, according to expression (10.48). The descriptive statistics of the 31 outcomes are gathered in Table 10.2. As a reading guide we look at the year 2007 relative to 2006. Average productivity change appeared to be 3.47%, technical efficiency change 2.21%, and technological change 0.19%; the average scale-including-input-mix effect appeared to be 0.94%, and the output mix effect 0.03%. The descriptive statistics of the input-orientated decompositions, according to expression (10.61), are gathered in Table 10.3. The productivity change figures in the two tables are by definition identical. The decompositions, however, are different. It now appears that on average technical efficiency change was 1.74%, technological change 6.29%—which is hugely different from the former 0.19%—input mix effect 0.78%, and scale-including-output-mix effect −2.83%— the negative sign is interesting. The next two tables concern the geometric mean Moorsteen-Bjurek index numbers  1/2 MBt (x k,t+1 , y k,t+1 , x kt , y kt ; x¯ t , y¯ t )MBt+1 (x k,t+1 , y k,t+1 , x kt , y kt ; x¯ t , y¯ t ) , for the same banks years. The conditioning variables chosen as sample K andk,t+1 K were kt )/2K and y¯ t ≡ k,t+1 + y kt )/2K. The means x¯ t ≡ (x + x (y k=1 k=1 summary results of the symmetric, output-orientated decompositions according to expression (10.78) are reported in Table 10.4, and those of the input-orientated counterparts, according to expression (10.79), in Table 10.5. Tables 10.6 and 10.7 concern the Fisher productivity index numbers  LWPROD(x k,t+1 , y k,t+1 , x kt , y kt ; w kt , pkt ) × LWPROD(x k,t+1 , y k,t+1 , x kt , y kt ; w k,t+1 , pk,t+1 )

1/2

,

for the banks and years mentioned above. Recall that the data-set contains firmspecific prices. The tables contain summary results of, respectively, the outputorientated decompositions according to expression (10.92) and the input-orientated decompositions according to expression (10.93). Tables 10.8 and 10.9 concern the Törnqvist productivity index numbers  CDPROD(x k,t+1 , y k,t+1 , x kt , y kt ; s kt , ukt ) × CDPROD(x k,t+1 , y k,t+1 , x kt , y kt ; s k,t+1 , uk,t+1 )

1/2

2009-08 1.0413 0.0528 1.2618 0.9637

SEC+IME 2007-06 1.0094 0.0596 1.2187 0.9010

All banks Average Std. dev. Maximum Minimum

2008-07 1.0457 0.0917 1.4236 0.9606

2009-08 1.0284 0.0985 1.1796 0.7643

Productivity change 2007-06 2008-07 1.0347 1.0274 0.0906 0.1302 1.2989 1.3029 0.8291 0.6939

All banks Average Std. dev. Maximum Minimum 2010-09 1.0282 0.0591 1.1413 0.8195

2010-09 1.1772 0.2232 1.7774 0.8493 OME 2007-06 1.0003 0.0376 1.1018 0.9030

EC 2007-06 1.0221 0.0750 1.1834 0.8758 2008-07 0.9857 0.0411 1.0482 0.8735

2008-07 0.9845 0.0923 1.2414 0.8233 2009-08 0.9880 0.0547 1.1275 0.8603

2009-08 0.9749 0.0547 1.0770 0.8242

Table 10.2 Geometric mean Malmquist productivity index: output orientated decomposition

2010-09 1.0367 0.0703 1.1917 0.9379

2010-09 1.0442 0.0790 1.2335 0.9304

TC 2007-06 1.0019 0.0537 1.1273 0.8509

2008-07 1.0148 0.0870 1.1967 0.6512

2009-08 1.0277 0.0949 1.2484 0.7190

2010-09 1.0581 0.1176 1.5152 0.9732

10.8 An Empirical Application 301

302

10 The Components of Total Factor Productivity Change

for the banks and years mentioned above. Recall that the data-set contains firmspecific prices so that firm-specific cost and revenue shares could be computed. The tables contain summary results of, respectively, the output-orientated decompositions according to expression (10.98) and the input-orientated decompositions according to expression (10.99). As all the decompositions discussed above share a geometric-mean CCD index, the EC and TC parts of the output-orientated decomposition tables are identical, as well as the EC and TC parts of the input-orientated decomposition tables. Between orientations, however, the outcomes are different.

10.9 Conclusion This chapter presented a broader view on total factor productivity (TFP) measurement than the usual neo-classical approach provides. We now recapitulate the main points and draw some lessons. For any production unit, be it an enterprise, an industry, or a country, TFP change between two time periods was here defined as the ratio of an output quantity index over an input quantity index, whereby ‘total’ means that all the inputs (aka factors) are taken into account. As we know, our economic-statistical toolbox provides several ways of defining such a TFP index. Assuming that in any time period considered there exists a production frontier satisfying mild regularity requirements, we can decompose any TFP index into mutually independent factors: efficiency change (aka catching-up), technological change (aka frontier shift), scale effect, input-mix effect, and output-mix effect. Such decompositions can be input orientated (that is, based on input distance functions) or output orientated (that is, based on output distance functions). The technological change component is what mainstream neo-classical economists mean when they refer to TFP change.28 We considered the four families of Malmquist, Moorsteen-Bjurek, Lowe, and Cobb-Douglas indices. As for any index from these families a specific viewpoint must be chosen—are we using the past or the current technology as benchmark?— it does not come as a surprise that we end up with numerous theoretically different measures and decompositions of TFP change. The literature provides us with a number of empirical implementations. However, most of these implementations, sometimes on microdata, sometimes on macrodata, appear to be partial, usually concerned with but one, two, or three decompositions at a time. The unique feature of this chapter is that all the theoretically possible decompositions are applied to the same data-set of a real-life panel of production units. Thus, in the particular case of this data-set, we are able to judge the extent to which the various measures and decompositions are empirically different. Moreover, there is a software toolbox so

28 Balk

and Zofío (2018) contains material on the incorporation of allocative efficiency change.

2009-08 1.0356 0.0629 1.2769 0.9576

IME 2007-06 1.0078 0.0360 1.0730 0.9074

All banks Average Std. Dev. Maximum Minimum

2008-07 1.0330 0.0593 1.1908 0.9672

2009-08 1.0284 0.0985 1.1796 0.7643

Productivity change 2007-06 2008-07 1.0347 1.0274 0.0906 0.1302 1.2989 1.3029 0.8291 0.6939

All banks Average Std. Dev. Maximum Minimum 2010-09 1.0278 0.0566 1.1332 0.8246

2010-09 1.1772 0.2232 1.7774 0.8493

2008-07 0.9622 0.1270 1.2266 0.6174

SEC+OME 2007-06 2008-07 0.9717 1.0182 0.1196 0.1263 1.1973 1.5495 0.5889 0.8225

EC 2007-06 1.0174 0.1177 1.4838 0.8257 2009-08 1.0099 0.0898 1.3582 0.8925

2009-08 0.9537 0.0905 1.0827 0.5833

Table 10.3 Geometric mean Malmquist productivity index: input orientated decomposition

2010-09 1.0023 0.0789 1.1645 0.8590

2010-09 1.0729 0.1037 1.3134 0.8762

TC 2007-06 1.0629 0.1500 1.6359 0.8776

2008-07 1.0376 0.1350 1.2482 0.4571

2009-08 1.0314 0.0726 1.1804 0.8630

2010-09 1.0587 0.1804 1.9133 0.9189

10.9 Conclusion 303

2009-08 1.0071 0.0210 1.0640 0.9701

SEC+IME 2007-06 1.0079 0.0260 1.0960 0.9630

All banks Average Std. Dev. Maximum Minimum

2008-07 1.0129 0.0685 1.3226 0.9459

2009-08 0.9908 0.1044 1.2648 0.7145

Productivity change 2007-06 2008-07 1.0252 1.0015 0.0850 0.1434 1.3026 1.4909 0.8837 0.7279

All banks Average Std. Dev. Maximum Minimum 2010-09 0.9992 0.0347 1.0942 0.9394

2010-09 1.1022 0.1419 1.5357 0.9105 OME 2007-06 0.9998 0.0037 1.0101 0.9857

EC 2007-06 1.0221 0.0750 1.1834 0.8758 2008-07 0.9999 0.0016 1.0047 0.9942

2008-07 0.9845 0.0923 1.2414 0.8233 2009-08 1.0000 0.0007 1.0034 0.9989

2009-08 0.9749 0.0547 1.0770 0.8242 2010-09 1.0006 0.0020 1.0074 0.9975

2010-09 1.0442 0.0790 1.2335 0.9304

Table 10.4 Geometric mean Moorsteen-Bjurek productivity index: output orientated decomposition TC 2007-06 1.0019 0.0537 1.1273 0.8509

2008-07 1.0148 0.0870 1.1967 0.6512

2009-08 1.0277 0.0949 1.2484 0.7190

2010-09 1.0581 0.1176 1.5152 0.9732

304 10 The Components of Total Factor Productivity Change

2009-08 0.8926 0.3092 1.0001 0.0000

IME 2007-06 0.9977 0.0113 1.0019 0.9392

All banks Average Std. Dev. Maximum Minimum

2008-07 0.9990 0.0049 1.0131 0.9801

2009-08 0.9908 0.1044 1.2648 0.7145

Productivity change 2007-06 2008-07 1.0252 1.0015 0.0850 0.1434 1.3026 1.4909 0.8837 0.7279

All banks Average Std. Dev. Maximum Minimum 2010-09 0.9642 0.1824 1.0238 0.0000

2010-09 1.1022 0.1419 1.5357 0.9105

2008-07 0.9622 0.1270 1.2266 0.6174

SEC+OME 2007-06 2008-07 0.9697 1.0352 0.1056 0.1343 1.0973 1.5718 0.5967 0.8977

EC 2007-06 1.0174 0.1177 1.4838 0.8257 2009-08 0.9019 0.3289 1.3582 0.0000

2009-08 0.9537 0.0905 1.0827 0.5833 2010-09 0.9356 0.1917 1.1110 0.0000

2010-09 1.0729 0.1037 1.3134 0.8762

Table 10.5 Geometric mean Moorsteen-Bjurek productivity index: input orientated decomposition TC 2007-06 1.0629 0.1500 1.6359 0.8776

2008-07 1.0376 0.1350 1.2482 0.4571

2009-08 1.0314 0.0726 1.1804 0.8630

2010-09 1.0563 0.1776 1.9133 0.9189

10.9 Conclusion 305

2009-08 1.0061 0.0348 1.1055 0.9561

SEC+IME 2007-06 0.9950 0.0658 1.2095 0.8492

All banks Average Std. Dev. Maximum Minimum

2008-07 1.0134 0.0475 1.0899 0.8860

2009-08 0.9827 0.1115 1.2579 0.5861

Productivity change 2007-06 2008-07 1.0132 0.9914 0.1005 0.1010 1.3272 1.1328 0.8228 0.7712

All banks Average Std. Dev. Maximum Minimum 2010-09 1.0015 0.0583 1.1359 0.7897

2010-09 1.1313 0.1980 1.8094 0.8495 OME 2007-06 0.9921 0.0673 1.1486 0.7724

EC 2007-06 1.0221 0.0750 1.1834 0.8758

Table 10.6 Fisher productivity index: output orientated decomposition

2008-07 0.9836 0.0747 1.1022 0.7543

2008-07 0.9845 0.0923 1.2414 0.8233 2009-08 0.9766 0.1068 1.2740 0.6492

2009-08 0.9749 0.0547 1.0770 0.8242 2010-09 1.0265 0.0808 1.2364 0.7831

2010-09 1.0442 0.0790 1.2335 0.9304

TC 2007-06 1.0019 0.0537 1.1273 0.8509

2008-07 1.0148 0.0870 1.1967 0.6512

2009-08 1.0277 0.0949 1.2484 0.7190

2010-09 1.0581 0.1176 1.5152 0.9732

306 10 The Components of Total Factor Productivity Change

2009-08 1.0063 0.0412 1.1188 0.9306

IME 2007-06 0.9902 0.0716 1.2279 0.8178

All banks Average Std. Dev. Maximum Minimum

2008-07 1.0076 0.0575 1.0892 0.8153

2009-08 0.9827 0.1115 1.2579 0.5861

Productivity change 2007-06 2008-07 1.0132 0.9914 0.1005 0.1010 1.3272 1.1328 0.8228 0.7712

All banks Average Std. Dev. Maximum Minimum 2010-09 1.0018 0.0734 1.1205 0.7437

2010-09 1.1313 0.1980 1.8094 0.8495

2008-07 0.9622 0.1270 1.2266 0.6174

SEC+OME 2007-06 2008-07 0.9641 1.0177 0.1312 0.1621 1.2075 1.7179 0.5791 0.7572

EC 2007-06 1.0174 0.1177 1.4838 0.8257

Table 10.7 Fisher productivity index: input orientated decomposition

2009-08 1.0055 0.1236 1.3935 0.6988

2009-08 0.9537 0.0905 1.0827 0.5833 2010-09 0.9969 0.0977 1.1706 0.7511

2010-09 1.0729 0.1037 1.3134 0.8762

TC 2007-06 1.0629 0.1500 1.6359 0.8776

2008-07 1.0376 0.1350 1.2482 0.4571

2009-08 1.0314 0.0726 1.1804 0.8630

2010-09 1.0563 0.1776 1.9133 0.9189

10.9 Conclusion 307

2009-08 1.0061 0.0348 1.1055 0.9560

SEC+IME 2007-06 0.9950 0.0656 1.2088 0.8496

All banks Average Std. Dev. Maximum Minimum

2008-07 1.0132 0.0475 1.0899 0.8859

2009-08 0.9814 0.1084 1.2209 0.5862

Productivity change 2007-06 2008-07 1.0147 0.9926 0.0977 0.0985 1.3263 1.1320 0.8732 0.7713

All banks Average Std. Dev. Maximum Minimum 2010-09 1.0015 0.0586 1.1370 0.7886

2010-09 1.1199 0.1716 1.6410 0.8495 OME 2007-06 0.9936 0.0627 1.1485 0.8197

EC 2007-06 1.0221 0.0750 1.1834 0.8758

Table 10.8 Törnqvist productivity index: output orientated decomposition

2008-07 0.9850 0.0713 1.1014 0.7809

2008-07 0.9845 0.0923 1.2414 0.8233 2009-08 0.9754 0.1034 1.2365 0.6494

2009-08 0.9749 0.0547 1.0770 0.8242 2010-09 1.0182 0.0739 1.2110 0.7831

2010-09 1.0442 0.0790 1.2335 0.9304

TC 2007-06 1.0019 0.0537 1.1273 0.8509

2008-07 1.0148 0.0870 1.1967 0.6512

2009-08 1.0277 0.0949 1.2484 0.7190

2010-09 1.0581 0.1176 1.5152 0.9732

308 10 The Components of Total Factor Productivity Change

2009-08 1.0063 0.0412 1.1188 0.9307

IME 2007-06 0.9902 0.0715 1.2272 0.8178

All banks Average Std. Dev. Maximum Minimum

2008-07 1.0074 0.0576 1.0886 0.8147

2009-08 0.9814 0.1084 1.2209 0.5862

Productivity change 2007-06 2008-07 1.0147 0.9926 0.0977 0.0985 1.3263 1.1320 0.8732 0.7713

All banks Average Std. Dev. Maximum Minimum 2010-09 1.0018 0.0736 1.1215 0.7426

2010-09 1.1199 0.1716 1.6410 0.8495

2008-07 0.9622 0.1270 1.2266 0.6174

SEC+OME 2007-06 2008-07 0.9660 1.0192 0.1299 0.1603 1.2077 1.7181 0.5791 0.7839

EC 2007-06 1.0174 0.1177 1.4838 0.8257

Table 10.9 Törnqvist productivity index: input orientated decomposition

2009-08 1.0042 0.1203 1.3931 0.6990

2009-08 0.9537 0.0905 1.0827 0.5833 2010-09 0.9893 0.1002 1.1231 0.7152

2010-09 1.0729 0.1037 1.3134 0.8762

TC 2007-06 1.0629 0.1500 1.6359 0.8776

2008-07 1.0376 0.1350 1.2482 0.4571

2009-08 1.0314 0.0726 1.1804 0.8630

2010-09 1.0563 0.1776 1.9133 0.9189

10.9 Conclusion 309

310

10 The Components of Total Factor Productivity Change

that researchers can replicate our work with their own data-set. This toolbox can be downloaded from www.tfptoolbox.com. A guide is provided by Balk et al. (2020).29 What are the main findings? • The common core of all the TFP indices appears to be a CCD index, capturing efficiency change and technological change. Although a CCD index in itself is not a proper productivity index, except in the case of technologies exhibiting global CRS, the remaining factors—scale, input mix, and output mix—are numerically less important. • Since the reference period (that is, the technology used as benchmark) matters empirically, researchers are advised to use geometric means of base period and comparison period viewpoint indices, unless there are compelling reasons for using a particular viewpoint. • Decompositions can be based on input or output distance functions. In a fourfactor decomposition, the scale effect is either combined with the input-mix effect or with the output-mix effect, but is not separately available. • The Malmquist and Moorsteen-Bjurek indices, based on quantities only, deliver higher productivity growth magnitudes than the Lowe and Cobb-Douglas indices which are based on quantities and (market) prices. To which extent this is due to the fact that in the empirical exercise the technologies were estimated by DEA remains to be seen.30 • The results of price-weighted (for example, Fisher) and value-share-weighted (for example, Törnqvist) indices are numerically very close. Thus the choice between the two families of indices is immaterial. • In our empirical example (local) technological change generally appeared to be the main component of TFP change.

ˇ o0 (x 1 , y 1 , x 0 , y 0 ) Along Path A Appendix A: Components of M Recall that productivity change, going from (x, y) to (x  , y  ), is measured by Mˇ o0 (x  , y  , x, y) = Dˇ o0 (x  , y  )/Dˇ o0 (x, y). Along the first segment of Path A productivity change is 1 Dˇ o0 (x 0 , y 0 /Do0 (x 0 , y 0 )) = 0 0 0 . 0 0 0 ˇ D (x ,y ) Do (x , y ) o

29 This

toolbox is MATLAB-based. An R-based software package, covering the same ground except for share-weighted geometric indices, was developed recently by Dakpo et al. (2018). The guide does not provide formulas, so that for precise definitions of the indices and specifications of the decompositions the user must consult the underlying literature. 30 Interestingly, Balk (1998) obtained a similar pattern.

Appendix A: Components of Mˇ o0 (x 1 , y 1 , x 0 , y 0 ) Along Path A

311

Along the second segment Dˇ o0 (λx 0 , y 0 /Do0 (λx 0 , y 0 )) Dˇ 0 (λx 0 , y 0 ) Do0 (x 0 , y 0 ) = o0 Do (λx 0 , y 0 ) Dˇ o0 (x 0 , y 0 ) Dˇ o0 (x 0 , y 0 /Do0 (x 0 , y 0 )) =

OSE0 (λx 0 , y 0 ) = SEC0o,M (λx 0 , x 0 , y 0 ). OSE0 (x 0 , y 0 )

Along the third segment Dˇ o0 (x 1 , y 0 /Do0 (x 1 , y 0 )) Dˇ 0 (x 1 , y 0 ) D 0 (λx 0 , y 0 ) = o0 1 0 o Do (x , y ) Dˇ o0 (λx 0 , y 0 ) Dˇ o0 (λx 0 , y 0 /Do0 (λx 0 , y 0 )) =

OSE0 (x 1 , y 0 ) = SEC0o,M (x 1 , λx 0 , y 0 ). OSE0 (λx 0 , y 0 )

Along the fourth segment Dˇ 0 (x 1 , y 1 ) D 0 (x 1 , y 0 ) Dˇ o0 (x 1 , y 1 /Do0 (x 1 , y 1 )) = o0 1 1 o Do (x , y ) Dˇ o0 (x 1 , y 0 ) Dˇ o0 (x 1 , y 0 /Do0 (x 1 , y 0 )) =

OSE0 (x 1 , y 1 ) = OME0M (x 1 , y 1 , y 0 ). OSE0 (x 1 , y 0 )

Along the fifth segment D 0 (x 1 , y 1 ) Dˇ o0 (x 1 , y 1 /Do1 (x 1 , y 1 )) 1 1 = o1 1 1 = TC1,0 o (x , y ). Do (x , y ) Dˇ o0 (x 1 , y 1 /Do0 (x 1 , y 1 )) Along the sixth segment Dˇ o0 (x 1 , y 1 )

Dˇ o0 (x 1 , y 1 /Do1 (x 1 , y 1 ))

= Do1 (x 1 , y 1 ).

Multiplication of the left-hand sides of these equations delivers productivity change between (x 0 , y 0 ) and (x 1 , y 1 ), Mˇ o0 (x 1 , y 1 , x 0 , y 0 ), whereas multiplication of the right-hand sides establishes the decomposition. Recall thereby that 1 × Do1 (x 1 , y 1 ) = ECo (x 1 , y 1 , x 0 , y 0 ). Do0 (x 0 , y 0 )

312

10 The Components of Total Factor Productivity Change

Appendix B: Data Envelopment Analysis (DEA) Suppose we are given panel data (w kt , x kt , pkt , y kt ) for production units k = 1, . . . , K and time periods t = 0, 1, . . . , T . These data allow us to calculate any productivity index we wish, cross-sectionally,31 intertemporally, or a combination of the two. If only quantity data are given, the choice is restricted to Malmquist indices. As we have shown in the foregoing sections, several options are available for a decomposition of productivity change. Although expressions such as (10.48) look quite intimidating, their computation should not be much of a problem, if one has knowledge of the functions involved. For example, for the computation of the various parts of expression (10.48) only knowledge of the output distance functions Dot (x, y) and Dˇ ot (x, y) is required, and for the various parts of expression (10.61) only knowledge of the input distance functions Dit (x, y) and Dˇ it (x, y), in both cases for t = 0, 1. Using the technique of Data Envelopment Analysis the period t technology S t is approximated by

S t ≈ {(x, y) |

K 

zk x kt ≤ x, y ≤

k=1

K 

zk y kt ,

(10.100)

k=1

zk ≥ 0 (k = 1, . . . , K),

K 

zk = 1},

k=1

whereas the associated cone technology Sˇ t is approximated by

Sˇ t ≈ {(x, y) |

K 

zk x kt ≤ x, y ≤

k=1

K 

zk y kt ,

(10.101)

k=1

zk ≥ 0 (k = 1, . . . , K)}. These approximations satisfy the axioms P.1–P.9. The right-hand side of expression (10.100) exhibits variable returns to scale (VRS). The right-hand side of expression (10.101) exhibits CRS. For proofs the reader is referred to Färe et al. (2015, Chapter 1). Based on these approximations, the computation of the required output or input distance functions is reduced to the solution of linear programming problems.

31 In

a cross-section productivity differences between production units are considered. It is then usually assumed that these units share the same technology, which implies that the cross-sectional analogue of technological change does not exist.

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Index

A Accounting ex post, 66, 165 Additivity, 241, 246, 249, 250 Asset price equilibrium equation, 75

B Baumol effect, 134, 144, 183 Baumol’s ‘Growth Disease’, 230, 233 Bortkiewicz relation, 150, 162 Bottom-up approach, 111

C Capital deepening, 30, 36 Capital input real, 119 Capital input cost, 66 Capital services, 72 Capital stock net, 71, 72, 78 productive, 71 used, 78 Capital utilization rate, 43 Cash flow, 38 net, 41 normal net, 42 Circularity Test, 121 Cobb-Douglas, 31, 36, 212, 224, 231 Consistent-in-Aggregation, 54 Constant prices, 101 Constant returns to scale (CRS), 258, 283–287, 291, 312 Consumer Price Index, 86

Cost, 15 Cost of waiting, 40

D Data Envelopment Analysis (DEA), 298, 312 Decomposition, 46 Baily, Hulten and Campbell (BHC), 130, 160 Baldwin and Gu (BG), 138 CSLS, 134, 144, 242, 247 Diewert, 135 Diewert-Fox-Balk (DFB), 143 Diewert and Fox (DF), 140 Foster, Haltiwanger and Krizan (FHK), 132, 160 GEAD, 134, 144, 183, 240, 245 Geometric DF, 146 Griliches and Regev (GR), 138, 160 Harmonic DF, 147 Melitz and Polanec, 152 Olley and Pakes, 150, 162 Petrin and Levinsohn, 232 Reinsdorf, 155 Sato-Vartia (generalized), 136, 143 Tang and Wang, 167, 179 TRAD, 134, 144, 184, 247 Deflation, 49 double, 58 single, 57 Denison effect, 134, 177, 183 Depreciation anticipated, 40, 77 cross-sectional, 90 time-series, 69, 90

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2021 B. M. Balk, Productivity, Contributions to Economics, https://doi.org/10.1007/978-3-030-75448-8

325

326 Disaggregation, 46 Distance function input, 258 output, 258 Domar factor, 35, 58, 62, 210, 227 Domar weight, 169, 170, 175, 210, 229, 233 Double deflator, 118 E Efficiency scale, 259 technical, 258 Ensemble, 83, 113 Equality Test, 60 Expectation, 76 G Gross output real, 178 Growth accounting, 22, 36, 211 H Homotheticity input, 283 output, 268, 283 I Identity test, 261 Input real, 251 Intermediate input real, 178 Investment Survey, 85 K K-CF format, 39, 66, 79 KLEMS-Y format, 12, 112 KL-NNVA format, 41 KL-NVA format, 41 KL-VA format, 33, 113 K-NCF format, 41, 79 K-NNCF format, 42, 80 L Labour input real, 119 Labour quality, 30, 37, 117 Logarithmic mean, 25, 52, 56, 59, 61, 96, 141, 143, 168, 188, 218

Index M Margin profit-cost, 16 profit-revenue, 17 Margin effect, 26 Markup, 16 Mix effect input, 260, 286, 292, 295 output, 260, 283, 291, 295

N Neo-classical assumptions, 5, 10, 35, 48, 62, 231 Non-market unit, 14, 20, 24 O Oulton hybrid approach, 82 P Paradox, 130, 138, 148 Perpetual Inventory Method, 85 Price index, 49 Dutot, 49 Fisher, 51, 59 GeoLaspeyres, 51 GeoPaasche, 51 Laspeyres, 50 Montgomery-Vartia, 52, 60, 169, 226 Paasche, 50, 241 Sato-Vartia, 52, 173, 176, 185, 193, 211, 220 Törnqvist, 52 two-stage, 53 Price indicator, 54 Bennet, 56 Laspeyres, 55 Montgomery, 56 Paasche, 55 Price recovery index total, 20 Price recovery indicator total, 24 Primary input real, 119, 217, 239 Producer Price Index, 85 Production function, 22, 38 Cobb-Douglas, 2 Production unit, 11, 112 continuing, 115 entering, 115 exiting, 115

Index Productivity aggregate, 237 labour, 244 multi factor, 125 of the aggregate, 237 physical, 128 revenue, 128, 202, 240 simple labour, 167, 245, 248 total factor, 2, 239, 247, 250 Productivity change aggregate, 128, 218 individual, 218 Productivity index between adjacent subperiods, 97 between corresponding subperiods, 97 between subperiod and period, 97 capital, 30, 38, 117 CCD input-orientated, 277, 278, 280, 286, 292 output-orientated, 266, 270, 272, 283, 291 Cobb-Douglas, 295 dual total factor, 20 Fisher, 293 labour, 29, 35, 63, 117 Laspeyres, 293 Lowe, 289 Malmquist input-orientated, 275 output-orientated, 262 Moorsteen-Bjurek (MB), 280, 294 multi factor, 28 Oulton TFP, 83 Paasche, 293 simple labour, 30, 35, 118 Törnqvist, 296 total factor, 19, 34, 39, 44, 61, 116, 190, 192, 217 Productivity indicator dual total factor, 24 multi factor, 28 total factor, 24 Productivity level labour, 120 simple labour, 120 total factor, 119 Product Test, 116 Profile age-efficiency, 72, 91 age-price, 73, 92 Profit, 15 Profitability, 16, 33, 39, 104, 188

327 Profit from normal operations, 80 Proportionality, 261

Q Quantity index, 49 Dutot, 30, 35, 49, 117 Fisher, 51, 59, 294 GeoLaspeyres, 51 GeoPaasche, 52 Laspeyres, 50, 58 Lowe, 103 Malmquist, 280 Montgomery-Vartia, 53, 60, 61, 169, 225 Paasche, 51, 58 Sato-Vartia, 52, 174, 185, 222 Törnqvist, 52 Quantity indicator, 54 Bennet, 56 Laspeyres, 55 Montgomery, 56 Paasche, 56 R Rate of return, 80 aggregate, 84 endogenous, 42, 80 exogenous, 80 normal endogenous, 81 Solow-Kuznets, 81 Reallocation, 151, 173, 174, 190, 221, 222 Regulation, 22 Return on Assets (ROA), 39, 81 Returns-to-scale effect, 266 Revaluation, 90 unanticipated, 40, 77 Revenue, 15 real, 224, 249 S Scale effect input orientated, 260, 286, 291 output orientated, 260, 283, 292 Second-hand assets, 71 Shift-share analysis, 110 Solow residual, 2, 5, 231, 289, 295 Steady growth model, 73 T Tax, 40, 69, 79 Technical efficiency change, 259

328 Technological change, 260 Hicks input-neutral, 273, 283 Hicks output-neutral, 283 Technology actual, 257 cone, 258 Transitivity, 261

U Unit-value index, 50 User cost

Index total, 70 unit, 68 V Value added, 32, 57 net, 41 normal net, 41 real, 118, 216, 238 W Waiting cost, 69