Rethinking Input-Output Analysis: A Spatial Perspective (Advances in Spatial Science) 303105086X, 9783031050862

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Advances in Spatial Science

Jan Oosterhaven

Rethinking Input-Output Analysis A Spatial Perspective Second Edition

Advances in Spatial Science The Regional Science Series

Series Editors Manfred M. Fischer , Vienna University of Economics and Business, Wien, Austria Jean-Claude Thill , University of North Carolina, Charlotte, NC, USA Jouke van Dijk , University of Groningen, Groningen, The Netherlands Hans Westlund , Jönköping University, Jönköping, Sweden Advisory Editors Geoffrey J.D. Hewings, University of Illinois, Urbana, IL, USA Peter Nijkamp, Free University, Amsterdam, The Netherlands Folke Snickars, Editorial Board, Heidelberg, Baden-Württemberg, Germany

This series contains scientific studies focusing on spatial phenomena, utilising theoretical frameworks, analytical methods, and empirical procedures specifically designed for spatial analysis. Advances in Spatial Science brings together innovative spatial research utilising concepts, perspectives, and methods relevant to both basic science and policy making. The aim is to present advances in spatial science to an informed readership in universities, research organisations, and policy-making institutions throughout the world. The type of material considered for publication in the series includes: Monographs of theoretical and applied research in spatial science; state-of-the-art volumes in areas of basic research; reports of innovative theories and methods in spatial science; tightly edited reports from specially organised research seminars. The series and the volumes published in it are indexed by Scopus. For further information on the series and to submit a proposal for consideration, please contact Johannes Glaeser (Senior Editor Economics) Johannes.glaeser@springer. com.

More information about this series at https://link.springer.com/bookseries/3302

Jan Oosterhaven

Rethinking Input-Output Analysis A Spatial Perspective Second Edition

Jan Oosterhaven University of Groningen Groningen, The Netherlands

ISSN 1430-9602 ISSN 2197-9375 (electronic) Advances in Spatial Science ISBN 978-3-031-05086-2 ISBN 978-3-031-05087-9 (eBook) https://doi.org/10.1007/978-3-031-05087-9 1st edition: © The Author(s), under exclusive license to Springer Nature Switzerland AG 2019 2nd edition: © The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2022, corrected publication 2022 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 of the First Edition

The origin of this book dates back to my lecture notes for the first regional economics course at the University of Groningen in 1976. The last revision of the resulting Dutch language Syllabus Ruimtelijke Economie appeared in 2003. An English language extension of its input–output (IO) part was written for the educational section of the Website of the IIOA with my dear colleague Dirk Stelder in 2007. The theoretical parts of this book benefited further from the Handbook articles I wrote with Karen Polenske in 2009 for Edward Elgar and with Geoff Hewings in 2014 for Springer, while all of this book benefited from my cooperation with a series of colleagues and Ph.D. students. … … … Finally, I thank an anonymous reviewer and Eva Mulder for useful comments on drafts of this book. Groningen, The Netherlands August 2019

Jan Oosterhaven

v

Preface of the Second Edition

An Addendum, explaining how markets function in the interacting IO price–quantity model, initiated this Second Edition. I thank Johannes Glaeser for stimulating me to be more ambitious than writing only an Addendum. This resulted in a new Chap. 3 about updating IO tables, which led to rewriting the present Chap. 4 about the construction of regional, interregional and international IO tables and supply–use models. The Addendum became integrated in Chap. 6. In Chap. 7 a re-interpretation of the Leontief model as a price model was added. Chapter 8 got an extra section on modelling the disaster reconstruction phase, while the treatment of IO complications in Chap. 9 was extended. An added Appendix on matrix algebra makes this edition more self-contained. Further explanations were added in all chapters. Just as the First, the Second Edition benefited from my cooperation with a series of colleagues and Ph.D. students. In the order in which my work and discussions with them are used in this edition, these are Dirk Stelder (Sect. 3.1) Theo Junius (Sect. 3.2), Roberto Mínguez (Sect. 3.3), Fernando Escobedo (Sect. 4.1.2), Piet Boomsma (Sect. 4.1.3), Jouke van Dijk, Henk Folmer and John Dewhurst (Sect. 5.2), Bjarne Madsen (Sect. 6.2), Maaike Bouwmeester and Johannes Többen (Sect. 8.2), Bert Steenge (Sect. 8.3), Umed Temursho (Sect. 9.1.1), Gerard Eding and Dirk Stelder (Sect. 9.1.2), Jan van der Linden, Jiansuo Pei and Erik Dietzenbacher (Sect. 9.2.2). Finally, I thank a second anonymous reviewer and Umed Temursho for comments on drafts of this edition. Naturally, the final text is my sole responsibility. Groningen, The Netherlands June 2022

Jan Oosterhaven

vii

Contents

1

2

Introduction: Importance Interindustry Relations and Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Basic, Demand-Driven IO Quantity Models . . . . . . . . . . . . . . . . . . . . . . 2.1 Single-Region IO Tables and Their Descriptive Power . . . . . . . . . . 2.2 Mathematics Versus Economics of the Closed Economy IO Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3 Open Economy Interregional and Multi-regional IO Models . . . . . 2.3.1 Separating Trade Origin Ratios and Technical Coefficients . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3.2 Underestimation of Interregional Spillovers and Feedbacks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1 3 5 5 7 11 12 17 18

3

Updating Different Types of IO Tables . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1 RAS and MR-RAS: Bi- and Multi-proportional Scaling . . . . . . . . . 3.2 GRAS and MR-GRAS: Including Negative Values . . . . . . . . . . . . . 3.3 CRAS: Advantage of Using Time Series of IOTs . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

21 22 27 30 33

4

From Regional IO Tables to Interregional SU Models . . . . . . . . . . . . . 4.1 Construction of Regional IO Tables: Towards Non-biased Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1.1 Most Non-survey Methods Overestimate Intra-regional Multipliers . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1.2 CRAS: Advantage of Using Cross Sections of RIOTs . . . . 4.1.3 DE-BRIOT: Advantage of Constructing Bi-regional IOTs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2 Construction of Interregional Supply–Use Tables and Models . . . . 4.2.1 Difficulty of Deriving an IO Model from a Supply–Use Table . . . . . . . . . . . . . . . . . . . . . . . . . . . .

35 35 36 39 42 45 45

ix

x

Contents

4.2.2 Family of Interregional Supply–Use Tables and Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3 Difference Between Constructing Interregional and International SUTs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

49 52 54

From Basic IO and SU Models to Demo-economic Models . . . . . . . . 5.1 Interregional Models with Endogenous Household Consumption . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2 Further Demo-economic Model Extensions . . . . . . . . . . . . . . . . . . . 5.3 Where to End with Endogenizing Final Demand? . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

57 64 69 70

6

Cost-Push IO Price Models and Interaction with Quantities . . . . . . . 6.1 Forward Causality of the Single-Region IO Price Model . . . . . . . . 6.2 Interregional IO Price Model with a Price–Wage–Price Spiral . . . . 6.3 Interacting IO Price and Quantity Models: Lower Multipliers . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

73 73 76 80 85

7

Supply-Driven IO Quantity Model and Its Dual, Price Model . . . . . . 87 7.1 On the Plausibility of the Supply-Driven IO Quantity Model . . . . . 87 7.1.1 Basic Supply-Driven IO Model: Factories May Run Without Labour . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88 7.1.2 Type II Supply-Driven IO Model: More Private Cars May Run on Less Gas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93 7.2 Revenue-Pull IO Price Model: Its Plausible Dual . . . . . . . . . . . . . . . 96 7.3 Markets: Why All Four IO Models Overestimate Their Typical Impacts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102

8

Negative IO Supply Shock Analyses: When Substitution Matters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.1 On the Limited Usability of the IO Model in Case of Supply Shocks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.2 Nonlinear SU Programming Alternative: Much Smaller Disaster Multipliers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.3 Disaster Reconstruction Phase: Adding the Dynamic Leontief Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

9

Other IO Applications with Complications . . . . . . . . . . . . . . . . . . . . . . 9.1 Key Sector and Linkage Analysis: A Half-Truth . . . . . . . . . . . . . . . 9.1.1 Analytical and Empirical Comparison of Key Sector Measures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.1.2 Cluster and Linkage Analysis for Three Dutch Spatial Policy Regions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.1.3 The Mostly Forgotten, Cost Side of the Coin . . . . . . . . . . . .

57

105 105 110 114 120 123 123 124 127 129

Contents

9.2 Structural Decomposition Analysis: Another Half-Truth . . . . . . . . 9.2.1 Shift and Share Analysis: Impact of Industry Mix . . . . . . . . 9.2.2 Structural Decomposition Analysis: A Demand-Side Story . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.2.3 Growth Accounting: The Other, Supply Side of the Coin . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.2.4 Solution: Econometric GA with SSA and SDA Components . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

xi

132 133 136 141 143 144

10 The Future: What to Forget, to Maintain and to Extend . . . . . . . . . . 149 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152 Correction to: Rethinking Input-Output Analysis . . . . . . . . . . . . . . . . . . . .

C1

Appendix: Matrix Algebra for Input–Output Analysis . . . . . . . . . . . . . . . . 153 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 157

List of Variables and Coefficients

a b

Intermediate input coefficients by industry Intermediate output coefficients, final demand bridge coefficients, capital/output ratios c Primary input coefficients by industry, commuting coefficients by industry d Final output coefficients by industry, total demand e External exports f Local final demand, final demand preference coefficients by category g Ghosh-inverse coefficients h Household consumption, product heterogeneity i Summation vector of ones, I unity matrix, investment expenditures j Employment (jobs) k Consumption expenditure coefficients, capital stock l Leontief-inverse coefficients, employment (labour) coefficients by industry m Import ratios and self-sufficiency ratios by product n Non-active people without benefits o Other value added p Prices, industry product mix ratios q Total product supply and ditto demand, household consumption/total output ratios r Industry market shares in product supply, replacement investments s Product supply by industry, industry sales ratios, savings rates t Trade volumes, export coefficients, tax rates u Product use by industry, unemployed people v Primary inputs (value added) by industry w Wage income, wage rate x Total industry input and ditto output y Total final demand by category, gross income z Intermediate inputs and ditto outputs by industry  Absolute change (first-order difference)

xiii

Mathematical Notation

In all chapters, columns are indicated by small bold cases x, rows as transposed columns x, matrices by bold capitals X and scalars by small italic cases x. Diagonal matrices are indicated by putting a hat on the vector that fills up the diagonal ˆi. The summation vector with ones is indicated by i and the unity matrix by I = ˆi. With super- or subscripted symbols, a dot indicates a summation over the super- or subscript at hand. With deliveries between industries or regions, the first index indicates the origin and the second index indicates the destination of the corresponding product flow. The Appendix to this book explains the minimally necessary matrix algebra.

xv

Chapter 1

Introduction: Importance Interindustry Relations and Overview

Abstract This introductory chapter outlines the type of research questions that you can answer with regional, national, interregional and international input–output analysis and gives an overview of the contents of this book. Keywords Globalization · Supply chains · Interindustry relations · Input–output analysis · Supply–use tables · Cumulative impacts · Exogenous final demand With the historic, continuous reduction of tariff and non-tariff barriers to international trade until the middle of the 2010s, firms became able to increasingly exploit international locational cost and revenue advantages. Fragmentation of production processes and lengthening of supply chains were the result, along with a globalization of the world economy and a steady increase in world welfare. International income differences predominantly declined (Sala-i-Martin 2006; ILO 2015), whereas interregional income differences often increased (Silva and Leichenko 2004; Wan et al. 2007). The latter mainly occurred because several regions in several countries lost their comparative advantages in the concomitant worldwide re-organization of interindustry relations. In the late 2010s, tariff and non-tariff barriers were raised again, and again regions within and between countries won and lost comparative advantages in the again changing global supply chains they participated in. Analysing such processes requires detailed data on interindustry relations, such as offered by interregional and international input–output tables (IOTs), as these tables show the transactions between say the Russian natural gas industry and the German energy distribution industry. An input–output (IO) model based on such data (Leontief 1936) can answer many questions related to these developments, such as those about the amount of Chinese value added embodied, both directly and indirectly, in American household consumption. Questions about consumer responsibility in environmental studies require comparable data and models to answer questions about, e.g. the direct and indirect amount of CO2 -emissions of the German car industry incorporated in French household investments. Besides, more mundane questions, such

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2022 J. Oosterhaven, Rethinking Input-Output Analysis, Advances in Spatial Science, https://doi.org/10.1007/978-3-031-05087-9_1

1

2

1 Introduction: Importance Interindustry Relations and Overview

as those about the likely employment, income and environmental impacts of organizing the Olympic Games or relocating central government offices,1 also require the information of such tables to build the models necessary to answer such questions. This book will help you to understand the social, economic and environmental importance of the relations between industries in the same and in different regions and countries and how to model these relations by means of regional, national, interregional and international IO models. While doing that, it will point out cases wherein interindustry computable general equilibrium models and econometrically extended IO models provide a further explanation, without going into details. An Appendix explains the matrix algebra you need for understanding the mathematics of this book. You will also learn how to update old IOTs, how to construct regional, interregional and international IOTs and how to use the modern IOTs called supply–use tables (SUTs), which explicitly distinguish the products produced per selling industry and those used per purchasing industry. Moreover, you will learn how to use social accounting matrices (SAMs) that additionally describe the generation, redistribution and spending of income by different types of households needed to endogenize their spending, which is especially important in case of smaller regions. And, you will learn to recognize situations wherein it is probably better to spend time on incorporating the supply side and price reactions into the standard IO model, instead of further endogenizing its remaining exogenous final demand. Besides the standard demand-driven IO quantity model, this book will also carefully lay out the economic assumptions of its supply-driven mirror image, indicate its limited usefulness and explain how its little known accompanying revenue-pull IO price model may be used to model demand-driven inflationary processes, just as the much better known cost-push IO price model has been used to model supply-driven inflation. After the mainly theoretical first chapters, the two final chapters discuss three well-known applications of the IO model, namely (1) economic impact analysis of negative supply shocks as caused by, for example, natural and man-made disasters, (2) regional and interregional forward and backward linkage analysis, better known as key sector analysis, and (3) structural decomposition analysis of regional, national, interregional and international economic growth. In all three cases, the standard IO approach is shown along with its problematic implications, such as producing misleadingly high multipliers in the first case and presenting policy makers with only half of the truth in the other two cases. A static nonlinear programming model and a dynamic nonlinear programming model are offered as integral alternatives in the first case, while additions to and changes in the standard approach are offered in the last two cases. 1

Estimating the impacts of relocating the head office of the national post and telecom company PTT, actually constituted my first IO application (FNEI 1975). The low multiplier for the origin region The Hague raised indignation there, whereas the also low multiplier for the destination region Groningen raised disappointment there and even to an attempt to block publication of the outcomes for precisely that reason (see Oosterhaven (1981, Chap. 7) for a review of this battle).

References

3

This book stands out with its emphasis on the behavioural foundations of the two IO quantity models and their accompanying two price models, and on the plausibility of the causal mechanisms of all four models. This leads to a far more critical evaluation of the usefulness of IO analysis than found in standard textbooks. This book also stands out by presenting examples of policy-relevant applications of IO analysis done in, especially, the Netherlands. It will thus be of relevance to both graduate and PhD students as well as to practitioners in research and consulting firms and agencies, as it provides a better understanding of the foundations, the power, the applicability and the limitations of input–output analysis.

References FNEI (1975) De komst van de Centrale Directie der P.T.T.: Enkele economische gevolgen voor het Noorden des Lands. Federatie van Noordelijke Economische Instituten, Groningen ILO (2015) Global Wage Report 2014/15: wages and income inequality. International Labour Office, Geneva Leontief WW (1936) Quantitative input and output relations in the economic system of the United States. Rev Econ Stat 18:105–125 Oosterhaven J (1981) Interregional input-output analysis and Dutch Regional Policy problems. Gower Publishing, Aldershot-Hampshire Sala-i-Martin X (2006) The world distribution of income: falling poverty and … convergence, period. Q J Econ 121:351–397 Silva J, Leichenko R (2004) Regional income inequality and international trade. Econ Geogr 80:261– 286 Wan G, Lu M, Chen Z (2007) Globalization and regional income inequality: empirical evidence from within China. Rev Income Wealth 53:35–59

Chapter 2

Basic, Demand-Driven IO Quantity Models

Abstract This chapter introduces the single-region, the interregional and the multiregional input–output (IO) table and the differences between the IO models based on these three sets of data. The crucial intermediate input coefficients are shown to represent the product of a technical coefficient and an intra-regional self-sufficiency ratio in case of the single-region IO model, plus interregional import coefficients in case of both types of multi-region models. Besides, it is shown that it is better to work with the dimensionless normalized income, employment, emissions and other impact multipliers than with the ordinary multipliers of exogenous final demand. Finally, the nature of interregional spillover and feedback effects is explained, and why both are underestimated by the single-region model. Keywords Input–output tables · Technical coefficients · Trade origin ratios · Leontief model · Normalized multipliers · Impact studies · Price elasticities · Interregional spillovers · Interregional feedbacks

2.1 Single-Region IO Tables and Their Descriptive Power Every model of an economy requires data to estimate the model’s coefficients. The data to estimate the coefficients of an IO model may be derived from a so-called input–output table (IOT). The core of every regional or national IOT consists of a square matrix with, along its rows, the sales of all industries i to all industries j (i.e. z i j in the first quadrant of Table 2.1, with i and j = 1, …, I). Looking along the columns of the IOT, these sales represent the purchases of intermediate inputs from industry i by industry j. The third quadrant of Table 2.1 contains the additional purchases of primary inputs of type p by industry j (i.e. v pj , with p = 1, …, P). For a closed economy, these consist of the payments for labour and capital, which is why they are called primary inputs. For an open economy, however, they also include imports of intermediate inputs from the Rest of the World (RoW). By including profits as payments for capital use, the The original version of this chapter was revised: The presentation of Tables 2.1 and 2.2 have been corrected. The correction to this chapter is available at https://doi.org/10.1007/978-3-031-050879_11 © The Author(s), under exclusive license to Springer Nature Switzerland AG 2022, corrected publication 2022 J. Oosterhaven, Rethinking Input-Output Analysis, Advances in Spatial Science, https://doi.org/10.1007/978-3-031-05087-9_2

5

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2 Basic, Demand-Driven IO Quantity Models

Table 2.1 Single-region input–output table with macro-economic totals Industry 1

Industry j

Industry I

Local final demand exports

1st quadrant

Total

2nd quadrant

Industry 1

z 11

…

z 1I

y11

…

y1Q

Industry i

…

zi j

…

…

yiq

…

xi

Industry I

zI1

…

zI I

yI 1

…

yI Q

xI

3rd quadrant

x1

4th quadrant

Imports

…

…

…

…

…

…

M

Value added

…

v pj

…

…

y pq

…

Y

Total

x1

xj

xI

C

I

G

E

Legend zij = intermediate sales from industry i to industry j,yiq = final sales from industry i to final demand category q, x i = total output/input of industry i, vpj = primary input of type p by industry j, ypq = primary input of type p by final demand category q, C = consumption, I = investments, G = government expenditures, E = exports, M = imports and Y = gross value added at market prices

overall column totals of the first and third quadrant equal the value of total production per purchasing industry, x j . Along the rows of the IOT, the second quadrant complements the intermediate outputs of industry i with its sales to the various categories of final demand q (i.e. yiq in Table 2.1, with q = 1, ..., Q). For a closed economy, these consist of the sales of consumer and investment goods to households, firms and government agencies, which is why they are called final outputs. For an open economy, however, they also include the exports of both intermediate and final outputs to RoW. By adding changes in stocks as a separate category of final demand, the overall row sums of the first and second quadrant of the IOT equal total production by selling industry, xi . By including profits as part of value added and changes in stocks as part of final demand, the row totals by industry equal the column totals by industry. The fourth quadrant with the purchases of primary inputs of type p by final demand category q completes the accounting framework of the IOT (i.e. y pq in Table 2.1). Summation along the columns of the second and the fourth quadrant gives the macroeconomic totals of consumption, investment, government expenditures and in case of an open economy also of exports to RoW. Summation along the rows of the third and the fourth quadrant gives the totals for the gross domestic product Y and in case of an open economy also the total for the imports from RoW. Reorganizing the row and column totals of the IOT produces the well-known macro-economic accounting identity for the gross domestic product (GDP): Y =C+I +G+E−M

(2.1)

With (2.1), it becomes clear that an IOT essentially represents a double sectoral disaggregation of the accounting identity for gross regional or gross national domestic product.

2.2 Mathematics Versus Economics of the Closed Economy IO Model

7

Any IOT offers a series of interesting possibilities for descriptive research at the industry level. Taking percentages along the upper rows of an IOT, for instance, enables you to make comparative analyses of the sales structures of various industries, while the lower rows of an IOT show the contributions of the various industries to total wage and capital income and thus to GDP. Moreover, taking percentages along the first columns of an IOT enables you to analyse the differences in cost structures of various industries, while taking percentages along the last columns of an IOT allows for a comparative analysis of the purchase structures of the various categories of final demand. Finally, combining the industries with the mutually strongest linkages allows for the identification of clusters of interwoven industries (see Sect. 9.1.2 for a policy application).

2.2 Mathematics Versus Economics of the Closed Economy IO Model The main use of IO tables, however, is to provide the data to build IO models; the first of which was formulated by Leontief (1941), who won the Nobel Prize in economics in 1972 for the development of IO analysis. The economics of the basic, closed economy IO model, i.e. without imports and exports, is a little complicated, but its mathematics is simple. It consists of only two sets of equations. The first set states that the production by industry is determined by the demand for its products, i.e. by the row totals of the matrices with intermediate and final demand:   zi j + yiq , ∀i, or in matrix notation: xi = j

q

x = Zi + Yi = Zi + y

(2.2)

The second set of equations states that the demand for intermediate and primary inputs is linearly dependent on and proportional to the size of total output: z i j = ai j x j , ∀i, j, or in matrix notation: Z = Aˆx

(2.3a)

v pj = c pj x j , ∀ p, j, or in matrix notation: V = Cˆx

(2.3b)

where ai j indicates the use of products from industry i per unit of output of industry j and where c pj indicates the use of primary inputs of type p again per unit of output of industry j. This second assumption (2.3) implies that substitution between inputs is absent as well as economies of scale. Instead, all inputs are assumed to be pure complements of one another.

8

2 Basic, Demand-Driven IO Quantity Models

When only one single IOT is available, point estimates of the intermediate and primary input coefficients ai j and c pj are made by simply dividing the cells of the columns with intermediate and primary inputs from the IOT by their column total, i.e. by total production. Hence, A = Z xˆ −1 and C = V xˆ −1 , which implies that i A + i C = i ; i.e., the overall column total of the input coefficients per purchasing industry j is equal to one. Only in case of a closed economy, the above-defined intermediate input coefficients may be called technical coefficients, as they describe the production recipe of the industry at hand. In case of an open economy, besides the production recipe, they also describe the spatial purchase recipes of that industry. More formally, for an ·r open economy intermediate input, coefficients equal airrj = m rr i j ai j , where the intrarr regional input coefficient ai j indicates the use of products from industry i in region r per unit of output of industry j in the same region r, whereas the technical coefficient ai·rj indicates the use of products from industry i summed over all its locations in the entire world per unit of output of j in r. The trade origin ratio m rr i j subsequently indicates the share of region r in the worldwide supply of products from industry i used by industry j in region r. Finally, note the opportunities that (2.3b) offers for various empirical applications. Instead of primary input coefficients, any other type of input or output that is technically or economically linked to the production level by industry may be modelled in the same way. The impact variable vpj may thus equally well represent employment, CO2 emissions or energy use of industry j, in which case the technical coefficients cpj will represent the employment, CO2 emissions or energy use of industry j per unit of output of industry j. This flexibility of (2.3b) explains the abundant use of IO analysis in all kinds of economic, social and environmental impact studies (see Oosterhaven et al. 2019, for an overview).1 The mathematical solution of the basic IO model is simple: (2.3a) is substituted into (2.2) and then transferred from its right-hand side (RHS) to its left-hand side (LHS), after which (2.2) is pre-multiplied with the so-called Leontief-inverse, L = (I − A)−1 . Hence, the solution for total output per industry is derived as follows: xi =



ai j x j + yi· ∈ x = Ax + y ⇒

j

x = (I − A)−1 y = Ly

(2.4)

The subsequent substitution of (2.4) into (2.3a) and (2.3b) gives the solution for the matrices with intermediate and primary inputs: z i j ∈ Z = ALy and v pj ∈ V = CLy

(2.5)

1 If you wish to calculate any kind of impacts with various IO models, the InterRegional Input– Output Software package IRIOS. https://www.rug.nl/research/reg/research/irios/irios-download? lang=en offers a handy tool (see Stelder et al. 2000).

2.2 Mathematics Versus Economics of the Closed Economy IO Model

9

where Ly indicates the diagonal matrix of Ly. This closes the mathematics of the IO model. The economic interpretation and practical application of (2.4) and (2.5) is straightforward. These equations have the general structure of the solution of any model: endogenous variable A = model X’s A multiplier of B * exogenous variable B. Whenever that information is clear from context at hand, “exogenous variable B” and “model X” do not need to be mentioned explicitly. In our case, it is most often not mentioned that the multipliers need to be applied to exogenous final demand nor that they are derived from an IO model. When value added is chosen as the impact variable of interest, the matrix with ci li j ∈ cˆ L derived from (2.5) contains the disaggregate income multipliers, indicating the value added of industry i embodied in (i.e. both directly and indirectly needed to produce) one unit of final output of industry j. The column sums of this multiplier matrix, i.e. i ci li j ∈ c L, subsequently, contain the aggregate income multipliers indicating the value added of the whole economy embodied in one unit of exogenous final output of industry j. Employment, energy use, CO2 emission, and any other impact multiplier may be defined and calculated analogously. Note that the total of all primary input multipliers equals one, as c L =  i (I−A)(I−A)−1 = i , which means that the value added multipliers equal one in case of a closed economy, which is logical as total value added (Y ) equals total domestic final demand (C + I + G) in case of a closed economy. For value added, it is, therefore, more informative to look that the direct and indirect income per unit of direct income embodied in the final demand of industry  j. Wecall this ratio the standardized or normalized impact multiplier. It equals i ci li j c j ∈ c L cˆ −1 . Normalized multipliers typically have values of 1.1–2.2. Larger and more closed economies with smaller imports leakages tend to have larger multipliers, while industries that are part of tightly interwoven local industrial clusters also tend to have larger multipliers. Normalized multipliers are also very useful when the impact variable has a nonmonetary value. In those cases, ordinary multipliers are numbers with a dimension, such as the number of direct and indirect jobs per unit of final demand in euros or the tonnes of directly and indirectly emitted CO2 per unit of final demand in dollars. These ordinary multipliers will inconveniently change over time, due to both price inflation and real labour productivity growth or real CO2 emission efficiency increases, in case of ordinary employment or ordinary CO2 multipliers, respectively. Normalized multipliers, however, are dimensionless numbers and will therefore be more stable over time. They signify, for instance, the number of direct and indirect jobs per direct job embodied in the final output of industry j or the amount of direct and indirect tonnes of CO2 emitted per tonne of CO2 directly embodied in the final output of industry j. Normalized multipliers have been used in a tremendous amount of impact analyses done with single-region IO models (see Oosterhaven et al. 2019, for an overview).

10

2 Basic, Demand-Driven IO Quantity Models Final demand

I

C

Total output

A

Primary inputs

I

Intermediate inputs

Fig. 2.1 Causal structure of the basic demand-driven IO quantity model

As the economics of the basic IO model is more complex than the mathematics, a further clarification is given in Fig. 2.1, which shows the economic causality of the model. The size of final demand y is exogenous, i.e. no arrows are entering that box. This means that the size of y needs to be determined outside the IO model. Both the level and any change in the level of y then endogenously, within the model, lead to an equally large level or change in total output of I y, called the direct effect or direct impact of that change, as indicated by the arrow with the unity matrix I that leaves the box with exogenous final demand. This direct effect on total output x subsequently needs A times I y of intermediate inputs and C times I y of primary inputs, as indicated by the arrows accompanied by the matrices A and C in Fig. 2.1. The change in primary inputs leads to no further changes in the basic IO model, as indicated by the absence of arrows leaving the box with primary inputs. The change in intermediate inputs, however, leads to a further effect on total output, as these inputs need to be produced, which leads to the first round indirect effect of A y on total output. This middle circle of arrows in Fig. 2.1 goes on and on, leading to increasingly higher round indirect effects of A2 y + A3 y + A4 y + . . . on total output. In this way, the total cumulative effect on production by industry may be derived by means of:   x = I + A + A2 + A3 + A4 + . . . y = (I − A)−1 y = L y

(2.6)

A sufficient condition for this Taylor expansion to converge to the Leontief-inverse is that column sums of A are all smaller than one, which is a specific case of the BrauerSolow row and column sum criteria (Nikaido 1970, p. 18). Note the similarity of this matrix convergence condition with the condition under which the simply algebraic  expansion 1 + a + a 2 + a 3 + . . . converges to (1 − a)−1 , namely a < 1. Since i A = i − i C, the sufficient condition for the Taylor expansion to converge is satisfied whenever the column sums of the primary input matrix C are positive, i.e. whenever gross value added in market prices (plus the external imports in case of an open economy) is strictly positive for all industries. In fact, some column sums of A may even be larger than one, implying negative value added at market prices, e.g. because of subsidies, as long as the other column sums are sufficiently smaller than one. The above economic explanation of the equilibrium process of the Leontief model is offered by almost all IO texts. It is, however, important to emphasize that (2.2) and (2.3) specify neither the length nor the nature of the equilibrium process. The IO

2.3 Open Economy Interregional and Multi-regional IO Models

11

model is a purely comparative static model. When firms correctly predict and anticipate future changes in demand, adaptation may be quite fast. Most IO applications, however, work with year-to-year changes. From Fig. 2.1, two more things become clear. First, it is demand that drives the model, while prices do not play a role. Second, the absence of price effects means that supply does not play an active role either. Its role is entirely passive: it follows any change in demand. This is why this model is further defined by labelling it the demand-driven IO quantity model. Demand is met without any restriction on the supply side; i.e., there are no capacity constraints nor shortages of any kind. Hence, by explicitly assuming that demand is always fully met at constant prices, it is implicitly assumed that the supply of primary and intermediate inputs has an infinite price elasticity. One important implication of this assumption is that the IO model will produce an overestimation of the production and employment effects of any increase in final demand whenever an economy is close to the top of its business cycle, i.e. whenever part of an increase in demand at least partly results in higher prices instead of higher production. Finally, consider the implied behaviour of industry j. The most general production function assumes heterogenous inputs and heterogenous outputs, as measured in column j and row j of an IOT. The basic IO quantity model simplifies this by assuming that the outputs constitute a single homogenous product j, while the heterogeneous intermediate inputs i and primary inputs p are combined according to the following Walras–Leontief production function:   x j = min z i j /ai j , ∀i; v pj /c pj , ∀ p

(2.7)

Under full competition (i.e. at given market prices) assuming   (2.7) for firms in industry j implies that maximizing profits ( p j x j − i pi z i j − p p p v pj ) is achieved by minimizing cost, which results in using the multiple inputs in the fixed proportions defined in (2.3). Under full competition, industry j will thus have a perfectly elastic supply of its single homogenous output and a perfectly inelastic demand for its heterogenous intermediate and primary inputs (Oosterhaven 1996).

2.3 Open Economy Interregional and Multi-regional IO Models The mathematics of the IO model does not change when an open interregional or international economy is considered instead of a closed regional or national economy. The economic interpretation of the interregionally extended IO model, however, becomes more convoluted, as do the data required to construct such a model.

12

2 Basic, Demand-Driven IO Quantity Models

2.3.1 Separating Trade Origin Ratios and Technical Coefficients Isard (1951), founder of the Regional Science Association, specified the “ideal” interregional input–output table (IRIOT) on which the “ideal” interregional extension of the basic IO model might be based (see Table 2.2). Its first and main quadrant again contains the intermediate output of the industries of the interregional economy at hand. However, now, not only the intra-regional sales of intermediate outputs are part of this quadrant, but also the interregional exports of intermediate outputs to all included regions. The typical element of this matrix z ri js ∈ Zr s , with r and s = 1, …, R, indicates the sales of industry i in region r to industry j in region s. The typical element of the second quadrant of the IRIOT, f iqr s ∈ Fr s , has a comparable interpretation, but now the second quadrant explicitly also contains eir ∈ er , ∀r , i.e. sub-columns with the combined export of both intermediate and final outputs to regions or nations external to the interregional economy at hand. This is done to clearly separate the local final demand of each region from the demand from outside the interregional economy at hand. The third and fourth quadrants have the same interpretation as in Table 2.1, but mr now matrices with intermediate imports from RoW, z imr j ∈ Z , have explicitly been added to the third quadrant, while matrices with imports of final products from RoW, f iqmr ∈ Fmr , have explicitly been added to the fourth quadrant. Again the row and column totals of the combined last quadrants contain the macroeconomic totals of the now R endogenous regions. Because total input equals total Table 2.2 “Ideal” interregional input–output table with macro-economic totals Intermediate demand

Final demand

Total

Region 1

Region s

Region R

Region 1

…

Region R

RoW exports

Region 1

Z11

…

Z1R

F11

…

F1R

e1

x1

Region r

…

Zrs

…

…

Frs

…

er

xr

Region R

ZR1

…

ZRR

FR1

…

FRR

eR

xR

RoW imports

…

Zms

…

…

Fms

…

Transit

M for

Value added

V1

…

VR

Y1

…

YR

0

Y nat

Total

x1´

xs´

xR´

C 1 I 1 G1

…

C R I R GR

E for

r s ∈ Fr s = IQ-matrix with final demand of type q of region s Legend Additional to Table 2.1: f iq

for products of industry i in region r, eir ∈ er = I-column with foreign exports of industry i in r, ms = II-matrix with foreign imports of intermediate products and f ms ∈ Fms = IQ-matrix z ims j ∈Z iq with foreign imports of final products of industry i by final demand of type q in region s

2.3 Open Economy Interregional and Multi-regional IO Models

13

output for each regional  sum of the column totals of the second industry,the overall and fourth quadrant ( r C r + r I r + r G r + E f or ) again equals the overall sum of the row totals of the third and the fourth quadrant (Y nat + M for , wherein for indicates foreign, while nat stands for nation). Although not directly clear from Table 2.2, a rearrangement of elements produces the macro-economic accounting identities of each of the R regions included, but now with much more detail than in Table 2.1: Y r = i Vr i + i Yr i = C r + I r + G r ⎛ ⎞   +⎝ i Zr s i + i Fr s i + i er ⎠ s =r

s =r





⎛ −⎝

i Zsr i +

s =r

⎞ i Fsr i + i Zmr i + i Fmr i⎠

s =r

= C + I + G + Er − Mr r

r

r

(2.8)

The added detail, of course, relates to regional exports (i.e. the first term between brackets) and regional imports (i.e. the second term between brackets) of both intermediate and final outputs. Table 2.2 represents the ideal IRIOT, but this amount of statistical detail is unavailable in practice. This is why a series of less data demanding interregional IO accounting frameworks has been developed (see Batten and Boyce 1986, for an overview). The most commonly used of these is the multi-regional input–output table (MRIOT, Chenery 1953; Moses 1955). A MRIOT is a straightforward aggregation of an IRIOT. Instead of all the submatrices with intermediate and final use distinguished by region of origin in Table 2.2, a MRIOT only contains the vertical aggregation of these submatrices, i.e.  Z·s = r Zr s + Zms and F·s = r Fr s + Fms , ∀s. This information on the type of products needed by firms, consumers, investors and government agencies, aggregated over all spatial origins, is relatively easily available from statistical industry and household surveys. To substitute for the lost information on the origin of the intermediate and final inputs, a MRIOT additionally contains columns with the total of all intra-regional trade and the total of all bi-regional trade per industry of origin; i.e., a MRIOT contains the horizontal aggregation over the destination industries and final demand categories of all intra-regional trade as well as of all bi-regional trade per region of origin and destination, i.e. tir s ∈ tr s = Zr s i + Fr s i , ∀r, s. An IO model may be based on both sets of data. Written with all submatrices of Table 2.2 separately, it reads as follows in case of the interregional IO model :

14

2 Basic, Demand-Driven IO Quantity Models

⎤ ⎡ 11 Z x1 ⎢ .. ⎥ ⎢ .. x=⎣ . ⎦=⎣ . ⎡

xR

Z R1

⎤⎡ ⎤ i ... Z1R .. .. ⎥⎢ .. ⎥ . . ⎦⎣ . ⎦ i ... Z R R

⎡

F11 ⎢ + ⎣ ... F R1

= Z i + F i + e = Ax + F i + e

⎤⎡ ⎤ ⎡ 1 ⎤ e i ... F1R .. .. ⎥ ⎢ .. ⎥ + ⎢ .. ⎥ . . ⎦⎣.⎦ ⎣ . ⎦ eR i ... F R R (2.9)

When only one single IRIOT is available, point estimates of the submatrices with intermediate input coefficients may be calculated directly per column of Table 2.2 by means of Ar s = Zr s (ˆxs )−1 . The multi-regional IO model has the same set-up and answers exactly the same set of research questions. It reads as follows: ⎡

⎤⎡ 1 ⎤ ˆ 1R A·R x ... m ⎥ ⎢ .. ⎥ .. .. ⎦⎣ . ⎦ . . R1 ·1 R R ·R ˆ A ... m ˆ A m xR

ˆ 11 A·1 m ⎢ .. x=⎣ .

= Ax + F i + e,

⎡

⎤⎡ ⎤ ⎡ 1 ⎤ e ˆ 1R F·R i ... m ⎥⎢ .. ⎥ ⎢ .. ⎥ .. .. ⎦⎣ . ⎦ + ⎣ . ⎦ . . R1 ·1 R R ·R ˆ F ... m ˆ F m eR i

ˆ 11 F·1 m ⎢ .. +⎣ .

(2.10)

Note that the summation dots in (2.10) include imports from RoW, which is why the coefficients of the submatrices A·r may be called technical coefficients. They may be calculated directly per column of a MRIOT by means of A·s = Z·s (ˆxs )−1 . The ˆ rr , and those with diagonal submatrices with intra-regional purchase coefficients m rs ˆ (with r = s), may also be calculated directly from interregional import coefficients m ˆ r s , ∀r, s. The position of the summation a MRIOT by means of m ri s = tir s /ti·s ∈ m dot is crucial. It sums over all regions of origin, which is why these two types of trade coefficients may best be labelled as trade origin ratios.2 The core difference between the interregional and the multi-regional IO model is found in the implicit assumptions about the trade origin ratios in both models: interregional IO model: airjs = m ri js ai·sj ; multi-regional IO model: airjs = m ri·s ai·sj (2.11) Equation 2.11 again shows that all intermediate input coefficients in an open economy IO model, in fact, consist of the product of a trade origin ratio and a technical coefficient. The difference between the two models is that the interregional model assumes that each cell of an IRIOT has its own cell-specific trade origin ratio, whereas the multi-regional model assumes that all cells along each sub-row of an IRIOT have one and the same trade origin ratio. The fixed cell-specific trade origin ratio assumption better fits with the situation in which say agriculture in each region produces its own unique product, in which 2 In the IO literature these trade origin ratios are often referred to as column trade coefficients. Export coefficients or trade destination ratios or row trade coefficients are used in the row coefficient IO model, which has been shown to perform worse than the standard model (Polenske 1970) and has not been used since. See (Oosterhaven 1984) for other members of the family of square interregional IO tables and corresponding IO models.

2.3 Open Economy Interregional and Multi-regional IO Models

15

case the trade origin ratios m ri js may be assumed to be fixed for technical reasons, as in the closed economy case of (2.7). The fixed row-uniform trade origin ratio assumption better fits with the situation in which say again agriculture in each region produces about the same product mix or, alternatively, when agriculture in each region produces a close substitute. In these cases, one may assume the trade origin ratios m ri·s to be fixed, as long as the relative prices of the products of different regions are more or less stable. In most multi-regional IOTs, however, the row-uniform trade origin ratio assumption is hidden, as most MRIOTs are published in an IRIOT format (see Sect. 4.3, for the reason why this is common practice). The mathematical solution of both multi-region models is derived in the same way as in the single-region case. The matrix with endogenous intermediate inputs is transferred from the RHS to the LHS of (2.9) and (2.10), whereupon both sides are pre-multiplied with the appropriate inverse. The solution of the interregional IO model then simply reads as: x = (I − A)−1 (F i + e) = L∗ (F i + e)

(2.12)

where airjs ∈ A and f iqr s ∈ F are equal to the corresponding block matrices in (2.9)and where L* indicates the interregional Leontief-inverse. The liss∗ j from the submatrices on the diagonal of L* indicates the intra-regional impact of the exogenous final demand for the products from industry j in s on the production of industry i in the same region s, whereas the lirjs∗ from the off-diagonal submatrices of L* indicates the interregional spillover effect of the final demand for the products from industry j in s on the production of industry i in a different region r. In the same way, the solution of the multi-regional IO model may be derived as:  −1   ˆt M Fˆ t i + e x = I −MA

(2.13)

ˆ t and m r s f ·s ∈ M Fˆ t equal the block matrices in (2.10), with A ˆt where m ri s aisj ∈ M A i· iq and Fˆ t indicating block diagonal matrices, with t standing for technical to indicate that the products at hand come from all over the world. Note that (2.10) and (2.13) show that the row-uniform trade origin ratio assumption not only applies to different destination industries, but also to different destination types of local final demand. Empirical research in case of international IOTs shows that international trade origin ratios for local final demand differ significantly from those for intermediate demand (Dietzenbacher et al. 2013). The same is most likely true for interregional IOTs, but has not been reported in the literature, as detailed interregional trade data are lacking for almost all countries. For the base year from which the model coefficients of (2.9) and (2.10) are calculated, the two multi-region IO model solutions (2.12) and (2.13) will produce exactly the same endogenous column with total output per regional industry xir ∈ x. However, for any other year than the base year or for any non-proportional exogenous final

16

2 Basic, Demand-Driven IO Quantity Models

demand impulse both models will produce a different outcome. This difference is caused by the aggregation error that is made with the multi-regional IO model compared to the “ideal” interregional IO model. Using hypothetical IRIOTs, as real data are lacking, Vali (1988) reports mean average percentage errors (MAPEs) of about 13% at the aggregate industry level, 4% at the aggregate regional level and 0.4% at the national level. Vali (1993), additionally, reports no systematic under- or overestimations, which were not expected, but does report average overestimations of aggregate trade flows of 20–30% and average underestimations of as much as 37–50%. To enhance the understanding of the economic working of the interregional IO model, Fig. 2.2 shows how two open economy single-region IO models are joined in one interregional IO model. The boxes and arrows with solid lines reproduce Fig. 2.1 for region r and region s, respectively. The boxes and arrows with dotted lines show the changes necessary to obtain the interregional model. We only discuss region r, as the changes for region s are identical. The single difference for region r is that its exports of intermediate outputs to region s, i.e. Zr s i, are exogenous in the singleregion model, whereas they become endogenously determined by the production levels of region s in the interregional IO model, i.e. by Zr s i = Ar s xs , as indicated by the upper dotted box with dotted arrows. This means that the exogenous final demand of the interregional model is smaller than that of the single-region model, as indicated in the upper left solid box of Fig. 2.2. To see what this implies, remember that reality does not change. Only the way in which reality is modelled changes. Thus, to get the same endogenous level of output in region r, the intra-regional part Lrr ∗ of the extended Leontief-inverse must become sufficiently larger to compensate for the smaller exogenous final demand. How is this possible? The answer is given by the circle of dotted arrows in the middle of Fig. 2.2. In the single-region model, the endogenous intermediate imports of region r, i.e. Zsr i, do not lead to any further endogenous effects, but in the interregional model they do. There, they constitute the endogenous intermediate exports of region s, which need to be produced there. This is called an interregional spillover effect. This spillover effect yr = f r +

Zrs i

ys = f s + Zrs

Zrr

Ars xs

xr Asr Vr

Zsr i

Zss

Zsr Vs

Fig. 2.2 Causal structure of the interregional IO model extension. Legend y = vector with exogenous final demand of the single-region IO model, f = vector with exogenous final demand of interregional IO model, Zrs = interindustry matrix with intermediate exports from region r to region s, x = vector with total output by sector and V = matrix with value added by type, by sector.

2.3 Open Economy Interregional and Multi-regional IO Models

17

in region s in turn requires endogenous intermediate imports from r, i.e. Zr s i. This second, reverse, interregional spillover effect rounds the dotted circle in the middle of Fig. 2.2. Together, these two spillover effects create what is called an interregional feedback effect. Following this circle shows that the size of the feedback effect in the two-region case can be determined as follows: interregional f eedback rr = interreg. spillover rs ∗ intra-regional multi plier ss ∗ interreg. spillover sr

See Oosterhaven and Hewings (2021, p. 408) for the proof of this two-region formula. Returning to the multi-region case of (2.12), the size of the interregional spillover effects of region s on say aggregate employment in region r can simply found  be r r s∗ c l ∈ by calculating the aggregate interregional employment multipliers i i ij (cr ) Lr s∗ . The size of the interregional feedback effects may, subsequently, be calculated by taking the difference between the intra-regional employment multipliers from the multi-region IO model and those from the single-region IO model, i.e. by calculating (cr ) [ Lrr ∗ − Lrr ], where the absence of * indicates that Lrr originates from the single-region or single-nation IO model.

2.3.2 Underestimation of Interregional Spillovers and Feedbacks Empirically, the size of these interregional feedback effects has been extensively studied in the 1970s. Miller and Blair (1985, p. 127) concluded in an overview of the literature that they could be disregarded, as they only added between 1 and 2% to the value of intra-regional multipliers when moving from a single-region model to a multi-region model. However, earlier on, Yamada and Ihara (1969) reported much larger errors of neglecting interregional feedbacks in case of Japan, while Greytak (1970) did the same for the USA. Unfortunately, the denominator of the error percentages reported always included the direct production effect of one unit of final demand. For this direct effect of 1.00, however, one does not need an IO model at all. The IO model is only needed to estimate the indirect effects, which is why the error measure should only have the indirect effect in its denominator. Consequently, most errors percentages reported in the literature should be multiplied with a factor of 2.00–3.00, assuming a single-region multiplier of 2.00 or 1.50, respectively. Comparing only the indirect effects, Oosterhaven (1981) found an underestimation of the regional indirect income effect of only 1.1% for the rural, peripheral Northern Netherlands and found a 3.4% underestimation for the urbanized greater Rotterdam area in the economic core of the Netherlands, confirming the conclusion of Miller and Blair (1985). When comparing Type II income multipliers with endogenous consumption expenditures (see Sect. 5.1), the underestimation increased to 3.1% in case of the Northern Netherlands and to as much as 6.6% in the case of the greater Rotterdam area. The reason for these larger interregional feedback effects

18

2 Basic, Demand-Driven IO Quantity Models

was the inclusion of interregional commuting and interregional shopping in the Type II input–output model. Recently, using intercountry supply–use tables at the global level (see Chap. 4), Termursho (2018) reported a weighted average error of neglecting spillovers and feedbacks as large as 7.9% of the global output multiplier of 1.9 in 1995, increasing to 11.5% in 2008 and decreasing to 9.8% in 2009, due to the 2008–2009 global economic decline. Note that the correct error calculated for only the indirect part of the global multiplier is about two times larger than the above reported percentages. A second difference between the single-region model and the multi-region model is hardly discussed in the literature, but is equally important. It appears in the interregional spillover effects of the own final demand f s on say the income vr of a different region. In the single-region model, the aggregate spillover effects equal (cr ) Ar s Lss , whereas they equal (cr ) Lrr ∗ Ar s Lss∗ in the two-region case (see Oosterhaven and Hewings 2021, for the proof). Hence, not only the intra-regional impacts but also the interregional spillovers of an exogenous impulse are larger when estimated with a multi-region model then when estimated with a single-region model. Bouwmeester et al. (2014) estimated the income and CO2 effects of the exports to third countries for the 27 members of the European Union (EU), both with 27 separate national IO models and with a single consolidated IO model for the EU27 as a whole, which includes all interregional spillovers and feedbacks between all 27 member states. They find an average first round intra-EU income spillover to the rest of the EU27 of 7.7%, when calculated with the 27 single-country models. The additional higher order intra-EU spillovers, as calculated with the consolidated EU27 model, appeared to be as large as an additional 10.7% of the domestic direct and indirect income effect. Note again that both percentages represent an underestimation of the importance of intra-EU spillovers as their denominator includes the direct domestic income effect for which no model is needed to estimate it. In sum, the underestimation of the interregional spillover effects as well as that of the interregional feedback effects with a single-region or a single-nation IO model appears to be much more serious than suggested in the early literature.

References Batten DF, Boyce DE (1986) Spatial interaction, transportation and interregional commodity flow models. In: Mills ES, Nijkamp P (eds) Handbook in urban and regional economics, vol 1. North Holland, Amsterdam Bouwmeester MC, Oosterhaven J, Rueda-Cantuche JM (2014) A new SUT consolidation method tested by a decomposition of value added and CO2 embodied in EU27 exports. Econ Syst Res 26:511–541 Chenery HB (1953) Regional analysis. In: Chenery HB, Clark PG, Vera VC (eds) The structure and growth of the Italian economy. U.S. Mutual Security Agency, Rome Dietzenbacher E, Los B, Stehrer R, Timmer M, de Vries G (2013) The construction of world input-output tables in the WIOD project. Econ Syst Res 25:71–98

References

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Greytak D (1970) Regional impact of interregional trade in input-output analysis. Pap Reg Sci Assoc 25:203–217 Isard W (1951) Interregional and regional input-output analysis, a model of the space economy. Rev Econ Stat 33:318–328 Leontief WW (1941) The structure of the American economy, 1919–1929: an empirical application of equilibrium analysis. Cambridge University Press, Cambridge Miller RE, Blair PD (1985) Input-output analysis: foundations and extensions. Prentice Hall, Englewood Cliffs, New Jersey Miller RE, Blair PD (2022) Input-output analysis: foundations and extensions, 3rd edn. Cambridge University Press, Cambridge Nikaido H (1970) Introduction to sets and mappings in modern economics. North-Holland, Amsterdam Moses LN (1955) The stability of interregional trading pattern and input-output analysis. Am Econ Rev 45:803–832 Oosterhaven J (1981) Interregional input-output analysis and Dutch regional policy problems. Gower Publishing, Aldershot-Hampshire Oosterhaven J (1984) A family of square and rectangular interregional input-output tables and models. Reg Sci Urban Econ 14:565–582 Oosterhaven J (1996) Leontief versus Ghoshian price and quantity models. South Econ J 62:750–759 Oosterhaven J, Polenske KR, Hewings GJD (2019) Modern regional input-output and impact analysis. In: Capello R, Nijkamp P (eds) Handbook of regional growth and development theories: revised and extended second edition. Edward Elgar, Cheltenham Oosterhaven J, Hewings GJD (2021) Interregional input-output models. In: Fischer MM, Nijkamp P (eds) Handbook of regional science. Springer-Verlag, Berlin Polenske KR (1970) An empirical test of interregional input-output models: estimate of 1963 Japanese production. Am Econ Rev 60:76–82 Stelder TM, Oosterhaven J, Eding GJ (2000) IRIOS: an InterRegional Input-Output Software approach to generalised input-output endogenisation, linkage, multiplier and impact analysis. Paper 40th ERSA congress, Barcelona, August 2000, and 47th NARSC Congress, Chicago, November 2000. https://www.rug.nl/research/reg/research/irios. Accessed 15 May 2022 Termursho U (2018) Intercountry feedback and spillover effects within the international supply and use framework: a Bayesian perspective. Econ Syst Res 30:337–358 Vali S (1988) Economic structure and errors in multiregional input-output model. Ric Econ 17:367– 390 Vali S (1993) Simulation evidence bearing on the structure of errors in MRIO analysis. Environ Plan A 25:159–178 Yamada H, Ihara T (1969) Input-output analysis of interregional repercussions. Pap Proc Third Far East Conf Reg Sci Assoc:3–31

Chapter 3

Updating Different Types of IO Tables

Abstract The iterative scaling and rescaling of the rows and columns of an old IOT until they equal the new row and column totals, known as RAS, outperforms alternative techniques for updating old IOTs. Still, the errors of this information gain minimizing technique remain large, which is why adding survey data for the target year is required. In case of updating bi-regional IOTs, adding the known values of new national IO cells, while using the multi-proportional scaling of MR-RAS, considerably improves the accuracy of the updates. When negative cells are present, RAS and MR-RAS need to be replaced by generalized RAS and MR-GRAS. Finally, it is shown how time series of old IOTs may be used in CRAS to improve the estimates of the cells in RAS or GRAS. Keywords RAS algorithm · Information gain · Sign preservation · Multi-proportional scaling · Bi-regional input–output tables · GRAS algorithm · Cell-corrected RAS IO models require the data from IO tables to estimate their parameters. For most countries, their national statistical office produces national supply–use tables (SUTs), which show the supply and use of products by industry (see Sect. 4.2 for details). SUTs are usually produced, for every calendar year with a delay of 3– 5 years, whereas national IOTs are usually produced only every five year—or not at all—with a delay that is at least as long. In most cases, these tables are based on extensive industry and household surveys, which are costly to carry out (UN 2018). In contrast, regional, interregional and international SUTs and IOTs are produced by a multitude of different organizations, much less frequently and with even larger delays. In most cases, these tables are only based on already existing statistics, while they use all kinds of non-survey methods to estimate the lacking data (see Oosterhaven et al. 2019, for the USA). Non-survey methods cope with two types of lacking data, namely data lacking in time and data lacking in space. In this chapter, we discuss updating techniques, which adapt old survey-based interindustry tables to the most recent macro-economic and industry totals. In the next chapter, we discuss the spatial type of non-survey methods.

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2022 J. Oosterhaven, Rethinking Input-Output Analysis, Advances in Spatial Science, https://doi.org/10.1007/978-3-031-05087-9_3

21

22

3 Updating Different Types of IO Tables

3.1 RAS and MR-RAS: Bi- and Multi-proportional Scaling The oldest and still most used technique to update old IOTs is the so-called RAS technique. RAS is a bi-proportional, iterative matrix balancing method, which sequentially scales and rescales the rows and columns of an old base year IOT until they equal the new row and column totals (called margins) of the target year IOT. Iterative scaling became available to social scientists through Deming and Stephan (1940) and was first used to update IOTs by Stone (1961; Stone and Brown 1962; see Lahr and de Mesnard 2004; Lenzen et al. 2009; Miller and Blair 2022, for overviews). The problem is: how to estimate a new matrix a i j ∈ A of which one only knows  the row and column totals j ai j = u j ∈ u and i ai j = v j ∈ v , along with an old matrix ai0j ∈ A0 that is considered to be the best available approximation of the unknown new A.1 Of course, for consistency reasons, the overall total of the new rows needs to equal that of the new columns, i.e. i u = v i, which means that one of the totals contains redundant information. The problem is to find a solution that adds as little information, i.e. as little unknown structural change, as possible to the structure of the old A0 . The RAS algorithm iteratively solves this problem in the following, easily programmable steps: 2 1. 2.

3. 4.

Initialization: set p = 0, define A0 , and define u and v . Row scaling: set p = p + 1, set row scaling vector at r p = u÷(A p−1 i), where ÷ = cell-by-cell division, and define new row-consistent cells A p−0.5 = rˆ p A p−1 , where rˆ p = diagonal matrix of r p , Column scaling: set column scaling vector at (s p ) = v ÷ (i A p−0.5 ), where v = transposition of v, and define new column-consistent cells A p = A p−0.5 sˆ p , Stopping: if |u − A p i| ≥ ε continue with Step 2, where ε = maximally allowable error per row; if |u − A p i| < ε stop and accept A p as the RAS estimate of the true A.3

Hence, the solution of RAS equals A p = ( p rˆ p ) A0 ( p sˆ p ) = rˆ A0 sˆ, which explains the label RAS. It is important to note that RAS preserves the signs of the old transaction matrix A0 . In most cases, the signs of A0 have an economic meaning (e.g. a – indicating subsidies, a 0 indicating impossible transactions or a + indicating regular transactions). This 1

Note that in this chapter we use the symbols that are usual in the RAS literature, which deviates for the meaning of the symbols used in other chapters. 2 The reader may wonder why this problem is not stated in terms of input coefficients instead of in transaction values. A good question! In the early days (e.g. Stone 1961), RAS was exclusively applied to input coefficients, as estimating them best was considered most important. Both problem statements, however, lead to the same result (see Dietzenbacher and Miller 2009), while the mathematics of the problem in terms of values is more elegant. So that is what we do here. 3 Note that one stopping rule suffices. Stopping after Step 3 assures that the column sums of the target IOT are exactly equal to their target values. Stopping after Step 3 better fits with the Leontief model, as it assures that the column sums of the input coefficients sum to exactly one (see Chap. 2). Any remaining small errors may best be incorporated into the mostly unreliable “changes in stocks” column of the IOT.

3.1 RAS and MR-RAS: Bi- and Multi-proportional Scaling

23

makes sign preservation, in general, a desired property. However, in the specific case of “changes in stocks” it is not desired, as stocks may easily turn from increasing (+) to decreasing (−) in size. Besides, when a 0 represents a rounded off small regular transaction, keeping it zero may not be desirable either, as it will be unable to grow with the growth of the economy, which may cause RAS to have an infeasible solution (see Miller and Blair 2022, p. 435, for an example). Interestingly, it may be proven that the rˆ A0 sˆ solution is identical to the solution of the following nonlinear programming problem (see Bacharach 1970; Macgill 1977): Minimize I G =

 ij 

  ai j ln ai j /ai0j , ∀ai0j > 0,

subject to A i = u, i A = v , with i u = v i

(3.1)

The goal function in (3.1) represents the principle of minimum information gain when going from A0 to A (see Kulback 1959; Theil 1967) as well as the principle of maximum entropy (see Jaynes 1957; Wilson 1970). Zero cells in A0 are excluded, as a division by zero is not defined. Negative cells in A0 are excluded, as making them more negative along with compensating increases in positive cells in the same row or column would mathematically reduce the information gain in cases wherein it actually increases (see the next section for the solution of this problem). Furthermore, note that taking the logarithm of the relative changes ai j /ai0j implies that the larger % changes weigh a little less than proportionally heavier, while adding ai j in front of the logarithm implies that changes in the larger new cells weigh exactly proportionally heavier. Still, one hopes that the purely mathematical outcome of (3.1) may also be given a sensible economic interpretation. Early on Stone (1961), considering cases wherein A0 only contains the intermediate transactions part of an IOT, interpreted the row multipliers rˆ as representing substitution effects and the column multipliers sˆ as representing fabrication effects, which—when positive—indicate that industries over time rely on more roundabout production processes with longer supply chains and less own value added. However, as noted above, one of the row or column sums in (3.1) contains redundant information. This makes any RAS solution non-unique, as also follows from rˆ A0 sˆ = (λ rˆ ) A0 (λ−1 sˆ). Van der Linden and Dietzenbacher (1995) provide a sensible solution that saves Stone’s economic interpretation; i.e. they propose to set the λ such that the weighted average of the positive and negative substitution effects equals unity (= no substitution at the aggregate level). Naturally, over time, economic change cannot be explained by substitution and fabrication effects alone; i.e., RAS will make errors, and—given the nature of RAS— these will be cell-specific. With the true transaction matrix equal to A, the additive (i.e. monetary) and multiplicative (i.e. %) errors of RAS per cell of A equal, respectively: δi j = ai j − ri ai0j s j and     εi j = ai j / ri ai0j s j − 1, ∀i j

(3.2)

24

3 Updating Different Types of IO Tables

There is a huge literature about the performance of RAS and the size of these errors (see Lecomber 1975; Polenske 1997, for excellent early overviews, and Temurshoev et al. 2011, for a recent comparison). Its conclusions may be summarized as follows: 1. 2. 3.

RAS performs equally well or better than alternative updating procedures that use the same information. Nevertheless, in most cases, RAS still results in unacceptably large errors at the cell level of the IOT (Polenske 1997). Hence, it is necessary to add as much additional survey data to the essentially non-survey RAS technique as is possible within the research budget (Lecomber 1975).

This additional data should preferably relate to the larger cells of A0 , and secondarily to so-called key sectors with large forward and backward linkages (see further Sect. 9.1). The way in which such additional data may be incorporated into RAS depends on their nature. If the additional data regard single cells of A, the standard four-step procedure is: (1) put these cells equal to zero in the old A0 , (2) subtract their values from the new margins u and v , (3) do RAS on the modified matrix and margins and (4) add the known data to this RAS result. This method became known as the modified RAS method (Paelinck and Waelbroeck 1963, see Miller and Blair 2022, p. 428, for a 3 × 3 numerical example). If the additional data regard a combination of cells, the solution is to add these combinations with their totals as additional scaling steps to the above four-step basic RAS algorithm or to add them to (3.1) if the nonlinear optimization part of, e.g. GAMS or MATLAB, is used to solve (3.1). Writing an RAS algorithm with arbitrary numbers of additional restrictions on arbitrary combinations of cells of A0 was probably done first by Dirk Stelder for FNEI (1986, see also Oosterhaven et al. 1986, p. 63). He called his algorithm AAP, but we will call it MR-RAS, for multiple restrictions RAS or multiple regions RAS (see also Gilchrist and St. Louis 1999, who present a comparable generalization, which they called TRAS). We close this section with a summary of the quite interesting comparison of construction cost, updating cost and statistical reliability of updating single-region IOTs by means of RAS versus updating bi-regional IOTs by means of MR-RAS (Oosterhaven et al. 1986). Below, we sequentially discuss the necessary type of data using the definitions of the full IRIOT shown in Table 2.2. When needed, we will additionally use Ar s = [Z r s Fr s ] to indicate the combination of the corresponding intermediate and local final demand sub-tables of Table 2.2. Base year table: Obviously, a survey-based construction of a single RIOT with Arr as its core matrix will be far less costly than the comparable construction of a full IRIOT with all Ar s . In fact, the difference in cost will be more or less proportional to the number of regions in the IRIOT, because the economies of scale in estimating the regional industry and final demand totals will be undone by the additionally needed interregional trade data. An exception to this proportional cost increase rule of thumb applies to the construction cost of a survey-based bi-regional IOT with the Rest of the Country (RoC) as the second region. These costs will not be twice as large as

3.1 RAS and MR-RAS: Bi- and Multi-proportional Scaling

25

those of constructing a single-region IOT. They will only be a little larger, as most data for RoC can be calculated as the difference between the national IOT of the country at hand and that of the single RIOT (for details, see Sect. 4.1.3). New row totals: Here, we see a reverse data cost story and a related reverse statistical reliability story. When updating an IRIOT, it is sufficient to have an estimate of the new total domestic sales for each regional industry, i.e. xr − er , ∀r in Table 2.2. To obtain estimates of er may sometimes be costly, but most firms are well informed about the size of their foreign exports. In the single RIOT case, however, the cost to obtain new row  totals will be larger, as RAS also needs an estimate of the exports to RoC, i.e. of s=r Ar s i. Regional domestic exports will be more costly to obtain, as firms are less well informed about their size, and when obtained they will be less reliable. In the IRIOT case, new domestic export data are not needed, as they are estimated with MR-RAS, but if available they can easily be added as an extra restriction. In the bi-regional case, estimating xr −er for RoC is simply done by taking the difference with the national total, whereas they need to be collected/estimated for all other regions separately in case of updating a full IRIOT. New sectoral column totals: Again, there is a reverse cost and reliability story. Whenever xr is available, regional industry value added i Vr will also be available. In both the single RIOT case and the full IRIOT case, the total of the foreign imports by regional industry i Zmr is additionally needed. These data are more costly to obtain and will be less reliable, as firms are less well informed about their foreign imports than about their foreign exports, partly because imports are often obtained indirectly through  retail and wholesale firms. In the single RIOT case, however, total domestic imports s=r i Zsr are also needed to obtain the necessary column totals in Table 2.2. Unfortunately, domestic import data are most difficult to come by. When available, they may also be used in the IRIOT case by simply adding them as an additional restriction in MR-RAS. Again, in the bi-regional case, the necessary (xs ) − i Vs − i Zms for RoC is simply calculated as the difference with the national total. New final demand column totals: Here, more or less the same cost and reliability observations about single-region, bi-regional and interregional IOTs may be made as in case of the sectoral column totals. New national cell totals: When available, national IO data can simply be added as additional restrictions in MR-RAS when updating a bi-regional or an interregional IOT, which means that national technology changes will be incorporated in these updates. The only changes that remain uncovered will be any changes in the structure of interregional trade. In case of updating a single RIOT, it is far more difficult to use national IO data for the target year (see Oosterhaven et al. 1986, for details). In sum, in the bi-regional IOT updating case it is evident that the small increase in base year construction cost is more than earned back in terms smaller updating cost and higher updating reliability, compared to the single-region updating case. Only in cases where the single region is much smaller and very different in economic structure compared to RoC, the advantage of constructing a base year bi-regional IOT instead of a single RIOT is not clear. In the full IRIOT case, i.e. with three or more regions, the same trade-off exists between higher initial construction cost, on

26

3 Updating Different Types of IO Tables

Table 3.1 Comparison of differences in bi-regional and single-regional updates of IOTs Comparisons

MAPE WAPE IG

Groningen: all cells, with vs. without national cell restrictions

32.3

16.6

0.189

Friesland: same

29.9

15.5

0.168

Drenthe: same

26.6

15.4

0.167

Groningen: only intra-reg. cells, with vs. without nat. cell restr.

20.5

13.7

0.137

Friesland: same

17.8

8.7

0.093

Drenthe: same

10.2

7.0

0.073

Groningen: intra-reg. cells, MR-RAS vs. RAS, without nat. cell restr.

21.1

22.1

0.223

Friesland: same

21.8

16.0

0.168

Drenthe: same

10.4

14.5

0.164

Source Adapted from Oosterhaven et al. (1986)

the one hand, and lower updating cost and higher updating reliability, on the other hand, but the outcome is not clear in that case. Table 3.1 gives an empirical impression of the above-discussed theoretical differences in the statistical reliability of updating single RIOTs with RAS versus updating bi-regional IOTs with MR-RAS. Table 3.1 is derived from the updates of the biregional IOTs of the relatively rural provinces of Groningen, Friesland and Drenthe, in the northeast of the Netherlands, from 1975 to 1980. Each province contains about 3–4% of the Dutch economy, implying that the three RoC regions each contain the remaining, slightly different 96–97% of the Dutch economy (see for further regional characteristics, Sect. 9.1.2). The three bi-regional base year IOTs for 1975 were constructed with a predecessor of the semi-survey, double-entry bi-regional IOT construction method (DE-BRIOT) discussed in Sect. 4.1.3. Hence, these base year tables with 59 industries may be considered as relatively reliable. The same holds to a lesser degree for the base bi-regional update with MR-RAS with national cell restrictions and several minor additional cell restrictions (see Oosterhaven et al. 1986, for details). As opposed to the base year bi-regional IOTs, however, these updates do not account for any changes in the structure of the intra-regional self-sufficiency ratios and the domestic import ratios discussed in Sect. 2.3.4 The first three rows of Table 3.1 show the importance of adding information about the values of the new national cells, where MAPE = mean absolute percentage error, WAPE = weighted mean absolute percentage error and IG = information gain, as measured in (3.1). Not adding the national cell restrictions results in unweighted 4

These updates also encountered the problem of having national cells that are positive in 1980, whereas the four corresponding bi-regional cells were zero in 1975. The solution was to (1) make an estimate of the 1980 spit-up of these national cells into the four constituent bi-regional cells, (2) insert this spilt-up into the base year bi-regional IOT for 1975 and (3) start the MR-RAS algorithm with the national cell restriction. Starting with this restriction is anyhow wise as it is the most stringent one (see Oosterhaven et al. 1986, for further details).

3.2 GRAS and MR-GRAS: Including Negative Values

27

errors of about 30% and weighted errors of about 15%, indicating that the larger percentage errors luckily tend to be made in the smaller cells. The second three rows show the same errors for only the intra-regional part of the bi-regional IOT. They are significantly smaller than those in the first three rows for Friesland and Drenthe, indicating that the larger errors luckily are concentrated in the submatrices for RoC. This is much less the case for Groningen, most likely because it has a much more deviating economic structure with its unique, huge natural gas production and exports to RoC and RoW. The third set of three rows shows the errors made if the three single-region IOTs are updated separately with RAS or whether they are updated as part of their biregional IOT with MR-RAS but without national cell constrains. These errors are surprisingly large, especially for Groningen, which in this comparison does not even enjoy the advantage of having its larger errors being concentrated in its less important smaller cells, which follows from its WAPE being larger than its MAPE as well as from its very large IG of 0.223. Aside from showing the great advantage of updating a single RIOT as part of a bi-regional IOT with MR-GRAS, Table 3.1 also confirms the first general conclusion from the literature, namely that using only RAS without additional survey-based data leads to unacceptably large errors.

3.2 GRAS and MR-GRAS: Including Negative Values The above-presented four-step RAS algorithm as well as its MR-RAS generalization only works well (i.e. only minimize the information gain) if both the base year matrix and the new margins contain only semi-positive cells. Whenever negative cells are present in the base year matrix, applying say a row scaler larger than one to a row with a positive target row total results with RAS in making the positive cells in that row more positive and the negative cells more negative, which might not lead to convergence and definitely results in unnecessarily large errors. The traditional solution to this problem is to use the modified RAS method, i.e. (1) remove the negative cells from the old A0 , (2) remove them from the new margins u and v , (3) do RAS on this modified combination and (4) set back the negative cells in its solution A p . As a result, the negative cells no longer hinder convergence and information gain minimization, but they do not help either. Intuition, however, suggests that the negative cells may help convergence and information gain minimization if they are reduced in size with the same percentage wherewith the positive cells are increased. Günlük-Senesen ¸ and Bates (1988) first proposed this solution, which was subsequently re-discovered and formalized by Junius and Oosterhaven (J&O) (2003). Lenzen et al (2007) corrected the goal function of J&O and Termurshoev et al. (2013) provide the MATLAB code for a slightly more general updating problem, allowing for rows and columns without positive elements. J&O prove that their Generalized RAS (GRAS) algorithm minimizes an adapted information gain formula with all weights positive:

28

3 Updating Different Types of IO Tables

Minimize I G ∗ =

      ai j  ln ai j /a 0 , ∀a 0 = 0, ij ij ij

subject to A i = u, i A = v , with i u = v i

(3.3)

To find a solution, A0 needs to be split-up into a matrix N0 that combines its negative cells and a matrix P0 that combines its positive cells, such that A0 = P0 − N0 . The GRAS algorithm that iteratively solves (3.3) consists of the following four steps (cf. the RAS algorithm in Sect. 3.1): 1. 2.

3.

4.

Initialization: set p = 0, define A0 = P0 − N0 , and define u and v , Row scaling: set p = p + 1, solve the second-order equations of  p p−1 − (ˆr p )−1 N p−1 i = u, put their positive roots in the row scaling rˆ P p matrix rˆ , and set P p−0.5 = rˆ p P p−1 and N p−0.5 = (ˆr p )−1 N p−1 , Column scaling: solve the second-order equations of   i P p−0.5 sˆ p − N p−0.5 (ˆs p )−1 = v , put their positive roots in the column scaling matrix sˆ p , and set P p = P p−0.5 sˆ p and N p = N p−0.5 (ˆs p )−1 , Stopping: if | u − (P p − N p ) i | ≥ ε continue with Step 2, where ε = maximally allowable error per row; if | u − (P p − N p ) i | < ε stop and accept A p = P p −N p as the RAS estimate of the true A.

Hence, the solution of GRAS is A p = rˆ P0 sˆ − (ˆr)−1 N0 (ˆs)−1 , with rˆ =  p rˆ p and sˆ =  p sˆ p . Note that the GRAS algorithm allows for negative and sign changing cells in the margins u and v . The cells of A0 , however, preserve their signs, as is the case with RAS. Testing the performance of GRAS started with J&O (2003). Using a 3 × 4 test matrix A0 , they find an information gain (IG*) as large as 9.19 with the modified RAS method, whereas IG* reduces to the lower value of 8.08 when GRAS is used instead. With the same test matrix, Jackson and Murray (J&M) (2004) compare GRAS with a sign preserving absolute differences (SPAD) optimization scheme and with a sign preserving squared differences (SPSD) optimization scheme, which both minimize differences in input coefficients, as opposed to GRAS that minimizes differences in transaction values. This distinction is of importance here, as minimizing the information gain defined on coefficients as opposed to transaction values does not lead to the same result when both positive and negative cells are present in A0 (see Temursho et al. 2020, for proof). J&M (2004) compare the three methods with the original IG from (3.1), which is not suited for summing logarithmic values with both positive and negative weights, as this unjustly leads to negatively weighted differences being compensated by positively weighted differences. Instead, we use the adapted IG* with only positive weights from (3.3) in the first three columns in Table 3.2. These IG*’s show that GRAS clearly outperforms the other two methods, even when differences in input coefficients are compared. This is all the more remarkable as SPAD and SPSD minimize precisely these last differences, whereas GRAS minimizes the differences in transaction values! The IG* value of −1.43, however, stands out because it is conceptually impossible to have a negative information gain. The probable reason for going unnoticed is that

3.2 GRAS and MR-GRAS: Including Negative Values

29

Table 3.2 Performance of alternative updating algorithms in terms of information gain (IG)

      Information gain measure I G ∗ =  ai j  ln ai j /ai0j I G ∗∗ =  ai j ln ai j /ai0j  ij

Optimization method

GRAS

IG measured on transactions

6.8a

IG measured on input coeff.

−1.43

ij

SPAD

SPSD

GRAS

SPAD

SPSD

19.7

19.9

11.2

19.7

21.3

0.24

0.23

1.70

0.24

0.42

a Only

case where GRAS performs best by definition Source Adapted from Oosterhaven (2005)

most economies grow over time. Hence, most ai j /ai0j will be larger than one, and most ln (ai j /ai0j ) will thus be positive. Only in the few empirical cases with some decreasing row and/or column totals, several ln (ai j /ai0j ) may become negative, but that remained unnoticed among the majority of positive values. However, if ai j represents input coefficients, they will be growing and declining equably. Hence, in about half of the cases ln (ai j /ai0j ) will then be negative! This explains the IG* value of −1.43 for GRAS in case of comparing input coefficients. In fact, when updating problems are tackled with constrained optimization solvers, these solvers will try to maximize the negative values in IG* and compensate them at little cost in terms of the goal function by increasing the related positive values. The solution is simple: instead of IG in RAS and instead of IG* in GRAS, one should in both cases use IG** from Table 3.2 that puts the absolute brackets around the whole term.5 With the IG** matrix distance measure used in the last three columns of Table 3.2, the outperformance of GRAS disappears in case of comparing input coefficients, as was to be expected. It, however, remarkably returns if not all 3 × 4 cells are compared with IG** but only the 2 × 3 cells that relate to intermediate and consumption expenditures, i.e. GRAS again outperforms its opponents when the row with net taxes and the column with net exports that contain the negative cells in the test IOT are excluded from the comparison (Oosterhaven 2005). The difference between the two methods that both minimize the distance between the matrices with old and new coefficients is also interesting. Here, minimizing absolute differences clearly outperforms minimizing squared differences. The reason is that minimizing squared differences puts too much weight on reducing the differences in the smaller input coefficients. Termursho et al. (2011) provide the first extensive empirical tests of GRAS and nine other updating methods that all handle negative values. The tests are done for supply and use tables (SUTs, see Sect. 4.2), which have much more zeros and negatives than IOTs, which is why using GRAS instead of RAS is more urgent in case of updating SUTs as opposed to updating IOTs. 5

In fact, replacing IG in (3.1) and IG* in (3.3) by IG** from Table 3.2 should also be done when constrained optimization solvers, from, e.g. GAMS or MATLAB, are used to minimize the information gain in (3.1) and (3.3), respectively. Inspecting the consequences of these changes in the goal function, however, is beyond the scope of this book.

30

3 Updating Different Types of IO Tables

The tests with Dutch SUTs with 59 industries and products show rather small WAPEs for GRAS of 3–9% and 6–18% for five and ten years projections, respectively. The squared differences methods perform extremely poorly with WAPEs as large as 30–60%, whereas the Harthoorn and van Dalen (1987) method and the Kuroda (1988) method have WAPEs about as large as those of GRAS. The five years projections with Spanish SUTs with 73 industries and products show even smaller WAPEs for GRAS, while the EUKLEMS method (Termursho et al. 2011) and the Euro method of Jörg Beutel (Eurostat 2008, Chap. 14) join the poor performance of the square differences methods with WAPEs of 8–20%. Although the WAPEs of GRAS in the above tests seem more or less acceptable, it still is advisable to add at least the easily available superior data to the new row and column sum totals, which will be all the more important in the case of updating interregional and international IOTs and SUTs. Hence, just like RAS, GRAS also needs a generalization that allows for an arbitrary number of additional survey-based restrictions on arbitrary combinations of cells. Temursho et al. (2020) extensively discuss the properties of their multi-regional GRAS method and present its complete analytical solution as well as a simple iterative algorithm for its computation. Moreover, the normalized adjustment multipliers of their algorithm have an economic interpretation and can be used to analyse problems with the convergence of the MR-GRAS algorithm. Such problems may, for example, be due to conflicting restrictions or to zeros in the base year table that need to become positive in the target year table. A limitation to their approach is that the additional restrictions need to be non-overlapping; i.e., each cell of A0 may only be constrained by a single additional restriction. An advantage, on the other hand, is that all restrictions may be non-exhaustive, i.e. leaving some cells unconstrained. This is, especially, important for the row and column totals. Not all of them are needed, which means that checking i u = v i is no longer needed. The presently most flexible matrix adjustment framework KRAS is also a generalization of GRAS (where K stands for Konfliktfreies, Lenzen et al. 2009). Additional to MR-GRAS, KRAS allows for overlapping and conflicting constraints, but additionally also requires numerical indications of the statistical reliability of both the cells of the base year matrix and the cells of the constraints on the target year matrix. These reliability indications are inter alia needed in the complex nonlinear optimizing approach to generate a compromise between conflicting constraints. Not surprisingly, these additional features come at the cost of substantial programming and computational requirements, transparency and the ability to give an economic interpretation to the outcome.

3.3 CRAS: Advantage of Using Time Series of IOTs The literature on updating IOTs has shown that bi-proportional scaling techniques (RAS and GRAS) outperform competing techniques when row and column totals

3.3 CRAS: Advantage of Using Time Series of IOTs

31

are the only available data for the target year, but the errors of such updates are still unacceptably large. The main strategy to cope with that problem has been to add more survey-based constraints on the cells of the target year IOT, which led to the development of multi-proportional scaling techniques (MR-RAS and MR-GRAS). Amazingly, hardly any use has been made of the nowadays easily accessible long time series of IOTs. Only in the early days of updating IOTs, there have been examples of projecting IO coefficients by means of time series of such coefficients, mostly by assuming a bi-proportional structure without knowledge of the future row and column totals (Tilanus 1966; Johanson et al. 1968; Lecomber 1969; Barker 1975). However, this did not lead to the projection of complete IOTs with given row and column totals from time series of IOTs. Moreover, Tilanus (1966) and Barker (1975) honestly reported that their projections were worse than using the most recent IO coefficients, which may be the reason why this approach has not been followed up (Miller and Blair 2022). The early approaches concentrate on improving the estimates of the row and column scalers of the bi-proportional RAS, whereas Mínguez et al. (2009) more recently propose to improve the estimates of the target year IOT at the level of its cells, which is why their method is known as the Cell-corrected RAS method (CRAS). Their point of departure is the logarithmic version of the relative errors of RAS in (3.2):   ln ait j /aibj = ln rit (b) + ln s tj(b) + ln εit j(b)

(3.4)

where t = true value for target year t, b = base year b, rit (b) , s tj(b) = row/column multiplier from the RAS estimate of ait j with aibj . The goal of CRAS is the reduction of the remaining cell-specific errors εit j(b) of RAS. The first stage of CRAS consists of calculating the actual errors εit j(b) for all combinations of survey-based IOTs b and t with the same length of projection period as that between the intended base and target year IOTs, followed by the calculation of the means μiεj and standard deviations σiεj of these errors: εit j(b) = ait j /ait j(b) , ∀i j ⇒ μiεj =

T  bt

εit j(b) / T and

 T

2  t (b) ε σi j = εi j − μiεj / (T − 1), ∀i j

(3.5)

bt

where ait j(b) = estimate of the target IOT t made by applying (3.1) to the base year IOT b, and T = number of comparable pairs of survey-based IOTs (see Mínguez et al. 2009, for details). In the second stage of CRAS, the outcomes of (3.5) are used to correct the projection of IOT t by means of IOT b by minimizing the sum of the squared errors weighted

32

3 Updating Different Types of IO Tables

by the inverse of the standard deviation, subject to the actual margins: Minimize

   2 εi j − μiεj /σiεj , ij

subject to

 j

εi j ait j(b) = u it ,



εi j ait j(b) = s tj and εi j ≥ 0

(3.6)

i

CRAS is finalized by substituting the resulting optimal error corrections εi∗j , ∀i j from (3.6) into (3.4). Note that these error corrections at the cell level for the target year IOT are dependent upon the choice of the base year IOT. In case of using a time series of IOTs, this choice hardly needs discussion: the most recent base year IOT obviously will be the best choice to predict the IOT of the target year. In case of using a cross section of regional or national IOTs to estimate an unknown regional or national IOT the choice is not evident at all (Oosterhaven and Escobedo 2011, see Sect. 4.1.2, for details). The first application of CRAS was a test whether using a time series of IOTs would improve a traditional RAS update. In the case of the Dutch IOTs for 1969–1986, it was shown that CRAS clearly outperforms RAS in making projections of five years or more, which is the most relevant period in practical IO work. This outperformance holds for all five distance measures used, but it holds most for the two squared error measures, and it holds more for the two weighted measures than for their two unweighted cousins, which is not surprising in view of the choice of the goal function in (3.6). This indicates that CRAS especially improves the more important larger percentage errors of RAS as well as the errors in the larger, more important cells. Besides, it was shown the CRAS performs better when larger time series were used, underscoring the very idea of using more survey data. Interestingly, however, when the smaller oil price decline of 1981–1982 had to covered, the outperformance of CRAS became smaller. Moreover, when the sharp oil price decline of 1985–86 had to be covered CRAS no longer outperformed RAS. The IOTs used by CRAS did cover the historic impacts of the large oil price hikes of 1973–74 and 1979–80, whereas RAS did not, but that price increase information was obviously not relevant for improving the RAS update from 1981 to 1986 that needed to predict the impacts of the sharp oil price decline. Hence, the preliminary conclusion seems to be that CRAS clearly outperforms RAS in situations of systematic structural change, but that its superiority reduces considerably when the impacts of unprecedented asymmetric (price) shocks have to be projected (see Mínguez et al. 2009, for further outcomes).

References

33

References Bacharach M (1970) Biproportional matrices and input-output change. Cambridge University Press, Cambridge Barker TS (1975) Some experiments in projecting intermediate demand. In: Allen RIG, Gossling WF (eds) Estimating and projecting input-output coefficients. Input-Output Publishing Company, London Deming WE, Stephan FF (1940) On a least-squares adjustment of a sampled frequency tables when the expected marginal totals are known. An Math Stat 11:427–444 Dietzenbacher E, Miller RE (2009) RAS-ing the transactions or the coefficients: It makes no difference. J Reg Sc 49:555–566 Eurostat (2008) European manual of supply, use and input-output tables, methodologies and working papers. Official publications of the EU, Luxembourg FNEI (1986) Input-output tabellen 1980 voor Groningen, Friesland en Drenthe: Actualisatie methoden en analyse. Federatie van Noordelijke Economische Instituten, Assen Gilchrist DA, St. Louis LV (1999) Completing input-output tables using partial information with an application to Canadian data. Econ Syst Res 11:185–193 Günlük-Senesen ¸ G, Bates JM (1988) Some experiments with methods of adjusting unbalanced data matrices. J Roy Stat Soc A 151: 473–490 Harthoorn J, Dalen J van (1987) On the adjustment of tables with Lagrange multipliers. NA-024, Central Bureau of Statistics, The Hague Jackson RW, Murray AT (2004) Alternative input-output updating formulations. Econ Syst Res 16:135–148 Jaynes ET (1957) Information theory and statistical mechanics. Phys Rev 108:171–180 Johansen L, Alstadheim H, Langsether A (1968) Explorations in long-term projections for the Norwegian Economy. Econ Plann 8:70–117 Junius T, Oosterhaven J (2003) The solution of updating or regionalizing a matrix with both positive and negative entries. Econ Syst Res 15:87–96 Kullback S (1959) Information theory and statistics. Wiley, New York Kuroda M (1988) A methods of estimation for the updating Transaction matrix in the input-output relationships. In: Uno K, Shishido S (eds) Statistical data bank systems, socio-economic database and model building in Japan. North-Holland, Amsterdam Lahr ML, de Mesnard L (2004) Biproportional techniques in input-output analysis: table updating and structural analysis. Econ Syst Res 16:115–134 Lecomber JRC (1969) RAS projections when two or more complete matrices are known. Econ. Plann. 8: 267–78 Lecomber JRC (1975) A critique of methods of adjusting, updating and projecting matrices. In: Allen RIG, Gossling WF (eds) Estimating and projecting input-output coefficients. Input-Output Publishing Company, London Lenzen M, Gallego B, Wood R (2007) Some comments on the GRAS method. Econ Syst Res 19:461–465 Lenzen M, Gallego B, Wood R et al (2009) Matrix balancing under conflicting information. Econ Syst Res 21:23–44 Linden J van der, Dietzenbacher E (1995) The nature of changes in the EU cost structure of production 1965–85: an RAS approach. In Armstrong HW, Vickerman HW (eds) Convergence and divergence among European regions. Pion, London Macgill SM (1977) Theoretical properties of biproportional matrix adjustments. Env Plan A 9:687– 701 Miller RE, Blair PD (2022) Input-output analysis: foundations and extensions, 3rd edn. Cambridge University Press, Cambridge Mínguez R, Oosterhaven J, Escobedo F (2009) Cell-corrected RAS method (CRAS) for updating or regionalizing an input-output matrix. J Reg Sci 49:329–348

34

3 Updating Different Types of IO Tables

Oosterhaven J, Piek JG, Stelder TM (1986) Theory and practice of updating regional versus interregional interindustry tables. Pap Reg Sci Ass 59: 57-72 Oosterhaven J (2005) GRAS versus minimizing absolute and squared differences: a comment. Econ Syst Res 17:327–331 Oosterhaven J, Escobedo F (2011) A new method to estimate input-output tables by means of structural lags, tested on Spanish regions. Pap Reg Sci 60:829–845 Oosterhaven J, Polenske KR, Hewings GJD (2019) Modern regional input-output and impact analysis. In: Capello R, Nijkamp P (eds) Handbook of regional growth and development theories: revised and extended second edition. Edward Elgar, Cheltenham Paelinck J, Waelbroeck J (1963) Etude empirique sur l’-évolution de coefficients “input-output.” Econ Appliqué 16:81–111 Polenske KR (1997) Current uses of the RAS technique: a critical review. In: Simonovits A, Steenge AE (eds) Prices, growth, and cycles. Macmillan Press, London Stone R (1961) Input-output and national accounts. Organization for European Economic Cooperation, Paris Stone R, Brown A (1962) A computable model of economic growth (vol 1 A programme for growth). Chapman and Hall, London Temurshoev U, Webb C, Yamano N (2011) Projection of supply and use tables: methods and their empirical assessment. Econ Syst Res 23:91–123 Temurshoev U, Miller RE, Bouwmeester MC (2013) A note on the GRAS method. Econ Syst Res 25:361–367 Temursho U, Oosterhaven J, Cardenete MA (2020) A multiregional generalized RAS updating technique. Spat Econ an 15:271–286 Theil H (1967) Economics and information theory. North-Holland, Amsterdam Tilanus CB (1966) Input-output experiments: The Netherlands 1948–1961. Rotterdam University Press, Rotterdam UN (2018) Handbook on supply, use and input-output tables with extensions and applications. United Nations Publication, New York Wilson AG (1970) Entropy in urban and regional modelling. Pion, London

Chapter 4

From Regional IO Tables to Interregional SU Models

Abstract An overview of non-survey construction methods for regional input– output tables (RIOTs) reveals a systematic overestimation of regional multipliers. The Cell-corrected RAS method uses the nowadays abundance of easily available survey-based RIOTs to improve the cell estimates of unknown RIOTs. Moreover, semi-survey bi-regional IOTs may be constructed with a double-entry construction method that requires minimal additional survey data about the spatial destination of total sales by regional industry. Rectangular supply–use tables (SUTs) have become prevalent over square IOTs. The possible set-ups of interregional SUTs are explained along with the models that may be based on them. Finally, large differences are shown to exist between the top-down interregional and the bottom-up international construction of SUTs. Keywords Location quotient methods · Cross-hauling · Cell-corrected RAS · Bi-regional input–output tables · Supply–use tables · Interregional supply–use models · International supply–use tables In Chap. 3, we discussed the non-survey temporal projection of IOTs. Here, we start with a discussion of the non-survey spatial projection of IOTs.

4.1 Construction of Regional IO Tables: Towards Non-biased Methods We start with a brief review of non-survey methods to estimate single-region IOTs and show that most of them minimize cross-hauling; i.e., they minimize the simultaneous import and export of comparable products, which leads to a systematic overestimation of all regional multipliers. Next, we discuss a non-survey method (CRAS) that avoids this problem and a semi-survey method (DE-BRIOT) that exploits the The original version of this chapter was revised: The presentation of Tables 4.2 and 4.3 have been corrected. The correction to this chapter is available at https://doi.org/10.1007/978-3-031-050879_11 © The Author(s), under exclusive license to Springer Nature Switzerland AG 2022, corrected publication 2022 J. Oosterhaven, Rethinking Input-Output Analysis, Advances in Spatial Science, https://doi.org/10.1007/978-3-031-05087-9_4

35

36

4 From Regional IO Tables to Interregional SU Models

double-entry strength of a bi-regional IOT with the rest of a country as the second region. Constructing the latter type of IOT is a prerequisite of achieving the cost and reliability advantages of updating bi-regional IOTs as opposed to single-region IOTs discussed in Sect. 3.1.

4.1.1 Most Non-survey Methods Overestimate Intra-regional Multipliers Depending on the amount of already available regional statistical data, additional surveys necessary to supplement the lacking data for the construction of a regional IOT will be expensive or very expensive. This is why the search for non-survey RIOT construction methods started early by Schaffer and Chu (1969). The first generation of non-survey methods all start with the assumption that the unknown regional “technical” coefficients equal the corresponding national coefficients, as in Miller and Blair (2022, Chap. 3). Since most of these methods were developed in the USA for US regions, the rather small US foreign imports of that time were ignored. Doing this, implicitly assumes that not only the regional real technical coefficients, but also the regional foreign import coefficients are equal to their national equivalents. In (4.1), these two basic, but mostly implicit non-survey assumptions are shown, together with the resulting explicitly made typical US assumption (consult Tables 2.1 and 2.2 for the meaning of the symbols):   mn mn n n aˆ i·rj = ai·nj = z i·nj /x nj and mˆ imr ˆ i∗rj = ai∗nj = z i·nj − z imn j = m i j = z i j /x j ⇒ a j /x j

(4.1)

where additionally, ^ = estimate, n = country at hand, m = foreign,· = summation over all regions in the whole world and * = summation over all regions in the country at hand. Multiplication of the coefficients of the last term of (4.1) with total output per regional industry results in the desired estimate of the matrix with intermediate inputs of domestic products aˆ i∗rj x rj = zˆ i∗rj ∈ Z∗r . Equivalent assumptions are usually made to estimate the use of domestic products by local final demand f iq∗r ∈ F∗r .1 The most frequently applied third non-survey assumption uses the well-known location quotient (LQ) to estimate regional self-sufficiency ratios, also known as regional purchase coefficients (RPCs):2 1 If countries only have statistical information on total employment by regional industry, regional industry output may be estimated with: regional industry employment * national industry output per unit of national employment. Using regional employment in this way implies making a strong additional assumption, namely that regional labour productivity by industry equals its national equivalent. 2 Stevens and Trainer (1980), who coined this term, show how RPCs may be estimated econometrically by means of secondary data. Stevens et al. (1989) show that this results in more reliable RPCs than those estimated by (4.2).

4.1 Construction of Regional IO Tables: Towards Non-biased Methods

 mˆ rr ij

=

 r r  xi /x· L Q ri if L Q ri < 1 r , ∀ j, with L Q i = 1 if L Q ri ≥ 1 xin /x·n

37

(4.2)

The LQ indicates whether industry i is regionally over-represented (LQ > 1) or under-represented (LQ < 1) compared to the national industry mix. Note the asymmetry in the use of the LQ in (4.2). The amount of underrepresentation matters, whereas the amount of over-representation does not. If an industry is absent in a region (LQ = 0), it is of course assumed that all inputs are imported, i.e. that the self-sufficiency is zero. If the relative presence of a regional industry becomes larger, it is assumed that the RPCs increase proportionally, while the import ratios decrease proportionally until LQ = 1. However, if the relative presence of a regional industry increases further (LQ > 1), this does not have a further impact on the amount of imports. In that case, imports are always assumed to be zero. Equivalent assumptions are usually again made for local final demand. The core of the problem of using (4.2) is that the LQ represents a reasonable approximation of the net exports of industry i, but not of its gross exports. Consider the equality of total regional supply and total regional demand of product i in region r: xir + m ri = dir + eir ⇒ xir − dir = eir − m ri

(4.3)

wherein dir = total local intermediate and final demand for product i. Note that the production surplus over local demand, xir − dir , and thus also net exports, eir − m ri , is nonlinearly proportional to the LQ if the local demand for product i is proportional to the size of the region.3 In that case, if LQ > 1, (4.2) sets gross imports equal to zero and gross exports equal to net exports. However, in reality gross exports will be larger than net exports, and imports will not be zero because of cross-hauling. A comparable conclusion holds if LQ < 1. In that case, (4.2) sets gross exports equal to zero and gross imports equal to net imports, but both will again be larger because of cross-hauling. Cross-hauling of the same product into—and out of—a region will, especially, be large for developed economies that produce and consume many close substitutes (e.g. different brands of the same product). Minimizing cross-hauling in this way results in a structural underestimation of both imports and exports and a subsequent overestimation of intra-regional transactions and thus of all regional multipliers. Willis (1987), while surveying the literature and adding own results for Staffordshire and Wales, reports average multiplier overestimations of more than 20%, which represents a misleadingly low percentage as it includes the direct effect of 1.0 for which no IO model is needed. With average regional multipliers of about 1.5, a correct error measure that is only defined on the 3

The association between the two measures would be linear if the LQ would be measured in an additive way as ALQri = xir /x·r − xin /x·n instead of the standard multiplicative definition. See Hoen and Oosterhaven (2006), for more reasons to use the additive definition instead of the multiplicative definition.

38

4 From Regional IO Tables to Interregional SU Models

indirect part of the multiplier above 1.0 would report these overestimations to be three times larger. In sum, almost any non-biased estimate of gross imports and gross exports will produce more reliable multipliers than applying the LQ method. Comparable conclusions hold for the purchases only LQ, the cross-industry quotient (CIQ), the semilogarithmic quotient, the supply–demand pool method and the commodity trade balance method, as they all share the asymmetric nature of the LQ method. See Round (1983) and Miller and Blair (2022, Chap. 10) for a further evaluation of the first-generation non-survey regionalization of national IOTs. The second-generation non-survey methods more or less start with Flegg’s et al. (1995) adaptation of the cross-industry quotient (CIQ):

δ    F L Q ri j = λr ∗ C I Q ri j = log2 1 + x·r /x·n ∗ L Q ri /L Q rj

(4.4)

Note that the CIQ equals the ratio of the LQ of the selling industry i over that of the purchasing industry j and is thus different at the cell level of the IOT, whereas the ordinary LQ is uniformly applied across entire rows of the IOT in (4.2). Most importantly, λr adapts the CIQ upwards with the size of the region. Since FLQ replaces LQ in (4.2), it inherits the asymmetric nature of LQ method: resulting in smaller imports when FLQ increases above zero, but not resulting in further smaller imports when FLQ increases above unity. The FLQ thus adds cross-hauling in case of industries that are regionally poorly represented, but it only partially adds crosshauling for industries that are regionally strongly represented. As expected when tested on a Finnish survey-based regional IOT, the FLQ with δ = 0.3 in (4.4) is reported to outperform both the LQ and the CIQ (Thomo 2004). More recently, Kronenberg’s (2009) cross-hauling adjusted regionalization method (CHARM) claims to explicitly take account of cross-hauling qir , which is sensibly assumed to increase with the heterogeneity hi of the products traded. His combined formulas equal:



qir = eir + m ri − eir − m ri = ttir − ntir = h i xir + dir

(4.5)

wherein tt = total trade and nt = net trade. His core assumption is that heterogeneity h i is product specific and invariant to the region at hand. Consequently, he assumes that 0 ≤ h ri = h in ≤ ∞, where h in is measured by means of national IO data. Többen and Kronenberg (2015), however, show that the CHARM formula only allocates international cross-hauling to the regional level, but still assumes interregional cross-hauling to be zero. Not surprisingly, when Flegg et al. (2015) test CHARM against a survey IOT for the Chinese province of Hubei, they find that it still systematically overestimates intra-regional transactions and regional multipliers. Even with an improved CHARM formula, Többen and Kronenberg (2015) come to the same conclusion.4 4 In view of the poor performance of almost all these non-survey methods, we do not pay attention to short-cut multiplier estimation methods that do not even use a non-survey IOT to calculate regional

4.1 Construction of Regional IO Tables: Towards Non-biased Methods

39

The last non-survey regionalization method we discuss is RAS. Remember that in case of updating old IOTs RAS outperforms other non-survey updating techniques, but still requires additional survey-based information to achieve acceptable levels of statistical reliability (see Polenske 1997 and Sect. 3.1 for details). Since RAS is very flexible, it may also be applied in situations wherein given regional margins (i.e. row and column totals) are combined with either a national IOT or an IOT of a different region, from either the same year or a different year. Hewings (1977; Hewings and Janson 1980) experimented with different parent regional IOTs and with different regional margins and concluded that the errors made by choosing the “wrong” parent region were far less serious than the errors made by using the “wrong” margins. Using the “right” margins, however, requires the availability of a survey-based or at least a semi-survey regional IOT, while estimating these margins is precisely the problem that has to be solved. Hence, RAS can only be applied if that problem is solved beforehand.5 In practice, luckily, there will always be all kind of ad hoc, region-specific survey information that should definitely be used to improve the statistical reliability of all the above-discussed non-survey regionalization methods. A systematic way to use this type of superior data is incorporated in the Generation of Regional Input– Output Tables (GRIT) procedure (see West 1990; see also Lahr 1993). Another way to incorporate such data is to use the multi-proportional iterative scaling technique MR-RAS, discussed in Sect. 3.1.

4.1.2 CRAS: Advantage of Using Cross Sections of RIOTs One way to solve the above-mentioned problem of estimating the internal margins of a RIOT for an application of RAS is to use the domestic import and export ratios of a comparable region. Nowadays, with electronic access to all kinds of data, a host of survey-based RIOTs is readily available to estimate the unknown margins of a specific region. But then: why would one only use such a cross section of comparable RIOTs to estimate the unknown margins, why not also use them to improve the RAS estimation of the unknown cells of the target IOT? The latter is exactly what the Cell-corrected RAS method does (CRAS, Mínguez et al. 2009). Section 3.3 summarizes the first test of CRAS on a time series of Dutch national IOTs for 1969–1986 (Mínguez multipliers. See Burford and Katz (1981) for a typical short-cut method, and Jensen and Hewings (1985) for a critical evaluation of a series of such methods. The same holds for the recent surge in non-IO estimations of local multipliers (Moretti 2010; van Dijk 2016). Note, however, that these short-cut methods have one advantage over most non-survey IO methods: they do not have the systematic bias of the asymmetric use of LQs, CIQs and so on. 5 Unfortunately, in the past it has been assumed that these margins were known exactly when applying RAS. As a consequence, it was unjustly reported that RAS outperformed several LQ methods (Czamanski and Malizia 1969; Morrison and Smith 1974; Sawyer and Miller 1983). RAS and LQ methods, however, apply to different data availabilities and may thus not be compared in this way.

40

4 From Regional IO Tables to Interregional SU Models

et al. 2009). Here, we summarize the far more complex second test of CRAS on a cross section of eleven survey-based Spanish regional IOTs by Oosterhaven and Escobedo-Cardeñoso (O&E) (2011). CRAS consists of two stages. In the first stage, for each of the eleven Spanish regions, RAS is used to make ten non-survey projections of its main matrix with the IOTs of the remaining ten regions as base matrices. The eleven target matrices all consist of the combination of the matrix with local intermediate demand, the matrix with local final demand, the column with domestic exports and the row with domestic imports (see Table 2.1). The estimation of their margins needs to simulate a real application of CRAS as closely as possible; i.e., these margins have to be estimated as if the survey-based IOT for the target region is not known. Obeying this restriction, the estimation of most of these margins is possible for all EU regions by means of Eurostat data and the respective national IOTs (see O&E 2011, for details). The margins with total domestic exports and total domestic imports, however, are not estimated as easily. O&E solve this problem by using the domestic import and export propensities of the survey-based IOT of the base region of the RAS estimation at hand. In the second stage, CRAS minimizes the weighted squared errors of the simulated spatial projections with the chosen, varying number of survey-based RIOTs used to improve the RAS projection of the eleventh IOT from the best base IOT (see O&E 2011, for details). In case of the temporal projection of an IOT with the help of a time series of comparable IOTs, the choice of the best base IOT is easy. That is of course the most recent survey-based IOT. Choosing the best base IOT in case of a spatial projection with the help of a cross section of RIOTs, is far more difficult, because space is two-dimensional, whereas time is one-dimensional. Moreover, time is uni-directional (from past to present), whereas space is bi-directional. The problem of choosing the best predictor in the regional case is not specific for CRAS. It also occurs when applying RAS as a non-survey method as discussed in Sect. 4.1.1. O&E solved this problem in their Spanish testing case by choosing the actually first best, second best, third, and so on best predicting base IOT when comparing the performance of RAS with a single best base IOT with that of CRAS with an increasing number of less-best base IOTs. Even when with the—in practice unknown—best predicting base IOT is chosen, RAS results in weighted average percentage errors (WAPEs) at the IO cell level of 20–40%, with outliers of about 50% for the Baleares islands’ region and the Madrid capital region, which both have a sector structure that is incomparable to that of the other nine regions. Galicia in the northwest of Spain, in contrast, is the region that most often appears to be the best predictor, indicating that it has the least peculiar economic structure. When the fourth or fifth best base IOT is chosen for RAS, instead of the first best, the WAPEs of RAS become about two times larger, confirming the conclusion of Polenske (1997) and Chap. 3 that RAS without additional survey-based, so-called superior data produces unacceptably large estimation errors. The performance of the best RAS is compared with the best CRAS with an increasing number of base regions in Table 4.1. Interestingly, CRAS with only the two best base IOTs performs the best for nine of the ten regions, with errors that

4.1 Construction of Regional IO Tables: Towards Non-biased Methods

41

Table 4.1 Normalized performance of best RAS, and best CRAS by number of Spanish regions Spanish Anda Arag Astur Bale Man C. y Vale Galic Madr Nava Vasc region lucia ón ias ares cha Leon ncia ia id rra o Best RASa

1.00

1.00

1.00

1.00

1.00

1.00

1.00

1.00

1.03

1.01

1.04

CRAS 2

0.53

0.40

0.38

0.52

0.37

0.35

0.22

0.45

0.38

0.31

0.39

CRAS 3

0.58

0.51

0.44

0.47

0.65

0.79

0.49

0.54

0.40

0.62

0.50

CRAS 4

0.75

0.77

0.50

0.46

0.60

0.85

0.55

0.73

0.79

0.74

0.63

CRAS 5

0.86

0.78

0.60

0.57

0.74

0.94

0.71

0.99

0.81

0.97

0.84

CRAS 6

1.04

1.02

0.77

0.48

0.76

1.69

0.98

1.88

0.85

0.99

0.92

CRAS 7

1.18

1.00

0.91

0.71

0.93

1.77

1.19

1.92

0.80

2.45

1.26

CRAS 8

1.15

1.17

1.40

0.63

0.88

2.25

1.10

2.48

0.76

3.35

1.43

CRAS 9

1.20

1.35

1.67

0.70

0.91

2.25

1.21

2.39

0.85

5.30

1.96

CRAS 10

1.63

2.25

1.71

0.78

1.64

3.80

1.34

3.02

0.90

5.80

1.99

a

The normalized “best” RAS may not be equal to 1, because it is based on the average of three matrix distance measures that are not always normalized with regard to the same base region Source Oosterhaven and Escobedo-Cardeñoso (2011)

are between 50 and 80% smaller than those of RAS with the single best base IOT (see the dark grey-coloured cells on the second row of Table 4.1). This much better performance of CRAS decreases when more next best base IOTs are added, but CRAS continues to outperform RAS until about the five best base IOTs are used (see the light grey-coloured cells in Table 4.1). Adding more, increasingly dissimilar IOTs leads to under-performance of CRAS compared to RAS with the single best base IOT, except for the Baleares and Madrid where using ten other RIOTs still leads to outperformance by CRAS. A further advantage of CRAS is that selecting the—in practice unknown—best two or three base IOTs from among a larger total of available base IOTs is more likely to happen than selecting the first or second best base region for RAS that only uses a single base IOT. Finally, note that the choice of the base IOT not only determines the size of the errors made in estimating the cell corrections of CRAS, but also the size of the errors made in estimating the margins of the target IOT, which according to Hewings (1977) represents the larger error. Hence, the only way to further improve the quality of the spatial projection of a regional IOT is by adding survey data on these margins, of which the totals for domestic exports and domestic imports are by far the most important margins.

42

4 From Regional IO Tables to Interregional SU Models

4.1.3 DE-BRIOT: Advantage of Constructing Bi-regional IOTs With limited resources, the next obvious question is what kind of additional domestic trade information may most easily be derived from a limited survey among local industries or even among local industry experts only. The international tradition is to ask for regional purchase coefficients (RPCs, i.e. self-sufficiency purchase ratios) and domestic import ratios (see Miller and Blair 2022; Oosterhaven et al. 2019, for overviews). Aside from tradition, this approach may be preferred for theoretical reasons, as it seems to better fit the Leontief model, but it proved already early on to produce unsatisfactory results (Isard and Longford 1971, p. 121). The extensive experience in the construction of semi-survey regional IOTs in the Netherlands led to the conclusion that asking for regional sales coefficients (RSCs, i.e. self-sufficiency sales ratios) and domestic export ratios produced higher response rates and higher quality data (Boomsma and Oosterhaven 1992). The reason is that firms, as a rule, are better informed about the spatial destination of their often few outputs than about the spatial origin of their often many inputs. This is even more the case, when firms sell and purchase through retail and wholesale channels. If they buy from wholesalers and, especially, if they buy from retailers, they have no clue about the primary spatial origin of their inputs, whereas they have a good idea about the final spatial destination of their outputs if they sell through wholesalers. If they sell through retailers, they know the ultimate spatial destination of their sales almost by definition. In fact, one of the easiest questions for firms to answer is: what percentage of your output do you sell within your own region, what percentage in the rest of your country (RoC), and what percentage abroad (RoW), with a pre-filled in total of 100%. Having this type of survey sales data, however, requires a change in the set-up of the rest of the construction of a regional IOT. The most important  that the  change is . construction process also needs a regional domestic sales table Zr n .. Fr n , next to   . the traditionally estimated regional domestic purchases table Znr .. Fnr (see Table

4.2). But with those two tables, it is only one extra step to construct a bi-regional IOT (BRIOT) with RoC as the second region, as shown by bold lined rectangle in Table 4.2. Doing that has the additional advantage that the analyst can benefit from the double-entry (DE) accounting identities of the BRIOT to check and double check all estimates. The full DE-BRIOT approach has the following six construction steps (see Boomsma and Oosterhaven 1992, for details): 1. The  non-survey estimation of the domestic purchases table of region  traditional, . . r, Znr . Fnr , by means of (4.1) and the calculation of the comparable table for RoC as the residual of the national and the regional domestic purchases table (see Table 4.2):

4.1 Construction of Regional IO Tables: Towards Non-biased Methods

43

Table 4.2 Components of the DE-BRIOT approach

Intra- and interregional transactions

To region r

To region s

Sum: to nation

From region r

Zrr

Frr

Zrs

Frs

Zrn

Frn

From region s

Zsr

Fsr

Zss

Fss

Zsn

Fsn

Sum: from nation

Znr

Fnr

Zns

Fns

Znn

Fnn

Legend Z = matrix with intermediate demand, F = matrix with local final demand, r + s = n = nation at hand. The row totals equal domestic sales per industry, i.e. x – e, while the column totals equal the use of domestic inputs, i.e. x − v − m , per industry and per category of local final demand (see also Table 2.1).



. Z .F

ns . ns





. = Z .F

nn . nn



  . nr . nr − Z .F

(4.6)

2. The non-traditional, non-survey estimation of a matrix with domestic sales ratios of region r as the weighted average of the demand structure of the region at   . . r r −1 nr . nr hand, i.e. (ˆx − eˆ ) Z . F , and that of RoC, i.e. (ˆxs − eˆ s )−1 Zns ..Fns , where the survey-based aggregate self-sufficiency sales ratios, i.e. ti·rr = (z rr i· + f i·rr )/(xir − eir ) ∈ trr , ∀i, and the survey-based aggregate domestic export ratios, i.e. ti·r s = (z ri·s + f i·r s )/(xir − eir ) ∈ tr s , ∀i, function as the weights per selling industry. This gives:    

−1 nr .. nr

−1 ns .. ns Sr n = tˆrr xˆ r − eˆ r Z .F + tˆr s xˆ s − eˆ s Z .F , with trr + tr s = i.

(4.7)

3. With these aggregate domestic sales ratios Sr n , the regional domestic sales table is then simply calculated row-by-row with: 



. rn Z .F = xˆ r − eˆ r Sr n rn.

(4.8)

and that of RoC as the residual of the comparable national and regional table (see Table 4.2): 

     . sn . . nn . nn rn. rn Z .F = Z .F − Z .F sn .

(4.9)

44

4 From Regional IO Tables to Interregional SU Models

4. The application of the survey-based domestic intra-regional sales ratios and domestic export ratios to the regional domestic sales table of Step 3 results in the core of the semi-survey IO table for region r, i.e. the intra-regional transactions table, and also in the domestic export table for region r: 

       . rr . . . rr rn. rn rs . rs rs rn. rn ˆ ˆ =t Z .F and Z . F =t Z .F . Z .F rr .

(4.10)

The combination of the above three tables for region r (see the light-shaded part of Table 4.2), in fact, represents a rows only full information RIOT on which one may already base a similarly named IO model (Oosterhaven 1984). It is, however, advisable not to stop the IOT construction process after this step, but to continue with: 5. The estimation of the semi-survey regional domestic import table as the residual of the regional domestic purchases table of Step 1 and the intra-regional transactions table of Step 4:      . . . Zsr .. Fsr = Znr .. Fnr − Zrr .. Frr



(4.11)

This residual domestic import table, together with the residual domestic exports table from Step 4, offer excellent opportunities for extensive checking for inconsistencies and improbabilities at the cell level of the RIOT (see Oosterhaven and Boomsma 1992, for details). The total of the light- and darker-shaded tables in Table 4.1 is called a dogleg full information RIOT (see Bourque and Conway 1977, for a good empirical example). Again, it is not advisable to stop here, but to continue with: 6. The calculation of the intra-regional transactions table for RoC as the final residual:           . . . . . (4.12) Zss .. Fss = Znn .. Fnn − Zrr .. Frr − Zr s .. Fr s − Zsr .. Fsr With this last step, all sub-tables of Table 4.2 are estimated. The resulting BRIOT offers the possibility to simulate not only the intra-regional impacts, but also the interregional spillovers and interregional feedbacks of any change in exogenous final demand, as explained in Sect. 2.3. In the Netherlands, fourteen BRIOTs were constructed simultaneously using a variant of the above DE-BRIOT method (see Eding et al. 1999). Using the column sum accounting identities for the corresponding fourteen bi-regional Type II employment multiplier matrices (see Sect. 5.1), the outcomes of a constrained regression on

4.2 Construction of Interregional Supply–Use Tables and Models

45

the fourteen intra-regional multipliers were used for the estimation, i.e. the spatial interpolation, of the intra-regional multipliers of forty smaller sub-regions. Using the same accounting identities, the subsequent use of distance sensitivities from a gravity model resulted in the spatial interpolation of the aggregate bi-regional spillovers into the disaggregate spatial spillovers between each of the forty subregions (see Oosterhaven 2005, for details). Due to the column sum accounting identities, such a non-survey spatial interpolation of multipliers does not suffer from the usual overestimation of intra-regional multipliers discussed in Sect. 4.1.1. Finally, note that the DE-BRIOT construction method may already be implemented with the minimal survey information on only three sales destination percentages, namely one for intra-regional sales, one for domestic sales to RoC and one for foreign sales to RoW. Collecting additional survey information is welcome, but will be difficult in the traditional industry-by-industry set-up of an IOT, as it would require firms to answer such questions as: to what industries in your own region are you selling your outputs and from what industries are you purchasing your inputs? Far more easy are questions such as what products are you selling to which spatial destinations and what products do you purchase from which spatial origins?

4.2 Construction of Interregional Supply–Use Tables and Models In fact, large-scale national statistical surveys among firms, households and government agencies do not ask for industries of origin or destination, because that is incomprehensible for most respondents, especially for households. Instead, they ask for products sold and products bought. When these types of data are processed into an IO type of table, one gets a so-called supply–use table (SUT). First, we discuss the set-up of a national SUT and the options to derive an IO type model from a national SUT. Next, we discuss ways to estimate the different types of interregional SUTs and which type of model belongs to each of them.

4.2.1 Difficulty of Deriving an IO Model from a Supply–Use Table The structure of a single-nation SUT, or a single-region SUT, for that matter, is shown in Table 4.3. The bold outlined block matrices represent four quadrants that are comparable with the four quadrants of the open economy IOT of Table 2.1. The macro-economic totals, the industry totals and the value added sub-matrices are   .. .. identical to those of Table 2.1. The first difference is that the use table U . F . e , with intermediate demand, u ci ∈ U, domestic final demand, f cq ∈ F, and exports to

46

4 From Regional IO Tables to Interregional SU Models

Table 4.3 Single-region supply–use table

Products Products Industries Imports

Domestic final demand

Exports

Total

uci ∈U

f cq ∈ F

ec ∈e

qc ∈q

sic ∈S

xi ∈x

s mc ∈ s m ′

M

Value added Total

Industries

qc ∈q′

xi ∈x′

Y

fpq

v pi ∈ V C

I

G

E

Legend See Table 2.1 and sic = supply of product c by industry i, smc = import of c, uci = use of c by industry i, f cq = use of c by final demand category q and ec = export of c.

RoW, ec ∈ e, has the use of separate products c on its rows instead of the product mix of a certain industry i. In the early IO literature, products were labelled commodities, and hence, we use c. The second difference is that a SUT contains a new matrix, namely the supply table, containing the supply of product c by domestic industry i, . The first rows of the supply table sic ∈ S, and the supply of c from RoW, smc ∈ sm indicate the product mix of each domestic industry, whereas its last row indicates the product mix of foreign imports. The empty product-by-product square in Table 4.3 should be larger than the empty industry-by-industry square. This reflects that a SUT usually distinguishes more products than industries. Hence, the domestic industry part of both the supply table and the use table is usually rectangular, whereas the domestic industry part of an IO table is always square. The additional product dimension means that a SUT has two accounting identities. The first, old identity states that total industry output equals total industry input xi :

c

sic = xi =

u ci +

c



v pi , ∀i or in matrices: S i = x = i U + i V

(4.13)

p

The second, new identity states that total product supply equals total product demand qc :

i

or in matrices:

sic + smc = qc =

i

u ci +

f cq + ec , ∀c

q

i S + s m = q = (U i + F i + e)

(4.14)

4.2 Construction of Interregional Supply–Use Tables and Models

47

In (4.14), the sum of the output of domestic industries i S and foreign imports s m equals total product supply q , which equals the sum of intermediate demand Ui, domestic final demand Fi and foreign export demand e by product. The most simple way to build a supply–use (SU) model is to fill the shaded subtables with zeros and to then use the first two quadrants of Table 4.2 mathematically in the same way as the first two quadrants in Table 2.1. This leads to the following formulation and solution of the basic supply–use model :             −1   q 0 A q Fi + e q I 0 0 A Fi + e = + ⇒ = − x R 0 x 0 x 0 I R 0 0

(4.15)

in which: A = matrix with assumingly fixed industry technology coefficients, point estimated column-wise by means of aci = u ci xi ∈ A = U xˆ −1 , while R = matrix with assumingly fixed industry market shares in the supply of product c, pointestimated column-wise with ric = sic qc ∈ R = S qˆ −1 . The solution in (4.15) serves to estimate the impact of any change in exogenous final product demand [F i + e] on total supply of product c, on total output of industry i and on any impact variable that may be linked to either one of them, such as foreign imports that may be linked to q, or value added and CO2 emissions that may be linked to x. Additional insight may be gained by decomposing (4.15) in the two separate solutions for total supply and total output, respectively: q = (I − A R)−1 (F i + e) and

(4.16)

x = R (I − A R)−1 (F i + e) = (I − R A)−1 R (F i + e)

(4.17)

Most interesting are the two alternative solutions for total output in (4.17). Note that the two matrices AR and RA in the two Leontief-inverses of (4.17) have different dimensions, namely product-by-product in case of AR and industry-by-industry in case of RA. This implies that the two assumptions underlying these two matrix combinations may be used to construct the standard two types of square, symmetric IO tables. The not yet discussed product-by-product IOT may be estimated by means of AR and the already familiar industry-by-industry IOT may be estimated by means of RA. In most IO impact and scenario studies, the change in exogenous final demand is given by product. With only an i-by-i IOT available, which delivers (I − RA)−1 needed in the last part of (4.17), the analyst has to allocate the exogenous changes in product demand, manually, to the domestic and foreign industries that are most likely to satisfy it. That is, the analyst needs to assemble the information that is contained in the industry market shares matrix R, which is absent in an i-by-i IOT. Many countries, however, only construct p-by-p IOTs and some construct them with AR. With only a p-by-p IOT available, (I − AR)−1 may be calculated to apply in the first part of (4.17). Doing an IO impact or scenario study, however, again,

48

4 From Regional IO Tables to Interregional SU Models

additionally requires the absent industry market shares matrix R. Only now, it is needed to allocate the predicted change in total supply to the domestic industries that supply part of that change in the first part of (4.17). The use of the basic, rectangular SU model (4.15), instead of one of the two, mathematically identical square IO models, provides the solution to both problems, as the industry market shares matrix R is an integral part of the SU model. This straightforward use of (4.15) for impact and scenario studies, however, ignores the problematic nature of assuming fixed industry technology coefficients in U i = A x. Fixing these coefficients implies making the assumption that each industry uses one and same technology for all of its different products, which is known as the industry technology assumption (Model A, Eurostat 2008). The alternative product technology assumption, in contrast, states that each product has unique input requirements irrespective of the industry that produces it (Model B, Eurostat 2008). Kop Jansen and ten Raa (1990) and ten Raa and RuedaCantuche (2003) show that the alternative assumption has superior theoretical properties, as it satisfies four desired axioms, namely material balance, financial balance, price invariance and scale invariance, whereas the industry technology assumption used in (4.15)–(4.17) only satisfies the first axiom. Unfortunately, using this theoretically superior alternative assumption to construct a p-by-p IOT requires the domestic part of the Supply table to be square instead of rectangular, because it requires calculating the inverse of the row ratios of the domestic supply table; i.e., it requires to calculate the inverse of the industry product mix ratios matrix, pic = sic /xi ∈ P = xˆ −1 S (Gigantes 1970). De Mesnard (2004), however, shows that the forward direction of causality of fixed product mix ratios is inconsistent with the backward direction of causality assumed in the demand-driven IO model, whereas it perfectly fits within the forward causality of the implausible supply-driven IO model (see further, Chap. 7). Moreover, using P−1 may lead to inacceptable negative product technology coefficients, which may be solved by a series of ad hoc methods (see Rueda-Cantuche 2017, for an overview). De Mesnard (2011), however, shows that even when no negatives occur among the product technology coefficients, negatives always occur in P−1 , and claims that this makes this approach unacceptable. To further complicate matters: there is not only a second way to construct a pby-p IOT, there is also a second way to construct an i-by-i IOT. The first way is to combine the industry technology assumption with the fixed market share assumption, as done in (4.15)–(4.17) (together labelled as the product sales structure assumption in Eurostat 2008, Model D). The second way is to combine the product technology assumption with the product mix ratios assumption (together labelled as the industry sales structure assumption in Eurostat 2008, Model C). Again, the first combination, used in the first part of (4.17), is shown to satisfy only the first of the above four axions, whereas the second combination satisfies all four axioms (Rueda-Cantuche and ten Raa 2009). Again, unfortunately, using fixed product mix ratios is in contradiction with the direction of causality of the Leontief model, while it again requires a square domestic supply table to be able to invert the matrix with product mix ratios P, which

4.2 Construction of Interregional Supply–Use Tables and Models

49

again delivers negatives in each row and column of P−1 , which de Mesnard (2011) again claims to be unacceptable. After an extensive weighing of advantages and disadvantages, Eurostat favours industry-by-industry IO tables based on fixed industry technology ratios and fixed industry market share ratios as in (4.15)–(4.17) (i.e. Model D, see Eurostat 2008, p. 310). Consult Miller and Blair (2022, Chap. 5) for an overview of the discussion with numerical examples as well as a presentation of the mixed technology assumption not discussed here. In practice, most countries nowadays construct SUTs far more regularly than symmetric IO tables, which may be the best reason to favour the basic SU model (4.15)–(4.17), as it may be based on more recent data than the IO model of Chap. 2.

4.2.2 Family of Interregional Supply–Use Tables and Models The obvious next question is how to regionalize the national SUT of Table 4.3. Oosterhaven (1984) discusses a whole family of regionalized national SUTs. Here, we only discuss the last phase of adding trade data to a national SUT that is already regionalized once. Adding trade data is required to obtain an interregional SU model. Being “already regionalized once” means that the national Supply table Sn· is already regionalized along its rows into Sr· + Ss· + . . ., while the national use table U·n is already regionalized along its columns into U·r + U·s + . . . (see Table 4.4, where the relevant core of the national SUT is indicated with double lines). The smallest amount of trade data is required in case of constructing a multiregional SUT (i.e. Table 4.4 sub A). In addition to the one-sided split-up of the national supply table and the national use table, a multi-regional SUT only requires aggregate bilateral trade data tcr s , which also represents the minimum trade data necessary to apply the DE-BRIOT approach in Sect. 4.1.3: tcr s =

i

rs sic =

i

u rcis +

rs f cq ∈ tr s ∀ r, s

(4.18)

q

The sales only interregional SUT (i.e. Table 4.4 sub B) requires the next smallest amount of additional trade data. On top of the data for Table A, it requires that all regional supply tables are regionalized a second time; i.e., the data in the second term in (4.18) need to be estimated for all combinations of r and s. If all industries in the same region have the same intra-regional sales ratios and the same export ratios for their supply of product c, this seconds split up simply requires calculating rs r· = (tcr s /tcr · ) sic , ∀i. sic The purchases only interregional SUT (i.e. Table 4.4 sub C, called “useregionalized” by Jackson and Schwarm 2011), instead, requires that all regional use tables are regionalized a second time; i.e., the data in the third term of (4.18) need to be estimated for all combinations of r and s. This requires much more effort, as intermediate demand inputs, in general, have trade origins that are different from

50

4 From Regional IO Tables to Interregional SU Models

Table 4.4 Four regionalized national SUTs that each have an interregional SU model A. Multi-regional SUT

B. Sales only interregional SUT

(trr )´

(trs )´

Sr

Srr

+

Srs

=

Sr

(tsr )´

(tss )´

Ss

Ssr

+

Sss

=

Ss

+

U

U

r

U

+

s

=

U

n

U

r

s

Sn

Urs

+

+

Sr

Usr

Uss

+

=

=

Ss

U

r

U

=

Sn

n

U

n

D. Full information interregional SUT

Urr

s

U

Sn

C. Purchases only interregional SUT

Urr

=

Srr

Urs Srs

+

=

Sr

Uss

+

Sss

=

Ss

+

U

Usr Ssr U

n

+

U

r

s

=

Sn

r s ∈ Sr s = sales of c by industry Legend t·cr s ∈ (tr s ) = total trade of product c from region r to s, sic

i in region r to s, u rcqs ∈ Ur s = use of c from r by producers and local final demand q in s. The national supply table Sn· is exclusive of foreign imports and the national use table U·n is exclusive of foreign exports Source Adapted from Oosterhaven (1984)

those of final demand inputs. However, if all industries and all final demand categories have the same intra-regional purchase ratios and the same import ratios, this rs ·s = (tcr s /tc·s ) f cq , ∀q for the local final second split up simply requires to calculate f cq use of product c (analogous for intermediate use). The FI interregional SUT (i.e. Table 4.4 sub D), at last, requires an estimate of all trade flows shown in (4.18). It is interesting to compare Table 4.4 with Table 4.2 and note that the first three bi-regional SUTs offer zero double-entry error checking possibilities, which is why they offer no data improvement opportunities, whereas bi-regional IOTs do. Only Table 4.4 sub D offers a double-entry checking possibility, not at the cell level as in Table 4.2, but at the level of the column sums and row sums

4.2 Construction of Interregional Supply–Use Tables and Models

51

of its doubly regionalized supply and use tables, respectively, as Table 4.4 sub D has to satisfy: i Sr s = (Ur s i) , ∀r, s. All four multi-region SUTs of Table 4.4 would suffice to build a multi-region version of (4.15). Here, we only present the solution of the two extremes with regard to data requirements (see Oosterhaven 1984, for the remaining two models). Without loss of generality as regards the number of regions, the multi-regional SU model reads as follows:  rr r s  ·r r   ·r   r   r   r· ˆ m ˆ R 0 m A x F i e x = + (4.19) + s ˆ sr m ˆ ss xs A·s xs e 0 Rs· F·s i m ˆ = diagonal matrix with trade origin where R = matrix with industry market shares, m ratios in total supply q and A = matrix with technical coefficients; all coefficients may be point-estimated directly from the data in Table 4.4 sub A. Note the rank order of the brackets used. Foreign exports er are already specified by region of production. Hence, foreign exports only need to be pre-multiplied with the regional industry market shares Rr· in order to determine which regional industry produces these exports. Intermediate and local final demands (Ax + Fi), on the other hand, are specified as the demand for products from anywhere and thus have to be ˆ to determine which regions satisfy pre-multiplied first with the trade origin ratios m this demand, before they are pre-multiplied with the regional industry market shares Rr · to determine which industries in those regions will produce these products. The solution of (4.19) is not shown as it is straightforward: transfer the term with x from the RHS to the LHS of (4.19) and pre-multiply both sides with the appropriate inverse. Again without loss of generality as regards the number of regions, the FI interregional SU model reads as follows: 

xr xs



 =

Rrr Arr Rr s Ar s Rsr Asr Rss Ass



xr xs



 +

Rrr Frr Rr s Fr s Rsr Fsr Rss Fss

   rr r  i R e + i Rss es

(4.20)

All coefficient matrices of (4.20) may be point-estimated directly from the data in Table 4.4 sub D. Note that the difference in treatment of foreign exports and regional demand in (4.19) disappears in (4.20), as information on the regions that satisfy the intermediate and local final demand is now available at the product level. As a consequence, (4.20) has become far simpler than (4.19), while its solution is even more straightforward. The amount of data required, however, has grown significantly, as is also evident from comparing Table 4.4 sub A with sub D. When exogenous final demand (Fi + e) of the base year is used, both models (4.19) and (4.20) will produce the same endogenous values for total industry output x and total product supply q. In case of a non-proportional change in exogenous final demand, however, both models will produce different endogenous outcomes. Such differences may be viewed as representing the aggregation error that is made by

52

4 From Regional IO Tables to Interregional SU Models

using the multi-regional SU model instead of the FI interregional SU model. Since FI interregional SU tables have not been constructed yet, it remains unknown how serious this aggregation error will be. The same holds for the FI international SU model with multiple countries.

4.3 Difference Between Constructing Interregional and International SUTs Although the mathematics and economics are exactly similar, the construction of international SUTs, noteworthy, requires a strategy that is quite different from the construction of within-country interregional SUTs. The construction of interregional SUTs essentially follows a top-down process. It starts with the one-sided regionalisation of a national supply table and a national use table and continues with adding trade data as discussed above.6 In most cases, aggregate survey or non-survey trade origin ratios are subsequently applied to the rows of the one-sidedly regionalized national use table resulting in a first estimate of an use-regionalized national SUT (i.e. Table 4.4 sub C). Next, iterative bi-proportional scaling of this first estimate by means of RAS is used to achieve consistency with the accounting identities. As a consequence of using RAS, each cell in the resulting table ends up with having its own different trade origin ratio. It is important to understand that this type of RAS-caused differentiation of trade ratios along the rows the resulting MRSUT does not have an empirical foundation. The resulting table, consequently, still suffers from the type of aggregation error that a survey-based MRIOT contains compared to a survey-based IRIOT, as discussed in Sect. 2.3. Unfortunately, most countries do not even have aggregate interregional trade data. Instead, they usually have transport data, mostly by product group and single transport mode, ignoring multi-modal transport chains, while they use a different product classification with a mostly large, miscellaneous category, mainly covering container transport. Moreover, most transport data are primarily measured in tonnes and only partially or not at all in monetary values. Integrating such quite different physical transport data into the monetary data and product classifications of a use-regionalized national SUT is rather complex and is best done by using an information gain minimizing or an entropy maximizing model (see Többen 2017b, for a very fine, recent example). Since a World SUT is not available, the construction of international SUTs, in contrast to that of interregional SUTs, follows an entirely bottom-up process. The two main data sources for this bottom-up construction process are national SUTs and national foreign trade data. The main international construction problem is not the lack of 6 See Többen (2017a) for a state of the art construction of a Purchases only interregional SUT, using the KRAS algorithm of Lenzen et al. (2009). See Madsen and Jensen-Butler (1999) for a general discussion on constructing interregional SUTs, and the actual construction of a Danish interregional SUT that has a more bottom-up character, due to the abundance of micro data in case of Denmark.

4.3 Difference Between Constructing Interregional …

53

trade data but the abundance of conflicting trade data, as for each international trade flow tcr s there are at least four values, namely the SUT and the export statistics of the origin country r, and the SUT and the import statistics of the destination country s. The differences between the four values may be as large as 10–30%, with a maximum of more than 200% for beverages, due to excise taxes (van der Linden and Oosterhaven 1995). Aside from measurement errors and transport chain timing differences, these flows mainly differ rather systematically because they are measured in different prices, as shown in Table 4.5. Table 4.5 also shows a second problem, namely, that most countries measure the values of the use table in purchasers’ prices, while the supply table is measured in basic prices. For modelling purposes, however, it is essential that the use table is also measured in basic prices, as only this valuation enables allocating economic and environmental impacts to the industries that actually produce them. Using purchasers’ prices would result in allocating impacts on taxes, subsidies, and trade and transport margins unjustly to the industries that produce the goods and services that carry these payments, instead of to the government that receives the taxes/subsidies and to the industries that produce the trade/transport margins. Van der Linden and Oosterhaven (1995) solved this problem probably for the first time. In case of the construction of their EU intercountry IOTs for 1965–1985, they applied RAS to an initial estimate of the off-diagonal blocks of their IRIOTs measured in purchasers’ prices, while using the exports to the rest of the EU measured in producers’ prices as row totals. Oosterhaven et al. (2008) use the same method with a predecessor of MR-GRAS (Stelder and Oosterhaven 2009) when estimating the ten country IRIOT for the Asian-Pacific region for 2000 by means of four non-survey construction methods. When comparing their outcomes with the semi-survey IRIOT of IDE (2006), they find errors in the disaggregate trade flows that are considerable, but decrease substantially when more and more detailed survey trade data are added. Table 4.5 Valuation of international commodity flows Country

Valuation layers

Statistical source

R

Basic price in R

Supply table

R

+ Taxes and subsidies

Supply table

R

+ Trade and transport margins Supply table

R

= Purchaser price in R

International(2x)

+ Trade and transport margins

S

= Basic price in S

Supply table and import stats (c.i.f. price)

S

+ Taxes and subsidies

Supply table

S

+ Trade and transport margins Supply table

Use table and export stats (f.o.b. price)

S

= Purchaser price in S

International

+ Trade and transport margins … and so on

Source Adapted from Bouwmeester (2014)

Use table and export stats (f.o.b. price)

54

4 From Regional IO Tables to Interregional SU Models

Bouwmeester (2014) describes in great detail how this trade pricing problem, along with several other problems, is solved in case of the EU Exiopol project (Tukker et al. 2013). In case of the EU WIOD project, Dietzenbacher et al. (2013) also favour to reprice the regionalized use table. A more demanding solution is to quantify the valuation layers specified in Table 4.5 at the cell level of either the regionalized use table or the regionalized supply table, as is done in the Eora project (Lenzen et al. 2013). See the special issues of Economic Systems Research of March 2013 and September 2014 for other international SUT databases and for several analyses of their methodological and empirical differences.

References Boomsma P, Oosterhaven J (1992) A double-entry method for the construction of bi-regional inputoutput tables. J Reg Sci 32:269–284 Bourque PJ, Conway RS (1977) The 1972 Washington input-output study. Graduate School of Business Administration, Seattle Bouwmeester MC (2014) Economics and environment—modelling global linkages. Dissertation, SOM Research School, University of Groningen Burford RL, Katz JL (1981) A method for estimation of input-output-type output multipliers when no I-O model exists. J Reg Sci 21:151–1621 Czamanski S, Malizia E (1969) Applicability and limitations in the use of national input-output tables for regional studies. Pap Reg Sci 23:65–78 de Mesnard L (2004) Understanding the shortcomings of commodity-based technology in inputoutput models: an economic circuit approach. J Reg Sci 44:125–141 de Mesnard L (2011) Negatives in symmetric input–output tables: the impossible quest for the Holy Grail. Ann Reg Sci 46:427–454 Dietzenbacher E, Los B, Stehrer R, Timmer M, de Vries G (2013) The construction of world input-output tables in the WIOD project. Econ Syst Res 25:71–98 Eding GJ, Oosterhaven J, de Vet B, Nijmeijer H (1999) Constructing regional supply and use tables: Dutch experiences. In: Hewings GJD, Sonis M, Madden M, Kimura Y (eds) Understanding and interpreting economic structure. Springer Verlag, Berlin Eurostat (2008) Eurostat manual on supply, use and input-output tables. European Communities, Luxemburg Flegg AT, Webber CB, Elliot MV (1995) On the appropriate use of location quotients in generating regional input-output tables. Reg Stud 29:547–561 Flegg AT, Huang Y, Tohmo T (2015) Using charm to adjust for cross-hauling: the case of the province of Hubei, China. Econ Syst Res 27:391–413 Gigantes T (1970) The representation of technology in input-output systems. In: Carter AP, Bródy A (eds) Contributions to input-output analysis. North-Holland, Amsterdam/London Hewings GJD (1977) Evaluating the possibilities for exchanging regional input-output coefficients. Environ Plan A 9:927–944 Hewings GJD, Janson BN (1980) Exchanging regional input-output coefficients: a reply and further comments. Environ Plan A 12:843–854 Hoen AR, Oosterhaven J (2006) On the measurement of comparative advantage. Ann Reg Sci 40:677–691 IDE (2006) How to make Asian input-output tables. Institute of Developing Economies, JETRO, Chiba Isard W, Langford TW (1971) Regional input-output study: recollections, reflections and diverse notes on the Philadelphia experience. M.I.T Press, Cambridge (MA)

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Jackson RW, Schwarm WR (2011) Accounting foundations for interregional commodity-byindustry input-output models. Lett Spat Resour Sci 4:187–196 Jensen RC, Hewings GJD (1985) Shortcut ‘input-output’ multipliers: a requiem. Environ Plan A 17:747–759 Junius T, Oosterhaven J (2003) The solution of updating or regionalizing a matrix with both positive and negative entries. Econ Syst Res 15:87–96 Kop Jansen P, ten Raa T (1990) The choice of model in the construction of input-output coefficients matrices. Int Econ Rev 31:31–45 Kronenberg T (2009) Construction of regional input-output tables using nonsurvey methods: the role of cross-hauling. Int Reg Sci Rev 32:40–64 Lahr ML (1993) A review of literature supporting the hybrid approach to constructing regional input-output models. Econ Syst Res 5:277–293 Lenzen M, Gallego B, Wood R (2009) Matrix balancing under conflicting information. Econ Syst Res 21:23–44 Lenzen M, Moran D, Kanemoto K, Geschke A (2013) Building EORA: a global multi-region input-output database at high country and sector resolution. Econ Syst Res 25:20–49 Madsen B, Jensen-Butler C (1999) Make and use approaches to regional and interregional accounts and models. Econ Syst Res 11:277–299 Miller RE, Blair PD (2022) Input-output analysis: foundations and extensions, 3rd edn. Cambridge University Press, Cambridge Mínguez R, Oosterhaven J, Escobedo F (2009) Cell-corrected RAS method (CRAS) for updating or regionalizing an input-output matrix. J Reg Sci 49:329–348 Moretti E (2010) Local multipliers. Am Econ Rev 100:373–377 Morrison WI, Smith P (1974) Nonsurvey input-output techniques at the small area level: an evaluation. J Reg Sci 14:1–14 Oosterhaven J (1984) A family of square and rectangular interregional input-output tables and models. Reg Sci Urban Econ 14:565–582 Oosterhaven J (2005) Spatial interpolation and disaggregation of multipliers. Geogr Anal 37:69–84 Oosterhaven J, Stelder D, Inomata S (2008) Estimating international interindustry linkages: nonsurvey simulations of the Asian-Pacific economy. Econ Syst Res 20:395–414 Oosterhaven J, Polenske KR, Hewings GJD (2019) Modern regional input-output and impact analysis. In: Capello R, Nijkamp P (eds) Handbook of regional growth and development theories: revised and extended second edition. Edward Elgar, Cheltenham Oosterhaven J, Escobedo-Cardeñoso F (2011) A new method to estimate input-output tables by means of structural lags, tested on Spanish regions. Pap Reg Sci 60:829–845 Polenske KR (1997) Current uses of the RAS technique: a critical review. In: Simonovits A, Steenge AE (eds) Prices, growth, and cycles. Macmillan Press, London Round JI (1983) Non-survey techniques: a critical review of the theory and the evidence. Int Reg Sci Rev 8:189–212 Rueda-Cantuche JM (2017) The construction of input-output coefficients. In: ten Raa T (ed) Handbook of input-output analysis. Edward Elgar, Cheltenham Rueda-Cantuche JM, ten Raa T (2009) The choice of model in the construction of industry inputoutput coefficient matrices. Econ Syst Res 21:363–376 Sawyer CH, Miller RE (1983) Experiments in the regionalization of national input-output table. Environ Plan A 15:1501–1520 Schaffer W, Chu K (1969) Nonsurvey techniques for constructing regional interindustry models. Pap Reg Sci 23:83–104 Stelder D, Oosterhaven J (2009) Non-survey international IO construction methods: a generalised RAS algorithm GRAS4. In Kuwamori H, Uchida Y, Inomata S (eds) Compilation and use of the 2005 international input-output tables. Institute of Developing Economies, JETRO, Chiba Stevens BH, Trainer GA (1980) Error generation in regional input-output analysis and its implications for nonsurvey models. In: Pleeter SP (ed) Economic impact analysis: methodology and applications. Martinus Nijhoff, Boston

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Stevens BH, Treyz GI, Lahr ML (1989) On the comparative accuracy of RPC estimation techniques. In: Miller RE, Polenske KR, Rose AZ (eds) Frontiers of input-output analysis. Oxford University Press, New York ten Raa T, Rueda-Cantuche JM (2003) The construction of input-output coefficient matrices in an axiomatic context: some further considerations. Econ Syst Res 14:439–455 Thomo T (2004) New developments in the use of location quotients to estimate regional input-output coefficients and multipliers. Reg Stud 38:43–54 Többen J (2017a) Effects of energy- and climate policy in Germany: a multiregional analysis. Dissertation, SOM research school, University of Groningen Többen J (2017b) On the simultaneous estimation of physical and monetary commodity flows. Econ Syst Res 29:1–24 Többen J, Kronenberg TH (2015) Construction of multi-regional input–output tables using the charm method. Econ Syst Res 27:487–507 Tukker A, De Koning A, Wood R, Hawkins T, Lutter S, Acosta J, Rueda-Cantuche JM, Bouwmeester MC, Oosterhaven J, Drosdowski T, Kuenen J (2013) Exiopol—development and illustrative analyses of a detailed global MR EE SUT/IOT. Econ Syst Res 25:50–70 van Dijk JJ (2016) Local employment multipliers in U.S. cities. J Econ Geogr 17:465–487 van der Linden JA, Oosterhaven J (1995) European community intercountry input-output relations: construction method and main results for 1965–1985. Econ Syst Res 7:249–269 West GR (1990) Regional trade estimation: a hybrid approach. Int Reg Sci Rev 13:103–118 Willis KG (1987) Spatially disaggregated input-output tables: an evaluation and comparison of survey and non-survey results. Environ Plan A 19:107–116

Chapter 5

From Basic IO and SU Models to Demo-economic Models

Abstract A social accounting matrix (SAM) is shown to represent the ideal data set to endogenize household consumption, as it contains a full description of the generation, redistribution and spending of income. The Type II multipliers and Type II spillovers of an interregional SAM model are both larger than those of the standard, Type I model, whereas exogenous final demand is smaller. They are shown to represent an upper limit for the true multipliers. Type III multipliers are smaller, as income growth of existing jobs needs to be multiplied with smaller marginal instead of average consumption/output ratios. Type IV multipliers are even smaller, as they include the feedback of employment growth on unemployment benefits. Endogenizing remaining final demand leads to ever larger, less plausible multipliers. Keywords Endogenous consumption · Social accounting matrices · Type II input–output model · Demo-economic models · Vacancy chains · Type IV multipliers · Infinite multipliers · Net multipliers Chapters 2 and 4 end with interregional input–output (IO) and interregional supply– use (SU) models, respectively. Extending single-region models into interregional models by endogenizing the export of intermediate outputs is shown to be methodologically interesting, but empirically—for most industries—it is much less important than endogenizing household expenditures. In this chapter, we explain the intricacies of endogenizing household consumption and conclude with a discussion of how far the analyst should go with endogenizing more and more components of exogenous final demand.

5.1 Interregional Models with Endogenous Household Consumption Figure 5.1 shows the causal nature of endogenizing that part of household consumption that may be tied directly to the size and growth of value added by industry. The solid arrows and boxes reproduce the causal structure of the interregional IO model shown in Fig. 2.2. The dotted separations in the top two boxes show which © The Author(s), under exclusive license to Springer Nature Switzerland AG 2022 J. Oosterhaven, Rethinking Input-Output Analysis, Advances in Spatial Science, https://doi.org/10.1007/978-3-031-05087-9_5

57

58

5 From Basic IO and SU Models to Demo-economic Models

f r = f r,ex +

f s = f s,ex +

hrr + hrs

hss + hsr

Zrs rr

Z

x

r

xs

Zss

Vs

hss

sr

Z hrr

Vr

hrs hsr

Fig. 5.1 Causal structure of the endogenous consumption extension. Legend See Fig. 2.2. In addition: f r,ex = vector with remaining exogenous final demand of the Type II interregional IO model, hrs = vector with the endogenous delivery of consumption goods and services by industries in region r to households in region s

part of exogenous final demand of the basic (Type I) interregional IO model remains exogenous (f r,ex and f s,ex ) and which part is made endogenous, i.e. hrr and hsr in case of region r, and hss and hsr in case of region s, where hri s ∈ hr s = endogenous consumption of products from industry i in region r by households in region s. In the Type I single-region IO model, only the consumption expenditures on regionally produced goods and services out of regionally earned value added may be made endogenous, as shown for region r by the dotted arrows entering and leaving the dotted box with hrr . For region r, this results in the so-called Type II single-region IO model, which has larger multipliers than the Type I model, as the additional causal loop (Vr ⇒ hrr ⇒ xr ) shown in the bottom-left part of Fig. 5.1 leads to additional impacts on regional total output, called intra-regional induced effects. As was the case with the interregional model extension, the larger multipliers of the singleregion model extension, when multiplied with its smaller base-year exogenous final demand, will exactly reproduce the base-year values for total output. Reality does not change, only the way in which it is modelled changes! In the Type II interregional IO model, not only the intra-regional consumption expenditures hrr and hss but also the interregional export of consumption goods and services that can directly be tied to the value added of other regions is made endogenous (i.e. hrs and hsr ). The result is that not only the intra-regional multipliers will be larger, but also the interregional spillover and feedback effects. These so-called induced spillovers and feedbacks of the Type II interregional model will enhance the indirect spillovers and feedbacks of the Type I interregional IO model, as illustrated by the dotted boxes and arrows in the middle part of Fig. 5.1, which provide a second connection back and forth between the two single-region Type II models on the LHS and the RHS of Fig. 5.1. How, exactly, can household consumption expenditures be endogenized? The columns with household consumption in an IOT (see Table 2.1) or in a SUT (see Table 4.3) usually relate to the consumption of all households living in the region or nation at hand. Only part of these expenditures will directly depend on the size and growth of the own value added of the region, and then especially on the size

5.1 Interregional Models with Endogenous Household Consumption

59

and growth of the labour income part of that value added, as assumed in Fig. 5.1. A large part of household consumption only indirectly depends on the value added of the own region and that of other regions via the redistribution of value added through interregional labour income flows (mainly via commuting), through interregional capital income flows (mainly via interregionally operating firms) and through central and local governments’ taxation and social security schemes. The empirical specification of the regional and interregional redistribution of value added requires the information that is available in interregional social accounting matrices (SAMs, see Pyatt & Thorbecke 1976; Pyatt & Round 1977, for the original concept). The fundamental difference between IOTs and SUTs, on the one hand, and SAMs, on the other hand, is that for each and every SAM all accounts balance, i.e. all successive row totals (=incoming payments) and column totals (=outgoing payments) in a SAM are equal. Besides balancing industry and product accounts as in a SUT, SAMs usually also have disaggregated, balancing household accounts, where all income sources and all expenditures are specified per type of household. Depending on the data, households may be disaggregated in many ways, but often they are disaggregated by income decile. When all household accounts balance, all household expenditures may be endogenized in exactly the same way as the commodity account in the basic SU model in (4.15). In that way, a SAM model may directly be derived from a SAM table. Table 5.1 shows a simplified single-region or single-nation SAM with a disaggregated household account and an aggregated account for all other institutions (i.e. the total of the government, capital and RoW accounts). In the SAM of Table 5.1, additional to the accounting identities by industry (4.13) and by product (4.14), total household expenditures equal total household receipts for each household type q:  c

h cq + tq = h qt =



wqi + yqex ∈

i H + t = (ht ) = (Wi + yex )

(5.1)

i

where hcq = expenditures on product c from everywhere by households of type q, t q = payments of households q to other accounts (mainly taxes, social security payments and savings, but not imports that are accounted for by smc in the supply table), hqt = total income (= total expenditures) of households q, wqi = incomes paid to households q by industry i, yqex = other incomes of households q (mainly social security benefits, domestic capital incomes, pensions and incomes from RoW). The SAM model that may directly be based on the data of Table 5.1 contains only three simple assumptions: 1. 2. 3.

The expenditures e, yex and T of the “other accounts” are exogenous. Any change in e and yex leads to an equally large change in the total receipts q and h of the endogenous product and household accounts, respectively. Any change in the total receipts of any endogenous account leads to proportional changes in all the expenditures of that account.

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5 From Basic IO and SU Models to Demo-economic Models

Table 5.1 Simplified single-region social accounting matrix

Legend See Table 4.3, and hcq = expenditure on product c by households of type q, ec = expenditure on c by other accounts, wqi = payments to q by industry i, yex = income from other accounts, hqt = total income/payments of q, oi = other value added of i, t q = payments by q to other accounts, T = transactions between other accounts, E = total receipts/payments of other accounts

In other words: all expenditures in the heavily outlined part of the SAM in Table 5.1 are determined by the size of their column totals and assumingly fixed expenditure coefficients. Note that this also makes the sm  , o and t receipts of the “other accounts” endogenous. With only a single available SAM, the model coefficients may be point-estimated by dividing all endogenous expenditures by their account’s base-year column total. This delivers the technical coefficient matrix A and the market share matrix R, already known from the basic SUT model (4.15), as well as the new household income shares in total industry input (i.e. cqi = wqi /xi ∈ C = W xˆ −1 ) and the new product shares in total household expenditures (or consumption package coefficients, 

−1

i.e. kcq = h cq / h qt ∈ K = Hht ). Note that the column sums of all expenditure coefficients adds to unity, as is the case with the column sums of the input coefficients of the IO and the SU model. Summarized by means of block matrices and stacked columns, as in (4.15), the SAM model directly follows from Table 5.1. Its subsequently solution is straight forward. The vector with the endogenous column totals q x ht is transferred from the RHS to the LHS of the first part of (5.2), and the result is pre-multiplied with the extended Leontief-inverse, i.e.

5.1 Interregional Models with Endogenous Household Consumption

⎤ ⎡ ⎤ ⎤⎡ ⎤ ⎡ q 0 AK e q ⎣ x ⎦ = ⎣ R 0 0 ⎦⎣ x ⎦ + ⎣ 0 ⎦ ⇒ ht 0 C 0 yex ht ⎡ ⎤ ⎡ ⎤ ⎡ ⎤⎡ ⎤ ⎤−1 ⎡ q I −A −K e LCC LC I LC Q e ⎣ x ⎦ = ⎣ −R I 0 ⎦ ⎣ 0 ⎦ = ⎣ L I C L I I L I Q ⎦⎣ 0 ⎦ L QC L Q I L Q Q ht 0 −C I yex yex

61

⎡

(5.2)

where the subscripts in the extended Leontief-inverse indicate the dimensions of the corresponding sub-matrices. LCC , for instance, indicates the impact of changes in exogenous demand for products e on the total supply of products q, while L I Q indicates the impact of changes in exogenous household income per type of household yex on total output per industry x. Pre-multiplication of the extended Leontief-inverse with a row vector with, for instance, employment coefficients [ 0 l I l Q ] delivers the ordinary employment multipliers indicating the impact of changes in the exogenous variables. lQ L QC , for instance, gives the total employment impact within the household sector, measured in say full-time equivalents (FTEs), due to changes in the exogenous demand for products e, measured in say euros. These multipliers will not be constant, but will decrease over time due to nominal increases in labour productivity. Post-multiplication of ordinary impact multipliers with the inverse of the impact coefficients, subsequently, delivers the dimensionless normalized impact multipliers, which have the advantage to be relatively constant. In case of employment impacts, lI L I C (ˆlC )−1 , for instance, gives the total increase of FTEs in all industries due to the increase in exogenous product demand that may be produced by one FTE. The decomposing the solution of (5.2) gives additional insight into the working of this most basic SAM model: q = (I − AR − KCR)−1 (Kyex + e).

(5.3a)

x = Rq

(5.3b)

ht = Cx + yex

(5.3c)

In (5.3a), changes in exogenous household income yex and in exogenous final demand e lead through the regional multiplier mechanism (I – AR – KCR)−1 to changes in endogenous product demand q, which leads in (5.3b) to changes in endogenous industry output x, which leads in (5.3c) to a further, now endogenous change in total household income ht . The overall multiplier effect of (I – AR – KCR)−1 consists of the sum of the direct I, the indirect AR and the induced KCR effect of a change in exogenous demand on all of the three sets of endogenous variables. When a single-nation version of Table 5.1 is doubly disaggregated by region, it directly provides the empirical basis for the coefficients of an interregional SAM

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5 From Basic IO and SU Models to Demo-economic Models

model. However, in the interregional case there are at least four ways in which a national SAM may be disaggregated by region, similar to the split up of a national SUT into four different interregional SUTs shown in Table 4.4 (see Madsen and Jensen-Butler 2005, for more detailed spatial SAMs). Here, we only discuss the multi-regional SAM, as it requires the least amount of additional data. The multi-regional SAM model that may be derived directly from these data reads as follows (see Eq. 4.19 for the comparable multi-regional SUT model):

xr xs





 Rr · 0 = 0 Rs· rr r s  ·r r  ·r r r  ·r r,ex  r  ˆ ˆ m m A x K Cx K y e + + + s (5.4) ˆ sr m ˆ ss m A·s xs K·s Cs xs K·s ys,ex e

The large term between {…} in (5.4) shows the four components of the total demand for product c from region r and s. Note that the demand of the exogenous account of region r (i.e. er ) is already specified by region of supply, whereas intermediate demand A·r xr , endogenous household demand K·r Cr xr and exogenous household demand K·r yr,ex , all of region r, need to be pre-multiplied with trade origin ratios ˆ sr to determine the region of supply of product c. Finally, all four terms are ˆ rr and m m pre-multiplied with Rr · to determine which industry i in r supplies product c from r. The solution of (5.4) is simple. The two terms with intermediate demand and endogenous household demand need to be moved from the RHS to the LHS of (5.4), and then both sides need to be pre-multiplied with the appropriate inverse. This gives:

xr xs





ˆ rr (A·r + K·r Cr ) I − Rr · m ˆ r s (A·s + K·s Cs ) I − Rr · m = ˆ sr (A·r + K·r Cr ) −Rs· m ˆ ss (A·s + K·s Cs ) −Rs· m  rr ·r r,ex 

r· ˆ r s K·s ys,ex + er ˆ K y +m m R 0 × ˆ sr K·r yr,ex + m ˆ ss K·s ys,ex + es 0 Rs· m

−1

(5.5)

The interpretation of (5.5) is relatively straightforward. Following the causal chain from the right to the left, the upper most right-hand term er describes the direct impact of exogenous final demand on the supply of products by region r. The second term ˆ r s K·s ys,ex describes the direct spillover of exogenous incomes of households in s on m ˆ rr K·r yr,ex describes the the supply of products by region r, whereas the third term m intra-regional impact of exogenous incomes of households in r on the same supply. Next, the matrix Rr · in the middle of (5.5) determines which industries in region r will satisfy these three types of exogenous product demand. Finally, the large inverse matrix determines the direct, indirect and induced impact of these three sets direct effects. Take, for instance, its bottom left-hand ˆ sr (A·r + K·r Cr ). Again along the causal chain from the right to the left, its term Rs· m sub-term between brackets describes the impact of total output by industry in region r on intermediate demand A·r and endogenous household demand K·r Cr for products

5.1 Interregional Models with Endogenous Household Consumption

63

ˆ sr determine which part of that demand from everywhere. Next, trade origin ratios m from region r is satisfied by region s, while market share ratios Rs· determine which part of those imports are delivered by which industry in s. When (5.5) is pre-multiplied with value added coefficients, the pre-multiplied inverse strongly resembles the sectorally and regionally disaggregated Keynesian income multipliers of Miyazawa (1976).1 When the inverse of (5.5) is pre-multiplied with, e.g. CO2 -emission coefficients, Type II interregional CO2 -emission multipliers of exogenous demand result. Note that the additional data required to estimate the multi-regional SAM model (5.4), compared to the multi-regional SUT model (4.19), is limited to a column-wise split up of the national consumption package coefficients per household type q (i.e. ·n n ·n ·r ·s = h ·n kcq cq / h qt ∈ K ) into the regional package coefficient matrices K and K , along with a column-wise split up of the national labour income/output ratios for n n = wqi /xin ∈ Cn ) into the corresponding regional households q per industry i (i.e. cqi r s matrices C and C . In both cases, this breakdown may be made by assuming that the regional coefficients equal the national coefficients, but using survey data is, of course, to be preferred. Also, note that the interregional labour income redistribution through commuting easily fits in this multi-regional disaggregation of the national SAM of Table 5.1. It only requires that the regional labour income/output ratios are given a second spatial dimension that accounts for the share of the labour income earned in industry j in region s that accrues to in-commuters from region r; i.e., the regional labour income/output ratios Cs then need to be disaggregated into cqr sj = wqr sj /x sj ∈ Cr s ∀r , wherein wqr sj = labour income earned by households q living in r and working in industry j in s. Finally, note that the other three types of interregional SAMs require a further split up of the simplified national SAM of Table 5.1, comparable to the further split up of the national SUT in Table 4.4. Hence, this requires the earlier discussed split up of the aggregate interregional trade data, as shown in (4.18), which is the most elusive interregional trade data to collect/construct with any degree of reliability.

1

See Miyazawa and Masegi (1963) for the original idea, and Sonis and Hewings (1999) for an overview of Miyazawa-type IO model extensions. Pyatt (2001) shows that Miyazawa-type multipliers can be viewed as special cases of interregional SAM multipliers. The core difference is that Miyazawa-type multipliers relate the generation of income by production factor, industry and region directly to the spending of that income on products produced by industry and region, whereas SAM multipliers offer additional detail and more understanding in that they add the spatial and governmental redistribution of income in-between its generation and its spending.

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5 From Basic IO and SU Models to Demo-economic Models

5.2 Further Demo-economic Model Extensions Unfortunately, there were and there still are only few single-region SAMs available, let alone interregional SAMs, which explains why most of the literature on endogenizing household consumption relates to single-region and interregional IO models, as opposed to the SAM models discussed above. In the absence of an interregional SAM, the interregional IO model extension that specifies the endogenous part of the household expenditures needs to be build-up piece-by-piece: h ri js = qirjs x sj = m ri hs ki·sh (1 − shs )(1 − ths ) chs j x sj ∈ H i = Qa x

(5.6)

where reading backwards, following the causal chain, chs j = gross household income earned in industry j in region s per unit of output of j in s, ths = tax rate for households in s, shs = savings rate for households in s, ki·sh = consumption of products from industry i from everywhere per unit total endogenous household consumption in s and m ri hs = share of consumption of i in s that originates from r. The columns of H thus represent the endogenous consumption expenditures by region of living s, while Qa contains average household consumption expenditures/industry output ratios. It should be noted that there are alternative ways to build a consumption function in the absence of a SAM. Emonts-Holley et al. (2021) distinguish five alternatives, which for Scotland result in Type II single-region multipliers ranging from 1.74 to 2.16, with the SAM benchmark multiplier having a value of 1.88. As an example of an additional alternative, note that (5.6) does not endogenize the consumption of commuters, which would require separate data on their specific consumption behaviour with a large share of interregional shopping in their region of work (see Madsen and Jensen-Butler 2004, for a SAM model with both interregional commuting and shopping). The addition of (5.6) to the interregional IO model of (2.9), as shown in Fig. 5.1, gives a Type II interregional IO model. Its solution is straightforward: ⎤ ⎡ ⎤ ⎡ 1,ex ⎤ i f h11 ... h1R ⎦ ⎣ ⎦ ⎣ ⎣ x = Zi + ... ... ... ... ⎦ = Ax + Qa x + f ex ⇒ ... + R1 RR R,ex h ... h f i ⎡

x = (I − A − Qa )−1 f ex = L∗∗ f ex

(5.7)

where L∗∗ represents the Type II interregional Leontief-inverse. When (5.7) is used for impact studies, any increase in endogenous total output will lead to proportional increases in employment, labour incomes and endogenous consumption. This proportional increase models what would happen if all actually necessary marginal input coefficients would be equal to the corresponding average input coefficients as point-estimated from the base-year IO table. Equality of marginal and average coefficients is a reasonable assumption for labour income coefficients,

5.2 Further Demo-economic Model Extensions

65

which equal one minus the total of the other input coefficients. It is, however, not a reasonable assumption for employment and consumption package coefficients, nor for tax and savings rates. Marginal employment coefficients are smaller than their average equivalents, whereas marginal tax and savings rates are larger, which together leads to marginal consumption/output ratios Qm that are smaller than the average Qa of (5.7). Miernyk et al. (1967) and Tiebout (1969) solved part of this problem by making a distinction between extensive income growth that accrues to people without an earlier income in the region (mainly school-leavers, non-active partners and new residents) and intensive income growth that accrues to people who stay with their jobs. To extensive income growth, they applied average consumption/output ratios, and to intensive income growth, they applied marginal consumption/output ratios. The indirect part of the resulting, so-called Type III multipliers was roughly about 20% lower than the comparable Type II multipliers.2 Additionally, Blackwell (1988) considered that new jobs would also go to formerly unemployed residents who would lose their unemployment benefits, which he labelled redistributive income growth. To this type of income growth either marginal consumption/output ratios, Qm need to be applied, or the difference between the average ratios of employed residents Qa and unemployed residents Qu , which delivers about the same result. Obviously, adding a negative feedback on unemployment benefits leads to substantially lower, so-called Type IV multipliers. Van Dijk and Oosterhaven (1986), with Dutch unemployment benefits and a vacancy chain submodel for the regional labour market, estimated Type IV multipliers to have values of 35–60% between the values of Type I and Type II multipliers per industry, which they therefore viewed as lower and upper limit for the true values of regional multipliers. Batey and Madden (1983) extended these models with a population block and labelled them demographic-economic or demo-economic models. Instead of the earlier ad hoc and iterative model specifications, they proposed the far more efficient commodity-activity framework to formulate them, which we already used in (4.15) and (5.2) (see also Batey and Rose 1990). Batey (1985) gave an overview of ten such models with increasing complexity, with the integration of the three types of income changes by Oosterhaven (1981) as the tenth model (see Batey and Hewings 2021, for a unique review of more historic demo-economic modelling endeavours). All ten models, however, still use a residual definition of intensive income growth, as these models do not make a distinction between the levels and the changes in

2

The 20% is derived from Miller and Blair (2009), who report a ratio of Type III to Type II multipliers of 0.87–0.91 on p.255. Their ratio, however, includes the direct effect, which we exclude, as no model is needed for its estimation. In the older IO literature, there is quite some discussion about fixed multiplicative relations between Type I, II and III normalized income multipliers (see Miller and Blair 2009, ch. 6). If one defines total endogenous income per regional industry to be equal to that part of it that is paid to regional households, then a fixed relation exists (see Miller and Blair 2009, for proof and empirical ratios). In the old days, this presented an important computational advantage, but nowadays this advantage is outweighed by the disadvantage of having to use a definition of income that only covers the labour part of value added.

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the levels of variables, which is necessary to adequately distinguish between cases where marginal ratios are needed and cases that require the use of average ratios. With an interregional labour market model, which included vacancy chains for job-to-job hoppers, social security and population growth, Oosterhaven and Folmer (1985) show how the distinction between levels and changes solves this remaining problem. To illustrate the nature of this solution, we finish this overview of demoeconomic modelling with a simplified version of their interregional model, skipping the vacancy chain sub-model and most of the social security and population submodel details. Changes in levels of, e.g. output, are indicated with x and lagged output levels with x−1 . Hence, x = x − x−1 . The nature of their approach is best understood by starting with endogenous employment growth, measured as the growth in the number of full-time equivalent (FTE) jobs per industry per region, jir ∈ j. For regular small changes, the growth of employment by definition equals: j = ˆl x + ˆl x−1

(5.8)

where lir = jir /xir ∈ ˆl = diagonal matrix with average employment coefficients, i.e. with inverse labour productivity levels, and ˆl = ˆl − ˆl−1 = diagonal matrix with the decrease in average employment coefficients, i.e. with the growth of nominal labour productivity. In general, the almost always negative last term of (5.8), with lagged output, will only be relevant in case of a projection of a whole economy, but not in case of impact studies with only limited exogenous changes. Note that in case of impact studies, the average employment coefficients ˆl are better replaced with marginal employment coefficients that will be smaller. Next, extensive wage income growth yext and intensive wage income growth yint by region of work may be defined using of the outcomes of (5.8): ˆ −1 j and yint = w ˆ j−1 yext = w

(5.9)

ˆ = diagonal matrix with wage rates per regional industry. Note that where wsj ∈ w the sum of extensive and intensive wage income growth equals total wage income ˆ ˆj−1 . Furthermore, ˆ ˆj) = w ˆ −1 ˆj + w growth, as for regular small changes (w r note that in many cases the wage income/production ratios chi = (wir jir )/xir = ˆ ˆl will be stable, in which cases the employment decreasing and wir lir ∈ cˆ h = w wage increasing effects of labour productivity growth cancel out. Next, to obtain extensive and intensive wage income growth by region of living, the outcomes of (5.9) have to be pre-multiplied with crj s ∈ Co , which denotes a Rby-IR matrix with commuting origin ratios, indicating the proportion of workers in industry j in region s that commutes in from region r. Note that i Co = i , if external (i.e. international) in-commuting is zero. To define redistributive income growth, first, the change in the number of unemployed people u and the number of inactive people without benefits n need to be

5.2 Further Demo-economic Model Extensions

67

explained by region of living: u = −Mu j + uex and n = −Mn j + nex

(5.10)

where m rusj ∈ Mu = R-by-IR matrix with re-employment probabilities, indicating the probability that new jobs in industry j in region s, either directly or indirectly after a vacancy chain,3 are taken up by unemployed formerly living in region r, while uex = exogenous change in number of unemployed per region of living. Comparable definitions hold for the second part of (5.10) with inactive people without benefits (mainly school-leavers, non-working partners and immigrants). Note that i Mu + i Mn = i , if there are no external (i.e. international) immigrants taking up part of the new jobs and if all job vacancies, either directly or indirectly after a vacancy chain, are filled up. Equation 5.10 makes explicit that only part of the new local jobs will be filled up by locally unemployed people, who will then lose their unemployment benefits and thus cause a negative feedback on employment in their own region. Another part will be filled up by unemployed from other regions, which will cause a negative feedback on unemployment benefits in those other regions! This will, especially, be the case for new jobs in industries that require relatively high-skilled workers or in the construction industry where workers are used to long commuting distances to work in often changing regions. The remainder of the new jobs, either directly or indirectly after a vacancy chain, will be filled up by either local or immigrating non-active people with no benefits to loose. Using the outcomes of (5.9)–(5.10), pre-multiplied with the commuting matrix Co , the change in total demand for products from industry i in region r may now be defined as: x = Ax + Ka Co yext + Km Co yint + Ku uˆ b u + f ex

(5.11)

rs where kia ∈ Ka = IR-by-R matrix with average consumption/gross income ratios, indicating the consumption of products from i in r per unit of gross income of households living in s, Km = comparable marginal consumption/gross income ratios, Ku = comparable average consumption/gross income ratios for unemployed, u sb ∈ uˆ b = diagonal matrix with unemployment benefits per region of living s, and f ir, ex ∈ f ex = column with changes in exogenous final demand for products of i in r. Substitution of (5.8) in (5.9)–(5.10) and subsequent substitution of the results for yext , yint and u in (5.11) give the following equation for the endogenous change in total output by industry and region:

ˆ −1 ˆl x + Km Co w ˆ ˆl x x = A x + Ka Co w 3

With twelve industries, nine occupations, males and females, in case of Queensland, Oosterhaven and Dewhurst (1990) report significant differences in re-employment and immigration probabilities when the vacancy-chains of people moving between jobs are ignored.

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5 From Basic IO and SU Models to Demo-economic Models

− Ku uˆ b ˆl x + Ku uˆ b uex + lagged v. + f ex

(5.12)

ˆ −1 ˆl x−1 + Km Co w ˆ ˆl x−1 − Ku uˆ b ˆl x−1 . where lagged variables = Ka Co w After moving the terms with x from the RHS to the LHS of (5.12) and premultiplying the result with the appropriate Leontief-inverse, the resulting solution of (5.12) may be summarized as follows: x = (I − A − Qa − Qm + Qu )−1 (Ku uˆ b uex + lagged v. + f ex )

(5.13)

In (5.13), the four coefficient matrices in the extended Leontief-inverse, when added cumulatively, represent the Types I, II, III and IV, direct, indirect and induced impacts of exogenous demand on total output and thus also on more policy-relevant variables such as employment, value added and CO2 -emissions. Note that the Type IV redistributive income growth effect (Qa − Qu )x is, in fact, estimated by taking the difference between the average consumption/output ratios of employed and unemployed people. Also note that (5.13), compared to the standard interregional IO model, has a much larger number of exogenous variables, most of them hidden in the lagged variables component. Finally, and most importantly, note that (5.10) for non-active people and the vacancy chain sub-model that is behind it, only indirectly influence the size of the Type IV regional multipliers and interregional spillovers. When more non-actives find jobs at the cost of local unemployed, the larger values for Mn and the consequently smaller values for Mu lead to smaller negative feedbacks of disappearing unemployment benefits and consequently to larger Type IV multipliers and spillover effects. A somewhat different effect occurs when, either directly or indirectly after a vacancy chain, more unemployed from other regions fill up new jobs at the cost of local unemployed. This also leads to larger Type IV multipliers, but it leads to smaller rather than larger Type IV interregional spillovers, as the negative feedback of disappearing benefits shifts from the own region to other regions. These two positive effects on the size of the intra-regional multipliers, namely that of more inactive people and that of more unemployed from other regions taking up new local jobs, lead to an interesting policy dilemma. Often, the goal of regional policy is to reduce local unemployment, while most policy instruments stimulate the growth of local employment (see van Dijk et al. 2019, for an overview of regional policy goals and instruments). The larger type IV multipliers that result from the two effects, in this regard, have contradictory effects: more new jobs, but relatively less local unemployed taking them up. Also, interesting from a policy point of view is the finding that in the case of Queensland a unit increase in exogenous final demand for some industries generated increased tax receipts and reduced unemployment benefits that actually outweighed the unit cost of increasing final demand (Oosterhaven and Dewhurst 1990). The more extensive interregional labour market model of Oosterhaven and Folmer (1985), of which (5.8)–(5.11) is a simplification, uses the commodity-activity framework of Batey and Madden (1983) to summarize the model, instead of the more

5.3 Where to End with Endogenizing Final Demand?

69

simple representation of (5.13). The advantage of (5.13) is that it nicely shows the three types of incomes growth and four types of multipliers that figure so prominently in the demo-economics literature. The disadvantage of this presentation is that it would result in an incomprehensible set single equation if this notation and solution would be applied to the full model.4

5.3 Where to End with Endogenizing Final Demand? After endogenizing most of household consumption demand, the obvious next question is: how far should an analyst go with endogenizing more and more components of final demand?5 Studying the regional impacts of plant close-downs with a singleregion SAM model, Cole (1989, 1997) advocates the fullest possible closure of the model to capture all possible short- and long-run impacts. The distinction between short-run and long-run impacts is made by means of expenditure lags, for which Cole introduces a very handy computational solution. The fullest possible closure is reached by linking the size of government expenditures to tax income, linking investment expenditures to the operating surplus and linking regional exports to regional imports. Especially adding that last causal relation led to a fierce debate with Jackson et al. (1997) who claimed that closing a single-region model with regard to RoW leads to inconsistencies, zero exogenous demand and infinitely large multipliers. Oosterhaven (2000) closed the debate on request of the editors of the journal at hand. Obviously, one cannot endogenize interregional feedbacks consistently without specifying the full interregional model. When that is done with regard to the whole RoW, as Cole advocates, the full closure of the extended model does result in zero exogenous demand and infinitely large multipliers, as in such a completely closed world model only the model’s coefficients remain exogenous, whereas all its variables become endogenous. Consequently, such a model may no longer be used to evaluate the impacts of changes in demand, as all demand has become endogenous. For a somewhat comparable reason, overestimation of impacts also occurs when total value added or total employment of an industry is multiplied by that industry’s already too large Type II normalized value added multiplier or normalized employment multiplier, c (I − A − Qa )−1 cˆ −1 , in order to indicate its importance for the economy at hand. This is a misuse of impact analysis for public relations purposes, as a multiplier may only be applied to exogenous final demand and never to endogenous value added or endogenous employment. Imagine that the average normalized multiplier equals 2.0 and that the analyst would apply this estimation procedure to 4

See van Dijk and Oosterhaven (1986) and Oosterhaven and Dewhurst (1990) for single-region applications of the above approach, and Oosterhaven et al. (2019) for an overview of extended IO models, especially for the USA. 5 Parts of this Section were written earlier for Oosterhaven (et al. 2019). I thank my transport economics colleague Jaap B. Polak for providing the data on the close-down of the Fokker aircraft company.

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all industries and sum the results. In that case, the predicted size of the total economy would be twice its actual size, which also numerically shows that this procedure is not allowed. One solution to the overestimation problem is to correct the calculated “gross impact” of a certain industry with that part of that industry that is endogenously dependent on the rest of the economy, in order to obtain the net impact of that industry (see Oosterhaven et al. 2003, for an example of this solution). A second solution is to downsize the standard “gross multiplier” with that part of an industry’s output that is exogenous, that is, to multiply the normalized Type II multiplier with the exogenous final demand/total output ratio yiex, r /xir ∈ yex x such that it becomes (see Oosterhaven and Stelder 2002, for a net multiplier c (I − A − Qa )−1 cˆ −1 yˆ ex x this solution).6 This also solves the overestimation problem, because when the net multiplier is multiplied with an industry’s total employment or total value added, the weighted average of all industries’ net multipliers does become one, which does result in the precise size of the whole economy, as it should. Finally, it is important to understand that even with exogenous exports to RoW, the fullest possible closure of the remaining components of final demand in a singleregion model exacerbates the theoretically already problematic one-sidedness of IO, SUT and SAM models. In case of the close-down of the Dutch aircraft producer Fokker, for instance, aside of the direct loss of 4500 jobs, not too much happened. After only three months, 60% of the mostly highly skilled former employees had found a new job, while most of Fokker’s many subcontractors suffered only a temporary setback in output growth. Some, in fact, were reported to have found new markets for their often high-tech goods and services: new markets that proved to grow much faster than the old secure, but low-growth Fokker demand. This illustrates that possible shortages on local labour markets, price and wage reactions and pressure to supply to new markets and to develop new products may strongly reduce the demand-driven multiplier estimation of the impact of a negative output shock. In such cases, endogenizing price/wage effects and supply reactions is more important than a further closure of the demand side, that is, if the consultant is interested in the most probable estimate of the real impacts instead of in reporting maximally large multipliers to serve the special interests of his or her principal.

References Batey PWJ (1985) Input-output models for regional demographic-economic analysis: some structural comparisons. Environ Plan A 17:77–93

6

Note that de Mesnard (2006) takes offense at the use of the word multiplier in this case. See Oosterhaven (2007) for a reply and Dietzenbacher (2005) for an independent evaluation. The conclusion of this debate is that the net multiplier is best viewed as a net key sector measure that takes into account the two-sided nature of an industry’s dependence on the rest of the economy versus the rest of the economy’s dependence on that industry. See for a further discussion Sect. 9.1.

References

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Batey PWJ, Madden M (1983) The modelling of demographic-economic change within the context of regional decline: Analytical procedures and empirical results. Socio-Econ Plan Sci 17:315–328 Batey PWJ, Rose A (1990) Extended input-output models: Progress and Potential. Int Reg Sci Rev 13:27–49 Batey PWJ, Hewings GJD (2021) Demo-economic modeling: review and prospects. Internat Reg Sc Rev 1–35 Blackwell J (1988) Disaggregation of the household sector in regional input-output analysis: some models specifying previous residence of worker. Reg Stud 12:367–377 Cole S (1989) Expenditure lags in impact analysis. Reg Stud 23:105–116 Cole S (1997) Closure in Cole’s reformulated Leontief model: A response to R.W. Jackson, M. Madden, and H.A. Bowman. Pap Reg Sci 76:29–42 de Mesnard L (2006) A critical comment on Oosterhaven-Stelder net multipliers. Ann Reg Sci 41:249–271 Dietzenbacher E (2005) More on multipliers. J Reg Sci 45:421–426 Emonts-Holley T, Ross A, Swales K (2021) Estimating induced effects in IO impact analysis: variation in the methods for calculating the Type II Leontief multipliers. Econ Syst Res 33:429– 445 Jackson RW, Madden M, Bowman HA (1997) Closure in Cole’s reformulated Leontief model. Pap Reg Sci 76:21–28 Madsen B, Jensen-Butler C (2004) Theoretical and operational issues in sub-regional economic modelling, illustrated through the development and application of the LINE model. Econ Model 21:471–508 Madsen B, Jensen-Butler C (2005) Spatial accounting methods and the construction of spatial social accounting matrices. Econ Syst Res 17:187–210 Miller RE, Blair PD (2009) Input-output analysis: foundations and extensions, 2nd edn. Cambridge University Press, Cambridge Miernyk WH, Bonner ER, Chapman JH, Shellhammer K (1967) Impact of the space program on a local economy: an input-output analysis. West Virginia University Library, Morgantown Miyazawa K (1976) Input-output analysis and the structure of the income distribution. Springer, Berlin Miyazawa K, Masegi S (1963) Interindustry analysis and the structure of income distribution. Metroecon 15:89–103 Oosterhaven J (1981) Interregional input-output analysis and Dutch regional policy problems. Gower Publishing, Aldershot-Hampshire Oosterhaven J (2000) Lessons from the debate on Cole’s model closure. Pap Reg Sci 79:233–242 Oosterhaven J (2007) The net multiplier is a new key sector indicator: reply to De Mesnard’s comment. Ann Reg Sci 41:249–271 Oosterhaven J, Dewhurst JHL (1990) A prototype demo-economic model with an application to Queensland. Int Reg Sci Rev 13:51–64 Oosterhaven J, Folmer H (1985) An interregional labour market model incorporating vacancy chains and social security. Pap Reg Sci Assoc 58:141–155 Oosterhaven J, Stelder TM (2002) Net multipliers avoid exaggerating impacts: with a bi-regional illustration for the Dutch transportation sector. J Reg Sci 42:533–543 Oosterhaven J, van der Knijff EC, Eding GJ (2003) Estimating interregional economic impacts: an evaluation of nonsurvey, semisurvey, and fullsurvey methods. Environ Plan A 35:5–18 Oosterhaven J, Polenske KR, Hewings GJD (2019) Modern regional input–output and impact analysis. In: Capello R, Nijkamp P (eds) Handbook of regional growth and development theories: Revised and extended second edition. Edward Elgar, Cheltenham Pyatt G (2001) Some early multiplier models and the relationship between income distribution and production structure. Econ Syst Res 13:139–163 Pyatt G, Round JI (1977) Social accounting matrices for development planning. Rev Income Wealth 23:339–364

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Pyatt G, Thorbecke E (1976) Planning techniques for a better future. International Labour Office, Geneva Sonis M, Hewings GJD (1999) Miyazawa’s contributions to understanding economic structure: Interpretation, evaluation and extensions. In Hewings GJD, Sonis M, Madden M, Kimura Y (eds) Understanding and interpreting economic structure. Springer, Berlin Tiebout CM (1969) An empirical regional input-output projection model: the State of Washington 1980. Rev Econ Stat 51:334–340 van Dijk J, Oosterhaven J (1986) Regional impacts of migrants’ expenditures: An inputoutput/vacancy-chain approach. In: Batey PWJ, Madden M (eds) Integrated analysis of regional systems. Pion, London van Dijk J, Folmer H, Oosterhaven J (2019) Regional policy: rationale, foundations and measurement of its effects. In: Capello R, Nijkamp P (eds) Handbook of regional growth and development theories: Revised and extended second edition. Edward Elgar, Cheltenham

Chapter 6

Cost-Push IO Price Models and Interaction with Quantities

Abstract In the Type I single-region, cost-push IO price model, under full competition, exogenous primary input price changes are fully passed on to all intermediate users that fully pass them on further, resulting in endogenous total and final output price changes. In the Type II interregional price model, additionally, consumption price changes are fully passed on in wage rates, while domestic export price changes are fully passed on to importing regions. Finally, it is shown how the IO price model may be combined with the IO quantity model by adding demand and supply price elasticities and how supply and demand shifts are passed on from market to market, resulting in lower, more realistic price and quantity multipliers. Keywords Cost-push IO price model · Cost shares · Price multipliers · Price–wage–price spiral · Consumption/output ratios · Demand and supply price elasticities · LINE model Chapter 5 ended with the conclusion that it may be more useful to add prices and the supply side to a Type IV interregional IO model, instead of further endogenizing final demand. In this chapter, we investigate of how prices figure in IO models. First, we discuss the basic IO price model, then its interregional extension with endogenous consumption expenditures and finally how it may be used together with its accompanying quantity model.

6.1 Forward Causality of the Single-Region IO Price Model The demand-driven IO quantity models of Chaps. 2, 4 and 5 are all accompanied by dual cost-push IO price models. Prices are entirely passive in the quantity models, whereas quantities are entirely passive in the price models. Figure 6.1 shows the causal structure of the basic IO price model. It is almost equal to Fig. 2.1, which shows the causal structure of the corresponding quantity model. Besides prices that replace quantities, the only but very important difference is the direction of causality that is completely reversed, which is why the position of the boxes with final output and primary input have been switched in Fig. 6.1 compared to Fig. 2.1. In the quantity © The Author(s), under exclusive license to Springer Nature Switzerland AG 2022 J. Oosterhaven, Rethinking Input-Output Analysis, Advances in Spatial Science, https://doi.org/10.1007/978-3-031-05087-9_6

73

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6 Cost-Push IO Price Models and Interaction with Quantities Primary input prices

C

I

Total output prices

I

Final output prices

A

Intermediate output prices

Fig. 6.1 Causal structure of the basic IO price model. Legend C = matrix with primary input cost shares, A = matrix with intermediate input cost shares

model, in Fig. 2.1, the causality runs backward from exogenous final demand => total output => intermediate demand, back => total output, and finally => primary inputs. In the IO price model the causality of runs in the opposite, forward direction. The I prices of the single homogeneous output of each industry i, pi ∈ p, are assumed to be endogenous and uniform along the corresponding rows of the first and second quadrant of the IOT in Table 2.1. The P prices of the single homogeneous primary inputs of type p, p p ∈ pv (i.e. capital, labour and import prices), on the other hand, are assumed to be exogenous and uniform along the corresponding rows of the third and fourth quadrant of the IOT, as shown in Fig. 6.1 where no arrows are entering the box with primary input prices. Any change in one of the P primary input prices is directly and fully passed on into the I output prices. The size of the subsequent change in output prices is, of course, determined by the size of the primary input cost shares in total output, c pj ∈ C, as indicated by the arrow accompanied by the matrix C. This direct effect on the price of total output, p v C, is fully passed on to all intermediate users (i.e. firms) along the rows of the first quadrant of the IOT and to all final users along the rows of its second quadrant, as indicated by the two arrows with the matrix I. Industries that use these more expensive intermediate outputs, in turn, pass these cost increases fully on to their clients. The size of these first-order indirect effects on the I output prices is, of course, determined by the intermediate input cost shares in total output, ai j ∈ A. These effects thus equal the direct impact p v C times A. These first-order indirect price changes are again fully passed on, which results in second-order indirect price effects p v C A2 and so on. The final users are subject to the same price changes of total output by industry, but in the basic model, they do not pass them on any further, as indicated by the absence of arrows leaving the box with final output prices. The cumulative effect of any change in primary input prices on total output prices therefore equals the outcome of the Taylor expansion p v (CI + CA + CA2 + CA3 + ...) = p v C (I − A)−1 = p . This total effect consists of the direct effect C and the indirect effect, the size of which is determined by the size of the Leontief-inverse (I − A)−1 . Note that the sum of both effects equals unity, i.e. i C (I − A)−1 = i , as the sum of the cost shares equals unity, i.e. i A + i C = i .

6.1 Forward Causality of the Single-Region IO Price Model

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The mathematics of this basic IO price model (Leontief 1951; Chenery and Clark 1959) formalizes the above explanation of its causality. Its accounting identities are based on the values of the cells of the columns of the IOT and distinguish quantities from prices, whereas those of the quantity model are based on the rows of the IOT and do not distinguish prices, because the quantity model assumes all prices are constant and equal to one (Schuman 1968). The accounting identities for the values of the cells of each column j of the IOT are equal to:   pi z i j + p p v pj , or in matrix algebra: pjxj = p i (6.1) p xˆ = p Z + p v V Second, just like the quantity model, the price model assumes that intermediate and primary input coefficients are fixed, and co-determine the size of Z and V, as explained in Sect. 2.2: Z = A xˆ and V = C xˆ

(6.2)

Substitution of both parts of (6.2) in (6.1) and post-multiplication with xˆ −1 shows that the prices of total output equal the sum of the prices for intermediate and primary inputs, weighted by their respective cost shares: p = p A + p v C

(6.3)

Finally, adding the assumption that, under full competition, all price changes are fully passed on to all users delivers the solution of the basic (also called Type I) IO price model. The solution shows how the endogenous prices for total output are determined by the exogenous prices of the primary inputs and both sets of cost shares: p = p v C (I − A)−1 = p v C L

(6.4)

Note that the  aggregate Type I final output price multipliers of exogenous primary input prices, i c pi li j ∈ C (I − A)−1 , equal the aggregate Type I primary input quantity multipliers of exogenous final output from Sect. 2.2. In case of the price model, they represent the total of the direct and indirect weight of the primary input prices in the final output prices, which equals one. In the quantity model, they equal the direct and indirect quantity of primary inputs embodied in final output, which also equals one, as all quantities are measured in base-year prices set equal to one. This primal–dual relationship between the IO quantity and price model may be further illustrated by post-multiplying (6.4) with final demand y, which gives: p y = p v C (I − A)−1 y = p v v

(6.5)

This confirms that the value of total final output p y equals the value of total primary input p v v, which also follows from the macroeconomic identity C + I + G + E = Y

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+ M in Table 2.1. Second, and more importantly, (6.5) shows the independency (or better duality) of the price and quantity model. Although their solutions are linked by means of (6.5), their variables move independently: with exogenous final demand quantities backwardly determining primary input quantities in the quantity model and exogenous primary input prices forwardly determining final output prices in the price model. Obviously, the IO price model is well suited to model the impact of any primary input price change (i.e. any change in product tax and subsidy rates, in wage rates, in capital return rates or in import prices) on the consumption, investment and export prices by delivering industry. This is why this model is also known as the cost-push IO price model. In (6.1), to keep the mathematics simply and the duality clear, it is assumed that the P exogenous primary input prices are equal for all purchasing industries. In practical applications, however, one may probably more realistically assume—precisely because they are exogenous—that the prices of these primary inputs behave differently for each purchasing industry. Early applications of this model show the price effects of pollution abatement policies (Evans 1973; Giarratani 1974), and the price effects of energy price increases in a single-region IO model (Miernyk 1976) and in a multi-regional IO model (Polenske 1979). A recent application, for example, uses the cost-push price model to chart the vulnerability of US industries to reaching peak oil production rates by simulating a 100% oil price hike (Kerscher et al. 2013).

6.2 Interregional IO Price Model with a Price–Wage–Price Spiral Naturally, both the Type I multi-regional extension and the Type I interregional extension of the basic IO quantity model in Sect. 2.3 have an accompanying (dual) cost-push price model. The mathematics and the economics of these two multi-region price models are straightforward. The dual of the Type II interregional IO quantity model of Sect. 5.2 is less straightforward, while it is more interesting economically, which is the first reason to explain it at length. The bottom part of Fig. 6.2 shows that the Type II interregional price model models the following causal chain: exogenous non-wage primary input prices (mainly capital and external import prices) => total output prices => intermediate input prices and consumption expenditure prices (=> wage rates), back => total output prices, finally => remaining final output prices. This price model thus allows for an interregional analysis of cost-push price–wage– price inflationary processes, which is the second reason to explain it at length. To discuss this price model properly, we first need to complete the Type II interregional IO quantity model. The causal structure of this model, shown in Fig. 5.1 in detail for two regions, is summarized in the upper part of Fig. 6.2. Its mathematics consist of four equations: three old ones and a new one that is needed to specify the price model.

6.2 Interregional IO Price Model with a Price–Wage–Price Spiral Exogenous in the quantity model

Zi

Hi

Qa

I

77

I

A

x

I

C

Demand-driven IO quantity model Cost-push IO price model I

Crem

p I

Qa I

A

Exogenous in the price model

Fig. 6.2 Interacting Type II interregional IO price and quantity models. Legend Fig. 5.1 and pz , ph and pex = vectors with IR identical prices for, respectively, intermediate output Zi, endogenous consumption output Hi, and exogenous final output f ex . pr em = vector with P prices of remaining primary inputs Vr em . Cr em = matrix with cost shares of the remaining primary inputs in total output. edp = vector with IR price elasticities of exogenous final demand. esp = vector with P price elasticities of exogenous supply of remaining primary inputs

First, the accounting identities for the rows of an IRIOT (see Table 2.2) express that total output/supply x, follows any change in the sum of intermediate demand Zi, endogenous consumption demand Hi and exogenous final demand f ex , without any change in prices: x = Z i + H i + f ex

(6.6)

Next, fixed intermediate input coefficients, airjs ∈ A, and fixed average consumption/output ratios, qirjs ∈ Qa , explain intermediate demand and endogenous consumption demand as a function of the output of industry j in region s: Z i = A x, H i = Qa x

(6.7)

Note that household consumption paid for by non-labour incomes, such as social security payments, remains exogenous. Finally, a new, third behavioural equation with fixed primary input coefficients, cspj ∈ Cr em , is needed to explain the remaining primary inputs of industry j in region s, v spj ∈ Vr em :

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Vr em = Cr em xˆ , with i Cr em xˆ = i C xˆ − i Qa xˆ , while i A + i Q + i C a

r em

= i (6.8)

Note that the remaining primary inputs do not have a feedback effect on industry output in the quantity model (see the lacking outgoing arrows in the upper part of Fig. 6.2). The middle  part of (6.8) clarifies that the remaining primary inputs per regional industry, p cspj x sj ∈ i Cr em xˆ , equal the difference between total primary inputs, i Cˆx, and the endogenous consumption of local households that is paid from their labour incomes by regional industry, i Qa xˆ , as explained in Sect. 5.2. In the Type II quantity model, the latter wage-related part of total primary inputs does have a feedback on total output by regional industry, as shown by (6.7) and the arrow with Qa in Fig. 6.2. The solution of the Type II quantity model for the remaining primary inputs and for any other impact variable, such as employment or CO2 emissions, is derived by substituting (6.7) into (6.6), moving the terms with endogenous total output from its RHS to its LHS, pre-multiplying the result with the Type II interregional Leontiefinverse L∗∗ = (I − A − Qa )−1 and substituting the result into the first term of (6.8). This gives, subsequently: −1  x = I − A − Qa f ex = L∗∗ f ex and Vr em = Cr em L∗∗ fˆ ex

(6.9)

An extensive overview of the type of regional and interregional impact studies that may be done with (6.9) is given in Oosterhaven et al. (2019). The price dual of the Type II quantity model (6.6)–(6.8) is relatively complex (Oosterhaven 1981). Here, without altering the economic mechanisms of the model, the mathematical presentation is simplified by ignoring the endogenous character of the taxes and savings from endogenous wage income specified in (5.6). This, especially, simplifies the presentation of accounting identities of the Type II interregional IO price model along the columns of the IRIOT (see Table 2.2). With this simplification, the total value of the inputs of industry i in region r may now be defined as: pir xir ∈ p xˆ = p Z + p H + p r em Vr em

(6.10)

All but one term of (6.10) already appeared in (6.1), but now they  are defined for multiple regions instead of for a single region. New is the term ri pir h ri js ∈ p H. It defines the total value of the endogenous consumption expenditures of households paid for by their labour incomes earned in industry j in region s. Note that wages are not explicitly defined in (6.10), but appear implicitly as the weighted average of the prices of the endogenous consumption expenditures. This means that the impact of increasing consumption prices on wages is also modelled implicitly. Finally, note that the precise definition of the last term of (6.10) is given in the middle term of (6.8).

6.2 Interregional IO Price Model with a Price–Wage–Price Spiral

79

Substitution of (6.7) and (6.8) into (6.10), and post-multiplication with xˆ −1 delivers (6.11), which shows that the prices of total output by regional industry equal the sum of the prices for intermediate inputs, endogenous consumption expenditures and remaining primary inputs, weighted by their respective cost shares in regional industry output: p = p A + p Q + p r em Cr em a

(6.11)

Note that wages implicitly have an impact on total cost and thus on output prices through the prices for endogenous consumption expenditures in the term p Qa . Again assuming full competition, any change in one of the prices will be fully passed on forwardly. The solution of the Type II interregional IO price model is obtained by moving the terms with p from the RHS to the LHS of (6.11) and post-multiplying the result with the Type II interregional Leontief-inverse L∗∗ . This gives: p = p r em Cr em (I − A − Qa )−1 = p r em Cr em L∗∗

(6.12)

Similar to the solution of the single-region Type I price model (6.4), the solution of the interregional Type II price model (6.12) alsodelivers final output price multipliers of the P exogenous primary input prices, i.e. ri crpi lirjs ∈ Cr em L∗∗ . They equal the corresponding primary input quantity multipliers of exogenous final demand of the Type II quantity model in (6.9). Both sets of multipliers again sum to one, now because of the last part of (6.8). In case of the Type II interregional price model, these multipliers represent the direct, indirect and consumption-induced weight of the remaining exogenous primary input prices (mainly tax and subsidy rates, capital return rates and external import prices) in the prices of final output per industry per region. The similar “equality to one” in both price models, in fact, hides two opposing changes. The exogenous part of the primary input prices has become smaller, because interregional intermediate import prices as well as wage rates have now become endogenous (see Eq. 6.10), whereas the column sums of the dimensionally larger, now interregional Leontief-inverse have become larger. The latter even holds for the intra-regional part of the dimensionally larger Leontief-inverse. In case of the price model, this has two causes. First, the intraregional price multipliers in the extended model will be larger because the following interregional feedback effect has been added: output prices home region = export prices home region = import prices other regions => output prices of other regions = export prices other regions = import prices home region => output prices home region, and so on. Second, the intra-regional price multipliers will be larger because the following price–wage–price feedback effect has been added: output prices home region = consumption prices home region and other regions = (implicit) wage rates home region and other regions => output prices home region and so on. Finally, it is important to understand that there is a difference in the use of price multipliers versus quantity multipliers in impact studies. In the quantity model all quantities are measured in base-year index prices that are all equal to one (Schuman

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1968). A change in the quantity of exogenous final demand thus represents an absolute change, i.e. if exogenous final demand of an industry is 50 million euro and it goes up with 10%,  f ind = 5 million euro. A change in an exogenous primary input price, however, represents a relative change in its base-year value of one, i.e. if the foreign import price goes up with 10%, p f or = 0.10. Taking this into account, the Type II interregional IO price model enables comparable types of cost-push price simulations as the Type I single-region model, but now including the price–wage–price causality chain in an interregional setting. In (6.8)– (6.10), to keep the mathematics simply and the duality clear, it is assumed that the P exogenous remaining primary input prices are equal for all purchasing industries in all regions. In practical applications, precisely because they are exogenous, one may probably more realistically assume that the prices of these primary inputs behave differently for each purchasing industry. The first application of the above Type II price model may be found in Oosterhaven (1981), who uses it to simulate the regionally different consumer price impacts of the increases in the international oil prices in the 1970s, in case of the Netherlands. Bazzazan and Batey (2003) show an interesting application of the single-region Type II price model to energy price increases in Iran and compare these impacts with those predicted by a Type IV price model (see Sect. 5.2 for the quantity version of this price model). Moreover, they present dynamic versions of the Type II and Type IV single-region price models, with price feedbacks for the use of capital goods (see also Sect. 8.3). Besides, the Type II interregional price model may, for instance, also be used to simulate the impact of the increasing US import tariffs on Chinese products in the late 2010s on US consumer prices and further forwards on the prices of US exports to RoW, including China.

6.3 Interacting IO Price and Quantity Models: Lower Multipliers Still, no interaction between prices and quantities appears in any of the applications of the IO quantity and price models of whatever type, with, as far as we know, with one major exception. Madsen and Jensen-Butler (2004, see also Madsen 2008, Chap. 6) show how the empirical richness of very detailed, interregional SAMs may be maintained while introducing various price–quantity interactions, which is usually only achieved for all prices in much smaller, but highly nonlinear computable general equilibrium (CGE) models (see Bröcker 1998; Shoven and Whalley 1992) or it is achieved more partially for restricted sets of prices in larger econometric IO models (see Kratena 2017). The quantity version of the LINE model of Madsen and Jensen-Butler is akin to the Type IV demo-economic model with endogenous unemployment, explicitly distinguishing stocks from flows, as discussed in Sect. 5.2. It has 12 industries, 20 products, 14 age/gender groups, 5 education levels, 4 household types, 13 public

6.3 Interacting IO Price and Quantity Models: Lower Multipliers

81

consumption types and 10 capital/investment types, all for 277 Danish municipalities. The price version of the LINE model is the mirror image of its quantity version. The combination of both models is solved by iteratively switching from the quantity model to the price model and back. This iterative method of model solution is summarized in Fig. 6.2, where the Type II interregional IO price and quantity model of Sect. 6.2 are linked by means of two sets of price elasticities, namely a set of demand elasticities for exogenous final demand edp = (f ex ÷ f ex ) ÷ (p ÷ p) and a set of supply elasticities for remaining primary inputs esp = (vr em ÷ vr em ) ÷ (pr em ÷ pr em ), with ÷ indicating a cell-by-cell division. For a better understanding of the economics of the combined model, Fig. 6.3 has been added. It is equivalent to Fig. 6.2. The difference is that Fig. 6.2 shows the direct causal relations between the variables of the combined model, whereas Fig. 6.3 shows how these causal relations operate through shifts of and shifts along the corresponding demand and supply curves.

Exogenous change

IO quantity model

D

Remaining primary input, inverse supply elasticities

D

D S

S

S

f ex

Remaining final output, demand elasticities

vrem

x, Zi, Hi

IO price model

Exogenous change

D S S S D f ex

D x, Zi, Hi

vrem

Fig. 6.3 Markets in interacting Type II interregional Leontief price and quantity models. Legend Horizontal axes = quantities, with f ex = remaining final demand, x = total output, Zi = endogenous intermediate demand, Hi = endogenous consumption demand and vr em = remaining primary inputs. Vertical axes = corresponding prices. Demand curves D are dashed, whereas supply curves S are solid

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The top left panel and the bottom right panel of Fig. 6.3 contain the two sets of possible exogenous changes of the combined model as well as their direct impacts. The light grey arrows indicate in which order both sets of exogenous changes are transferred from market to market. The bold broken lines indicate demand curves, whereas the bold solid lines indicate supply curves. The top left panel shows the direct impact of an exogenous increase of one of the IR remaining final demand curves. In the demand-driven IO quantity model, any shift in demand is fully satisfied by an equally large shift along the supply curve of the corresponding type of total output, at prices that remain constant. These constant prices are the result of the assumption of the demand-driven model that the supply of total output by regional industry is perfectly price elastic, i.e. that their supply curves are horizontal (see further Sect. 2.2). The bottom right panel of Fig. 6.3 shows the direct impact of an exogenous increase of one of the P supply curves of remaining primary inputs. In the cost-push IO price model, any change in supply/price is confronted by a perfectly price-inelastic demand; i.e., in the price model, quantities do not react to price changes (see further Sect. 6.1). Hence, any increase in the supply/price of one of the inputs, shown in the middle bottom panel of Fig. 6.3, leads to an upward shift along its corresponding vertical demand curve. An impact study with the combined model may start with either an exogenous remaining primary input price change or an exogenous remaining final demand quantity change. Without loss of generality, we start the description of the causal changes from market to market with the top left panel of Fig. 6.3. The direct, one-to-one impact of a shift of one or more remaining final demand curves on total output at the going price—in the IO quantity model—leads to subsequent shifts of both the endogenous intermediate demand curves (Zi = Ax) and the endogenous consumption demand curves (Hi = Qa x) in the top middle panel of Fig. 6.3. The semi-circle with light grey arrows in Fig. 6.3 symbolizes the subsequent multiplier process detailed in Fig. 6.2. Both types of demand curves are vertical, indicating that both types of demand in the IO model are perfectly priceinelastic, whereas the supply of total output that satisfies these demands in the IO model is perfectly elastic. After the quantity model is run for the first time, any change in the IR types of total output in the quantity model also leads to shifts of the demand curves for the P remaining primary inputs (vr em = Cr em x), one of which is shown in the top right panel of Fig. 6.3. These demand curves are also vertical, as all endogenous demand in the IO model is inelastic to price changes. The top right panel, however, contains the first deviation from the basic IO model. Without changing any other of the IO quantity model’s assumptions, one may assume that the supply of remaining primary inputs is not perfectly price elastic. This does not change the working nor the outcomes of the model, because these inputs do not feedback into the IO quantity model. This first deviation entails the assumption of upward sloping supply curves for remaining primary inputs in the top right panel of Fig. 6.3. The consequence is that the prices of remaining primary inputs do not remain constant anymore, but move upward along the supply curves of these inputs

6.3 Interacting IO Price and Quantity Models: Lower Multipliers

83

with pr em = (ˆesp )−1 vr em . Note that this direction of causality requires the use of the inverse of the supply elasticities eˆ sp . With the thus estimated increases in remaining primary input prices, the Type II interregional price model is run, starting in the bottom right part of Fig. 6.2 and the bottom middle panel of Fig. 6.3. There, with the usual assumptions of the IO price model, these price increases lead to increases in the prices of total input/output, along the—already introduced—perfectly inelastic demand curves for endogenous intermediate and consumption expenditures, i.e. p = p r em Cr em (I − A − Qa )−1 . The semi-circle of light grey arrows in the bottom middle panel of Fig. 6.3 symbolizes the multiplier process of the IO price model, shown in more detail in the bottom middle panel of Fig. 6.2. Note that any exogenous change in one of the P primary inputs prices in the bottom right panel of Fig. 6.3 needs to be added to the endogenous change in those prices from the top right panel, but only in the first round of the iterative solution process. The last, bottom left panel of Fig. 6.3 does more or less the same as its top right panel and thus contains the second deviation from the basic IO model. The top right panel links the IO quantity model to the IO price model. Here, the opposite happens. Remaining final demand is the only type of demand that does not need to be perfectly price inelastic to preserve the working of the IO quantity model, as it is exogenous in that model. Instead, the bottom left panel assumes downward sloping demand curves for remaining final demand. In formula, the bottom left panel thus assumes f ex f ex = eˆ p p. Note that, as opposed to the supply elasticities of top right panel, the bottom left panel’s demand elasticities are used in their usual direction, i.e. from price change to quantity change. More importantly, note that demand elasticities have a negative sign. This implies that the first round feedbacks from the IO price model back into the IO quantity model reduce all positive first round quantity impacts in each of the three top panels of Fig. 6.3. Subsequently, in the second round of the iterative solution procedure, this results into a negative second round quantity effect in the top right panel, reducing the positive first round effect on the price of the remaining primary inputs. Consequently, the second round feedback from the IO quantity model into the IO price model will diminish the first round positive price impacts in all three bottom panels. Both sets of opposite second round signs show that both the Leontief quantity multipliers and the Leontief price multipliers overestimate their typical impacts (see Sect. 5.2 for other reasons why Type II quantity multipliers are too high). Note that the size of these multiplier-reducing effects of adding demand and supply elasticities to the combination of the two models is only indirectly related to the question whether or not the iterative solution procedure converges. The size of the elasticities is decisive in this respect.1 1

To see that, consider that all remaining primary input quantity multipliers of exogenous final demand are equal to one, i.e. i Cr em (I − A − Qa )−1 = i , because i Cr em + i A + i Qa = i . The same “equal to one” property holds for the identical remaining final demand price multipliers of remaining primary input prices. This implies that convergence is solely determined by the size of

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To keep the mathematics as well as the parameter requirements as minimal as possible, both elasticity-induced feedbacks are specified at their most aggregate level. In fact, the above-defined feedback from the quantity model into the price model only requires P primary supply elasticities, while the reverse feedback only requires IR final demand elasticities. When the data are available and/or when the type of application requires it, these two feedbacks may easily be applied at more disaggregate levels. Instead of only using demand elasticities for the row totals of the matrix with remaining final demand, the combined Type II interregional IO model may equally well use different elasticities for each individual cell of the remaining final demand matrix Fex (see Table 2.2), which represents the other extreme as regards elasticity data. In this way, demand elasticities for foreign exports may be different from those for domestic sales to the various regions, while government demand elasticities may be different from investment and exogenous household demand elasticities. Analogously, instead of only using P supply elasticities for the row totals of the matrix with remaining primary inputs, the combined model may equally well use elasticities at the cell level of the matrix Vr em (see Table 2.2). In this way, a distinction may be made between the supply elasticities for foreign imports and those for other primary inputs and between imports of different products as well as between imports of different domestic industries and different types of households, while all may also be made different by region of destination. A simple application of the basic idea of the combined IO price–quantity model would be to do just one iteration, starting with the solution of a single-region Type I price model, via negative demand elasticities, back to the solution of the singleregion Type I quantity model, as done in Choi et al. (2010) for an analysis of the impacts of a carbon tax on output, resource use and CO2 emissions by US industries. However, their interpretation that such an analysis shows the short-run impacts of a carbon tax, whereas doing more iterations would show the longer run impacts is false, as the length of the market equilibrium process has no relation with the length of iterative solution of the combined model. In fact, Figs. 6.2 and 6.3 still represent a comparative static model. The duration of the market equilibrium process may be quick if people and firms have perfect expectations about future price and quantity changes. It may also take a long time with temporary adaptions in various levels of stocks when information about future changes is not perfect (Romanoff and Levine 1986). In fact, a better way to distinguish short-run from long-run impacts would be to use different elasticities, as short-run price elasticities are usually much smaller than longer run price elasticities. Finally, note that notwithstanding the presence of two price–quantity interfaces between the Type II interregional price and quantity model, the combined model still does not serve to simulate the impact of supply-side quantity shocks to the the elasticities. A sufficient condition for convergence is that all elasticities are smaller than unity. Consequently, the necessary condition for convergence will be that some weighted average of the demand elasticities as well as some weighted average of the supply elasticities is smaller than one. In an email of 24 April 2021 Bjarne Madsen reports that he never had a problem with convergence in the many versions and the many applications of LINE, which uses the same modelling approach.

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interregional economy, such as those caused by natural or man-made disasters. It only serves to simulate the interregional, interindustry price and quantity impacts of supply-side price shocks, such as international oil price hikes, the introduction of carbon taxes, the imposition of import tariffs and the like.

References Bazzazan F, Batey PWJ (2003) The development and empirical testing of extended input-output price models. Econ Syst Res 15:69–86 Bröcker J (1998) Operational spatial computable general equilibrium modelling. An Reg Sci 32:367–387 Chenery HB, Clark PG (1959) Interindustry economics. Wiley, New York/London Choi J-K, Bakshi BR, Haab T (2010) Effects of a carbon price in the U.S. on economic sectors, resource use, and emissions: an input-output approach. Energy Pol 38:3527–3536 Evans MK (1973) A forecasting model applied to pollution control cost. Pap Proc Eighty-Fifth Annu Meet Am Econ Assoc 63:244–252 Giarratani F (1974) The effect on relative prices of air pollution abatement: a regional input-output simulation. Model Simul 5:165–170 Kerscher K, Prell C, Feng K, Hubacek K (2013) Economic vulnerability to peak oil. Global Environ Change 23:1424–1433 Kratena K (2017) General equilibrium analysis. In: ten Raa T (ed) Handbook of input-output analysis. Edward Elgar, Cheltenham Leontief WW (1951) The structure of the American economy: 1919–1939, 2nd edn. Oxford University Press, New York Madsen B (2008) Regional economic development from a local economic perspective—a general accounting and modelling approach. Habilitation Thesis, University of Copenhagen Madsen B, Jensen-Butler C (2004) Theoretical and operational issues in sub-regional economic modelling, illustrated through the development and application of the LINE model. Econ Model 21:471–508 Miernyk WH (1976) Some regional impacts of the rising costs of energy. Pap Reg Sc Assoc 37:213– 227 Oosterhaven J (1981) Export stagnation and import price inflation in an interregional input-output model. In: Buhr W, Friedrich P (eds) Regional development under stagnation. Nomos-Verlag, Baden-Baden Oosterhaven J, Polenske KR, Hewings GJD (2019) Modern regional input-output and impact analysis. In: Capello R, Nijkamp P (eds) Handbook of regional growth and development theories: revised and extended second edition. Edward Elgar, Cheltenham Polenske KR (1979) Energy analyses and the determination of multiregional prices. Pap Reg Sci Assoc 43:83–97 Romanoff E, Levine SH (1986) Capacity limitations, inventory, and time-phased production in the sequential interindustry model. Pap Reg Sci Assoc 59:73–91 Schuman J (1968) Input-output analyse. Springer Verlag, Berlin Shoven JB, Whalley J (1992) Applying general equilibrium. Cambridge University Press, New York

Chapter 7

Supply-Driven IO Quantity Model and Its Dual, Price Model

Abstract The supply-driven IO quantity model is shown to be the mirror image of the standard IO model. In this Ghosh model, any change in the exogenous supply of primary inputs is passed on forwardly to purchasers that pass it on further with fixed intermediate and fixed final output coefficients. The Ghosh model assumes a single homogeneous input, which means that factories may work without labour. The Type II supply-driven model, additionally, has a supply-driven consumption function, which allows kitchen appliances to run without electricity. The dual of the Ghosh quantity model, the revenue-pull IO price model, simulates the backward passing on, under full competition, of any final output price change to the suppliers of intermediate inputs who pass them on further, to end up in changes in the endogenous prices of the primary inputs. Finally, the functioning of markets in all four basic IO models is compared, which shows that the price and quantity impacts in all four models are overestimated. Keywords Supply-driven IO quantity model · Allocation coefficients · Trade destination ratios · Ghosh-inverse · Processing coefficients · Revenue-pull IO price model · Revenue shares · Price multipliers Chapter 5 ended with the conclusion that it may be more useful to add prices and the supply side to a Type IV interregional IO model, instead of further endogenizing its final demand. Chapter 6 dealt with the role of prices in IO models. Here, we continue with the role of the supply side in IO models.

7.1 On the Plausibility of the Supply-Driven IO Quantity Model In the 1970s and 1980s, it was thought that the supply-driven IO model (Ghosh 1958) would be suited to simulate the impacts of quantity shocks to the supply side of the economy.

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2022 J. Oosterhaven, Rethinking Input-Output Analysis, Advances in Spatial Science, https://doi.org/10.1007/978-3-031-05087-9_7

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7 Supply-Driven IO Quantity Model and Its Dual, Price Model

7.1.1 Basic Supply-Driven IO Model: Factories May Run Without Labour The core assumption of the basic (Type I) supply-driven IO quantity model is that the intermediate and final output ratios (also called allocation coefficients) are fixed along the rows of the IOT and determine the size of the supply of intermediate and final outputs by industry: z i j = xi bi j ∈ Z = xˆ B and yiq = xi diq ∈ Y = xˆ D

(7.1)

When only one single IOT is available, point estimates of these output ratios are calculated along the rows of the IOT by means of B = xˆ −1 Z and D = xˆ −1 Y. Note that (7.1) represents the exact opposite of the assumption of fixed intermediate and fixed primary input ratios of the demand-driven quantity IO model (2.3). The second assumption of the demand-driven model, namely that total output/production follows the demand for outputs along the rows of the IOT, also has its exact opposite in the supply-driven IO model (SDIOM); therein total input/production follows the supply of primary and intermediate inputs along the columns of the IOT:   zi j + v pj ∈ x = i Z + i V = i Z + v (7.2) xj = p

i

Hence, instead of final demand, the SDIOM has the supply of primary inputs as its exogenous driving force. This is illustrated by Fig. 7.1, where no arrows are entering the box with primary inputs. Any increase in exogenous primary inputs leads to an equally large increase in total input, indicated by the outgoing arrow with the I. The equally large absolute increase in total output is then distributed to intermediate and final users, in accordance with the fixed intermediate and final output ratios, as indicated by the arrows with B and D, respectively. Hence, the direct supply effect of an unit increase in primary inputs equals I, the first round indirect supply effect equals IB, the second round indirect supply effect equals IB2 and so on. Like in the demand-driven model, the cumulative supply effect of a unit increase in exogenous primary inputs thus equals the outcome of a Taylor expansion, namely I + B + B2 + B3 + … = (I −

Primary input

I

D

Total output

B

Final output

I

Intermediate output

Fig. 7.1 Causal structure of the supply-driven IO quantity model. Legend B = matrix with intermediate output coefficients, D = matrix with final output coefficients

7.1 On the Plausibility of the Supply-Driven IO Quantity Model

89

B)−1 = G, where G is the so-called Ghosh-inverse. A sufficient condition for this Taylor expansion to converge is that the row sums of the output ratios are all smaller than one, i.e. B i < i, which is a specific case of the Brauer-Solow row and column sum criteria (Nikaido 1970, p. 18). This will almost always be the case, because B i + D i = i, while final output will almost always be positive. The economics of the SDIOM may be problematic: its mathematical solution is simple. It follows from substituting the first part of (7.1) into (7.2), moving the term with x B to the LHS of (7.2), and instead of pre-multiplying with the Leontiefinverse, the result now has to be post-multiplied with the Ghosh-inverse G. This gives the following solution for the row with total input: x = v (I − B)−1 = v G

(7.3)

Additionally, instead of the endogenous intermediate and primary inputs of the demand-driven model, the SDIOM has endogenous intermediate and final outputs. Their solution follows from substituting (7.3) into (7.1), which gives: Z = vˆ G B and Y = vˆ G D

(7.4)

 In (7.4), the cells of j gi j d jq ∈ G D represent the final output (i.e. consumption, investments and export) quantity multipliers of the exogenous primary input of industry i. Originally, Ghosh (1958) formulated his model for the—then—rather centrally planned Indian economy, which suffered from excess demand and multiple shortages. Distributing the scarce supply of state-controlled industries according to historical allocations seemed a wise first approach to planning the economy. On second thought, however, the consequences of applying the SDIOM in centrally planned economies may be severe, as this model ignores the complementarities of inputs along the columns of the IOT, which may easily lead to the, only seemingly implausible cooccurrence of stocks of redundant supplies of some products with shortages of other products (see Oosterhaven 1988). Early on, the SDIOM also became regarded as a serious alternative for projections and impact studies in case of market economies, partly because Ehret (1970) for Germany, and Giarratani (1981) and Bon (1986) for the USA, reported a similar temporal stability of input coefficients and output coefficients. Helmstädter and Richtering (1982), for Germany, for 1960–75, even found output coefficients to be significantly more stable and reported smaller prediction errors for the SDIOM than for the demand-driven IO model. These, hardly discriminating empirical findings are not surprising in view of the following close relationship between the two sets of coefficients: B = xˆ −1 Z = xˆ −1 A xˆ and A = Z xˆ −1 = xˆ B xˆ −1

(7.5)

Consequently, only under conditions of uneven growth by industry one may expect a significant difference in stability of the two sets of coefficients. This follows even

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more clearly from the specification of the temporal change in one set of coefficients whenever the other set is assumed to be stable (Chen and Rose 1986): Bt+1 = gˆ −1 At gˆ for a stable A matrix and At+1 = gˆ Bt gˆ −1 for a stable B matrix

(7.6)

where gˆ = xˆ t+1 xˆ t−1 = a diagonal matrix with the growth rates of total output by industry. Note that (7.6) will, especially, be discriminating if only one or a few industries receive a supply or demand shock, as will be the case with impact studies, but not with projections for an entire economy. Early applications of the SDIOM were done by Augustinovics (1970) and Giarratani (1976). Augustinovics inventively used modifications of (2.5) and (7.4) for international and intertemporal comparisons of economic structure, while Giarratani  used the total supply multipliers j gi j ∈ G i to indicate the potential, economywide forward or downstream impact of changing the supply of energy producing industries i. Later on, the model also became used to simulate the impacts of specific product shortages and supply disruptions (Davis and Salkin 1984; Chen and Rose 1986), such despite early warnings about the implausibility of the model (Giarratani 1980; Oosterhaven 1981, 140–41). This led to a sharp debate (Oosterhaven 1988; Gruver 1989; Rose and Allison 1989; Oosterhaven 1989) leading to the conclusion that the SDIOM is unsuited to do impact studies of quantity shocks to the supply side of the economy. In an otherwise fine article (Rose et al. 2018), it was recently suggested that the SDIOM would be less implausible in case of negative supply shocks as opposed to positive supply shocks. This is clearly not the case. Why? Consider a sudden drop in the production of aluminium of say 50%. Before using the SDIOM, this drop first needs to be translated into an equivalent drop in the primary input of the aluminium industry, as only primary input is exogenous in the SDIOM. However, what is “equivalent” is not evident. The most plausible drop in primary inputs would be a corresponding drop of 50%, which may be set equal to say an absolute drop of 50 million dollars in the value of primary inputs. What would common sense predict regarding the impact of a 50% drop in the supply of primary aluminium inputs? The direct effect would, of course, be an—in value terms—much larger absolute drop in total output, about equal to the inverse of the primary input coefficient times the exogenous drop in primary aluminium input. This inverse may best be labelled as a processing coefficient (Oosterhaven 1988). Besides, one would directly also expect a drop of about 50% in all other inputs of the aluminium industry, as these are no longer needed. Neither of these two most likely direct effects will be predicted by the SDIOM, which only predicts an output effect that will be much too low, as the direct output effect is wrongly assumed to exactly equal the exogenous primary input effect (see Fig. 7.1). Next, consider the first round indirect effects. Aluminium using industries will be confronted with a reduction of their supply and will look for substitutes, either spatially (imports) or technically (aluminium substituting products). However, these

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two logical behavioural reactions are not predicted by the SDIOM. Imports cannot change, as they are exogenous in the SDIOM, whereas local industries that are able to produce substitutes are predicted by the SDIOM to decrease their output, instead of to increase it.1 Moreover, cumulatively, all other industries will decrease their production, either directly as aluminium users or indirectly as users of users, but they will not proportionally decrease their value added nor increase their imports of aluminium substitutes, as these are exogenous in the SDIOM (see again Fig. 7.1). The reasons for these false predictions of the SDIOM are found in its theoretical assumptions. The micro-economic foundation of the demand-driven IO model started in Sect. 2.2 with simplifying the most general production function, by assuming a single homogeneous output and multiple heterogenous inputs. The comparable foundation the SDIOM starts with the opposite simplification of assuming a single homogeneous input and multiple heterogeneous outputs that are produced according to the following “Ghosh” production function, which is the opposite of (2.7):2   xi = max z i j /bi j , ∀ j; yiq /diq , ∀q

(7.7)

  With (7.7) maximizing profits ( j z i j p j + q yiq pq − xi pi ) under full competition (i.e. at given market prices) comes down to maximizing revenues and producing the heterogenous outputs in fixed proportions according to (7.1). Under full competition, industry i will thus have a perfectly elastic demand for its single homogenous input (it buys everything that is supplied to it at the going price) and have a perfectly inelastic supply of its multiple and mutually complementary (called joined) intermediate and final outputs (Oosterhaven 1996). Obviously, the assumption of a production process with a single homogeneous input is ludicrous, as it implies that all inputs are perfect substitutes for each other; i.e., factories may run without labour, cars may drive without gasoline and so on. This alone should be sufficient reason to not use this model at all. The perfect jointness of all intermediate and final outputs is another extreme assumption.3 Note that, comparable to the intermediate input coefficients of the 1

Note that Chen and Rose (1986) do report “counter intuitive results” and talk about “absurd machinations of the supply model” in case of a 50% reduction of aluminium production in Taiwan. However, at that time, they did not conclude that the model could not be used for impact studies of such type of events. 2 An alternative interpretation in which the Ghosh model is not the exact opposite of the Leontief model is presented by de Mesnard (2009). His point of departure is a physical IO table that has homogenous outputs along its rows and heterogeneous inputs along its columns. With this asymmetric base assumption naturally only asymmetric results can be derived. In reality, however, even tons of steel have different qualities and different prices and cannot be simply added in physical units. In reality, any IOT will have heterogeneous outputs along its rows as well as heterogenous inputs along its columns. This more realistic situation is our point of departure. This is also the reason why we do not discuss IO models based on physical data (as in Miller and Blair 2022, Chap. 2), as such data are impossible for entire economies. 3 The equivalent of this assumption in case of a supply-driven supply-use model consists of the product of two separate assumptions (de Mesnard 2004), namely (1) the fixed product mix assumption that applies to the rows of the Supply table and (2) the fixed (intermediate and final) product

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7 Supply-Driven IO Quantity Model and Its Dual, Price Model

open economy Leontief model (2.11), the intermediate output coefficients of the open economy SDIOM also need to be viewed as the product of a technical Ghosh coefficient (7.7) and a trade destination ratio, instead of the trade origin ratio in the demand-driven IO model. The fixedness of the technical Ghosh coefficients may apply to some parts of some industries for technical reasons, such as chemical formulas in the chemical industry. It may further apply for economic reasons to those industries that in the short run want to keep all their customers equally supplied (Giarratani 1981). In most cases, however, ignoring differential developments in the willingness to pay on the demand side will result in wrong projections of the economy. In fact, only by a smart combination of processing coefficients (i.e. inverse input coefficients) and intermediate output coefficients, while adapting intra-regional purchase coefficients to accommodate for import and export substitution, one may still use the basic forward linkages causality of the SDIOM (see Oosterhaven 1981, Chap. 8, and Oosterhaven 1988, for the model, and FNEI 1977, for its first application to a 4000 ha land reclamation plan, see Cartwright et al. 1982, for a nuclear disaster application, and Rose and Wei 2013, for a port shutdown application).4 Finally, have a second look at Fig. 7.1 and note that the (forward or downstream) causal structure of the SDIOM is exactly equal to that of the cost-push IO price model shown in Fig. 6.1. The only difference is in the content of the effects. In Fig. 6.1, prices are moving, whereas quantities move in Fig. 7.1. Hence, it is not surprising that the SDIOM may also be interpreted as the cost-push price model expressed in values, instead of in prices (Dietzenbacher 1997). The proof is easy. Substitute A = xˆ B xˆ −1 into the solution of the cost-push price model (6.3) and post-multiply the result with xˆ . This gives: p xˆ = p xˆ B xˆ −1 xˆ + p v C xˆ

(7.8)

Next, substitute V = C xˆ in (7.8) and simplify and solve the result as follows: p xˆ = p xˆ B + p v V ⇒ p xˆ = p v V (I − B)−1 = p v V G

(7.9)

sales ratios assumption that applies to the rows of the Use table (see Table 4.3). Assumption (1) seems less implausible than the comparable SDIOM assumption, but assumption (2) is equally implausible. 4 FNEI (1977) was my second IO application. In total at least eleven (!) employment estimates were made. The lowest with a negative (!) estimate 9–60 full time jobs in 1986 was made by an environmental protection group, and was based on the assumption of a rationalisation of the use of old land because of its assumed combined exploitation with the new land. The highest positive estimate, made by an agricultural interests’ group, amounted to 480–555 full time jobs for 1975. Our own estimate came to 135–170 jobs for 1975 and 100–130 for 1985. Long articles in local newspapers went as far as discussing the technical details of the various estimations, inter alia the basic idea of IO analysis, and welcomed our estimate because we were considered to have no direct interests in this matter (see Oosterhaven 1981, Chap. 9, for a full review).

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This shows that the cost-push price model is identical to the Ghosh quantity model, except that the quantities of the Ghosh model (7.3) are replaced with their corresponding values in (7.9). Thanks to this economically far more plausible price interpretation, as opposed to the original implausible quantity interpretation, the row sums of the Ghosh-inverse (I − B)−1 may be used to indicate the size of an industry’s forward linkages, just as the column sums of the Leontief-inverse (I − A)−1 may be used as a measure of the size of the backward linkages of an industry (for a further discussion, see Sect. 9.1).

7.1.2 Type II Supply-Driven IO Model: More Private Cars May Run on Less Gas The economic causality and the mathematics of the interregional extension of the single-region or single-nation SDIOM is straightforward. The multi-regional extension is mathematically somewhat more complicated (Bon 1988), but is not of much interest, as most MRIOTs have been subject to RAS procedures, which is why they are usually published and used as IRIOTs (see further Sect. 4.3). The Type II extension of the single-region SDIOM (Davis and Salkin 1984) is more complicated and more interesting, particularly since Guerra and Sancho (2011) claim that this extension makes the SDIOM more plausible in case of centrally planned economies. Oosterhaven (2012), however, argues that the Type II SDIOM is even more implausible for market economies than the Type I SDIOM, while it becomes even more problematic as a guide for centrally planned economies. Why? The causality of the Type II Ghosh model runs in the same forward direction as that of the extended cost-push price model shown in bottom part of Fig. 6.2. Therefore, its basic accounting identity is very similar to that of (6.10). It expresses that total input/production is determined column-wise, by the exogenous supply of the remaining (non-labour) primary inputs v r em = i Vr em , by the endogenous supply of consumption good to workers i H, and by the endogenous supply of intermediate inputs i Z: x = i Z + i H + v r em

(7.10)

This, additional to the basic SDIOM, further implies that any increase in household consumption of workers (i.e. in their wage sums) leads to an absolutely equally large direct increase in total input of firms, without the direct need of any other intermediate or primary inputs, as all inputs are perfect substitutes for one another. The Type II Ghosh model, furthermore, requires a split up of the endogenous final output in (7.1) into H and remaining final output Yrem , and requires the subsequent use of now three sets of fixed output coefficients (i.e. allocation ratios): Z = xˆ B, h i j = xi dihj ∈ H = xˆ Dh and Yr em = xˆ Dr em

(7.11)

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The new, middle term of (7.11) expresses that any increase in the total supply of industry i leads to a percentage-wise equally large increase in the consumption of i, without the need of any other consumption goods and services. This implies that now consumers may, just like firms in the Type I model, also drive their cars without gasoline as well as run their kitchen appliances without electricity. The solution of the Type II Ghosh model is derived as usual. The terms with Z and H from (7.11) are substituted into (7.10). Then, the terms with x on the RHS of (7.10) are moved to its LHS and the result is post-multiplied the Type II Ghosh-inverse G∗ = (I − B − Dh )−1 , which results in the solution for total inputs/production:   −1  r em  ∗  = v G x = vr em I − B − Dh

(7.12)

and the solution for the endogenous remaining final outputs:  −1 Yr em = vˆ r em I − B − Dh Dr em = vˆ r em G∗ Dr em

(7.13)

Up till now the Type II Ghosh model has only been used in some numerical exercises that discuss its plausibility (Guerra and Sancho 2011; Oosterhaven 2012; Manresa and Sancho 2013). The additional assumptions of the supply-driven consumption function are shown above to be as implausible as those of the supplydriven intermediate output function. Whether or not the combination of both sets of implausible assumptions makes the Type II model more or less implausible than the Type I model is not directly clear. The impacts of four hypothetical exogenous impulses (scenarios), shown in Table 7.1 answer this question. The impacts shown are calculated by applying the Type II Leontief model to two exogenous final demand scenarios and by applying the Type II Ghosh model to the two corresponding exogenous primary input scenarios. The Table 7.1 Output, value added and consumption impacts of twoa scenarios, with two modelsb Type II demand-driven Leontief model with a shift in government expenditures

Type II supply-driven Ghosh model with a shift in product taxes and subsidies

Scenario +5 in i1 & −5 in i3 +5 in i2 & −5 in i3 +5 in i1 & −5 in i3 vi

h i

xi

vi

h i

xi

vi

h i

+5 in i2 & −5 in i3

Industry

xi

xi

vi

h i

i1

+ 4.7 + 0.9 −0.8 −2.4 −0.5 −1,4 + 8.2 + 0.6 + 2.9 + 0.6 −0.2 + 0.2

i2

−1.6 −0.2 −0.1 + 2.0 + 0.2 −0,2 + 1.7 + 0.3 + 0.1 + 4.8 −0.1 + 0.2

i3

−5.8 −2.3 −0.7 −6.4 −2.5 −1,2 −2.4 + 1.3 −0.7 −3.5 −0.3 −1.1

Totalc

−2.7 −1.5 −1.5 −6.8 −2.8 −2.8 + 7.5 + 2.2 + 2.2 + 1.9 −0.6 −0.6

The results for a third scenario of “+5 in i1 & −5 in i2” may easily be derived by deducting the results of the first scenario from those of the second scenario b All coefficients and values used are derived from Table 2a in Guerra and Sancho (2011) c Due to individual rounding, the numbers do not necessarily add up to the rounded totals Source Oosterhaven (2012) a

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first, upper left number in Table 7.1 (+4.7) indicates that the Type II Leontief model predicts that total output in industry i1 increases with 4.7, say billion dollar, if the government shifts 5 billion dollar of its expenditures from industry i3 to industry i1. The interpretation of the remaining numbers in Table 7.1 is similar. First, note that the changes in output xi and the changes in value added vi , for all three industries, move in the same direction when the Type II Leontief model is used in the first two scenarios of Table 7.1. When the Type II Ghosh model is used in the last two scenarios of Table 7.1, however, the changes in output and value added move in opposite directions in three of the six cases, which is definitely worse than the already implausible “no change in value added” of the Type I Ghosh model.5 Second, the pattern of the changes in household consumption h i in the two models leads to a similarly strong conclusion. In the Type II Leontief model, all changes have the same sign, whereas in the Type II Ghosh model the changes in i3 have a sign that is opposite to the changes in i1 and i2. This means that more cars in the Type II Ghosh model may actually require less gasoline to run, while additional kitchen appliances may actually require less electricity, which also is an outcome that is worse than the “zero change” of the Type I model. Finally, have a look at the totals of the three impact variables shown in the last row of Table 7.1. First, note that the changes in the totals for value added and for household consumption are exactly equal in both models, regardless of their opposite role in the causal chain. In the Type II Leontief model, it is total value added that drives the column with consumption expenditures, whereas in the Type II Ghosh model it is total consumption that drives the row with value added by industry, leading to the same total impact. So, we only need to compare the totals for production and value added to get one more impression of the comparative plausibility of the two extensions. In case of the extended Leontief model both totals move in the same direction, as they almost always do in real life. In the case of the extended Ghosh model, they move in an opposite direction in one of the two scenarios, which is highly unlikely. That is enough on the plausibility of the SDIOM. What about the plausibility of its hardly known dual, price model?

5

Manresa and Sancho (2013) rightfully point out, partly in reaction to Oosterhaven (2012), that the mathematical symmetry between the Ghosh and Leontief models regards all outcomes. They, subsequently, suggest that it is equally problematic when production and consumption move in a different direction in case of the Type II Leontief model, as they do in Table 7.1 in two out of four cases, whereas they always move in the same direction in case of the Type II Ghosh model. I disagree. Mathematical symmetry does not imply symmetry in economic plausibility. There is no reason why moving in a different direction should be implausible in case of production and consumption, definitely not by industry and not even at the aggregate level.

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7.2 Revenue-Pull IO Price Model: Its Plausible Dual Literally, in the footnotes of the debate on the plausibility of the supply-driven IO model, the question was raised (Oosterhaven 1988) and answered positively (Oosterhaven 1989) whether or not the supply-driven IO quantity model also has a dual, price model. Davar (1989) presents this dual in detail, but does not discuss the plausibility of the underlying economic assumptions nor their implications. Interestingly, the dual of the Ghosh quantity model has not been used for price impact studies yet, even though this is clearly possible, as follows from both its behavioural assumptions and its solution. Here, we skip the single-region price model (see Oosterhaven 1996) and directly present its interregional extension, which is mathematically identical but has matrices and vectors with dimension IR instead of I. In all respects, this dual is the mirror image of the cost-push price dual of Sect. 6.1. Not primary input prices, but final output prices are the driving exogenous factor in this model (see Fig. 7.2, where no arrows are flowing into the final output prices box). In empirical price impact analyses, it may often be useful to assume that each cell of the final output matrix Y has its own exogenous price change. However, in presenting the basic revenue-pull IO price model, we assume that each of the QR final output columns y = i Y has its own uniform price (change), just like in the cost-push model where we assumed that each of the P primary input rows V i = v had its own uniform price (change). The direct effect of an increase in these QR prices (i.e. pqs ∈ p y  ) will be an increase the total revenue of all industries in all regions that supply to final demand type q in region s. The size of these revenue increases, of course, equals p y  times the final revenue shares of q in s in the total sales of each industry i in each region r rs rs rs = yiq /xir ∈ D). These direct price effects thus equal diq pqs ∈ D p y  , as (i.e. diq indicated by the arrow with the D in Fig. 7.2. Since quantities, being determined by the quantity model, are constant in the price model, the assumption of full competition ensures that these revenue increases will entirely be passed on backwardly in the prices of the single homogeneous input of each industry i in each region r. They thus apply column-wise to all suppliers of primary and intermediate inputs to each industry j in each region s, as indicated by the two arrows with the matrix I in Fig. 7.2.

Final output prices

D

Total input prices

I

I

Primary input prices

B

Intermediate input prices rs ∈ D = Fig. 7.2 Causal structure of the interregional revenue-pull IO price model. Legend diq IRxIQ matrix indicating the share of final demand of type q in region s in the total revenues of industry i in region r and birjs ∈ B = IRxIR matrix with indicating the share of industry j in region s in the total revenues of i in r

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97

For the suppliers of primary inputs to j in s, this will not lead to any further causal effects, as indicated by the absence of arrows flowing out of the box with primary input prices in Fig. 7.2. For the suppliers of intermediate inputs to j in s, however, these price increases represent revenue increases, which equal the direct price increase Dp y  times the intermediate revenue shares of the sales of i in r to j in s (i.e. birjs = z ri js /xir ∈ B), as indicated by the arrow with the B in Fig. 7.2 These indirect revenue increases, under full competition, with constant quantities, are again be fully passed on in the prices of the homogenous inputs of the suppliers j in s. The corresponding first round indirect price effects thus equal B D p y  , which are again fully passed on, leading to second round indirect price effects of B2 D p y  , and so on. The cumulative price effect for the single homogeneous input by regional industry, consequently, equals (I + B + B2 + B3 + ...) Dp y  = (I − B)−1 Dp y  = G∗∗ Dp y  = p, in which G** = interregional Ghosh-inverse: This description of the causality of this price model clarifies why it may best be labelled as the revenue-pull IO price model, which more accurately summarized its nature than the “demand-pull” label used in Oosterhaven (1996). The mathematics of the interregional revenue-pull price model starts, not with the column-wise identities for total cost, as in the cost-push price model (6.1), but with the row-wise identities for total revenue: xir pir =

s  j

z ri js p sj +

s 

rs s yiq pq , or in matrix algebra:

q 

(7.14)



xˆ p = Z p + Yp y  , with y = i Y In (7.14), the QR prices for each column with homogeneous final inputs ( pqs ∈ p y  ) are exogenous, whereas the IR prices for each column with homogeneous industry inputs ( p sj ∈ p) are endogenous. Note the difference with the cost-push price model. There we had prices for the single homogeneous outputs. Here we have prices for the single homogeneous inputs. However, in both price models quantities do not change, which means that the quantity implications of the implausible single homogeneous input assumption do not occur in the Ghosh price model. The unchanging quantities of intermediate and final outputs in (7.14) are determined by the fixed output ratios of the interregional Ghosh quantity model. Substitution of (7.1) in (7.14) and pre-multiplication with xˆ −1 gives: p = B p + D py

(7.15)

which shows that, in case of the revenue-pull price model, total output prices equal the sum of all column-wise uniform intermediate and final input prices, weighed along the rows of the IRIOT (see Table 2.2) with their respective intermediate and final revenue shares (i.e. allocation coefficients). Note that the interregional intermediate and final revenue shares add to unity, i.e. Bi + Di = i, as the accounting identities (7.14) are measured in base-year prices equal to one.

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The solution of the interregional revenue-pull price model is obtained by moving the endogenous total input prices p from the RHS to the LHS of (7.15) and premultiplying the result with the interregional Ghosh-inverse G∗∗ : p = (I − B)−1 D p y  = G∗∗ D p y 

(7.16)

 In (7.16), tj girjt d tsjq ∈ G∗∗ D contains the primary input price multipliers of industry i in region r with respect to the exogenous price of final demand of type q in region s. They represent the opposite of the final output price multipliers of exogenous primary input prices in the solution of the cost-push price model in (6.4). Moreover, they equal the final output quantity multipliers of exogenous primary input of the Ghosh quantity model in (7.4). Comparable to the primal–dual relationship between the Leontief quantity and price model of (6.5), the primal–dual relationship between the Ghosh quantity and price model may be further illustrated by pre-multiplying (7.16) with total primary input v, which gives: v p = v (I − B)−1 D p y  = x D p y  = y p y 

(7.17)

Obviously, in the Ghosh models, the equality between the value of total primary input v p and the value of total final output y p y  is preserved, just as in the Leontief models. More importantly, (7.17) shows the independence (or better duality) of the two Ghosh models. Although their solutions are linked by means of (7.17), their variables move independently, with exogenous primary input quantities forwardly determining final output quantities in the quantity model, and exogenous final output prices backwardly determining primary input prices in the price model. In view of the observed implausibility of the SDIOM, one would expect a comparable implausibility of its dual, revenue-pull price model. However, this is not the case. In the revenue-pull price model, quantities are assumed to be constant and revenue gains pull the prices backwardly up, just as cost increases forwardly push prices up in the cost-push price model. The major difference between the two price models is the assumption about what changes exogenously, final output prices or primary input prices. After that, the forward or backward nature of the causality chain more or less logically follows from that starting assumption. Furthermore, and for completeness sake, note that the causality of the Ghosh price model in Fig. 7.2 runs in exactly the same, backward/upstream direction as that of the Leontief quantity model in Fig. 2.1. This suggests that the Leontief quantity model (2.4) may also be interpreted as the revenue-pull price model expressed in values, instead of in prices, just like the Ghosh quantity model (7.3) could be interpreted as the cost-push price model expressed in values, as shown in (7.9). The proof is comparable (Oosterhaven 2022): B = xˆ A xˆ −1 is substituted into the revenue-pull price model (7.15) and the result is pre-multiplied with xˆ , which gives: xˆ p = xˆ xˆ −1 A xˆ p + xˆ D p y 

(7.18)

7.3 Markets: Why All Four IO Models Overestimate …

99

Next, Y = xˆ D is substituted into (7.18), and the result is simplified and solved as follows: xˆ p = A xˆ p + Y p y  ⇒ xˆ p = (I − A)−1 Y p y  = L Y p y 

(7.19)

This proves that the Leontief quantity model may also be regarded as the revenue-pull price model expressed in values, instead of in prices. Finally, to our knowledge, the revenue-pull price model has not been applied yet. Single-country or single-region applications, however, would be straightforward and could simulate the backward total industry input price, wage rate and import price impacts, under full competition, of, e.g. exogenous world market increases of specific export prices. Type II extensions of the above interregional price model could expand such simulations to the interregional or international price-wage-price backward impacts of specific external export price changes on the prices of total industry inputs, wages rates and tax revenues in the same as well as in different regions or countries.

7.3 Markets: Why All Four IO Models Overestimate Their Typical Impacts After having specified and discussed the fourth and last basic IO model, it is time to take stock of the economics and the applicability of all four basic IO models. Table 7.2 collects all assumptions that have been made previously. First, note that both sets of models may be derived as special cases (i.e. simplifications) of a general equilibrium interindustry model, for a regional or national economy, with profit maximizing firms, operating under full competition, subject to the most general production function with heterogeneous inputs and heterogenous outputs, as measured by either an IOT, a SUT or a SAM (ten Raa 2004; Kratena 2017).6 The most important simplification appears in the first line of Table 7.2. It alternatively, assumes either a single homogeneous output (Leontief) or a single homogeneous input (Ghosh). Assuming a single homogeneous output implies the perfect substitution among all outputs in case of Leontief, which clearly represents a simplification, but a minor one, whereas assuming perfect substitution among all inputs in case of Ghosh represents a major simplification, namely, one that allows factories to run without labour and cars to drive without gas. Note that this simplification only 6

Note that Dietzenbacher (1997), while showing that the Ghosh quantity model may be interpreted as the Leontief price model measured in values, decided to rename it the Ghosh price model and to name the price interpretation of the Leontief quantity model (7.19) the Ghosh quantity model. This unfortunate terminology leads to defining away two of the four basic IO models, i.e. the original Ghosh quantity model and its original price dual, as is evident in Rueda-Cantuche (2011) and Miller and Blair (2022, p. 301).

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7 Supply-Driven IO Quantity Model and Its Dual, Price Model

Table 7.2 Assumptions and solutions of the four basic input–output models Demand-driven quantity and cost-push price model:

Supply-driven quantity and revenue-pull price model:

For the individual firm: • given demand for its single homogeneous output, i.e. perfect substitution among all outputs • full complementarity of all inputs (fixed input ratios) • cost minimization at given input prices • derived demand for inputs (backward linkages) • full competition, i.e. forward passing on of all input price changes into the single output price

• given supply of its single homogeneous input, i.e. perfect substitution among all inputs • perfect jointness of all outputs (fixed output ratios) • revenue maximization at given output prices • derived supply of outputs (forward linkages) • full competition, i.e. backward passing on of all output price changes into the single input price

For the economy as a whole: • exogenous demand for final outputs per industry • endogenous demand for all inputs per industry • perfectly elastic supply of all primary inputs, i.e. exogenous primary input prices • endogenous total output prices and quantities

• exogenous supply of primary inputs per industry • endogenous supply of all outputs per industry • perfectly elastic demand for all final outputs, i.e. exogenous final output prices • endogenous total input prices and quantities

Solution of the two Leontief models:

Solution of the two Ghosh models:

• v = C (I •

p

y

=

p

− A)−1 y, =

p

with y = Y i and v = V i • y = v (I − B)−1 D, with y = i Y and −1 v  = i V v C (I − A) • pv  = p = (I − B)−1 D p y 

Source Extension of Nieuwenhuis (1981) and Oosterhaven (1989, 2012)

makes the Ghosh quantity model implausible, but not its dual, price model, as all quantities are constant in both price models. The next simplifying assumption is fixing the ratios on the other side of the general production function. In case of the Leontief model, this implies assuming full complementarity of all inputs, and in case of the Ghosh model this implies assuming full jointness of all outputs. Both assumptions are serious simplifications of reality, and one might argue that Leontief’s simplification might be more severe than that of Ghosh, but that very much depends on which variables change exogenously in the application of either model. In case of a quantity shock to the demand side of the economy, assuming fixed input ratios in the Leontief model does not seem problematic, but assuming fixed output ratios in that case implies that all sales in a row of the IOT are assumed to change proportionally to the exogenous shock, which is highly implausible in case of a specific demand shock. In case of a specific supply shock, assuming fixed input ratios is very implausible as firms will directly look for substitutes (see further Sect. 8.1), whereas assuming

7.3 Markets: Why All Four IO Models Overestimate …

101

fixed output ratios in case of a supply shock is less of a problem. Note again, that either simplification does not influence the relative plausibility of the two price models, as they both assume all quantities to remain constant in face of a price change. The non-interaction between prices and quantities in both sets of models is illustrated in Fig. 7.3. In the Leontief quantity model, the perfectly inelastic demand for the single homogeneous output of industry i shifts (either exogenously or endogenously) to the left and right along a perfectly elastic supply curve, not causing any price reaction (Fig. 7.3a). In the Leontief price model, the opposite happens. The price of output i (i.e. its supply curve, either exogenously or endogenously) shifts up and down along a perfectly inelastic demand curve, without any impact on the quantity demanded. The same holds for the demand and supply of primary inputs of type p (not shown in Fig. 7.3a). In case of an under-utilization of production capacities and factor supplies, i.e. around the bottom of the business cycle, these assumptions are more or less reasonable, but at the top of the business cycle, the Leontief models will overestimate the quantity impacts of positive demand shocks, but not those of negative demand shocks. In the Ghosh models, opposite assumptions are made. In the Ghosh quantity model, the perfectly inelastic supply of the single homogeneous input of industry j, shifts (either exogenously or endogenously) to the left and right along a perfectly elastic demand curve, not causing any price reaction (Fig. 7.3b) (i.e. consumers consume whatever is supplied at the going price). This was more or less what happened under the old EU agricultural policy with its fixed product prices and its fluctuating milk lakes and butter mountains. Independently, in the revenue-pull price model, the demand curve (i.e. the price of the single homogeneous input of industry j) shifts up and down along a perfectly inelastic supply curve, not causing any quantity reaction of the supply of input j. In this case, the same holds for the supply and demand of final outputs of type q (not shown in Fig. 7.3b).

Fig. 7.3 Functioning of product markets in the four basic input–output models. Source Refinement of Oosterhaven (1996)

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7 Supply-Driven IO Quantity Model and Its Dual, Price Model

The fact that the two IO quantity models both overestimate the quantity impacts of exogenous changes may also be shown in an informal way. Every economist has seen the standard diagonal market equilibrium cross with an upward sloping supply curve and a downward sloping demand curve. An horizontal quantity shift of one of the two diagonal curves always produces an equilibrium quantity impact that is smaller than the horizontal shift of the curve, because a price reaction will dampen the quantity impact of the shift (see also McGregor et al. 1999). The same holds for the two IO price models. A vertical shift of one of the two diagonal curves always produces an equilibrium price impact that is smaller than the shift of the curve, because a quantity reaction will dampen the price impact. So, both price models overestimate the price impacts of their exogenous changes. From this summary evaluation of the four basic IO models, it is clear that, especially, the Leontief quantity model is far more plausible than its Ghoshian opposite, but it is also clear that both sets of IO models represent extreme cases of a general equilibrium model. Clearly, implementing a CGE model at the combined interindustry and interregional level is more complicated and far more data demanding than a comparable IO model (see Bröcker et al. 2004). For this reason, most developments in IO analysis seek to modify the basic Leontief model by introducing more flexible (e.g. translog) production functions for capital, labour and intermediate inputs (e.g. KLEM), and by introducing econometrically estimated consumption, investment and export functions, while sticking to the Leontief specification for the matrix of intermediate demand only (see Almon 1991; Kratena 2005).

References Almon C (1991) The INFORUM approach to interindustry modeling. Econ Syst Res 3:1–7 Augustinovics M (1970) Methods of international and intertemporal comparison of structure. In: Carter A, Bródy A (eds) Contributions to input-output analysis. North-Holland, Amsterdam Bon R (1986) Comparative stability analysis of demand-side and supply-side input-output models. Int J Forecast 2:231–235 Bon R (1988) Supply-side multiregional input-output models. J Reg Sci 28:41–50 Bröcker J, Meyer R, Schneekloth N, Schürmann C, Spierkemann K, Wegener M (2004) Modelling the socio-economic and spatial impacts of EU transport policy. IASON deliverable 6. ChristianAlbrechts-Universität Kiel/Universität Dortmund Cartwright JV, Beemiller RM, Trott EA, Younger JM (1982) Estimating the potential impacts of a nuclear reactor accident. Bureau of Economic Analysis, Washington D.C. Chen CY, Rose A (1986) The joint stability of input-output production and allocation coefficients. Model Simul 17:251–255 Davar E (1989) Input-output and general equilibrium. Econ Syst Res 1:331–344 Davis HC, Salkin EL (1984) Alternative approaches to the estimation on economic impacts resulting from supply constraints. Ann Reg Sci 18:25–34 de Mesnard L (2004) understanding the shortcomings of commodity-based technology in inputoutput models: an economic-circuit approach. J Reg Sci 44:125–141 de Mesnard L (2009) Is the Ghosh model interesting? J Reg Sci 49:361–372 Dietzenbacher E (1997) In vindication of the Ghosh model: a reinterpretation as a price model. J Reg Sci 37:629–651

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Ehret H (1970) Die Anwendbarkeit von input-output Modellen als prognose Instrument. Dunkler & Humbolt, Berlin FNEI (1977) Wadinpoldering tussen Zwarte Haan en Ternaarderpolder: Werkgelegenheidseffecten van blijvende aard. Federatie van Noordelijke Economische Instituten, Leeuwarden Ghosh A (1958) Input-output approach in an allocation system. Econ 25:58–64 Giarratani F (1976) Application of an interindustry supply model to energy issues. Environ Plan A 8:44754 Giarratani F (1980) The scientific basis for explanation in regional analysis. Pap Reg Sci Assoc 45:185–196 Giarratani F (1981) A supply-constrained interindustry model: forecasting performance and an evaluation. In: Burh W, Friedrich P (eds) Regional development under stagnation. Nomos Verlag, Baden-Baden Gruver GW (1989) On the plausibility of the supply-driven input-output model: a theoretical basis for input-coefficient change. J Reg Sci 29:441–450 Guerra AI, Sancho F (2011) Revisiting the original Ghosh model: can it be made more plausible? Econ Syst Res 23:319–328 Helmstädter E, Richtering J (1982) Input coefficients and output coefficients types models and empirical findings. In: Proceedings of the Hungarian conference on input-output techniques. Statistical Publishing House, Budapest Kratena K (2005) Prices and factor demand in an endogenized input-output model. Econ Syst Res 17:47–56 Kratena K (2017) General equilibrium analysis. In: ten Raa T (ed) Handbook of input-output analysis. Edward Elgar, Cheltenham Manresa A, Sancho F (2013) Supply and demand biases in linear interindustry models. Econ Model 33:94–100 McGregor P, Swales JK, Yin YP (1999) Spillover and feedback effects in general equilibrium models of the national economy: a requiem for interregional input-output? In: Hewings GJD, Sonis M, Madden M, Kimura Y (eds) Understanding and interpreting economic structure. Springer-Verlag, Berlin Miller RE, Blair PD (2022) Input-output analysis: foundations and extensions, 3rd edn. Cambridge University Press, Cambridge Nieuwenhuis A (1981) Vraag, aanbod en input-output tabellen. Centraal Planbureau, Notitie nr. 8, The Hague Nikaido H (1970) Introduction to sets and mappings in modern economics. North-Holland, Amsterdam Oosterhaven J (1981) Interregional input-output analysis and Dutch regional policy problems. Gower Publishing, Aldershot-Hampshire Oosterhaven J (1988) On the plausibility of the supply-driven input-output model. J Reg Sci 28:203– 217 Oosterhaven J (1989) The supply-driven input-output model: a new interpretation but still implausible. J Reg Sci 29:459–465 Oosterhaven J (1996) Leontief versus Ghoshian price and quantity models. South Econ J 62:750–759 Oosterhaven J (2012) Adding supply-driven consumption makes the Ghosh model even more implausible. Econ Syst Res 24:101–111 Oosterhaven J (2022) A price reinterpretation of the Leontief quantity model. SOM Research Reports 2022001-GEM, University of Groningen Rose A, Allision T (1989) On the plausibility of the supply-driven input-output model: empirical evidence on joint stability. J Reg Sci 29:451–458 Rose A, Wei D (2013) Estimating the economic consequences of a port shutdown: the special role of resilience. Econ Syst Res 25:212–232 Rose A, Wei D, Paul D (2018) Economic consequences of and resilience to a disruption of petroleum trade: the role of seaports in U.S. energy security. Energy Pol 115:584–615

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Rueda-Cantuche JM (2011) The choice of type of input-output table revisited: moving towards the use of supply-use tables in impact analysis. Stat Oper Res Transact 35: 21-38 ten Raa T (2004) A neoclassical analysis of total factor productivity using input-output prices. In: Dietzenbacher E, Lahr ML (eds) Wassily Leontief and input-output economics. Cambridge University Press, Cambridge

Chapter 8

Negative IO Supply Shock Analyses: When Substitution Matters

Abstract The inoperability IO model is one of the most used approaches to estimate the indirect impacts of negative supply shocks. It is a regular IO model formulated in relative changes that inadequately estimates only part of only the negative demandside impacts of disasters, while it completely ignores the positive substitution effects on the supply side. Other IO approaches are also shown to be unsuited to this task. An information minimizing interindustry programming model is presented as an alternative. Its basic assumption is that economic actors, after a disaster, primarily try to restore their old pattern of economic transactions. By adding the usual fixed ratio assumptions of SU models, an indication is given of the heavy overestimation of the negative impacts of a supply shock when demand-driven IO models are used. Finally, to model the reconstruction phase of major disasters the dynamic IO model is added to this approach. Keywords Disaster impact analysis · Inoperability input–output model · CGE models · Nonlinear programming model · Danube and Elbe floods · Capital/output ratios · Dynamic input–output model Chapter 7 concluded, among others that the supply-driven IO quantity model may not be used to simulate the impacts of quantity shocks to the supply side of the economy. This raises the question what alternative approaches are suitable to estimate the interindustry and interregional impacts of negative supply shocks, of which natural and man-made disasters are about the most dire examples.

8.1 On the Limited Usability of the IO Model in Case of Supply Shocks The use of demand-driven IO models to estimate the indirect economic losses of disasters has gained increasing popularity, as evidenced by two special issues of Economic Systems Research (2007/2 and 2014/1) and two edited volumes (Okuyama The original version of this chapter was revised: Crucial brackets in Eq. 8.9 has been updated. The correction to this chapter is available at https://doi.org/10.1007/978-3-031-05087-9_11 © The Author(s), under exclusive license to Springer Nature Switzerland AG 2022, corrected publication 2022 J. Oosterhaven, Rethinking Input-Output Analysis, Advances in Spatial Science, https://doi.org/10.1007/978-3-031-05087-9_8

105

106

8 Negative IO Supply Shock Analyses: When Substitution Matters

and Chang 2004; Okuyama and Rose 2019). Of the IO approaches, the Inoperability IO Model (IIM, Santos and Haimes 2004) constitutes the single most used model.1 To calculate the absolute loss of output by industry, the IIM follows the standard IO model: x = A x + y ⇒ x = (I − A)−1 y

(8.1)

To this, the IIM adds the normalization of (8.1) by the lagged level of output xˆ −1 ,2 to obtain the relative loss of output by industry q = (ˆx−1 )−1 x, and innovatively labels this the inoperability by industry:  −1  −1 q = xˆ −1 Aˆx−1 q + xˆ −1 y ⇒  −1    −1 −1 q = I − xˆ −1 Aˆx−1 xˆ −1 y = (I − B)−1 y∗

(8.2)

wherein (I − B)−1 = Ghosh-inverse from (7.3). From the last part of (8.2) the IIM literature incorrectly concludes that the IIM is a new model with supply-side relations. Instead, it is just a demand-driven IO quantity model “with a small tweak” (Dietzenbacher and Miller 2015). The direct losses in case of a natural disaster, essentially, represent the destruction of stocks of capital, labour and infrastructure, whereas the wider, indirect effects, essentially, represent the subsequent changes in the flows of production and consumption (Okuyama and Santos 2014). The latter may be negative as well as positive, may occur in the short run and in the longer run, and may be due to both supply and demand effects. In the specific case of terrorist attacks, the direct destruction of capital and labour and the related direct loss of demand and supply is mostly minimal. The main impact is psychological. Its subsequent economic impact will be a—mainly locational— redistribution of private consumption demand (Galea et al. 2002). The IIM, as any IO model, is well suited to estimate the further positive and negative backward impacts of such a redistribution, especially, when the economy is operating below capacity. Note that not only in the case of disasters, but in case of all impact studies, the honest analyst should try to make an estimate of the net impacts of the event or industry at hand, i.e. an estimate of the difference between the positive and negative impacts (cf. Oosterhaven et al. 2003). In the more general case of natural disasters, on the demand side of the economy, the destruction of stocks will cause an almost direct loss of intermediate and final demand in the disaster region. The short-run backward impacts of the direct reduction in final demand may be modelled by means of the IIM without any problem. Estimating the backward impact of the direct reduction of intermediate demand by 1

On March 10, 2022, “inoperability input–output model” scored 1150 hits on Google Scholar. Dietzenbacher and Miller (2015) show that it is far simpler to normalize the solution of (8.1) instead of normalizing its separate terms, as done by the IIM in (8.2). This results in: q = (ˆx−1 )−1 (I − A)−1 y. 2

8.1 On the Limited Usability of the IO Model in Case of Supply Shocks

107

means of the IIM is problematic, as it requires the analyst to translate this shock, which is endogenous in the IIM, into an exogenous shock to final demand that, with the IIM, exactly reproduces the exogenous shock to intermediate demand (see Rose 2004, on avoiding double counting exogenous and endogenous impacts, and Oosterhaven 2017, about what went wrong in IIM applications while doing so in practise). The destruction of stocks will also cause an almost direct loss in the supply of intermediate and final outputs from the disaster region. Estimating the short-run forward impacts of this shock by means of the IIM or any other IO or SU model is impossible for several reasons. First and foremost, firms and households will not react to a negative supply shock by proportionally reducing all their other purchases, as is implied by the fixed ratio assumptions of the Type I and Type II, IO and SU models discussed in earlier chapters. Instead, they will look for replacements. Three broad types of substitutes are possible: 1. Firms and households may look for different firms in the same region that produce the same product. This may lead to changes of the industry market shares in the supply of the product at hand. The assumption that these shares are fixed is hidden in the construction of most symmetric IO tables, but is explicit in the SU model (see Sect. 4.2). 2. Firms and households may look for suppliers from different regions. This may lead to changes in the self-sufficiency and the imports ratios for the product at hand, i.e. in changes in the trade origin ratios. The assumption that these ratios are fixed is mostly made implicitly, but is well recognized in the IO and SU literature (see Sects. 2.3 and 4.2). 3. Firms and households may look for different products that perform the same function, e.g. plastic parts instead of metal parts. This implies a change in the real technical coefficients for firms and the real preference coefficients for households. Especially in case of firms, such changes represent the least likely reaction in the short run, as it implies changing the production process.3 In all three cases, replacement of fallen off inputs by firms and households will lead to positive instead of negative impacts elsewhere in the economy: impacts that cannot be estimated by the IIM nor by any other IO or SU model. Only if an input is truly irreplaceable, the lack of its supply may force purchasing firms to shut down their production when the stocks of this input are depleted. The reduction in the supply of the intermediate input at hand, in that case, needs to be multiplied with processing coefficients (i.e. reciprocal real technical coefficients (ai·sj )−1 , see Oosterhaven 1988).4 If the negative shock relates to the supply of an— in value terms—insignificant but irreplaceable essential input, the reduction in the 3 Kujawski’s early (2006) critique of the IIM only related to this assumption of fixed technical coefficients and excess supply in all industries. It did not have an impact on the proliferation of the IIM. 4 Unfortunately, readily available IOTs and SUTs are usually too aggregated to calculate the necessary processing coefficients, with exception of agricultural industries based on a single major input (see Oosterhaven 1981, Chap. 8, for an application in case of a new polder). See Klaassen (1967)

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8 Negative IO Supply Shock Analyses: When Substitution Matters

output of the purchasing firm at hand will be many, many times larger than the value of the drop in the intermediate input! A fine example of very large processing coefficients could be observed during the corona pandemic of 2020–2021. A shortage of computer chips led to the complete temporal close-down of several car manufacturing plants. This impact was especially large because of the use of “just in time” logistics, which implies the almost absence of buffers of parts and components. Aside from the above-discussed impacts of the destruction of stocks of capital, labour and infrastructure, there will also be short- and long-run impacts of private and public aid and reconstruction activities. The positive backward impacts of such activities may well be estimated by means of interregional or, need be, international IO models. These backward impacts will be spatially concentrated in case of the reconstruction of buildings and infrastructure, but will be spatially spread in case of the reconstruction of machinery, as that is delivered from all over the world. In conjunction, financing these reconstruction activities requires higher insurance premiums and higher taxes, which will lead to longer run negative forward macroeconomic impacts, which will be spatially spread, and which cannot be estimated by any IO or SU model. Obviously, since the IIM is able to estimate only part of only the demand-side impacts of a disaster, (8.1) and (8.2) may not be used as a risk management instrument to prioritize the public support for industry resilience programmes as advocated by Santos and Haimes (2004, also Anderson et al. 2007, and Barker and Santos 2010). In fact, the results from all kinds of IIM applications show total simulated indirect losses of economic activity that are larger than the direct losses. Santos and Haimes (2004), for example, report ratios of total to direct losses due terrorist attacks that vary between 2.5 and 3.6. Santos (2006) reports a disaster multiplier of about 2.0 for 9/11, while Anderson et al. (2007) find a disaster multiplier of about 2.2 in case of the 2003 blackout in the northwest of the USA. Such large multipliers are highly implausible, if only because the IIM completely ignores the positive impacts of a disaster on the supply side of the economy. If regular IO and SU models are unsuited to estimate most of the indirect impacts of a disaster, what alternative modelling approaches are available? Up till now, the positive substitution effects could only be estimated by means of spatial computable general equilibrium (CGE) models (Tsuchiya et al. 2007; Kajitani and Tatano 2018). In fact, different versions of such a model are needed to model the short run as opposed to the longer run impacts, because short-run substitution elasticities are much closer to zero than their longer run equivalents (Rose and Guha 2004). Moreover, in longer run simulations, more variables need to be modelled endogenously than in short-run simulations. Consequently, CGE models are difficult and rather costly to estimate, even if the necessary data, such as interregional social accounting matrices (SAMs)

for the probably first use of processing coefficients in the so-called regional attraction model. See Klaassen (1974) for its interregional generalization. See Oosterhaven (1981, pp. 78–81 and 142–8) for a critical evaluation of the regional attraction model and, especially, of its claimed suitability to simultaneously model forward and backward linkages in interindustry analyses.

8.1 On the Limited Usability of the IO Model in Case of Supply Shocks

109

and all kinds of elasticities, are available (see Albala-Bertrand 2013, for a further critique). The much simpler hypothetical extraction method (see Sect. 9.1, for details) is advocated by Dietzenbacher and Miller (2015) as an alternative to the IIM and tested as such by Muldrow and Robinson (2014). The HE approach does circumvent the problem of assuming fixed trade origin ratios, as the HE method (implicitly) assumes that the lost sales of the extracted industry are compensated by an increase of imports from external regions. However, contrary to what was originally suggested (Paelinck et al. 1965; Strassert 1968), the complete or partial extraction of a row from an IOT does not simulate the forward, supply effects of that HE on its customers. In contrast, extracting a row simply measures the backward, demand effects of a drop (or complete disappearance) of the intermediate outputs of the extracted industry. Moreover, HE does not estimate the above-discussed three types of positive forward substitution effects either. The mixed exogenous–endogenous variables IO model (see Miller and Blair, 2022, Chap. 14, for details) also is not suitable to estimate the forward impacts of supply shocks. It only estimates the backward impacts of supply shocks to industries with exogenous outputs by means of fixed intermediate input coefficients, but this is done at the cost of making final demand for the products of these industries an endogenous variable with a residual character (see Surís-Regueiroa and Santiago 2018, for an interacting price-quantity version of this model with one iteration step, cf. Sect. 6.3). Positive substitution effects of negative supply shocks are absent, while residual final demand may—unacceptably—become negative whenever the negative supply shocks are large enough (see Steinback 2004, for an overview of agricultural applications). Though not relevant for the present discussion, note that residual endogenous final demand also becomes negative when a positive shock to the exogenous final demand of industries with endogenous outputs is sufficiently large. Finally, the supply-driven IO quantity model does not constitute a plausible model for studying the forward impacts of disasters either (see Sects. 7.1 and 7.3, for the why). This negative verdict, of course, also holds for the supply-driven version of the IIM (Crowther and Haimes 2005), as this model simply equals the basic supply-driven IO model expressed in percentage changes. Without a spatial CGE, a more or less plausible estimation of the impacts of a negative supply shock requires hypothetical rationing schemes and ad hoc assumptions regarding the adaptation behaviour of upstream and downstream industries, as theoretically indicated by Oosterhaven (1988) and implemented by Hallegate (2008, 2014) and Rose and Wei (2013). The only other alternative to the complex construction of a spatial CGE is the nonlinear programming approach of Oosterhaven and Bouwmeester (2016), which according to the authors “combines the simplicity of the IRIO model with the plausibility of the CGE approach” (p. 585).

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8 Negative IO Supply Shock Analyses: When Substitution Matters

8.2 Nonlinear SU Programming Alternative: Much Smaller Disaster Multipliers The basic behavioural assumption of this alternative to CGE is that both firms, households and governments, in the short run after a disruptive event, as much as possible, try to return to their old pattern of sales and purchases. This basic assumption is made operational by minimizing the weighted total distance between the cells of a simulated post-disaster interregional IOT and an actual pre-disaster IRIOT. The distance between the two IRIOTs is measured by means of the information gain measure of Kullback (1959, see also Theil 1967). Here, we show how the model may be specified for a use-regionalized interregional SUT (see Table 4.4 sub C), as that type of IRSUT represents a more detailed and a more frequently used accounting scheme than an IRIOT. The objective function of the nonlinear interregional SU model equals: Minimize

r 

r,ex sic

r,ex r ln sic /sic

ic

+

r  c

ecr,ex ln ecr /ecr,ex +

+

rs 

u rcis,ex

ln u rcis /u rcis,ex

ci s 

v·is,ex ln v·is /v·is,ex

+

rs 

yc·r s,ex ln yc·r s /yc·r s,ex

c

(8.3)

i

The symbols of (8.3) are defined in Tables 4.3 and 4.4 sub C. The five separate terms of (8.3) specify the information gain of changes in, sequentially: (1) the regional supply tables, (2) the doubly regionalized intermediate use tables, (3) the doubly regionalized total local final use columns, (4) the regional external export columns and (5) the regional total value added rows. The superscript ex, additionally, indicates the exogenous values of cells of the pre-disaster IRSUT, whereas the transactions without an ex define the endogenous values of the cells of the post-disaster IRSUT. The further idea of this approach is to only add the minimally necessary behavioural restrictions to (8.3). The first of these requires that all economic transactions are non-negative. This implies that the ultra-short run depletion of available stocks to cope with supply shortages cannot be part of the objection function (8.3). Adding this adaptation possibility, as in Hallegate (2008) and in Mackenzie et al. (2012), requires very hard to obtain data on the pre-disaster level of stocks. IOTs and SUTs, for example, only contain information on historic changes in stocks, but never on their actual levels. However, if data on actual stock levels are available from other sources, the values of the stocks may easily be added as a constant to the RHS of the next equation. Second, and foremost, it is assumed that prices change in such a fashion that the economy remains in short-run equilibrium; i.e., it is assumed that demand equals supply, per product, per region:

8.2 Nonlinear SU Programming Alternative: Much … s 

u rcis +

s 

yc·r s + ecr =

i

111



r sic , ∀ c, r

(8.4)

i

A great advantage of this specification is that there is no need to specify the underlying supply and demand elasticities nor the corresponding price changes. Instead, it is possible to concentrate on modelling the volume changes. Consequently, all variables in this model are measured in base year prices that all equal unity; i.e., they all represent quantities. Third, we assume cost minimization under the Walras–Leontief production function (2.7), per industry, per region, which results in: r 

·s u rcjs = acj



s sjc and v·sj = c·sj

c



s sjc , ∀ j, s

(8.5)

c

·s wherein acj and c·sj denote fixed technical coefficients that specify the per unit use of inputs regardless of their spatial origin. These are calculated from the coefficients ·s +c·sj = 1, ∀ j, s. Note that this columns of the pre-disaster use table and satisfy c acj “adding to unity” property of (8.5) elegantly and automatically also  secures that total  output equals total input, per industry, per region, i.e. c s sjc = rc u rcjs + v·sj , ∀ j, s. Fourth, the same cost minimization assumption may also be used for final demand, which results in: r 

yc·r s

=

f c··s

r 

yc·r s , ∀c, s

(8.6)

c

wherein f c··s denote fixed preference coefficients that specify the per unit use of products delivered from all over the world. These coefficients  are also calculated from the columns of the pre-disaster use table and satisfy c f c··s = 1, ∀s. This minimal set of equations completes the nonlinear SU model. The first test that any programming model of this type has to satisfy is that it reproduces the pre-disaster IRIOT or IRSUT from which its ex values and its fixed coefficients are derived. After satisfying that test, additional disaster-specific restrictions can be added. Oosterhaven and Bouwmeester (2016) did that with a two region (core/periphery) and a two industry (goods/services) hypothetical IRIOT, which was small enough to inspect all cell-by-cell changes that were simulated for two types of disasters, namely the total shut down of the production of a single region, and the total shut down of all transport in a single interregional direction. Both the signs of—and the largest differences in—the impacts of the four disasters could be explained by the structural economic differences between the two regions and the two industries and by the not explicitly modelled, corresponding price changes. Bouwmeester and Oosterhaven (2017) used the corresponding nonlinear IRIO model, calibrated it on the EXIOPOL international IOT (Tukker et al. 2013) and

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8 Negative IO Supply Shock Analyses: When Substitution Matters

added one-sided trade capacity constraints with which they simulated the possible impacts of four Russian natural gas export boycotts of—parts of—the European Union. The simulations showed considerable effects at lower levels of aggregation, but negligible aggregate economic impacts for the EU and only a little larger than negligible aggregate economic impacts for Russia. Interestingly, the effects on GDP and on welfare, as measured by the size of domestic final demand, proved to have opposite signs due to changes in trade balances; with the EU experiencing a very small positive aggregate GDP effect due to increased domestic production of natural gas and its substitutes, while a small positive welfare effect is predicted for Russia mainly due to the enlarged domestic consumption of natural gas at lower internal Russian market prices. The nonlinear IRSU model of (8.3)–(8.6) was calibrated on the interregional SUT for Germany for 2007 (Többen 2017, Chap. 4) and used by Oosterhaven and Többen (2017) to simulate the interregional impacts of the 2013 massive floods of the Elbe and the Danube rivers by adding production capacity constraints. They find regional and national disaster multipliers that are all smaller than 1.14 (see the first row of Table 8.1). Moreover, they examined the sensitivity of their outcomes to varying economic environments. These sensitivity analyses show that central government support of regional final demand (i.e. disaster aid) substantially reduces the already small indirect losses, whereas being at the top of the business cycle considerably increases them. Their most interesting sensitivity analysis, however, regards the implications of imposing the two remaining fixed ratio assumptions of the standard demand-driven IO or SU model, namely the fixed industry market share assumption and the fixed trade origin ratio assumption. Investigating the impacts of adding these two assumptions (1) enables an examination of the size of the indirect economic losses that are avoided because of the ability of industries and final consumers to find alternative suppliers when faced with a negative supply shock, and (2) it enables an assessment of the overestimation of the indirect disaster impacts when IO and SU models are used that do not allow for these substitution possibilities. Table 8.1 Comparison of indirect disaster impacts in million e and in percentage of the base impacts Assumptions

Bayern

Sachsen

Thüringen

All of Germany

Base model (fixed technology)

13.9

4.1

4.6

11.0

+ fixed industry market 31 = + 220% shares*

8 = + 200%

5.7 = + 24%

19 = + 180%

+ fixed trade origin sharesa

21 = + 150%

18 = + 430%

13 = + 270%

33 = + 300%

+ both shares fixed*

59 = + 420%

83 = + 2,030%

42 = + 910%

97 = + 880%

The figure behind = gives the increase in the base estimate in % of the first row Source Adapted from Oosterhaven and Többen (2017)

a

8.2 Nonlinear SU Programming Alternative: Much …

113

The first assumption added to the basic nonlinear model of (8.3)–(8.6) is that the industry market shares in regional product supply are fixed, i.e. r r = ric sic



r sic , ∀i, c, r

(8.7)

i r wherein ric denotes the fixed industry market shares, which are calculated from the columns of the pre-disaster supply table. This fixed ratio assumption is explicitly made in standard SU models and is mostly implicitly made when deriving a symmetric IOT from a standard rectangular SUT (see Sect. 4.2). Adding this assumption is more or less sensible in case of a negative demand shock, but it is highly implausible in case of a negative supply shock. This can easily be shown with an example. Assume the extreme case where a certain product is produced by two industries only. The first provides 90% of the total supply, whereas the second provides the remaining 10%. If this second industry shuts down because of a disaster, assuming fixed industry market shares implies that the first industry will also shut down its 90% share. In the German flooding case, adding this assumption inflates the indirect disaster impact estimates with as much as 24–220% (see the second row of Table 8.1). The second additional assumption is that the trade origin ratios are fixed, i.e.

u rcis = m rcis



u rcis , ∀ c, i, r, s and

r

yc·r s = m rcsf



yc·r s , ∀c, r, s

(8.8)

r

wherein m rcis and m rcsf denote the fixed trade origin ratios for intermediate demand and for final demand, respectively. These ratios are calculated from the columns of pre-disaster use table. The assumption of fixed trade origin ratios extends the fixed technology assumption (8.5) to the geographical origin of intermediate and final inputs. In the context of a negative demand shock, it is more or less plausible to assume that firms proportionally purchase less inputs from all their suppliers. In the case of a negative supply shock, however, firms will immediately search for different sources for their disrupted inputs. In an extreme case, assuming fixed trade origin ratios implies that firms have to shut down all of their production if a single specific region is not able to deliver the required input, notwithstanding that other regions are able to supply the same input. Hence, this fixed ratio assumption also leads to overstating the indirect impacts of disasters considerably. In case of a demand shock to household income, assuming fixed preference coefficients (8.6) is already a bit problematic, as households will not proportionally reduce all their expenditures. Instead, they will maintain their consumption of basic needs and reduce their other consumption. In case of a negative supply shock, however,

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8 Negative IO Supply Shock Analyses: When Substitution Matters

households will look for substitutes as much as firms, and will show even more flexibility in changing, first, their trade origin ratios (8.8) and, second, even their preference coefficients (8.6). In the German flooding case, adding (8.8) for both firms and households inflates the indirect impact estimates with as much as 150–430% (see the third row of Table 8.1). When both ratios are fixed in combination, as in the extended IIM and all other extended IO and SU models, the indirect German flooding loss estimates were amplified with as much as 420–2030% (see the last row of Table 8.1). Disregarding the precise size of the percentages, this clearly shows that almost all disaster multipliers are overestimated heavily. This has an important policy implication, namely that the IIM literature emphasis on stimulating the resilience of the economic system as a whole is not justified. Instead, much more attention needs to be paid to preventing and mitigating the direct damages of natural and man-made disasters.

8.3 Disaster Reconstruction Phase: Adding the Dynamic Leontief Model The basic behavioural assumption of Eq. 8.3, i.e. that economic actors try to stay as close as possible to the pre-disaster size of their transactions, is fine for short-run behavioural reactions and is fine for most transactions. It is, however, problematic in case of the transactions in the columns with regional investment expenditures, as those columns describe the gross fixed capital formation (GFCF) of an economy that intends to change the economy, not to keep it constant. Purchasers of capital goods will consciously try to deviate from their historic purchases of capital goods, as their demand will primarily be determined by the location and the nature of direct damages of the disaster at hand. A better behavioural assumption is that firms will try to restore these damages as quickly as possible. The investment expenditures needed to restore production capacity are technically ·r ∈ B that indicate the amount determined by interindustry capital/output ratios bcj of capital goods c needed to produce one unit of output of industry j in region r. The matrix B is known as the capital coefficients matrix. As opposed to the technical ·r and crpj , the technical capital/output intermediate and primary input coefficients acj ·r ratios bcj cannot be derived directly from an IOT or a SUT. Most countries that publish SUTs may regularly also publish a disaggregation by purchasing industry of the SUT column with private investment expenditures; i.e., they may publish a GFCF matrix with gross investments Ig . This gross investments matrix, however, contains the sum of the expenditures on capital goods needed to replace worn-out old capital goods and the expenditures on capital goods needed to enlarge production capacity. The column structure of the unknown sub-matrix with replacement investments R is quite different from that of the unknown sub-matrix with net investments In , as replacement rates considerable

8.3 Disaster Reconstruction Phase: Adding the Dynamic Leontief Model

115

differ between different types of capital goods. Computers, for example, are replaced within a few years, whereas factory and office buildings, for example, endure for tens of years. This leads to relatively more computers in R, and relatively more factory and office buildings in In . The columns of R may be estimated by multiplying depreciation rates for capital r ∈ D with the composition of the old capital goods c used in industry j in region r dcj r stock of j in r kcj ∈ K, if both types of data are known or can be estimated. The columns of In can then be calculated as the residual, and will equal the product of the capital coefficients matrix and the planned capacity enlargement from x0 to x1 . The capital coefficients matrix B may then be estimated assuming that the measurable growth of actual output (x1 − x0 ) equals the non-measurable planned growth of production capacity (x1 − x0 ), in formula:    −1 Ig − D ⊗ K ≈ In = B xˆ 1 − xˆ 0 ⇒ B ≈ (Ig − D ⊗ K) xˆ 1 − xˆ 0

(8.9)

See Södersten and Lenzen (2020) for a further discussion of this estimation problem, among others using investment vintages of capital goods. Setting aside the treatment of replacement investments, Leontief (1953, Chap. 3) used the accounting framework of Table 2.1 together with the capital coefficients matrix B to formulate the dynamic IO model for a single region or single nation: xt = A xt + B(xt+1 − xt ) + ytex ⇒ B xt+1 = (I − A + B)xt − ytex

(8.10)

where the regional origin and destination superscripts have been dropped, as in (8.9), to keep the notation simple. Note that not only the cells of A (see Sect. 2.2) but also those of B need to be interpreted as the product of an intra-regional purchase ·r coefficient m rr i j and the above-defined technical capital/output ratio bi j that relates to rr ·r capital goods delivered by industry i from all over the world, i.e. bi j = m rr i j bi j ∈ B. More importantly, also note that (8.10) has I equations with 2*I unknown variables. Hence, besides exogenous final demand ytex , either current xt or future xt+1 needs to be assumed exogenous to obtain a solution. In the original forward-looking dynamic IO model, current xt is assumed to be given, which leads to the following solution for future industry output: xt+1 = B−1 (I − A + B)xt − B−1 ytex = B−1 G xt − B−1 ytex

(8.11)

where G = (I−A+B). This result is problematic, because B will empirically almost always have multiple rows with zeros for service industries that do not produce capital goods. Consequently, the inverse of B will not exist. In addition, the model is unstable for realistic values of the variables (see Steenge and Reyes 2020, for a brief summary of more reasons why this model is problematic).

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8 Negative IO Supply Shock Analyses: When Substitution Matters

In the alternative backward-looking dynamic IO model, it is assumed that future xt+1 is given, which leads to the solution for current industry output (Leontief 1970; Lieuw 1977):     xt = (I − A + B)−1 B xt+1 + ytex = G−1 B xt+1 + ytex

(8.12)

The existence of the inverse of G is no problem, as the presence of the unit matrix ensures that G is non-singular for acceptable values of A and B. However, when more and more periods are added the assumption of a fixed final year size as well as composition of industry output might have been sensible for less developed, and therefore simple, centrally planned economies that used to work with five-year plans, but it does not make sense for complex market economies. Interestingly, there is a new third solution of (8.10) assuming that both current and future industry output levels are given exogenously (see Steenge and Boˇckarjova 2007, for the origin of this model and the need for extensive contingency planning). Not unexpectedly, this solution leads to endogenous and residual final demand: yten = (I − A + B)xt − B xt+1 = G xt − B xt+1

(8.13)

where ytex has been relabelled yten to indicate that remaining final demand in (8.13) is endogenous. Steenge and Reyes (2020) propose to use (8.13) for the reconstruction phase of an economy hit by a major disaster, as they assume that under such circumstances central government planning will take over to steer the reconstruction of the disaster economy by setting a target recovery path for industry output. For further clarification, the single period (8.13) is therefore extended to its fouryear equivalent: ⎡ ⎢ ⎢ ⎣

y1en y2en y3en y4en

⎤

⎡

⎡

⎤

G −B 0 0 0 ⎢ ⎢ ⎥ ⎢ 0 G −B 0 0⎥ ⎥=⎢ ⎥⎢ ⎦ ⎣ 0 0 G −B 0 ⎦⎢ ⎢ ⎣ 0 0 0 G −B

x1 x2 x3 x4 x5

⎤ ⎥ ⎥   ⎥ ⎥, with x1 = I − qˆ x0 ⎥ ⎦

(8.14)

  The target recovery path is defined by x1 x2 x3 x4 x5 , while qˆ = diagonal matrix with the relative loss of industry output due to the disaster in t = 0, which constrains the production capacity in the first period after the major disaster. Note that the periods in (8.14) do not necessarily have to equal one calendar year. They may be shorter depending on the length of time needed to rebuild and re-install the damaged factories and offices.5 5

Note that the input coefficients in A in that case will not change, as they are dimensionless. The values of the exogenous variables, however, will need to be decrease proportionally, whereas the capital/output ratios will have to be increased proportionally, as the latter have a dimension, namely stock/flow.

8.3 Disaster Reconstruction Phase: Adding the Dynamic Leontief Model

117

Obviously, very fast recovery paths will not be feasible, as negative final demand is impossible. In fact, some minimum level of endogenous final demand, especially, for food, health and shelter will be required in all cases, which limits the maximum speed of reconstruction. Nevertheless, Steenge and Reyes (2020) explicitly do not see and do not want to use (8.14) as an optimization model. To counter the problem of getting unacceptable outcomes, they suggest that central planners should be left free to revise their target recovery path period by period in the light of the development of among others endogenous final demand. However, Sect. 8.2 shows that a mathematical programming model need not necessarily be seen as an optimization tool. It may also be used as a tool to model the behavioural reactions to a major disaster, without optimizing anything. So, we close this section by presenting an alternative solution that shows how the nonlinear interregional supply–use model of (8.3)–(8.6) may be extended with the dynamic Leontief model of (8.10). Instead of the sweeping central planning assumption of Steenge and Reyes (2020), we believe that it is more realistic to assume that central planners only set a recovery path for the regional investments in public infrastructure and public services. As far as private investments are concerned, we assume that firms themselves choose the recovery path for their investments such that they may re-service their old customers as soon as possible. This implies that final consumers and firms, interacting through the market price processes of demand and supply, determine the speed of the recovery of private production capacity. To model these additional behavioural assumptions, the goal function (8.3) needs to be extended from a single period to multiple periods, while its local final demand part needs to shrink, as follows: Minimize (8.3), ∀ t, with

rs 

  yc·r s,ex ln yc·r s /yc·r s,ex

c

⇒

rs 

  yrcr s,ex ln yrcr s /yrcr s,ex , ∀t

(8.15)

c

where yrcr s,ex and yrcr s = pre-disaster and post-disaster remaining local final consumption of product c in region s delivered by industries in region r, i.e. exclusive of public and private investments in region s. Thus, yrcr s mainly consists of household consumption expenditures in s, which is why the summation dot for different categories of local final demand has been dropped. Note that the transactions in all periods in (8.15) have an equal weight. It would, however, hardly complicate the modelling effort if all future transactions in (8.15) would be discounted with a time preference coefficient that decreases over time. More importantly, the only slightly revised behavioural goal function (8.15) becomes subject to revised constraints needed to reflect the addition of (8.10). First, all variables, of course, remain semi-positive. The second constraint of the original

118

8 Negative IO Supply Shock Analyses: When Substitution Matters

model secures the equality of supply and demand. It now needs to explicitly include the two separate SUT investment columns next to the SUT column with remaining local final demand: s 

u rcis

+

s 

rs ii c

+

s 

i

ni cir s +

s 

yrcr s + ecr =



i

r sic , ∀c, r, t

(8.16)

i rs

Note that the time indices are dropped to keep the notation simple. The new ii c indicates the capital goods c delivered from region r and invested in the public infrastructure of region s, where the bar indicates that this term is fixed by central planning. The new ni cir s indicates the net investment in capital goods c from r by industry i in s. The next equation of the original model fixes the technical input coefficients per industry. It, of course, does not change, except that it needs to be specified for all periods, i.e. r 

·s u rcjs = acj



s sjc and v·sj = c·sj



c

s sjc , ∀ j, s, t

(8.17)

c

Note again that the adding to unity of the technical coefficients in (8.17) elegantly and automatically secures that total output equals total input for each regional industry. The final equation of the original model fixes the local final demand preference coefficients. It now needs to be limited to remaining local final demand, exclusive of private and public investments in capital goods: r 

yrcr s = f rc·s

r 

yrcr s , ∀c, s, t

(8.18)

c

where f rc·s = fraction of remaining local final demand of region  s that is spend on product c delivered by firms from over the whole world, with c f rc·s = 1. ·s and c·sj may It stands to reason that not only the technical input coefficients acj ·s be fixed, but also the technical capital/output ratios bcj . Note that fixing only the technical capital coefficients and not the trade origin ratios m rcjs allows for the spatial substitution of the same capital goods produced in different origin regions r, just like (8.17) and (8.18) enable the spatial substitution of intermediate and consumption goods, respectively. For capital goods, this is done by adding (8.10), as follows: r 

 rs ni cj

=

·s bcj

 c

s sjc,t+1

−

 c

 s sjc

, ∀c, s, j, t

(8.19)

8.3 Disaster Reconstruction Phase: Adding the Dynamic Leontief Model

119

Note that (8.19) does not use the growth of total regional industry output (x sj,t+1 − x sj ) to determine the purchases of capital goods by industry j in s, but the equivalent growth of the row totals of the regional supply tables. This minimizes the number of variables and the number of equations needed. Finally, note that all variables are still measured in base year prices of one. Before adding disaster-related constraints, (8.15)–(8.19) needs to be tested with rs rs rs ii c, t added to the goal function and ii c, t replacing ii c, t in (8.16). Running this nondisaster version of the dynamic programming model (8.15)–(8.19) should result in a stationary equilibrium in which all variables in all periods maintain their predisaster levels. After that, running (8.15)–(8.19) with the centrally planned postrs disaster infrastructure investments ii c, t plus the earlier mentioned reduced producˆ 0 plus any other disaster-related constraints should tion capacities x1 = (I − q)x deliver the recovery growth path of the interregional economy. Along this growth path, all variables ultimately return to their pre-disaster equilibrium levels. A side-path: note that to end up on a continuing growth path instead of a stationary equilibrium, it is sufficient to give all transactions in (8.15) an increasing time preference coefficient expressing the expectation of all actors that the economy will grow forever. This opens up many interesting questions and further extensions that are, however, beyond the scope of this book. The crucial equation of the above dynamic nonlinear programming model is (8.19), as it connects the different time periods in the model. To rebuild and reinstall their destroyed factories and offices in order to re-service their old customers, as stipulated in the goal function (8.15), firms need to bid up the prices of capital goods sufficiently such that purchasers of competing products “voluntarily” agree to buy smaller amounts of them. In fact, the period-by-period market processes behind (8.15) and (8.19) force the firms to more or less mediate between the desires of their customers to return to normal in each separate period. Equation (8.16) effectively constrains the current purchases of intermediate, capital, consumption and export products in each period, but it does allow for a reshuffling through changing prices of which industries and which regions, exactly, produce these products. This double substitution process between the alternative suppliers, however, is dampened by the assumed behaviour of all regional industries to maintain their products´ market shares as much as possible, as modelled in the first term of (8.15). In addition, (8.19) effectively constraints the future purchases of intermediate, capital, consumption and export products for each period. Likewise, an equation comparable to (8.19) could be added that links the investments in damaged transport infrastructure to the future capacity for intra-regional and interregional trade, which could be done separately for different types of products that require different types of transport infrastructure. In sum, as noted in Sect. 8.2, the outcomes of the nonlinear SU model (8.3)–(8.6) are driven by market processes and changing prices, without modelling those changes explicitly. The same holds for the dynamic nonlinear SU model (8.15)–(8.19). This dynamic extension eloquently shows the flexibility of the nonlinear programming approach to modelling economic behaviour, without any claim of market economic or central planning optimality.

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8 Negative IO Supply Shock Analyses: When Substitution Matters

References Albala-Bertrand JM (2013) Disasters and the networked economy. Routledge, Oxon Anderson CW, Santos JR, Haimes YY (2007) A risk-based input-output methodology for measuring the effects of the August 2003 Northeast blackout. Econ Syst Res 19:183–204 Barker K, Santos JR (2010) Measuring the efficacy of inventory with a dynamic input-output model. Int J Product Econ 126:130–143 Bouwmeester MC, Oosterhaven J (2017) Economic impacts of natural gas flow disruptions between Russia and the EU. Energy Pol 106:288–297 Crowther KG, Haimes YY (2005) Application of the inoperability input-output model (IIM) for systemic risk assessment and management of interdependent infrastructures. Syst Eng 8:323–341 Dietzenbacher E, Miller RE (2015) Reflections on the inoperability input-output model. Econ Syst Res 27:478–486 Galea SJ, Ahern J, Resnick H, Kilpatrick D, Ducuvalas M, Gold J, Vlahov D (2002) Psychological sequelae of the September 11 terrorist attacks in New York city. New England J Medicine 346:982– 987 Hallegate S (2008) An adaptive regional input-output model and its application to the assessment of the economic cost of Katrina. Risk an 28:779–799 Hallegate S (2014) Modelling the role of inventories and heterogeneity in the assessment of economic cost of natural disasters. Risk an 34:152–167 Kajitani Y, Tatano H (2018) Applicability of a spatial computable general equilibrium model to assess the short-term economic impact of natural disasters. Econ Syst Res 30:289–312 Klaassen LH (1967) Methods of selecting industries for depressed regions. OECD, Paris Klaassen LH (1974) Some further considerations on attraction analysis. Netherlands Economic Institute, Rotterdam Kujawski E (2006) Multi-period model for disruptive events in interdependent systems. Syst Eng 9:281–295 Kullback S (1959) Information theory and statistics. Wiley, New York Leontief WW (1953) Studies in the structure of the American economy. Oxford University Press, New York Leontief WW (1970) The dynamic inverse. In: Carter AP, Bródy A (eds) Contribution to input-output analysis, vol 1. North-Holland, Amsterdam Lieuw CK (977) Dynamic multipliers for a regional input-output model. An Reg Sci 11:94–106 MacKenzie CA, Santos JR, Barker K (2012) Measuring changes in international production from a disruption: case study of the Japanese earthquake and tsunami. Int J Prod Econ 138:293–302 Miller RE, Blair PD (2022) Input-output analysis: foundations and extensions, 3rd edn. Cambridge University Press, Cambridge Muldrow M, Robinson DP (2014) Three models of structural vulnerability: methods, issues and empirical comparisons. Pap Annu Meet South Reg Sci Assoc, San Antonio, Texas Okuyama Y, Chang SE (eds) (2004) Modelling spatial and economic impacts of disasters. Springer, New York Okuyama Y, Rose A (eds) (2019) Modelling spatial and economic impacts of disasters. Springer, New York Okuyama Y, Santos JR (2014) Disaster impact and input-output analysis. Econ Syst Res 26:1–12 Oosterhaven J (1981) Interregional input-output analysis and Dutch regional policy problems. Gower Aldershot Oosterhaven J (1988) On the plausibility of the supply-driven input-output model. J Reg Sci 28:203– 217 Oosterhaven J (2017) On the limited usability of the Inoperability IO model. Econ Syst Res 29:452– 461 Oosterhaven J, Bouwmeester MC (2016) A new approach to modelling the impact of disruptive events. J Reg Sci 56:583–595

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Oosterhaven J, Többen J (2017) Regional economic impacts of heavy flooding in Germany: a nonlinear programming approach. Spat Econ an 12:404–428 Oosterhaven J, van der Knijff EC, Eding GJ (2003) Estimating interregional economic impacts: an evaluation of nonsurvey, semisurvey, and fullsurvey methods. Environ Plan A 35:5–18 Paelinck J, De Caevel J, Degueldre DJ (1965) Analyse quantitative de certaines phénomènes du développement régional polarisé: Essai de simulation statique d’itérarires de propogation. In: No. 7, Problémes de Conversion Économique: Analyses Théoretiques et Études Appliquées, M.-Th. Génin, Paris Rose A (2004) Economic principles, issues and research priorities in hazard loss estimation. In: Okuyama Y, Chang SE (eds) Modelling spatial and economic impacts of disaster. Springer-Verlag, Berlin Rose A, Guha GS (2004) Computable general equilibrium modelling of electric utility lifeline losses from earthquakes. In: Okuyama Y, Chang SE (eds) Modelling spatial and economic impacts of disaster. Springer-Verlag, Berlin Rose A, Wei D (2013) Estimating the economic consequences of a port shutdown: the special role of resilience. Econ Syst Res 25:212–232 Santos JR (2006) Inoperability input-output modelling of disruptions to interdependent economic systems. Syst Eng 9:20–34 Santos JR, Haimes YY (2004) Modeling the demand reduction input-output (I-O) inoperability due to terrorism of connected infrastructures. Risk Anal 24:1437–1451 Södersten CJH, Lenzen M (2020) A supply-use approach to capital endogenization in input-output analysis. Econ Syst Res 32:451–475 Steenge AE, Boˇckarjova M (2007) Thinking about imbalances in post-catastrophe economies: an input-output based proposition. Econ Syst Res 19:205–223 Steenge AE, Reyes RC (2020) Return of the capital coefficients matrix (2020). Econ Syst Res 32:439–450 Steinback S (2004) Using ready-made regional input-output models to estimate backward linkage effects of exogenous output shocks. Rev Reg St 34:57–71 Strassert G (1968) Zur bestimmung strategischer sektoren mit hilfe von von input-output modellen. Jahrb Nationalök Stat 182:211–215 Surís-Regueiroa JC, Santiago JL (2018) A methodological approach to quantifying socioeconomic impacts linked to supply shocks. Env Imp Ass Rev 69:104–110 Theil H (1967) Economics and information theory. North-Holland, Amsterdam Többen J (2017) Effects of energy and climate policy in germany: a multiregional analysis. Ph.D., Faculty of Economics and Business, University of Groningen Tsuchiya S, Tatana H, Okada N (2007) Economic loss assessment due to railroad and highway disruptions. Econ Syst Res 19:147–162 Tukker A, De Koning A, Wood R, Hawkins T, Lutter S, Acosta J, Rueda-Cantuche JM, Bouwmeester MC, Oosterhaven J, Drosdowski T, Kuenen J (2013) Exiopol—development and illustrative analyses of a detailed global MR EE SUT/IOT. Econ Syst Res 25:50–70

Chapter 9

Other IO Applications with Complications

Abstract This chapter deals with two other applications of IO analysis that regularly appear in the literature without consideration of their limitations. Regional and interregional, forward and backward linkage analysis, also known as key sector analysis, only looks at the benefits while ignoring the policy cost of stimulating the sector chosen. Structural decomposition analysis (SDA) of national and interregional economic growth only looks at demand-side explanations of growth, while ignoring the supply side. Hence, in both types of studies, policy-makers are shown only half of the truth. Keywords Forward and backward linkages · Key sector analysis · Dutch mainport regions · Net multipliers · Shift and share analysis · Structural decomposition analysis · Growth accounting In Sect. 7.3, it was concluded that, although the demand-driven IO model is far more plausible than the supply-driven model, all four basic IO models need to be applied with great care to prevent wrong policy advice or to suggest too large quantity and price impacts of, respectively, exogenous quantity and exogenous price shocks. In this chapter, we will further observe that this advice is more easily given than applied.

9.1 Key Sector and Linkage Analysis: A Half-Truth The formulation and empirical calculation of linkage measures represents one of the earliest applications of IO analysis (Rasmussen 1956; Chenery and Watanabe 1958). Hundreds of such studies have been done since,1 mostly with the aim to identify so-called key sectors—usually defined as sectors with a high potential of spreading their own growth to the whole of the economy (see Hirschman 1958, for 1 On March 10, 2022, “backward and forward linkages” scored about 19,700 hits with Google Scholar, and “key sector analysis” about 687 hits.

The original version of this chapter was revised: The presentation of Table 9.5 has been corrected. The correction to this chapter is available at https://doi.org/10.1007/978-3-031-05087-9_11 © The Author(s), under exclusive license to Springer Nature Switzerland AG 2022, corrected publication 2022 J. Oosterhaven, Rethinking Input-Output Analysis, Advances in Spatial Science, https://doi.org/10.1007/978-3-031-05087-9_9

123

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9 Other IO Applications with Complications

the original formulation, and Perroux 1961, for the first spatial interpretation). The basic idea is that sectors with relatively large intermediate purchases (i.e. backward linkages) as well as relatively large intermediate sales (i.e. forward linkages) do so most effectively.

9.1.1 Analytical and Empirical Comparison of Key Sector Measures Temursho and Oosterhaven (2014) give an extensive overview of the different forward and backward linkages proposed in the literature and conclude that this delivers ten sensibly different measures. The first eight of them (in the first eight rows of Table 9.1) try to capture the same basic concept, namely the one-sided dependence of the Whole of the Economy (WoE) on the sector at hand. The only exception is the net backward linkage interpretation (Oosterhaven 2004) of the net multiplier concept (Oosterhaven and Stelder 2002). The reason for this exception is that net linkages intend to capture the two-sided nature of sectoral dependence, as Table 9.1 Ten generalized and normalized key sector measures Name of measure

Total backward linkage

Linkage formula  i ci ai j /c j  d f i = j bi j c j /ci  tb j = i ci li j /c j

Total forward linkage

t fi =

Direct backward linkage Direct forward linkage

db j =

 j

gi j c j /ci

Complete HE backw. linkage cb j = tb j / l j j

Explanation with jobs as impact variable Direct jobs with suppliers to j per job in j Direct jobs with buyers from i per job in i Direct + indirect jobs with suppliers to j per job in j Direct + indirect jobs with buyers from i per job in i Extraction of complete j from Leontief model

Complete HE forw. linkage

c f i = t f i /gii

Extraction of complete i from Ghosh model

Partial HE backward linkage

pb j = (tb j − 1)/ l j j

Extraction of input column j from Leontief model

Partial HE forward linkage

p f i = (t f i − 1)/ gii

Extraction of output row i from Ghosh model

Net backward linkage

nb j = tb j (y j / x j )

Total backward linkage × final output ratio j

Net forward linkage

n f i = t f i (vi / xi )

Total forward linkage × primary input ratio i

Legend HE = hypothetical extraction, aij = intermediate input coefficient, bij = intermediate output coefficient, ci = impact variable per unit of output, l ij = cell from Leontief-inverse, gij = cell from Ghosh-inverse. Source Adaptation of Temursho and Oosterhaven (2014)

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they equal the dependence of the WoE on the sector at hand divided by the dependence of the sector at hand on the WoE (Dietzenbacher 2005). To complete the list, Oosterhaven (2008) defined a tenth measure, i.e. the net forward linkage equivalent of the net backward linkage (see the last two rows of Table 9.1 for both). Most linkage measures use output as the impact variable. Hazari (1970), however, already early advocated weighing (or better rescaling) the Rasmussen (1956) linkages with the planner’s preference function, which he operationalized as each sector’s share in total final demand. Whereupon Loviscek (1982) much later reacted with the proposal to weigh the forward linkages with each sector’s share in total primary inputs, instead of its share in final demand. In fact, to be really relevant for policy formulation, linkage measures should not reflect the impacts on total output, whether weighed or unweighted, but the impacts on the policy goal at hand, such as income generation, job creation or CO2 emission reduction (see Oosterhaven 1981, and Diamond 1985, for early examples of employment linkages). This requires that linkage measures should be multiplied with the relevant impact variable per unit of output. If this ci in Table 9.1 is replaced by 1.0, the traditional output-based linkage measure results, which is why the ten linkages of Table 9.1 are labelled generalized linkages. They are also labelled normalized linkages, as they are all made independent of the size of the sector at hand.  Table 9.1 does not include the row sums of the Leontief-inverse, j li j ∈ L i = (I − A)−1 i, as a forward linkage measure (Rasmussen 1956), as these row sums measure the backward impact of a meaningless unit vector of  exogenous final demand. Instead the list uses the row sums of the Ghosh-inverse, j gi j ∈ G i = (I − B)−1 i, as first proposed by Beyers (1976) and Jones (1976). This choice implies that the forward linkage measures should not be interpreted as representing the WoE’s total output impact of a primary supply quantity shock to the sector at hand, as the quantity interpretation of Ghosh multipliers has been shown to be highly implausible (Sect. 7.1). Instead, these measures should be interpreted as the impact on the value of the WoE’s total output of the complete passing on of a primary supply price shock to the sector at hand; i.e., they should use Dietzenbacher’s (1997) re-interpretation of the Ghosh model as a price model. Table 9.1 also does not include the output-to-output multiplier (see Miller and Blair 2022, Chap. 6) as that multiplier equals the total flow multiplier (Szymer 1984), which on its turn is not included as it equals the earlier hypothetical extraction (HE) of complete sectors (Paelinck et al. 1965; Strassert 1969; Schultz 1977), as first indicated by Szyrmer (1992) and recently proven by Gallego and Lenzen (2005). The hypothetical extraction method is included instead, because it offers much more flexibility, as it allows for the extraction of any subset of transactions from an IOT. All HE variants suggested in the literature (Miller and Lahr 2001), however, require a cumbersome three-step calculation process and all result in linkage measures that primarily tell the analyst that deleting monetary large parts of an IOT has large impacts, which is trivial. Hence, HE outcomes need to be rescaled (normalized) with the size of the industry at hand to provide meaningful information. Including this normalization, the following generalized three-step HE procedure gives the desired result:

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1. Define A−s as the matrix with intermediate input coefficients with the selected cells equal to zero, c−s as the vector with the impact variable per unit of output with the selected cells equal to zero, and y−s as the vector with exogenous final demand with the selected cells equal to zero. 2. Calculate the impact vector v, without and with the selected cells equal to zero, as vi ∈ v = cˆ (I − A)−1 y and vi−s ∈ v−s = cˆ −s (I − A−s )−1 y−s , respectively. 3. Calculate the generalized and normalized HE backward linkages of the selected IO transactions for industry i as H E ib,v,s = (vi − vi−s )/vi . f,y,s

are calculated analogously, but The corresponding HE forward linkages H E i with the Ghosh model of Sect. 7.1. Fortunately, the two most obvious HE key sector measures both have an analytical solution that makes their calculation and comparison with the non-HE measures far more easy. These two HE measures result from, respectively, the complete HE of an individual sector and a column-only or row-only partial HE of an individual sector to calculate the partial HE backward and partial HE forward linkages, respectively. See Table 9.1 for the two analytical solutions, and Temursho and Oosterhaven (T&O) (2014) for the mathematical proofs. T&O, furthermore, prove that the diagonals of the Leontief-inverse and the Ghoshinverse are equal (lii = gii , ∀i), and show that the values of the diagonal cells vary empirically between about 1.01 and 1.20. This means that the corresponding HE linkages and total linkages will be very strongly correlated, as these diagonal elements constitute the only difference between them (see the formulas in Table 9.1). Moreover, T&O show that the final output ratios of the net backward linkages empirically vary between about 0.35 and 0.85, whereas the primary input ratios of the net forward linkages vary less from 0.50 to 0.70. Consequently, after inspection of the formulas of Table 9.1, one may predict that the net backward linkages will show a larger deviation from their corresponding total linkages than the net forward linkages from their total linkages. Finally, T&O analyse the sector-by-sector similarities of both income, employment and CO2 backward and forward linkages for 34 sectors in 33 countries. In general, the rank order of sectors per country is very dissimilar between the group of backward linkages, on the one hand, and the group of forward linkages, on the other hand, which is not too surprising as these two sets of linkages measure fundamentally different impact mechanisms, namely the backward passing on of final demand quantity growth and the forward passing on of primary input price increases. Within each of these two groups of linkages, the rank orders of the 34 sectors in case of the total linkage and the two HE linkages are very similar for almost every country, while the direct linkages are similar to these three linkages, but a little less so. Within the group of forward linkages the net forward linkages are only weakly correlated with the other (gross) forward linkages, while the net backward linkages within their group are even weaker correlated with the other (gross) backward linkages. The quite different sectoral rank orders for the two net linkages are not too surprising either, as they represent the only measures that try to capture both sides of sectoral dependence.

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9.1.2 Cluster and Linkage Analysis for Three Dutch Spatial Policy Regions A typical example of a key sector analysis is the cluster and linkage analysis for three Dutch spatial policy regions by Oosterhaven et al. (2001). It was an investigation commissioned by the Dutch Ministry of Economic Affairs, among others, responsible for Dutch regional policy. At that time, Dutch infrastructure policy and Dutch regional policy was based on strongly held beliefs about the supposedly large national economic importance of the transport and distribution sectors in the two so-called mainport regions—the greater Rotterdam Harbour area and greater Amsterdam Airport/Harbour area. The peripheral and more rural Northern Netherlands, on the other hand, was viewed as a weak region in need of regional policy help, with never mentioned, and thus implicitly believed little importance for the national economy. The outcomes of the study, however, did not confirm these beliefs, which is why the Ministry initially refused to publish the report. Thanks to a well-informed journalist who threatened to go to court, the study was published with a few months delay (see Oosterhaven et al. 1999, for the conclusion of the fierce, subsequent Dutch policy debate). The cluster analysis of this study is based on the absolutely and relatively largest direct linkages of the 48 sectors distinguished in each of the three bi-regional IOTs used (RUG/CBS 1999). Clusters are defined as groups of interdependent sectors with linkages in both directions. Against popular belief, the study shows that the transport and distribution sectors of the two mainport regions do not constitute the core of an important cluster of sectors. Instead, the core of the strongest cluster in the greater Amsterdam region consists of printing and publishing, business services and trade services, whereas the core of the strongest cluster in the greater Rotterdam region consists of the chemical sector. Both clusters also have strong direct linkages with comparable sectors in the rest of the country. Against popular belief, having strong links with the rest of the country is also the case for the strongest cluster in the Northern Netherlands with agriculture and food processing as its core industries. The second strongest cluster in all three regions has construction as its core sector, but in that case the relation with comparable sectors in the rest of the country is different. The construction cluster in the two much smaller, more densely populated mainport regions is strongly tied to comparable sectors in the rest of the country. In contrast, the construction cluster in the spatially larger and more rural Northern Netherlands only has strong direct linkages within its own region. The overall totals of the direct linkages of these three regions, i.e. the direct spatial linkages, are summarized in Table 9.2. As expected, the spatially larger North has the strongest intra-regional direct linkages, but those of the two mainport regions are surprisingly strong too. Besides, Greater Amsterdam has the strongest direct linkages with the Rest of the Country, whereas Greater Rotterdam has the strongest direct ties with the Rest of the World.2 2 The own region origin percentages and the rest of the country origin percentages in Table 9.2 represent weighted improvements for the two unweighted direct backward spatial linkages in Miller

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Table 9.2 Overall trade relations, i.e. direct spatial linkages per region Northern Netherlands

Greater Amsterdam

Greater Rotterdam

Origin of purchases in % // Destination of sales in % Own region

56 // 51

46 // 44

41 // 39

Rest of Country

20 // 25

29 // 33

25 // 27

Rest of World

24 // 24

25 // 23

34 // 34

Source RUG/CBS (1999)

Table 9.3 Weighted average total spatial linkages per region, times 100 Northern Netherlands

Greater Amsterdam

Greater Rotterdam

Total backward linkage // Total forward linkage Intra-regional linkages

26 // 24

16 // 16

19 // 19

Bi-regional spillovers

21 // 29

25 // 24

26 // 25

47 // 53

41 // 40

45 // 44

Total: national

linkagesa

a

The total forward and the total backward linkage of the weighted average national sector times 100 both equal 50 (see Oosterhaven et al. 2001, footnote 6, for the proof of the equality). Source RUG/TNO (1999)

It is interesting to compare the implicitly weighted direct spatial linkages of Table 9.2 with the explicitly weighted total, i.e. direct plus indirect, spatial linkages of Table 9.3. In case of the backward linkages the weighing is done by means of the share of each regional sector in regional exogenous final demand in the bi-regional Leontief model, yi·r · /i yr· , while the weighing in case of the forward linkages is done by each sector’s share in regional exogenous primary input in the bi-regional Ghosh model, v·ir /i vr . This gives the following formulas for the weighted intra-regional total backward and total forward linkages, respectively3 : i (Lrr − I) yr· /i yr· and (vr ) (Grr − I)i/i vr

(9.1)

Note that both intra-regional linkages are exclusive of the meaningless direct impact I, whose removal is necessary to be able to compare the intra-regional linkages with the bi-regional total backward and total forward spillovers, which are calculated likewise: i Lsr yr· /i yr· and (vr ) Gr s i/i vr

(9.2)

and Blair (2009, p. 563), whereas the corresponding destination percentages represent weighted improvements for the two unweighted direct forward spatial linkages in Miller and Blair (op cit). 3 Note that this weighing serves to aggregate the individual sector linkages to a weighted total spatial linkage. It should, therefore, not be confused with the rescaling of individual sector´s total linkages to better represent the planner’s preference as advocated by Hazari (1970) and Loviscek (1982).

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Note that the backward spatial spillovers relate to imports, whereas the forward spillovers relate to exports.4 As one may expect from inspecting the direct spatial linkages in Table 9.2, the total intra-regional linkages of the Northern Netherlands in Table 9.3 are significantly larger than those of the two mainport regions. The mutual rank order of the two mainport regions, however, reverses. Inspecting only the direct linkages disregards the greater strength of the clusters of greater Rotterdam. To a lesser extent, the same holds for the spillovers of the Rotterdam clusters into the Rest of the Country. The direct spillovers in Table 9.2 are clearly smaller than those of Greater Amsterdam, but the total spatial spillovers in Table 9.3 are a little larger. Consequently, the total national linkages in Table 9.3, being the total of the intraregional linkages and the bi-regional spillovers, are also larger for Greater Rotterdam than for Greater Amsterdam, such despite the clearly larger foreign import and foreign export leakages of Greater Rotterdam in Table 9.2. Obviously, only looking at direct linkages may lead to wrong conclusions, despite their positive correlation with the total linkages for practically all countries observed by Temursho and Oosterhaven (2014). The large size of the total national forward and backward linkages of the Northern Netherlands, represented the most shocking outcome of Oosterhaven et al. (2001), as all national policy-makers were convinced that those of the two mainport regions would be significantly larger than those of the peripheral North. An interesting side result was the rather strong relation found between the relative size of a regional sector, as measured by means of its Location Quotient (LQ, see Sect. 4.1), and the relative size of its total linkages. A clear specialization bonus seems to be present: sectors with large LQs systematically have larger total forward and backward linkages than the corresponding sector in other regions, as first noted in Oosterhaven (1981).

9.1.3 The Mostly Forgotten, Cost Side of the Coin The core question, however, is the relevance of the above type of outcomes in selecting sectors that will stimulate regional or national economic growth most. Schaffer (1973) already early points at the rather big gap between selecting key sectors and the problem of choosing which instruments to use to stimulate which sectors. The problem of bridging the gap with policy-making, to a large extent, depends on which linkages are largest. If a sector is selected because of its large forward linkages, the causality of the cost-push IO price model (see Sect. 6.1), which underlies these linkages, suggests that the appropriate instruments should operate on the primary cost side of that sector: making its outputs cheaper such that its processing sectors are stimulated too. However, if both this key sector and its processing sectors completely pass these 4

The two intra-regional total linkages and two bi-regional total spillovers in Table 9.3 represent weighted improvements for the unweighted total spatial linkages in Miller and Blair (2009, p. 563).

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price reductions on to their customers, as is assumed in the model that underlies the forward linkages, the whole primary cost decrease will end up in lower final output prices. To still get a positive quantity effect that is as large as the value decrease of final demand, i.e. that is as large as the forward linkage of that sector, one needs to assume that all final demand elasticities equal minus one, which runs against one of the very assumptions on which the underlying model is based (see Table 7.2). Nevertheless, this “minus one” assumption is more plausible than the “infinite demand elasticity” assumption of the underlying model. The “minus one” assumption, however, requires that the estimation of the impacts subsequently goes through the circular process described in Fig. 7.2, and this results in ex post forward linkages that are considerably smaller than the ones on which the selection of this sector was based! The opposite case is the selection of a sector because of its large backward linkages. Then, the causality of the underlying IO model suggests the use of instruments that increase the volume of that sector’s final demand in order to stimulate its supplying sectors. There are, however, many ways to do so, each with quite different policy cost. And, again the question is how the stimulated sector and its supplying sectors will react. If the stimulated sector is subject to supply restrictions, it most likely will raise its prices and hardly raise its output, whereupon imports will have to increase to satisfy the larger final demand, implying ex post backward linkages that are smaller than the ones on which this sector was selected! The same reaction may occur with the supplying sectors if they face supply restrictions, also leading to smaller ex post backward linkages. McGilvray (1977) emphasizes that Hirschman (1958), in fact, was looking for potentially growing sectors that could play a leading role in creating disequilibrium, which would induce investment, especially, in supplying sectors that have a minimum operating capacity that is small compared to the additional demand of the stimulated key sector, which implies that the standard total backward linkage measure needs to be rescaled with this ratio in order to be able to select the right key sectors. Both he and Bulmer-Thomas (1978), furthermore, argue that the difference between the total backward linkages based on technical coefficients and those based on intra-regional input coefficients, i.e. i (I − A·r )−1 − i (I − Arr )−1 , should be used to indicate the potential backward impacts of stimulating the sector at hand, as Hirschman’s disequilibrium approach is based on substituting imports by domestic production as a development strategy. As a complement, one may advocate to add a development strategy that is based on substituting exports by domestic processing, which would imply selecting sectors with the largest difference between the total forward linkages measured by what one might call technical output coefficients and those measured with the standard intra-regional output coefficients, i.e. by (I − Br · )−1 i − (I − Brr )−1 i. Hewings (1982) comments that measures of import substitution potential, especially for smaller developing countries and for most regions, should include the potential of import substitution of consumer goods and not only that of intermediate goods. The same comment may be made for the above-suggested measure of the potential impacts of export substitution. Note that both comments may easily be

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accommodated by using the Type II Leontief-inverse and the Type II Ghosh-inverse, respectively, instead of the above two Type I inverses. However, the likelihood of actually achieving the thus estimated impacts of import and export substitution strongly depends on the possibilities to increase the comparative advantage of the region or country at hand in, respectively, the domestic production of the targeted imports and the domestic processing of the targeted exports. Obviously, this requires more serious research, than only calculating the above two formulas; i.e., it requires research such as that advocated for a target industry analysis (McLean 2018). Finally, let us assume that all the above qualifications are solved and that total backward and total forward linkages are specified in such a way that they correctly predict the benefits in terms of the chosen goal variable per unit of the exogenous impulse that belongs to the linkage at hand. Does that then deliver the key sectors that stimulate the growth of the chosen goal variable most effectively? The answer to this rhetorical question is of course NO: but why? Well simply because not one of the gross linkages (see the first eight lines of Table 9.1) tells the user anything about the policy cost of creating one additional unit of the presumed exogenous impulse. Take the case of the key sectors found for the Northern Netherlands (see Fig. 9.1). Agriculture has, especially, large forward linkages. Creating a new polder would increase the quantity of its primary inputs, and would lead to large positive impacts on its processing sectors, but at large financial and environmental cost (see Oosterhaven 1983). Stimulating agriculture by directly subsidizing its primary inputs is

Fig. 9.1 Forward and backward linkages of industries in the Northern Netherlands. Legend The size of the circles is proportional to gross value added of the industry at hand. The dotted cross in the middle indicates the size of the weighted average forward and backward linkage Source Oosterhaven et al. (2001)

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forbidden by the EU, but doing that indirectly by training of labour or stimulating agricultural research is allowed, but involves different cost and different production effects for each measure. The second key sector (food processing) has, especially, large backward linkages. Directly stimulating the size of its final demand is only possible and only allowed by the EU by means of lowering, for instance, the value added tax on food products. Doing that offers follow-up opportunities, such as stimulating the development of new food products, but each policy instrument will have different policy cost and different effectiveness in terms of its impact on final demand. Oosterhaven (2017) suggested that using net linkages (see the last two lines of Table 9.1) might solve the problem of specifying the policy cost of stimulating a key sector with one unit, as having a relatively large final demand offers more opportunities for stimulating backward linkages, while having relatively large primary costs makes stimulating forward linkages easier. This sounds reasonable. Take. e.g. the large raw materials industry in the Northern Netherlands, mainly consisting of the winning of natural gas from the largest gas field in Europe. It has rather large forward linkages, but no backward linkages to speak of (see Fig. 9.1). Increasing its relatively large value added (mainly profits and taxes) seems easy. Simply increase production, but that increase can only be sold if its price is lowered, leading to lower instead of larger value added as the demand elasticity for natural gas is lower than one. Hence, more generally, Oosterhaven’s claim about net linkages being able to correct for policy cost needs to be rejected as the relative size of final demand or primary cost has little bearing on the policy cost of creating an additional unit of it. In all, the above discussion makes one thing very clear: developing yet another new backward or new forward linkage measure, e.g. by means of qualitative IO analysis or graph theory or neural network analysis, is rather pointless.5 Instead, key sector analysis needs to be based on interindustry models with price–quantity interaction, as well as a specification of both the benefits and the costs of using different policy measures. Calculating whatever linkages only shows the half-truth of one side of the coin.

9.2 Structural Decomposition Analysis: Another Half-Truth The decomposition of output growth into the growth of final demand and changes of the Leontief-inverse is another important field of IO applications dating back to Leontief (1941) himself. Since then, thousands of such structural decomposition analyses (SDAs) have been done (see Miller and Blair 2022, Ch. 8, for a recent overview). In their extensive overview, Rose and Cassler (1996) emphasise the similarities between

5

A comparable conclusion holds for the concept of the Average Propagation Length (Dietzenbacher et al. 2005) be it for different reasons (Oosterhaven and Bouwmeester 2013).

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133

SDA, shift and share analysis (SSA) and growth accounting (GA). The communality is that all three approaches decompose an economic growth equation into its constituent parts.6 The explanation of economic growth and its variation between regions and nations is one of the most important theoretical and empirical issues in economics. In empirical analyses, two quite different approaches are used, namely the deterministic/decomposition methods mentioned above and stochastic/econometric methods. In deterministic approaches total growth or the total growth difference between regions/nations is decomposed into and fully attributed to different components according to some formula/theory. In stochastic approaches, a set of possible explanatory variables is econometrically tested to determine which variables contribute most to the explanation of growth or growth differences and which do not: always leaving an unexplained residual. Here, we aim at a comparison and evaluation of the three deterministic/decomposition approaches that are used in empirical analyses of economic growth. Contrary to Rose and Cassler (1996), who emphasize the similarities, we will emphasise the clarifying, fundamental differences between these three techniques.

9.2.1 Shift and Share Analysis: Impact of Industry Mix We start with the decomposition technique that has no theoretical foundation, except for the notion that the mix of industries is important in explaining the difference between a region and the nation or the world with which one wants to compare that region; i.e., we start with shift and share analysis (Creamer 1942, popularized by Dunn in Perloff et al. 1960, see Lahr and Ferreira 2020, for a recent overview). Its most simple application decomposes the following identity: vr − v n =

 i

sir vir −

 i

sin vin with

 i

sir =



sin = 1

(9.3)

i

where v = variable of interest (e.g. total GDP growth, total job growth, average wage level, total energy use or total CO2 emission level) for some unit r (e.g. region) that is to be compared with some norm n (e.g.  nation) and that is aggregated over some index i (e.g. industry), with sir = vir / i vir = share of i in r for variable v, and sin = the analogous share of i in n.7 From his description follows that SSA may be

6 On March 10, 2022, “shift and share analysis” scored about 1,200 hits on Google Scholar, “structural decomposition analysis” and “input–output” combined scored about 5,400 hits, and “growth accounting” and “production function” combined scored about 19,700 hits. 7 Note that the definition of the share may need to be adapted to the definition of the variable, as in case of labour productivity growth (see Oosterhaven and Broersma 2007). In international economics, when v = export growth, r = some country, n = all of the world, and i = products, shift

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Table 9.4 Possible shift and share decompositions of Eq. (9.3) No. Structural component, with:

Combined differences (specialization) regional growth national growth regional industry national industry component rates rates shares shares  r     r n r n n n n r n n n r r r r i (si − si )vi i (si − si )vi i si (vi − vi ) i si (vi − vi ) i (si − si )(vi −vi )

1

+

2 3a

½

4

+

5

Growth component, with:

+ +

+

½

½

½

+ +

– +

+

a This decomposition results from taking the average of decomposition 1 and 2 as well as that of 4 and 5. Source Oosterhaven and van Loon (1979)

used to analyse a multitude of problems. Here, we only discuss its oldest and most frequently used application to regional economic growth. Table 9.4 shows the five ways in which (9.3) may be decomposed. The first decomposition shows the classical SSA of regional growth vr into the regional sharein national growth v n , plus a proportional  shift due to a different industry mix i (sir − sin ) vir , plus a differential shift i sin (vir − vin ) (the italics indicate the origin of the term “shift and share” analysis). This last component gives an indication of the impact of regional competitiveness, as it measures whether the weighted average regional industry grows faster of slower than its national counterpart. Capello (2007) further clarifies that the industry mix component will primarily be related to demand-side growth factors and the residual competitiveness component primarily to supply-side factors. Both the industry mix component and the competitiveness component may be measured—respectively weighted—differently, as is evident from a comparison of the first and the second decomposition in Table 9.4. Taking the average of the first two decompositions delivers the third decomposition. Taking the simple average is the typical solution of SDA and GA to the problem of choosing between components measured in base year terms and those measured in end year terms. In SSA, this is not the preferred choice. When the research interest is in comparing different regions, each component needs to be measured/weighted in the same way. This argument makes the first three, and especially the fourth decomposition inacceptable for interregional comparisons. Luckily, there is a fifth decomposition that measures both the industry mix component and the competitiveness component in the same way. To reach this result, a third combined differences component needs to be added (see the last row of Table 9.4). This third component is theoretically interesting on its own account, as it measures whether the industries in which the region is specialized have a higher or lower score on the variable of interest: in our case whether they grow faster or slower than their and share analysis is known as constant market share analysis (see Jepma 1986, for an overview and several applications).

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national counterparts. This third component thus measures the impact of regional specialization; i.e., it signals so-called localization economies or diseconomies when it proves to be negative (see Oosterhaven and Broersma 2008, for the difference with cluster, urbanization and agglomeration economies). In view of its theoretical interest, the fifth decomposition should be considered to represent the preferred decomposition, even when the research interest only regards a single region.8 In case of regional wage differences (Oosterhaven and van Loon 1979) and in case of regional labour productivity differences (Oosterhaven and Broersma 2007) specialization clearly pays off, in the sense that the industries in which a region is specialized have higher levels of labour productivity and pay higher wages than their national counterparts, indicating positive localization economies. In the case of labour volume growth and value added growth, however, the specialization component proved to be negative for all Dutch regions, which was interpreted as representing diminishing returns to these positive localization economies (Oosterhaven and Broersma 2007). The same result was found for earlier periods, for different regions and for different industry classifications (WWR 1980; Oosterhaven and Stol 1985). In these studies, the industry mix component proved to exhibit a stable regional pattern over eight periods between 1951 and 1983, with a slowly diminishing importance, starting with “explaining” a halve, and ending with “explaining” only a quarter of the regional differences in job growth. The residual competitiveness component, on the other hand, gained importance, but with an unstable regional pattern with sometimes changing signs between subperiods. Interestingly, the changing of signs appeared to be related to changes in national economic growth. Core regions showed a relative slowdown of their residual growth during periods of national growth, probably due to local congestion and supply shortages, whereas peripheral regions reduced part of their economic arrears during periods of national growth, probably due to picking up part of the core regions’ choked off growth. In fact, SSA may be considered as a special case of index decomposition analysis (IDA, see Hoekstra and van den Berg 2003, for an overview). The difference is that IDA links the variable of interest (e.g. energy use or job growth) explicitly to the output by industry, which results in at least one extra component related to the change in the corresponding coefficient, whereas SSA directly decomposes the change in the variable of interest by industry without linking it to the output of that industry. In case of an IDA of job growth, the additional component will show the impact of regional labour productivity growing faster or slower than its national equivalent (see Rigby and Anderson 1993, for an early application).

8 The fifth decomposition also proved to be superior from an empirical point of view in case of regional wage differences in The Netherlands, as its three components were mutually uncorrelated, as opposed to the components of the other decompositions (Oosterhaven and van Loon 1979).

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The most attractive properties of SSA, and IDA for that matter, is its versatility and the limited amount of aggregate sectoral data needed. The most mentioned objections against SSA are (1) its lack of a theoretical foundation (Richardson 1978) and (2) the impossibility to determine the statistical significance of its components (Stillwell 1969; Chalmers and Beckhelm 1976; Stevens and Moore 1980). Besides, it was already noted early on that (3) the industry mix component is sensitive to sectoral aggregation, being more important at lower levels of aggregation, while (4) its size is underestimated due to ignored interindustry interdependences (Mackay 1968). The lack of a theoretical foundation, however, may be turned into an advantage when, the industry mix and the specialization component are used as regular, but composite variables in an econometric estimation of the LHS of (9.3). This, in fact, simultaneously solves the second objection, as it provides a measure of the statistical significance of these two components in explaining the LHS of (9.3) (see Graham and Spence 1998, for a first application of this principle). In the case of Dutch regional labour productivity levels and growth rates, using this econometric approach, Broersma and Oosterhaven (2009) find that both the industry mix and the specialization component from their SSA are highly significant, along with the regional capital/labour ratio, a regional diversity index, and the own and the neighbouring regions’ job density as indicators of agglomeration economies or diseconomies.

9.2.2 Structural Decomposition Analysis: A Demand-Side Story Next, consider the oldest and most simple structural decomposition analysis (SDA), which uses the solution of the basic input–output model (see Sect. 2.2) to split up output growth by industry: x ≡ x1 − x0 = (I − A1 )−1 y1 − (I − A0 )−1 y0 = L1 y1 − L0 y0

(9.4)

An SDA of (9.4) represents a comparative static analysis that sequentially looks at the impact on the variable of interest of changes in each set of parameters, holding the other parameters constant. Note that SDA may be used to decompose any first order difference in a matrix equation, such as the difference between national and regional embodied CO2 emissions or the growth of energy use (see Hoekstra and van den Berg 2002, for an overview such SDAs). However, here, we only discuss its most common application, namely, to economic growth. Just like the decomposition of (9.3), there are also five comparable decompositions of (9.4) (see Table 9.5). Skolka (1989) presents four of them, while Decomposition 4 is added by Oosterhaven and van der Linden (1997). In choosing between the first two decompositions, neither Skolka (1989) nor Dietzenbacher et al (2004) nor Miller and Blair (2022, Ch. 8) see any preference, which is why they all prefer taking

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Table 9.5 Possible structural decompositions of industry output growth x No.

1

Leontief-inverse change, with:

Final demand change, with:

base year y

end year y

base year L

end year L

L y0

L y1

L0 y

L1 y

+

2 3a

½

4 5 a

L y

+ +

+

½

½

+ +

Interaction component

½ +

+

– +

This decomposition results from taking the average of decompositions 1 and 2 as well as of 4 and

5

the average, i.e. Decomposition 3. This choice neglects the interaction component L y. However, this is not a real loss, as (1) the interaction component is empirically found to be rather small (Uno 1989), while (2) it is theoretically considered to have no clear economic interpretation (Miller and Blair 2022). Here, SDA clearly deviates from SSA.9 De Boer and Rodrigues (2020), in an excellent overview, show how both IDA and SDA may benefit from the outcomes of index number theory, which also divides the (price–quantity) interaction term over the other components (i.e. over the price and quantity indices). They furthermore present the corresponding multiplicative IDAs and SDAs, wherein geometric averages are taken instead of the arithmetic averages taken in the additive, third decomposition of Table 9.5. Departing from the most simple IO model used in (9.4), many, more sophisticated variants with an increasing number of components have been developed (see Rose and Casler 1996, and Miller and Blair 2022, for overviews). To increase the understanding of the type of outcomes that a SDA may generate, we showcase the decomposition of value added growth in the EU by Oosterhaven and van der Linden (1997), as it includes most of the individual components proposed in the literature. They use the interregional IO model of Sect. 2.3, with each of the intermediate input coefficients rs ∈ B (often called bridge airjs and each of the final demand input coefficients biq coefficients) split up into: (1) a technical or preference coefficient and (2) a trade origin ratio:

9 Elements of a SSA may be integrated into an SDA, as suggested by Lahr and Dietzenbacher (2017). They show that in case of a regional SDA – with the added data from two national IOTs – both y and L may be split up further into changes in the levels and structures of the regional and national y and L. Such a further split up of the decomposition of Table 9.5, however, does not change its demand-driven nature nor its other properties to be discussed next. The reverse, an integration of the IO model into a SSA, is equally possible (e.g. Shaw & Spence 1998), but that does not change the basic nature of SSA either.

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v = cˆ L B y ≡ cˆ (I − Ma ⊗ A)−1 (M f ⊗ F) y

(9.5)

In (9.5), ⊗ = Hadamar product (i.e. cell-by-cell  r s matrix multiplication), and going ∈ y = a QR column with macrobackwards along the causal chain, y·q·s = ri yiq  economic levels of final demand of type q in country s, f iq·s = r f iqr s ∈ F = an IR × QR block matrix with R mutually identical I × QR matrices with final demand preference coefficients, indicating the total use of product i delivered from all over the world per unit of final demand, m riqs ∈ M f = an IR × IQ matrix with cell-specific trade origin  ratios, indicating which fraction of that total originates from region r, ai·sj = r airjs ∈ A = an IR × IR matrix with R mutually identical I × IR matrices with technical coefficients, indicating the total use of product i delivered from all over the world per unit of output of industry j in s, m ri js ∈ Ma = IR × IR matrix with cell-specific trade origin ratios, indicating which fraction of that total originates from country r, cˆ = IR × IR diagonal matrix with gross value added coefficients and v = IR column with gross value added per industry, per region. The comparative static decomposition of the change in (9.5) is laborious, but straightforward, except for the decomposition of the change in the intercountry Leontief-inverse L into its constituent parts: L = L1 (Ma ⊗ A)L0 = 0.5 L1 (M0a + M1a )

⊗ AL0 + 0.5 L1 Ma ⊗ (A0 + A1 )L 0

(9.6)

The crucial first equality of (9.6) can be proven by pre-multiplication and postmultiplication of the first two terms of (9.6) with (I − M1a ⊗ A1 ) and (I − M0a ⊗ A0 ), respectively. The last term of (9.6) gives the average of the two possible decompositions of the change in the Leontief-inverse without an interaction term. Using (9.6), the laborious overall decomposition of (9.5) may be done in three steps: 1. The standard decomposition is applied to the four terms of the first part of (9.5). That is done in such a way that the variable with the  moves from the left to the right for the first decomposition, whereas it moves from the right to the left for the second decomposition, after which the average of these two, so-called polar decompositions is taken. This delivers the component for ˆc in (9.7a) and that for y in (9.7f), where each of the two sets of weights nicely shows the polar nature of the two decompositions that are averaged. 2. Next, the last part of (9.6) is substituted into 0.50 (ˆc0 L B1 y1 + cˆ 1 L B0 y0 ), i.e. the component for the change in the Leontief-inverse from Step 1. This gives the component for Ma in (9.7b) and that for A in (9.7c). 3. Finally, 0.50 (ˆc0 L0 B y1 + cˆ 1 L1 B y0 ), the component for the change in the matrix with final demand input or bridge coefficients from Step 1, is subdivided into a component for M f in (9.7d) and one for F in (9.7e).

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139

These three steps result to the following decomposition of the changes in the last part of (9.5):10 v = 0.50 ˆc (L0 B0 y0 + L1 B1 y1 )

(9.7a)

  +0.25 cˆ 0 L1 Ma ⊗ (A0 + A1 )L0 B1 y1 + cˆ 1 L1 Ma ⊗ (A0 + A1 )L0 B0 y0 (9.7b)   +0.25 cˆ 0 L1 (M0a + M1a ) ⊗ A L0 B1 y1 + cˆ 1 L1 (M0a + M1a ) ⊗ A L0 B0 y0 (9.7c)   (9.7d) +0.25 cˆ 0 L0 M f ⊗ (F0 + F1 )y1 + cˆ 1 L1 M f ⊗ (F0 + F1 )y0   f f f f +0.25 cˆ 0 L0 (M0 + M1 ) ⊗ F y1 + cˆ 1 L1 (M0 + M1 ) ⊗ F y0 +0.50 (ˆc 0 L0 B0 + cˆ 1 L1 B1 ) y

(9.7e) (9.7f)

The above average of two polar decompositions, however, represents only one of many possible decompositions. Dietzenbacher and Los (1998), also ignoring interaction components, show that the number of possible basic decompositions equals the faculty of the number of components (n); i.e. it equals n! = 1*2*3 … *(n − 1)*n. They, luckily, also show that decompositions like that of (9.7), being the average of two polar decompositions, have outcomes that are very close to the average of all n! possible basic decompositions. When used to analyse economic growth, SDA is usually applied to longer time periods and practically always reports that changes in the level of final demand constitute by far the most important component.11 Feldman et al. (1987), following the seminal study of Anne Carter (1970) with more recent and more detailed IO data, analyse a decomposition of output growth with x = LB y for the USA over the period 1963–1978. They find that changes in y are far more important than changes in either L or B, for some 80% of the 400 American industries distinguished. Coefficient changes were only important in case of the fastest and the slowest growing industries (see Fujimagari 1989, for very comparable results for Canada). From those outcomes,

Note that an IDA of the change in v = cˆ s x, with s = x x −1 = vector with country industry shares in total EU output x, would have (8.7a) as its first components, while the last five components of (9.7) would be replaced by an regional industry mix change component and a total EU output growth component. Such an IDA, of course, provides much less information, but also requires much less data. 11 In case of an SDA of GDP growth, part of the reason of the dominance of the final demand change component is that any change in value added coefficients has to be compensated by an equally large but opposite change in the sum of the technical coefficients, because the sum of both equals unity, i.e. c + i A = i . Note that the sum of the value added coefficients and the input coefficients, however, is smaller than unity whenever foreign imports are positive, i.e. c + i (Ma ⊗ A) ≤ i . 10

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they conclude that the best growth policy is a good macro-economic policy. The question is whether that conclusion is justified by these analyses. Applying (9.7) to their EU IRIOTs for 1975–1985, Oosterhaven and van der Linden (1997) also report final demand growth, especially of household consumption, to be by far the most important component for all eight countries and for almost all of the 25 industries distinguished. The combined effect of the five types of coefficient changes in (9.7) proved to be rather small and predominantly negative, which was mainly caused by a systematic decline in value added coefficients in (9.7a), indicating more roundabout production processes with longer supply chains incorporating more non-EU value added. At the industry level for individual countries, however, they did find larger impacts of different types of coefficient changes, which leads them to conclude that sector policies may be more important for economic growth than indicated by Feldman et al. (1978), also because the economically much more open EU countries have less scope for macro-economic policies than the economically more closed USA. Again, the question is whether that conclusion is justified by the analysis. Finally, consider SDA results for the third large international trading unit, i.e. China. Andreosso-O’Callaghan and Yue (2002) find that the growth of total final demand, and specifically the export growth of “high-tech” industries, constitutes by far the largest contribution to Chinese output growth for 1987–1997. They, however, do not make a distinction between ordinary exports and processing exports that add only limited amounts of domestic value added to mainly imported materials. This distinction is important as processing exports—in contrast to ordinary exports— hardly have any indirect impacts on domestic value added. Pei et al. (2012), using Chinese IOTs with both kinds of exports separated for 2002–2007, conclude that the contribution of exports to domestic value added is overestimated with 32% if the two types of exports are aggregated, while the contribution of exports to the value added of the “high-tech” telecommunication industry is even overestimated with 63%. Still, they too report that the growth of domestic final demand “explains” as much as 70% of Chinese GDP growth, whereas changes in coefficients “explain” only minus 5%. The remainder of about 35% is “explained” by the growth of both types of exports. In the last paragraph, the word explained has been put between quotation marks. The phrase “deterministically attributed to” would have been more correct, be it more cumbersome. As opposed to SSA, its generalization IDA as well as SDA are seldom criticized. The main critique of SDA (Rose and Casler 1996; Dietzenbacher and Los 1998; Miller and Blair 2022) regards (1) the non-uniqueness of each decomposition and (2) the weak theoretical foundation for taking averages. However, just like SSA, SDA also needs to be criticized because of (3) the impossibility to determine the statistical significance of its components and (4) its sensitivity to sectoral aggregation. As opposed to SSA and IDA, however, SDA does have a theoretical foundation, namely the demand-driven IO quantity model. In case of SSA, the lack of a theoretical foundation and the related presence of a residual component can be turned into an advantage that solves the problem of establishing the statistical significance of its non-residual components.

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In contrast, having a theoretical foundation may easily be considered to represent the weakest aspect of SDA, for two reasons. (1) As opposed to SSA, and precisely because of its theoretical foundation, SDA does not have a residual component that, by dropping it in an econometric estimation, may be used to establish the statistical significance of the other components. (2) Depending upon the type of application, the assumptions of the underlying demand-driven IO quantity model may represent a major problem. This is, especially, the case in the largest area of SDA applications, i.e. the decomposition of industry output growth and GDP growth. In case of short-run, year-to-year economic fluctuations, especially when the economy operates below full capacity, the Leontief model more or less adequately captures the, under those conditions dominant demand-side causes of short-run economic growth and decline (cf. Sects. 2.2 and 6.3). In case of the usual analysis of longer run changes over five or more years, however, SDA unjustly ignores the impact of changes on the supply side of the economy, such as the growth of the labour supply, the growth of the capital stock and technological progress.

9.2.3 Growth Accounting: The Other, Supply Side of the Coin In contrast, both neo-classical growth theory and new growth theory (Solow 1999; Sengupta 1998), as well as empirical analyses of long-run economic growth (e.g. Durlauf et al. 1996) only look at supply-side factors to explain growth differences between regions and nations. The third decomposition approach, growth accounting (GA) perfectly fits into this literature, as it ignores the demand side entirely and decomposes the growth of industry output and GDP exclusively into the contributions of supply-side components. GA may be founded in production theory (Diewert 1976; Caves et al. 1982). Using a translog function of production possibility frontiers, and assuming competitive markets, full input utilization and constant returns to scale, the relative growth of multi-factor productivity of industry j ( ln R j ) may be defined as the residual of the relative growth of the total output of industry j ( ln x j ) and the sum of the relative growth of its inputs, i.e. growth of its use of capital (k j ), labour (l j ) and intermediate inputs (z j ), weighted with their respective cost shares wcj (Timmer et al. 2010, ch. 2):  ln R j =  ln x j − wk j  ln k j − wl j  ln l j − wz j  ln z j

(9.8a)

with: wk j = pk j k j / p j x j , wl j = pl j l j / p j x j , wz j = pz j z j / p j x j , with: wk j + wl j + wz j = 1

(9.8b)

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wherein: R = level of multi-factor productivity, w = respective weights and p = respective prices. In empirical applications, capital, labour and total intermediate inputs are often split up further, mostly by means of IOT or SUT data, while the weights are mostly calculated as the average of the begin year weight and the end year weight, as in the well-known and often used EU KLEMS database (see Timmer et al. 2010, Chap. 3). Note that (9.8) may also be calculated with two IOTs or two SUTs, in which case the weights will equal average of the primary and intermediate input coefficients of the begin year and the end year of the analysis. This results in decompositions of aggregate factor productivity growth that also attribute part of its growth to changes in industry mix of final demand and changes in IO coefficients (e.g. Wolff 1985, Casler and Gallatin 1997, see Kuroda and Nomura 2004, for a fine application to Japan). The use of the same IO data seems to suggest that SDA and GA are just two extreme cases of a single, integrated approach. The same conclusion might be drawn from the fact that the IO model on which SDA is based may also be founded in production theory (cf. Sect. 2.2). The IO model, however, uses a far more simpler production function and, more importantly in this context, SDA assumes that the demand for outputs drives production while supply follows, as opposed to GA that assumes that the supply of inputs drives production while demand follows. This fundamental difference is most telling in the role that investment plays in both decomposition methods. In SDA, it is the year-to-year fluctuations in the demand for investment goods that co-determine the fluctuations in total output: a mechanism known as the multiplier (Samuelson 1939; Puu 1986). In GA, it is the level of investments that co-determines the growth in the supply of capital and therewith the growth of output: a mechanism known as the accelerator (Samuelson 1939; Puu 1986, see also Sect. 8.3). The primary field of application of GA is not the analysis of industry output and GDP growth, but that of productivity growth. Comparing the USA and Europe, van Ark et al. (2008, see also Timmer et al. 2010) show that Europe was catching up in labour productivity until about 1995, after which it experienced a slowdown, whereas the USA significantly accelerated its productivity growth, at least until 2006. At the detailed industry level, traditional manufacturing no longer acted as the productivity engine of Europe, probably due to exhausted catching-up possibilities, while Europe’s industries lagged in participating in the new knowledge economy, lagged in investing in information and communication technology, and lagged in keeping up their multi-factor productivity growth. These differences, especially, led to an increasing gap in the productivity of European trade and business services: of course with variations from industry to industry and from country to country. Also, in the case of China, GA tells a story that is completely different from that of SDA, where the growth of final demand is the dominant “explanation”. Wu (2016) decomposes China’s 9.16% annual GDP growth over the period 1980–2000 into 6.61% due to the growth of capital, 1.32% due to the growth of labour and 1.24% due to the growth of total factor productivity (TFP). Of the 1.32% due to labour

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growth, 75% is attributed to quality improvement and 25% to the growth of hours worked. Of the 1.24% due to TFP growth, 70% is attributed to TFP growth at the industry level and 30% to the reallocation of capital and labour between industries. Differences in the contribution of the individual industries to these aggregate results are mainly explained by industry differences in market structures and policy interventions, running from being essentially centrally planned to being open to world competition. Comparable to SDA, the above type of GA analyses also suffers from their deterministic nature. In both cases, the statistical significance of the components cannot tested. GA simply believes the assumptions of the production function model underlying (9.8), just as SDA simply believes in its IO model, be it (9.4) or (9.5). In the case of GA, solely by assumption, demand does not play a role. Only the supplyside components matter. However, looking at only the supply side is as one-sided as looking at only the demand side. As said before, the latter may be more or less acceptable when analysing short-run, year-to-year changes in economic growth for economies operating below full capacity. Looking only at the supply side is far more appropriate when analysing longer run economic growth, but it is also one-sided.

9.2.4 Solution: Econometric GA with SSA and SDA Components Luckily, in the case of GA there is no reason to restrict the analysis to supply-side factors. The reason is that GA contains a residual component ( ln R j in Eq. 9.8a), just like shift and share analysis. And just like SSA, dropping the residual component allows for an econometric estimation of industry output growth (i.e.  ln x j ) or GDP growth (i.e.  ln (G D P) =  ln ( j c j x j )). And, just like SSA, this simultaneously delivers an estimate of the statistical significance of the, in that case, estimated weights of the remaining GA components. This econometric GA approach, furthermore, enables the analysist to estimate the real importance (if any) of demand-side factors in explaining shorter or longer run economic growth, namely by adding components from either a SSA and/or a SDA as composite explanatory variables. These components may, for example, be used to test the significance of industry mix changes, be it in total output as measured by means of a SSA or in final demand as measured by a SDA. Additionally, they may, e.g. be used to test the significance of the total of all changes in intermediate input coefficients from a SDA or the significance of cluster or localization economies as measured by the specialization component from a SSA.

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Chapter 10

The Future: What to Forget, to Maintain and to Extend

Abstract The four basic IO models are essentially unsuited as prediction models. Demand-driven IO models, however, represent a perfect descriptive device to measure the direct and indirect value added or natural resources embodied per unit consumption or exports, etc., whereas IOTs, or better SUTs, or even better SAMs, have proven to provide the indispensable data for ever more sophisticated, econometrically extended IO models and interindustry CGE models, both for single and for multiple regions and nations. Keywords IO quantity models · IO price models · Descriptive statistics · Consumer responsibility · Trade in value added · Social accounting matrices · Econometric IO models · Interindustry CGE models This book has shown that the use of input–output (IO) as a causal model of the working of an economy is best left aside, as it may only be used as a predictive model with the upmost care, and in many cases not at all. This holds, in the extreme, for the supply-driven IO quantity model (Ghosh 1958), which cannot be used to predict the forward impacts of quantity shocks to the primary supply side of the economy. It also holds for the demand-driven IO quantity model (Leontief 1941), but to a much lesser extent. The Leontief model may be used to estimate the backward impacts of quantity shocks to the final demand side of the economy whenever the economy is functioning below full capacity. Nevertheless, the absence of price reactions, even in that case, leads to an overestimation of the backward impacts of such shocks. This will, especially, be the case when the Leontief model is extended into a Type II demand-driven IO model with endogenous consumption expenditures. The two accompanying price models do something comparable. They systematically overestimate the price impacts of their exogenous price shocks, as they do not take the quantity reactions into account. Nevertheless, both models could, in fact, be used much more. The cost-push IO price model (Leontief 1951) simulates how primary input (capital, labour, import) price shocks, under full competition, will be fully passed on forwardly to end up in final output prices. Hence, this model may well be used to simulate, e.g. the maximum consumer price increases that may result from the © The Author(s), under exclusive license to Springer Nature Switzerland AG 2022 J. Oosterhaven, Rethinking Input-Output Analysis, Advances in Spatial Science, https://doi.org/10.1007/978-3-031-05087-9_10

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increases in tariffs between the USA and China in the late 2010s, while international versions of this price model may be used to simulate the further forward impacts of the indirect increase of export prices on the consumer and investment prices of other countries. The revenue-pull IO price model (Davar 1989; Oosterhaven 1989), in turn, may be used to simulate how final output (consumption, investment, export) price shocks, under full competition, may be fully passed on backwardly to end up in primary input prices. Hence, this as yet unused model may well be used to simulate, e.g. the maximum wage increases that may result from the backward passing on of exogenous increases in export prices, such as that of oil and ITC-services, while international versions of this price model may be used to simulate the further backward impact of the indirect increase in import prices on the wages and capital incomes of other countries. However, especially, the basic demand-driven IO model offers much more than only a one-sided causal interpretation of the working of the economy. It may also, and very fruitfully, be used as a descriptive device. Over the last two to three decades, possibly a majority of the applications of IO analysis regard all kinds of calculations of the direct and indirect use of natural resources embodied per unit of final demand, as evidenced by the special issues of Economic Systems Research of 2005/4, 2009/3, 2011/4 and 2016/2. These calculations of consumer responsibility for environmental problems may be as detailed as the indirect CO2 emissions by specific Japanese industries embodied per unit of the consumption of specific goods by EU consumers. Moreover, adding the dynamic IO model (Leontief 1970) allows to incorporate the natural resources embodied in capital goods next to those embodied in intermediate goods (Södersten and Lenzen 2020). More recently, over the last decade or more, the basic interregional IO model has become comparably popular as a descriptive device in the area of international trade, where it is used to quantify such concepts as vertical specialization (Hummels et al. 2001), trade in value added (Johnson and Noguera 2012) and global value chain income (Timmer et al. 2013) (see Koopman et al. 2014, for an integration of some of these concepts). However, all these descriptive applications need to be done very carefully too, as shown by Bouwmeester and Oosterhaven (2013) who report serious sectoral and spatial aggregation errors when such calculations are made with too aggregate international IO models. They also report serious specification errors when worldwide CO2 or value added footprints are calculated with national IO coefficients instead of with the appropriate international ones. More specifically, there is still much scope for structural decomposition analyses (SDA), not of industry and GDP growth, but of such questions as why do some countries, directly and indirectly, use far less fossil fuels or why do they emit far more CO2 than others? Is it because they have a different composition of demand? Is it because they outsource more of their polluting activities to other countries? Is it because they use cleaner technologies? To analyse longer run industry and GDP growth, instead of SDA, econometric estimates of growth accounting equations with panels of IOTs or supply–use tables (SUTs), such as WIOD (Dietzenbacher et al. 2013), offer much more analytic opportunities than used hitherto.

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Finally and may be most importantly, IOTs, or better SUTs, of even better social accounting matrices (SAMs), constitute the indispensable databases that are needed to build more realistic interregional interindustry models, i.e. models with price– quantity interactions and spatial and technical substitution of intermediate inputs as well as of goods and services for final use. The promising nonlinear programming approach to modelling the behavioural reactions to major disasters (Oosterhaven and Bouwmeester 2016) handles price–quantity interactions implicitly. The most simple approach that handles these interactions explicitly combines an interregional SAM quantity model with its dual price model and solves this combination iteratively using final demand price elasticities and primary supply price elasticities as the links between the two models (Madsen 2008). An older and more convoluted tradition (Almon 1991) extends the basic IO model with econometrically estimated functions for final demand (consumption, investments, exports) and ditto for primary inputs (capital, labour, imports). It is, especially suited to generate interindustry projections of regional and national economic growth. This tradition continues until today (e.g. Kratena 2005) and will definitely flourish further into the future: not only at the level of regional and national economies but also at the international level, linking environmental and energy issues through international trade to what happens at the regional and national level (e.g. Többen et al. 2022). A second older and even more convoluted tradition (Shoven and Whalley 1992; Bröcker 1998) calibrates computable general equilibrium (CGE) models, with profit maximizing representative firms and utility maximizing representative households, on (inter)regional and (inter)national SAMs, often using behavioural coefficients from different studies. This tradition will most certainly also continue to strive (e.g. Anderstig and Sundberg 2013), as it mostly contains social welfare measures, while it is very suited to simulate the impacts of all kind of policy measures. A nice example is provided by the evaluation of transport infrastructure projects by means of New Economic Geography (NEG) models (Fujita et al 2001). In NEG models, different regions sell varieties of the output of each industry on monopolistically competitive regional markets linked by transport cost. CES-aggregates of these varieties are combined in Cobb–Douglas consumption and production functions. As freight and passenger transport cost reductions impact different industries differently, detailed interregional SAMs are needed to calibrate them. In NEG models, transport cost reductions (i.e. positive supply shocks) increase each region’s exports (demand) as well as its imports (supply). The net economic impact may well be negative for some industries in some regions, while causing the agglomeration of other industries in other regions (see Venables and Gasiorek 1998; Knaap and Oosterhaven 2011, for seminal applications). In all, there is plenty of future for IO-based and SAM-based analyses of environmental consumer responsibility, trade in value added, projections of future developments of the economy, and all kinds of regional, interregional, national and international policy simulations.

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References Almon C (1991) The INFORUM approach to interindustry modeling. Econ Syst Res 3:1–7 Anderstig C, Sundberg M (2013) Integrating SCGE and IO in multiregional modelling. In: Pagliara F, de Bok M, Simmonds D, Wilson A (eds) Employment location in cities and regions. Springer, Berlin Bouwmeester MC, Oosterhaven J (2013) Specification and aggregation errors in environmentallyextended input-output models. Environ Resour Econ 56:307–335 Bröcker J (1998) Operational spatial computable general equilibrium modelling. An Reg Sci 32(3):367–387 Davar E (1989) Input-output and general equilibrium. Econ Syst Res 1:331–344 Dietzenbacher E, Los B, Stehrer R, Timmer M, de Vries G (2013) The construction of world input-output tables in the WIOD project. Econ Syst Res 25:71–98 Fujita M, Krugman P, Venables AJ (2001) The spatial economy: cities, regions, and international trade. The MIT Press, Cambridge Ghosh A (1958) Input-output approach in an allocation system. Economica 25:58–64 Hummels D, Ishii J, Yi K-M (2001) The nature and growth of vertical specialization in world trade. J Int Econ 54:75–96 Johnson RC, Noguera G (2012) Accounting for intermediates: production sharing and trade in value added. J Int Econ 86:224–236 Knaap T, Oosterhaven J (2011) Measuring the welfare effects of infrastructure: a simple spatial equilibrium evaluation of Dutch railway proposals. Res Transp Econ 31:19–28 Koopman R, Wang Z, Wei S-J (2014) Tracing value-added and double counting in gross exports. Am Econ Rev 104:459–494 Kratena K (2005) Prices and factor demand in an endogenized input-output model. Econ Syst Res 17:47–56 Leontief WW (1951) The structure of the American economy: 1919–1939, 2nd edn. Oxford University Press, New York Leontief WW (1941) The structure of the American economy, 1919–1929: an empirical application of equilibrium analysis. Cambridge University Press, Cambridge Leontief WW (1970) The dynamic inverse. In: Carter AP, Bródy A (eds) Contribution to input-output analysis, vol 1. North-Holland, Amsterdam Madsen B (2008) Regional economic development from a local economic perspective—a general accounting and modelling approach. Habilitation thesis, University of Copenhagen Oosterhaven J (1989) The supply-driven input-output model: a new interpretation but still implausible. J Reg Sci 29:459–465 Oosterhaven J, Bouwmeester MC (2016) A new approach to modelling the impact of disruptive events. J Reg Sci 56:583–595 Shoven JB, Whalley J (1992) Applying general equilibrium. Cambridge University Press, New York Södersten CJH, Lenzen M (2020) A supply-use approach to capital endogenization in input-output analysis. Econ Syst Res 32:451–475 Timmer MP, Los B, Stehrer R, De Vries GJ (2013) Fragmentation, incomes and jobs: an analysis of European competitiveness. Econ Pol 28:613–661 Többen JR, Distelkamp M, Stöver B, Reuschel S, Ahmann L, Lutz C (2022) Global land use impacts of bioeconomy: an econometric input–output approach. Sustainability 14:1976 Venables AJ, Gasiorek M (1998) The welfare implications of transport improvements in the presence of market failure. Reports to SACTRA, Department of Environment, Transport and Regions, London

Correction to: Rethinking Input-Output Analysis

Correction to: J. Oosterhaven, Rethinking Input-Output Analysis, Advances in Spatial Science, https://doi.org/10.1007/978-3-031-05087-9 The original version of this book was published with errors, which have now been corrected as follows: The missing crucial brackets in Eq. 8.9 have been added in Chap. 8; the incorrect presentation of Tables 2.2, 4.2, 4.3, and 9.5 has been corrected in Chaps. 2, 4 and 9, respectively. The book and the chapters have been updated with the changes.

The updated original version of these chapters can be found at https://doi.org/10.1007/978-3-031-05087-9_2 https://doi.org/10.1007/978-3-031-05087-9_4 https://doi.org/10.1007/978-3-031-05087-9_8 https://doi.org/10.1007/978-3-031-05087-9_9 © The Author(s), under exclusive license to Springer Nature Switzerland AG 2022 J. Oosterhaven, Rethinking Input-Output Analysis, Advances in Spatial Science, https://doi.org/10.1007/978-3-031-05087-9_11

C1

Appendix

Matrix Algebra for Input–Output Analysis

A matrix is a square or a rectangular block,   or  honeycomb if you wish, filled with 4 −5 0 a11 a12 a13 , where A is called a 2 numbers, for example A = = a21 a22 a23 1 3 −2 × 3 matrix. The first subscript i of the typical cell of A with scalar ai j indicates the row number of that cell (in the main text, often the origin of the product flow), and the second subscript j indicates its column number (in the main text, often the destination of the product flow). Most matrices as well as operations on matrices are defined in the main text by means of their typical element as ai j ∈ A, where the ∈ indicates that ai j is the typical element of A. A matrix that consists of only one row or only one column is called a vector, for example ai ∈ a. A specific matrix that is often used in the main text is the diagonal matrix that has only zero’s in its off-diagonal cells. A diagonal matrix is, therefore, uniquely defined by the vector that fills up its diagonal. Hence, a diagonal matrix may be and most often is defined by putting a hat on top of that vector, as in  ai= j = 0, aii = ai ∈ aˆ . From its definition, it is clear that a diagonal matrix can only be a square matrix, not a rectangular one. A very special diagonal matrix is the unity matrix I that has the unity vector with only ones, i.e. 1.0 ∈ i, on its diagonal, i.e. I = ˆi.   a11 a12 a13  = The transposition of the above example matrix A equals A = a21 a22 a23 ⎡ ⎤ 4 1 ⎣ −5 3 ⎦. Hence, the transposition of a column ai ∈ a delivers the row ai ∈ a , 0 −2 where the transposition sign ´ is always attached to the row and not to the column. The addition of the matrices B and C is accomplished by adding their corresponding elements, i.e. ai j = bi j + ci j ∈ A = B + C. The subtraction of matrix C from matrix B is defined analogously as ai j = bi j − ci j ∈ A = B − C. From both expressions, it is clear that addition and subtraction of matrices is only possible if

© The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2022 J. Oosterhaven, Rethinking Input-Output Analysis, Advances in Spatial Science, https://doi.org/10.1007/978-3-031-05087-9

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the matrices at hand have the same dimension, i.e. if they have the same number of rows and columns. The cell-by-cell multiplication of matrix B and C is indicated by a ⊗, and is defined as ai j = bi j ci j ∈ A = B ⊗ C. The cell-by-cell division of B by C is indicated by a ÷, and defined analogously as ai j = bi j /ci j ∈ A = B ÷ C. Cell-bycell multiplication and division is sometimes handy, but is seldom used. The same holds for the multiplication of all cells with the same scalar b, i.e. b ∗ ai j = ai j ∗ b ∈ b A = A b. The standard multiplication of a matrix B with a matrix C is one of the more complex operations in matrix algebra and is used to create multiple equation systems. The cells of the resulting matrix A are calculated by multiplying the cells of row i of the first matrix B with the corresponding cells of the column j of the second matrix C and summing the result. This obviously requires that the number of cells in row i of B is equal to the number of cells in column j of C. This is a fine case where matrix algebra

K provides a much more efficient way to define this operation, namely bik ck j ∈ A = B C, where the necessary use of the same running by ai j = k=1 subscript k directly indicates that the number of cells in row i of B needs to be equal to that of column j in C. This also directly shows that the product C B is not even defined when I = J . Moreover, even if defined

when I = J , it will not result in the K H bik ck j = h=1 ci h bh j = ai∗j for two reasons. same outcome, because ai j = k=1 (1) The dimensions of B C and C B may not be the same. (2) Even if they are the same, the outcome will be different. Some specific multiplications are used quite often in the main text. (1)

(2)

(3)

Summation of cells. Post-multiplication of a matrix z i j ∈ Z with the unity z i· =

column i results in a column with the row sums of Z, i.e.  z ∗ 1.0 ∈ z = Z i, while pre-multiplication with the unity row i delivers j j ij

a row with the column sums of Z, i.e. z · j = i 1.0i ∗ z i j ∈ z = i Z. The tran sition sign indicates that z is a row and, therefore, it must contain the column sums of Z. In the main text, for obvious reasons, this operation is never applied to P that indicates a matrix with prices. Vector division. Consider the post-multiplication of Z with the inverse of the   diagonal matrix of vector x, i.e. with 1/x j ∈ xˆ −1 , for which of course holds xˆ −1 xˆ = I = xˆ xˆ −1 . This post-multiplication results in a matrix, say A, in which the columns of Z are divided by the corresponding cells of x, i.e. ai j = z i j /x j ∈ A = Z xˆ −1 . Pre-multiplication of Z by the inverse diagonal matrix xˆ −1 , analogously, results in a matrix, say B, in which the rows of Z are divided by the corresponding cells of x, i.e. bi j = z i j /xi ∈ B = xˆ −1 Z. Matrix division. Post-multiplication or pre-multiplication of a square matrix A with its square inverse matrix A−1 delivers the unity matrix I, i.e. A A−1 = I = A−1 A. This mysterious multiple equation system is the matrix equivalent of the following ordinary algebra a ∗ a −1 = 1.0 = a −1 ∗ a. In the ordinary algebra case, it is necessary that a = 0, as a division by zero is not defined. In the matrix division case each matrix A has a so-called determinant |A|, a single number that should also be unequal to zero, in which case the matrix

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A is called non-singular. A singular matrix with |A| = 0 does not have an inverse. Going into further complex details (see Takayama 1985 or Miller and Blair 2022) is not necessary to follow the main text, except

that the inverse matrix is needed to solve systems of linear equations, such as j ai j x j = bi ∈ A x = b. This is done by means of pre-multiplication with A−1 , as follows from A−1 A x = A−1 b => I x = x = A−1 b. If the system of equations has a transposed form, as in

  i x i bi j = c j ∈ x B = c , it needs a post-multiplication with the inverse of B to get solved, as follows from x B B−1 = c B−1 => x = c B−1 . Finally, it is handy to know that the relationship (A B)−1 = B−1 A−1 holds for non-singular and equally large square matrices A and B. Analogously, (A B) = B A holds for equally large square matrices A and B, which may be singular.

References Miller RE, Blair PD (2022) Input-output analysis, foundations and extensions, 3rd edn. Cambridge University Press, Cambridge Takayama A (1985) Mathematical economics. Macmillan, New York