Capital Theory and Political Economy: Prices, Income Distribution and Stability 2020054002, 2020054003, 9780815375654, 9781032006253, 9781351239424

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Capital Theory and Political Economy

In recent years, there have been a number of new developments in what came to be known as the “Capital Theory Debates”. The debates took place mainly during the 1960s as a result of Piero Sraffa’s critique of the neoclassical theory according to which the prices of factors of production directly depend on their relative scarcities. Sraffa showed that when income distribution changes, there are many complexities developed within the economic system impacting on prices in ways which are not possible to predict. These debates were revisited in the 1980s and again more recently, along with a parallel literature that has developed among neoclassical economists and has also looked at the impact of shocks on an economy. This book summarizes the debates and issues around the theory of capital and brings to the fore the more recent developments. It also pinpoints the similarities and differences between the various approaches and critically evaluates them in light of available empirical evidence. The focus of the book is on the price trajectories induced by changes in income distribution and the resulting shape of the wage rates of profit curves and frontier. These issues are central to areas such as microeconomics, international trade, growth, technological change and macro stability analysis. Each chapter starts with the theoretical issues involved, followed by their formalization and subsequently with their operationalization. More specifically, the variables of the classical theory of value and distribution are rigorously defined and quantified using actual input–output data from a number of major economies, but mainly from the USA, over long stretches of time. The empirical results are not only consistent with the anticipations of the theory but also further inform and therefore strengthen its predictive content raising new significant questions. Lefteris Tsoulfidis holds a Ph.D. and an M.A. in economics from the New School for Social Research in New York, and a B.A. in economics from the University of Macedonia. He is currently a Professor in the Department of Economics at the University of Macedonia in Thessaloniki, where he teaches courses in the history of economic thought, economic history, political economy, mathematical economics and macroeconomics. He is the author of Competing Schools of Economic Thought (2010) and co-author of Modern Classical Economics and Reality. A Spectral Analysis of the Theory of Value and Distribution (2016) and Classical Political Economics and Modern Capitalism: Theories of Value, Competition, Trade and Long Cycles (2019).

8. Peter Beaumont, Israel says EU is emboldening its enemies with labelling plans , The Guardian, 10/11/2015, http://www.theguardian.com/world/2015/nov/10/israel-accuses-euof-emboldening-iraels-enemies-with-labelling-plans (accessed 11/11/2015).

Routledge Frontiers of Political Economy

Foundations of Post-Schumpeterian Economics Innovation, Institutions and Finance Beniamino Callegari Distributive Justice and Taxation Jørgen Pedersen The China–US Trade War and South Asian Economies Edited by Rahul Nath Choudhury Politics and the Theory of Spontaneous Order Piotr Szafruga Preventing the Next Financial Crisis Victor A. Beker Capital Theory and Political Economy Prices, Income Distribution and Stability Lefteris Tsoulfidis Value and Unequal Exchange in International Trade The Geography of Global Capitalist Exploitation Andrea Ricci Inf lation, Unemployment and Capital Malformations Bernard Schmitt Edited and Translated in English by Alvaro Cencini and Xavier Bradley

For more information about this series, please visit: www.routledge.com/books/ series/SE0345

Capital Theory and Political Economy Prices, Income Distribution and Stability

Lefteris Tsoulfidis

First published 2021 by Routledge 2 Park Square, Milton Park, Abingdon, Oxon OX14 4RN and by Routledge 52 Vanderbilt Avenue, New York, NY 10017 Routledge is an imprint of the Taylor & Francis Group, an informa business © 2021 Lefteris Tsoulfidis The right of Lefteris Tsoulfidis to be identified as author of this work has been asserted by him in accordance with sections 77 and 78 of the Copyright, Designs and Patents Act 1988. All rights reserved. No part of this book may be reprinted or reproduced or utilised in any form or by any electronic, mechanical, or other means, now known or hereafter invented, including photocopying and recording, or in any information storage or retrieval system, without permission in writing from the publishers. Trademark notice: Product or corporate names may be trademarks or registered trademarks, and are used only for identification and explanation without intent to infringe. British Library Cataloguing-in-Publication Data A catalogue record for this book is available from the British Library Library of Congress Cataloging-in-Publication Data Names: Tsoulfidis, Lefteris, author. Title: Capital theory and political economy : prices, income distribution and stability / Lefteris Tsoulfidis. Description: 1 Edition. | New York : Routledge, 2021. | Series: Routledge frontiers of political economy | Includes bibliographical references and index. Identifiers: LCCN 2020054002 (print) | LCCN 2020054003 (ebook) Subjects: LCSH: Capital. | Income distribution. | Wages. | Economic development. Classification: LCC HB501.T756 2021 (print) | LCC HB501 (ebook) | DDC 339.2/1—dc23 LC record available at https://lccn.loc.gov/2020054002 LC ebook record available at https://lccn.loc.gov/2020054003 ISBN: 978-0-815-37565-4 (hbk) ISBN: 978-1-032-00625-3 (pbk) ISBN: 978-1-351-23942-4 (ebk) Typeset in Bembo by codeMantra

Contents

List of figures List of tables List of abbreviations Preface and acknowledgments

ix xiii xv xvii

1 Preliminaries 1.1 Introduction 1 1.2  Structure of the book 4

1

2 Theory of capital in historical perspective 2.1 Introduction 6 2.2  Capital in theories of value and distribution 6 2.3  Capital in the broad classical approach 9 2.4  Capital in the neoclassical theory 17 2.5  Production with capital and the early neoclassical economists 21 2.6  Summary and conclusions 29

6

3 Capital theory controversies 3.1 Introduction 31 3.2  Cambridge capital theory controversies 31 3.3  Production with produced means of production 37 3.4  Samuelson’s one-commodity world 39 3.5  From the one-commodity to the world of many commodities 46 3.6  Reswitching of techniques and the demand for capital 50 3.7  Summary and conclusions 53 Appendix 3.A: Samuelson’s and Pasinetti’s reswitching examples 53

31

4 Price trajectories and the rate of profit 4.1 Introduction 61 4.2  Prices of production and their determinants 63

61

vi

Contents 4.2.1 The mathematics of the linear model of production 65

4.3 Price paths using input–output data, BEA 2018 70 4.3.1 The fixed capital model, BEA 2018 70 4.3.2 The circulating capital model, BEA 2018 74 4.4 Price paths using input–output data, USA 2014 78 4.4.1 The fixed capital model, USA 2014, WIOD (2016) 79 4.4.2 The circulating capital model, USA 2014 84 4.5 Summary and conclusions 88 Appendix 4.A: Sources of data and estimating methods 90 93

5

Wage rate of profit curves and technological change 5.1 Introduction 93 5.2 Survey of the first empirical studies 94 5.3 Technological change and wage rate of profit curves 96 5.4 The choice of numéraires and the standard commodity 104 5.5 WRP curves in 49 industries, USA 2007 and 2014 108 5.5.1 WRP curves, circulating capital model 109 5.5.2 WRP curves, fixed capital model 114 5.6 Discussion of empirical findings 119 5.6.1 Some recent developments 123 5.7 Summary and conclusions 127 Appendix 5.A: Further testing of the randomness hypothesis 129

6

Distribution of eigenvalues and the shape of price and wage rates of profit curves 132 6.1 Introduction 132 6.2 Bródy’s conjecture 133 6.2.1 Evolution of spectral ratio and the size of matrices 135 6.3 Eigendecomposition, price trajectories and dimensionality reduction 140 6.4 Eigendecomposition and the circulating capital model 144 6.4.1 Second-order approximation of PRP trajectories 147 6.4.2 The distribution of eigenvalues and the near linearities in PRP trajectories 150 6.5 Factorization and the construction of a hyper-basic sector 159 6.5.1 The Application of the SVD factorization method 162 6.5.2 The fixed capital model and the hyper-basic industry 163 6.6 Summary and conclusions 166 Appendix 6.A: Digression in the effective rank 168

7

A simple but realistic linear model of production 171 7.1 Introduction 171 7.2 Input–output data and the estimation of relative equilibrium prices 172

Contents vii

7.3 A Realistic numerical example and the estimation of direct prices 173 7.3.1 Direct prices, prices of production and market prices 175 7.3.2 The Sraffian standard commodity 176 7.4 Prices of production in circulating and fixed capital models I and II 177 7.4.1 Prices of production in a circulating capital model 177 7.4.2 Prices of production in a fixed capital model I 179 7.4.3 Prices of production in a fixed capital model II 180 7.5 Prices of production/direct prices trajectories 182 7.5.1 PP/DP trajectories, circulating capital model 183 7.5.2 PP/DP trajectories, fixed capital model I 184 7.5.3 PP/DP trajectories, fixed capital model II 185 7.6 Wage rate of profit curves 187 7.7 The eigendecomposition and the approximations of price trajectories 190 7.8 Hyper-basic industry 195 7.9 Summary and conclusions 198 8

Summing-up 8.1 Introduction 200 8.2 Capital theory debates in retrospect 201 8.3 Capital theory debates in prospect 205 8.4 Concluding remarks 210

200

References Index

213 221

Figures

2.1 Labor commanded theory of value 11 3.1 Isocost and isoquant curves 38 3.2 WRP curves and their WRP frontier 41 3.3 Derivation of the WRP frontier 42 3.4 Isocost-isoquant and the WRP curve 43 3.5 Different techniques and the demand schedule for capital 44 3.6 Derivation of isoquants through WRP curves 45 3.7 WRP curves of different shapes 48 3.8 A concave WRP curve with two goods 49 3.9 WRP curves with many goods 50 3.10 Switching points and the demand schedule for capital 51 3.A1 Samuelson 1966: Relative price w.r.t. the interest rate 55 3.A2 Pasinetti 1966: Relative price w.r.t. the rate of profit 57 3.A3 Pasinetti 1966: WRP curves 59 4.1 Typical trajectories of prices and relative rate of profit 64 4.2 Price trajectories, fixed capital model, BEA 2018 73 4.3 Capital–output ratios and the standard ratio, fixed capital model, BEA 2018 75 4.4 Price trajectories, circulating capital model, USA 2018 76 4.5 Capital–output ratios and the standard ratio, circulating capital model, BEA 2018 77 4.6 Price trajectories, fixed capital model, USA 2014 82 4.7 Capital intensities, fixed capital model, USA 2014 83 4.8 Monotonic PP trajectories, circulating capital model, USA 2014 85 4.9 Capital intensities, circulating capital model, USA 2014 86 4.10 PP and capital intensities, non-monotonic trajectories, circulating capital model, USA 2014 87 5.1 WRP curve, fixed capital model, Greece 1970 98 5.2 WRP curves, circulating and fixed capital models, S. Korea 1995 and 2000 99 5.3 WRP curves, circulating and fixed capital models, UK 1990 100 5.4 WRP curves, circulating and fixed capital models, USA 2018 101

x Figures

5.5 WRP curves, fixed capital model, USA 2000, 2005, 2007, 2010 and 2014 102 5.6 WRP curves, circulating capital model, USA 2000, 2005, 2007, 2010 and 2014 102 5.7 Standard vs. actual output, fixed and circulating capital models, USA 2007 and 2014 106 5.8 Employment vector vs. l.h.s. eigenvector of matrix A, USA 2007 and 2014 107 5.9 WRP curves, circulating capital, USA 2007 and 2014 110 5.10 Industries displaying switching, circulating capital model, USA 2007 and 2014 112 5.11 Differences in WRP curves with switching, circulating capital model, USA 2007 and 2014 114 5.12 WRP curves infixed capital model, USA 2007 and 2014 115 5.13 Industries displaying switching, fixed capital model, USA 2007 and 2014 117 5.14 Differences in WRP curves with switching, fixed capital model, USA 2007 and 2014 118 5.A1 OLS regressions between m and v, 2007 and 2014 130 6.1 Eigenvalue distribution, 15, 34, 71 and 490 industry detail, USA 1997 137 6.2 Eigenvalue distribution, 15, 54, 71 and 426 industry detail, USA 2002 137 6.3 Eigenvalue distribution, 15, 54, 71 and 405 industry detail, USA 2007 138 6.4 Eigenvalue distribution, 15, 54, 71 and 405 industry detail, USA 2012 138 6.5 Singular values and their sum of squares, USA 2018 143 6.6 Linear approximations of price trajectories, sample of industries, USA 2018 145 6.7 Economy-wide MAD resulting from the first order approximation 148 6.8 Linear and quadratic approximations, USA 2018 149 6.9 MAD% of approximations from the actual price trajectories, USA 2018 150 6.10 Eigenspectrum, large dimensions input–output tables, USA 1997, 2002, 2007 and 2012 152 6.11 The evolution of the moduli of the eigenratios, USA 1997–2018, 71 industries 155 6.12 Distributional patterns of absolute normalized eigenvalues, circulating capital: AUS, CAN, PRC, FRC, GRC and JPN, 2000 and 2014 158 6.13 Hyper-basic industries, first rows of matrix, SH vs. the approximation SH1 , Schur factorization method 162

Figures  xi

6.14 Hyper-basic industries, first rows of matrix, SH k vs. approximated SHk1 165 7.1 Price trajectories and the relative rate of profit, circulating capital model, USA 2014 183 7.2 Capital intensities and the relative rate of profit, circulating capital model, USA 2014 184 7.3 Price trajectories and the relative rate of profit, fixed capital model I, USA 2014 185 7.4 Capital intensities and relative rate of profit, fixed capital model I, USA 2014 185 7.5 Price trajectories and the relative rate of profit, fixed capital model II, USA 2014 186 7.6 Capital intensities and relative rate of profit, fixed capital II 186 7.7 WRP curves of circulating and fixed capital models, USA 2014 188 7.8 WRP curves in circulating and fixed capital models I and II, USA 2014 189 7.9 Price trajectories and their linear approximation, circulating capital model, USA 2014 191 7.10 Deviation of actual and linear approximation price trajectories, USA 2014 192 7.11 Price trajectories and their linear, quadratic and cubic approximations 193 7.12 MADs of linear, quadratic and cubic approximations of price trajectories, USA 2014 194 7.13 First row of matrix SH vs. approximation SH1 197

Tables

3.A1 Samuelson 1966: Relative price w.r.t. the interest rate 55 3.A2 Pasinetti 1966: Relative price w.r.t. the rate of profit 57 3.A3 Pasinetti 1966: WRP curves 58 4.1 Direct prices, prices of production and capital intensities, BEA 2018 71 4.2 Direct prices, prices of production and capital intensities, USA 2014 80 5.1 Maximum real wage and rate of profit in circulating and fixed capital models 120 6.1 Eigenvalue distribution of matrices A and H, 15 sectors, USA 1997 136 6.2 Eigenratios of various size matrices HR, USA five benchmark years, circulating capital 139 6.3 Distribution of absolute normalized eigenvalues, circulating capital, various countries, 2000 and 2014 156 6.4 Column vector u1 of matrix H1, Schur factorization method 160 6.5 Row vector of matrix SH1 , Schur factorization method 161 6.6 First row of matrix SH, Schur factorization method 161 6.7 Column vector u1 of matrix H1, SVD factorization method 162 6.8 Column sums of matrix eH1 163 6.9 Column sums of matrix eHR 163 6.10 Column vector u1 of matrix H k1, Schur method 164 6.11 Row vector of matrix Sk1, Schur method 164 6.12 Column vector u1 of matrix H k, SVD method 164 6.13 First row of matrix SH k , SVD method 165 6.14 First row of matrix SH k , Schur method 165 7.1 Input–output table 172 7.2 Aggregated into five industries input–output table, USA 2014 174 7.3 The matrix SH1 , Schur method 196 7.4 The first row of the matrix SH 197 7.5 The matrix SH1 , SVD method 198 7.6 The first row of the matrix SH, SVD method 198

Abbreviations

APP BEA CCC DP l.h.s. MAD MAWD MP PP PRP r.h.s. SEA SF SVD w.r.t. WIOD WRP

Average Period of Production Bureau of Economic Analysis Cambridge Capital Controversies Direct Prices Left Hand Side Mean Absolute Deviation Mean Absolute Weighted Deviation Market Prices Prices of Production Price Rate of Profit Right Hand Side SocioEconomic Accounts Spectral Flatness Singular Value Decomposition With Respect To World Input–Output Database Wage Rate of Profit

Preface and acknowledgments

I met with Andy Humphries of Routledge during the Eastern Economic Association Conference in New York (2017). He proposed the idea of summarizing the various new developments within the classical political economy approach that were underway with a focus on capital theory. In fact, in the past decade, there have been many articles or books written within the classical tradition of economic theory, and similar but parallel developments started taking place in the neoclassical side without actual interaction and debate between the two sides. I was glad to accept, especially, because I would have team worked with Theodore Mariolis with whom I had had a long and very productive collaboration since 2005. Unfortunately, though, the collaboration could not bear fruit this time for reasons that superseded us. This is, perhaps, the reason why the completion of the book comes with some, I feel, reasonable delay. Clearly, our joint theoretical and empirical works, in one way or another, permeate the whole book. Meanwhile, there have been many other relevant papers published in major journals by Scott Carter, Harald Hagemann, Heinz Kurz, Bertram Schefold, Anwar Shaikh, Robert Solow, Dimitris Paitaridis and Torres-Gonzales. With all of them, at some point, I happened to exchange views and received helpful comments. I thank them all. Above all, I wholeheartedly thank my partner in life and academia Persefoni Tsaliki, who read the whole manuscript, helped in the restructuring of the text and suggested significant improvements; without her invaluable help, this book would not have been completed. Of course, any problems with the book are the author’s responsibility. Lefteris Tsoulfidis Thessaloniki, October 2020

1

Preliminaries

1.1 Introduction In recent years, there have been a number of developments in what came to be known as “Cambridge Capital Controversies” (CCC). The debates sparked by Robinson’s (1953–1954) important article inspired, as she admitted, mainly by Sraffa’s (1951) introduction to Ricardo’s Principles (Works I), in which she laid bare the inconsistencies in the neoclassical theorization of capital as a “factor of production”. The controversy, however, essentially began after the publication of Sraffa’s (1960) book in which, besides the development of his own version of the classical theory of value and distribution, leveled a critique on the dominant neoclassical theory by raising questions about the conceptualization of capital and its measurement units. Sraffa’s major positive contribution was that he fundamentally changed the way of thinking of the notion of capital intensity and the movement in prices in the face of changes in income distribution. He argued that variations in income distribution elicit intricate price movements to all different directions, which become extremely difficult, if not impossible, to predict. Furthermore, Sraffa (1960) argued that the intricate movement of prices gives rise to wage rate profit (WRP) curves of various curvatures and not straight or monotonic lines as anticipated by the neoclassical theory. In so doing, Sraffa (1960) challenged the neoclassical principle that the prices of factors of production depend directly on their relative scarcities. In the neoclassical perspective, straight-line WRP curves, as we will show in Chapter 3, are absolutely necessary for the derivation of well-behaved demand schedules for both capital and labor. The demand schedule for capital is hypothesized to be inversely related to interest rate while that for labor is inversely related to real wage. Based on these well-behaved demand schedules for capital and labor, the neoclassical economics derives a host of other schedules, which in their turn are indispensable in dealing with macroeconomic questions. Sraffa and economists inspired by him opined that the price trajectories are non-linear and the associated with these WRP curves, or in neoclassical economics, the price-factor frontier are also non-linear, thereby rendering unlike the construction of a well-behaved demand schedule for

2

Preliminaries

capital. The non-linearity in WRP curves indicates that substitution between capital and labor takes place in a way different from what the neoclassical principle of scarcity prices dictates. Samuelson (1962) assumed the defense of the fundamental propositions of the neoclassical theory in the context of a one-commodity world through his “surrogate production function” and claimed that his conclusions apply to economies with many commodities. The assumption of a one-commodity world is no different from the assumption of a uniform capital–labor ratio across industries, and essentially from an economy without prices! Samuelson’s assumption was criticized for its lack of realism by Piereangelo Garegnani (1930–2012), initially as a participant in a seminar at MIT, where Samuelson presented for the first time his “surrogate production function”, and later in an article (1962). Garegnani (1970, 1976, 1990) criticized Samuelson’s efforts to rescue the neoclassical theory of value and distribution. Similarly, Pasinetti (1966) indicated that once we hypothesize different capital intensities across industries, the neoclassical principles no longer hold. The idea is that as relative prices change consequent upon changes in income distribution, the resulting revaluation of capital can go either way, and it is possible for an industry to start as capital-intensive in one income distribution and to become labor-intensive in another (see Pasinetti 1977, ch. 5). Consequently, we no longer derive Samuelson’s straight-line WRP curves (see Chapter 3), which are consistent with the cost-minimizing choice of technique and give rise to well-behaved demand schedules for capital and other related schedules. In the presence of many capital goods and various capital-intensities across industries, the WRP curves become non-linear and may cross over each other more than once. This is equivalent to saying that a capital-intensive technique may be selected, for both low and high rates of profit, a result that runs contrary to the neoclassical theory of value and income distribution, whose quintessence is the notion of scarcity prices. Samuelson gave an intellectual struggle and, a few years later (Samuelson 1966), admitted the limitations of neoclassical theory, without pursuing further the implications of the findings. In other words, he conceded that in the more realistic case of the multi-commodity world, we could not rule out the issue of the reswitching of techniques; that is, the same technique might be the most profitable at several rates of profit (or real wages). Under these circumstances, there is no way to construct a consistent demand schedule for capital and the same is true for labor; as a result, it is not possible to determine long-run equilibrium prices and the associated with these equilibrium rate of interest (profit) or wage rate, as the rewards for the marginal productivities of capital and labor, respectively. Therefore, it becomes meaningless to discuss short-run deviations from the long-run equilibrium because such a long-run equilibrium does not exist in the first place. Sraffa’s critique also casts doubt on the old classical economists’ view that prices change with income redistribution, according to their capital intensity vis-a-vis the economy’s average. In fact, Sraffa argued that the concept

Preliminaries

3

of capital intensity is frivolous because of the intricate price complexities developed from changes in income distribution. In other words, the capital intensity of an industry may be found higher than the economy-wide average in one income distribution, and in another, its ranking may be the other way around. It is important to stress that these conclusions were derived only at the theoretical level ranging from patently unrealistic numerical examples, starting from Samuelson’s (1962) parable of a one-commodity world, to a seemingly more realistic world of two commodities produced by different techniques as discussed by Samuelson (1966) and Pasinetti (1966), confirming reswitching scenarios. It is important to stress that in the utilized numerical examples no attention was paid to what extent, if any, they were representative of the operation of actual economies. The occurrence of reswitching along with the other problems related to the measurement of capital led, from the 1970s onward, to the development of an inf luential variant of the neoclassical theory known as the intertemporal equilibrium approach. The latter surpasses the problem of the theoretical inconsistencies in the measurement of capital goods, however, at the expense of spiriting away the long-run character of the analysis associated with the equalization of the rate of profit and, therefore, the determination of equilibrium prices (Garegnani 1970; Eatwell 1990, 2019; Petri 2017, 2020; Fratini 2019). While this has been the state of knowledge up until recently, lately, we have witnessed many interesting developments in this crucial for the logical consistency of economic theories area. More specifically, already from the 1980s, there have been many studies showing that the price rate of profit (PRP) curves display monotonic trajectories in the face of changes in income distribution and the resulting WRP curves are nearly straight lines, thereby ruling out, in most cases, the reswitching scenarios of the Sraffian economists. These empirical findings are certainly less damaging for neoclassical economics, but by no means negate the remaining inconsistencies in its theory of value and distribution, which must be addressed and resolved. By contrast, in the classical approach, such findings become thought-provoking questions and enrich the approach adding additional qualifications and calling for further research efforts on the theoretical front but by no means cast doubt on the very core tenets of the classical theory of value and distribution. It should be stressed though that these explorations became possible thanks to Sraffa’s seminal contributions and especially with the aid of his conception of standard commodity, that is, the output of an industry that remains invariant to changes in income distribution (Sraffa 1960, ch. 5). The latter became instrumental in making meaningful comparisons and derive statements of general validity about the movement of prices and the related to these capital intensities of industries. Recent studies have shown that technologies described by the input–output structure of the economies share similar properties persisting over time; more specifically, the particular distribution of eigenvalues is what gives rise to the observed quasi-linearities. In effect, the research has shown that in many

4 Preliminaries

countries and over the years, the distribution of eigenvalues follows an exponential pattern. Figuratively speaking, the eigenvalues form an “elbow” taking place at eigenvalues much lower than the dominant; that is, the second eigenvalue is by far lower than the dominant (or Perron-Frobenius) one. Our theoretical discussion and empirical findings lend support to the view that the ratio of the second to the dominant eigenvalue, the so-called “spectral ratio”, is chief ly responsible for the observed quasi-linearities of relative price trajectories and of WRP curves and resulting frontiers.

1.2 Structure of the book Chapter 2 of the book begins with the notion of capital as it has been conceptualized by the old classical economists (Smith, Ricardo and J.S. Mill) and Marx. The broad classical economists’ notion of capital is contrasted with that of the first neoclassical economists ( Jevons, Menger, Walras, Bohm-Bawerk, J. B. Clark and Wicksell). The two alternative views moved parallel to each other and, certainly, we cannot say that there has been any debate, since their differences were obvious and irreconcilable. Along the way, the reasons why the first neoclassical economists were dissatisfied with the classical definition of capital and thought of the need to theorize it in a way consistent with the requirements of their theory are explained. In so doing, we point out the differences in these two competing approaches and the challenges that they face in the presence of capital and changes in income distribution. Chapter 3 deals with the capital theory controversies of the 1960s initiated by Robinson’s (1952–1953) article and Sraffa’s (1960) writings and teachings, whose very purpose was the critique of the economic orthodoxy of the time and the revelation of its inconsistencies in the measurement of capital. The chapter continues with Samuelson’s (1962) valiant efforts to provide satisfactory answers to the issues raised by Sraffa and his followers regarding the neoclassical theory of value and distribution and the treatment of capital goods. Chapter 4 deals with the issues raised by Cambridge economists on both sides of the Atlantic. In fact, the empirical research, mainly conducted for the US economy, has shown that the actual PRP curves are not far from linearity and the case of reswitching of techniques, that so much ink was spilled on, is in effect derived from numerical examples that are not representative of the actual economies. The subsequent research for a number of diverse economies over time strengthened the view of the near linearities in price trajectories. Chapter 5 is about the shape of the WRP curves in actual economies and for this reason presents estimates of such curves for the US economy using input–output data of various years and industry detail. In this chapter, we make an effort to fill the gap between the theoretical research and the compelling need for relevant empirical support of the actual shape of the WRP curves. The WRP curves from the USA and many economies show that the case of reswitching cannot be excluded, but at the same time by no means

Preliminaries

5

is the general case. The WRP curves are characterized by near linearity, a result that does not vindicate the neoclassical theory of value for reasons that we explain and does not dump the classical one as one may hasten to point out. On the contrary, these findings give credence to the explanatory content of the classical theory of value and distribution, and open new directions in grappling with issues relating to technological change. The near linearities in both PRP trajectories and WRP curves, which by now have already become a stylized fact, compel an explanation of this lawlike regularity. Chapter 6 deals with the empirically repeatedly established near linearities in PRP paths and WRP curves through the spectral or eigendecomposition of the economic system matrices and their characteristic skew distribution of both eigenvalues and singular values that explain the low effective rank of the system matrices. Furthermore, we discuss the nature of technology as ref lected in the technological coefficients matrices and, particularly, in their vertical integration form, which indicates the presence of quasi-linear dependence of the columns of the matrices. The theoretical and empirical research presented in the previous chapters suggests the use of both the Schur triangularization theorem and the singular value decomposition method for the construction of a hyper-basic industry from the input–output structure of the entire economy. Such a hyper-basic industry, in combination with hyper-non-basic industries, as derived from a similarity transformation ref lects the intrinsic properties characterizing the behavior of the economic system. Chapter 7 reviews the estimation methods of Chapters 4–6 using an aggregated input–output table of five sectors of the US economy and the year 2014. In so doing, the interested reader can keep better track of the utilized estimating methods and techniques, and have a better understanding of the possible meaning of the characteristic distribution of eigenvalues and triangularization methods, by inspecting this simple but insightful representation of the economy. Chapter 8 summarizes the findings and concludes with some general remarks about the current stage of the CCC and its future prospects and directions.

2

Theory of capital in historical perspective

2.1 Introduction This chapter is about the evolution of the notion of capital in the history of economic thought starting with the works of classical economists and Marx and continuing with the ideas of the first neoclassical economists and in particular their Austrian variant. Furthermore, we present and critically evaluate the different approaches on capital as a produced means of production, a concept that has haunted economic theory ever since its inception as a scientific inquiry. The controversies between leading economists on the role of capital not only have not settled the issue but also have raised new questions making the capital theory and its intricacies even more interesting and challenging. This chapter presents and critically evaluates alternative perspectives on the concept of capital. The rest of the chapter is organized as follows: Section 2.2 deals with the importance of capital in economic theory and the problems for its definition in a theoretically consistent way. Section 2.3 introduces the fundamental ideas of the old classical economists (Smith, Ricardo, J.S. Mill) and Marx. Section 2.4 is about the efforts of the first neoclassical economists to introduce capital in a way, which is consistent with their theory of value. Section 2.5 presents the ideas of the pioneer neoclassical economists, Jevons, Menger, Bohm-Bawerk and Wicksell, among others. Finally, Section 2.6 summarizes and concludes.

2.2 Capital in theories of value and distribution In recent decades, there have been several developments in what came to be known as “Capital Theory Debates” or Cambridge Capital Controversies (CCC) between Cambridge University, UK and MIT Cambridge MA, USA. These developments answer, if not completely, nevertheless, to a greater extent the issues regarding the theory of capital that for many years pitted the sharpest minds of Cambridge, Massachusetts, against the brightest theoretical lights of Cambridge, England. (Leontief 1986, p. 410)

Capital in historical perspective

7

We may add, not accidentally, but precisely because the notion of capital is of paramount importance for it is intrinsically connected to the economic theory of value and distribution. Thus, the notion of capital permeates economic theory starting with the Physiocrats going to the old classical economists (Smith and Ricardo in the main) and Marx continuing with the major neoclassical economists of the late nineteenth century and the modern ones of our times. Capital goods actually form a vector, whose elements are the various produced means of production employed in the production process; these goods by their very nature are heterogeneous, and their economically meaningful addition requires their homogeneity, that is, the capital goods must have an economically meaningful common property, which will enable their representation by a scalar. In other words, the aggregation of capital goods requires a common economically meaningful substance characterizing all capital goods and this substance must be amenable to optimization (cost minimization). Classical economists argued that for the evaluation of capital goods one needs an estimate of their worthiness ref lected in either labor values or natural (equilibrium) prices that is price incorporating the economy-wide average rate of profit and between the two, of course, natural or prices of production (PP) are preferred. Market prices (MP), ref lecting all the “noise” in the market and as such are disequilibrium prices, could be used for the evaluation of capital goods. Businesspeople have no other option but to evaluate in terms of MP both their capital goods and wealth in general and make intertemporal comparisons and decisions based on anticipated future profits or losses. However, the trouble with MP is their ephemeral and ever-changing character, and so they are not amenable to abstract theorization. Furthermore, to estimate the value of capital based on the discounted expected future profits requires the prior knowledge of interest rate; consequently, the analysis is entrapped in a vicious cycle. The MP can only become the starting point of economic theory as a scientific discipline, whose purpose is to explain observed phenomena and their transitory character by penetrating underneath them toward the more stable and permanent determinants. Consequently, businesspeople through MP may have a good first and rough approximation of the worth of their capital and a helpful decision-making guide; however, the economic theory must go beyond the ephemeral in its way to the more permanent characteristics that regulate economic life. Therefore, MP remain in a state of f lux rendering them inappropriate for the required evaluating tasks in economic theory regardless of its standpoint, classical or neoclassical. The question, therefore, is not whether one can measure the worthiness of capital goods, because, one way or another, businesspeople do evaluate their capital goods; for example, in terms of MP. However, it would be a complete failure of the economic theory, if it could not somehow consistently quantify the very substance of the capitalist system, that is, the presence of capital goods and their utilization for profit-making purposes. Piero Sraffa,

8 Capital in historical perspective

already in 1958 on the Corfu Conference, commenting on Hicks’s (1958) paper pointed out that there are […] two types of measurement. First, there was the one in which the statisticians were mainly interested. Second there was measurement in theory. The statisticians’ measures were only approximate and provided a suitable field for work in solving index number problems. The theoretical measures required absolute precision. Any imperfections in these theoretical measures were not merely upsetting, but knocked down the whole theoretical basis. One could measure capital in pounds or dollars and introduce this into a production function. The definition in this case must be absolutely water-tight, for with a given quantity of capital one had a certain rate of interest […]. The work of J. B. Clark, Böhm-Bawerk and others was intended to produce pure definitions of capital, as required by their theories, not as a guide to actual measurement. If we found contradictions, then these pointed to defects in the theory, and an inability to define measures of capital accurately. It was on this—the chief failing of capital theory—that we should concentrate rather than on problems of measurement. (Piero Sraffa, Interventions in the debate at the Corfu Conference on the Lutz and Hague 1961, 305–306, emphasis added) Joan Robinson, some years later, argued along the same lines: The real dispute is not about the measurement of capital but about the meaning of capital. (Robinson 1975, vi, emphasis in the original) The trouble with capital goods has always been their evaluation not in any prices but in equilibrium prices, that is, prices incorporating the economy-wide rate of profit that act as centers of gravitation for the ever-f luctuating MP. The derivation of long-run (or equilibrium) prices becomes the object of economic theory; classical and neoclassical theories are competitive to each other precisely because they have the same object of analysis, that is, the long-run (equilibrium) prices. If they had a different object of analysis, then there would not be any reason for their contrast and comparison of their explanatory content. However, as we will argue, it is one thing to state what is the object of analysis and quite another to deliver logically consistent generalizations about it. The difficulties in the development of a consistent theory of capital are what make neoclassical economics keep redefining capital by expanding its definition to include a plethora of characterizations of labor- and other nonlabor-based concepts as “capital”. In these redefinitions, having demoted labor from what in fact is, that is, the creator of the new value-added, give to capital names such as “human capital” and “social capital” among a long list of different kinds, further obscuring the issue at hand. These new “capitals”

Capital in historical perspective

9

do not stand critique, and they are employed to justify phenomena or findings that are inconsistent with the neoclassical theory. For example, the concept of human capital might be meaningful in slave-type societies, where the slaves, along with their stock of knowledge and skills, that is, their human capital, are part of one’s property and can be sold in the market for slaves. In fact, human capital is utilized in the intent to spread confusion and downplay the creative powers of labor for the production of new value-added. From a multitude of capitals, popular enough is the case of social capital, which in our view is an even worse conceptualization than that of human capital, primarily because everyone, and in a sense, no one owns this kind of “capital”, let alone its consistent measurement. The last two examples show that when it comes to the notion of capital, neoclassical economics has taken a blind route and it would be better to return to Smith’s original conceptualization, where our attention turns next.

2.3 Capital in the broad classical approach The notion of capital and its treatment in the theory of value and distribution has always been a challenging and still remains an open issue since the time of the Physiocrats. To trace out this concept historically, we start with the father of political economy Adam Smith (1723–1790), who by “capital goods” denotes both, physical capital and social relations, which are combined to produce output within a specific economic system. Smith (1776) notes, The annual produce of the land and labor of any nation can be increased in its value by no other means, but by increasing either the number of its productive laborers, or the productive powers of those laborers who had before been employed. The number of productive laborers, it is evident, can never be much increased, but in consequence of an increase of capital, or of the funds destined for maintaining them. The productive powers of the same number of laborers cannot be increased, but in consequence either of some addition and improvement to those machines and instruments which facilitate and abridge labor; or of a more proper division and distribution of employment. In either case the additional capital is almost always required. It is by means of an additional capital only that the undertaker of any work can either provide his workmen with better machinery, or make a more proper distribution of employment among them. (Wealth, p. 114, italics added) that part of man’s stock [=capital] which he expects to afford him revenue. (Wealth, p. 363) It is interesting to note that Smith’s usage of the word “capital” (or stock) is not limited to physical objects and does not apply to all societies. Smith is

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categorical when he notices that there is no capital in societies that are not characterized by a systematic division of labor (Wealth, p. 267). Systematic division of labor is only possible with additional capital, and that requires strong profit motivation existing regularly only in modern society, that is, in the emerging out of feudalism, capitalism. This implies that Smith conceives capital not as a mere object but also as a social relation that gives rise to profits and emerges exclusively in capitalism; however, he did not fully theorize capital as a social relation of production, an aspect of capital that Marx expanded later. Smith’s conceptualization of capital arises from his need to develop a consistent theory of value and distribution. Whatever weaknesses in the theorization of capital were derived precisely from his lack of a consistent theory of value. As is well known, he initially adopted a theory of value (or relative prices) of commodities dependent on the relative labor times spent on their production. The real value of a good for the person who wants to buy it is the torture and exertion of that good […]. The thing that is bought with money or good is actually bought by the labor […]. Labour is the money paid as the first and real purchase cost for everything; labor is the only unit of measure that the value of each kind of good can be compared; labor is the source of all wealth in the world. (Wealth, 32–39) This came to be known as labor theory of value or a more appropriate characterization would have been labor theory of relative prices according to which the relative prices of commodities are determined by their relative labor times spent on their production. Smith realized that this theory, when applied in modern society characterized by the presence of capital goods and rate of profit, runs into insuperable difficulties because the relative prices are no longer equal to relative labor times. Clearly, Smith had had a very good grasp of the concept of capital; nevertheless, he thought that its presence invalidates his labor theory of value and for this reason, he reverted to the labor commanded theory of value, that is, the quantity of labor time, l, that a commodity can purchase. That is, the ratio of the price of a commodity i, pi, over the uniform wage rate, w, (unit of labor), pi /w, tells us how much of labor can commodity i buy.1 At first sight, this seemingly different theory, when applied in the presence of capital, runs through the same “drawbacks” of Smith’s labor theory of value. On closer examination, however, we notice that the labor commanded theory of value might be of great help in dealing with issues related to the presence of capital. More specifically, the relative prices in the presence of capital, on the one hand, may not change very much for reasonable variations in the profit rate and, on the other hand, may give solid estimations of the absolute prices because the money wage serves as a reliable numéraire.

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Figuratively speaking, the deviations of relative prices from the respective relative labor times are controlled, to a great extent, for reasonable rates of profit (see Figure 2.1). The idea is that all relative prices are expected to deviate from their relative labor times, as the rate of profit takes on values varying from zero to its theoretical maximum. More specifically, all prices will be rising, albeit not necessarily proportionally, due to a rising rate of profit. The deviations in prices are controlled for rates of profit not very far away from the economy’s natural rate of profit, r *. The relative prices are not exactly equal to the labor times, but one does not expect to find on average and for reasonable rates of profit significant deviations. Hence, from these rather small, expected and to a great extent anticipated deviations, it would be a mistake to arrive at the conclusion that the labor theory of relative prices must be abandoned. On the contrary, the movement in relative prices depends, largely, on relative labor times and variations in the rate of profit change the price level and slightly, on average, the relative prices. The above proximity of relative price to relative labor time hold to the extent that the rate of profit is reasonably high. From the above discussion, it follows that neither Smith nor most of his commentators had noticed that starting from zero rates of profit until the theoretically reasonable or maximum, the deviations of each commodity’s absolute price from its respective labor time required for its production increase. However, this is true for all commodity prices; thus, their relative deviations may not be so high as one, at first sight, would have expected. Consequently, the likelihood of deviations to be large enough to invalidate Smith’s labor theory of value (or relative prices) becomes, on average, slim. This issue from a point onward is also an empirical one, which, of course, could not be pursued in the times of classical economists, not only due to the lack of appropriate data but in addition to a host of other theoretical and mathematical issues that had to be addressed and solved. Smith, in his quest for a fully satisfactory theory of value and facing difficulties with both the labor and commanded theories of value, finally settled to the third theory of value according to which the natural (equilibrium) p w

pi w li pj w lj r

r*

Figure 2.1 Labor commanded theory of value.

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price is equal to the sum of the three natural incomes, namely, wages, profits and rents. This so-called, according to Dobb (1973), “adding-up theory of value” suffers from two critical issues of consistency. First, the circularity issue is that the determination of natural incomes requires the prior knowledge of natural prices and vice versa. Second, the logical conclusions regarding the price level depending on changes in the constituent components of natural prices. Smith, in so doing, violated the inverse relationship between wages and profits, which came to be known as the “fundamental principle of distribution” of the broad classical approach. Thus, Smith, despite the advanced (for his time) theory of capital, did not manage to overcome the difficulties and to offer an equally advanced theory of value. A task that was taken up by Ricardo, whose view on capital we discuss below. David Ricardo (1772–1823), in his first efforts to evaluate capital and determine the rate of profit, utilized one commodity, corn, that could be used as both input (wages and capital) and output; thus, the difference between output and input over the input of corn would give the rate of profit in terms of corn. The subsequent competition and the equalization of profit rates across sectors in the economy would give rise to the economy-wide rate of profit. This is the famous Ricardo’s corn model, which has been discussed in the literature (for details, see Eatwell 1975). Soon, however, Ricardo realized the limitations of the corn model and turned to the labor time as the common substance of capital goods, which enables their aggregation. This becomes clear in his numerical examples, where machines are produced by labor and then used along with labor in the production of other goods. For example, Chapter 31 of Ricardo’s (1821) Principles (Works I) explains how the construction of a machine by labor ends up replacing labor, thereby increasing the unemployment rate. Besides, capital goods depreciate and this is something Ricardo acknowledges although, for simplicity reasons, he does not treat depreciation explicitly. Ricardo further advanced the labor theory of value by dealing effectively with the exact same issues that made Smith abandon it. Notwithstanding, when it comes to the question of what is capital, Ricardo’s view is not on par with Smith’s as he limits the definition of capital to a mix of objects without necessarily referring to hidden social relations. Thus, according to Ricardo, capital is that part of the wealth of a country which is employed in production, and consists of food, clothing, raw materials, machinery, &c. necessary to give effect to labor. (Works I, p. 95) In other parts of Works I (ch. 33), Ricardo describes capital as “accumulated past labor”, which, during the production process, transfers this stored up labor to the value of commodities. Ricardo criticized Smith for not sticking to the originally correctly stated principle that the relative prices of commodities depend on their relative labor times spent on their production. The

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presence of capital goods, Ricardo argued, simply modifies, but only to a limited extent an initially correctly stated principle. Puzzled by the presence of capital, he pursued more vigorously the labor theory of value by arguing that the deviations of relative prices from the respective labor times are not only minimal but, what is perhaps more important, they are amenable to abstract theorization. According to Ricardo, three are the sources of deviation of relative prices from their respective labor times; namely, the presence of capital and of rate of profit, the redistribution of income between wages and profits and, finally, the required time for the completion of the production process, that is, the turnover time (Tsoulfidis and Tsaliki 2019, ch. 1). It is worth noticing and in connection with the next chapters, that Ricardo, unlike the neo-Ricardians or Sraffian economists, assumed that the capital intensities of industries are invariable to changes in the distributive variables. In the languages of mathematics, the first derivative of capital stock with respect to (w.r.t.) the wage or rate of profit is assumed, and reasonably so, equal to zero. The word “assumed” is used because Ricardo appears that he is fully aware of the development of intricate price feedback effects emanating from changes in the distributive variables and reverberating throughout the entire economic system.2 In his numerical model, he assumed that the possible feedback effects are of minimal importance and they are not expected to change the ranking of industries and their characterization as capital or labor-intensive. After all, Ricardo’s economic models consist of two industries with such large differences in capital intensities that no income redistribution can change their ranking (see Tsoulfidis and Tsaliki 2019, ch. 1). It is important to note that in Ricardo’s often-cited numerical example (Works I, ch. 1) of two commodities, the notion of the average industry and its capital intensity are implicit but not developed. The reason is that Ricardo’s interest was in the search of an invariable measure of value defined either analytically or practically, whose prerequisites included invariance to both changes in income distribution and technology. He gave an intellectual struggle to derive such an invariable measure of value, but in vain. This search was continued and partially accomplished by Sraffa through his novel concept of standard commodity, which is invariable only to changes in distribution but not to technological change. Very similar to Ricardo’s is John S. Mill’s (1806–1873) definition of capital, who notes This accumulated stock of the produce of labor is termed Capital. What capital does for production is, to afford the shelter, protection, tools, and materials which the work requires, and to feed and otherwise maintain the laborers during the process. These are the services which present labor requires from past, and from the produce of past, labor. Whatever things are destined for this use—destined to supply productive labor with these various prerequisites—are Capital. (Mill 1848, p. 65)

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This definition of capital prevailed during the nineteenth century, and it was also adopted by the first neoclassical economists. Characteristically different is the definition of capital in Karl Marx (1818– 1884) who, unlike the other economists, placed his emphasis on capital as a specific form of social relation between capitalists and laborers and not merely as produced means of production. His analysis of capitalism begins with commodities and their labor values, that is, the abstract socially necessary labor time embodied in them and its monetary expression, that is, the direct price (DP). Capital, as means of production, has been produced by past labor and so one can use the monetary expression of value, that is, the DP to evaluate capital and reduce the vector of heterogeneous goods to a single number (scalar). This reduction is reliable and gives theoretically consistent and realistic estimates of the worthiness of capital goods in each particular period. The DP, in Marx’s analysis, are only a first approximation of the center of gravitation around which market prices (MP) persistently gravitate. Having derived the DP, Marx probes further his analysis in the nature and logic of the capitalist system by introducing a more concrete center of gravitation, the price of production (PP), that is, Ricardo’s and Smith’s natural prices. This type of price incorporates the economy-wide average rate of profit on the invested (fixed and circulating) capital. Marx shows that one can start from DP to arrive at PP, which constitute a more concrete center of gravitation of the ever-f luctuating MP. The transition from DP to PP took Marx two volumes; and only in the ninth chapter of volume III of Capital, he introduced the, what is now famous, “transformation problem” (Capital  III, ch. 9). Since then, a lot of ink has been spilled over this “problem” and not a few solutions have been proposed, whose discussion, however, is beyond the scope of the present book. In fact, the movement from DP to PP and the problems associated with this movement were effectively resolved in the works by Shaikh (1973, 1977), while Morishima (1973) and Okishio (1974) independently proposed a similar mathematical solution (see Tsoulfidis and Tsaliki 2019, ch. 3). The difference is that Shaikh in his works brings into the analysis the various conceptual issues associated with the movement from one set of prices to the other in a series of steps insisting that Marx’s solution was on the right track and approximated the final solution to the problem satisfactorily well. Suffice to say that the passage from labor values to their monetary expression, that is, the DP and from them to PP is not without its problems. DP and PP are strictly related to each other and PP change with the distribution of income in ways consistent with the capital intensity of each industry relative to an economy-wide average capital intensity. Hence, Marx differs from Ricardo in that he introduces the concept of “average” in his analysis, which does not appear explicitly in Ricardo’s usually two commodities (industries) numerical examples and in his search for the invariable measure of value. However, Marx, like Ricardo, was fully aware of the feedback effects and complexities developed going from one set of prices (i.e., DP) to the other

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(i.e., PP), but he anticipated, like Ricardo, that the final results are not qualitatively different from those in the first step. For example, Marx notes, The foregoing statements have at any rate modified the original assumption concerning the determination of the cost-price of commodities. We had originally assumed that the cost-price of a commodity equaled the value of the commodities consumed in its production. But for the buyer the price of production of a specific commodity is its cost-price, and may thus pass as cost-price into the prices of other commodities. Since the price of production may differ from the value of a commodity, it follows that the cost-price of a commodity containing this price of production of another commodity may also stand above or below that portion of its total value derived from the value of the means of production consumed by it. It is necessary to remember this modified significance of the costprice, and to bear in mind that there is always the possibility of an error if the cost-price of a commodity in any particular sphere is identified with the value of the means of production consumed by it. Our present analysis does not necessitate a closer examination of this point. (Capital III, pp. 164–165) That was perhaps the best that Marx and Ricardo could do given that they did not have the necessary mathematical tools to delve into the full solutions of their system of equations. The broad classical approach through the labor theory value managed to express “capital”, a vector of heterogeneous use-values, in terms of labor time. Marx went further in expressing capital goods in terms of equilibrium prices, that is, in terms of PP. It is important to note that classical economists share the long-period method of analysis in which the natural prices or PP are viewed as the center of attraction of the continuously f luctuating MP. The broad classical approach to determine the long-run (natural) prices starts with the following data: the real wage, the output produced and its distribution along with technology.3 According to some followers of the classical tradition, these givens may be differentiated regarding the distributive variable that may be the profit-wage ratio (or the wage share) instead of the real wage. On the other hand, Marx’s discussion of the schemes of simple reproduction in Capital II is based on the idea of a given rate of surplus value. In our view, the uniform rate of surplus value across sectors (departments) is the result of a given real wage as the basket of goods normally consumed by workers to reproduce their capacity to work. The uniform rate of surplus value is derived from the assumption of exchange in terms of DP, and the allocation of surplus value across sectors takes place in proportion to variable and not constant capital, which is in the analysis in Capital III (ch. 9). In Ricardo, there are hints that a given profit–wage ratio might be taken as a viable alternative to the given real wage (see Tsoulfidis and Tsaliki 2019, ch. 2). We also encounter the case of an exogenous rate of profit, which may replace the

16 Capital in historical perspective

real wage as the given in the system of price determination. The given rate of profit in combination, for instance, with the vector of private consumption expenditures from input–output tables allows for the determination of the consumer price index, with the aid of which we can def late the given money wage and, finally, arrive at the real wage. This is the line of research followed by Leontief (1986) and others, whereas Sraffa (1960) also proposed the rate of profit as the exogenously determined variable on the grounds that is known through the rate of interest. Ricardo’s labor theory of value was met with acceptance in the UK during the 1820s and also by economists with socialist ideas known as Ricardian Socialists. Unlike Ricardo, however, these economists of socialist persuasion argued that labor is the only source of wealth and labor is therefore entitled to all it produces.4 Even Ricardo himself did not escape from Ricardian Socialists’ polemic, because he attributed income to capital, something that was unacceptable by his followers. For them, capital is nothing but past labor; therefore, all income must belong exclusively to labor and there is neither economic nor ethical justification for profit, rent and interest incomes. The Ricardian Socialists further argued that the private property of the means of production should be replaced by cooperatives owned and run by workers. This movement was particularly active in England in the early to midnineteenth century, and it was no less active in France with the Utopian Socialists. In the German-speaking world, we had had the rise of the revolutionary then Social Democratic Party, the SPD, in the last quarter of the nineteenth and early twentieth centuries. The different versions of labor theory of value, separate and in combination, were a threat for the status quo of the system; therefore, an alternative explanation for capital and its income found fertile ground to f lourish despite its subjective nature and, as we will argue, persistent problems of internal consistency. Although not explicitly stated by the first neoclassical economists (the trio Jevons, Menger and Walras), J.B. Clark could not be more explicit by arguing that the newly emerging theory was more on the ideology side and less on the science side. He opined that in capitalism, everyone gets what one produces and everything is “fair and square”. There is no doubt that J.B. Clark was very conscientious of what was at stake. The new theory that was to replace the classical theory of value and distribution had to break away from a labor theory of value, which inescapably leads to the exploitation of labor as the source of surplus and profit type incomes.5 Other reasons for the emergence of neoclassical economics may also include the feebleness of the supporters of the labor theory of value to give satisfactory answers to a number of attacks targeting seemingly weak aspects of their theory (see Blaug 1997; Tsoulfidis 2010). Among these aspects is the role of demand in the price determination and, above all, there are issues that relate to what came to be known as the “transformation problem”. The discussion about motivations is not meant to imply that there is a covert coordination between the new needs of the ruling class or classes of their vested

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interests and the new direction of economic theory. By no means are the motivations necessarily dependent on each other, but rather their common route is derived from their common interests.

2.4 Capital in the neoclassical theory A summary history of the concept of capital in neoclassical economics would be a volume in itself. For this reason, in this section, we discuss the essential ideas of the pioneer neoclassical economists, mainly Stanley Jevons, Carl Menger, Leon Walras, Eugen Böhm-Bawerk, John B. Clark and Knut Wicksell, who discarded the theory of value based on labor time mainly because of its association with exploitation and the possible social consequences. It is quite striking that the necessity to consistently define capital compelled the first neoclassical economists to utilize the labor time as the physical unit of measuring capital. However, the new theory of value could not come to terms with the issues related to the measurement of capital in a way consistent with its own premises. The analysis of the structure of the neoclassical theory takes place in three analytical stages. In the first stage, the discussion is limited to a pure exchange economy, where there is no production at all and the individuals (or households) are endowed with various commodities as their initially given endowment. The differences in preferences among individuals prompt them to change the mix of their commodities by keeping some of them for their own use and exchanging the rest with others. The purpose of these exchanges is one and the same for all rationally behaving participants, the maximization of their utility. Walras’s (1874) contribution was that he managed, better than any of his contemporaries, to incorporate the (new) utility theory (independently and nearly simultaneously introduced by Jevons and Menger in the early 1870s) into an explicit model of a pure exchange economy. In such a model, with given the preferences of individuals and their initial endowment of goods, the difference in preferences and the substitutability in the elements of endowment gives rise to individual demand curves whose aggregation forms the total social demand curves. It is important to point out that for each demand curve, there is an implicit offer or supply curve, whose intersection with the demand curve determines the equilibrium price and quantity at the same time. The model of pure exchange economy is used only for analytical purposes and is restricted to showing the attainment of general equilibrium. The next analytical step is to generalize the pure exchange model to one with production assuming that individuals, besides the endowment of goods they are also endowed with factors of production. The transition from the pure exchange economy to the economy with production was not simple or straightforward and along the way, some asymmetries and obstacles had to be overcome. The analysis at first is restricted to non-capitalist production, that is, a model with the services provided by the non-produced factors of

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production (i.e., land and labor) as a transitory step toward a full-f ledged capitalist production, which requires a theory of value capable of addressing the issue of the logically consistent measurement of capital goods. The issues associated with the theoretically consistent measurement of capital goods were pointed out from the first major neoclassical economists, who tried to offer plausible solutions brief ly outlined in Section 2.5. We could characterize these early discussions as the first capital theory controversies while the CCC appear as their second round. In the non-capitalist production (second analytical step), we hypothesize that individuals have as givens their preferences and their endowments, a portion of which they offer in exchange for goods or other endowments. For example, individuals do not offer all of their potential labor services but retain some portion to use for leisure activities, while some of their land services may also be kept, say for kitchen gardening. The difference with the analysis of pure exchange economy is that the endowment of resources includes the productive services of the (non-produced) means of production, that is, the services of land and labor. The analysis of an economy with non-produced means of production becomes an extension of the analysis of pure exchange in that the goods that people consume are the result of the use of labor and land. Consequently, instead of exchanging directly with each other the services of the factors of production, people do it indirectly through their demand for goods. There are some asymmetries between the direct and indirect exchanges, which however are not particularly difficult to be resolved. For example, the goods that individuals demand are not exactly commensurate with the services of the factors of production offered by the individuals. In short, individuals do not demand (or consume) the services of the factors of production in any direct way. The services of the factors of production that the individuals are endowed with must be transformed into an offer of goods that will be matched by their respective demands. Hence, there is a need to connect the demand for final goods with the supply of services of the factors of production. This becomes possible by adding, in the data of the neoclassical model, the technology that describes the way in which the demand for factors of production is utilized in the production of goods and services. In other words, the demand for factors of production is a “derived demand” in the sense that consumers through their demand for specific final goods essentially activate the demand for the services of particular factors of production employed in the production of these goods. In effect, the analysis of production with non-produced means of production is simply a generalization of the pure exchange economy and does not present unsurmountable problems (Metcalfe and Steedman 1972). One would have expected that the analysis of production with nonproduced means of production could be straightforwardly generalized to include produced means of production (third analytical step); however, this is not the case, as the first neoclassical economists have already pointed out. The

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reason is that the unit of measurement of a factor of production must fulfill two basic requirements: – –

must be suitable to cost minimization, which is another way to say that the selected measurement unit must be economically meaningful must be independent of equilibrium prices.

Clearly, in the case of non-producible means of production, their measurement units fulfill the above two requirements; measuring arable land in terms of acres of uniform fertility and labor in terms of hours of work of simple labor posits no, insoluble at least, problems regarding their aggregation and determination of their equilibrium prices. Turning to capital, we realize that the two requirements are hardly met, because capital goods are heterogeneous and, to aggregate them, one needs a common unit of measurement independent of (equilibrium) prices, which we need to estimate in the first place. Paradoxically how, although the marginal theory of value and distribution was developed mainly from the dissatisfaction of the first neoclassical economists about the various aspects of the labor theory of value; nevertheless in their solutions, they theorized and evaluated capital goods in terms of labor time (see Tsoulfidis 2010, ch. 8). Hence, we could argue that as with Marx’s theory of value the issue of seeming inconsistency was brought about through the “transformation problem”, a similar critique was leveled against the neoclassical theory of value through the “capital theory critique”. Regarding to the transformation problem, it has been argued, time and again, that the issues of consistency not only have been effectively addressed but in addition have opened new ways of theoretical and empirical investigations (see Shaikh 2016; Tsoulfidis and Tsaliki 2019 and the literature cited therein). However, we cannot say the same with the neoclassical theory of value and the capital theory critique associated with it. It is important to stress that the measurement of capital has created a (popular) misconception according to which capital cannot be measured at all. This is not true since capitalists do indeed evaluate their capital goods in MP. Moreover, if it were true that capital cannot be measured at all, then it would not be possible to theorize credibly the reality of capitalism. In the CCC, the real issue with the measurement of capital that surfaces, time and again, is that the capital goods cannot be evaluated in equilibrium prices that are consistent with the requirements of the neoclassical theory of value and distribution. The option of measuring capital in terms of value is not available to neoclassical economics, because it accepts that the marginal physical productivity of capital determines the value of capital goods; we cannot reverse the arrow of causality, that is, to assume the value of capital and with its aid to determine its marginal productivity. Thus, in order for the neoclassical theory to be consistent (and circumvent the circularity issue), it is required to measure the quantity of capital independent of prices. This is an issue specific to the neoclassical theory of value and it does not appear in the classical

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Capital in historical perspective

theory of value, because capital can be measured either in terms of labor values or in PP. The idea is that PP and the rate of profit are determined having as givens the size and composition of output, the level of real wages and the state of technology and pose no problems of logical consistency. Thus, the difficulty is not the measurement of capital per se but the logical consistency of its measurement in regard to the requirements of the utilized theory of value and distribution. According to the neoclassical approach, both the profit rate and the normal prices are determined by the forces of supply and demand. The crucial issue is that in the models of pure exchange and production by employing only non-reproducible means of production (land and labor), the available goods are divided into two broad categories. The first includes the non-reproducible means of production, whose price is co-determined by the forces of supply and demand. The second includes consumer goods, whose equilibrium price is determined by their cost of production. If in the neoclassical model, we hypothesize produced means of production (i.e., capital), then their price is determined in both ways. First, as with consumer goods, the price of capital goods is determined by their cost of production. Second, as with the case of land or labor, the price of capital is determined by the capitalized income generated during its useful life, which is equivalent to saying that the prices of capital goods are determined by the forces of supply and demand. In other words, in neoclassical theory, the price of capital goods is overdetermined; that is, we have more equations than unknowns and, therefore, there is no unique solution. Sraffa (1951) pointed out an in-built measurement problem in the neoclassical theory of value and income distribution when it comes to the quantity of capital. The reason is that the estimation of the rate of profit requires the prior measurement of capital, which is a set of heterogeneously produced goods that someway must be added as to enable a cost-minimizing choice of techniques. On further consideration, the exact same problem exists also in labor and land whose characteristic difference from capital goods is that, despite known difficulties, they are reducible into homogeneous units stated in their own terms (for example, hours of the same skill and intensity of labor or land of the same fertility). From the various alternatives, the standard neoclassical theory opts to measure capital goods in value terms; that is, the product of physical units (buildings, machines, etc.) times their respective (equilibrium) prices. Sraffian economists have argued, time and again, that the measurement of capital goods, within the neoclassical theory, requires the prior knowledge of equilibrium prices, which in turn requires an equilibrium rate of profit that cannot be obtained unless we have estimated the value of capital and, as a result, a vicious cycle is in order. Sraffa notes that “[t]he reversals in the direction of the movement of relative prices, in the face of unchanged methods of production, cannot be reconciled with any notion of capital as a measurable quantity independent of distribution and prices” (Sraffa 1960, p. 38).

Capital in historical perspective

21

2.5 Production with capital and the early neoclassical economists The measurement of capital within the neoclassical theory of value and distribution has not been an easy project and the first neoclassical economists made valiant efforts for its consistent theorization. The problem is that the determination of the rate of profit (or interest rate) and the equilibrium prices associated with it requires the specification of capital in units of measurement independent of prices. However, the neoclassical theory specifies capital as a quantity of value, which does not allow the estimation of the marginal physical productivity of capital and its respective reward. This is the fundamental riddle built-in the very core of the neoclassical theory of value that the early neoclassical economists had noticed and tried to resolve. Starting from Stanley W. Jevons (1835–1882), in his incomplete studies of capital, we find that he essentially accepted the Ricardian notion of capital as a stock of goods that can be consumed either directly (like food) or indirectly (like machines). Jevons in his effort to restate the definition of capital goods notes, The views which I shall endeavour to establish on this subject are in fundamental agreement with those adopted by Ricardo; but I shall try to put the Theory of Capital in a more simple and consistent manner than has been the case with some later economists. ( Jevons 1871, p. 221) Hence, Jevons admits that the concept of capital will be reintroduced in a consistent way with the emerging new theory, which would dispense with Ricardo’s conceptualization of capital as accumulated labor and its social status quo implications. According to Jevons, capital is intrinsically connected to the passage of time between the input of labor and the final act of consumption. He notes, Capital, as I regard it, consists merely in the aggregate of those commodities which are required for sustaining laborers of any kind or class engaged in work. A stock of food is the main element of capital; but supplies of clothes, furniture, and all the other articles in common daily use are also necessary parts of capital. The current means of sustenance constitute capital in its free or uninvested form. The single and all-important function of capital is to enable the laborer to await the result of any long-lasting work,—to put an interval between the beginning and the end of an enterprise. ( Jevons 1871, p. 214) To measure the length of time between the direct and indirect consumptions, Jevons introduced the concept of an “average interval of investment”, which

22

Capital in historical perspective

anticipated, in many respects, the Austrian notion of “average period of production” that we deal with below. Formally, let q = f (t ) be the output of a production process lasting let t years, having a positive first derivative f ˛ (t ) , which is another way of saying that “roundaboutness” makes inputs more productive. The interest rate, which is assumed not to be different from the rate of return on capital, is equal to the growth rate of output resulting from waiting ( Jevons 1871, p.  267),6 and can be written as r=

f ˝ (t ) f (t )

or in Jevons’s words, the interest on capital is […] the rate of increase of the produce divided by the whole produce. ( Jevons 1871, p. 267) Of course, q = f (t ), as a function of time, can be optimized and its maximum is attained if the second derivative f ˜˜(t ) < 0 is negative as a result of the law of diminishing marginal productivity and the falling rate of profit associated with it. Interestingly enough, the long-run falling rate of profit had the status of an economic law in the classical school and continued, with the same status, in the writings of the major neoclassical economists. For example, It is one of the favourite doctrines of economists since the time of Adam Smith, that as society progresses and capital accumulates, the rate of profit, or more strictly speaking, the rate of interest, tends to fall. The rate will always ultimately sink so low, they think, that the inducements to further accumulation will cease. This doctrine is in striking agreement with the result of the somewhat abstract analytical investigation given above. Our formula for the rate of interest shows that unless there be constant progress in the arts, the rate must tend to sink towards zero, supposing accumulation of capital to go on. There are sufficient statistical facts, too, to confirm this conclusion historically. The only question that can arise is as to the actual cause of this tendency. ( Jevons 1871, p. 94) Jevons came too close to the notion of marginal productivity of capital. Had he pursued more vigorously his analysis, he could have derived a demonstration of the relationship between the rate of interest and the production period from the perspective of a producer’s maximizing behavior. The optimal point could have been derived by invoking the usual formula of the present value, v, of a stream of income in continuous time v (t, r ) = f (t )(1+ r )−t

Capital in historical perspective  23

Given the rate of interest, maximization requires v ′ (t, r ) = 0 and so v ′ = f ′ (t )(1 + r )−t − f (t )(1 + r )-t ln (1 + r ) = 0 and after some manipulation, we get ln (1 + r ) =

f ′ (t ) f (t )

The second-order conditions are fulfilled, since the second derivative of v gives v ′′ = f ′′ (t )(1 + r )−t − f ′ (t )(1 + r )−t ln (1 + r ) − f ′ (t )(1 + r )−t ln (1 + r ) + f (t )(1 + r )−t ln (1 + r )2 < 0 From the first-order conditions, we get −  f ′ (t )(1 + r )−t − f (t )(1 + r )-t ln (1 + r ) ln (1 + r ) = 0 Assuming that f (t ) is well-behaved, that is, it is twice differentiable with positive the its first and negative the its second derivative, for a positive interest rate, we have f ′′ (t )(1 + r )−t − f ′ (t )(1 + r )−t ln (1 + r ) < 0 With the aid of the above equation, we can determine the time, t, that maximizes the present value of output, when the interest rate is given, r = r . Carl Menger (1840–1921) argued that there is a “time structure” to the production process that classifies goods into those of lower and higher order: the former are less productive identified with the consumer goods, while the latter with capital goods (Menger 1871, p. 153). Interesting enough in Menger and neoclassical economics in general, and contrary to the popular view, the direction of causality runs from lower- to higher-order goods, which amounts to the same thing as saying that the demand determines the supply side of the market. Hence, we have a complete reversal of the causal relation of the classical economics; it is not the cost of production that determines the price of a good but rather the intensity of satisfaction. Hence, the magnitude of utility enjoyed by the consumer dictates both the price of the good and the payment to the factors of production that went into its production. In short, the cost of production, which is an economic category objectively defined both by the classical economists as well as by the businesspeople, surprisingly enough becomes subjective by the emerging neoclassical theory. In effect, Menger set as one of the goals of his studies to bridge the gap between what businesspeople thought about the cost of production and how this view could come to terms with the utility theory. And he thought that the ranking of goods into those of lower and higher order answers this inconsistency.

24 Capital in historical perspective

Eugen von Böhm-Bawerk (1851–1914), a student of Menger, made the study of the interest rate and capital the areas of his specialization. He began by adopting his teacher’s ranking of goods according to which capital goods are used in the production of consumer goods and so they are called goods of “higher order” that take time to mature, whereas the consumer goods are immediately available for consumption and are called “goods of the first order”. Clearly, the time dimension pervades both types of goods and having made time the salient feature of goods, Böhm-Bawerk introduced the notion of the “average period of production” (APP) to quantify capital in terms of time and independently of prices. The APP is the weighted average of all intervals of time during which the quantity of labor is spent to acquire a certain amount of output, where the weights are given by the respective quantities of labor. By way of an example, to produce a certain capital good, if we applied 240 units of labor, l, 11 years, t, ago and 60 units of labor 1 year ago; the total sum of past labor is 300 labor units and the average period of production, τ, will be

˜=

240l ˙11t + 60l ˙1t 2,700l ˙ t = = 9t 300l ( 240 + 60) l

Hence, the physical measure of capital is nine years. The so estimated APP raises three issues: first, the absence of land; second, the use of a simple average instead of the more appropriate weighted average; and third, the exclusion of interest rate from the estimations. Regarding the land, the answer is that its estimation, although not trouble-free, certainly is not more intricate than that of capital, as we explained above. The choice of the simple average presents no serious problems because the more inclusive weighted average will not give a wholly different answer. Finally, the inclusion of the simple interest rate is not difficult at all, and the answer for the APP will be the same because the interest rate, so long it remains simple, will appear in both numerator and denominator of the fraction canceling each other out. We can illustrate this by referring to our numerical example, where we include a uniform interest rate. Thus, we may write 240l (1 + 11ti ) + 60l (1 + i ) ( 240 + 60) l (1+ ˜ i ) =   300l ( 240 + 60) l and

˜=

9i = 9 years i

We observe that the simple interest rate does not change the picture; in both cases, the APP will be 9 years or 108 months, and the measurement of capital stock is totally independent of the simple rate of interest. Naturally, one wonders about the effects of the more realistic compounded (annually or any fraction of the year) interest rate on the value of capital and why did Böhm-Bawerk and other Austrian economists (not all) not pay the required attention? Is it that they ignored the compound interest rate? Or were

Capital in historical perspective

25

they concerned with the intricacies of the compound interest rate in the estimation of capital stock and the possible negative consequences for their theory of capital? There is no doubt that Böhm-Bawerk (1884), a renowned economist and an expert on the theory of the interest rate, as this can also be judged by his monograph on Capital and Interest was aware of the compound interest rate and its non-linear consequences (Kurz and Salvadori 1995, pp. 436–437). Let us now examine these issues by experimenting in our simple, but fair, numerical example with a compounded annual interest rate. The issue of compound interest rate was brought about by Wicksell (1893, p. 184) who argued that the APP depends on the rate of interest provided it is compounded annually. In terms of our numerical example and by using compounding to estimate the new APP, we get 240l (1+ i )11 + 60l (1+ i ) 300l (1+ i )˙ = 300l 300l and solving for ˜ , we get

˜=

ln ˆˇ240 (1+ i )11 + 60 (1 + i )˘ − ln ( 300) ln (1+ i )

For given i = 5%, the APP is ˜ ˛ 9.353 years. In this particular example in terms of a reasonable time period and reasonable interest rate, the difference between the APP estimated in simple interest rate and the APP estimated in compound in the same nominal interest rate is only 0.353 and, therefore, not too large in absolute terms, while in relative terms, it is about 3.97%. Thus, the APP depends also on the interest rate, when it is compounded annually but one does not expect wild differences between these two estimates for reasonable periods of time and interest rates. If, for example, the interest rate doubles, then the new APP will be equal to 9.624 and the deviation from the estimate based on simple interest rate will be equal to 6.49%, which is higher than six months. In both cases, we have quite different interest rates and yet the deviation is not equally large. If, however, we repeat the same experiment, increasing the years to maturity, say from 11 to 15, then the results are substantially different. For 5% interest rate, we get 12.86, and for 10% interest rate, the APP becomes 13.33. Such differences simply cannot be ignored. The non-linear relationship between APP and the compounded rate of interest makes us reluctant to say a priori whether the APP of one good, ceteris paribus, is higher than another, unless we know the rate of interest and how frequently it is compounded within the year. From the above, it follows that for all practical purposes, one does not expect that the compound interest rate gives completely different results in terms of rankings of the various APPs. Hence, we need to exclude the cases of APPs too close to each other in the first place rendering their rankings f lexible or unpredictable in the face of different interest rates.

26

Capital in historical perspective

Let us now express the labor productivity, q = f (˜ ), as a function of the APP while the usual assumptions hold, that is, the function is twice differentiable with f ˝ (˜ ) > 0 and f ˝˝ (˜ ) < 0. The wage rate, w, is determined in the labor market and profits, π, are residually determined by the difference between productivity and wage rate, that is, π = q ‒ w. Furthermore, in order for the entrepreneur to complete the production period, τ, the capital outlays will be equal to the product of the APP times the wage rate, τw. This is equivalent to saying that capital in Böhm-Bawerk’s conceptualization of production process is actually a wage fund, provided that the land has been put aside in these estimations. It follows, therefore, that the rate of return on capital will be r=

f (˜ ) − w w˜

Thus, the problem for the entrepreneur is with given the wage rate, w, to find the APP, τ, such that the rate of interest, r, to be maximized. The first-order condition of the above function is f ˝ (˜ ) ˜ = f (˜ ) − w and so the maximum (optimum) interest rate will be r=

f ˛(˜ ) w

In the above expression, Böhm-Bawerk, unlike Jevons, managed to express the rate of interest (return) as the marginal product of capital relative to the wage rate. Furthermore, turning to the economy as a whole, we see that while the wage rate is a variable, the labor force is considered to be fixed. Thus, if by k = K / L , we represent the capital–labor ratio of the economy, then it follows that the equilibrium wage will be the one at which all labor is being employed during the period of production by the entrepreneurs. Then w = k / ˜ and by substituting in the first-order condition, we get f ˝ (˜ ) ˜ 2 = f (˜ ) ˜ − k where the left term of the equation stands for total profits, that is, the product of unit profits, f ˝ (˜ ) ˜ times APP, ˜ . Total output is f (˜ ) ˜ , and total wages are equal to k = w˜ . The above equation essentially states that equilibrium is attained at the point where, given the total employment in the economy, the wage is at a level where labor is fully employed. Böhm-Bawerk with the introduction of the APP managed to homogenize capital goods by expressing them in terms of time (a physical measure amenable to quantification) and to show that the marginal product of capital equals to the rate of profit (interest). However, the conceptualization of the quantity of capital through the APP is very difficult to operationalize in the real economy even in the case of a simple rate of interest, let alone the complication arising with compound interest rates and multiple equilibria.

Capital in historical perspective

27

There was another strand of neoclassical theory that conceptualized capital in an altogether different way. Léon Walras developed his theory of capital starting with the distinction of an initial endowment of factors of production and their services provided by them. In this sense, the endowment in labor and land could be classified as capital. Walras (1874) treats all intermediate and  consumer goods as final goods and as capital the durable goods. The analysis of capital, although too short in the Elements (40 pages), is much too involved with complexities and the interested in reader can find critical presentations of Walras’s theory of capital in Eatwell (1990), Garegnani (1990, 2012) and Petri (2017, 2020). By contrast, J.B. Clark’s theory of income distribution treats capital as a value magnitude. For J.B. Clark, the equilibrium rate of profit is determined by technology, in particular by a production function, where the various factors of production (capital and labor) are employed to produce a given amount of value of output. There is substitution between the factors of production and each and every factor of production is being paid according to its marginal contribution to the value of output produced. Capital is paid by profit according to its marginal contribution exactly as wages are being paid according to the marginal productivity (contribution to production) of labor. If perfect competition prevails and the factors of production are being paid according to their marginal contribution to the value of output produced, nothing is left to be received as additional income by any social class. There is no surplus and, therefore, there is no exploitation. Hence, J.B. Clark only verbally explains this point while later neoclassical economists, for example, Weeksteed (1910) offered rigorous formulations of the “product exhaustion hypotheses”. In so doing, J.B. Clark thought that not only a satisfactory answer was provided for the payment of capital but also the neoclassical economic theory dispensed with the “indictment that hangs over society” and that is the “exploiting labor”. J.B. Clark notes, […] the distribution of the income of society is controlled by a natural law, and that this law, if it worked without friction, would give to every agent of production the amount of wealth which that agent creates. However wages may be adjusted by bargains freely made between individual men, the rates of pay that result from such transactions tend, it is here claimed, to equal that part of the product of industry which is traceable to the labor itself; and however interest may be adjusted by similarly free bargaining, it naturally tends to equal the fractional product that is separately traceable to capital. (Clark 1899, p. 3) Consequently, profits are viewed as rewards for abstaining from current consumption; that is, saving leads to the creation of the capital goods. In this view, profit income is a reward for those who value future income highly and are thus willing to sacrifice current consumption. It is important to emphasize that

28 Capital in historical perspective

unlike J.B. Clark’s view, the neoclassical theory nowadays does “not” claim that capital’s or labor’s income is “deserved” in some moral or normative sense. Knut Wicksell’s (1851–1926) contributions to the theory of capital are based partly on Walras’s notion of capital as a vector of heterogeneous goods and partly on Böhm-Bawerk’s idea of the APP. Wicksell analyzed how the structure of capital goods depends not only on the number of units of investment but also on the length of time during which the inputs are invested. He argued that the importance of the time-element in production was never properly appreciated by Walras and his school. The idea of a period of production or of capital-investment does not […] exist in the Walras-Pareto theory; in it capital and interest rank equally with land and rent; in other words, it remains a theory of production under essentially non-capitalistic conditions, even though the existence of durable, but apparently indestructible instruments, is taken into account. (Wicksell 1893, p. 171) According to Wicksell, capital should be treated in a way similar to that of labor and land, which is equivalent to saying that capital should be transformed into a homogeneous and, therefore, measurable factor of production. Wicksell was critical to the Walrasian “solution”, which was to express capital as a bundle of heterogeneous goods, because in this case, we derive the rate of profit for each type of capital good and not the general (uniform) rate of profit that the neoclassical theory proper would require. Wicksell (1893, p. 150) argued that the homogenization of capital is possible once we express it in terms of the original and indestructible factors of production of labor and land together with their time dimension. Capital, therefore in Wicksell’s view, is a derived concept resulting from the accumulated labor and land, a single coherent mass of saved-up labor and saved-up land. Assuming, for a moment, that land is a free good, we are left with accumulated (dead) labor versus current (living) labor with the former being more productive than the latter. In general, the productivity of accumulated labor or what amounts to the same thing, the “productivity” of capital depends on the stages of production (roundabout production), while the time element of production is responsible, other things equal, for the increase in efficiency, which is a condition sine qua non for the rate of profit (interest). According to Wicksell, Capital is saved-up labour and saved-up land. Interest is the difference between the marginal productivity of saved-up labour and land and of current labour and land. (Wicksell 1893, p. 154) Having conceived the physical quantity of capital as a sum of past quantities of labor and land, Wicksell could proceed with the estimation of the value of the

Capital in historical perspective

29

marginal product of each of the two factors of production, which added would give the total value of the capital stock. In other words, the value of the capital stock is the amount invested in labor and land compounded by the interest rate over the average investment period. He clearly understood that the heterogeneous nature of capital goods did not allow their expression into a scalar magnitude unless they are defined in value terms. Wicksell’s (1893) discussion on capital theory in value terms is only in a few pages (178–181) and his presentation is in terms of calculus. In these few pages, Wicksell made an effort to deal with capital goods in a way compatible with the neoclassical theory of value and distribution. However, he discovered that unfortunately, the marginal productivity of capital is not always equal to the rate of interest (Kurz 1990, p. 84). An inequality, which he attributed to the revaluation of capital as an effect of income redistribution. Fisher (1930) introduced the intertemporal analysis in a general equilibrium framework. In this analysis, a composite commodity could be produced and consumed at different dates. In the determination of interest rate, prices and distributional variables were assumed fixed. Fisher further introduced diminishing returns and arrived at the well-known demand for saving; that is, investment, whose intersection with the supply of savings determines the equilibrium interest rate. The latter is equal, as Keynes (1936, ch. 11, p. 140) notes, to the marginal efficiency of capital (see Tsoulfidis 2008a, 2010, ch. 8). The discussion continued during the 1930s with the neoclassical economists Frank Knight and Friedrich Hayek, who argued that there was no need to refer to capital and, in so doing, the vicious cycle with prices disappear because the investment is not the same as capital stock. But it has been argued that what is true for the stock variable must also be true for the f low variable.

2.6 Summary and conclusions The classical economists conceived capital goods as past (or dead) labor and the measurement unit, naturally, was labor time. Smith understood pretty well that capital is associated with the emerging, out of feudalism, capitalism. He did not see capital only as physical goods but rather as a social relation and it appears only if there is a division of labor, which is promoted systematically in conditions of capitalism. Ricardo although expanded Smith’s labor theory of value and applied it consistently when it came to capital, his view was rather narrow as he downplayed the social relation aspect of it. J.S. Mill was somewhere between the two with less insistence on the labor theory of value and more on the social relations aspects of capital. Marx, on the other hand, theorized capital as a specific to capitalism social relation of production, where labor is the creator of new value from which is being paid only a portion of the total labor time spent in production and the rest goes to profit type of incomes. This exploitative nature of the system is embedded in the social relation of capital, something that is more explicitly analyzed in Marx but it is also found, at least partly, in Smith. The socialist followers of Ricardo in England, unlike Ricardo and Marx, thought that all wealth is created exclusively by labor

30 Capital in historical perspective

and its exploitation gives rise to profit for which there is neither moral nor economic justification. Arguments of this sort, along with the strong socialist movement of the second half of the nineteenth century, created the need for an alternative to the classical political economy perspective and gave rise to neoclassical economics in the last quarter of the nineteenth century. The first neoclassical economists ( Jevons, Menger, Walras) utilized time, albeit not exactly labor time, in their effort to measure capital to determine equilibrium prices and the rate of profit, exactly as the classical economists did before them. The neoclassical economists nowadays employ capital in value terms (see Chapter 3) in spite of the fact that the pioneer neoclassical economists realized that the measurement of capital must be independent of (equilibrium) prices because the latter is what, in the first place, must be determined through the endowment of capital. Consequently, the measurement of capital in physical terms attracted the attention from very early and became a subject of debate among the neoclassical economists of the late nineteenth century and early twentieth century without, however, reaching any judgement in this less well-known first round of capital theory controversies. To make a long story short, the price of avoiding the labor theory of value and especially its Marxian version was very high and paid dearly by the first neoclassical economists Jevons, Böhm Bawerk, Walras and Wicksell, among many others. Such a high price is always being paid whenever straightforward ideas are rejected or circumvented for ideological reasons; they are bound to be rediscovered some other time only in distorted ways. This is the early history of the evolution of the ghost-like concept of capital that haunts the structure of neoclassical economic theory until the present day.

Notes 1 The idea of a labor commanded theory of value reappears in Keynes’s (1936, ch. 19) concept of “wage units”. 2 For example, he criticized Smith’s ‘adding-up theory’ of value with regard to the effects of taxation on wages. Ricardo characterized the adding-up theory of value “absurd”, and he notes that “this rise in the price of goods will again operate on wages, and the action and re-action first of wages on goods, and then of goods on wages, will be extended without any assignable limits” (Works I, p. 225). 3 Kurz and Salvadori (2002) in the classical long-period analysis include the following data: (i) the set of available techniques, (ii) the size and composition of the social product, (iii) the ruling real wage rate for common labor, and (iv) the quantities of the different qualities of land available and the known stocks of depletable resources. In contrast, the neoclassical long-run analysis includes the following data: preferences, technology, and endowments, whose details we discuss in the next section. 4 It is interesting to note that Marx (1875, p. 341) did not agree with this view, as he argued that “Labor is not the source of all wealth. Nature is just as much the source of use-values […] as labor”; the emphasis is placed in the original. 5 For various views on these issues, see Blaug (1997). 6 Wicksell (1893) calls “natural” the rate of interest, which is equal to the growth rate of output.

3

Capital theory controversies

3.1 Introduction The focus of our analysis now is on the famous Cambridge Capital Controversies (CCC) that involved economists of the caliber of Piero Sraffa and his followers Joan Robinson, Piero Garegnani, Luigi Pasinetti and Geoffrey Harcourt at the University of Cambridge in England and Paul Samuelson and Robert Solow along with their students at the Massachusetts Institute of Technology in Cambridge, Massachusetts. The debates were about the consistency of the neoclassical theory of value measurement of capital goods. As we have already indicated, there is a problem arising from the dual nature of capital goods of being both produced goods and means of production at the same time. The dual nature of capital goods imposes two ways for their evaluation: first, as produced goods by their cost of production as is the case with consumer goods and second, as produced means of production by their expected returns during their useful life, exactly as is the case with the non-produced means of production (land or labor). The structure of the remainder of the chapter is as follows: Section 3.2 provides a summary of a historical account of the debate and deals with the essence of the debate, which is more about the theoretical consistency of the neoclassical theory and less about the measurement of capital goods. Section 3.3 discusses the derivation of the wage rates of profit (WRP) curves and the associated WRP frontiers. Section 3.4 applies the WRP curves and frontiers in the context of one commodity world. Section 3.5 continues with Samuelson’s surrogate production function and derives the demand schedules for capital. Section 3.6 brings into discussion the possibility of reswitching of techniques. Section 3.7 concludes the discussion of this not necessarily a long-lasting but a very intense debate. In the Appendix, Samuelson’s and Pasinetti’s numerical examples of double switching are presented and critically evaluated.

3.2 Cambridge capital theory controversies The neoclassical theory of value is based on marginal productivity theory according to which the incomes of factors of production are determined by

32

Capital theory controversies

their marginal contribution to production. In particular, the marginal product of labor is equal to the wage rate, the marginal product of capital is equal to the rate of profit (or interest rate) and the marginal product of land is equal to the rent. In all cases, the logical consistency of the theory compels that the marginal products of all factors of production ought to be measured in physical terms. This is quite plausible in the case of labor input, which can be homogenized and measured in terms of labor time; however, the homogenization even of this factor of production is by no means without its problems, if we think of the thorny question of differences in skills. In the classical approach, the skilled labor is reduced to simple or unskilled labor according to the labor time required for the acquisition of the necessary skills. Thus, for all practical purposes and assuming everything else equal, skilled labor is viewed as a multiple of the unskilled and so the marginal productivity of each kind of labor, skilled or unskilled is expected to be ref lected in wage differentials.1 Similarly, the heterogeneous land input, in principle at least, can be homogenized and measured in cultivated areas of the same quality of land and fertility. Hence, for both labor and land, we can stipulate economically meaningful physical units of measurement, in the sense that they are amenable to cost minimization. By contrast, the measurement of the marginal physical productivity of capital is subjected to a number of limitations. The reason is that capital as a factor of production consists of a multitude of heterogeneous goods, whose aggregation to a single entity is fraught with problems of consistency. In fact, the problem is that capital—unlike labor or land—is an amalgam of heterogeneously produced commodities, which must be added in such a way as to enable a cost-minimizing choice of technique. In other words, there is a need to devise a kind of a yardstick with the aid of which the aggregation of the different components of capital (various tools, machines and structures) to an economically meaningful entity becomes possible. In addition, the marginal product of capital must be expressed in physical and not in monetary terms, a task that becomes nearly impossible to successfully carry out. From the various available alternatives (e.g., labor time, market prices), neoclassical theory from J.B. Clark and Wicksell and others opts to measure capital goods in terms of values; in particular, the capital goods in terms of physical units (buildings, machines, etc.) times their respective market prices (MP). Hence, the neoclassical theory is in the logic and spirit of businesspeople, who evaluate every component of their entire endowment of capital stock by assigning to it a MP and arrive at a single value by summing them up, thereby expressing their capital stock in market value terms. The trouble, however, in expressing capital in monetary terms, is its dependence on the interest rate through which future revenues are discounted to the present. In this way, the value of capital is derived on the ever-f luctuating MP, which by their very nature are disequilibrium prices and as such cannot be the foundation of neoclassical or any economic theory. Moreover, if for a moment

Capital theory controversies 33

assume away the long-run character of economic analysis and the associated natural or equilibrium prices and like businesspeople use MP for the measurement of capital, we are definitely subjected to circularity issues. The idea is that there is no discounting and measurement of capital without knowing the rate of interest; therefore, it is odd to use the value of capital in a production function to determine the interest rate. Simply put, the interest rate cannot be determined from capital measured in MP because the value of capital has been previously determined based on the already known interest rate. In addition, the valuation of capital in terms of any kind of prices (equilibrium or market) is not really an option to the neoclassical theory of income distribution, whose consistency to its own premises requires the quantification of the endowment of capital goods without resorting to any value measure. This is equivalent to saying that we need firstly to measure capital goods in physical terms such that to derive their marginal productivity and the associated equilibrium prices and secondly, by assigning these prices to each and every one of the components of the entire capital stock, to arrive finally at an estimate of its total value. The theoretically consistent measurement of capital, as being of cardinal importance, was pointed out from the first neoclassical economists, who tried to offer plausible solutions. More specifically, Jevons and Menger tried to measure capital goods in physical terms and, surprisingly enough, in terms of labor time. Walras, on the other hand, attempted to measure capital goods in terms of a vector of heterogeneous entities, however, without success (see Eatwell 1990; Garegnani 1990, 2012; Petri 2020 and the literature cited). Finally, Wicksell, who fully understood the problem of specifying the endowment of capital goods in a way consistent with the requirements of neoclassical theory, gave up after a long intellectual struggle and, in his search for scientific truth and intellectual honesty, he proposed to measure the endowment of capital goods as a given quantity of value. In Chapter 2, we argued that these first neoclassical economists went through an intellectual struggle, on the one hand, to arrive at a new theory of value and distribution and in so doing to discard the labor theory of value for its explicit or implicit social implications threatening the status quo of the system. On the other hand, regarding the notion of capital to device a common substance, other than labor time, characterizing all capital goods to aggregate them in an economically meaningful manner. Surprisingly enough in the measurement of capital, they utilized labor time and the evaluation of capital became possible through the use of turnover time; a concept of minor importance in Ricardo’s numerical examples (see Tsoulfidis and Tsaliki 2019, ch. 1). The problem of consistent evaluation of capital goods does not appear in the classical approach because in the estimation of equilibrium prices and rate of profit, it is assumed that the size and composition of output are given together with the real wage and the state of technology. In other words, the classical analysis assumes one of the distributive variables, usually the real wage, as given and determines the other distributive variable, the rate profit.

34

Capital theory controversies

Alternatively, we could hold as given the rate of profit and the money wage and through them to determine the relative prices as well as the price level (Leontief 1986). Thus, the evaluation of capital goods (assuming them in physical terms in an input–output setting) can be made without the consistency problems of the neoclassical theory. By contrast, in the neoclassical approach, by determining the profit rate and equilibrium prices through the forces of demand and supply, theoretical concerns arise about the logical consistency and the analytical structure of the neoclassical theory, known as logical or internal critique. This logical critique arises because of the heterogeneity of the endowment of capital goods and the lack of a single physical measure that rules out the assessment of the marginal physical productivity and the associated with it rate of profit. The issue of capital and its measurement in a way consistent with long-run equilibrium prices resurfaced in Robinson’s (1953–1954) important article inspired by the teaching and writings of Piero Sraffa, especially by his introduction in Ricardo’s Principles. In this seminal article, Robinson raised three interrelated issues: first, the meaning of capital within the neoclassical theory of income distribution; second, the measurement units of the entity capital such that to be independent of equilibrium prices; and third, the existence of long-run equilibrium position of the economy and its attainment. Robinson was particularly interested in the theoretically consistent measurement of capital goods as to whether and to what extent it is possible to employ capital as a pure physical magnitude in a production function and in particular to be used in a theory of value and income distribution. Robinson notes, [T]he production function has been a powerful instrument of miseducation. The student of economic theory is taught to write Q = f (L, C) where L is a quantity of labor, C a quantity of capital and Q a rate of output of commodities. He is instructed to assume all workers alike, and to measure L in man-hours of labor; he is told something about the index number problem involved in choosing a unit of output; and then he is hurried on to the next question, in the hope that he will forget to ask in what units C is measured. Before ever he does ask, he has become a professor, and so sloppy habits of thought are handed on from one generation to the next. (Robinson 1953–1954, p. 81) The background of all these discussions on capital theory is Piero Sraffa and his path-breaking book (1960) that made more explicit the internal consistency problems in the measurement of capital within the neoclassical framework and gave a much deeper content and perspective in the debates that followed. In particular, Sraffa argued that within the neoclassical theory, it is not possible to obtain a relationship between wage and profit rate (interest), described by the WRP frontier, with negative and simultaneously monotonic or even better constant slope, as required by the neoclassical

Capital theory controversies 35

production function. Hence, it is not possible to rank the different techniques according to their capital intensity, simply because capital intensity is not constant, but it varies with changes in income distribution in ways impossible to theorize and, therefore, predict. Moreover, if the characterization of an industry as capital- or labor-intensive changes with income distribution, it follows that the edifice of neoclassical theory of value is not based on solid foundations. The debate started in the early 1960s, when Samuelson (1962) presented a model based on the heroic assumption that capital intensity is uniform across sectors, which is no different to saying that there is a one-commodity world. In such an economy, as income distribution varies, the subsequent revaluation of capital gives rise to results that are absolutely consistent with the requirements of neoclassical theory. In fact, Samuelson derived a straight-line WRP frontier or in Samuelson’s terminology factor-price frontier, the mirror image of the usual convex isoquant curves, which enables the derivation of a well-behaved demand-for-capital schedule. It is important to point out that Samuelson (1971) attacked Marxian value theory for its alleged inability to explain relative prices. However, if one applies Samuelson’s heroic assumption of an equal capital intensity across all industries to Marx’s labor theory of value, then the labor values become equal to prices of production (PP) and profits to surplus values. Consequently, all of Samuelson’s (1971) criticisms of Marx on the so-called transformation problem become untenable. The Cambridge UK participants in the capital debates noticed this irony. Samuelson’s assumption was criticized for its simplicity and, above all, for the lack of realism by Garegnani (1970, 1990) and Pasinetti (1966, 1977), among others, who showed that once we hypothesize different capital intensities across industries, the neoclassical results do not necessarily hold. The idea is that as relative prices change, the revaluation of capital can go either way. Thus, it is possible for an industry to begin as capital intensive in one income distribution and to become labor intensive in another. Consequently, we no longer derive Samuelson’s straight-line WRP frontiers, which are absolutely required for the cost-minimizing choice of technique and give rise to well-behaved demand-for-capital schedules. Samuelson’s doctorate student and research associate Levhari (1965) opined that reswitching is not possible in an indecomposable production system and, therefore, reswitching is only applicable to a single industry and not to the totality of the economy. The article stimulated a debate and a symposium organized by the Quarterly Journal of Economics. Among those that refuted Levhari’s claim were after Sraffa’s encouragement his student Pasinetti (1966). Samuelson (1966) conceded that his one-commodity-world description of the operation of the economy, the only way that one may derive consistent demand schedule for capital, was patently unrealistic. The Sraffian economists of the UK Cambridge with Samuelson (1966), this time on their side, claimed using numerical examples that the neoclassical demand and supply schedules determining equilibrium prices for goods and produced factors of production may not hold.

36 Capital theory controversies

The internal theoretical inconsistencies of the neoclassical theory in its long-period method of analysis that were revealed in the CCC led to the development of its new variant known as intertemporal equilibrium approach. The new variant of neoclassical theory surpasses the problem of inconsistency in the measurement of capital at the expense of spiriting away the long-run character of the analysis associated with the equalization of the rate of profit and, therefore, equilibrium prices. That is, this strand of the neoclassical theory has abandoned the long-period natural price concept as a center of gravity and it deploys the concept of intertemporal equilibria, where future prices are determined simultaneously with relative current prices, through an assumption either of the existence of complete markets for all future goods, or of perfect foresight.2 It is important to stress that Sraffa’s primary interest was in the price movement induced by changes in income distribution and the subsequent reordering of techniques. In effect, the standard commodity, a major Sraffa’s innovation, was designed to shed light and settle the very old question that troubled Ricardo. That is, the discovery of an industry, theoretically or practically, whose price is not affected by changes in income distribution, and therefore it can be used as a reference or standard for the evaluation of changes in relative prices of all other industries. Hence, Sraffa’s primary concern was on the shape of the price rate of profit (PRP) trajectories and not the shape of WRP curves. Nevertheless, the latter, which are the mirror image of isoquant curves through which the neoclassical demand schedules of capital and labor are derived, attracted most of the attention of Sraffian economists. The idea was to bring neoclassical economists on their own playfield by invoking production functions and the isoquants associated with them and by applying their logical critique to reveal the internal inconsistencies of the neoclassical theory. The trouble with the logical critique, however, is that once one makes-believe that accepts, as if they were realistic, assumptions such as perfect substitutability and perfect competition, may end up entrapped in the framework of the theory whose internal inconsistencies seeks to reveal. While neoclassical theory may be criticized for many unrealistic assumptions, such as perfect foresight and perfect rationality, there is a more fundamental critique, and this is its inability to deal consistently with a world that contains heterogeneous capital goods. In conclusion, economists of the Cambridge University in the UK adopted Sraffa’s work and insights. Among his many students in the 1960s, we distinguish Joan Robinson, Pierangelo Garegnani, Geoffrey Harcourt and Luigi Pasinetti. On the other side of the Atlantic Ocean, in Cambridge Massachusetts, top neoclassical economists, like Paul Samuelson, Robert Solow, and many others, defended the neoclassical theory and, as we will argue, without success. The exchange of views on this issue has been established in the literature as the famous Capital Theory Controversy between Cambridge University in England and the MIT in the Cambridge region of Massachusetts. We now turn to CCC and we bring the internal theoretical inconsistency of neoclassical economics to deal with capital as a means of production in the framework of the production function.

Capital theory controversies 37

3.3 Production with produced means of production The neoclassical theory of value and distribution as we discussed in Section 2.4 has been advanced in three stages: In the first stage, the discussion is limited to pure exchange, where the individuals (or households) are endowed with various commodities and their differences in preferences induce them to exchange in their effort to maximize their utility. The next stage was to generalize the pure exchange model to one with production with nonproduced means of production (land and labor). The transition from the pure exchange model to non-capitalist production was not without problems but these could be superseded by homogenizing both labor and land in their own economically meaningful units of measurement. However, the same is not true, when the analysis was further generalized to a third stage to include produced means of production (capital goods) in full-f ledged capitalist conditions. The reason is that the price of capital goods is determined by their cost of production as in any other produced good. But to define the cost of production of capital goods, we need to know their value that requires the prior knowledge of the cost of production, thereby falling to the characterization of what came to be known after Joan Robinson as “impregnated circularity”. According to the neoclassical theory of value and distribution, the value of each factor of production is determined by its marginal contribution to production, and the presence of perfect substitutability between inputs leads to diminishing returns to scale. In short, the price of a factor of production ref lects its relative scarcity; so does the rate of profit (or interest) as being the price of capital input. More specifically, the relative abundance of capital in conjunction with the law of diminishing returns3 leads to a lower rate of profit; the converse applies to the case of its relative scarcity. Within the neoclassical analytical framework, the shape of isoquant curves depicts the various combinations of two inputs used to produce a given amount of output. The usual consistent with the neoclassical theory shape of isoquants is convex, a shape that stems from the law of diminishing returns to factors of production and the idea of their nearly perfect substitutability giving rise to many possible techniques.4 In addition, the isocost curve depicts the price ratio of the two inputs and its tangency with the isoquant determines the optimum combination of land and labor. Figure 3.1 presents a set of convex isoquant curves combined with the isocost line for the simple case in which the only factors of production are the non-reproducible inputs of labor and land. The vertical axis depicts the quantity of land measured in acres of the same quality while on the horizontal axis, we place the quantity of the labor input measured in hours of homogeneous labor. Hence, both units of measurement of the two non-reproducible factors of production are independent of prices. When we refer to an amount of land expressed in hectares of the same quality (fertility) and to a quantity of labor expressed in hours of homogeneous labor, it means that both variables can be

38 Capital theory controversies

Land

Isoquants

Isocost

Labor

Figure 3.1 Isocost and isoquant curves.

expressed in an appropriate economically meaningful way to form quantity indices that minimize the production cost of a specific good. In other words, the measurement units of the two variables are amenable to the selection of their optimal economical combination. That is, the point of tangency of the isocost curve with the highest attainable isoquant curve determines the optimal land–labor combination for a given level of output. Although the analysis of production with non-reproducible inputs presents no particular problems, when capital goods (or produced means of production) enter into the production function, we note that we are dealing with a reproducible input and, therefore, it differs from non-reproducible ones, because – –

depreciates, and its price is equal to the cost of its production, as is the case with the final or intermediate goods.

The first characteristic needs no further discussion, while the second needs some further qualifications. The price of capital is determined as follows: PK =

XK r

(3.1)

where PK is the price of capital, X K is its net (from depreciation) annual revenues and r is the rate of profit (or interest).5 In equilibrium, the demand price for capital goods should be equal to their production cost. Because this condition should apply to all capital goods, it follows that the rate of profit on production cost for each capital asset must be equal to the economy’s average rate of profit: r=

XK CK

(3.2)

Capital theory controversies 39

where CK is the production cost of a capital good. Therefore, the presence of a uniform rate of profit dictates that the demand price for capital goods is identical to their production cost: PK = C K

(3.3)

which is a condition that should be met by all reproducible goods. However, capital goods besides being reproducible possess another important feature; they are an endowment owned by individuals. According to this second feature, the price for its services, PK, should be determined in the capital market. But, while labor’s and land’s prices are specified in the relevant markets by their respective supply and demand forces and the prices of consumer goods are determined by their production costs, the prices of capital goods are determined simultaneously (Eatwell 1990, p. xii): – –

Firstly, by the conditions of supply and demand in the capital market as being part of the endowment of the individuals. Secondly, by their production costs in the relevant market, since capital goods are produced means of production.

In other words, within the neoclassical theory of value, there is a peculiarity about capital goods, since there are two conditions that identify the same set of values. Put in mathematical terms, the system is over-determined and as such may have one, zero or infinite number of solutions.

3.4 Samuelson’s one-commodity world Samuelson (1962) essentially addressed the issues raised by Robinson (1953– 1954) in her seminal article on capital theory and, in his effort to defend the neoclassical theory of value and distribution; he created a prime example of genuine scientific debate on the two sides of the Atlantic. The essence of this debate revolved around the fundamental principles governing the neoclassical theory of value and distribution as well as economic growth, which depend on an aggregate production function whose inputs (mainly capital and labor) enter into production prior to the determination of their price, rate of profit and wage, respectively.6 In dealing with the above riddle, Samuelson (1962) proposed a parable economy producing only two goods (a consumer and a capital good), which employs the same technique for the production of both goods. In equilibrium, the price of the consumer good should be equal to its production cost: PC = wLC + rPK K C

(3.4)

where PC is the consumer good’s price, w is the wage rate, and LC and KC stand for the labor employed and capital invested in the production of the

40

Capital theory controversies

consumer good, respectively. PK is the equilibrium price of the capital good equal to PK = wL K + rPK K K

(3.5)

Given that the subscript K refers to capital goods, the rest of the symbols of the above equation bear the exact same meaning as in the equation of consumer goods.7 Consequently, we end up with two equations which include the wage cost, wLC  and wL K , and profits, rPK K C  and rPK K K of each sector. Such a simplification does not affect our theoretical results, in any qualitative way, and we can form a system of two simultaneous equations with three unknowns, the two prices, PC , PK and the rate of profit, r. For a unique and, therefore, economically meaningful solution, the price of the consumer good (equation 3.4) is selected as the numéraire (or a third equation PC = 1 is being added) and the system can be rewritten as 1 = wLC + rPK K K PK = wL K + rPK K K Solving the second equation for the price of capital good PK, we get PK =

wL K 1− rK K

(3.6)

Substituting PK in equation 3.4, we get ˝ wL K ˇ 1 = wLC + rK C ˆ  ˙ 1− rK K ˘ or 1− rK C = wLC (1− rK K ) + rK C wL K And solving for w, we get the WRP curve w=

1− rK K 1− rK K   = LC (1− rK K ) + rK C L K LC + r ( K C L K − K K LC )

(3.7)

which is a quadratic equation. Positing the constraint that the capital–labor ratio is the same in both sectors, that is K C / LC − K K / L K = K C L K − K K LC = 0

(3.8)

the following linear equation is derived: w=

1− rK K LC

(3.9)

Capital theory controversies 41

whose plot is displayed in Figure 3.2a. The maximum real wage, w max (when r = 0) is equal to 1/ LC , which is also the productivity of labor, that is the real value-added per unit of labor input. The maximum rate of profit, rmax (when w = 0) is 1/ K K , the real value added per unit of capital input or “productivity” of capital. This relation can be generalized for a variety of techniques, where each and every one is characterized by a different capital–labor ratio measured by the slope of each of the techniques. Theoretically, there are many possible techniques that can be expressed through the following relation: w = f (r ) =

1− rK L

(3.10).

From equation 3.10, we have removed the subscripts indicating the characterization of goods, ending up with a one-commodity economy since we assume the same capital–labor ratio for the production of both types of goods. Let us now assume that for the production of a good, two techniques, A and B, are employed (Figure 3.2b); further suppose that the above techniques A and B intersect at point C. For r < r1 technique A is selected; that is, with given the rate of profit, the technique with the higher real wage is selected or, what is the same, with given the real wage the technique with the higher rate of profit is selected. Technique A is associated with a higher capital–labor ratio, indicating that capital is abundant and so the technique with the less scarce factor is chosen. The exact opposite will occur if r > r1. Technique B with a lower capital–labor ratio will be selected as the more profitable one. w

w 1/Lc=wmax

Technique A

w1

C Technique B

˜ r max

r1

r

(a)

Figure 3.2 WRP curves and their WRP frontier.

r (b)

42 Capital theory controversies

The idea is that capitalists always select the technique with the higher rate of profit, or what is the same as the technique with the minimum cost and therefore lower price. This entails higher real wage, which is the side effect of the competitive pressure and not of capitalists’ pro-labor sentimentality. As a rule of thumb, the technique farther out from the origin of the graph, for any given real wage, is associated with the higher rate of profit and so is selected. Alternatively, at any given rate of profit the technique that lies farther out from the origin of the graph is characterized by the lower unit cost or price and, therefore, is preferred over the other ones. In our example above, we utilized two techniques but one can hypothesize many techniques like the four (A, B, C and D) displayed in Figure 3.3. Clearly, the choice of technique takes place on the curve that lies farther out of the origin and with the aid of these techniques we may construct an envelope encompassing all available techniques (curves) and, in so doing, it becomes the WRP frontier. The WRP frontier is formed by the outer segments of the WRP curves and represents the locus of points of wage–profit rate combinations derived from selecting the optimal segments of the available techniques. The slope of each and every WRP curve is derived by differentiating the wage rate with respect to the rate of profit: f ˛ (r ) =

dw K = − = constant  dr L

(3.11)

that is, the WRP curves are linear with negative slope equal to the capital– labor ratio. Hence, by assuming a uniform capital–labor ratio across all industries, the relation between the rate of profit and the wage rate becomes linear.

w A

B

C D

r Figure 3.3 Derivation of the WRP frontier.

Capital theory controversies 43

This linear and negative relation depicted in WRP curves and their frontier gives rise to the dual presentation of the familiar from the standard microeconomics textbooks as the isocost and isoquant curves, as shown in Figure 3.4 whose left-hand-side (l.h.s.) panel depicts labor on the vertical axis and capital on the horizontal axis. The isoquant curve of the same figure represents the locus of points of different combinations of capital and labor producing a given quantity of output. The tangents to this curve represent the price ratios of the two factors of production, which in the case of labor and capital are the wage and profit rates, respectively. The tangent of the angle ˜ of the WRP curves, displayed on the right-hand side (r.h.s.) panel of Figure 3.4, represents also the capital–labor ratio. L

w

dw =− d

= constant slope

r (a)

(b)

Figure 3.4 Isocost–isoquant and the WRP curve.

The above relations between isoquants and the WRP frontiers as well as between isocosts and WRP curves are extended by the introduction of a third or more techniques as shown on the top left panel of Figure 3.5; the lower left panel of the same figure represents the value of capital for each of the three techniques. Technique A shown on the upper part of the l.h.s. panel of Figure 3.5 has the highest capital–labor ratio and thus the lower rate of profit as it is indicated in the lower left panel. Continuing with technique B, we observe that the tangent of its WRP curve is lower than A’s and the rate of profit associated with it, with the given real wage, is higher. These relationships are repeated in the shape of technique C characterized by an even lower capital–labor ratio and, therefore, a higher rate of profit. The results are absolutely consistent with the requirements of the neoclassical theory of scarcity prices, since the scarcer (more abundant) the capital, the higher (lower) its price. At switch points, the two techniques share the same price for the produced output and so capitalists are indifferent as to which technique to

44 Capital theory controversies w

w WRP curve technique A

Infinite number of techniques and the WRP frontier

WRP curve technique B WRP curve technique C

r K

r K The demand schedule for capital

r

r

Figure 3.5 Different techniques and the demand schedule for capital.

choose. If the interest rate increases, technique B is preferred because, for a given real wage, a higher rate of profit than technique A is obtained or, similarly, for a given rate of profit, the capitalist chooses the technique with the higher real wage. Producers, due to competition, choose the technique that minimizes the cost of production or alternatively maximizes their profit and select points on the envelope formed by the outer segments of each WRP curves representing the three techniques; hence, profit-maximizing producers choose points on the envelope (the WRP frontier), which represent the optimal technological capabilities. Because by construction, the various techniques are represented by straight lines, only one intersection point exists for each pair of lines. Consequently, we derive the negative relationship between the value of capital (or its scarcity) and the rate of profit exactly as theorized by the neoclassical theory and shown on the lower left panel of Figure 3.5 with the step-like demand schedule for capital. If the number of techniques increases indefinitely, as is represented on the upper right panel of Figure 3.5, the relationship between capital and rate of profit tends to become the usual continuous demand schedule for capital commonly displayed in the macroeconomic textbooks and here is presented on the lower right panel of Figure 3.5. In Figure 3.6, we can put together the various techniques, not too many for visual clarity, and derive the demand schedules for capital and labor associated with these isoquants, the neoclassical mirror image of the WRP curves. A prerequisite for a well-behaved isoquant is the assumption of perfect substitutability, which allows the choice from many alternative combinations of

Capital theory controversies 45 w

w Technique A Technique B

Demand for labour

Technique C Technique D

L

r K

K

Demand for capital

Isoquant

r

(a)

L (b)

Figure 3.6 Derivation of isoquants through WRP curves.

factors of production. The latter makes possible the construction of a smooth WRP frontier and, therefore, well-behaved demand schedules for capital and labor giving rise to well-behaved production functions. These were the relationships that led Samuelson to the conclusion that if the capital–labor ratio is uniform across industries, or what amounts to the same thing as if we deal with a one-commodity world, the profit rate is determined by the relationship between the cost of production and demand for this single capital good. Moreover, the rate of profit and the value of capital are inversely related, a result that is fully consistent with the neoclassical theory, where the payments for the services of the factors of production ref lect their respective relative scarcity. It might be noticed in passing that it is ironic that Samuelson criticized Marx’s theory of value for its logical inconsistency, because labor values are equal to PP, if and only if, the capital–labor ratio (the value composition of capital or capital intensity) is the same across sectors (see Tsoulfidis and Tsaliki 2019, chs. 1 and 3). Samuelson in defending the neoclassical theory of value did not hesitate, although he was alerted against,8 to utilize in his analysis the unrealistic assumption of a uniform capital–labor ratio in the effort to rescue the neoclassical theory of value from problems of internal consistency. It is important to note though that Marx’s theory of value need not make such a counterfactual assumption to establish the consistency between DP and PP (see Tsoulfidis and Tsaliki 2019, ch. 3 and the literature cited therein).

46 Capital theory controversies

In short, the neoclassical theory of value and distribution not only is subjective as it is based on preferences, but more importantly, it exaggerates by assuming perfect substitutability between the factors of production. Both subjectivity and substitutability are exemplified in the shape of isoquant curves, which represent the extension and application of the ideas implied in the indifference curves while production is an extension of the pure exchange model through the indirect exchange. In production, costs are also subjective in that they ref lect the disutility from suffering by the individuals in parting with the services of their endowments in producing output. Hence, factors’ rewards are determined by their marginal physical productivity or what is the same, by their marginal contribution to production.9

3.5 From the one-commodity to the world of many commodities Samuelson claimed that the conclusions drawn from the analysis of a one-commodity world can be generalized to represent the operation of real economies in which there is production of a large variety of commodities. What I propose to do here is to show that a new concept, the ‘surrogate production function’, can provide some rationalization for the validity of the simple J. B. Clark parables which pretend there is a single thing called “capital” that can be put into a single production function and along with labor will produce total output. (Samuelson 1962) One is wondering for the reason(s) why the neoclassical economists delved into this discussion. The answer relates to the quintessence of neoclassical theory of value and distribution, that is, the consistency of the price of a commodity and its cost of production. Starting with the super simple case of the price in a one-commodity world, we write P = wL + rPK

(3.12)

If w = 1, then for each P > 0 corresponds to a positive profit rate, r > 0. Hence, in the case of a one-good economy, we find that the price of the good is determined by market forces and the size of the r is determined indirectly in the factor markets through the associated with it derived demand schedule for capital. If we have two commodities, free competition will lead to equalization of profit rates and the establishment of equilibrium relative prices. We should bear in mind that if two commodities share the same K / L , they are essentially the same commodity and the analysis actually refers to a one, all purposes, commodity world. If more commodities (industries) are introduced, more price ratios must be determined all of which should be consistent with a uniform rate of profit and different capital intensities. From

Capital theory controversies 47

the  above, the reasonable question to ask is the extent to which the analysis can be generalized to a multi-commodity economy with the employment of heterogeneous capital goods. The complications that may arise from the production of two or more commodities are examined starting first with the introduction of a second commodity and then continuing with further generalizations. The case of two commodities, which for their production require different capital–labor ratios, may be described by invoking the quadratic equation 3.7 of the WPR curve, which may be either convex or concave depending on the sign of the second derivative. The curve is convex, that is, looking upward with respect to the origin, if the second derivative of the above relation is positive while it is concave, that is, looking downward toward the origin, if the second derivative is negative. Consequently, we estimate the sign of the derivatives, which are necessary to state the conditions that must be fulfilled for the different shapes of the WRP curves. The first derivative of equation 3.7 with respect to (w.r.t.) profit rate will be dw (1− rK K )˙ (LC + rZ ) − (LC + rZ )˙ (1− rK K ) = dr (LC + rZ )2 −K K ((LC + rZ )) − Z (1− rK K ) = (LC + rZ )2 −  K K L K = (LC + rZ )2 where Z = K C L K − K K LC = K C / LC − K K / L K . The second derivative of the above will be 2 ˇ ˙ d 2w ˆ(LC + rZ ) ˘ K K L K = dr 2 (LC + rZ )4 2 (LC + rZ ) ZK K L K = (LC + rZ )4 2ZK K L K   = (LC + rZ )3

Since all the coefficients of the above relation are positive, it follows that the shape of the WRP curves depends only on the sign of the Z factor. In particular, KC K K d 2w > and 2 > 0, the WRP curve is convex. LC LK dr KC K K d 2w If Z < 0, that is, < and 2 < 0, the WRP curve is concave. LC LK dr KC K K d 2w If Z = 0, that is, = and 2 = 0, the WRP curve is linear. LC LK dr If Z > 0, that is,

48 Capital theory controversies

The third case refers to a one-commodity world, or of an economy with uniform capital–labor ratio, which we examined in the previous section. Figure 3.7 summarizes the respective shapes of the three possible WRP curves. r 1/Lc Z0

r 1/K

Figure 3.7 WRP curves of different shapes.

Our attention now focuses on an economy with two sectors and we compare the results with those derived from a one-all purposes commodity world. In Figure 3.8, we display a WRP curve of a two-commodities economy with a concave shape as the more interesting case because it indicates that as the rate of profit increases, the capital intensity increases as well. The slope of the concave WRP curve is equal to the tangent of angle ˜ for each particular rate of profit along the curve and going from left to right increases. The converse is true if the WRP curve is convex. The area below the WRP curve measures the per capita physical output. We distinguish the following cases: If r = 0, then all output goes to labor and therefore the real wage is at its maximum, w = w max . Clearly, the vertical intercept gives us the value added per capita or, what is the same thing, the productivity of labor. In the other extreme case, where w = 0, all output goes to capital and the rate of profit is at its maximum, r = rmax , and the horizontal intercept gives the “productivity” of capital or the output–capital ratio. The profit rate is estimated from r = 

profits ˜ ˜ /L ˛ = = = capital stock ° K / L K / L

where ˜ is profits per capita or unit of labor. Hence, the capital–labor ratio can be rewritten as K ˜ = L r

Capital theory controversies 49 w

wmax=1/ L Profits per capita =  ˜

Real wage = w

K

rmax =1/KK

r

Figure 3.8 A concave WRP curve with two goods.

which is equal to the tangent of angle ˜ −

K = tan ˜ L

(3.13)

However, unlike the case of a one-commodity world, we observe that the tangent of angle ˜ changes, as we move along the curve. This is equivalent to saying that the capital–labor ratio changes, whenever the distribution between wages and the rate of profit changes. In Figure 3.8, we observe that in the case of a concave WRP curve (Z < 0), if the rate of profit increases so does the capital–labor ratio, or what amounts to the same thing, the tangent of angle ˜ increases; result which prima facie contradicts the neoclassical theory according to which an inverse relationship between the two variables is expected. The idea is that prices in neoclassical economics are indexes of relative scarcity in factors of production; so as the rate of profit increases, one expects a pari pasu falling capital–labor ratio indicating the scarcity of capital and the abundance of labor. Hence, the so-called “Wicksell effect” is “perverse”, because the capital–labor ratio increases following the increase in the rate of profit and we witness the so-called “capital deepening paradox”. Capital controversies often refer to Wicksell’s price and real effects. The “Wicksell price effect” relates to changes in the value of capital caused by changes in distribution (between wages and profits) with a given technique. The “Wicksell real effect” refers to the changes in the value of capital caused not only by changes in distribution but also by changes in the technique.10 In reverse “capital deepening”, the capital–labor ratio rises (falls) and the rate of profit over the real wage ratio rises (falls) (Kurz and Salvadori 1995, pp.  120–124; Shaikh 2016, p. 437). In case that the WRP curve is convex (Z > 0), capital’s value falls as the rate of profit rises and Wicksell’s price effect is positive or “normal”; in that way, it enhances Wicksell’s real effect. Therefore, a convex WRP curve is in accordance to neoclassical premises, provided

50

Capital theory controversies w

r

Figure 3.9 WRP curves with many goods.

that we deal with a single technique. Finally, if the WRP curve is linear, the price effect completely vanishes and the rate of profit becomes equal to the marginal physical product of capital. We have shown that in a one-commodity world, two or more techniques are depicted with straight lines, which intersect with each other only once. By taking the outer segments of the WRP curves, we construct the factor price or WRP frontier as shown in Figure 3.3. However, in moving from the economy of one to the multi-commodity world, the many available techniques may be represented by WRP curves, as these are shown in Figure 3.9. The shapes of these WRP curves depend on the number of sectors. Mathematically speaking, a single sector model entails a WRP curve, which is a straight line, a two sector model entails one extreme at most while the inclusion of a third sector gives rise to two extremes and one inf lection point. Extending the analysis to four sectors model, we expect three extreme and two inf lection points, and so forth. The maximum number of curvatures in WRP curves will depend on the number of industries (see Garegnani 1970, p. 474). Consequently, moving from one technique to another, the value of capital and the rate of profit may display any possible and not necessarily the inverse relationship, as expected in the neoclassical theory. If the number of available techniques increases and there are many produced goods, then the argument about the uncertain and complex relationship between the relative scarcity of capital and the rate of profit is strengthened rather than weakened.

3.6 Reswitching of techniques and the demand for capital Having derived the WRP curves and their possible shapes, it follows that the demand schedules for capital might not necessarily be well-behaved as

Capital theory controversies 51

a result of reswitching of techniques. For the sake of brevity and clarity of presentation, we distinguish the following analytical properties associated with the possible shapes and locations of the WRP curves and the respective WRP frontier (for details and proofs of these properties, see Pasinetti 1977, pp. 158–159). – – – –

At the switch points between two techniques, the commodity sells at the same price. If, for a given rate of profit, one technique dominates another, then it will yield prices, in terms of the wage rate, that are strictly lower than those yielded by the other technique. The switch points are independent of the choice of numéraire. The technological frontier is strictly decreasing as the rate of profit increases.

In Figure 3.10, we represent an economy producing two goods, whose techniques are displayed in the shape of their capital–labor ratios. Let us suppose that one of these two techniques (technique A) displays a concave WRP curve, which makes it a candidate for the appearance of “perverse” Wicksell’s effects. Let us further suppose that the technique B is a straight or a convex line. Putting together these two WRP curves in Figure 3.10, we observe

w Technique A

Technique B

r

r1

0

r2

kA A kB B

B

k

Figure 3.10 Switching points and the demand schedule for capital.

52 Capital theory controversies

two switching points, which of course could not exist in the case of straight WRP curves of a one-commodity world. In addition, independently of dealing with a single or a multi-commodity world and in the presence of many alternative techniques of production, the cost-minimizing technique dictates the choice of points on the outer WRP curve. Let us now assume that the rate of profit is very low, lower than r1, which makes technique B to be preferred since it is characterized by a higher capital– labor ratio, k, than that of technique A. As the rate of profit increases and reaches the first intersection of the two techniques (first switching point at r 1), the two techniques become equally profitable; therefore, we are indifferent as to which technique to choose. However, if the rate of profit increases furthermore, the most profitable technique and, therefore, the preferred one is technique A; this technique is characterized by a lower capital–labor ratio (or the tangent of angle ˜ is lower for technique A rather than for technique B). In other words, as the rate of profit increases we select the technique with the lower capital–labor ratio (the discontinuity in the lower part of the Figure 3.10 is no problem and can be fixed, if we have more techniques). A capital–labor which keeps rising with the rate of profit until we reach the next intersection of switching point at r 2; and once again, we become indifferent whether to choose technique A or B. As the rate of profit increases, even more, we no longer become indifferent and switch, this time, to technique B, that is, a technique with the higher capital–labor ratio than that of technique A. That is, from the technique with the lower capital– labor ratio, we return back to the technique with the higher capital–labor ratio at a much higher rate of profit, which is absolutely inconsistent with the neoclassical theory of scarcity prices. A higher capital–labor is consistent with a lower not higher rate of profit. Theoretically speaking, one cannot rule out the case of many switching points, as shown in Figure 3.9.11 The importance of this result (reswitching of techniques) is that Samuelson’s (1962) parable of one-commodity world cannot be generalized to many commodities. Practically, this means that we simply cannot estimate, for example, the capital–labor ratios in constant or current prices and, based on these estimates, to derive some form of demand function for the overall economy. The reason is that, if the rate of profit is not equal to the marginal product of capital, we cannot derive a consistent demand schedule for capital goods and, by extension, we cannot derive basic tools of modern macroeconomic analysis, such as the IS curve. Furthermore, it is futile to estimate the scarcity of capital via another variable, such as, for example, saving by assuming it a well-behaved function of interest (profit) rate. Finally, if we cannot formulate the demand schedule for capital goods, we cannot define the demand for labor or almost every other schedule. In effect, the hypothesis of substitutability between the factors of production precludes the consistent construction of the demand curve of any factor of production. Hence, if the demand for capital, so central in the neoclassical economic analysis behaves “abnormally”, it imparts its abnormal behavior to other schedules rendering the analysis out of touch with the real economy (Garegnani 2012; Fratini 2019; Petri 2020).

Capital theory controversies 53

3.7 Summary and conclusions The neoclassical “transformation problem”, that is, the determination of the value of capital in a logically consistent way was discussed intensively mainly in the 1960s in the famous capital theory controversies. We showed that theoretically speaking, all the questions that were raised within this debate remain open issues for the neoclassical theory. The problem at hand is not the measurement of capital per se, but its definition in a consistent way with the requirements of the neoclassical theory of value and income distribution. The same issue does not appear in the classical approach, as one of the distributive variables, the real wage (or rarely, the rate of profit), is taken among the givens of their theoretical model. While we explained the neoclassical views on the requirements of the neoclassical well-behaved curves, the Sraffian economists have argued that the arrow of causality must run from the marginal productivities to factor payments and that this arrow of causality does not necessarily hold when the analysis includes produced means of production. Hence, the estimated curves are not well-behaved and give rise to reswitching such as a typical case described in Figure 3.10. One wonders under what circumstances does reswitching appear mathematically and how realistic is its probability of occurrence in the face of data from actual economies. This is equivalent to asking by how much does the capital–labor ratio or rather the capital intensity differ across sectors, and are the resulting differences large enough to create more than two switch points? This is an empirical question in the main and, as we will see in Chapter 4, there are significant differences in capital intensities across industries; nevertheless, these differences, whatever they might be, do not generate effects akin to those described in the capital theory debates. More specifically, the WRP curves are not far from straight or convex lines and, at the same time, rarely display the curvatures and switching points that are usually described by the Cambridge UK-side economists. However, the empirical results in and of themselves do not lend support to the one-commodity description of the economic world and the consistency problems within the neoclassical theory remain unresolved.

Appendix 3.A: Samuelson’s and Pasinetti’s reswitching examples It is important to stress at the outset that what characterized Marx’s and Ricardo’s numerical models is that their construction was based under extreme or unfavorable conditions for their examined hypotheses. For example, the famous 93% labor theory of value in Ricardo’s 2×2 model was based on the assumption that the capital–labor ratio in one sector was much higher (55 times in our slightly modified Ricardian example, see Tsoulfidis and Tsaliki 2019, ch. 1) than that of the labor-intensive sector with zero capital. Marx (1885) also in his schemes of simple reproduction (Capital II) assumes uniform

54 Capital theory controversies

capital intensity between sectors so as to conduct the analysis in terms of labor values and direct prices, the simplest possible monetary expression of value. In his next analytical stage, however, in the schemes of expanded reproduction, the capital intensity differs reasonably (twice higher in the department of capital goods). Hence, both Ricardo and Marx in their numerical example used the power of abstraction in the construction of their economic models and their analytical steps were based on the idea of the proof being under unfavorable for their argument circumstances. This methodological approach featuring in Ricardo’s and Marx’s analyses is not followed neither by Samuelson’s (1966) nor by Pasinetti’s (1966) numerical examples, as we show below. More specifically, Samuelson initially utilized an unrealistic description of the economy (one-commodity world), although he was informed about the problem and its consequences. Four years later, he returned to the question by stipulating a seemingly more realistic two commodities world admitting that reswitching is possible and the Cambridge UK side of the debate was right about the possibility of reswitching. Samuelson’s numerical model from which he derived reswitching is extreme and therefore unlikely to be representative of a real economy. Hence, he argued that reswitching is a rare case and therefore the usual production functions could be used to derive the well-behaved demand schedules for capital. Samuelson’s (1966) numerical example starts with the following two equations describing techniques A and B for the production of the same good (bottles of champagne), where PA stands for the price of technique A and PB the price of technique B and i is the common interest rate. Technique A : PA = 7 (1+ i )2 and Technique B : PB = 2 (1+ i )3 + 6 (1+ i ) Technique A employs seven units of labor, two periods before the bottle of champagne is made and no labor in other periods. Technique B requires two units of labor three periods before and six units of labor one period before the champagne is made. Table 3.A1 below displays estimates of prices of the two alternative methods of production. The last column gives the relative price or cost of production. At switch points, the commodity sells at the same price, that is the relative price is equal to one. In Table 3.A1, there are two such switch points, while the method of production that is chosen in the spectrum of interest (profit) rates is indicated in bold figures. In Figure 3.A1, we depict the relative price on the vertical axis against the interest rate on the horizontal axis.

Capital theory controversies 55 Table 3.A1 Samuelson 1966: relative price w.r.t. the interest rate Interest Rate 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 2

PA = 7 (1 + i )2

Technique A

PB = 2 (1 + i )3 + 6 (1 + i )

Technique B

Relative Price

7 8.47 10.08 11.83 13.72 15.75 17.92 20.23 22.68 25.27 28 30.87 33.88 37.03 40.32 43.75 47.32 51.03 54.88 58.87 63

8 9.262 10.656 12.194 13.888 15.75 17.792 20.026 22.464 25.118 28 31.122 34.496 38.134 42.048 46.25 50.752 55.566 60.704 66.178 72

1.142857 1.093506 1.057143 1.030769 1.012245 1 0.992857 0.989916 0.990476 0.993985 1 1.008163 1.018182 1.029814 1.042857 1.057143 1.072527 1.088889 1.106122 1.124138 1.142857

1.15

1.1

1.05

1

0.95

0

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

1

1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9

Figure 3.A1 Samuelson 1966: Relative price w.r.t. the interest rate.

2

56 Capital theory controversies

Clearly, the two techniques do not differ very much in terms of their price (cost), which is the result of differences of time and interest rate compounded. If prices are so close, variations in the interest rate may give rise to switching and reswitching as is the case of Samuelson’s (1966) example. Technique A is the dominant up until the interest rate approaches the 50%, from 50% until 100% technique B is chosen and to return to technique A for the interest rates higher than 100%! Clearly, we have a hyperbolic example because interest rates of 50% and 100% are by no means typical to the business world. Furthermore, the two prices, even though interest rates change so much (from 50% to 100%!), differ only by 1%! to make someone move from mainly one technique to the alternative. Pasinetti’s (1966) numerical example has the advantage over Samuelson’s in that the interest rate varies, as we will see, in a realistic (unlike Samuelson’s exotic) range. The two alternative techniques are described by the following WRP equations: Technique A :  W A =

1 − 0.8 (1+ i ) 20 (1+ i )8

and 1 − 0.8 (1+ i ) 24 + (1 + i )25 The techniques A and B are comparable since they share the same rate of interest (profit) and the wage is expressed in terms of the same physical commodity. Rational producers always choose the technique with the maximum profit or the lower cost of production (see Pasinetti 1966, p. 507). Let us now hypothesize that the two techniques entail different cost and, therefore, price for the production of the same good; hence, we can estimate their relative price assuming a uniform real wage equal to one, W A = WB = 1. Subsequently, we assign different interest rates and solve for the prices (costs) in both techniques. The results are displayed in Table 3.A2. The similarity with Samuelson’s results is striking. We observe that when the relative price is below one and this occurs for a rate of interest approximately equal to 3.7% and 16.3%, technique B is being used and technique A is chosen for all the other interest rates. Figure 3.A2 paints the shape of the path of relative price, which is similar to Samuelson’s that we examined above, but they are better in that the switch points are at realistic range of interest rates. This by no means renders the example typical and realistic for the modeling of actual economies. To the contrary, the two techniques do not differ dramatically over these realistic range of interest rates and one wonders if there is really a choice of techniques when prices differ so little. In effect, slight variations in interest rate is not possible to change the order of techniques provided that they are not too close in the first place. Technique B:  WB =

Capital theory controversies 57 Table 3.A2 Pasinetti 1966: Relative price w.r.t. the rate of profit Interest Technique A Technique B Relative Rate PA = 20 (1+ i )8 + 0.8 (1+ i ) − 1 PB = (1 + i )25 + 0.8 (1 + i ) + 23 Price 0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1 0.11 0.12 0.13 0.14 0.15 0.16 0.17 0.18 0.19 0.2

19.8 21.46513 23.24919 25.1594 27.20338 29.38911 31.72496 34.21972 36.8826 39.72325 42.75178 45.97876 49.41526 53.07288 56.96373 61.10046 65.4963 70.16507 75.12118 80.37971 85.95634

24.8 25.09043 25.45661 25.91778 26.49784 27.22635 28.13987 29.28343 30.71248 32.49508 34.71471 37.47346 40.89606 45.13454 50.37392 56.83895 64.80224 74.59383 86.61263 101.3401 119.3562

1.252525 1.168892 1.094946 1.030143 0.974064 0.92641 0.886995 0.855747 0.832709 0.818037 0.812006 0.815017 0.8276 0.850426 0.884316 0.930254 0.989403 1.063119 1.152972 1.260767 1.388568

1.4

1.3

1.2

1.1

1

0.9

0.8

0

0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1 0.11 0.12 0.13 0.14 0.15 0.16 0.17 0.18 0.19 0.2

Figure 3.A2 Pasinetti 1966: Relative price w.r.t. the rate of profit.

58 Capital theory controversies

A very similar view is obtained by deriving the WRP curves, whose estimation is in the Table 3.A3. In Figure 3.A3, we display the WRP curves. Clearly, there are switch points and so the neoclassical view about scarcity prices and selection of techniques according to their cost does not necessarily hold. In Pasinetti’s words, […] the curves representing WA and W B intersect each other three times. There are three distinct levels of the rate of [interest], namely 3.6 per cent, 16.2 per cent, and 25 per cent, at which WA = W B , i.e., at which the two technologies are equally profitable. These three points of intersection correspond to the switching from one technology to the other as the rate of [interest] is increased from zero to its maximum. (Pasinetti 1966, p. 507) However, once again, one may cast doubt to numerical examples like the above because of its unstable nature meaning that, with a slight change in the parameters, the switch points do not really exist let alone its lack of realism. Table 3.A3 Pasinetti 1966: WRP curves Interest Rate

0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1 0.11 0.12 0.13 0.14 0.15 0.16 0.17 0.18 0.19 0.2 0.21 0.22 0.23 0.24 0.25

Technique A

Technique B

Difference between WRPs

0.008865439 0.007852111 0.006946801 0.006137798 0.005414715 0.004768334 0.004190466 0.003673828 0.003211944 0.002799044 0.002429988 0.002100193 0.001805567 0.00154246 0.001307607 0.001098092 0.000911304 0.000744907 0.000596809 0.000465136 0.000348207 0.000244513 0.000152704 7.15627E-05 0

0.007594 0.007176 0.006745 0.0063 0.005842 0.005373 0.004893 0.004409 0.003924 0.003445 0.00298 0.002537 0.002122 0.001744 0.001406 0.00111 0.000857 0.000646 0.000473 0.000335 0.000226 0.000143 7.97E-05 3.33E-05 0

0.001271 0.000676 0.000202 ‒0.00016 ‒0.00043 ‒0.0006 ‒0.0007 ‒0.00073 ‒0.00071 ‒0.00065 ‒0.00055 ‒0.00044 ‒0.00032 ‒0.0002 ‒9.8E-05 ‒1.2E-05 5.41E-05 9.88E-05 0.000123 0.00013 0.000122 0.000102 7.3E-05 3.83E-05 0

1 − 0.8 (1+ i ) 1 − 0.8 (1+ i ) WA = WB = 8 20 (1+ i ) 24 + (1 + i )25

Capital theory controversies 59 0.012

w 0.01

0.008

0.006

0.004

0.002

0

0

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

0.18

0.2

0.22

0.24

r

0.26

Figure 3.A3 Pasinetti 1966: WRP curves.

In evaluating these two numerical examples, which display reswitching, we observe that they are both based on assumptions not in conduct to the reality of the possible structure of technologies and how technologies appear and firms are compelled, upon extinction, to “choose”. Regarding Pasinetti’s description of technology, we observe that only employment is different while the technological coefficients are represented by two sparse matrices. Technique A is described by an extreme example of 9×9 matrix whose first element is 0.8, the other elements on the main diagonal are ones and the remaining cells of the matrix are filled with zeros. Zeros are also the eight first elements of the row vector of employment coefficients while the ninth element is equal to 20. The alternative Technique B is also described by a nearly identity 25×25 matrix with the exception of its first element, which happens to be equal again to 0.8. Hence, the two “alternative” technology matrices are not alternative at all, since they share the exact same maximal eigenvalue and have equal to one all of the remaining eigenvalues. The techniques differ in the row vector of employment coefficients with technique B, whose first element is 24 the last is one and all the others are zero. The above description of technologies is particularly sensitive to parameters selected and there is no particular reason to hypothesize the first element of the above matrix equal to 0.8, whose slight deviation to the upward or downward direction may not give rise to reswitching of techniques. Reswitching is restored, if we suppose uniform parameter reductions, although the shapes of the curves are not so emphatic. In general, the results are not robust to different parameter values.

60 Capital theory controversies

Notes 1 The empirical research lends support to the view that wage differentials ref lect differences in skills (Botwinick 1993; Mokre and Rehm 2020). 2 For more about the intertemporal approach of the neoclassical economics, see Garegnani (2012), Eatwell (2019), Petri (2020) and the literature cited therein. 3 According to the law of diminishing returns, the more extensive use of an input implies lower marginal product, holding constant all the other factors of production. 4 The idea of substitutability is specific to the neoclassical theory; by contrast, the classical theory does not recognize substitutability of any degree other than zero, which is equivalent to saying that the isoquant curve forms a 90 degrees right angle stressing the fact of a single optimizing technique of production. 5 Here, capital is treated like a bond with infinite lifetime; that is, an asset whose net income, for simplicity reasons, remains the same year after year forever. 6 For a comprehensive survey of the CCC, the reader may refer to Harcourt (1972) and Cohen and Harcourt (2003). 7 For reasons of simplicity and clarity of presentation, we assume away the cost of circulating capital as well as of depreciation expenses; hence, they are assumed equal to 1 (see Garegnani 1970). 8 Garegnani in the seminar at MIT taught by Samuelson had commended about the problem and the consequences of the assumption of a one-commodity world. Samuelson (1962) acknowledged the comment by Garegnani, nevertheless downplayed its importance. 9 For the subjective nature of the neoclassical theory, see Tsoulfidis (2010, ch. 7, 2017). 10 Ricardo and Marx were aware of these feedback effects but downplayed their quantitative importance. Their intuition was their only available option in the lack of empirical data and advanced computational methods. 11 In Appendix 3A, we present the case of reswitching of techniques reproducing the rather exotic numerical examples by Samuelson (1966) and Pasinetti (1966).

4

Price trajectories and the rate of profit

4.1 Introduction The Cambridge capital controversies (CCC) were predominantly about the consistency of the measurement of capital with the theoretical requirements of the neoclassical theory of value and distribution. The focus of the debate was on the use of capital in production functions and the extent to which the marginal productivities of factors of production are equal to their respective payments. As a consequence, the CCC were more about the precise shape of the wage rate of profit (WRP) curves and whether or not they cross each other once, twice or even more, as we discussed in Chapter 3. The idea is that the WRP curves and frontiers are directly related to the neoclassical isoquants and isocosts through which the demand schedules for capital and labor as well as the marginal productivities of the respective factors of production are derived in accordance with the neoclassical theory of value and distribution. Lurking behind these debates, however, is the movement of relative prices and how they are affected by changes in income distribution. The economists of the Cambridge, UK side, had already adopted the Sraffian version of the classical theory of value and distribution according to which equilibrium prices may be determined objectively. More specifically, the relative equilibrium prices and the rate of profit can be determined with givens: the technology as described by the matrix of technological and the vector of employment coefficients, the real wage or the rate of profit and the level of output with its distribution among industries. It is important to note that the above givens of the classical theory of value and distribution are objectively measured and their estimates are provided on a regular and systematic basis by the official national statistical services and international institutions such as the OECD, the World Input-Output Database, WIOD, Timmer et al. (2015), among others. The Sraffian economists thought that such an objective determination of relative prices is superior to the subjective neoclassical approach based on given preferences, endowment and technological alternative and soon, the neoclassicals would follow the Sraffian variant of the classical approach

62

Price trajectories and rate of profit

seemingly “free” from the disturbing social consequences of the labor theory of value. According to the Sraffian strand of the classical approach, there is no need to refer to labor values or to their monetary expression, direct prices (DP), because DP do not offer anything more other than a nuance and unnecessary ideological, in the main, objections.1 Moreover, Sraffians thought that once their theory of prices is established, the second step would be its integration with Keynes’s theory of effective demand into a unified theory for the determination of the equilibrium level of output and the employment associated with it. Such a project has been the “holy grail” of the Sraffians in the 1980s and later, but little progress until today has been made toward this line of research (Tsoulfidis 2017). A further issue in the movement of relative prices, according to Sraffa and his followers, is that one cannot theorize their particular direction induced by changes in income distribution. We may recall that Ricardo and Marx argued that the movement in prices is predetermined according to the capital intensity of an industry as measured by the capital–labor ratio or rather the capital–wage ratio (the value composition of capital). Ricardo’s numerical examples were usually of the production of two commodities with wide differences in capital intensities, whereas Marx’s examples usually involve the production of two or more commodities and the movement of relative prices is theorized on the basis of meaningfully estimated economy’s average capital intensity. Sraffa (1960) argued that the changes in income distribution generate such complex feedback effects that make the price paths impossible to predict and such a movement in prices may change even the capital intensities of industries; one industry’s capital intensity may be higher than the average at one income distribution and at another below the newly formulated average. This novel development concerning the movement in prices became the foundation of the critique of the Cambridge, UK side economists as the WRP curves are derived and explained based on such movements in relative prices. While theoretically, the feedback effects may lead prices toward different directions, the empirical findings do not support Sraffa’s conjecture of price movements. Even when the feedback effects are strong, they do not change the capital intensity of industries in any qualitatively different way. Exceptions may apply chief ly in industries, whose capital intensity is too close to the economy’s average in the first place. This is the reason why in this chapter we present the recent developments on price trajectories and the WRP curves and frontiers are discussed in Chapter 5. The remainder of the chapter is structured as follows. Section 4.2 deals with the theoretical developments on the movement of prices induced by changes in income distribution. Section 4.3 is an application of the theoretical issues by introducing a fixed capital model based on data of the most recent input–output table of the USA for the year 2018; hence, the matrices of depreciation and fixed capital stock are constructed based on the last available capital f low table of the year 1997. The results of the fixed capital model are then contrasted to those derived from the circulating capital model. Section

Price trajectories and rate of profit

63

4.4 repeats the estimations using the input–output data from the WIOD of the year 2014 (Timmer et al. 2015) deriving very similar results with the difference that the matrix of capital stock coefficients is constructed using a simple, albeit not simplistic, methodology. Section 4.5 summarizes the empirical findings and makes some concluding remarks connected to the next chapter on WRP curves.

4.2 Prices of production and their determinants In both, the classical and neoclassical analysis, the centrality of the effects of income distribution on relative prices is the litmus test for the internal consistency of their theories. Classical economists paid particular attention to the issue of relative prices and their movement. Starting with the classical approach, we invoke Ricardo’s “fundamental theorem of distribution”, that is, the inverse relationship between profits and wages, as this can be judged from the following quotations from the Principles the proportion which might be paid for wages is of the utmost importance in the question of profits; for it must at once be seen that profits would be high or low, exactly in proportion as wages were low or high. (Works I, p. 27) […] in proportion then as wages rose, would profits fall. (Works I, p. 111) Profits depend on high or low wages. (Works I, p. 119) In the classical approach, the effect of changes in distribution on relative prices depends chief ly on the capital intensity and the price elasticity with respect to (w.r.t.) the distributive variables. This particular relative price elasticity w.r.t. the wage or profit rates is highly inelastic (less than 0.1 in Ricardo’s numerical examples), which is another way to say that the effect of changes in distribution on relative prices is not only very small but, in general, theoretically predictable (see Tsoulfidis and Tsaliki 2019, ch. 1). In effect, the presence of capital and distribution variables does not lead to significant deviations of the resulting prices from their respective labor times. Marx’s (1894) position was remarkably similar, as this can be judged by the numerical examples that he utilized in Capital III (ch. 9). There is no doubt that both Ricardo and Marx argued that the changes in relative prices brought about by changes in income distribution are both small and predictable, in general. Moreover, in classical analysis, and in particular in Ricardo (Works I, pp. 30–43) and Marx (Capital III, ch. 11), the sign and size of changes in relative prices, as an effect of changes in income distribution depend decisively on the capital intensity relative to the economy-wide average. The argument usually

64 Price trajectories and rate of profit

starts by hypothesizing a change in the wage rate across industries, which leads to varying rates of profit and then prices move up or down so as to restore equality in the rates of profit across industries. These typical changes in prices of production (PP) are portrayed in Figure 4.1 along with DP, the monetary expression of labor values, which remain invariant to changes in distribution as shown by the relative rate of profit, ρ. The latter is essentially the share of profits in income as this is shown below:

˜=

r ° /K ° profits = =  = R Y /K Y income

where r is the economy-wide average rate of profit, R is the maximum rate of profit which we find to be no different from the Sraffian standard ratio, Π stands for total profits, Y is the value-added, and K is the fixed capital stock. In Figure 4.1, the PP and DP are shown on the vertical axis while on the horizontal axis is the relative rate of profit, ˜ = rR −1, which takes on values ranging between zero (minimum) and one (maximum). For ρ = 0, PP and DP are equal to each other. For example, if the wage rate falls, the capital-intensive industries’ profits increase, but by less compared with the gains in profits of the labor-intensive ones. This inequality in profit rates is a disequilibrium situation and, therefore, it cannot last; capital accumulation increases the supply in labor-intensive industries and their prices fall restoring the displaced equilibrium. By contrast, the supply in the capital-intensive industries decreases and their prices increase so as to restore equilibrium. DP, PP PP in capital-intensive industries

DP

PP in labor-intensive industries

0

max

=1

Figure 4.1 Typical trajectories of prices and relative rate of profit.

Price trajectories and rate of profit

65

Lastly, if an industry displays the economy’s average composition of capital, its DP will be equal to its PP, because the gains in profits will be equal with the other industries and so there is no need for changes in relative prices. In other words, the industry is neutral to changes in income redistribution. Both Ricardo and Marx shared the idea of monotonicity in price changes in the upward or downward direction. Sraffa (1960, pp. 37–38), on the other hand, argued that prices are expected to show by far more complex trajectories than those hypothesized by Ricardo and Marx in their usual and simple (2×2 or 3×3 industries) numerical examples. The idea is that in actual economies of many interdependent industries as prices change, consequent upon changes in income distribution, give rise to various complex feedback effects making exceedingly more difficult to anticipate their exact direction. The reason is that theoretically, at least, the development of complex feedback price effects leads to the revaluation of capital, and an industry starting as labor-intensive (mathematically speaking) may end up as capital intensive and vice versa. Therefore, according to Sraffa, we cannot a priori determine the direction of the movements in the new prices brought about by changes in income distribution; and, therefore, we cannot a priori characterize an industry as capital- or labor-intensive. Hence, the relation between income distribution and relative prices is non-linear and, therefore, there is uncertainty in the prediction of the particular direction of prices and the definitive characterization of an industry as capital- or labor-intensive (Pasinetti 1977, ch. 5). The hitherto literature has repeatedly shown that the movement of PP trajectories in the fixed capital model is monotonic in the upward or downward direction and displays nearly no curvature. By contrast, in the circulating capital model, the movement of PP is, in general, monotonic but with slightly curved lines indicating accelerating or decelerating behavior. The case of PP to cross the line of equality with the DP is not excluded; however, this occurs rarely in the fixed capital model and only in a few cases in the circulating capital model, as the empirical research of many authors has shown. In the recent years, further research on this issue arrived at interesting results that lend support to new interpretations that we discuss in Chapters 6 and 7. 4.2.1 The mathematics of the linear model of production In the interest of brevity, we hypothesize a fixed capital model with depreciation. In our estimations, we excluded the wages advanced because such capital is relatively very small and usually is obtained by businesses through their available credit lines and the same is true, to a lesser extent, with the materials advanced which require turnover times.2 So we may write π = wl + πA + πD +  rπK

(4.1)

where upper-case bold-faced letters refer to square matrices, lower-case bold-faced letters refer to vectors of dimensions conformable to pre- or

66

Price trajectories and rate of profit

post-multiplication by matrices; finally, scalars are indicated by lower-case letters in italics. The notation in Equation 4.1 is as follows, π = 1×n vector of relative PP defined up to multiplication by a scalar λ = 1×n vector of prices of unit values l = 1×n vector of employment coefficients A = n×n matrix of input–output coefficients D = n×n matrix of depreciation coefficients I = n×n identity matrix of the same dimensions with the matrix A K = n×n matrix of fixed capital coefficients e = 1×n vector of ones or market prices. This vector also serves as the summation vector x = n×1 vector of gross output r = the economy-wide rate of profit and w = the wage rate After some manipulation of Equation 4.1, we arrive at the equation of relative PP, which need to be normalized. Thus, we have π = wl [ I − A − D]−1 + rπK [ I − A − D]−1 or π = w 1 [ I − A − D]−1 + rπ K [ I − A − D]−1 ˜˛˛°˛˛ ˝ ˜˛˛°˛˛˝ ˆ

H

or π = w˜ + rπH

(4.2)

It is important to note that the vector of unit labor values, λ, is normalized through its multiplication by the monetary expression of labor time (MELT), which in our case is the ratio of the column vector of gross output, x, evaluated at market prices (ex) over the same gross output evaluated in unit labor values λx. Thus, the monetary expression of labor values called “direct prices” (Shaikh 1977) and symbolized by v are defined as follows: ˛ ex ˆ v = ˜˙ ˘ ˝ ˜x ˇ MELT

We post-multiply Equation 4.2 by the standard commodity, σ, which is derived from the right-hand-side (r.h.s.) eigenvector of the matrix of vertically integrated capital coefficients H (see Shaikh 1998):

˜ = RH˜

(4.3)

where R = 1/ ˜max is the maximum rate of profit or the reciprocal of capital– output ratio derived from the maximal eigenvalue, ˜max, of Equation 4.3 and

Price trajectories and rate of profit  67

σ the column vector of output proportions corresponding to the maximal eigenvalue. The standard proportions or standard commodity must be normalized when multiplied by the ratio of gross output evaluated in market prices (MP) over the DP multiplied by the output proportions such that  ex  s =σ    vσ  and in so doing, we establish the following equality (see Shaikh 1998): sx = vx = ex The next step is to fix the relative prices by the normalized standard commodity, s, and derive the normalized row vector of PP, p,  ex  p = π   πs  with the property ps = vs = ex So, Equation 4.2 can be rewritten as p = wv + rpH (4.4) We post-multiply 4.4 by the normalized standard commodity, and we get ps = wvs + rpHs It follows that vs = wvs + rR −1vs We divide through by vs, and we end up with 1 = w + rR −1 which solves for the linear WRP curve w = 1 − rR −1 = 1 − ρ (4.5) where ρ ≡ r / R , with 0 ≤  ρ  ≤ 1.3 By taking into account Equation 4.4, we arrive at −1

p = (1 − ρ ) v [ I − HRρ ] (4.6) We know that the maximum eigenvalue of the matrix HR equals to one and, therefore, the matrix HRρ has an eigenvalue less than one, which makes it a convergent matrix. Hence, the matrix H represents the vertically integrated fixed capital coefficients and, if data are available, could include the circulating (besides the fixed) capital advanced, that is, both materials and wages. The lack of data, but also the simplicity of presentation, limits our exposition to matrix of fixed capital coefficients. In the case of the circulating capital

68  Price trajectories and rate of profit

model, the matrix H consists of the intermediate inputs per unit of output, A, as well as the wage goods input per unit of output, bl, or H = ( A + bl ) [ I − A − D]−1 We post-multiply Equation 4.6 by the inverse of the diagonal matrix of DP, v, and we get p v

−1

= pH v

−1

Rρ + e (1 – ρ ) (4.7)

–1

where p denotes the ratio of PP to labor values or DP. Equation 4.7 can be restated in the case of a single industry j, as follows:   pH   p  – R −1   (4.8)   – 1 = Rρ   v j  v  j  where H j is the j-th column of matrix H, pH j / v j is the so-called capital intensity of the vertically integrated industry producing commodity j, and R −1 is the capital intensity of the standard system (which is independent of prices and distribution). Finally, the derivative of Equation 4.8 w.r.t. ρ gives  p d   v j dρ or  p d   v j dρ

=

  d   pH   – R −1  + 1 Rρ  dρ   v  j  

 dk j  = R ρ + k j − R −1   dρ 

(

)

where k j ≡ ( pH /v ) j. We factor out k j from the bracketed term, and we get    k j − R −1  dk j ρ   (4.9)  +    k dρ k j j   = Rk j     dρ Ricardo-Marx Effect  Sraffian   or    Wicksellian Effect  The change in PP/DP w.r.t. ρ depends on the elasticity of industry’s j capital intensity w.r.t. the relative rate of profit, ρ (the first bracketed term of Equation 4.9), which can be positive, negative or even zero; this term might be called “Sraffian or Wicksellian effect”. According to Ricardo and Marx, the sign of change in PP/DP w.r.t. ρ depends mainly on the second bracketed term, which for obvious reasons, we call the “Ricardo-Marx effect”. Hence, relative prices move according to industry’s capital intensity relative to the economy’s average.4 Sraffa’s great contribution is the introduction and the accounting of the complex price effects, which may not let the Ricardo-Marx effect to play  p d   v j

Price trajectories and rate of profit

69

its dominant role. It is possible, therefore, the Sraffian effect to enhance, lessen or even supersede the Ricardo-Marx effect for particularly high relative rates of profit. These possibilities become extremely important in the capital theory controversies, whose theoretical implications were presented in Chapter 3, while their empirical dimensions are discussed in the next sections. For the two extreme and hypothetical relative rates of profit, Equation 4.9 becomes for ˜ = 0 ˝ pˇ dˆ  ˙ v ˘j = R k j (0) − R −1  d˜

(4.10)

and for ˜ = 1 ˛ pˆ ˙ ˘ = Rk j (1) ˝ v ˇj

(4.11)

From Equations 4.10 and 4.11, we may conclude the following: 1

2

3 4

If the Ricardo-Marx effect is positive, then the capital intensity of industry j is higher than the standard ratio, while the Sraffian effect can be positive, negative or even zero. If it is positive, it may strengthen the upward direction of the price movement; however, if the elasticity is negative, then the outcome depends on the relative strength of these two effects. The elasticity of capital intensity w.r.t. ˜ is very close to zero (especially for low values of ˜) and so the likelihood for the change in the direction of price movement for realistic ˜ depends on how close is the industry’s capital intensity, k j , to the standard capital intensity, R −1. The capital intensities of industries in the fixed capital model are usually quite distant from the standard capital intensity; therefore, a counteractive elasticity effect cannot make anything quite different. The situation becomes less rigid, when we move to circulating capital models, where the elasticity of capital intensity w.r.t. ˜ remains low, but the differences in capital intensities of industries relative to the standard capital intensity are much smaller than those of the fixed capital model. As a consequence, we cannot rule out the case that a change in ˜ may give rise to an elasticity of capital intensity with a sign opposite to that of the Ricardo-Marx effect; furthermore, the value of this elasticity may exceed that of the Ricardo-Marx effect and thus may even overturn the direction of the price trajectory displaying extremes. Thus, one may not exclude results, which are opposite to those theorized by Ricardo and Marx and for low values of ˜, a vertically integrated industry may start as capital- (labor-) intensive industry relative to the standard ratio, R, but as ˜ increases, it may be transformed to a labor- (capital-) intensive industry.

70

Price trajectories and rate of profit

From the above discussion, it follows that the claims of Sraffa (1960) and his followers for the development of quite complex price-feedback effects because of income redistribution that change the characterization of an industry is most likely an exaggeration. It goes without saying that, to a great extent, this is an empirical question, which we explore in the sections below.

4.3 Price paths using input–output data, BEA 2018 In what follows, our focus is mainly empirical, and it is restricted to showing the results of relevant analyses of the U.S. economy for the year 2018, the most recent input–output data released from the Bureau of Economic Analysis (BEA, www.bea.gov/data/economic-accounts/industry). We selected this particular input–output data and year because it has not been tested yet and, in our view, it is detailed enough for conducting the relevant analysis in both the circulating and fixed capital models. 4.3.1 The fixed capital model, BEA 2018 We start first with the case of a fixed capital model, which is a more realistic representation of the economy, and subsequently, we continue with a circulating capital model utilizing data of the U.S. economy of the year 2018.5 It is important to point out that in the creation of the matrices of depreciation and capital stock coefficients, we utilized the capital f lows (investment) matrix of 1997 of 65×65 dimensions, the last capital f lows table published by the BEA. Although this matrix was constructed 20 years earlier than the input–output matrix of 2018, we feel that a more recent capital f lows table would not be different in its characteristics. For the details of the construction of the matrix of fixed capital coefficients, see Appendix 4.A. Before we start with the plotting of trajectories of PP relative to DP, it is important to show the proximity of DP and PP to each other and to MP using both circulating and fixed capital models. The first column of Table 4.1 presents the DP, while the next two columns present the PP for both models. The measures of price deviations are shown at the end of Table 4.1. The Mean Absolute Weighted Deviation (MAWD) is computed as the absolute difference of estimated prices relative to the MP, which are by definition equal to one (million dollars worth of output, Miller and Blair 2009, ch. 2) multiplied by the weight of each industry’s output relative to the economy’s total. In the same spirit and independent of the chosen numéraire metric of deviation is the d − statistic = 2 (1 − cos ˜ ) , where the cosine of ˜ is equal to the arcsine of the tangent of the two vectors estimated by their coefficient of variation (Steedman and Tomkins 1997). Both statistics show reasonable deviations supportive of the proximity of the estimated prices from the MP (Mariolis and Tsoulfidis 2010). The last two columns of Table 4.1 stand for the capital intensities of industries in both circulating and fixed capital models. The estimations of

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34

Farms Forestry, fishing and related activities Oil and gas extraction Mining, except oil and gas Support activities for mining Utilities Construction Wood products Nonmetallic mineral products Primary metals Fabricated metal products Machinery Computer and electronic products Electrical equipment, appliances and components Motor vehicles, bodies and trailers and parts Other transportation equipment Furniture and related products Miscellaneous manufacturing Food and beverage and tobacco products Textile mills and textile product mills Apparel and leather and allied products Paper products Printing and related support activities Petroleum and coal products Chemical products Plastics and rubber products Wholesale trade Retail trade Air transportation Rail transportation Water transportation Truck transportation Transit and ground passenger transportation Pipeline transportation

Industries

0.834 1.186 0.759 0.857 1.057 0.725 0.978 1.000 0.927 0.970 1.041 1.068 1.137 0.991 1.009 0.972 1.080 1.025 0.876 1.011 1.250 0.985 1.032 0.669 0.817 0.980 0.845 1.009 0.864 0.918 1.076 1.033 0.879 0.563

DP

0.968 1.103 0.804 0.893 1.024 0.726 0.959 1.060 0.954 1.102 1.094 1.123 1.023 1.014 1.202 0.993 1.122 1.017 1.024 1.084 1.239 1.087 1.032 0.792 0.885 1.050 0.804 0.976 0.838 0.919 1.139 1.037 0.845 0.526

PP Circulating Capital Model

Table 4.1  Direct prices, prices of production and capital intensities, BEA 2018

0.976 1.112 0.810 0.900 1.032 0.732 0.967 1.069 0.961 1.111 1.103 1.132 1.031 1.022 1.212 1.002 1.131 1.026 1.032 1.092 1.249 1.096 1.040 0.799 0.893 1.059 0.811 0.984 0.845 0.926 1.148 1.046 0.852 0.531

5.462 2.208 9.365 4.543 2.554 9.766 1.607 2.200 2.850 3.291 2.247 2.192 2.394 2.168 2.440 2.018 1.891 2.023 3.274 2.880 1.991 2.971 2.206 7.515 4.176 2.724 2.025 2.791 2.941 7.148 3.485 2.346 2.047 11.577

2.669 1.457 2.168 2.064 1.673 1.850 1.724 2.171 1.995 2.544 2.092 2.101 1.293 1.946 2.792 1.956 2.034 1.792 2.681 2.209 1.793 2.380 1.829 2.829 2.277 2.209 1.590 1.668 1.686 1.842 2.199 1.872 1.640 1.483 (Continued)

PP Fixed Capital Intensity Fixed Capital Intensity Capital Model Capital 1/R = 1.984 Circulating Capital 1/R=1.834

Price trajectories and rate of profit  71

35 Other transportation and support activities 36 Warehousing and storage 37 Publishing industries (includes software) 38 Motion picture and sound recording industries 39 Broadcasting and telecommunications 40 Information and data processing services 41 Financial Institutions 42 Securities, commodity contracts and investments 43 Insurance carriers and related activities 44 Funds, trusts and other financial vehicles 45 Real estate 46 Rental and leasing services and lessors of intangible assets 47 Legal services 48 Computer systems design and related services 49 Miscellaneous professional, scientific and technical services 50 Management of companies and enterprises 51 Administrative and support services 52 Waste management and remediation services 53 Educational services 54 Ambulatory health care services 55 Hospitals and nursing and residential care facilities 56 Social assistance 57 Performing arts, spectator sports, museums, etc. 58 Amusements, gambling and recreation industries 59 Accommodation 60 Food services and drinking places 61 Other services, except government 62 Federal general government 63 Federal government enterprises 64 State and local general government 65 State and local government enterprise Mean absolute weighted deviation d-Statistic

Industries

1.076 1.144 1.053 0.932 0.836 0.892 0.903 1.150 0.841 0.960 0.942 0.688 0.870 1.191 1.060 1.248 1.124 1.356 1.116 1.129 1.261 1.221 0.899 1.002 0.899 1.045 1.053 1.031 1.478 1.250 1.098 0.119 0.161

DP

1.050 1.102 0.951 0.914 0.860 0.874 0.862 1.098 0.840 1.091 1.011 0.725 0.787 1.021 0.981 1.129 1.043 1.354 1.006 1.012 1.166 1.127 0.848 0.954 0.851 1.008 0.978 0.937 1.334 1.131 1.097 0.091 0.144

PP Circulating Capital Model 1.059 1.111 0.959 0.922 0.868 0.881 0.869 1.107 0.847 1.100 1.020 0.731 0.793 1.029 0.990 1.138 1.051 1.365 1.014 1.020 1.176 1.136 0.855 0.962 0.858 1.016 0.986 0.945 1.345 1.141 1.106 0.092 0.144

2.093 2.653 1.967 3.888 4.336 2.590 2.726 1.817 1.868 2.227 9.841 4.465 1.417 0.701 1.664 1.706 1.262 2.384 3.012 1.320 2.260 1.646 2.694 3.239 3.324 2.120 2.217 2.295 8.656 4.646 8.071 SD 2.35 Avg. 3.36 CV 0.70

1.706 1.635 1.323 1.752 2.023 1.758 1.609 1.625 1.846 2.658 2.269 2.131 1.323 1.062 1.447 1.325 1.459 1.828 1.293 1.275 1.425 1.409 1.542 1.574 1.551 1.646 1.446 1.344 1.300 1.306 1.834 SD 0.41 Avg. 1.82 CV 0.23

PP Fixed Capital Intensity Fixed Capital Intensity Capital Model Capital 1/R = 1.984 Circulating Capital 1/R=1.834

72 Price trajectories and rate of profit

Price trajectories and rate of profit

73

capital intensities in both models are at a relative rate of profit equal to zero and so PP=DP=MP=1. This gives us an initial grasp of the deviations between capital intensities at the starting point of the trajectories of PPs. The standard ratios are also reported in the top two right cells of Table 4.1, and they are equal to 1.984 and 1.834 for fixed and circulating capital models, respectively. The standard deviations and the mean of these capital intensities are displayed in the last two rows of the table. Their respective ratios, that is, the coefficients of variation are 2.35 / 3.36 = 0.70 and 0.41/ 1.82 = 0.23 for the fixed and circulating capital intensities, respectively. Clearly, the coefficient of variation in the fixed capital model is at least three times higher (=3.11) than that of the circulating capital model. This outcome makes more unlikely the case of crossing the PP–DP line of equality by the PP in the fixed capital model in which PP are expected to move monotonically to the upward or downward direction according to their capital intensity relative to the standard ratio, R. Figure 4.2 displays the price trajectories of industries of each and every of our 65 industries for 2018. Hence, we used Equation 4.8 and gave the relative rate of profit ˜ prices from zero up to 1. The vector of PP according to 6

1.7

34,4

5 4 3 2

24

1.4 1.3

1 4 25

1.1

1.2

0

0.2

0.4

0.6

0.8

1

1

1.8

1.2

1.4

32

0.2

0.4

0.6

0.8

1

1.5

0.8

1

1.2 39

1.2

1.2

41 36 40 35 33

0.9

0

0.2

0.4

0.6

0.8

1

0.7 0

0.2

0.4

0.6

0.8

1

1.2

1

0.2

0.4

0.6

0.8

1

1.2 45

44 42 47 49 48

1 1.2

0

0

0.2

0.4

0.6

0.8

1

1.2

1.06 1.05

62

1.04 1.03 1.02

64

1.5

54 51

0

2

65

2

50 56

0.8

0.9

37

2.5

1

13 14 18, 21 17

1.1

46

1.4

3

55

1.1

15

1.2

3

4

52

1.3

4

38

1.6

0.8

20

5

3.5

1.3

0.6

0.6

1

53

1.4

0.4

1.2

27 0

0.2

2

23 1

0

1.8

26, 28

1.2

1.4

1

2.2

29 22

1.5

5

7

2.4

31

1.6

0.8

9

2 11, 8 12

0.9 1

19

1.6

1.5

63 30

1.7

10

1.6

6

0

0.2

0.4

0.6

0.8

1

58 59 57 61 60

1.01 1 16

0.99 1.2

0.98

0

Figure 4.2 Price trajectories, fixed capital model, BEA 2018.

0.2

0.4

0.6

0.8

1

1.2

74

Price trajectories and rate of profit

Equation 4.8 is divided, element-by-element, by the corresponding vector of DP, which of course is not affected by changes in ˜. Clearly, the movement of relative prices is monotonic with only two exceptions; namely industries, 16 (other transportation equipment) and industry 62 (federal general government) presented in the last right graph of the panel of nine graphs in Figure 4.2 (for the nomenclature of other industries, see Table 4.1). Both exceptional industries attain their maximum at a relative rate of profit much higher than the equilibrium relative rate of profit, ˜ * , which is ˜ = ˜ * = r / R = 0.129 / 0.504 ˛ 25.6%. More specifically, the PP/ DP ratio of industry 16 attains its maximum at ˜ = 40% and crosses the line of PP–DP equality at a relative rate of profit of ˜ = 70%. Industry 62 displays non-monotonic behavior and a maximum at a relative rate of profit of ˜ = 70%. Despite their non-monotonic movement, the price trajectories of both industries are too close to PP–DP line of equality, something that indicates that the capital intensities of these two industries will not be too different from the standard ratio, which is equal to the reciprocal of the maximum eigenvalue 1/ R ˜ 1.984 and is no different from the maximum rate of profit. In the remainder of the graphs, we observe that although the price rate of profit (PRP) curves of some industries are too close to the PP–DP line of equality, do not cross it, and, therefore, do not change the characterization of their intensity. On closer examination, we observe that in all relative rates of profit, the PP–DP deviations are in the neighborhood of one percent, and for the industry 16, the difference is trivially small, even less than 1%! In the next set of graphs in Figure 4.3 that paint the capital intensities of the 65 industries, we observe that capital intensities move nearly parallel to the standard ratio, in general, or they are so far away that crossing the standard ratio appears as a remote possibility. The trajectories of the capital–output ratios usually move parallel to the standard ratio as is shown in Figure 4.3. In the last right graph, we present the movement of the capital–output ratio of industry 16, which intersects the standard ratio at a relative rate of profit of 70% and of industry 62, which is approaching the standard ratio but does not intersect it. The capital–output ratios of these two industries start and remain too close to the standard ratio. These findings along with similar others, as we will discuss below, lend support to the view that switching (for positive relative rates of profit) is a remote possibility. Moreover, it is only possible in cases where the initial differences in capital–output ratios from the standard ratio are relatively small; under these rare circumstances, the price feedback effects are quite possible to change the PP–DP differences from positive to negative and vice versa. 4.3.2 The circulating capital model, BEA 2018 Having examined the case of a fixed capital model and taking into account other similar studies such as those of Ochoa (1984, 1989) as well as

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Figure 4.3 Capital–output ratios and the standard ratio, fixed capital model, BEA 2018.

Shaikh (1984, 1998, 2016), our interest now turns to the circulating capital model that is used extensively in the Sraffian literature.6 The graphs of Figure 4.4 illustrate the price trajectories of the circulating capital model utilizing input–output data of the U.S. economy reduced to 65×65 dimensions of 2018. We observe that in the circulating capital model, the PP maintain their monotonic movement as in the case of the fixed capital model. However, five industries shown in the last graph of Figure 4.4 display extrema and cross the line of their equality with the DP. More specifically, industry 6 (utilities) reaches its maximum at the relative rate of profit of 40% and crosses the line of equality at the rate of profit of 80%. Industry 23 (printing and related support activities) displays a minimum at a relative rate of profit of 30% while at 70% is above one. Industries 30 (rail transportation), 32 (truck transportation) and 43 (insurance carriers and related activities) display maximum. As in the case of fixed capital, it is important to have an idea of the equilibrium relative

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Figure 4.4 Price trajectories, circulating capital model, USA 2018.

rate of profit, which, in this case, is ˜ = ˜ * = r / R = 0.180 / 0.545 ˛ 33.0%. The common characteristic of these five industries (5/65 = 7.7% of the total) is that their price trajectories are, surprisingly close, in the vicinity of the PP–DP line of equality. In all five industries and for reasonable relative rates of profit, the PP trajectories display trivially small deviations from the PP–DP line of equality; the deviations are usually in the third decimal! More specifically, the maximum PP–DP difference is 3.4% in industry 43 and occurs at the maximum relative rate of profit, ˜ = 1! while its crossing is at the low ˜ = 0.20. For the remaining four industries, the crossings take place at too high relative rates of profit, ˜ ° 0.60. The empirical findings suggest that the capital intensities of these particular industries will be almost indistinguishable from the standard ratio, as this becomes obvious in the next panel of graphs of Figure 4.5. In fact, the last right graph in Figure 4.5 displays the trajectories of capital–output ratios

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of the five exceptional industries as well as the standard ratio 1/R, which is equal to 1.834. Finally, we observe that the price trajectories of industries 18 and 52 are not far from one and their price paths are in accordance with Ricardo’s invariable measure of value and the Sraffian standard commodity. In effect, the capital intensities of these two industries are almost equal to the standard ratio. More specifically, industry 18 (miscellaneous manufacturing) whose capital intensity is 1.80 differs only slightly from the standard ratio of 1.834, while industry’s 52 (waste management) standard ratio is remarkably equal to 1.833! The trajectories of capital–output ratios in Figure 4.5 are consistent with the premises of the broad classical theory of Ricardo and Marx. This amount to the same, that is, the Sraffian-Wicksellian effect in Equation 4.9, regardless of its sign, is relatively weak to give rise to non-monotonic movement in relative prices, let alone for the PP to cross over the line of PP–DP equality.

78

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In general, the circulating capital model by virtue of the intermediate inputs has more interindustry interactions than the fixed capital model where not all industries produce capital goods, as this can be judged by the ditribution of eigenvalues of technology matrix A or rather by the matrix of vertically integrated input–output coefficients A [ I − A ]−1 whose details are discussed in Chapter 6. As a result, the changes in the relative rate of profit propagate and may impart further changes throughout the economy. Furthermore, the elasticity of capital–output ratio w.r.t. ˜ may be higher than the Ricardo-Marx term; thus, as the ˜ increases, the quantitative importance of the Sraffa-Wicksell term (see equation 4.9) rises to a point that supersedes the Marx-Ricardo effect. By contrast, the movement in prices induced by changes in relative rate of profit (in the presence of fixed capital stock) are monotonic going either to the upward or to the downward directions drifting further and further away from the line of DP–PP line of equality. A cursory consideration of graphs in Figure 4.3 reveals that the particular directions depend mainly upon the vertically integrated capital intensity of an industry relative to the standard ratio. Hence, the elasticity of the capital intensity w.r.t. the relative rate of profit is negligible (not very different from zero) and its effect may strengthen the difference in the capital–output ratio from the invariant to changes in income distribution standard ratio. Almost, all relative prices move monotonically, and the capital intensity, whatever its starting value, displays no significant variability by virtue of its very construction. More specifically, the capital stock matrix has many linearities developed and so the interactions that one expects from the multitude of industries do not exist. The results from other similar studies agree that in the case of the fixed capital stock model, the trajectories of PP induced by changes in the relative rate of profit give rise to the following empirical regularities: – – – –

There are no complex patterns in the movement of PP as expected by Sraffian economists. Monotonic trajectories of PP are the rule and non-monotonic price trajectories are the exception Non-monotonic movement of PP display much less variability than the monotonically moving PP Non-monotonically moving PP are closer to their DP than the monotonically moving PP.

4.4 Price paths using input–output data, USA 2014 Our experiments continue with the U.S. economy using an alternative set of data derived from the WIOD (Timmer et al. 2015) for the year 2014, the most recent input–output data released from this source. The reason for these additional estimations is not only to ascertain the general findings of proximity between different price vectors but mainly because we use data of 2014 to

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extend our research (in Chapter 5) by making intertemporal comparisons of the WRP curves of the USA. In addition, in Chapter 6, we experiment with these data in our effort to ascertain additional properties of the linear model of production and to reduce large-scale description of the economies into a few or even a single hyper-industry. Finally, in Chapter 7, we aggregate the 2014 input–output data into five sectors and proceed with all of our computations in Chapters 4–6 in our effort to show the techniques and the matrices utilized in the estimations. 4.4.1 The fixed capital model, USA 2014, WIOD (2016) Based on data provided by WIOD (Timmer et al. 2015), we extract the input–output table of the U.S. economy for the year 2014, the last available input–output table supplemented with data for total compensation, employment and capital stock from the Socio Economic Accounts (SEA, http:// www.wiod.org/database/seas16). Both sources of data became available in 2016 (see Timer et al. 2015). The lack of capital f low tables compels the use of the best available alternative estimating method of the matrix of capital stock coefficients. This alternative method utilizes the investment data in the final demand column of the input–output table (see Appendix 4.A) to construct the matrix of the vertically integrated capital stock coefficients whose maximum eigenvalue is equal to one and zero is of its subdominant ones. The properties of the so-derived matrix of the vertically integrated capital stock coefficients are not too different had we employed a capital f lows matrix, as we did with BEA data whose second eigenvalue is particularly small, equal to 0.16. In earlier studies, for the U.S. economy and 39-industry detail, gave similar results. More specifically, for the benchmark years, the second eigenvalue shown in parenthesis are as follows: 1947 (0.41), 1958 (0.31), 1963 (0.31), 1967 (0.47), 1972 (0.55) and 1977 (0.46) while using respective data from the OECD for the year 1990 and 33 industries input–output structure gave a second eigenvalue equal to 0.20 (Mariolis and Tsoulfidis 2016a, 2018). The OECD data for the UK of the year 1990 gave a second eigenvalue equal to 0.19 (Mariolis and Tsoulfidis 2016b). The research for the Greek economy for the year 1970 gave a second eigenvalue equal to 0.04 and for S. Korea for the years 1995 and 2000 a second eigenvalues equal to 0.08 and 0.06, respectively.7 Finally, for Germany, the input–output data and capital stock matrices aggregated into seven sectors (see Cogliano et al. 2018, pp. 240–244 for details) gave a maximum eigenvalue equal to 1.981 and the second eigenvalue was found to be trivially small. These reported results confirm that both methods give rise to similar matrices of capital stock coefficients. In Table 4.2, we display, as in Table 4.1, the results regarding the PP and DP relative to MP, which by definition is equal to one (million USD). The resulting deviations of PP and DP from MP are relatively small as this can be judged by our usual statistic of deviation, which are in the neighborhood

Crop and animal production, hunting and related service activities Forestry and logging Fishing and aquaculture Mining and quarrying Manufacture of food products, beverages and tobacco products Manufacture of textiles, wearing apparel and leather products Manufacture of wood and of products of wood and cork Manufacture of paper and paper products Printing and reproduction of recorded media Manufacture of coke and refined petroleum products Manufacture of chemicals and chemical products Manufacture of basic pharmaceutical products, etc. Manufacture of rubber and plastic products Manufacture of other non-metallic mineral products Manufacture of basic metals Manufacture of fabricated metal products, except machinery and equipment 17 Manufacture of computer, electronic and optical products 18 Manufacture of electrical equipment 19 Manufacture of machinery and equipment n.e.c. 20 Manufacture of motor vehicles, trailers and semi-trailers 21 Manufacture of other transport equipment 22 Manufacture of furniture; other manufacturing 23 Repair and installation of machinery and equipment 24 Electricity, gas, steam and air conditioning supply 25 Water collection, treatment and supply 26 Sewerage; waste collection, treatment and disposal activities; etc.

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16

Industries

Table 4.2 Direct prices, prices of production and capital intensities, USA 2014

0.770 0.957 0.957 0.632 1.150 1.353 1.303 1.328 1.297 0.720 0.868 0.868 1.179 1.116 1.471 1.386 0.938 1.317 1.344 1.500 1.328 1.280 1.123 0.636 0.636 1.184

0.999 1.092 1.079 0.998 1.085 1.129 1.206 0.618 0.618 1.087

PP Circulating Capital Model

0.546 1.097 1.097 0.583 0.782 1.115 1.076 0.997 1.171 0.506 0.680 0.680 0.934 0.972 1.004 1.114

DP

1.033 1.056 1.032 1.092 1.249 1.096 1.040 0.799 0.893 1.059

0.864 1.053 1.053 1.084 0.902 1.120 1.035 1.092 1.108 0.824 0.912 0.912 0.979 1.069 1.176 1.072

PP Fixed Capital Model

1.108 1.818 1.881 2.259 1.845 1.741 1.350 0.911 0.911 1.587

1.355 1.078 1.078 0.892 1.996 1.984 1.876 2.029 1.778 1.411 1.415 1.415 1.840 1.633 2.298 1.935

Capital Intensity Circulating Capital 1/R = 1.354

1.935 1.762 1.755 1.776 1.589 1.672 2.192 6.158 6.158 2.327

2.516 1.731 1.731 3.467 1.972 2.007 1.711 2.234 1.775 2.441 2.333 2.333 1.877 2.195 2.617 1.771

Capital Intensity Fixed Capital 1/R = 1.846

80 Price trajectories and rate of profit

27 Construction 28 Wholesale and retail trade and repair of motor vehicles, etc. 29 Wholesale trade, except of motor vehicles and motorcycles 30 Retail trade, except of motor vehicles and motorcycles 31 Land transport and transport via pipelines 32 Water transport 33 Air transport 34 Warehousing and support activities for transportation 35 Postal and courier activities 36 Accommodation and food service activities 37 Publishing activities 38 Motion picture, video and tv production, publishing activities, etc. 39 Telecommunications 40 Computer programming, consultancy and related activities, etc. 41 Financial service activities, except insurance and pension funding 42 Insurance, reinsurance and pension funding, except compulsory SS 43 Activities auxiliary to financial services and insurance activities 44 Real estate activities 45 Legal and accounting activities, activities of head offices, etc. 46 Architectural and engineering activities, technical testing and analysis 47 Scientific research and development 48 Advertising and market research 49 Other professional, scientific and technical activities, etc. 50 Administrative and support service activities 51 Public administration and defense, compulsory social security 52 Education 53 Human health and social work activities 54 Other service activities Mean absolute weighted deviation d-Statistic

1.103 1.108 0.958 1.024 0.962 0.884 0.888 1.217 1.094 1.113 1.023 0.689 0.735 1.352 0.942 1.042 1.409 0.298 1.273 1.245 1.245 1.245 1.245 1.094 1.348 1.361 1.359 1.144 0.249 0.235

1.114 0.996 0.898 0.957 1.044 1.131 0.980 1.210 1.094 1.124 0.961 0.730 0.847 1.228 0.852 1.142 1.283 0.368 1.149 1.161 1.161 1.161 1.161 1.035 1.182 1.189 1.231 1.082 0.220 0. 212

0.811 0.984 0.845 0.926 1.148 1.046 0.852 0.531 1.059 1.111 0.959 0.922 0.868 0.881 0.869 1.107 0.847 1.100 1.020 0.731 0.793 1.029 0.990 1.138 1.051 1.365 1.014 1.020 0.265 0.195

1.423 1.145 1.083 1.177 1.500 1.848 1.495 1.577 1.412 1.464 1.162 1.032 1.281 1.427 0.977 1.507 1.482 0.595 1.331 1.389 1.389 1.389 1.389 1.257 1.330 1.389 1.449 1.333 SD 0.356 Mean 1.457 CV 0.244

1.117 1.246 1.166 2.127 2.738 2.551 2.482 2.074 1.686 1.820 1.214 2.729 2.945 1.386 1.731 1.031 1.295 8.105 1.402 1.385 1.385 1.385 1.385 1.417 4.653 3.305 1.851 2.268 SD 1.309 Mean 2.258 CV 0.580

Price trajectories and rate of profit 81

82 Price trajectories and rate of profit

of 20%. The last two columns of Table 4.2 stand for the capital intensities of industries in both circulating capital and fixed capital models. The standard ratios are also reported in the top two right cells of Table 4.2, and they are equal to 0.931 for the circulating capital (A [ I − A ]−1) and 1.605 for the fixed capital model K [ I − A ]−1. The standard deviations and the mean of these capital intensities are displayed in the last two rows of the table. Their respective ratios, that is, the coefficients of variation are 0.36 / 1.46 = 0.24 and 1.31/ 2.26 = 0.58 for the circulating and fixed capital intensities, respectively. Clearly, the coefficient of variation in the fixed capital model is at least twice higher (=2.38) than that of the circulating capital model making more unlikely the crossing of the PP–DP line of equality in the fixed capital model. In the graphs of Figure 4.6, we display the price trajectories for the fixed capital model utilizing actual input–output data of the U.S. economy. It is interesting to note that the WIOD Timmer et al. (2015) gives both the 3.5

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domestic input–output tables as well as the imported along with the row of total output or input, which is the sum of domestic and imported parts. In adding up these two input–output tables (domestic and imported) and deriving the (total) input–output coefficients, we get repeated industries. These are industries 11 (chemicals) and 12 (pharmaceutical products) as well as 24 (electricity, gas, etc.) and 25 (water supply). The similar industries include also the following four: namely, industry 46 (architectural and engineering activities), industry 47 (research and development), industry 48 (advertising) and industry 49 (other scientific, technical and related activities). The same is true for the employment input and the capital–output coefficients. Consequently, the actual number of industries in the input–output table is 49. The monotonic movement of prices is in line with Ricardo and Marx’s views about the changes in prices induced by changes in income distribution. This movement is consistent with that of capital intensities. Figure 4.7

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Figure 4.7 Capital intensities, fixed capital model, USA 2014.

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displays the movement of the capital intensities of industries relative to the standard ratio, which is R −1 = 1.846. The graphs in Figure 4.7 ascertain the parallel movement of capital intensities, as expected by construction in that the subdominant eigenvalues are all equal to zero. The reason is that the columns of the matrix K are linearly dependent derived from a multiplication of two vectors. It is important to stress that had we had the actual capital f lows tables, we would not observe anything quite different, unless an industry’s capital–output ratio is near the standard ratio and the elasticity term moves in the opposite direction to the Ricardo-Marx effect. As we have already mentioned, under these circumstances, it is possible for the feedback effects to change the trajectories of capital intensities and to observe crossings, but not too many as in the case of fixed capital model of the year 2018. Schefold (2013, p. 1177) opined in the case of a circulating capital model that the near linearities in prices could be explained by the small subdominant eigenvalues of the matrix A whose elements he considered as randomly distributed. By contrast, the same configuration of eigenvalue distribution and the resulting near linearities in PRP and WRP curves is explained by the low effective rank of the economic system matrices (Mariolis and Tsoulfidis 2011, 2014, 2016a, 2016b, 2018; Iliadi et al. 2014; Shaikh 2016, ch. 9). In the next chapters, we explore more thoroughly the underlying system’s matrices and the resulting eigenvalue distributions and their consequences. 4.4.2 The circulating capital model, USA 2014 The price trajectories of the circulating capital model using the same 2014 input–output data, as expected, are monotonic albeit they have a distinct second derivative indicating acceleration or deceleration in the direction of prices trajectories; thereby enhancing their monotonic movement as shown in 49 actual price trajectories in the panels of six graphs in Figure 4.8. It is important to point out that our estimations are carried out by assuming that wages are paid ex ante, and so our PP in our circulating capital model without depreciation are estimated from equation 4.6 provided that the matrix H = ( A + bl ) [ I − A ]−1 . The price paths of industries 38 (motion pictures, etc.) and 42 (insurance, etc.) are the ones that display switching at the exceptionally high relative rates of profit 80% and 100%, respectively. All other industries display monotonic behavior, while industries 26 (sewerage, waste, etc.), 34 (warehousing), and 35 (postal and courier) are near one, which means that they are only lightly responsive to changes in income redistribution. It is equally intriguing to note that even the industries that cross the line of PP–DP of equality display minimal variability, and they remain near one. These examples are consistent with Sraffa’s conjecture about the possible price feedback effects that could lead to crossing(s) although the empirical findings show that the price feedback may become effective to those industries that are in the vicinity of the PP–DP line of equality. Furthermore, the crossing occurs because these particular industries have a

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capital intensity near the standard ratio, as this is shown in the next panel of graphs in Figure 4.9. The respective capital–output ratios of the 49 industries whose change is in accordance with the movement of PP, while for the two industries with “perverse” behavior, we observe that they are not far from the standard ratio of 1.354. We repeated the experiment by invoking equation 4.6 provided that the matrix H = A [ I − A ]−1 with no qualitative differences in the results, which we have displayed in Figure 4.10. In fact, on the l.h.s. graph we display only six industries, which display “regular” behavior, that is, non-monotonicity. We observe that industries 39 and 44 display maximum at a relative rate of profit of

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0.60 and 0.70, respectively. Industries 4 and 31 cross the PP–DP line of equality at very low relative rates of profit, the converse is true for industry 33. The common characteristic of these three industries is that their price trajectories are in the vicinity of PP–DP line of equality, which means that their capital intensities are not expected to be too different from the standard ratio. Finally, industry 42 crosses the line of equality at a relative rate of profit of ˜ = 50%. All six price trajectories display maximum and none minimum. This does not mean that the case of minimum is excluded; quite the contrary, we may have occurrences of minimum as in industry 23 of the input–output table of the U.S. economy of the year 2018 (see Figure 4.4). We also have found a

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minimum in our previous studies of Japan and Greek economies along with many others (see Tsoulfidis and Mariolis 2007; Tsoulfidis 2008b; Shaikh 2016; Tsoulfidis and Tsaliki 2019). It is also important to point out that the shapes of most capital–output curves are rigid and, more often than not, move nearly parallel to the horizontal axis. A result indicating the negligible effect of the capital elasticity w.r.t. the relative rate of profit term, or the SraffianWicksellian effect of Equation 4.9. It is worth stressing that the PP–DP of 4 out of 6 industries with non-monotonic behavior remains too close to one for all relative rates of profit and despite crossings. The paths of PP of the remaining two industries, for reasonable rates of profit, are not far from the PP–DP line of equality. A result indicated that the capital intensities of these industries behave accordingly. Thus, the industries 33 and 44 from capital intensive that they start, they become labor intensive when they cross the standard ratio at the relative rates of profit 50% and 100%, respectively. The remaining industries remain below although, near the standard ratio, as shown in Figure 4.10. In an overall comparison of the fixed capital and the circulating capital models, we observe that in both models, the movement of PP relative to their respective DP display monotonicity. Consequently, the rankings of the capital–output ratios relative to the standard ratio only rarely and under particular circumstances are affected by changes in income distribution. In such a comparison, we further observe that in the circulating capital model, the paths of relative PP display not too many curvatures and switch points. More specifically, in our 65×65 industry detail of the year 2018, we had only five switching cases while a sixth case, industry 52 (waste management and remediation services), its PP/DP trajectory was almost indistinguishable from one (see Figure 4.4). At the same time, its capital–output ratio was different from the standard only in its fourth decimal point! (Figure 4.5). Hence, Ricardo’s search for a practical invariance measure of value and Sraffa’s analytically

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derived invariance to distribution standard commodity (industry) came to be almost one and the same thing! The movement of vertically integrated capital–output ratios is analogous, and only in a relatively few cases display switch points with the standard one giving rise to the appearance of curvatures to the trajectories of PP, thereby increasing the likelihood of crossing the line of equality with the DP. As for the case of fixed capital stock, the interindustry capital–output ratios are far more distant from each other and from the standard ratio. Furthermore, the elements of matrix K are less connected to each other; that is, there are many zero elements and particularly small or exceptional large capital stock coefficients. Consequently, the change in the relative rate of profit lead to revaluations of fixed capital stock, which are by far weaker than those of the circulating capital model where industries are more connected to each other. Thus, it must come as no surprise that in the fixed capital model there is no switch of orderings between industries and changes in their characterizations from capital- to labor-intensive, and vice versa. In effect, we found that the capital–output ratios in the circulating capital model, using BEA 2018 data, as these are estimated from vA [ I  – A − D]−1 gave an average capital–output ratio equal to 1.82 and a standard deviation of 0.41 with a coefficient of variation of 0.23. The fixed capital model vK [ I  –  A − D]−1 gave a mean equal to 3.36, a standard deviation of 2.35 and a coefficient of variation of 0.70, which is at least three times higher than that of the circulating capital model (see Table 4.1). These typical results show that the case of switching is nearly excluded in the case of a fixed capital model (see also Shaikh 2016, p. 413).

4.5 Summary and conclusions In this chapter, we showed that the classical theory of value, when cast in a linear model of production contains explanatory power that has not been appreciated as much as it deserves. We say that because not only the equilibrium prices estimated within this approach, predict extremely well the MP but also enables us to make theoretical statements of general validity about the technology and its change over time. We have shown in Equation 4.9 and the discussion that followed that the movement of PP relative to DP depends on the workings of two effects, the Sraffa-Wicksell and the Ricardo-Marx, which may move to the same or different directions. From these two effects, the results overwhelmingly show that the Ricardo-Marx effect is the dominant one, which is enhanced in the case of the fixed capital model. Furthermore, the research so far has shown that the reswitching scenario is rare in the data of actual economies, and it occurs for unrealistic relative rates of profit. Fewer are the cases in which crossing occurs at realistic relative rates of profit and whenever it occurs, it might be attributed to the proximity of the capital intensities of these industries to the standard ratio. Consequently, the price feedback effects and the revaluation of capital may even change the

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characterization of the industry from the capital to labor intensive and vice versa. Under no circumstances, however, we rule out the strength of the price feedback effects but in no way say that they are decisive enough to make us agnostic concerning the characterization of an industry. In an overall evaluation of the circulating capital with that of the fixed capital model for the U.S. economy, we observe that in both models the paths of PP relative to DP display more often than not monotonicity. There is no complex pattern in PP as anticipated by what can be called Sraffa’s (1960) conjecture. Non-monotonic prices are actually observed, but they are very few and what is quite interesting is that their variability is significantly less than that of the monotonically moving prices. Furthermore, the nonmonotonically moving PP are closer to DP than the monotonically moving PP. At the same time, the positions of the capital–output ratios relative to the standard ratio, which is not very different from the economy-wide average capital–output ratio, are maintained in most cases. The repeated occurrence of these interesting empirical findings, virtually in every country and year tested, rightfully renders them the characterization of “law-like regularities”. In comparison, we observe that in the circulating capital model, the paths of PP in a relatively few cases display extremes and, in fewer cases, they cross over the line of equality with the DP. In the fixed capital model, there has been only a single crossing; of course, crossings are rare but exist as the dimensions of input–output tables increase. Ochoa (1989) and Shaikh (1998) find few crossings in the US input–output data of 71 industries spanning the period 1953–1972. The time period extends to 1997 in Shaikh (2016) with similar findings. The more recent study by Shaikh et al. (2020) where the dimensions of input–output tables for the years 2002 and 2007 vary from 15 industries to up to 425 confirms that the non-monotonic movement in prices are rare and the crossings are even more exceptional. Analogous is the movement of the vertically integrated capital–output ratios, which, in relatively few cases, display switch positions with the standard ratio giving rise to the appearance of curvatures and the likelihood of switching of the PP with the line of DP. It is worth noting that the interest in the Sraffian literature is more on the circulating capital whereas the fixed capital case is relegated to future investigation within the joint production framework. The latter not only is much more difficult to carry out analytically, but its treatment requires the presence of a secondary market, which, we know it does not exist, in any systematic way, in the actual economy. The fixed capital model, however, is one in which the interest must be laid out because the presence of fixed capital in the production process is the salient feature of capitalism. There is no doubt that a fuller discussion of issues related to the actual trajectories of PP requires the inclusion of fixed capital, whose data are hard to come by. This is the reason why, by any means, we seek to employ the fixed capital stock model and its results to be contrasted to those of the circulating capital model. Such a comparison can only lead to fruitful conclusions about the nature of the

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fundamental relations of the system. The lack of detailed data on capital stock and capital f low matrices is no excuse for researchers for whom the inclusion of the matrix of fixed capital stock is compelling. There are always published industry data on investment and capital stock, which can be creatively used through the application of indirect methods to construct matrices of capital stock that can be reliably employed for the needs of analysis like the present one.

Appendix 4.A: Sources of data and estimating methods The input–output table of 2018 is, as of this writing, the last available in the site of BEA as a direct requirement input–output matrix, that is, the Leontief inverse. The initial data include 72 industries and we brought down to 65 industries to make it suitable to 65×65 capital f lows (investment) matrix of the year 1997.8 Some of the eliminated industries have zero rows and the entries in their columns are trivially small; thus, we removed these industries because their aggregation leads to more distortions. After all, aggregation of industries would require the initial input–output table (not Leontief ’s inverse), which is not available unless we constructed it through the Use and Make matrices as is suggested in the BEA site. The changes from the elimination of the seven industries are only marginal, and our estimates are not affected in any empirically significant way. Data on depreciation and fixed capital stock matrices for our 2018 come from BEA’s industry data for both the private sector and the government, federal and local. We allocated the total capital stock and depreciation of government enterprises, following information provided in past practices. In particular, we allocated 70% of the total capital stock and depreciation to local government enterprises and the remainder to the federal government enterprises. For the estimation of the matrix of capital stock coefficients for the year 2018, we proceeded as follows: we formed weights derived from the capital f lows matrix, which is post multiplied by the diagonal matrix of depreciation per unit of output. In so doing, we derive estimates of the matrix of depreciation coefficients per unit of output, D, while the post-multiplication of the capital f lows matrix of weights by the diagonal matrix of capital stock per unit of output gives us the matrix of capital stock coefficients, K. The idea is that despite the old data of the matrix of capital (or investment) f lows, one does not expect wild changes in its structure over the years, since the investment goods industries produce intermediate and capital goods. By contrast, the consumption goods industries and services do not produce investment goods. Consequently, only the investment goods industries will have their rows filled with data, while the remaining have many zero elements. In general, the resulting matrices of depreciation and capital stock coefficients are of the sparse kind: that is, they contain by far more zeros or near zero elements compared with the matrix of input–output coefficients.

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The last investment matrix for the USA regrettably is available for the year 1997, and this is what we have used for our estimates of PRP trajectories and WRP curves for the year 2018. It is important to note the other available alternative was the OECD investment matrix for the year 1990, at the 34 input–output industry detail, which could not be used for our 65-industry structure of 2018. However, we used the 34×34 capital f low (investment) matrix of the USA of the year 1990 in Chapter 7, where we explicate our methods of estimations on the basis of 5 sectors input–output table of the year 2014. For the estimation of the matrix of capital stock coefficients for the year 2014 and the other years, we proceed as follows: The vector of investment expenditures for the 54 (actually 49) industries for the period 2000–2014 is provided in the world input–output database (WIOD) whose link is: http:// www.wiod.org and is accompanied by the necessary documentation (Timmer et al. 2015). The vector of capital stock in current prices is obtained from WIOD the Socio-Economic Accounts (SEA) whose link is: http://www. wiod.org/database/seas16 and it has been def lated by the investment def lator (with base year 2010). The real capital stock of each industry is divided by the respective industry’s real output; in so doing, we get the row vector of real capital stock per unit of output. The matrix of fixed capital stock coefficients is derived from the product of the column vector of investment shares of each industry times the row vector of capital stock per unit of output (see also Montibeler and Sánchez 2014; Tsoulfidis and Paitaridis 2017; Tsoulfidis and Tsaliki 2019; Cheng and Li 2020). Hence, it is important to note for the accuracy of our estimations that the column sums of the resulting square matrix are the same as those that we would have derived had we utilized the more accurate capital f low tables. The rank of the resulting matrix is one as the product of multiplication of two vectors and because of the presence of linear dependence, the maximum eigenvalue of the resulting matrix K [ I − A ]−1 R −1 is equal to one with zero for all the subdominant eigenvalues. The resulting new matrix of capital stock coefficients, K, possesses the properties of the usual capital stock matrices derived and employed in the hitherto empirical studies (see Tsoulfidis and Paitaridis 2017; Mariolis and Tsoulfidis 2016a, and the literature cited therein). The idea is that the investment matrices contain many rows with zero elements (consumer goods and service industries do not produce investment goods) and so the subdominant eigenvalues will be substantially lower (indistinguishable from zero) than the dominant; this is another way to say that the equilibrium prices are determined almost exclusively by the dominant eigenvalue. In similar fashion, the matrix of depreciation coefficient, D, was estimated as the product of the column vector of investment shares of each industry times the row vector of depreciation per unit of output. Data for depreciation by industry is not available in the world input–output database, so we used data from other sources, such as the database of Structural Analysis of the OECD (STAN) https://stats.oecd.org/Index.aspx?DataSetCode= STAN08BIS. To minimize the effects of any possible methodological

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differences between databases, we estimated the ratio of depreciation to gross value added by industry from the OECD data sets and then we multiplied it by the corresponding gross value added data that is available in the WIOD. The total wages are also derived from the industry data available in the BEA site. Each industry’s total wages for full-time equivalent are divided by the economy-wide average wage estimate at 57,000 USD. The employment coefficients are derived by dividing the industry employment by its respective real output available also in the same database. The vector of consumption expenditures of workers is derived by dividing each industry’s personal consumption expenditures by the total personal consumption expenditures. The derived vector of relative weights is multiplied by the economy-wide average real wage. The estimates of A, l, b and K for the year 2014 are all in constant prices of 2010. So, the deviations of PP and DP from MP are expected to be somewhat higher because of the probable mistakes in the construction of price indexes.

Notes 1 In the 1970s and early 1980s, there were many Sraffians arguing about the redundancy of labor theory of value although over the years the objections faded away. The often-cited book on this issue is Steedman’s (1977). 2 The available official data do not include industry turnover times and one must estimate them from limited information. For example, Ochoa (1984, 1989) defined as turnover time the ratio of inventories to output for each of the 71 industries of his study. The trouble with this estimation is that the data are not readily available for all years and industries and he was forced to use the turnover time of 1972 to the other benchmark years as 1947, 1953, 1958, 1963 and 1967. 3 This approach draws on Tsoulfidis and Mariolis (2007), for other similarly motivated approaches, see Parys (1982) and Caravale and Tosato (1980, pp. 85–87). 4 The standard ratio is not too different from the economy-wide average capital– output ratio. The characterization of the effects is somewhat unfair to Sraffa, whose contributions inspired the distinction of the above two effects. 5 The dimensions of 2018 BEA input–output table are 71×71 industries but we reduce it to 65×65, so as to make it compatible with the capital f lows matrix. For details, see Appendix 4.A. 6 The fixed capital model is typically studied within the joint production analytical framework, where fixed capital is considered along with the produced output. For the treatment of fixed capital in the Sraffian literature and its critique, see Semmler (1983, ch. 6), Shaikh (2016, pp. 804–807) and Cogliano et al. (2018, chs. 10 and 11). 7 Details for the data and methods of derivation see Tsoulfidis and Tsaliki (2019 and the literature therein). 8 Capital f low tables for the USA are no longer published by the BEA. The last capital f lows table refers to the year 1997. The capital f lows table for the year 2002 was not completed due to the lack of funding (Meade 2010).

5

Wage rate of profit curves and technological change

5.1 Introduction In Chapter 3, we have argued that in the Cambridge capital controversies (CCC), the U.K. side of the debate prevailed by showing the presence of the reswitching of techniques and Samuelson (1966) from the U.S. side, in an act of intellectual honesty, admitted. The possibility of reswitching of techniques occurs when a capital-intensive technique is used at lower interest (profit) rates, in the intermediate interest rates is withdrawn in favor of a labor-intensive technique and, at much higher interest rates, the capital-intensive technique returns once again, rendering the idea of scarcity prices, the quintessence of neoclassical economics, under question. Samuelson, along with other neoclassical economists, argued that although the occurrence of reswitching cannot be excluded, however, in reality, must be rare and, therefore, it might take its place along with other paradoxical results of the neoclassical theory. For example, the Giffen paradox in microeconomics did not rule out the neoclassical “law of demand”, which maintains its status as the least disputed law in neoclassical economics. Very similar was the treatment of the so-called “Leontief paradox” (Leontief 1953; Paraskevopoulou et al. 2016), whose empirical findings in international trade challenged the central premise of neoclassical economics and its theory of international trade epitomized in the Heckscher-Ohlin-Samuelson model. These findings and phenomena are considered exceptional, and for this reason, are characterized as paradoxes, and, by demoting their significance, the neoclassical theory continues to dominate. In this chapter, we are dealing with the shape of the wage rate of profit (WRP) curves and their significance in economic theory. Our interest in the shape of the WRP curves stems from the need to derive positive, rather than negative, results that contribute to the advancement of the classical economic theory through the study of the structure of the economy and its change over time. In particular, we seek to enhance our understanding of the character of technological change and its overall effects on the economy. The remainder of the chapter is organized as follows. Section 5.2 begins with a survey of the first empirical studies concerning the WRP curves and their particular shapes. Section 5.3 explains the derivation and the different expressions of the WRP curves and applies them to a number of economies. Section

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5.4 discusses issues related to the choice of the numéraire and examines the possible relationship of the various numéraires to the Sraffian standard commodity. Section 5.5 gives estimates of the WRP curves for 54 (actually 49) industries of the U.S. economy for the years 2007 and 2014 using both circulating and fixed capital models and expands the analysis by bringing to the fore the nature of technological change. Section 5.6 presents and critically evaluates similar analyses and results from more recent literature. Finally, Section 5.7 summarizes and makes some concluding remarks.

5.2 Survey of the first empirical studies The theoretical analysis of the movement of relative prices and the shape of the WRP curves has not been backed up by the necessary empirical documentation. The first empirical results on this issue were reported by Krelle (1977), a neoclassical economist, whose research on the former West German economy showed that the shape of the WRP curves is quasi-linear. Krelle in his research utilized input–output data and estimating techniques; however, his results did not attract the attention that they deserved. Probably because neoclassical economists, admitting the weakness of their theory (see Samuelson 1966), over the years, lost interest in the questions at hand and, in any case, considered the theoretical findings exceptional and, as such, did not deserve any further investigation. Sraffians, on the other hand, assumed that the debates were about the logical consistency of the neoclassical theory and thus needed not to be tested empirically. There is no doubt that if a theory is logically inconsistent then it cannot be saved empirically! The trouble, however, with the Sraffian approach was that the bright idea of the price feedback effects was tested using numerical examples detached from the reality of the capitalist economies (see Appendix 3.A). Had the Sraffians tested their insights utilizing data from the actual economies, they would have ascertained on the one hand the presence of the limited price feedback effects and, on the other hand, the boundaries of their critique of the neoclassical theory. As a result, they might have directed their research efforts to issues such as the subjective character of the neoclassical theory, the nature of competition and the lack of causality from the marginal physical productivity to factor rewards, among others. Krelle’s (1977) estimates of WRP curves of Western Germany were not rigorously formulated, and regrettably, they were not so much motivated by the search for scientific truth, but rather by ideology to provide the support that the neoclassical theory was in desperate need right after the CCC. Based on his empirical findings, Krelle (1977) had no hesitation to arrive at the following verdict with direct hints about the realism of the Sraffian assumptions and their distance from reality: some of the arguments of the reswitching debate are similar to the arguments of a physicist inventing the ether and “proving” that Einstein’s relativity theory is wrong. (Krelle 1977, p. 301)

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On further consideration, however, the negligence of empirical findings such as those presented by Krelle and others should not be particularly surprising, if one takes into account that even Leontief ’s (1986) important contribution, designed, among other things, to settle the debate on both camps, passed almost unnoticed. This negligence is peculiar, if we think that not too many years ago people on both sides of the Atlantic were debating fiercely about the shape of the WRP curves. Only a few scholars cite Leontief ’s, important for the debate, results that show quasi-linearities in the WRP curves in the case of the U.S. economy for the 1979 input–output table, at the 85-industry level. Running the risk of indulging in more historical repetition, we begin by citing the very first studies of Ochoa (1984, 1989), Shaikh (1984, 1998), Özol (1984), Cekota (1988), Bienenfeld (1988), Silva (1991), Silva and Rosinger (1992), Petrović (1991) and Tsoulfidis and Maniatis (2002) and continue with the more recent studies that attracted attention and renewed interest. In these more recent studies, we include those of Han and Schefold (2006), Mariolis and Tsoulfidis (2011, 2016a, 2016b, 2018), Shaikh (2012, 2016), Schefold (2013, 2018, 2020), Li (2014) and Tsoulfidis and Tsaliki (2019), all of which lend not only the empirical but also the required theoretical support of the existence of quasi-linear WRP curves. These studies have overwhelmingly showed that the phenomena of reswitching, although they do exist, nevertheless, they are rare and in and of themselves do not, in general, lend support to the view of capital-intensity reversals. For example, Petrović (1991, p. 166) pointed out that the capital reversal and reswitching are highly improbable events in actual economies. While these are the findings on the empirical front of the debates on capital theory, on the theoretical front, Pertz (1980), using simulations, argued that the likelihood of reswitching is inversely related to the number of industries examined. By contrast, Ahmad (1991) opined that the converse might be true; that is, the likelihood of reswitching increases with the number of industries. Mainwaring and Steedman (2000), by utilizing a simple two-sector single product model, argued that the highest reswitching probabilities are observed in the cases where the WRP curves exhibit low curvature and conclude by noting that this is so in a two-sector model should make us particularly wary of claiming a simple relationship between probability and curvature in theoretical or actual multi-sector economies. (Mainwaring and Steedman 2000, p. 346) Bidard (2020), in his search for the conditions ensuring the linearity of the WRP curves in a two-sector model, argued that the nominal rank of the system matrix must be one to ensure linearity in the WRP curves. A condition, he argued, very unlikely to be fulfilled in actual economies, however, without giving us any clue of what kind of rank to expect in actual economies. Mariolis and Tsoulfidis (2016a, 2016b, 2018) have argued that the nominal rank of a matrix might be high depending on the number of linearly independent vectors of the system matrices. However, the effective rank is what

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really counts and it is determined by the first few eigenvalues or singular values that compress most of the “energy” of the system matrices and, in so doing, rule out the cases of what might be called pseudo-linear independence. Hence, the effective rank of the circulating or fixed capital models matrices is responsible for the particular shape and location of the WRP curves. Theoretical discussions and findings such as the above compel the empirical testing and experimentation with various scenarios and, by utilizing more recent and complete data sets, to derive results shedding further light on the shape of the WRP curves in the hope of arriving at theoretical statements of general validity. We grapple with these issues in Chapter 6, where we show that the effective rank is significantly lower than the nominal rank of an economy’s matrices, a result that explains the near linearities in the PRP and WRP curves. The empirical research since the early 1980s has shown that direct prices (DP) are too close to prices of production (PP) as their proximity is measured by the various non-parametric statistics of deviation (Tsoulfidis and Tsaliki 2019 and the literature therein). Consequently, estimates in terms of DP are not expected to be too different from those in terms of PP. Moreover, while it is true that the mathematical structure of the problem allows the theoretical possibility for many curvatures in the WRP curves and, therefore, the Sraffa’s-inspired critique appears to be justified from a purely mathematical point of view, however, from an empirical point of view such a reswitching remains a theoretical possibility. The shape of the WRP curves is approximate-linear, a result which has been observed in every single economy tested. More specifically, the empirical results from the economies of West Germany (Krelle 1977), USA (Ochoa 1984; Özol 1984; Leontief 1986; Shaikh 1998, 2016; Shaikh et al. 2020; Mariolis and Tsoulfidis 2016a, 2018; Tsoulfidis and Tsaliki 2019), Canada (Cekota 1988), former Yugoslavia (Petrović 1987, 1991), Greece (Tsoulfidis and Maniatis 2002), Korea (Tsoulfidis and Rieu 2006), Columbia (Molina and Garzón 2011), Brazil (Silva 1991), China (Li 2014) and the UK (Mariolis and Tsoulfidis 2016a, 2016b) confirm the quasi-linearity of the WRP curves.

5.3 Technological change and wage rate of profit curves Sraffa’s (1960) main interest was on the possible curvatures of the price trajectories caused by changes in the distribution variables, a corollary of which was the WRP curves, whose shape but also location are rich in insights about the nature of technological change and the complexity of the price trajectories in the face of changes in income distribution. However, the CCC were more about the WRP curves and only implicitly about the PRP curves for reasons that have been explained in Chapter 3 and relate mainly to the direct relation between the WRP curves and frontiers with the neoclassical isoquants and isocosts, whose shape is of strategic importance in defining the well-behaved demand schedules for capital and labor. If there is reswitching

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97

between the WRP curves, it follows that the neoclassical theory of scarcity prices no longer holds. The estimation of WRP curves, in the case of the presence of the matrix of fixed capital coefficients, K, the matrix of depreciation coefficients, D, as well as the diagonal matrix of indirect tax coefficients, t (i.e., indirect business taxes per unit of output), is carried out starting from the normalization of the monetary expression of labor time (MELT) PP (see Chapter 4)1 p = pbl + pA + pD + p t + rpK

(5.1)

Substituting the money wage, w = pb, into the equation of PP and after some manipulation, we get p [ I − A − D − t − rK ] = wl The definitions of the other variables remain the same as in Chapter 4. Solving for p, we get p = wl [ I − A − D − t − rK ]

−1

We post multiply by x (the column vector of the gross output of each sector) both sides of the above and by invoking (from Chapter 4) our normalization condition px = ex , we arrive at the WRP relation w=

ex

−1

l [ I − A − D − t − rK ] x

(5.2)

If we consider one of the variables, for example, the rate of profit, as the independent one and we assign to it different hypothetical rates starting from zero (corresponding to the maximum wage) up until we reach the maximum rate of profit (which corresponds to zero wage), we can generate the WRP curve of the totality of the economy. Such a curve, of course, refers to a multi-commodity world and, from a mathematical point of view, is expected to display many curvatures or in mathematical terms inf lection points (Garegnani 1970, p. 417). Consequently, if the system behaves well or “regularly”, according to Schefold (1976), the trajectories of WRP curves should cross each other many times. When this does not happen the system, according to Schefold (1976), behaves “irregularly”, a misnomer, in our view, as we will argue below. Figure 5.1 portrays the WRP curve derived from data of the Greek economy for the year 1970 using a fixed capital model along with the matrices of depreciation and indirect tax coefficients (Tsoulfidis and Maniatis 2002). The gross output was used for normalization purposes but the results would not have been affected, had we used the net output or even the vector of the basket of wage goods, b (Bidard 2020). In Figure 5.1, we have also plotted a hypothetical straight dashed-line connecting the maximum

Wage rate of profit curves

98 0.05

Real wage 0.04

w-r curv e, fi xed capi tal m ode l, G reec e 19 70

0.03

0.02

0.01

0

Rate of proft –0.01

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0.5

0.55

0.6

Figure 5.1 WRP curve, fixed capital model, Greece 1970.

real wage to the maximum rate of profit. From the graph, we observe that the WRP curve crosses the straight line at the critical rate of profit of 45%, which is near the maximum rate of profit equal to 53%. Thus, for 0 < r ° 0.45 the WRP curve is slightly convex, a shape typically associated with the presence of fixed capital in the utilized model, and from that critical rate of profit onwards and until the maximum rate of profit that is, for 0.45 < r ° 0.53, the WRP curve becomes slightly concave. In any case, we can characterize the shape of the WRP curve of the Greek economy of the year 1970 near-linear and daresay “irregular” following Schefold’s (1976) taxonomy of shapes. In a similar fashion, we have estimated the WRP curve using both, circulating and fixed capital models for the South Korean economy (Tsoulfidis and Rieu 2006). Hence, we do not use the matrix of indirect tax coefficients as in the case of Greece above and in the circulating capital model, we assume that wages are paid ex-ante and so profits are estimated on both circulating capital and real wages. Furthermore, for reasons of simplicity and geometrical clarity, we utilize the relative rate of profit, ˜ = rR −1, as our distributive variable on the horizontal axis, which increases by tenths starting from zero until we arrive at its maximum equal to one. Figure 5.2 describes the WRP curves in the case of circulating and fixed capital models for the years 1995 and 2000, the two years that we managed to collate reliable data on capital stock of the South Korean economy. More specifically, for the circulating capital model, we utilized the following relationship: w=

ex

−1

l ˇ˘I − A − D − ˜ R ( A + bl ) x

(5.3)

Wage rate of profit curves

99

30

w-˛ curv e, fi xed cap ital mod el, S .

25

20

15

w-˛ cur ve, cir cul atin gc api tal mo del ,S .K ore a2 000

Kor ea 2 000

10

5

0

w-˛ curve,

0

0.1

0.2

w-˛ curve, cir culating capit al model, S. Korea

fixed capi tal 0.3

model, S. Korea 0.4

1995

1995

0.5

0.6

0.7

0.8

0.9

1

Figure 5.2 WRP curves, circulating and fixed capital models, S. Korea 1995 and 2000.

and for the fixed capital model, Equation 5.3 becomes w=

ex −1 l [ I − A − D − ˜ RK ] x

(5.4)

Both models, as expected, have a common intercept on the horizontal axis, ˜max = 1 and w = 0 and the same on the vertical axis, ˜ = 0 and w = w max. In the case of the fixed capital model, the rate of profit is estimated only on the fixed capital for the lack of adequate data. We observe that the economy-wide WRP curve in the circulating capital model is concave (dashed line) while that of the fixed capital model is, more often than not, convex (solid lines). In both cases, the difference from a straight line is minimal and so we can characterize the WRP curve as “quasilinear”, which means that alternative techniques must be in the interior of the WRP curves. The case of near linearity is so frequent that it might be considered as a stylized fact or a law-like regularity. The curvature of the WRP curves as in Figure 5.2 precludes the case even of a single switch point, indicative that the new technique is of much higher productivity. According to the selected numéraire vector, the shape of the WRP curves changes from concave to convex and vice versa while preserving its near linear shape. The idea of an arbitrarily chosen numéraire does not find us in agreement because of the lack of any kind of “physical significance”. In our research, the numéraire is the gross output by industry available in the input– output tables. In setting vx = px = ex, we essentially set the purchasing power of a monetary unit (dollar, euro or whatever this might be) estimated by DP or PP equal to that of a monetary unit at MP through the MELT. Hence, the basket of goods used for this purpose is not any randomly selected vector but rather the real basket of goods contained in x, that is, the gross output vector.

100 Wage rate of profit curves

In the case of the Korean economy (see Figure 5.2) of the years 1995 and 2000, the vector of net output was also employed as the numéraire without any particular change in the shape of the WRP curves. The net value added as the numéraire commodity was selected in some other studies; namely, Fitrady (2017) for Indonesia, García and Garzón (2011) for Colombia, and Silva (1990) and Silva and Rosinger (1991) for Brazil with similar results concerning the shape of the WRP curves. The WRP curves presented in Figure 5.3 have been derived from the input–output data of the UK for the year 1990. Details of construction of the various matrices including the capital stock and vectors, see Mariolis and Tsoulfidis (2016b), where the vector z was derived on the basis of the vector of output proportions x such that its product by v (the DP) or e (the unit vector, which is also the MP) to give one, vz = ez = exn −1 = 1 where n is an appropriately selected scalar. It is important to point out that the vector z can be any output vector with the above property and this is usually the suggestion in the Sraffian literature. The estimations are carried out in a way similar to those of the Korean economy with the difference that we fixed the numerator using the normalization condition vz = pz = 1. We observe that the fixed capital model gives a convex WRP curve, while the circulating capital model gives a concave one, which in our case, is only trivially different from a straight line, as shown by the dotted line. This can also be thought of as no different from the Sraffian WRP curve derived from the standard commodity (see Equation 4.5). The next set of graphs refers to the U.S. economy, which is important in its own right, and it has been studied extensively starting from the works of

1

0.8

w-˛ cur ve, fix ed cap ital mo del ,U K1 990

0.6

0.4

w-˛ cur ve, circ ulat ing

cap ital mo del, UK

199 0

0.2

0

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

Figure 5.3 WRP curves, circulating and fixed capital models, UK 1990.

0.9

1

Wage rate of profit curves

101

Shaikh (1984), Ochoa (1984, 1989) and Leontief (1986) and was resumed by Mariolis and Tsoulfidis (2009, 2016a, 2018), Shaikh (2016) and Tsoulfidis and Tsaliki (2019). In Figure 5.4, we display our results for the total U.S. economy of the year 2018, the last available from the BEA input–output table. The estimated WRP curve of the circulating capital model is lightly concave and only slightly different from a straight line indicated by the dashed line in Figure 5.4, while the WRP curve of the fixed capital model displays the expected convex shaped pattern. A straight chord-like dashed line is drawn between the two WRP curves with the purpose of underscoring their exact curvature and shape. Very similar are the results in the case of the U.S. economy obtained for the selected years 2000, 2005, 2007, 2010 and 2014 using the report by WIOD (2016) 54, actually 49, industries (because some industries are simply repeated, as explained in Chapter 4). In Figure 5.5, we observe the convex shape of the WRP curves for the fixed capital model, while for the circulating capital model, the WRP curve is lightly concave as shown in Figure 5.6. The results accord with previous findings for many countries and years, which rule out or, at least, make very unlikely the case of reswitching for the total economy. The numéraire that we choose, once again, is the vector of gross output. From the shape of WRP curves in the fixed capital model (see Figure 5.5), the WRP curve of 2014, the most recent year in the WIOD database, represents a more effective technique than those of the previous years, as this is indicated by its higher vertical intercept that denotes labor productivity. Techniques of both 2007 and 2010 are preferred over the technique of the year 2005 for relative rates of profit higher than 30% and 10%, respectively. 120

100

w-˛ cur ve, circ ula ting

80

60

cap ital mo del ,U SA 201 8

w-˛ cur ve, fixe dc api tal mo del ,U SA 201 8

40

20

0

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

Figure 5.4 WRP curves, circulating and fixed capital models, USA 2018.

0.9

1

102 Wage rate of profit curves 140

201

120

100

0

2007

80

200

201

0

60

200

4

5

40

20

0

0

0.1

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0.5

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1

Figure 5.5 WRP curves, fixed capital model, USA 2000, 2005, 2007, 2010 and 2014.

140

2010 120

100

2000 80

2014 2007 2005

60

40

20

0

0

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1

Figure 5.6 WRP curves, circulating capital model, USA 2000, 2005, 2007, 2010 and 2014.

However, when techniques are close to each other, for example, those of the years 2005 and 2007 or 2007 and 2010, it is possible for particular rates of profit, an earlier technique to appear somewhat more efficient than the more recent one. An explanation of this change in the unexpected order might be the Great Recession of 2007–2009 and for reasons that perhaps

Wage rate of profit curves

103

require further research efforts. Nevertheless, there is no significant technological change to be captured by the location of the WRP curves, and the supposed superiority of the WRP curve of 2007 over that of 2010 is minimal and might be attributed to the construction of price indexes or of other non-quantitatively significant reasons. As these two techniques stand, the 2007 WRP curve shows that the respective technique is more efficient than that of 2010 for almost all relative rates of profit! The same is true of the 2007 technique over that of 2005 or 2000 and those in the intermediate years that do not appear in Figure 5.5. As it has been argued, time and again, technological change is a rather slow and certainly not a linear process depending upon a long gestation period to work itself out and diffuse throughout the economy. Consequently, on examining the impact of technological change, the comparisons of WRP curves distant from each year appears to be more relevant, for example, comparing 2014 with 2007 or 2000. However, 2000 is far too distant to be relevant in a “choice of technique” discussion, as we can confer from a cursory consideration of the WRP curves in Figure 5.5 above. To undertake meaningful intertemporal comparisons, in Section 5.5, we utilize the data for 2007 and 2014, two years not too close to each other but apart enough to allow for technological change to become visible and measurable. Furthermore, the selected period is sufficiently lengthy, so the impact of the Great Recession, especially through 2008–2009 may be ref lected in the location of the WRP curves. The idea is that during the period 2007–2014, we had the elimination of a great deal of technologically backward firms and, at the same time, the introduction of new technologies started showing in the outward shift of the WRP curve of the year 2014. In Figure 5.6, we display the WRP curves for the exact same years in the case of the circulating capital model, so as to get a fuller picture of the way in which the economy operates. The WRP curves in the circulating capital model for the U.S. economy, as expected, are slightly concave, but not curved enough, and so they look more like straight (or rather lightly curved) lines, as we found for the UK, Korean, Greek and a host of other economies. In a way similar to the fixed capital model, the latest technique of the year 2014 is more efficient than the older techniques and when it comes to techniques that are not chronologically far from each other, the differences become indistinguishable. For example, the WRP curves of 2007 and 2010 that are nearly indistinguishable from each other, the crossings, regardless of their number, would not inform us about any “choice of technique” because there is none on an average: a result that might be attributed to a number of factors from which, speculatively speaking, the Great Recession might be the most important one. After a thorough inspection of the data, we can tell that the WRP curve of the year 2010 for relative rates of profit up to 25% is (slightly) more efficient than the WRP curve of the year 2007; nevertheless, the difference is negligible and,

104 Wage rate of profit curves

therefore, indistinguishable. Clearly, the WRP curve of the year 2014 dominates absolutely over the other years, indicating that in the meantime visible technological change has taken place. Near-linear WRP curves make the switching of WRP curves particularly difficult to occur while the double or re-switching is exceedingly more difficult and of course triple switching has not been seen in the relevant empirical research. The occurrence of a single switching of WRP curves is typically associated with consistently higher productivity of labor and lower “productivity” of capital. In Chapter 3, we have shown that the vertical intercept of the WRP curve stands for the productivity of labor, or maximum real wage and the horizontal axis intercept stands for the standard ratio, which is not very different from the maximum rate of profit or “productivity” of capital. The expectation is that labor productivity increases with time and capital “productivity” to decrease. If the capital “productivity” also increases then, we have no switch points at all, and this is not a case to be underestimated for relatively short periods. We say, “relatively short periods” because in the long-run, the rising mechanization of production (the rising organic composition of capital) eventually leads to falling maximum rate of profit (see Tsoulfidis and Tsaliki 2019, ch. 8).2 In our experiments for the year 2014, we found both, the overall productivity of labor and the standard ratio are higher than those of the other years and, therefore, there is no crossing. However, this may not always be the case in individual industries, as our results in Section 5.5 indicate.

5.4 The choice of numéraires and the standard commodity The results of the WRP curves, using as numéraire the gross output vector of the industries comprising the total economy, suggest the near linearity in the circulating capital models, whose slightly concave shape makes them often indistinguishable from the straight line. By contrast, the fixed capital model gives convex WRP curves, close enough to linearity but not so much compared with the circulating capital model. These findings may be challenged as artifacts derived from the possible proximity between the vectors of actual gross output to the standard output, which, as we discussed in Chapter 4, is constructed in such a way as to give rise to a linear WRP curve. The near linearity of the WRP curves may be also attributed to the vectors of employment, l, or consumption expenditures coefficients, b, when utilized as numeraires; because both vectors are suspected as being very close to the lefthand-side (l.h.s.) and the right-hand-side (r.h.s.) eigenvectors of matrix A, respectively (see also Bidard 2020). Hence, we have an interesting hypothesis deserving further investigation. That is, to what extent, if any, the observed near linearity in WRP curves represent a result derived from the input– output structure of the economy? Or is an “artifact” resulting from the choice of numéraire, not too different from the Sraffian standard commodity?

Wage rate of profit curves

105

By focusing on the data of the years 2007 and 2014, and starting with testing the hypothesis of proximity between the vectors of gross output and standard commodity (output), we find that the two vectors, after their normalization in the unit simplex, are quite distant from each other, thereby ruling out the hypothesis of their closeness. More specifically, for the year 2007 and the fixed capital model, we found a mean absolute deviation (MAD) of the two in comparison vectors equal to 93.14%, the d-statistic equal to 112.15%, and the correlation coefficient equal to 35.81%. The circulating capital model gave very similar results, also ruling out the hypothesis of the proximity between the standard and actual output vectors; the MAD of the two vectors is estimated at 83.05% with their d-statistic at 70% while their correlation coefficient is in the low of 30.40%. For the year 2014, we got results comparable to those of 2007. More specifically, the MAD of the standard output and the actual output in the fixed capital model is 128.49%, the d-statistic is 93.81% and the correlation coefficient is in the poor 19.36%. For the circulating capital model of the year 2014, the MAD of the two in comparison vectors is 83.76%, the d-statistic is 64.23% while their correlation coefficient is 34.22%. These results rule out the hypothesis of the proximity of the output vector to the standard product. Moreover, such results, if held in the comparisons between the DP, PP and MP, we would certainly have rejected the old classical and Marxian labor theories of value. The normalized actual and standard output vectors are plotted into two pairs (fixed and circulating capital models) for the years 2007 on the top panel of Figure 5.7 and the respective 2014 graphs on the lower panel. The linear regressions along with the OLS results of the standard commodity (s) against the gross output (x) are presented in each of the graphs with the t-statistics indicated in parentheses and the R-square. The visual inspection of the four graphs indicates that there is no proximity or significant correlation between the standard commodity and the actual output vector of the economy in the two selected years. The regression results do not indicate any strong association between the two vectors in comparison. The linear regression of the year 2007 is weak with an R-square equal to 0.128 for the fixed capital model and only 0.092 for the circulating capital model. Similarly, the linear regression of the year 2014 is also weak with an R-square equal to 0.037 for the fixed capital model and 0.11 for the circulating capital model. The results suggest that the standard commodity bears no proximity with the gross output vector from which it is derived; in effect, the pairs of vectors in comparison are, at best, slightly related with each other. Consequently, the near linearity of the WRP curves must be attributed to reasons other than the proximity of our numéraire to the standard commodity. In effect, the absence, or rather, the weak relationship between the two vectors that we visualize in Figure 5.7 is consistent with the quasi-linear WRP curves.

106 Wage rate of profit curves 0.14

0.14

Fixed Capital Model, USA 2007

0.12 0.1

R² = 0.128

0.08

0.1

s = 0.012 +0.336x (3.15) (2.30)

0.08

R² = 0.092

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0

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Circula˜ng Capital Model, USA 2007

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s = 0.016 +0.106x (5.81) (2.77)

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Fixed Capital Model, USA 2014

0.10

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0.1

s = 0.011 + 0.393x (2.86) (2.63)

0.08

s = 0.016 + 0.097x (5.22) (1.42)

0.08

Circula˜ng Capital Model, USA 2014

0.1

0.08

0.06

R² = 0.117

0.06

R² = 0.037 0.04

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Figure 5.7 Standard vs. actual output, fixed and circulating capital models, USA 2007 and 2014.

It is important to note that for the standard product we use the r.h.s. vector of the matrix K [ I − A ]−1 (Shaikh 1984, 1998, 2016). In fact, the so derived eigenvector may be interpreted as the von Neumann ray and the eigenvalue as the maximum profit rate or the standard ratio. We have also experimented with Sraffa’s standard output vector, that is, the r.h.s. eigenvector of [ I − A ]−1 K (Pasinetti 1977, pp. 92–99) whose results were no different; in fact, it displayed higher deviations (see Ochoa 1984). Hence, our chosen numéraire, the gross output, is quite different from the standard product regardless of the way the latter is estimated. We continue testing our hypotheses by examining, this time, the proximity of the row vector of employment coefficients, l, with the l.h.s. vector, π, of the matrix A or, what is the same, the eigenvector derived from the matrix H = A [ I − A ]−1 . Hence, if the two in comparison vectors are close to each other, we end up with the equality of unit labor values l [ I − A ]−1 to relative PP. The last equality implies uniform capital intensity (or value compositions of capital) across industries and the WRP curves are straight lines. The results of our testing (after our usual normalization in the unit simplex) are plotted in Figure 5.8 in a panel of two graphs placed side-by-side using only the circulating capital models for the years 2007 and 2014 for the USA. The deviation between the two vectors is far

Wage rate of profit curves 0.04

0.04

Circulating Capital Model, USA 2007

0.035

0.03

˜ = -0.0997l + 0.0204 (1.7) (13.0)

0.025

0.02

0.015

0.015

0.01

0.01

0.005

0.005

0

0.02

0.04

0.06

0.08

˜ = -0.4019l + 0.026 (3.95) (12.04)

0.025

R² = 0.0527

0.02

0

Circulating Capital Model, USA 2014

0.035

0.03

107

0.1

0

R² = 0.2313

0

0.01

0.02

0.03

0.04

0.05

0.06

Figure 5.8 Employment vector vs. l.h.s. eigenvector of matrix A, USA 2007 and 2014.

too large and by no means explains the near linearity of the WRP curves, which, as in the case of gross output, must be attributed to the interconnection of industries and, in general, in the input–output structure of the economy. The scatter graphs of the two years in Figure 5.8 show a rather weak relationship between the two vectors. The relation is weaker in the year 2007 (hence, we have estimated the WRP curves for twentieths of the maximum ˜ = 1) and stronger in the year 2014 (hence, we have estimated the WRP curves for tenths of the maximum ˜ = 1). We have also estimated the MADs and the d-statistics, which for the year 2007 were 123% and 111.4%, respectively; the correlation coefficient of the two vectors was –22.9%. The results for the year 2014 gave a somewhat closer relation; MAD = 113.1%, d-statistic  = 86.5% while the correlation coefficient = –48.1%. There appears to be a relation between the two in comparison vectors, but they are not close enough to attribute to their proximity the observed linearity in the WRP curves for the USA and, most likely, for a host of other economies. 3 We have also tested the circulating capital model of the UK for the year 1998, with results quite similar to those of the USA. In particular, the MAD and the d-statistic between the vector of output and the standard output (both normalized in the unit simplex) is 80.27% and 70.18%, respectively. The correlation coefficient between the two vectors was found to be quite high at 63.04%. Similarly, the MAD of the normalized vector of employment coefficients and the l.h.s. eigenvector of the matrix of technological coefficients, A, is 50.40% while the d-statistic is 55.78% and the correlation coefficient was particularly low, equal to –5.41%. Finally, we tested the vector of workers’ consumption coefficients and the standard output (both normalized in the unit simplex), and the results obtained for the MAD and the d-statistic are 165.07% and 103.89%, respectively, while

108 Wage rate of profit curves

the correlation coefficient is at 30.07% (Bidard 2020). The above results rule out the case of the proximity of vectors of output, employment and consumption coefficients with the standard vectors in the UK input–output data for the year 1990. In conclusion, the results derived from the U.S. economy suggest a weak or the lack of any relationship between the vectors of the standard commodity and actual output. There appears to exist a relationship but is far too weak to lend support to the hypothesis that the proximity of the two vectors should be held responsible for the near linearity of the WRP curves. The latter must be attributed to factors other than the proximity of the chosen numéraire and the standard commodity. Similar were the results of the vector of employment coefficients with the l.h.s. eigenvector of the matrix of technological coefficients; after all, the employed technology depends on the productivity of each sector or what amounts to the same thing, the reciprocal of the employment coefficients. Hence, one cannot base the observed shape of the near linearity of the WRP curves on a relationship between the employment coefficients and the standard ratios. Finally, we tested the vector of goods on which workers spend their money wage and it was not found to be associated with the standard output in the case of the UK data for the year 1998. The results suggest that the near linearity of the WRP curves are independent of the chosen numéraires. Another hypothesis to explain the near linearity of the WRP curves is that industries comprising the economy may have different capital–output ratios or value compositions of capital but when their vertically integrated counterparts are compared, the differences in capital intensities get much smaller. The issue raised with this hypothesis is that the estimation of capital intensities requires the prior knowledge of PP while the capital intensities are required for the estimation of PP. Hence, we discern  the presence of a vicious circle. The circularity issue may be circumvented by measuring the vertically integrated capital intensities in terms of values, but this may give rise to other issues. For this reason, in Chapter 6, we make an effort for an alternative interpretation of the WRP approximate linearity based on a non-price explanation. More specifically, we seek to find the extent to which the columns of the vertically integrated technological coefficients are, to a large extent, linearly dependent as this is indicated by the particular distribution of the eigenvalues and singular values of the economic system matrices.

5.5 WRP curves in 49 industries, USA 2007 and 2014 Up until now, we have estimated the WRP curves of the economy using as numéraire the vector of the economy’s gross output, which we found to be quite different from the standard output. In similar fashion, the vector of employment coefficients was found to be quite different from the l.h.s.

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eigenvector of the matrix of the vertically integrated input–output coefficients. Consequently, we cannot attribute the observed near linearity in the WRP curves to the alleged closeness of the output and employment vectors to their respective counterparts. In this section, using as our numéraire not the gross output vector of the total economy, but rather each industry’s output, we get as many different WRP curves as the number of our effective 49 industries in the effort to shed further light on developments taking place in each individual industry in connection with the rest of the economy. We experiment first within the more popular circulating capital model and subsequently with the more realistic fixed capital model. By selecting two years quite distant from each other, such as 2007 and 2014, we make meaningful comparisons with respect to technological change. 5.5.1 WRP curves, circulating capital model By assuming away the matrices of depreciation and indirect tax coefficients, we get from Equation 5.3 the following dot division: −1 w = e x . / ˇ˘l ( I − A − ˜ R ( A + bl )) x 

(5.5)

which gives two row vectors on which we apply an element-by-element division, ./, to get the WRP curves of each of the 49 industries of the U.S. economy. We say 49 industries, because, as we explained in Chapter 4, some similar industries are simply repeated in the WIOD (2016) database. In the interest of simplicity and clarity of presentation, we select two years, 2007 and 2014, which are quite apart from each other. A technological change, even though might be significant, does not appear immediately in the input–output data because only a relatively few establishments adopt the new technique in one or just a few industries. The estimated techniques, as these are ref lected in the WRP curves, are average techniques. Consequently, only after the passage of a sufficiently long time, we start to distinguish differences in productivity and the techniques in use inasmuch the majority of establishments have adopted or have been affected by the new technology. We believe that a seven-year period is enough to allow these processes to take place. In Figure 5.9, we present the WRP curves of those industries using a circulating capital model, which displays the expected or usual shape. These curves represent techniques, which are chosen from the hypothetical book of blueprints available in the U.S. economy. The graphs below, as Ochoa (1984) admits, are very difficult to explicate. In our view, each of the graphs below shows WRP curves of individual industries, whose connections with the others is depicted by the input–output coefficients in Equation 5.5. The equation might be described as the unit cost of production, which is expected

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to decrease with the passage of time because, other things constant, the technological change increases labor productivity and reduces the unit cost of production. If the relative rate of profit is zero, we get the unit labor values per industry, whose reciprocal is equal to the (gross) productivity of labor or the maximum real wage, the intercept of the vertical axis. If the relative rate of profit increases, it follows that the unit cost increases as well until the attainment of the standard ratio (or “productivity” of capital approximately equal to the maximum rate of profit) at a relative rate of profit equal to one. In short, the further out to the right from the origin of the WRP curve of industry, the more efficient the technology in use, the higher the productivity and the lower the unit cost of production. 270

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The industries whose WRP curves of the two years cross each other and, therefore, are characterized by switching points are displayed in a separate set of graphs in Figure 5.10. The outer dotted WRP curve stands for the year 2014, while the inner solid line represents the year 2007 in all of the 39 graphs in Figure 5.9. The

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estimations are in constant prices (2010). A cursory look at the graphs reveals that the system indeed behaves more “irregularly” rather than “regularly” in Schefold’s usage of the terms. During these seven years, the U.S. economy experienced technological change and, therefore, improvements in productivity as this stands out in the shape of the 39 WRP curves. The effect of technological change is ref lected in the difference of the size of the area under the two curves. In defining technological change by the difference in the areas under the WRP curves, we observe industries experiencing significant technological change, for example, industry 1 (crop and animal production, hunting, etc.), industry 5 (manufacture of food, beverages and tobacco products) and industry 31 (land transport and transport via pipelines), among others. However, we also notice industries that are only marginally affected by technological change, for instance, industry 32 (water transport), industry 34 (warehousing and support activities for transportation) and industry 42 (insurance, reinsurance and pension funding, except social security), among others. For the nomenclature of the other industries, see Table 4.2. In the next set of graphs in Figure 5.10, we display the industries in which crossing in their WRP curves has taken place. We observe ten occurrences of switching, that is, 10/49=20.41% of the cases and none of the double switching. A careful examination of all the graphs shows the near-linear nature of the WRP curves that rules out the case of the many different curvatures and more than one switch points. The shape and location and the trajectories of WRP curves reveal the absence of any significant technological change in these ten industries. We have essentially, on average, the same technology in use during this period of seven years, and naturally, one expects that there will be crossings in quite many cases. If, however, we repeat the experiment comparing the year 2014 with the more distant 2000, it is very unlikely to find industries with the productivity of labor higher than that of the year 2014. In other words, the technology of the year 2014 will dominate absolutely in all industries of such distant years. In examining each and every one of these graphs in Figure 5.10, it is important to bear in mind the following: for the year 2007, the equilibrium relative rate of profit is ˜ = 31.69% and the standard ratio (maximum rate of profit) R = 1.06. By taking into account the equilibrium relative rate of profit in the case of the circulating capital model, we observe that only in industry 37 (publishing industries and software) crossing takes place at a relative rate of profit ˜ = 30%, that is, at a point near the equilibrium. In all other cases, crossing takes place at rates of profit too high (near the maximum), or too low (near the minimum) and rarely around the equilibrium relative rate of profit. It is also interesting to note that by construction, the intercept on the horizontal axis is the same for the two techniques. The difference between the two curves is negligible in terms of the maximum rate of profit since for the year 2014, the maximum rate of profit R = 1.07 is slightly higher than that of the year 2007. By contrast, the maximum real wage or the productivity of labor of the year 2014 is usually much higher than that of the year 2007 in all industries except three (industries 4, 15 and 37).

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In Figure 5.11, we place together all the more “regularly” (in the mathematical sense of the term) behaving WRP curves by taking their difference for the two years. If the difference is on the positive (negative) side, it means that the WRP curve of the year 2014 is above (below) that of the year 2007. We observe that there is at most one crossing and, therefore, the sign alternates from positive to negative or vice versa, only once. This is equivalent to saying that the case of double-switching does not appear like a realistic possibility in the actual data of the U.S. economy in these two years. Of course, we do not exclude a priori the cases of double switching; we only ascertain the near-linear nature of the WRP curves rendering the case of re-switching a remote possibility. It is important to note that we also used the actual maximum rate of profit, R, for each year and not the maximum relative rate of profit, ρ. The results were, as expected, the same; that is, the signs changed only once. In constructing the WRP curves, we decided to use the relative rate of profit because the maximum rates of profit of the years 2007 and 2014, as we have already noted, although appear to be different, their difference is only in the second decimal, 1.06 vs. 1.07. 5.5.2 WRP curves, fixed capital model In Figures 5.12 and 5.13, we display the WRP curves of the same two years, which have been derived by applying an element by element division of the two vectors as shown in Equation 5.5; the derived WRP curves are quite similar to those of the circulating capital model.4

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In the above set of 42 graphs, we observe no crossings and the curves, with only a few exceptions (with industry 44, the real estate, as the more emphatic) are not far from linearity. On further thought and having excluded the case of reswitching, we end up with the conclusion that the WRP curves and their location indicate the presence of technological change in these industries during these seven years. The seven graphs in Figure 5.13 convey a somewhat different picture. The near-linear WRP curves cross each other indicating switching casting doubt on both, the neoclassical theory of value and the old classical economists’ and Marx’s theorization of price movements based on industries’ capital intensities relative to the economy-wide average capital intensity. On a closer examination of the graphs, however, we realize that their shape indicates more of the lack of any significant technological change. This is another way to say that the change in distributive variables, under these circumstances, may lead to crossings but as the vertical and horizontal intercepts remain approximately the same, it is more accurate to say there has not been any noticeable technological change in these seven industries during the seven years. It is also important to note that crossings occur at very low relative rates

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of profit, as we observed in the circulating capital model with the possible exception of industry 37 (publishing activities). A thorough examination of these “regular” industries starting from number 4 (mining and quarrying) reveals that the crossing occurs at a very low relative rate of profit and also the two curves move close to each other. Such movement of the WRP curves indicates that technological change is slow and this particular industry may be in a state of decay because of internal (extension of the margin of extraction) or external (environmental) pressures. Similar is the finding and explanation in the remaining industries, that is, industries 9 (printing and recorded media) and 15 (manufacturing metals), while industries 24 (electricity and gas) and 25 (water collection, treatment and supply) are the usual utilities, whose productivity is not expected and it does not change in any significant way during those seven years. Industry 37 (publishing industries and software) is also characterized by low productivity improvements, which are not visible in time periods such as the one under examination. The same might be true with industry 40 (computer and consultancy related activities). Finally, industries 46–49 (architectural and engineering, research and development, advertising, other professional activities)

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are subsumed together in the WIOD (2016) precisely because they are involved in services whose productivity either is not easily measurable or it does not change very much during the seven-year period of our analysis. This is equivalent to saying that in these industries, there are virtually no alternative techniques to “choose” from. It is important to note that what we see in the above graphs is what happened to these industries on average over a specified period of time. The average is usually an adequate index of what happened technologically in each industry; however, there might be cases where the technological change may affect only a few establishments in a particular industry. Thus, the advances in productivity by these few establishments are overshadowed by the far larger number of establishments that continue using the old technology. Hence, it is possible the technique employed in an industry a few years ago appear to be more efficient than the more recent one. These particular cases may become the focus of interesting research efforts within the field of industrial organization. Finally, as in the case of the circulating capital model, we fixed the horizontal intercept at the maximum relative rate of profit ˜max = 1, for both years of our analysis. So in all WRP curves comparisons, we have a common horizontal intercept at ˜max = 1 and different intercept on the vertical axis, that is, the maximum real wage in the two years, 2007 and 2014, under comparison. The year 2014, even in the above seven exceptional cases, nearly always displays maximum real wages higher than those of the year 2007. In the exceptions are industries 4, 15 and 37 whose value of output when used as numéraire for the year 2007 gives rise to a real maximum wage somewhat higher than that of the year 2014 indicative of the stagnant productivity 20

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growth. Finally, yet importantly, the shape or rather the curvatures in all WRP curves regardless of the kind of capital (circulating or fixed) is quite similar. This result indicates that particular features and idiosyncrasies of industries are preserved over the years. In any case, the similarities in shapes and curvatures of the WRP curves might become the subject of further study. In a way similar to the circulating capital, we display the “regular”, in a sense, WRP curves in Figure 5.14, which shows the difference of the WRP curves of the two years with switching behavior. As expected, we have a sign change only once indicating that a technique of the past might be the preferred one up to a rate of profit; that is, there might be a technique of the past, which continues to be the most profitable in all relative rates of profit. We stress once again that the same results in regard to WRP curves were derived utilizing the actual maximum rates of profits whose difference is only in the second decimal, 0.542 (for 2014) vs. 0.531 (for 2007).

5.6 Discussion of empirical findings The empirical findings from our study of the U.S. input–output data as well as the results from other researchers suggest the following: a

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The WRP curves display, more often than not, an “irregular” or what is the same “well-behaved” shape in the aggregate economy. Slightly convex in the case of a fixed capital model, and more linear and imperceptibly concave shape in the circulating capital model. The same conclusion does not hold at the level of particular industries as we showed in Figures 5.9–5.14. Both curvatures (convex or concave) are possible independently of the employed (circulating or fixed capital) model. There is a clear trend in rising maximum real wage or productivity of labor for the aggregate economy and to individual industries with only a few exceptions indicating idiosyncratic characteristics of particular industries. Consequently, a rising maximum real wage amounts to falling unit costs and prices, as a result of rising productivity induced by technological change. For low profit rates, the WRP curves rotate upward induced by technical change. We just cannot see this effect in our graphs, simply because by construction and convenience in presentation, the curves share the same horizontal intercept, which is the standard ratio or maximum relative rate of profit, for ˜ = 1. The maximum rate of profit, R, or what is the same thing, the output–capital ratio or “productivity” of capital over long stretches of time is expected to be falling. If the vertical intersection of a recent WRP curve is above that of the past years and, this is the usual case, we have technological progress, which potentially can be measured and compared with that of the curves of past years by the area under the respective curves.

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e f

The WRP curves either do not cross at all, the usual case or, cross only once in the relevant (positive) region. Given all of the above, it follows for reswitching to be possible, the two WRP curves must be too close to each other and the more convex WRP curve must have both ends (the maximum real wage and maximum rate of profit) higher than the other WRP curve. The above requirements make reswitching a remote possibility.

Table 5.1 reports the maximum real wages and the maximum rates of profit for both circulating and fixed capital models for the U.S. economy and selected years. These findings may be of help in our judgment about the possibilities of reswitching, provided the approximate linear character of the WRP curves. In Table 5.1, we observe that both models share the same maximum real wage (or productivity of labor), which increases over time indicating technological progress. This combined with falling, as expected from the classical analysis, maximum rate of profit limits the possibility of reswitching; this, however, does not preclude the case, for short time periods, the maximum rate of profit to be higher (as is the case of the years 2007 and 2014) than that of a previous year against which is compared. Under these circumstances, the old technique is inferior and is absolutely dominated by the more recent; so the case of reswitching may be excluded. Consequently, Samuelson’s (1966) “facts of life” do not include the general occurrence of “capital reversals” on which Sraffians have invested so much over the years in the hope that neoclassical economists would ultimately recognize the “truth” and turn to the strand of the classical theory of value and distribution that makes no use of the upsetting, for its social repercussions, labor theory of value. The idea of switching is not necessarily supported by the empirically found near-linear shape of the WRP curves or the monotonic shape of the PRP curves. The above empirical findings do not, however, rule completely out the case of switching, which, time and again, has been found to be not only exceptional but, whenever it happens, it takes place at unusually high or Table 5.1 Maximum real wage and rate of profit in circulating and fixed capital models Circulating Capital Model

Fixed Capital Model

Years

Maximum Real Wage Maximum Maximum Real Wage Maximum or Labor Productivity Rate of Profit or Labor Productivity Rate of Profit

2000 2005 2007 2010 2014

112.68 126.28 128.47 128.00 139.93

1.068 1.101 1.062 1.164 1.074

112.68 126.28 128.47 128.00 139.93

0.623 0.515 0.531 0.529 0.542

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low rates of profits. Consequently, the rare occasions of switching do not cast doubt on the classical political economy theorization of price trajectories consequent upon changes in income distribution as Sraffian economists in the late 1970s hastened to claim (Steedman 1977). From the Sraffian perspective, Kurz and Salvadori (1995, p. 450) were from the first if not the first that commented on the empirical findings of the 1970s and 1980s, all of which were supportive of the near linearity WRP curves. They characterized the empirical works “fundamentally mistaken” by arguing that the WRP curves because they were constructed from input–output tables of different years they did not allow for the effective choice of techniques to take place over the examined years. In other words, the so derived WRP curves represented the only technique applicable in a single year, whereas the theoretical relation “refers to technical knowledge at a given moment of time”. However, one cannot downplay the fact that when empirical data are given at quite distant times, say for every five years, as for example in the U.S. benchmark input–output tables, then apparently there is no real “choice” of alternative techniques from those employed in the past. The reason is that firms are forward-looking and, neither, choose techniques from a hypothetical book of blueprints of various past alternatives nor they just look at the distributional variables and switch to a new technique, which suits best to the new combination of factor payments and so forth. The near linearity of WRP curves for the various economies, however, should not lead to the conclusion that the neoclassical theory escapes criticism. The marginal productivity theory of income distribution, the cornerstone of neoclassical economics, is based on the assumption of perfect competition in all markets, which allows for the perfect substitution of factors of production that leads to the choice of technique and the optimization behavior of the firm. However, the idea that firms freely choose between alternative techniques available in a book of blueprints specifying all possible input combinations for the production of a given f low of output, and select the most efficient combination, is meaningless in conditions of real (or classical) competition, a dynamic process of rivalry among units of capital over market shares (Shaikh 1978; Tsoulfidis and Tsaliki 2005). The idea is that in real competition, the so-called “choice” of technique is not realized in any smooth way, as is usually claimed in the neoclassical approach, according to which even a slight increase, for example, in wage, is enough to lead to the immediate substitution of labor for another relatively cheaper productive resource. In real competition, techniques change after the passage of relatively long stretches of time in order for their effects to become visible. Moreover, the substitution between factors of production is more limited than is usually thought, in the short period, at least, since techniques, which are actually in use, are not so sensitive to price changes. Consequently, between the neoclassical notion of perfect substitution and Leontief ’s conception of fixed proportions (the famous “cooking recipe”

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analogy), we would say that the latter represents a closer to reality description of the economy. Sraffian economists (not all) have silently adopted an approach to competition no different from perfect competition and along with a static time framework attempt to derive the reswitching results. It seems that followers of the Sraffian tradition would rather exclude these near linearity WRP, disturbing for their approach, curves and for this reason, they pose a Kuhnian kind of “protective belts” around their core proposition of “reswitching of techniques”. If reswitching is true, not only theoretically but also empirically, then the neoclassical theory is in big trouble and the Sraffian alternative is there waiting to become the new theory of value and distribution adopted by economists because of its logical consistency and, therefore, superiority. Because the required data are available, there is no choice for the Sraffians but to utilize them in testing the validity of their core theoretical proposition. The testing terrain for the Sraffians and their theory of value is the derivation of the exact shape of both the PRP trajectories and the WRP curves associated with them. It is important to emphasize that the WRP curves that we derive for each year and industry are just the weighted average technique employed in each industry, where many firms or rather establishments according to the BEA and other official statistical services, or “units of capital” according to Marx, are activated. However, each one of the establishments or firms is employing its own technique. Usually, relatively few capitals or firms activated in the industry utilize the very advanced techniques and other less advanced capitals are stuck with their older and perhaps outdated technique, which probably should be abandoned any time soon; however, capitals are bound to carry on their old technologies because of their past investment decisions, which become compelling under the present circumstances. The weighted average technique is the one ref lected in the WRP curves of each industry and not necessarily the dominant or the “best practice” in the sense that it attracts new investment and determines prices and profit rates (Salter 1969, ch. 2; Tsoulfidis 2015). This does not mean that the average techniques are necessarily by far too different, both quantitatively and qualitatively, from the dominant (especially in most manufacturing and service industries); thus, by utilizing the average technique, we do not get quite different results from those that we would have derived had we had the option of using the dominant techniques. However, there are industries such as in agriculture and mining where the “dominant” producers (or “regulating capitals”) happen to be the least efficient ones and over time, as demand increases, we may have the old “least efficient” producers to become the more efficient and the new average technique may even fall below the old, provided that there is only increase in demand and no technological change. In other industries, such as the high-tech ones, the most advanced firms (the dominant ones) set the pace for the whole industry and operating under these conditions becomes extremely difficult for the outdated technological firms to catch up (Tsaliki and Tsoulfidis 2015).

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From the above discussion, it does not follow that the various other techniques, even the dominant ones, activated within an industry and their WRP curves are of an altogether different shape from the specific year’s weighted average. Moreover, if we could make a selection of dominant techniques over the years, we would not obtain any different WRP curves and frontiers from those derived from the available input–output data average technique based on benchmark years during which there is a sufficiently long time for technological change to take place. After all, the more advanced techniques of today, employed by a relatively limited number of establishments in the industry after a few years, will be diffused and they will become the new industry’s average. It follows, therefore, that if the shape (quasi-linear) of the WRP curves in every year tested are quite similar to each other, one does not expect any different WRP curves had the choice of techniques been made over the years.

5.6.1 Some recent developments Han and Schefold (2006) shed new light on the question regarding the shape of the WRP curves and stimulated new research efforts in capital theory and its controversies. They utilized data from the OECD database of thirty-two 36-sectoral input–output tables for nine OECD countries spanning the period 1986–1990 in order to test empirically the possibility of reswitching. For this purpose, they make pairwise comparisons between the WRP curves of ˜ 32 ˝ pairs of countries, and they find ˛ ˆ = 496 envelopes of WRP curves de° 2 ˙ rived from 32 input–output tables with total 4389 switch points, from which only one involves reswitching while reverse capital deepening is involved in about 3.65% of cases.5 More specifically, in 96.35% of the cases, the capital intensity and rate of profit (interest) are inversely related, as expected by neoclassical economics. In 2.28%, a fall in capital intensity is accompanied by a fall in profit–wage ratio, meaning that as capital intensity decreases, the wage rate increases along with employment. Capital intensity increases by 0.73% of the cases and so does employment despite the fall in the rate of profit and the rising wage. Finally, in 0.64% of the case, the capital intensity increases, reducing employment. Han and Schefold’s (2006) findings suggest that the case of reswitching of techniques (or capital reversals) is unrealistic, which finds us in agreement; however, their method of analysis raises various important issues, which we present next –

Pairwise comparisons of industries and countries and the techniques associated with them do not make much sense even in the ideal case of the perfect competition. For example, in comparing the transportation industry of Switzerland with that of Italy or the electricity industry of

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–

– – –

Greece with that of France and so forth are questionable. The reason is that Italy’s transportation industry includes shipping, which is not applicable to Switzerland, and France’s electricity industry uses a great deal of nuclear energy, which is not available in Greece. Thus, there is not really a choice of techniques, especially when the level of disaggregation is limited to only 36 industries. A more industry detailed input–output structure is not necessarily a more realistic description of the so-called “choice” of techniques, because even within the same industry, country and year, there are actually no alternatives and, what is more important, available techniques. The comparisons between seemingly alternative techniques are static and stay inactive until the movement of prices re-activates them by restoring their profitability. The absence of fixed capital in both theoretical and empirical analysis, probably because its treatment is postponed for future research efforts. Finally, Han and Schefold (2006), but also Schefold, in his more recent works (2009, 2013, 2020), deliberately assume perfect competition in order to prove the thesis of near-linear WRP curves, under the least favorable circumstances. If the results support the near linearities of the WRP curves, then the no reswitching hypothesis stands on firmer grounds.

The trouble with this line of research, however, is the nature of technological change, which is slow, non-linear and irreversible on many occasions. Han and Schefold (2006), however, opine that what [technique] is used today can be used tomorrow, and what was used yesterday might be used again. (Han and Schefold 2006) In so doing, they essentially downplay the notion competition proper of classical economists and Marx and utilized instead the neoclassical idea of perfect competition and the idea of choice of techniques associated with it. Furthermore, technological change is inconceivable without the presence of fixed capital and the rising capital intensity in the long run. All these realistic features of the operation of actual economies require the analysis to be conducted in conditions of real or classical and by no means perfect competition. Zambelli et al.’s (2017) simulations are also based on the assumption of perfect competition and on a circulating capital model; the results are against the reswitching. In their article, however, paradoxically, the likelihood of the (circulating) capital intensity to be inversely related to the rate of profit (that is, there is no capital reversing) is limited to only 40% of the cases. Therefore, they conclude that “real Wicksell effects […] do hold” (p. 115) rendering the neoclassical analysis unsustainable. Zambelli et al.’s (2017) study, unlike Han and Schefold’s (2006), assumes that the choice of techniques is not restricted to a single or at most pair of economies but rather extends to include all

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available in the sample economies and associated technologies. Such an extension, technically more advanced and at first sight, appears more pragmatic than that of Han and Schefold’s (2006) simple pairwise comparisons; however, on further thought, we find that Zambelli et al.’s (2017) analysis suffers from the exact same problem of issues of technological change and the nature of competition proper and the lack of fixed capital stock. The neglect of fixed capital in the studies in Han and Schefold’s (2006) and Zambelli et al.’s (2017) approaches is not without consequences because oneway or another it is associated with the assumption of perfect competition. All firms are alike and the presence of fixed capital should not contaminate the results. Technological change without accounting for the presence and consequences of fixed capital is deceptive and is out of touch with the economic reality, because it gives the idea that firms merely choose technologies from the book of blueprints. Cost-minimizing technique, by and large, come with more fixed capital per unit of output, which firms bear for long periods under given circumstances. Their actual “choice” may in effect become the technique with a higher unit cost. Hence, the analysis and derived results in both empirical researches are arguably misleading from a realistic perspective, because their WRP curves are not derived by taking into account the presence of fixed capital. By not considering fixed capital, Han and Schefold (2006) and Zambelli et al. (2017) end up with a rate of profit, which actually is the profit margin on circulating cost with properties and features quite different from those of the rate of profit proper. As already pointed out, the technological coefficients in an input–output table describe the average of all employed techniques. The structure of input– output coefficients is a snap-shot of an ongoing process of technological change. As time goes by, the technological improvements, which are brought about mainly through the introduction of fixed capital, are ref lected in the rising productivity of labor or what is the same, the slowly but steadily falling unit labor values of commodities associated with rising capital intensities. The latter takes on the status of a “law of diminishing labor content of commodities” (see Carter 1980; Farjoun and Machover 1983; Cockshott et al. 1995; Seretis and Tsaliki 2016; Tsaliki et al. 2018; Cogliano et al. 2018). This law-like regularity is ref lected in the slowly falling input–output coefficients, as it has been shown in the works of Carter (1980), Ochoa (1984, 1986, 1989) and more recently with input–output data and capital stock for the USA for the period 1995–2009 (Tsoulfidis and Tsaliki 2019, pp. 154–155). The results of the above studies show falling unit labor values of industries for the U.S. economy accompanied by rising capital intensity. In order to illustrate the above, let us hypothesize that a new technique, characterized by higher productivity and, therefore, lower unit cost of production, is introduced in a particular industry. The new more efficient technique is not freely available to all participating firms in the industry and this may be true not only in the current but, also, in the years to follow. The high initial cost of the new technique usually prevents firms (rather establishments)

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from adopting it for reasons that relate to risk and mainly because firms are locked-in by their older technology and capital equipment. As a result, the new technique will not be ref lected in the input–output coefficients and its productivity effect will be overshadowed by the numerous other less efficient firms comprising the industry that are bound to employ the old technologies. The more recent techniques will shape the new average some years later and then, in their turn, will overshadow the future new technologies, and so forth. This dynamic evolutionary process is certainly not captured by the assumption of perfect competition. Furthermore, the absence of fixed capital or, rather more generally, of capital advanced, may lead to bias results and probably to mistaken conclusions concerning the pragmatic use of alternative technologies especially when the analysis extends in the long run. In the same spirit, Gandolfo (2008) commenting on Schefold’s (2008) article argued that substitutability is only possible ex ante; once a technique is installed, there is no substitutability, and for this reason, he criticizes the models that assume that capitalists freely choose techniques from a book of blueprints. He further argues that the data used, being based on real-life observations, presumably include nonlinearities and non- uniformity, as well as putty-clay phenomena. If so, they are not suitable to either confirm or disprove the theory behind the [capital] controversy. In particular, […] the debate on reswitching and reverse capital deepening makes sense only ex ante. Ex post, changes in the real wage rate (for example determined by bargains between firms and trade unions) do not lead to changes in technique, but to changes in the rate of profit, at least in the short run. (Gandolfo 2008, p. 799) Cogliano et al. (2018) have also raised the issue of fixed capital and its effects on the WRP curves and by utilizing input–output data and fixed capital stock matrices for the German economy spanning the period 1991–1999, they derived quasi-linear WRP curves pretty much like the ones that have been presented for the USA and a host of other economies. They opined that the appropriate description of alternative techniques and the derivation of realistic WRP curves should necessarily include fixed capital and depreciation expenses associated with it. Furthermore, they argued that by taking into account the presence of fixed capital, the estimated rates of profit are no longer exotic and so the WRP curves are more realistic,6 a result that justifies their remark that “a sufficiently big decline in wages would lead the economy ‘back in time’ to older techniques”. Consequently, these authors acknowledge that factor substitution of the neoclassical type may exist in actual economies only as an exception to accommodate unusual circumstances (externally or internally generated shocks). Cogliano et al. (2018), however, continue by taking issue with the classical treatment of the real wage and rate of profit by arguing that economies

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are characterized by both, differential wages and rates of profit. They further claim that neither the wage rates, nor profit rates are attracted to uniformity and by doing so, the classical concept of “turbulent equalization” and attraction to uniform real wage and rate of profit is spirited away. In short, they accept the monopolistic or oligopolistic and not the perfect competition side of the same neoclassical conceptualization of competition. These authors mistook the concept of tendential equalization of classical economists and Marx with the concept of “convergence”, a purely neoclassical conception as it has been argued by Tsoulfidis and Tsaliki (2019, and the literature therein). Moreover, this view of competition is the other side of the same as Zambelli et al.’s (2017) and Han and Schefold’s (2006) conceptualizations of the static neoclassical competition. The latter is created to accommodate the neoclassical theoretical needs and is characteristically different from the classical competition, which ref lects a real time process by modeling a turbulent equalization process of both profit and wage rates (Botwinick 1993; Shaikh 2016; Tsoulfidis and Tsaliki 2005, 2019, ch 6; Mokre and Rehm 2020).

5.7 Summary and conclusions The capital theory controversies took place in a rather compressed period of time, during the 1960s. In the controversy, the Cambridge U.K. side prevailed by presenting stronger and more convincing arguments, as this has been admitted by Samuelson (1966) and later by a chorus of neoclassical economists. For example, Stiglitz (1974) opined that the debate showed the possibility of reswitching, but this should not be interpreted as that neoclassical economics should be abandoned. On the contrary, Stiglitz, following the traditional neoclassical line of downplaying any result that contradicts the neoclassical theory argued that the possibility of reswitching obtains a status very similar to that of Giffen’s goods and the associated paradox. Mas-Colell (1989) admitted that the relationship between capital and labor and between the rate of profit and real wage may take any possible shape. This is equivalent to saying that the demand schedule for capital is not well-behaved, that is, it is not inversely related to the interest rate; similarly, the demand schedule for labor is not necessarily inversely related to the wage rate. Another inf luential neoclassical economist Lucas, Jr. (1988) acknowledged that to the extent that the Cambridge controversy was about whether capital consists of heterogeneous goods (what else? one may wonder), the Cambridge, U.K. side won the debate. As we have argued in Chapter 3, in the case of linearity, the slope of the WRP curve is equal to the capital–labor ratio, whose slope with regard to the rate of profit is constant and negative. Our empirical research has shown that indeed the WRP curves are approximately linear; however, from this finding one should not hasten to claim that the neoclassical postulates hold and that in the CCC the U.S. side won this time the debate. The truth is that

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the neoclassical microeconomic theory requires the testing of the output produced (and not of income) through the marginal physical contributions of factors of production (Shaikh 2016, pp. 429–431; Kurz 2020).7 In such a testing, the arrow of causality stems from the marginal physical product to the payments of the factors of production according to their marginal contribution to the total output and not the other way around. However, neoclassical economists, not rarely, reverse the arrow of causality by taking the value of output, whose estimation presupposes prices and, in doing so, they are forced to assume constant income shares. Empirically, this was quite convenient because of the approximate constancy of income shares, which for the U.S. economy was true up until the mid- to late 1970s.8 Since then, the profit share is rising and so the constancy of income shares no longer holds; thus, the marginal products of capital or labor are no longer equal to their respective payments. In contrast to neoclassical economics, the old classical political economy microeconomic theory is based on real and, at the same time, operational conceptualization of competition with objective givens (the technology, the real wage and the level of output with its allocation to industries), all of which can be quantified. It is important to emphasize that the character of technology is such that the above givens are rigid and do not allow substitution of inputs unless there are dramatic externally or internally generated shocks in the economy. Leontief ’s description of technology as a “cooking recipe” proves to be true, as we know from Leontief himself and the relevant empirical literature. The equilibrium prices derived from the above givens are pragmatic and so is the conceptualization of competition. From the preceding discussion, one thing is certain that Sraffa’s agnosticism regarding the movement of prices induced by changes in income distribution and the WRP curves associated with these are justified theoretically under particular conditions. However, practically, the actual data of the economies that have been studied, so far, show, overwhelmingly, the quasi-linearity of the WRP curves and potential frontiers; more importantly, these results are derived from the internal input–output structure of the economy and are independent of the utilized numéraire. Our results derived mainly from the U.S. economy, over many years, and from other major economies have the required generality and persistence that we may assign them the status of “stylized facts”. These empirical typical results, however, follow a law-like regularity, which, as we argue in the next chapter, is derived by the configuration of eigenvalues and singular values of the economic system matrices. This particular configuration, we argue, has to do with the effective rank of the economy’s matrices ref lecting the true degree of connectedness and the input–output internal structure of industries and it is repetitive over time and countries. The next chapters grapple with this particular issue of eigenvalue and singular values distribution that has occupied the attention of many economists in recent years.

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Appendix 5.A: Further testing of the randomness hypothesis The lack of proximity between the actual output vector x with the standard output, s, the employment coefficients, l and the l.h.s. eigenvector, π, of the matrix A pave the way to examine the extent to which the matrices A or H under study are random or not. Following Schefold (2019, 2020) and also his discussion with Mori (2019) and Morioka (2019), the randomness hypothesis requires the following condition: the correlation and covariance coefficients of the vectors m = s − x  and v = π − l are equal to each other and both are equal to zero. That is, cor ( m, v ) = cov ( m, v ) = 0 The testing was for the years 2007 and 2014 in the case of circulating capital model using data from the WIOD (2016). We did not test the fixed capital model because the matrix of the capital stock coefficients, in our case, by construction has a rank equal to one, and so all eigenvalues except the first are zero. This is a special case of randomness, because random matrices generically have full rank and their eigenvalues are small, but not all zero. This means that in our fixed capital model, the paths of PP are linear, and the same is true with the capital–output ratios. Our data gave the following correlation coefficients and covariance of vectors m and v, along with the t-ratio, for the year 2007 cor ( m, v ) = 0.562 with t = 4.89 and cov ( m, v ) = 0.0006. Similar were the results for the year 2014; in particular, we got cor ( m, v ) = 0.304 with t = 2.30 and cov ( m, v ) = 0.0001 Clearly, the correlation coefficients and the covariances are not the same; hence, the requirement for the randomness hypothesis of matrix A is violated. The intuitive idea is that if m and v are totally unrelated, then it follows that the elements of the matrix A or H are randomly distributed. If, however, they are related, it follows that the elements of the matrix A or H are characterized by some patterns indicative of the presence of regularities. For example, the columns of these matrices are close to each other or they are multiples, as a result the row vectors π and l are too close to each other and the same is true with the column vectors s and x. If there is no correlation then there is no covariance; however, the converse is not true. The idea is that the correlation coefficient is independent of the normalization condition, or the scaling of the vectors in comparison, whereas the covariance depends on both the normalization and the scaling factor. In the next Figure 5.A1, we display the m and v vectors along with the R-square provided that the slopes are statistically significant, and the intercept in both cases are no different than zero.

130 Wage rate of profit curves 0.05

0.1 2007 m = 0.531v R² = 0.315

-0.1500

-0.1000

2014 m = 0.425v R² = 0.093

0.08 0.06

0.04 0.03

0.04

0.02

0.02

0.01

0 -0.0500 0.0000 -0.02

0.0500

0.1000

-0.1

-0.05

0 -0.01

-0.04

-0.02

-0.06

-0.03

0

0.05

0.1

Figure 5.A1 OLS regressions between m and v, 2007 and 2014.

The results indicate a rather weak but certainly not zero correlation between the two vectors in comparison and certainly, the covariance although is not different from zero, nevertheless, is dependent on the normalization condition. In any case, since correlation and covariance are not equal, they cast doubt on the randomness hypothesis. Perhaps there is need for more testing from other countries and years before something more definitive is stated. It is worth noting that the randomness hypothesis is also under question by the fact that the empirical findings show that the second and the third eigenvalues of the system matrices that we examine in the next chapter get closer rather than further from the maximal eigenvalue. In our view, the quasilinearities of PRP and WRP are of paramount importance in this research and the explanation for that is to be found in the structural characteristics and the repeated patterns, to the extent that they exist, in the economic system matrices. Torres-Gonzalez and Yang (2018) argue that intertemporal changes of the technical coefficients are not random but follow deterministic trends. We grapple with this issue along with related others in the next chapters.

Notes 1 The capital stock, in principle, should be augmented to become capital advanced, that is, to include circulating capital and wages advanced. Both require the estimation of turnover times. The interested reader in these estimations may find details in Ochoa (1984, 1989) and Shaikh (1998, 2016). Finally, the fixed capital, in principle, must be adjusted by the degree of capacity utilization. We need not argue how demanding are the data for the estimation of both, turnover times and capacity utilization for all industries in a single year, let alone many years and countries. However, the difficulties in estimations should not discourage the empirical research whose results, in our view, would be strengthened by using the missing variables.

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2 It is important to note that J. Robinson in her response to a question by Schefold indicated precisely this scenario (Kersting and Schefold 2020). Salvadori and Steedman (1988) have argued that if all techniques are [near] linear, then one of them will necessarily dominate over all the others. In this case, the frontier is a single linear system with relative prices equal to relative labor values (Shaikh 2016, p. 431). 3 The same vectors have been invoked to support the hypothesis of randomness (Schefold 2013, 2019, 2020) that we subject to a preliminary empirical testing in the appendix to this chapter. 4 For the methods of construction of the matrix of fixed capital per unit of output coefficients, see Appendix 4.A. 5 Capital deepening paradox in the capital theory refers to the property according to which it might be efficient to have a lower (higher) capital–labor ratio and a lower (higher) rate of profit. This is inconsistent with the neoclassical theory according to which prices ref lect relative scarcity and so if the capital–labor ratio or intensity decreases, it follows that the rate of profit increases and vice versa. 6 The rate of profit in Han and Schefold (2006) varies reasonably within the low range, whereas in Zambelli et al. (2017), the maximum rate of profit is 250%! Which is quite distant from the reality of the standard ratio for circulating capital in the range of 100%. 7 After all, we should bear in mind that the marginal physical productivity theory of income distribution, as advanced by J.B. Clark and other major neoclassical economists, sought to sanctify the origins of profit income on sources other than exploitation of labor. The marginal physical productivity of income distribution was particularly appealing as a first-rate explanation of profit income derived from capital’s marginal physical productivity. In short, capital is placed on par with labor as being a factor of production contributing to the total output produced and its reward for that is equal to its marginal physical contribution (see Chapter 3 for details). 8 The use of value of output together with constant income shares renders the production function to a tautological equation. For further discussions on this issue, and the “humbug” production function, see Shaikh (1974, 1981, 1990, 2005, 2016) as well as Solow’s (1974) response while his article with the estimation of production function (1957). The issues with the estimations of production functions are also discussed by Felipe and McCombie (2013).

6

Distribution of eigenvalues and the shape of price and wage rates of profit curves

6.1 Introduction The empirical research in the last two chapters has shown that the actual price rate of profit (PRP) trajectories and the resulting wage rate of profit (WRP) curves are approximately linear. The case of reswitching of techniques was in effect derived from numerical examples, which, as argued in Chapter 3, were anything but in contact with the reality of the economies that were supposed to represent. The availability of detailed input–output data and estimations of both PRP and WRP curves have shown quasi-linear shapes over time not only for the U.S. economy but also for many other diverse economies, thereby strengthening the view for the presence of empirical regularities in the economies and compelling the research on the theoretical explanations of these observed quasi-linearities. The initial explanation for the quasi-linearities was based on the notion of vertical integration. In particular, the simple capital intensities of industries estimated by the (circulating or fixed) capital–labor ratios may be distant enough from the average; however, their vertically integrated counterparts get much closer to each other and, therefore, close enough to the average which is not too different from the standard ratio. In the language of statistics, the coefficient of variation in the simple capital intensities is much higher than that of the vertically integrated ones (Shaikh 1984, 1998, 2016; Ochoa 1984; Petrović 1991).1 We need to state at the outset that although this explanation intuitively is in the right direction, it suffers from a shortcoming, namely, the estimated vertically integrated compositions of capital are evaluated in terms of prices, and the movement in prices is what we want to estimate. Consequently, a non-price explanation is in principle required, to shed further light on the movement of both prices and the proximity of vertically integrated capital intensities of industries to their average. In this chapter, we pursue such an explanation of the observed quasilinearities based on the distribution of eigenvalues, which is a solid basis and, as we argue, a quite promising alternative which may open new directions in theoretical and empirical research. This particular explanation has its roots in Mariolis and Tsoulfidis (2011) and was brought about after a stimulating comment by Robert Solow on an earlier version of the article. He, in fact, agreed that the distribution of eigenvalues explains the linearities in WRP

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curves but he raised questions with regard to the particular distribution of eigenvalues and their abruptly falling shape.2 The answer to this question came up later in a number of papers (Mariolis and Tsoulfidis 2016b, 2018) and two books (Mariolis and Tsoulfidis 2016a; Tsoulfidis and Tsaliki 2019) and relates to the way technology is described in the input–output structure of the economy. In particular, the input–output structure of the economy is characterized by the presence of many zeros and near zero elements, which is another way to say that most industries are only weakly connected to each other and their connectedness is inversely related to the input–output industry detail. The discussions about the internal structures of input–output matrices was initiated by Bródy’s (1997) path-breaking work, where he showed that the eigenvalues of a system matrix follow particular patterns intimately relating to the stability of an economic system. Although the stability properties of the system were the focus of Bródy’s analysis, his article stimulated many thought-provoking questions that promoted the research in other related areas including the shape of the PRP and WRP curves. The analysis that follows in this chapter is based largely on Mariolis and Tsoulfidis (2009, 2010, 2011, 2014, 2016a, 2016b). However, it extends to encompass additional literature and some recent developments in the field (Schefold 2013, 2019, 2020; Zambelli et al. 2017; Shaikh 2016; Shaikh et al. 2020) and to provide some further empirical evidence lending support to the previous results and, in so doing, strengthening the theoretical explanations of the empirical regularities suggested in current research. The remainder of the chapter continues as follows: Section 6.2 presents and critically evaluates Bródy’s conjecture and brings to the fore new empirical findings supplemented by the results of other researchers paying particular attention to the evolution of spectral ratios as indicators of the stability properties of economies over time. Section 6.3 presents an eigendecomposition, and through that explores the paths of prices of production (PP) and makes a dimensionality reduction paving the way for the construction of a hyperbasic industry that compresses the basic properties of the entire economic system. Section 6.4 applies the various terms of the eigendecomposition in the effort to mimic the actual paths of the PP. Section 5 uses all the previous information and approximations and through the Schur or, alternatively, the Singular Value Decomposition (SVD) methods construct the hyper-basic industry using the economy’s input–output data. It also brings to the fore the similarities and differences of the two estimating methods and makes some suggestions for the direction of future research efforts. Finally, Section 6.6 summarizes and concludes.

6.2 Bródy’s conjecture Adras Bródy (1924–2010) argued that as the dimensions of the matrix of input–output coefficients increases over time, it becomes more and more probable to obtain a random-like matrix. In such a matrix, the spectral ratio, defined as the second eigenvalue to the maximal, decreases, thereby

134 Distribution of eigenvalues

enhancing the stability of the entire system. In other words, the lower is the spectral ratio; the smaller the number of required iterations for the economic system to attain its equilibrium position. The spectral ratio, although it has been used to infer about the stability properties of the economic system in Bródy’s sense, on further thought, we discover that the size of the spectral ratio and the particular distribution of eigenvalues may shed more light on the shape of PRP trajectories as well as of WRP curves.3 The stability properties of the economic system can be illustrated starting with the technology matrix, A, which is a semi-positive matrix, diagonalizable and also with a dominant eigenvalue. Thus, by invoking the notation in Chapters 4 and 5, and denoting by ˜ the diagonal matrix of eigenvalues, we may write π−1Aπ = ˜  or A = π ˜ π−1

(6.1)

Let ˜1,  ˜2 ,… ˜m be the m eigenvalues (counted with multiplicity) of A and y1,  y 2 ,…,  ym be the corresponding eigenvectors. Suppose that ˜1 is the dominant eigenvalue, so that   ˜1   > ˜ j for j > 1. For the initial approximation, let the start vector be the nonzero vector x 0 , such that, the linear combination is x 0 = c1y1 + c 2 y 2 +  + cm ym

(6.2)

with coefficient c1 ˜ 0. By pre-multiplying both sides of Equation 6.2 by the matrix A, we yield Ax 0 = c1Ay1 + c 2 Ay 2 +  + cm Aym = c1˜1y1 + c 2 ˜2 y 2 +  + cm ˜m ym

(6.3)

Repeated (k-times) multiplication of both sides of Equation 6.3 by A gives rise to the following  ˝ c ˇ˝ ˜ k ˇ ˝ c ˇ˝ ˜ k ˇ  (6.4) Ak x 0 = c1˜1k y1 + ˆ 2  ˆ 2k  y 2 +  + ˆ m  ˆ mk  ym   ˙ c1 ˘ ˙ ˜1 ˘ ˙ c1 ˘ ˙ ˜1 ˘   The ratios ˜2 / ˜1 , ˜3 / ˜1 ,…, ˜ j / ˜1 are smaller than 1 and, in effect, they become negligible as k ° ˜. Consequently, we may write

˜j 1  and

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convergence and the other ratios determine its acceleration. In our analysis, we normalize the system matrices such that to give maximal eigenvalue equal to one, and the spectral gap is defined as the difference between the first and the second eigenvalue.4 6.2.1 Evolution of spectral ratio and the size of matrices Bródy’s discussion and findings were about hypothetical matrices and naturally, one would like to know what happens in studying the actual economies. In such a research, the question of stability in Bródy’s sense is certainly important and the evolution of the spectral ratio would shed further light on the stability properties of real economies. There are, however, other related questions dealing with the character of input–output data and the extent to which the technology matrices follow a random pattern or other more specific patterns, which need further investigation. In this context, the key question to be addressed is whether the second eigenvalue tends to zero as the size of the technology matrix increases and what happens to the subdominant ones with the passage of time. These questions, in one way or another, relate to the stability of the economy in Bródy’s sense. However, such investigation inescapably deals with the distribution of eigenvalues and, in so doing, we inevitably grapple with issues more directly related to the PRP trajectories and the particular shape of the WRP curves. In testing Bródy’s conjecture, we put together input–output matrices from BEA of the USA of varying dimensions (15, 71 and 400 plus industries) starting from the year 1997 and continuing every five years until the year 2012, the last from the four benchmark years utilized in our analysis. We supplemented these data with those of the WIOD database with 34 industries for the year 1997 and 54 industry detail for the remaining years. Bródy’s conjecture relies on the elements of the matrix A whose maximal eigenvalue is less than one. In our analysis, we prefer to use the matrix   H = A [ I − A ]−1 (instead of A) normalized by its respective maximal eigenvalue, such that to give rise to a maximum eigenvalue equal to one. The rationale for this selection is that the distribution of eigenvalues of the matrix H shapes the price trajectories of our circulating capital model described by Equation 4.4 as follows: p = wv + rpH where v stands for the direct price (DP), that is the monetary expression of the vertically integrated labor time, l [ I − A ]−1. Hence, we leave out of the analysis the effect of depreciation matrix, which has shown that when it is included in the estimations, the results are enhanced, in that we get a larger eigengap and also the DP and PP get closer to each other and both closer to market prices (MP) (see also Chilcote 1997).5 The subdominant absolute eigenvalues of the matrix H will be lower than those of the matrix A, although both matrices are normalized (i.e., divided by their maximum eigenvalues) and share the exact same maximal eigenvalue equal to one. The elements of

136 Distribution of eigenvalues

the matrix A are less than one and so is their column norms; thus, the Leontief inverse compresses the subdominant moduli of eigenvalues of the matrix even further down, as shown below: H = A [ I − A ]−1 = A ˆˇI + A + A 2 + + A n ˘

(6.5)

Apparently, the moduli of subdominant eigenvalues of the normalized matrix H will be smaller than those of the matrix A, thereby increasing the eigengap, that is, the difference between the first and second eigenvalues and, in so doing, further reducing the eigenratio. By way of a super simple but realistic example, let us take the moduli of eigenvalues of the arbitrary selected year 1997 of the 15 sectors input–output table of the U.S. economy available from the BEA. Both matrices A and H are normalized by their maximal eigenvalues 0.48 and 0.93, respectively. Their results are displayed in Table 6.1 and give us a sense of the differences in the eigenvalues of the normalized matrix H, which are much (about twice) smaller than those of the matrix A. The choice of the matrix H instead of A becomes more useful not only by providing additional information in exploring the properties of the price trajectories but also by giving further support to Bródy’s conjecture rendering a much more difficult task its falsification. The resulting moduli of the matrix H / ˜1 = HR are displayed in the panel of four graphs in each of Figures 6.1– 6.4. Hence, we display the eigenvalues of matrices of various dimensions over five years periods starting from 1997 until 2012, the last year that we have large dimensions input–output data from the BEA. The 34 and 54 dimensions matrices of the USA come from the WIOD base (Timmer et al. 2015). The results show that the eigengap gets progressively smaller going from the smaller to the larger dimensions input–output tables. “Bródy’s conjecture” hypothesizes that the eigengap increases with the size of the matrix and in so doing enhances the stability of the system and renders the elements of the matrix randomly distributed. The actual input–output data (see Table 6.2) show that the eigengap decreases with the passage of time indicating that the attainment of equilibrium once the economy is displaced out of it, because of external or internally generated shocks, takes longer time to return to its equilibrium position. The distribution of eigenvalues in Figures 6.1–6.4 is consistent with the idea that as the matrix size increases from 15 to 400 plus, the eigengap gets progressively smaller, that is, the second eigenvalue Table 6.1 Eigenvalue distribution of matrices A and H, 15 sectors, USA 1997 Eigen Values

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10th 11th 12th 13th 14th 15th

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Figure 6.1 Eigenvalue distribution, 15, 34, 71 and 490 industry detail, USA 1997.

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Figure 6.2 Eigenvalue distribution, 15, 54, 71 and 426 industry detail, USA 2002.

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Figure 6.4 Eigenvalue distribution, 15, 54, 71 and 405 industry detail, USA 2012.

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gets closer to the maximal eigenvalue normalized to one. Therefore, the eigenratio increases rendering the attainment of equilibrium in the face of an external or internal shock more difficult to achieve. Moreover, if the system happens to be already out of equilibrium, a disturbance, other things equal, will further delay the attainment of equilibrium. In the case of the matrix of capital stock coefficients, the second eigenvalue is found to be much smaller than that of the circulating capital model that we have already examined and it is reasonable to expect that the second eigenvalue will increase with the size of the total matrix. The reason is that the matrix of fixed capital coefficients is characterized by relative sparsity and this affects the entire matrix of vertically integrated capital stock coefficients; so the issue of stability in the sense of Bródy does not appear in the same way as in the case of the circulating capital model. However, on further thought, we find that the trouble with the capital stock is its estimation not only for the totality of the economy but also for each and every industry. Hence, we have two major issues, first, gross versus net fixed capital stock and second its treatment in the estimation of equilibrium prices for which there are two lines of research; the first is the one that we follow and can be traced in Leontief (1953, 1986) but also in Bródy (1970). The second is the joint production treatment of fixed capital as in Sraffa (1960, ch. X) and von Neumann (1945–1946), which, however, to the extent that we know the literature, has not been pursued empirically by Sraffian economists.6 In Table 6.2, we display the second and third eigenvalues of benchmark input–output tables of the U.S. economy. We observe that the second eigenvalue increases with the size of the input– output tables, which is equivalent to saying that the speed toward convergence slows down; the same is true with the third eigenvalue. Looking at the second and third eigenvalues over time, we observe that they tend to increase indicating that the system becomes less and less stable in Bródy’s sense. It is Table 6.2 Eigenratios of various size matrices HR, USA five benchmark years, circulating capital 1997 15 industries 2nd Eigenvalue 3rd Eigenvalue 34 (1997) and 54 industries 2nd Eigenvalue 3rd Eigenvalue 71 industries 2nd Eigenvalue 3rd Eigenvalue 490, 426, 405 and 405 industries 2nd Eigenvalue 3rd Eigenvalue

2002

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2012

2017

0.334 0.377 0.378 0.359 0.380 0.214 0.246 0.217 0.221 0.226 0.491 0.491

0.457 0.513 0.551 0.445 0.469 0.497

0.523 0.520 0.540 0.616 0.615 0.411 0.463 0.475 0.533 0.509 0.829 0.802 0.608 0.839 0.506 0.559 0.576 0.532

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important to point out at this juncture that the collected data and dimensions of actual input–output tables are much smaller than those theorized by Bródy to ascertain the view of rising instability. Thus, our findings suggest that for realistic dimensions of input–output matrices, there is a rising trend in the second and third eigenvalues undermining the stability of the system always in Bródy’s sense of the term.

6.3 Eigendecomposition, price trajectories and dimensionality reduction The evolution of the second eigenvalue and its hypothetical dependence on the size of the respective matrix prompt us to speculate that the subdominant eigenvalues and their effect on equilibrium and stability is not so negligible, as one would tend to think based on Bródy’s conjecture. This is the reason why in this section, we attempt a breakdown of PP through their eigenvalues starting from the maximal and going down to the second, third and so forth eigenvalues and their respective eigenvectors. We apply what is called eigen- or spectral decomposition, a method to break down a square matrix into terms consisting of its eigenvalues and eigenvectors. As expected, the first term by virtue of the maximal eigenvalue (provided that the subdominant eigenvalues are by far smaller), exerts most of the inf luence on PP and on their trajectories and, by extension, on the shape of the WRP curves. Therefore, the idea to identify the number of terms, which are important for a tolerably good approximation of PP and the eigenvalues associated with it or, sometimes, singular values might be useful in our search for the selection of the critical number among the top eigenvalues. To this end, we invoke the matrix H, which can be rewritten into the following eigen- or spectral decomposition form (Meyer 2001, pp. 233–234):

(

H = y1x1ˆ

)

−1 ˆ x1y1 +

˜2 y 2 xˆ2 ˜1

(

)

−1 ˆ x2 y2

++

˜n yn xˆn ˜1

(

)

−1 ˆ x n yn

(6.6)

where, lambdas, ˜i , i = 1,  2,  ...,  n, stand for the eigenvalues of the matrix H and y and x are the left-hand-side (l.h.s.) and right-hand-side (r.h.s.) eigenvectors, respectively. The prime over the vector x indicates its transpose. The first or maximal eigenvalue is denoted by ˜1 , whereas the second eigenvalue by ˜2 and the remainder or subdominant eigenvalues by ˜n . Invoking the Perron-Frobenius theorem, the maximal or dominant eigenvalue is the only one associated with a semi-positive eigenvector uniquely defined when multiplied by a scalar; thus, the first term of the above decomposition is

(

H ˛ H1 = y1x1˙

)

−1 ˙ x1y1  

which may be a tolerably good approximation of the matrix H, without necessarily referring to the remainder terms (Meyer 2001, 517–518). The matrix

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H1 has only one eigenvalue equal to one by definition (˜1 / ˜1 = 1, and all the other eigenvalues are zero) and as the product of two vectors its rank is equal to one. We may test the extent to which the above linear approximation to relative prices, through the first term, H1, of the above eigendecomposition is tolerably good; in the sense that the trajectories of the resulting PP induced by changes in the relative rate of profit are not too different from those of the actual trajectories. If the first term (the linear one) does not approximate well the actual curved trajectories, the approximation can be improved by continuing with the second eigenvalue and the second (quadratic) term associated with it, and so forth. The merit of this approach is that it is up to the researcher to decide on the desired percentage accuracy by adding terms, accordingly. If the PRP trajectories are curvilinear, the first term might make a good approximation indicating a matrix of nominal and effective rank equal to one. In general, though, the number of linearly independent vectors of an actual matrix defines its nominal or numerical rank, which might be considerably higher than its effective rank, especially in cases where the price trajectories are featuring slight curvatures.7 The difference between the nominal and effective ranks becomes particularly pronounced in the case of the fixed capital model, whose price trajectories, as our findings in Chapter 4 indicate, are slightly linear as in our data for the year 2018 or, exactly linear as in our data for the year 2014. It is important to point out that in the 2014 case, the effective rank was equal to one due to the unit rank of the matrix of fixed capital stock and so is the rank of the matrix of vertically integrated capital stock coefficients K [ I − A − D]−1; hence, effective and nominal rank are both equal to one. Furthermore, the spiked eigenvalues in their exponential-shaped distribution that we observed in Chapters 4 and 5 and in sections below as well as in previous works by Mariolis and Tsoulfidis (2016a, 2016b) lend overwhelming support to the view that the effective rank of the matrix H that we are dealing with is significantly smaller than its nominal rank. From the above, we arrive at the following steps that need to be taken starting with the presence of quasi-linear price trajectories and WRP curves indicating that the effective rank is significantly smaller than the nominal rank of the economy’s matrices. Once we establish that only a few terms from the eigendecomposition 6.6 are enough for a good approximation, it follows that the effective rank of the system matrices is equal to the number of utilized terms. Once such an approximation has been successively carried out, the seemingly very complex economies, as were thought by the theoretical economists in the Sraffian tradition, are in fact much simpler. The low effective rank further indicates that the columns of the estimated matrices are highly associated to each other, as there are many zero elements and trivially small numbers. Consequently, there is a kind of pseudo-linear independence between the columns of the matrices under investigation, which

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amounts to lighter connections between most of industries. Hence, only a few of industries become hubs connecting each other and the rest of the economy, lending support to the view that the operation of the entire economy may be captured by a few sectors or even a single one as we attempt to show in Section 6.5. From the above, it follows that the method of eigendecomposition is the best available alternative to identify the effective rank of matrices, such as those constructed by input–output data and capital f low matrices. In so doing, we not only simplify the complex structure of the initial matrix, but we may also raise new questions about the properties of the economic system and its structural characteristics. In the kind of analysis and related questions that we are interested in, the eigendecomposition is perhaps the most certain available method of estimating the effective rank of the circulating capital model matrices that we are dealing with.8 The fixed capital model cannot inform us more, especially when the matrix of fixed capital is derived as the product of two vectors; that is, the column vector of investment shares times the row vector of capital– output ratio gives a matrix of rank equal to one while the rest of eigenvalues are equal to zero. In addition, the available literature deals mostly with the circulating capital model; thus, our attention now turns to the intricacies of the case of circulating capital model using input–output data for the U.S. economy for the year 2018. A practical rule of thumb is to retain enough singular values to make up say 90% of the energy contained in the matrix under study. That is, the sum of the squares of the retained singular values should be at least 90% of the sum of the squares of all the singular values of the matrix H˜H. The intuition behind this practical rule of squaring the singular values is to assign more weight to the larger eigenvalues and further minimize the importance of the smaller ones and in, particular, those singular values, which are less than one in the purpose of extracting the simplest but, at the same time, not simplistic representation of reality. This practical rule, when applied to our matrix, H˜H, suggests that 7, or at most 8 singular values make up the 90% threshold value. In Figure 6.5, we display on the l.h.s. axis the singular values, which, as expected, follow an exponential pattern remarkably similar to the eigenvalues while on the r.h.s. axis, the dotted curve stands for the accumulated sum of singular values. We observe that the 90% usual threshold is attained between the seventh and eighth singular value. The results, however, change dramatically, when we applied this to the matrix H augmented by the matrix of consumption coefficients, which gave just a single term, since the first singular value made up the 94% of the energy contained in the technology matrix.9 We also tested the case of the fixed capital model whose results, as expected, are overwhelmingly supportive of one term arrangement while the possible inclusion of further terms would not improve the estimations. In large-data applications, as is the case with the large size matrices of technological coefficients, it is expected that a great deal of entries of the matrix H are zeros or trivially small. Hence, the matrix H

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is of sparse type and its eigendecomposition cannot but become even sparser. The idea is that the input–output matrix, whose rows represent the sales of industries to themselves and the others and their columns the cost of their operation, will very likely become of the sparse type as their size of input– output tables increases, because the interactions (sales and purchases) of most industries get progressively weaker. The number of zero-valued elements divided by the total number of elements of the matrix forms its sparsity index of the matrix. A matrix is considered sparse, if more than fifty percent of its elements are zero. The sparsity index for the usual size of matrices in the circulating capital models is not expected to be particularly high, that is, much lower than fifty percent of their elements are zero. The sparsity index is much higher in the case of the fixed capital model, precisely because not all industries produce capital goods and so naturally the consumer goods industries and services will have zero output. The capital stock matrix K multiplied by the Leontief inverse gives the vertically integrated fixed capital coefficients, which take on the form of the matrix K. In short, the matrix K imposes its form in its product by the Leontief inverse. In our numerical example, the nominal rank of the matrix K of the year 2018 is 26 and the product of K times the Leontief inverse has the exact same nominal rank equal to 26. The index of sparsity in the resulting vertically integrated fixed capital coefficients K [ I − A − D]−1 is equal to 60. From the above discussion, it follows that the rank of a matrix is very important in determining the shape of the PRP paths as well as the WRP curves, which are so crucial in the discussion of the CCC. Our analysis

144 Distribution of eigenvalues

showed that the nominal rank is not so useful as the effective rank of the system’s matrix. Furthermore, the usual decisions for a threshold higher than 90% contains a subjective character that can change to either direction. Although all these efforts did not give us a definitive answer, they gave us additional insights for an indirect, parsimonious and, at the same time, economically meaningful estimation of the effective rank through the eigendecomposition of the system matrix, which we experiment with in the next section.

6.4 Eigendecomposition and the circulating capital model As shown in Chapter 4, the PRP trajectories are, in most cases, monotonic for the 65-industry detail circulating capital model for the USA in 2018. For reasons of visual clarity and economy in space, we do not put all 65 industries and respective graphs in the same Figure 6.6, but only those industries with curved PRP trajectories as well as those, very few ones, whose price paths cross the line of equality between PP and DP (see Figure 4.4). With respect to the fixed capital model, as shown in Chapter 4, its price trajectories are no different from straight lines and there are only two industries with non-monotonic behavior for exceptionally high relative rates of profit (see Figure 4.2). Consequently, the approximation of price trajectories, in the fixed capital model, would not be any different from what is to be approximated by the suggested eigendecomposition. In Figure 6.6, the curved (dotted) lines are those of the actual trajectories derived from the circulating capital model for the USA (2018) and the matrix H; the straight lines, connecting the start and endpoints, are derived from the first term of the above eigendecomposition of the matrix H, that is, the matrix H1. As a first-order approximation, the trajectories of respective relative prices derived from H1 will be linear while those of matrix H will have some curvature, although, in most cases, they are monotonic. It goes without saying that adding additional terms, the approximation will be, in general, improving; however, as is the case with meaningful approximations, just a few terms should be adequate. The exact number of the required terms for a satisfactory approximation is determined by the observer’s view or the nature of the problem at hand. For the kinds of problems that we are dealing with, an overall approximation of 90% or 95%, or what is the same deviation up to 5%, although not exactly perfect, nevertheless is satisfactory, provided that our approximation remains parsimonious. In experimenting with the US data of the year 2018, the currently last available input–output table from the BEA, we found that the results derived using the matrix H1 is a surprisingly good approximation of those derived by the matrix H, as this can be judged by each and every of the 65 industries that we utilized in our analysis. Clearly, the two price trajectories of an industry estimated from H1 and H coincide at ˜ = 0, where the relative

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Figure 6.6 Linear approximations of price trajectories, sample of industries, USA 2018.

146 Distribution of eigenvalues 1.007

1.05

32

1.004

39

1.0375

1.001 1.025 0.998 1.0125

0.995

1

0.992 0

0.2

1.01

0.4

0.6

0.8

0

0.2

0.4

1

43

1

0.6

0.8

1

0.6

0.8

1

0.6

0.8

1

0.6

0.8

1

49

0.93

0.99

0.86

0.98 0.79

0.97

0.72

0.96 0

0.2

1

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0.6

0.8

0

1

0.2

1

50

0.4 52

0.9995

0.93

0.999

0.86

0.9985 0.79

0.998

0.72

0.9975 0.997

0.65 0

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1

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0.8

0

1

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1

61

65

0.95 0.995

0.9 0.85

0.99

0.8 0.985

0.75 0

0.2

0.4

Figure 6.6 (Continued)

0.6

0.8

1

0

0.2

0.4

Distribution of eigenvalues

147

PP of the two matrices are equal to each other and also equal to the DP. As ˜ increases, the two estimates of relative PP depart from both DP and themselves moving, in most cases, to the same direction; at the end, that is, when ˜ ° 1 the two estimates, that is the relative PP and its approximation, of each and every of 65 industries tend to become equal to each other, once again. As explained above, in Figure 6.6 we display a sample of representative industries with seemingly extreme behavior, such as crossings and high curvatures. The dotted line is the actual PP and the solid straight line is its linear approximation. The estimation of deviations between the two trajectories in all 65 industries shows that the mean absolute deviation (MAD), computed by first summing the absolute values of 65 pairs of deviations (errors) for each ˜ = 0, 0.1, …, 1 and then dividing by their number, is maximized in the middle range of relative rates of profit. That is, at ˜ = 0.5, the MAD is 1.04%, while at ˜ = 0.6, the MAD falls slightly to 1.02%; of course, the MAD at the start and endpoints of the curves is zero. On the vertical axis in Figure 6.7, we display the percentage MAD and on the horizontal axis, the relative rate of profit. The maximum percentage deviation is in industry 15 (motor vehicles, bodies and trailers and parts) with a deviation of 5.03%, for ˜ = 0.5. These results although derived from the US input–output data of 2018, it is reasonable to assume their general applicability to other years for the USA and to other countries. The results suggest that the linear approximation is extremely good precisely because the maximal eigenvalue contains most of the explanatory content and leaves little to be explained by higher-order approximations. The idea is that, in general, the second eigenvalue is substantially lower than the maximal; this characteristic distribution of eigenvalues generalizes our results and the applicability of the above eigendecomposition. 6.4.1 Second-order approximation of PRP trajectories The above-derived results suggest that the first order of approximation is particularly good even in the circulating capital model let alone the fixed capital model, where the deviations of the approximation will be trivially small. One wonders whether the second-order approximation, using the second eigenvalue relative to the first, would be of any improvement. For this purpose, we invoke the second term in our estimations, and so we get −1 ˜2 y 2 x '2 x '2 y 2 ˜1 As expected, the approximation is no longer linear but of a second order and so, it displays some curvature. For reasons of economy in space and better visual inspection, once again, we present just a few from the industries, those with the greatest curvature in WRP curves, which happen to be those that cross the line of price-value equality as shown in Figure 6.8.

(

H ˝ H 2 = y1x1'

)

−1 ' x1y1 +

(

)

148

Distribution of eigenvalues

MAD %

1.2

1 0.8 0.6 0.4 0.2 0

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Figure 6.7 Economy-wide MAD resulting from the first-order approximation.

The first-order approximation (solid line) is good, the second-order approximation (dashed line) is better coming closer to the true curvature of the PRP paths (dotted line). The maximum percentage deviation is again observed in industry 15; this time is 3.45%, which is somewhat smaller and, therefore, more accurate than that of the first approximation (5.03%). The MAD of the two approximations from the actual price trajectories are shown in Figure 6.9, where on the vertical axis we display the percentage MAD and on the horizontal axis the relative rate of profit. The MAD is maximized at the relative rate of profit in the middle. Thus, for ˜ = 0.5, the first-order approximation gives a MAD equal to 1.04%, which as we discussed is a pretty satisfactory approximation; however, one might argue that when the PRP curves display curvatures or reswitching, then the first-order approximation may not be so good. We therefore need a second-order approximation, which in our case reduced the MAD from 1.04% to 0.84%. Of course, a third term might be picked but we feel that a somewhat more accurate approximation will not add much to our knowledge about the behavior of the system. We tried third- and fourth-order approximations, which are displayed in Figure 6.9. The results show that for reasonable relative rates of profit, the third- and fourth-order approximations are indistinguishable to the second-order approximation and, whatever differences exist are trivially small. However, as the relative rate of profit takes on values above 40%, the third- and fourth-order approximations, surprisingly enough, were outperformed by the second-order approximation. This may be attributed to the presence of rounding-off errors in the estimates of both eigenvalues and eigenvectors, especially in the third- and fourth-order approximations.

Distribution of eigenvalues 1

1

52

65

0.998

0.996

0.996

0.992

0.994

0.988

0.992

0

0.2

0.4

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0.8

1

1.002

149

0.984

0

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1.8

0.6

0.8

1

15

6 1.6

1

1.4 0.998

0.996

1.2

0

0.2

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1.003

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0.8

1

1

0

0.2

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1.02

23

1.002

0.6

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1

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43

1

1.001 0.98

1 0.999

0

0.2

0.4

0.6

0.8

1

0.96

0

0.2

0.4

Figure 6.8 Linear and quadratic approximations, USA 2018.

From the above, it follows that in our case and in this particular data, the first two eigenvalues contain all the necessary information about the “laws of motion” governing the PRP curves of the system. It follows, therefore, that the observer may decide the size of the error in approximation and accordingly select terms from the above eigendecomposition. For example, if we choose a realistic relative rate of profit of 50%, which at the same time happens to be the one that maximizes the MAD, it is in the observer’s discretion (or rather to the statistics of deviation such as the MAD) to decide on the precise number of terms to be selected from the eigendecomposition. In so doing, the number of selected terms essentially determines the effective rank of the matrix.

150 Distribution of eigenvalues MAD % 1.2

4th approxima˜on 1st approxima˜on

1

3nd approxima˜on

0.8 0.6 2nd approxima˜on

0.4 0.2 0

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1 =r/R

Figure 6.9 MAD% of approximations from the actual price trajectories, USA 2018.

To the extent that we know the literature from the very early to recent studies starting from Bienenfeld (1988), a second-order approximation is adequate to mimic quite satisfactorily the actual PRP curves. Furthermore, the first few eigenvalues are much higher than the rest and constellate at much lower values. This is equivalent to saying that only the first few eigenvalues might be crucial in determining the trajectories of PRP curves. The rest may not have any noticeable effect; they merely add some, although not too much noise, in the behavior of the entire economic system. The subsequent research (summary may be found in Mariolis and Tsoulfidis 2016a, 2016b, 2018) shows, beyond any doubt, that the distribution of eigenvalues of the first few terms is definitely distinct from the rest. If the matrix is of large dimensions, the subdominant eigenvalues f lock together at trivially small values and do not change in any quantitatively significant way the structural properties of the input–output matrices. This is invaluable information if we want to present the complexities of a real economy in the simplest terms possible. 6.4.2 The distribution of eigenvalues and the near linearities in PRP trajectories In what follows, we examine the extent to which the moduli of eigenvalues follow the exponential shape that we described above with the second eigenvalue being substantially lower than the dominant and the rest at values low enough to have any noticeable effect on the movement of prices and on the behavior of the entire system. We start with the US economy examining mainly its benchmark input–output data and then the study extends to

Distribution of eigenvalues

151

include some other major economies. The results of such research are general, although we restrict our investigation to just a relatively small sample of countries. Starting with the location of the eigenvalues in the complex plane for the U.S. economy of the benchmark years 1997, 2002, 2007 and 2012 in connection with the results in Table 6.2, leads us to the following propositions: First: the maximal eigenvalues do not differ as we move from one aggregation to another or the same aggregations over time. The eigengap decreases with the size of the matrix, and the converse is true with the eigenratio. Consequently, “Bródy’s conjecture” does not really hold when the matrices are derived from actual input–output data. The third eigenvalue, that is, the acceleration in the speed, keeps track of the second eigenvalue as we conclude by inspecting the data displayed in Table 6.2. It is important to stress that Bródy (1997) added the following cautionary note concerning the coefficients of input–output matrices which are not evenly distributed and do not seem to follow a clear-cut distribution. Their pattern is skew, with a few large and many small and zero elements. […] A special distribution and/or a special structure of the matrix may still permit exceptions. (Bródy 1997, p. 255) Bródy (1997) apparently refers to his conjecture, that is “the greater the (random) matrix A is, the more elements (or sectors, or branches) it possesses, the smaller will be its second eigenvalue in relation to the maximal eigenvalue, and the faster will be its convergence to equilibrium” (Bródy 1997, 255). As we will see, this is not exactly right, at least with the rather dispersed distribution of eigenvalues, which clearly form a very specific pattern. And in this pattern, the second and especially the subdominant eigenvalues stand at a significant distance away from the maximal eigenvalue. Similarly, the distribution of input–output coefficients forms patterns as the research by Torres-Gonzales (2017) and Torres-Gonzales and Yang (2019) has shown.10 Second: the moduli of the first non-dominant eigenvalues fall quite rapidly in an elbow-like manner and the rest constellate in much lower values forming a “long tail”. Figure 6.10 indicates that the moduli of super large input–output data follow an exponential path of the following form:

( )

˜ = a + b ˆ exp x −c with a < 0, b > 0 and c < 0 Figure 6.10 displays the absolute values of eigenvalues of the matrix H of our large scale benchmark input–output data of the years 1997, 2002, 2007 and 2012 of the U.S. economy. In each set of data, we apply a non-linear regression where ˜ stands for the eigenvalues and x for their ranking. Consequently, the exponential distribution of the absolute eigenvalues represents quite accurately the exponentially falling pattern of the eigenvalues.

152 Distribution of eigenvalues 1.2

1997, n = 491

1 0.8

° = -0.797+0.684exp(x-0.309),

R2 =0.99

0.6 0.4 0.2 0

0

50

100

150

200

250

300

350

400

450

500

-0.2 1

2002, n=426

0.8

° = -0.749 + 0.617exp(x-0.276),

R2 = 0.992

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0.4

0.2

0

0

50

100

150

200

250

300

350

400

450

350

400

450

-0.2 1.2

2007, n=426

1.0

° = -0.845 + 0.724exp(x-0.316),

0.8

R2=0.993

0.6

0.4

0.2

0.0

0

50

100

150

200

250

300

-0.2 1.2

2012, n=426 1

0.8

° = -0.810 + 0.677exp(x-0.270),

0.6

R2 = 0.995

0.4

0.2

0

0

50

100

150

200

250

300

350

400

450

-0.2

Figure 6.10 Eigenspectrum, large dimensions input–output tables, USA 1997, 2002, 2007 and 2012.

Distribution of eigenvalues

153

This functional form is similar to that associated with the findings of previous studies for a number of diverse economies and years quite distant from each other (see Mariolis and Tsoulfidis 2016a, 2016b); so, as one can axiomatically assume, the fall in eigenvalues is pretty much uniform obeying a law-like regularity. We stipulate a rate of fall b ˜ −0.3, while in past studies, we tried b = −0.2. What is important is the rapidity of the fall until the attainment of the “elbow”, which is located much better by stipulating a falling rate equal to −0.3 as this can be judged by the higher R-square. Such a generality in eigenconfiguration suggests that the effective rank of matrix H is much smaller than its nominal rank. The trouble is that the concept of effective rank, so useful in this kind of analysis, nevertheless in praxis, is left to “the eyes of the observer”. The observer, under these circumstances, should be interested in identifying the eigenvalue, which indicates the formation of the so-called “elbow” from which onward the bulk of eigenvalues exert no appreciable inf luence on the behavior of the entire system; that is, the remainder eigenvalues do not impact on the trajectories of both the PRP and WRP curves. Bidard and Schatteman (2001) opine that the crucial hypothesis in Bródy’s conjecture is that the entries of I-O tables can be considered as i.i.d. random variables. The hypothesis is but the expression of our a priori ignorance. The difficulty does not come from the presence of many zeros in I-O tables but from specific linkages between some industries. It would be interesting to check whether the existence of patches of intense relationships between some sectors resists disaggregation and is sufficient to reverse the result established when the entries are chosen at random. […] Since the randomness hypothesis is economically unrealistic, an application of the theorem to actual I-O tables remains subject to practical tests. (Bidard and Schatteman 2001, p. 297) Schefold (2013) opines that the random hypothesis is very similar to the hypothesis of perfect competition in neoclassical economics. More specifically, he argues that the actual input–output tables are not exactly characterized by randomness but rather by a skew-kind pattern of distribution of eigenvalues in which the first or maximal eigenvalue is far distant from the rest and essentially compresses most, if not all, of the information contained in an actual input– output table. In his most recent work, Schefold (2016, 2019, 2020) opines that such a distribution of eigenvalues characterizes matrices not too different from those derived from the random distribution hypothesis, where the maximal eigenvalue is equal to one and the subdominant ones approximate zero. By specifying the extent to which the utilized matrices follow a random distribution, in effect, Schefold assumes a kind of deviation very similar to that between perfect in comparison with monopolistic competition, in the sense that the former is the ideal form of market and the latter, a deviation from it. To put it succinctly, the perfect competition model in neoclassical economics

154

Distribution of eigenvalues

plays the role of the yardstick through which we can measure the deviation of the actual from the ideal. In similar fashion, the random hypothesis serves as a kind of measure of the deviation of the actual distribution of input–output coefficients and the skew distribution of eigenvalues associated with them. Clearly, although the assumption of randomness shares properties of the actual input–output tables, it does not fully apply to them. In effect, the randomness hypothesis is imposed in a way quite similar to the assumption of perfect competition in neoclassical economics. Perfect competition is not derived from the observation of the way in which firms actually organize and compete with each other but rather is a theoretical assumption imposed so as to make the requirements of the neoclassical theory consistent with each other. Similarly, the assumption of randomness, which may serve as an explanation, is imposed to explain the observed near linearities, in that if the economy’s matrices are random, then the maximal eigenvalue is equal to one and all the rest are near zero. However, randomness is an assumption imposed from outside to fit the facts and not a property characterizing the internal structure of the economy’s system matrices that are not and cannot be random. What we actually observe is that the subdominant eigenvalues are usually less than fifty percent of the dominant eigenvalue and their effect might be considered negligible from a practical point of view as we discussed above (see Mariolis and Tsoulfidis 2011). Consequently, not only the relatively small dimension matrices but also the extra-large matrices can be stripped down to their absolute essentials corresponding to economies of no more than a few sectors. Certainly, this is an alternative way of looking at the economy from a mathematical perspective. Third: the occurrence of complex eigenvalues may even be ignored in this kind of research. The idea is that (whenever) they appear at very small eigenvalues as this can be judged by a cursory look in Figures 6.1–6.4, where the eigenvalues of vertically integrated input–output coefficients of matrices of varying sizes are displayed. Thus, it is safe to rely on the absolute eigenvalues (moduli) in the few occurrences of complex eigenvalues. We do not exclude the case that the real part is smaller than the imaginary part, but this occurs at the very low ranks of eigenvalues, and so for studies like ours, we may safely rely on absolute eigenvalues. Because of these properties of the complex eigenvalues; that is, the imaginary is very small if the real part of complex eigenvalues is large and they are relatively large when the real part is small, we can safely rule out the case of the cyclical patterns in the PRP trajectories. Apart from the benchmark years 1997 and 2002, 2007 and 2012, the BEA provides input–output data spanning the period 1997–2018, and the dimensions of these tables are of 15 and 71 industries. The industry structure and the methods of assembling the data is the same and so allows for meaningful comparisons. In Figure 6.11, we display the first and second eigenratios data of the 71-industries input–output structure, provided that the 15-industry structure gives quite similar results. More specifically, in Figure 6.11, we present the first (i.e., the ratio of the second eigenvalue to the maximal, ˜2 / ˜1) and the second eigenratio (i.e., the ratio of the third eigenvalue to the maximal, ˜3 / ˜1), in order to observe their evolution

Distribution of eigenvalues

155

over the long enough period of 22 years, where t stands for time and all coefficients are statistically significant. In Figure 6.11, we also display the linear regression trend lines along with their R-squares. Clearly, the slopes of the two lines are almost identical indicating the strong correlation between the two eigenratios. In Figure 6.11, we observe that the time series results, spanning the period 1997–2018, are consistent with the findings for the years 1997, 2002, 2007 and 2012; both the second and third eigenratios of the matrix H follow an upward trend pretty much parallel to each other. Such trends indicate the instability of the system in that once it is set out of equilibrium as a result of an external (e.g., an oil crisis or pandemic) and/or internal (e.g., major innovation), the system remains at disequilibrium longer and also becomes more prone to f luctuations. Our results are in absolute accordance, both qualitatively and quantitatively, with those of many diverse economies (i.e. Canada, China, India, Germany, Greece, and Japan, among others) as shown in Mariolis and Tsoulfidis (2016a) using the input–output structure of 34 industries (WIOD, 2013) and other national sources of various industry detail. We ascertain those findings by taking another sample of countries using more recent input–output data and 54 input–output structure of the first (2000) and the last years (2014) of the available input–output data WIOD (2016); hence, the analysis is contacted using data for the same years, and the same source and methods of compilation of the input–output tables. The results concerning the ranking of absolute eigenvalue distribution are displayed in Table 6.3. The last rows of the Table 6.3 refer to the average of the subdominant eigenvalues and the more representative, under this skew configuration of eigenvalues, geometric mean (the square root of the products of eigenvalues). The advantage of the geometric mean over the simple average 0.65

0.6

˜2 /˜1 = 0.487 + 0.006t R² = 0.651 0.55

0.5

0.45

˜ 3/˜1= 0.414 + 0.005t R² = 0.718 0.4 1996

1998

2000

2002

2004

2006

2008

2010

2012

2014

2016

2018

Figure 6.11 Evolution of the moduli of the eigenratios, USA 1997–2018, 71 industries.

AUS 2014

1.000 0.489 0.360 0.360 0.327 0.303 0.270 0.248 0.248 0.209 0.209 0.180 0.166 0.165 0.165 0.144 0.144 0.129 0.129 0.104 0.103 0.098 0.098 0.095 0.095 0.090 0.087 0.073 0.073 0.064 0.064

AUS 2000

1.000 0.435 0.435 0.380 0.293 0.275 0.259 0.259 0.241 0.241 0.202 0.202 0.176 0.165 0.163 0.162 0.162 0.153 0.132 0.128 0.128 0.121 0.106 0.106 0.099 0.093 0.093 0.080 0.080 0.078 0.068

Rank

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31

1.000 0.501 0.419 0.355 0.342 0.342 0.329 0.314 0.299 0.238 0.219 0.218 0.218 0.183 0.177 0.177 0.166 0.154 0.154 0.138 0.111 0.110 0.110 0.089 0.089 0.064 0.064 0.052 0.052 0.050 0.050

CAN 2000 1.000 0.524 0.432 0.358 0.321 0.321 0.276 0.246 0.246 0.238 0.222 0.222 0.199 0.184 0.166 0.152 0.152 0.134 0.121 0.121 0.114 0.114 0.100 0.093 0.093 0.065 0.065 0.058 0.058 0.052 0.052

CAN 2014 1.000 0.341 0.315 0.286 0.239 0.216 0.177 0.177 0.148 0.148 0.111 0.079 0.079 0.068 0.068 0.060 0.051 0.051 0.049 0.044 0.040 0.040 0.038 0.031 0.027 0.025 0.025 0.025 0.022 0.020 0.020

PRC 2000 1.000 0.420 0.358 0.330 0.306 0.297 0.230 0.181 0.181 0.144 0.144 0.098 0.098 0.092 0.086 0.086 0.061 0.059 0.059 0.049 0.049 0.049 0.044 0.036 0.036 0.030 0.030 0.028 0.028 0.026 0.022

PRC 2014 1.000 0.435 0.435 0.380 0.293 0.275 0.259 0.259 0.241 0.241 0.202 0.202 0.176 0.165 0.163 0.162 0.162 0.153 0.132 0.128 0.128 0.121 0.106 0.106 0.099 0.093 0.093 0.080 0.080 0.078 0.068

FRC 2000 1.000 0.489 0.360 0.360 0.327 0.303 0.270 0.248 0.248 0.209 0.209 0.180 0.166 0.165 0.165 0.144 0.144 0.129 0.129 0.104 0.103 0.098 0.098 0.095 0.095 0.090 0.087 0.073 0.073 0.064 0.064

FRC 2014 1.000 0.504 0.409 0.409 0.409 0.336 0.336 0.293 0.276 0.254 0.251 0.251 0.213 0.213 0.192 0.173 0.173 0.167 0.167 0.142 0.138 0.126 0.124 0.124 0.123 0.123 0.103 0.100 0.100 0.087 0.084

DEU 2000 1.000 0.485 0.401 0.401 0.322 0.322 0.321 0.321 0.247 0.247 0.239 0.239 0.222 0.193 0.193 0.178 0.172 0.159 0.150 0.150 0.134 0.133 0.116 0.110 0.110 0.106 0.106 0.097 0.097 0.096 0.076

DEU 2014 1.000 0.621 0.348 0.323 0.319 0.288 0.209 0.201 0.201 0.182 0.182 0.143 0.128 0.113 0.113 0.101 0.101 0.084 0.084 0.081 0.070 0.070 0.069 0.069 0.068 0.068 0.054 0.049 0.042 0.042 0.041

GRC 2000

GRC 2014 1.000 0.515 0.350 0.350 0.305 0.233 0.233 0.228 0.228 0.171 0.171 0.138 0.099 0.099 0.098 0.093 0.091 0.091 0.088 0.088 0.086 0.084 0.084 0.075 0.075 0.070 0.064 0.055 0.055 0.046 0.040

Table 6.3 Distribution of absolute normalized eigenvalues, circulating capital, various countries, 2000 and 2014

1.000 0.652 0.564 0.358 0.358 0.353 0.353 0.281 0.281 0.260 0.223 0.189 0.189 0.151 0.130 0.130 0.121 0.121 0.108 0.094 0.094 0.086 0.082 0.079 0.079 0.068 0.060 0.052 0.038 0.038 0.038

JPN 2000

1.000 0.464 0.464 0.415 0.415 0.340 0.266 0.238 0.226 0.226 0.200 0.173 0.156 0.106 0.106 0.101 0.082 0.066 0.064 0.064 0.064 0.049 0.049 0.046 0.046 0.041 0.028 0.028 0.027 0.027 0.026

JPN 2014

156 Distribution of eigenvalues

32 0.051 33 0.051 34 0.046 35 0.037 36 0.037 37 0.036 38 0.036 39 0.036 40 0.031 41 0.031 42 0.023 43 0.021 44 0.021 45 0.018 46 0.018 47 0.012 48 0.006 49 0.006 50 0.006 51 0.006 52 0.002 53 0.001 54 0.000 Mean 0.076 G. Mean 0.044 SF 0.579

0.058 0.039 0.039 0.038 0.038 0.037 0.037 0.037 0.037 0.028 0.027 0.027 0.024 0.024 0.023 0.023 0.011 0.010 0.010 0.006 0.001 0.001 0.000 0.081 0.051 0.630

0.049 0.046 0.046 0.045 0.037 0.035 0.035 0.030 0.030 0.023 0.023 0.023 0.021 0.015 0.006 0.006 0.002 0.002 0.000 0.000 0.000 0.000 0.000 0.118 0.075 0.636

0.050 0.044 0.044 0.044 0.044 0.037 0.033 0.028 0.028 0.026 0.024 0.015 0.015 0.013 0.009 0.003 0.001 0.001 0.000 0.000 0.000 0.000 0.000 0.112 0.070 0.625

0.019 0.016 0.016 0.011 0.011 0.010 0.010 0.007 0.007 0.005 0.005 0.004 0.004 0.002 0.002 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.059 0.034 0.576

0.022 0.019 0.019 0.017 0.017 0.014 0.014 0.014 0.011 0.011 0.010 0.007 0.007 0.004 0.004 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.073 0.041 0.562

0.051 0.051 0.046 0.037 0.037 0.036 0.036 0.036 0.031 0.031 0.023 0.021 0.021 0.018 0.018 0.012 0.006 0.006 0.006 0.006 0.002 0.001 0.000 0.114 0.065 0.570

0.058 0.039 0.039 0.038 0.038 0.037 0.037 0.037 0.037 0.028 0.027 0.027 0.024 0.024 0.023 0.023 0.011 0.010 0.010 0.006 0.001 0.001 0.000 0.111 0.065 0.586

0.064 0.064 0.061 0.053 0.053 0.049 0.049 0.046 0.045 0.035 0.035 0.030 0.017 0.015 0.015 0.013 0.013 0.009 0.008 0.006 0.006 0.004 0.001 0.134 0.075 0.560

0.065 0.048 0.048 0.044 0.044 0.036 0.036 0.033 0.033 0.030 0.029 0.028 0.028 0.025 0.025 0.021 0.011 0.009 0.009 0.005 0.005 0.003 0.001 0.128 0.070 0.547

0.041 0.033 0.031 0.031 0.022 0.020 0.018 0.013 0.013 0.012 0.012 0.010 0.008 0.004 0.004 0.004 0.002 0.002 0.001 0.001 0.001 0.001 0.000 0.089 0.037 0.416

0.040 0.035 0.035 0.034 0.025 0.025 0.017 0.017 0.014 0.014 0.014 0.014 0.012 0.006 0.006 0.006 0.004 0.004 0.003 0.002 0.001 0.001 0.000 0.089 0.041 0.461

0.034 0.034 0.028 0.028 0.027 0.026 0.026 0.022 0.021 0.021 0.019 0.019 0.012 0.008 0.007 0.007 0.005 0.005 0.002 0.000 0.000 0.000 0.000 0.113 0.062 0.549

0.026 0.022 0.022 0.020 0.020 0.016 0.016 0.013 0.013 0.012 0.012 0.012 0.012 0.011 0.006 0.004 0.004 0.002 0.002 0.000 0.000 0.000 0.000 0.091 0.044 0.484

Distribution of eigenvalues 157

158

Distribution of eigenvalues 1.2

1.2

1

1.0

AUS = -0.904 + 0.703*exp(x-0.3) R2 = 0.934

0.8 0.6

0.6

0.4

0.4

0.2 0

0.2 0

20

40

60

80

100

120

0.0 0

-0.2 1.2

20

40

60

80

100

120

1.2

PRC = -0.961 + 0.734 * exp(x-0.3) R2 = 0.945

1

FRC = -0.957 + 0.763 * exp(x-0.3) R2 = 0.957

1.0

0.8

0.8

0.6

0.6

0.4

0.4

0.2 0

CAN = -1.007 + 0.801 * exp(x-0.3) R2 = 0.953

0.8

0.2 0

20

40

60

80

100

120

0.0

-0.2

0

20

40

60

80

100

120

1.2

1.2

GRC = -1.008 + 0.784 * exp(x-0.3) R2 = 0.969

1.0

DEU = -0.973 + 0.788 * exp(x-0.3) R2 = 0.941

1.0

0.8

0.8

0.6

0.6

0.4

0.4

0.2 0.2 0.0 0.0

0

20

40

60

80

100

120

0

20

40

60

80

100

120

-0.2

1.4 1.2

JPN=-1.098+0.856*exp(x-0.3) R2=0.964

1 0.8 0.6 0.4 0.2 0

0

20

40

60

80

100

120

-0.2

Figure 6.12 Distributional patterns of absolute normalized eigenvalues, circulating capital: AUS, CAN, PRC, FRC, GRC and JPN, 2000 and 2014.

Distribution of eigenvalues

159

is that the ratio of the smaller eigenvalue to the geometric mean is close, if not equal, to the ratio of the geometric mean to the highest eigenvalue. Furthermore, the ratio of the geometric mean to the arithmetic mean gives the spectral f latness (SF), that is to say, an index of how dissimilar the distribution of eigenvalues is.11 Finally, the graphs along with their trend lines are displayed in Figure 6.12. Of particular interest is the sharp decay of the top eigenvalues lending support to the view that the effective rank of the vertically integrated input– output coefficients is much lower than the nominal (n = 54). Algebraically, this means that the span of the rows of the H matrix exists primarily (but not entirely) in a subspace of dimension much lower than the nominal rank of H. From a practical perspective, this means that a relatively accurate recovery of matrix H can be derived from significantly fewer eigenvalues or singular values than those suggested by the nominal rank of matrix H. The trouble with the estimation of the effective rank of a system of matrices is that the threshold between significantly small and insignificantly large singular or eigenvalues, to the extent we know the literature, cannot be theoretically designated. The decision depends on the nature of the problem at hand and the target set by the investigator. An approximation for example of 90% of the full rank of matrix H might be tolerably good and so the eigenvalues that will be used will correspond to the target set. If two eigenvalues or singular values give the desired or planned approximation, then we can say that the effective rank of the matrix is two, even though its nominal rank might be four hundred and two! It is important to stress at this juncture that moving from the f low to the more realistic capital stock input–output data and matrices, the results suggest that the second eigenvalue falls even more abruptly. The third eigenvalue becomes “indistinguishable” from the rest, trivially small near zero (see Mariolis and Tsoulfidis 2011), indicating that in Bródy’s conjecture, there is a need for further investigation. Our results lend overwhelming support to the idea of factorization, an idea based on the low rank of the economic system matrices, which give rise to near linearities in both PRP trajectories and WRP curves. The curvatures are usually of low order and in some cases, we observe one at most extreme. In these rare cases, as for example, industries 6, 23, 43 or 52 shown in Figure 6.8, we observe that their high curvature does not imply large deviations from one. The maximum deviation might be in the range of 0.2% (in industry 6) or about 0.6% (in industry 52) and so forth.

6.5 Factorization and the construction of a hyper-basic sector Having established that a good approximation of PP can be attained with just the first eigenvalue and the eigenvectors associated with it, we may proceed in the construction of a hyper-basic industry, which in principle encompasses the salient features of the economy’s matrices. For this purpose, we apply the Schur factorization method initially on the matrix H1 instead of H, in

160 Distribution of eigenvalues

order to derive an economically meaningful matrix of eigenvalues and eigenvectors precisely because the rank of the matrix H1 = 1. By utilizing the Schur factorization method, we derive, along with the matrix of eigenvalues with maximal eigenvalue equal to one, the rest of the eigenvalues, which are semi-positive, and less than one. We also derive the matrix U of eigenvectors. Hence, it is important to note that the matrix U is orthogonal meaning that UU˜ = I = U˜U , where the prime indicates transpose. This result occurs because the columns of the matrix U are orthonormal, which is equivalent to saying that the inverse of U and its transpose are one and the same matrix, that is, U −1 = U°. From the matrix U, only one vector is economically meaningful, and it is associated with the maximal eigenvalue, which in our case is equal to one. The next step is the estimation of the matrix S, defined as S = U°H1U or S = U −1H1U From this, we can derive the following similarity transformation: H1 = USU° The real Schur form is an upper quasi-triangular matrix, that is, it is block triangular where the blocks are 2×2 submatrices that correspond to complex eigenvalues of H. If H has no complex eigenvalues, S will be strictly upper triangular (Meyer 2001, pp. 508–509). Based on the particular distribution of eigenvalues of the matrix H and having experimenting with the input–output data for a number of countries, we have established that the H1 matrix is a good approximation of matrix H (see Section 6.4); in the sense, that both matrices give rise to price trajectories not very different from each other. Therefore, we can decompose the matrix H1 into two matrices (factors) by applying the Schur factorization method, whose useful properties, in our view, have not found the applications that they deserve in economic theory.12 The Schur factorization gives us the matrix of eigenvalues S, which is a diagonal matrix with maximal eigenvalue equal to one and the eigenvector u1 of matrix H1 is displaced in Table 6.4 below Table 6.4 Column vector u1 of matrix H1, Schur factorization method 0.163 0.135 0.086 0.099 0.095

0.115 0.219 0.110 0.092 0.093

0.110 0.130 0.095 0.116 0.119

0.120 0.152 0.113 0.101 0.116

0.117 0.125 0.150 0.155 0.089

0.090 0.179 0.127 0.138 0.105

0.115 0.157 0.094 0.100 0.091

0.146 0.151 0.055 0.073 0.115

0.125 0.165 0.123 0.080 0.101

0.178 0.129 0.125 0.098 0.089

0.154 0.131 0.088 0.105 0.128

0.158 0.130 0.106 0.104 0.110

0.095 0.152 0.109 0.168 0.135

With the aid of the vector u1 and the identity matrix, I, we construct the matrix U of dimensions 65x65 which is the result of replacing the first column of the identity matrix by the above column vector u1, which is literally the l.h.s. economically meaningful eigenvector of the matrix H1 obtained

Distribution of eigenvalues

161

through the application of the Schur method. The matrix S can be now estimated as follows: SH1 =  U −1H1U The first row of the resulting matrix SH1 is the economically significant vector, which is displayed in Table 6.5. Table 6.5 Row vector of matrix SH1 , Schur factorization method 1.000 0.080 0.370 0.089 0.096

0.033 0.253 0.035 0.216 0.278

0.164 0.022 0.062 0.156 0.290

0.042 0.041 0.028 0.357 0.066

0.010 0.083 0.011 0.050 0.087

0.166 0.369 0.084 0.312 0.022

0.062 0.055 0.020 0.099 0.075

0.048 0.118 0.019 0.094 0.099

0.054 0.083 0.075 0.073 0.270

0.267 0.021 0.043 0.368 0.207

0.190 0.196 0.054 0.205 0.020

0.098 0.429 0.034 0.262 0.027

0.143 0.102 0.189 0.030 0.066

We observe that the first element of the so-estimated vector S is (approximately) equal to one and the rest are all positive or rather of the same sign. Finally, all the elements of the other vectors of matrix S are in principle zero.13 Thus, with the aid of the Schur decomposition, we managed to transform the matrix H1 into the matrix S and since matrix H1 is not very different from H; thus, both matrices H1 and H are quite similar to the matrix S. This similarity among the three matrices essentially amounts to the cells of the first row corresponding to a single industry of the matrix S. The composition of this so to speak hyper-basic industry, as the direct result of a similarity transformation is designed to encompass essentially the structural properties of the entire economy. This approximation of the matrix H through the matrix H1 is further enhanced, the lower the spectral ratio. In short, if the second eigenvalue tends to zero, the more accurate the approximation, and this is the case of the fixed capital model that we examine latter. In the case of the circulating capital model, the application of Schur decomposition method to the matrix H gives the same matrix U1 as that already obtained from the matrix H1. The first column of the matrix U1 replaces the first column of the identity matrix that multiplied by the matrix H gives SH =  U −1HU The first row of the matrix S gives us the economically significant vector displayed in Table 6.6. Table 6.6 First row of matrix SH, Schur factorization method 1.000 0.080 0.643 0.044 0.060

0.264 0.203 0.042 0.148 0.175

0.114 0.013 0.043 0.073 0.184

0.042 0.026 0.040 0.190 0.042

0.005 0.051 0.012 0.029 0.047

0.145 0.609 0.148 0.456 0.014

0.064 0.028 0.011 0.091 0.044

0.032 0.068 0.012 0.054 0.056

0.035 0.056 0.058 0.038 0.175

0.150 0.012 0.033 0.222 0.136

0.151 0.214 0.032 0.133 0.015

0.115 0.502 0.012 0.164 0.020

0.088 0.086 0.110 0.020 0.057

162

Distribution of eigenvalues

1

0.8

0.6

y = 1,030x - 0,013 R² = 0,809 0.4

0.2

0

0

0.2

0.4

0.6

0.8

1

Figure 6.13 Hyper-basic industries, first rows of matrix SH vs. the approximation SH1 , Schur factorization method.

One wonders to what extent, if any, the two row vectors, displayed in Tables 6.5 and 6.6 derived from the rank 1 matrix SH1 and the rank 65 matrix SH , respectively, relate close enough to each other. For this reason, we plot them in scatter graph along with their linear regression line in Figure 6.13. The R-square of the two vectors is quite high 80.9% indicating that the two estimated vectors move pretty much together. The MAD of the two estimated basic vectors is only 4.91%. To check the robustness of the above-derived results, we resort to another method of eigendecomposition, that is, the SVD method that is presented in the next section. 6.5.1 The Application of the SVD factorization method The column vector u1 that we derived from the application of the SVD on H1 is slightly different, as the reader can identify from a cursory look at Table 6.7 contrasting it with Table 6.4. Table 6.7 Column vector u1 of matrix H1, SVD factorization method 0.162 0.134 0.089 0.102 0.097

0.114 0.201 0.114 0.099 0.100

0.109 0.114 0.093 0.126 0.124

0.118 0.144 0.111 0.124 0.119

0.119 0.127 0.148 0.162 0.089

0.087 0.168 0.122 0.145 0.109

0.109 0.163 0.100 0.103 0.092

0.130 0.148 0.056 0.077 0.120

0.118 0.153 0.116 0.085 0.103

0.173 0.128 0.124 0.105 0.099

0.155 0.125 0.090 0.112 0.128

0.152 0.154 0.101 0.108 0.111

0.097 0.170 0.110 0.159 0.129

Distribution of eigenvalues  163

The two column vectors, displayed in Tables 6.4 and 6.7, may differ in the third decimal and so their MAD is less than one percent, only 0.551% to be exact. Thus, it comes as no surprise that the SVD method gave very similar results with respect to the extraction of the first row of the matrix H, which we do not display in the table for economy of space. One wonders to what extent, if any, these two row vectors derived from the rank 1 matrix SH1 and the rank 65 matrix SH using the SVD method, respectively, relate close enough to each other. The MAD of the matrix SH1 and of the matrix SH obtained through the application of the SVD is 4.92%, which is very small and differs only by 0.01% from that derived through the application of the Schur method. These results should not come as a surprise since the column norms of the two matrices eH1 and eHR (where e is the summation vector and R = 1/ λ1, the maximum rate of profit of the Sraffian system) is very small, as this can be judged by the following column sums of H1 displayed in Table 6.8, below Table 6.8  Column sums of matrix eH1 1.34 1.11 0.71 0.81 0.78

0.95 1.80 0.90 0.76 0.76

0.90 1.07 0.78 0.95 0.98

0.99 1.25 0.93 0.83 0.95

0.96 1.03 1.23 1.27 0.73

0.74 1.47 1.05 1.14 0.86

0.94 1.29 0.77 0.82 0.75

1.20 1.24 0.45 0.60 0.94

1.03 1.36 1.01 0.66 0.83

1.46 1.06 1.03 0.80 0.73

1.27 1.08 0.73 0.86 1.05

1.29 1.07 0.87 0.85 0.91

0.78 1.24 0.89 1.38 1.10

The matrix H normalized by its maximal eigenvalue gives the following column sums (Table 6.9): Table 6.9  Column sums of matrix eHR 1.28 1.06 0.73 0.85 0.80

0.95 1.55 0.92 0.80 0.79

0.93 1.04 0.81 1.01 0.99

0.99 1.21 0.96 0.89 0.95

0.98 1.01 1.27 1.35 0.77

0.77 1.35 1.06 1.19 0.87

0.94 1.26 0.80 0.81 0.76

1.19 1.21 0.46 0.63 0.94

1.04 1.30 1.02 0.68 0.83

1.37 1.04 1.03 0.84 0.78

1.20 1.14 0.75 0.91 1.07

1.23 1.07 0.90 0.89 0.91

0.79 1.23 0.94 1.34 1.14

The total sum of the matrices eH1e′ and eHRe′ are 64.35 and 64.58, respectively. Their MAD is equal to 3.39% and indicates that the two matrices share the same fundamental properties. 6.5.2  The fixed capital model and the hyper-basic industry In a similar fashion, the hyper-basic industry was estimated in the case of the fixed capital model. If the second eigenvalue in the circulating capital model was much lower than the first, in the fixed capital model, the gap between the first and second eigenvalues is many times bigger. In effect, in the particular year 2018 that we study, we found that the second eigenvalue of the

164

Distribution of eigenvalues

fixed capital model was nearly ten times lower than the maximal! An eigengap, which essentially underscores that the behavior of the entire economy is regulated by the first eigenvalue and far too little remains to be regulated by the subdominant eigenvalues. As expected, the approximation of the actual trajectories of the matrix H k = K [ I − A − D]−1 through the matrix H k1 is extremely good. The estimated l.h.s. vector u1 of the H k1 derived through the Schur method is displayed in Table 6.10 while the estimated first row of the matrix Sk1 is displayed in the next Table 6.11.14 Table 6.10 Column vector u1 of matrix H k1, Schur method 0.157 0.070 0.056 0.072 0.107

0.091 0.081 0.090 0.081 0.049

0.281 0.063 0.086 0.066 0.091

0.137 0.067 0.210 0.049 0.064

0.094 0.068 0.124 0.068 0.076

0.240 0.097 0.082 0.291 0.103

0.054 0.096 0.062 0.112 0.091

0.074 0.083 0.211 0.040 0.072

0.088 0.097 0.079 0.025 0.076

0.105 0.076 0.098 0.055 0.070

0.077 0.191 0.065 0.067 0.371

0.076 0.112 0.121 0.046 0.174

0.085 0.088 0.126 0.107 0.269

0.000 0.000 0.000 1.185 0.000

0.000 0.000 0.000 0.277 0.000

0.104 0.000 0.007 0.000 0.000

2.933 0.020 0.388 0.000 0.000

1.463 0.003 0.014 0.000 0.000

Table 6.11 Row vector of matrix Sk1 , Schur method 1.000 0.219 0.769 0.000 0.000

0.000 1.166 0.278 0.000 0.000

0.000 0.230 0.043 0.000 0.000

0.000 0.236 0.013 0.000 0.000

0.478 0.087 0.000 0.000 0.000

0.000 0.000 0.060 0.040 0.000

4.808 0.023 0.000 0.000 0.000

0.074 0.000 0.000 0.000 0.000

The vector u1 and the first row of matrix U that we derived from the application of the SVD to the matrix H k is somewhat different but not very different as the reader can identify from the entries in Table 6.12. Table 6.12 Column vector u1 of matrix H k, SVD method 0.100 0.041 0.050 0.049 0.124

0.046 0.051 0.100 0.064 0.041

0.085 0.035 0.039 0.064 0.091

0.055 0.047 0.167 0.051 0.070

0.037 0.041 0.081 0.069 0.090

0.196 0.062 0.059 0.433 0.113

0.035 0.059 0.036 0.050 0.119

0.048 0.048 0.201 0.042 0.073

0.050 0.059 0.045 0.019 0.072

0.067 0.049 0.104 0.047 0.088

0.049 0.068 0.037 0.056 0.507

0.046 0.062 0.091 0.036 0.249

0.044 0.055 0.074 0.082 0.370

The first column of the matrix U1 replaces the first column of the identity matrix that multiplied by the matrix H gives SH k =  U −1HU The first row of the matrix SH k gives us the economically significant vector displayed in Table 6.13 derived from SVD. The Schur method gave similar results, which are displayed in Table 6.14.

Distribution of eigenvalues

165

Table 6.13 First row of matrix SH k , SVD method 1.000 0.205 0.891 0.000 0.000

0.000 1.237 0.181 0.000 0.000

0.000 0.564 0.040 0.000 0.000

0.000 0.122 0.020 0.000 0.000

1.552 0.036 0.000 0.000 0.000

0.000 0.000 0.092 0.013 0.000

2.40 0.010 0.000 0.000 0.000

0.024 0.000 0.000 0.000 0.000

0.000 0.000 0.000 0.613 0.000

0.000 0.000 0.000 0.343 0.000

0.106 0.000 0.004 0.000 0.000

3.472 0.022 0.236 0.000 0.000

0.940 0.001 0.006 0.000 0.000

0.000 0.000 0.000 0.551 0.000

0.000 0.000 0.000 0.324 0.000

0.177 0.000 0.007 0.000 0.000

4.808 0.027 0.244 0.000 0.000

0.998 0.002 0.009 0.000 0.000

Table 6.14 First row of matrix SH k , Schur method 1.000 0.177 1.438 0.000 0.000

0.000 3.812 0.584 0.000 0.000

0.000 0.326 0.031 0.000 0.000

0.001 0.142 0.039 0.000 0.000

0.551 0.053 0.001 0.000 0.000

0.000 0.000 0.196 0.076 0.000

7.090 0.022 0.000 0.000 0.000

0.134 0.000 0.000 0.000 0.000

The MAD for the Schur was 1.32% while for the SVD was 15.01%; the difference is magnified because of some obvious outliers and the fact that there are many zero elements in the vectors indicative of the sparsity of the matrix H k or what amounts to the same, the low effective rank of the matrix H k. The results for the actual matrix H k that are displayed above are indicative of its sparsity characteristics ref lecting weak or absence of connections between the industries. Slightly different were the estimates of the Schur method, but in any case, they convey the same picture. In Figure 6.14, we plot the vectors SH k and SHk1 , where we can see the strong association between the two estimates. 8 7

y = 1,486x - 0,012 R² = 0,906

6 5 4 3 2

y = 0,704x + 0,044 R² = 0,779

1 0

0

1

2

3

4

5

Figure 6.14 Hyper-basic industries, first rows of matrix, SH k vs. approximated SHk1.

166 Distribution of eigenvalues

This strong association between the two estimates indicates that there is proximity. The coefficient of determination is at the 96.6% indicating that the two vectors are closely related and so if we approximated the price trajectories through the first term of the above spectral breakdown, the results of the trajectories would be indistinguishable. It is important to note that the first row of the S matrix encompasses the major features of the economy as this is described by its technology matrix. Such a hyper-basic industry may become the first major one in the construction of a simple representation of the workings of the entire economy whose prototype may be found in the Physiocratic Tableau Économique and the Marxian Schemes of Reproduction.

6.6 Summary and conclusions In this chapter, we showed that the PRP trajectories resulting from changes in the relative rate of profit are described by the configuration of the eigenvalues contained in its input–output structure. In the case of a circulating capital model, we found that the distribution of eigenvalues follows a rectangular hyperbola-like pattern. In the benchmark input–output tables of the U.S. economy, we observe that the second eigenvalue is much smaller than the first while their gap decreases with the size of the matrix. A result that does not accord with Bródy’s conjecture derived from randomly generated matrices. This conclusion is enhanced in the case of a fixed capital model, whose second eigenvalue is found to be much smaller than the first and, therefore, the eigengap is much higher in the fixed capital rather than the circulating capital model. This particular distribution of eigenvalues implies the quasi linearity of PRP and WRP curves, as we show in the case of the U.S. economy using the 65×65 industry structure of the BEA. In addition, we showed that we need not have all the information contained in the input– output data of an economy, but the fundamental structure of the economy can be stripped down to a much simpler but compressed structure containing most of the vital information generating the system’s behavior. The input–output data of the USA, as well as of many other economies that we have examined, suggested that the majority of the non-dominant eigenvalues of the normalized vertically integrated technical coefficient matrices are crowded in a relatively small region around zero and far from the maximal, which is equal to one. This repeated and, in a sense, “stylized fact” of the economies that we have examined, suggests that the trajectories of relative prices consequent upon changes in the relative rate of profit follow rather consistent and, more often than not, near linear and in general monotonic patterns. The approximation through low-order terms (ranging from linear to quadratic, at most cubic), as we have ascertained, give quite accurate results, which do not further improve by including higher-order terms. The near linearities that we derived for the price trajectories suggest that the hitherto tested economies behave “irregularly” in Schefold’s (1976) sense of the term. At the same time, the price paths are not fully reduced

Distribution of eigenvalues

167

to controllable systems in that a change in one variable leads precisely and always to the theoretically expected result; such deviations render the system less than fully controlled in Kalman’s (1974) sense. These dual characteristic properties of system matrices, the “irregularity” cum “non-controllability”, justify the reduction of the actual economies to a single or just a few hyperbasic sectors compressing the structural characteristics of the economies. In short, we need not have many equations to derive the major features that regulate the motion of the economic system. The steps to arrive at a compressed structure of an economy are the following: – –

First, we start with the eigendecomposition of the matrix of vertically integrated input–output coefficients in the circulating or vertically integrated capital stock coefficients in the fixed capital model. Second, we apply factorization (Schur or/and SVD) method in order to apply a similarity transformation of the complex system matrix and to construct a hyper-basic industry, which as a result of a similarity transformation contains all the essential properties of the entire economic system.

The derivation of such a sector on further thought enlightens the efforts of economists like the Physiocrats whose Tableau Économique utilizes essentially one all purposes commodity. The idea of a one-commodity world is also found in Ricardo’s notorious corn model discovered and made it widely known by Sraffa in his introduction to Ricardo’s Principles (Works I). Marx’s schemes of reproduction are in the same line of thought. In the schemes of simple reproduction, the two departments produce two, nominally only, different goods, namely, investment and consumption goods. However, on further thought, the distinction between these two types of goods is essentially upon their use. Moreover, the assumption of equal organic composition of capital across departments enhances the conceptual identification of the two goods. Hence, we have one-all purposes commodity. Continuing the analysis in the schemes of expanded reproduction, the organic compositions of capital (capital intensities) across departments are reasonably different. Nevertheless, the deviations between DP and PP are quite small without necessarily implying the near equality of the capital intensities of the departments (see Tsoulfidis and Tsaliki 2019, ch. 2). Such a result lends support to our argument of near linearities of capital intensities in the structure of the actual economies, which is manifested in the low effective rank of the system matrices. Coming closer to the years of the CCC, we observe that Sraffa’s standard commodity is in the same train of thought, because of its association with the standard ratio, which is no different from the average economy-wide capital intensity. Finally yet importantly, Samuelson’s “surrogate production function” perhaps could be interpreted in the same spirit and not necessarily that Samuelson was not aware of that his one-commodity-world model implied equal capital intensities.15 The theoretical and empirical findings indicate that many of issues still lie hidden underneath the surface of capital theory debates and may be

168 Distribution of eigenvalues

discovered through a combination of proper economic theory and use of input–output and socio-economic data describing the details of the structure of economic system. The core properties of the latter may be described, in satisfactory ways, by much simpler structures. In such a direction, it appears that super parsimonious structures such as a hyper-basic industry may entail losses in information but compensates by encompassing the structural properties of the entire economic system and showing the way for improvements.

Appendix 6.A: Digression in the effective rank Roy and Vetterli (2007) are from the first that proposed a metric for the estimation of the effective rank of a matrix. For this purpose, they suggested the application of the SVD method to the matrix H. The singular values ˜ differ from the eigenvalues of the same matrix H, in that they are the positive square roots of the eigenvalues of the matrix H˜H, which are no different from those of the matrix HH˜. If the rank of the matrix H is equal to k (< n), then only the k singular values are positive (with zero all the rest). In case that the singular values are relatively small in that they contain very little of the information governing the behavior of the economic system most of which is compressed by the uppermost singular values, we can ignore them (by setting them equal to zero) and reconstruct the new matrix H1 whose rank is k. This new matrix H1 constructed by fewer singular values is considered a very good approximation of the matrix H. The SVD allows an exact representation of any matrix, and makes it easy to eliminate its less important terms and, by so doing, to give rise to an approximate configuration with any desired number of dimensions. It goes without saying that the fewer the dimensions, the less accurate will be the approximation. In order to pass judgment on the accuracy of a representation, Roy and Vetterli (2007) suggest the concept of the effective rank. In order to find the required number of terms to be included in the representation, they employ Shannon’s (1948) entropy index or the spectral entropy defined as n

n

∑q ln q where q = σ / ∑σ  

S=−

i

i

i

i

i

i

σi = σ1 ≥ σ 2 ≥ ≥ σ n ≥ 0

i

where ˜ i stands for the singular values of the matrix H.16 Assuming that 0ln0=0, the effective rank (erank) becomes erank ( H )  = e S In applying this formula to our 2018 BEA 65×65 matrix of vertically integrated circulating capital coefficients, we get an erank = 27. Of course, such a high rank is way far from what can be called “desired approximation”. From a mathematical point of view, the idea of the estimation of the effective rank in the above formula is sound, in that, it examines the extent

Distribution of eigenvalues

169

to which the singular values are different from the majority of eigenvalues. Consequently, just a few singular values compress a lot more information than the rest of them. However, the estimates of effective rank according to the suggested formula were not of any great help in our eliminating process. We only got the assurance that the above estimated effective rank is not equal to the nominal and there is a significant difference between nominal and effective rank, a result that is particularly pronounced in the case of fixed capital model. In our example of the U.S. economy, the matrix of capital stock coefficients of 2018 gave a nominal rank 27, which is no different from the rank of the vertically integrated capital stock coefficients. Furthermore, in estimating the number of zeros of our 65×65 matrix K, we found 39 rows containing zero elements together with the zeros scattered to the rest of cells with zeros constitutes the 61% of total figures of the matrix K, without counting the near-zero or not very different from zero elements. In short, the applied factorization method revealed that the structure of the economies is simpler than originally thought and a lot of information is compressed in the maximal eigenvalue of the system matrices while the remaining eigenvalues add little additional information. Thus, by limiting ourselves to a first-order term, we obtain a very good approximation of the price trajectories in the face of a change in income distribution. In so doing, we end up with the view that the economies, although they are not similar to a single commodity world, the capital intensities of industries comprising the economy would be exactly equal to each other; so, the system’s matrix would be of nominal and effective rank equal to one.

Notes 1 Shaikh’s (1984) article was available at least two years before its publication. 2 See https://library.duke.edu/rubenstein/findingaids/solowrm/ 3 It is important to note that the discussion of Bródy’s kind of description of the economy is limited to circulating capital models. The fixed capital model is not investigated in this particular literature, which is usually left to a later stage of the analysis. Our treatment of fixed capital follows that suggested by Leontief (1953) and Bródy (1970). 4 In general, the spectral gap is defined as the difference between the moduli (the absolute values) of the two largest eigenvalues of a matrix. 5 In experimenting with the US input–output data of dimensions of 71×71 and the years 1958, 1962, 1967, 1972, 1977, 1982 and 1987, Chilcote (1997, pp. 176–177) concludes that the depreciation matrix plays an important role in the measurement of DP and brings them 20% to 25% closer to MP as this can be judged by the various statistics of deviation. The matrix of fixed capital improved the predictability of PP. However, the inclusion of turnover time, capacity utilization and we could add indirect business taxes exert only a relatively minor effect to either direction. 6 For the differences and the implications between the net and gross fixed capital stock, see Shaikh (2016, pp. 801–806), Malikane (2017) and Tsoulfidis and Paitaridis (2019). For the joint production treatment of fixed capital, see Schefold (1971), and for its critique, see Shaikh (2016, pp. 804–806) and Congliano et al. (2018, ch. 11).

170  Distribution of eigenvalues 7 The nominal or numerical rank of a matrix is equal to the number of nonzero eigenvalues. The effective rank is defined as the dimensionality of the matrix determined by the number of eigenvalues or singular values that exert most of the inf luence in a matrix and these may end up to be only very few. 8 From the very early attempts to arrive at a metric of effective rank is in Roy and Vetterli (2007). For further discussion, see the Appendix of this chapter. 9 In the case of circulating capital model, we found that the more detailed the Leontief inverse (by including the workers’ consumption expenditures coefficients), the smaller the number of singular values required to approximate the 90 percent borderline. The idea is that the matrix of workers consumption coefficients, bl , as the product of two vectors, is linearly dependent and its presence in the Leontief inverse increases the maximal eigenvalue and its gap with the subdominant eigenvalues. 10 For further discussions of eigenvalue distribution when changing the aggregation level, see Mariolis and Tsoulfidis (2014, 2016a), Gurgul and Wojtowicz (2015). Shaikh and Nassif-Pirez (2018) and Shaikh et al. (2020) show that the spectral ratio decreases with the disaggregation level. 11 The spectral f latness (SF) is indicated in the last row of Table 6.3 and it can be at most equal to one, since the geometric mean will be always less or equal to arithmetic mean. Of course, the closer to one the more distinct are the eigenvalues. In Table 6.3, we find that the SF varies from 0.416 for GRC 2000 (the lowest) to 0.636 for CAN 2000 (the highest). 12 We used the Matlab for the estimation of the vector u1, which is no different in the case of the estimation of the l.h.s. eigenvector of the matrix H1or of similar experiment with the SVD method (see Section 6.5.2). 13 Practically, this might not be exactly the case; the other rows of S may contain cells containing near zero numbers most likely due to rounding. In the subroutine of Gauss (or Matlab) that we utilized in our estimations, we found that the elements in the rows other than the first were trivial in size. Thus, the rank of H1 although we know is equal to 1, the answer that we derive from Gauss or Matlab may be different. 14 We present the estimates of the l.h.s. eigenvector corresponding to the maximum singular value. The results with the application of the Schur method were the same with the difference in the 39th element. The reason perhaps is that the Schur gives the maximal eigenvalue not necessarily as its first eigenvalue and respective eigenvector; thus, we need to make the appropriate multiplications and then interchange with the first element which is zero in our case with the 39th element which is equal to 1. This is the reason that we opted to display the SVD estimations whose 39th element is equal to 0.014 and all other elements are equal in both estimations. 15 Samuelson (1962) thanked Garegnani ‘for saving me from asserting the false conjecture that my extreme assumption of equi-proportional inputs in the consumption and machine trades could be lightened and still leave one with many of the surrogate propositions’. 16 It follows that the more similar are the singular values, the higher is the entropy which is maximized when Smax =-n ∗ m ∗ ln(q ), where m = ∑ σ i /n , and in our case for n = 65 are of the same value, the entropy will be m = 0.157117. The exponential of the term Smax gives an effective rank of approximately equal to one depending on the number of terms, whereas the maximal nominal rank might be n, that is the number of linearly independent rows or columns. In the case of a random matrix, its effective rank will be 1! Although the nominal rank may be quite higher, because the subdominant eigenvalues tend to becoming negligibly small near but not necessarily zero.

7

A simple but realistic linear model of production

7.1 Introduction In this chapter, we use a realistic numerical example in the effort to illustrate most of our empirical findings. Realistic in the sense that the data come from an actual input–output table that of the U.S. economy of 2014. The choice of 2014 from the WIOD instead of the published by the BEA more recent input–output tables of 2018 has to do with the availability of most of the required information. We use two versions of the model, one with circulating capital commonly used in the literature on capital theory and the other with fixed capital, which is a more concrete and, therefore, a more realistic approach. Furthermore, for reasons of simplicity and clarity of presentation, in both models, we do not include the matrices of taxation, depreciation and circulating capital advanced, whose potential presence, as experience has shown, it does not change the results in any significant quantitative and certainly not qualitative way. It is understood that all three matrices, along with the vector of capacity utilization for the adjustment of capital stock matrix to each normal use, could be very easily included (if available) at a later and more concrete stage of analysis. The remainder of the chapter is structured as follows: Section 7.2 illustrates some of the fundamentals of an input–output analysis and the derivation of the matrix of technological coefficients with the aid of which we determine the relative prices and the equilibrium rate of profit. Section 7.3 expands the analysis by utilizing the aggregated into five sectors input–output table of the U.S. economy of the year 2014 to estimate labor values and their monetary expression, direct prices (DP). Section 7.4 uses the technology matrix along with the money wage and the vector of the basket of wage goods for the estimation of prices of production (PP) within circulating and fixed capital models using two alternative methods for the estimation of the matrix of fixed capital coefficients. Section 7.5 traces the price rate of profit (PRP) trajectories and the wage rate profit (WRP) curves in both models. Section 7.7 applies the eigendecomposition to our aggregated model. Section 7.8 derives the hyper basic industry for illustrative purposes. Finally, Section 7.9 concludes the chapter with a summary and remarks for future research efforts.

172 Linear model of production

7.2 Input–output data and the estimation of relative equilibrium prices An input–output table describes the relations between industries whose total output must equal to their total input. Quesnay’s Tableau Économique and Marx’s schemes of reproduction are the prototypes of modern input–output tables. The columns of an input–output table describe the inputs of each industry from itself and from the others and the column sum gives the total cost of production of each industry, which in addition includes the payments to the primary inputs; that is, wages and operating surplus that includes profits, depreciation, taxes, etc. The rows of an input–output table refer to the sales of an industry to itself and to the other industries. A portion of total output produced is absorbed by the final demand; that is, consumption, investment, government expenditures and net exports. A summary view of an input–output table is given below: The classical assumption of given technology implies that if for some reason, the output, xj, of industry j increases, it follows that the inputs xij of the industry j must increase proportionally. The technological coefficients are determined by the following relationship aij = xij / x j , where aij stands for the technological or input–output coefficients. Once computed, input–output coefficients are treated as constant (fixed) indicating the lack of substitution between inputs. Moreover, the column sum of technological coefficients, for a viable economy, must be less than one, which is equivalent to saying that the value of output produced must exceed the value of inputs that were used in its production. The input–output coefficients can be converted from a descriptive device to a useful analytical tool with the aid of linear algebra. In the interest of brevity and clarity of presentation, we restrict the analysis into two industries. Thus, we have x1 = a11 + a12 + d1 x 2 = a 21 + a 22 + d 2 where d1 and d2 are the final demands of industries 1 and 2, respectively. Hence, we have a system of equations, which can be written in terms of matrices as follows: ˜ x ˛ 1 ˛° x 2

˝ ˜ a ˆ = ˛ 11 ˆ˙ ˛° a21

a12 a22

˝˜ x ˆ˛ 1 ˆ˙ ˛° x 2

˝ ˜ d ˆ+˛ 1 ˆ˙ ˛° d2

˝ ˆ ˆ˙

Table 7.1 Input–output table Outputs Inputs

Intermediate Demand

Industry 1 Industry 2 …. Industry n Wages Operating Surplus Total

Industry 1 x11 x 21 … xn1 w1 v1 x1

Industry 2 x12 x 22 …. xn2 w2 v2 x2

Final Demand Total Output … … … … … … …

Industry n x1n x 2n … xnn wn vn xn

y1 y2 … yn

x1 x2 … xn

Linear model of production 173

By denoting matrices in bold capital letters, vectors in bold lowercase letters and scalars in italics, we can write the above system of equations in compact form as follows: x = Ax + d which solves for the output vector x = [ I − A ]−1 d The matrix [ I − A ]−1 is known as the “Leontief inverse” whose columns show the input requirements, both direct and indirect, of all producers, generated by one unit of output. Having derived the technology matrix, we can estimate the prices of the two industries p1 and p2, which correspond to the outputs of industries 1 and 2, respectively. The total cost of each industry is Cost of industry 1: ( p1a11 + p2a21 ) x1 Cost of industry 2: ( p1a12 + p2a22 ) x 2 If we further assume a uniform profit rate r, we get the prices of each industry p1 = p1a11 + p2a21 + r ( p1a11 + p2a21 ) = (1 + r ) ( p1a11 + p2a21 ) p2 = p1a12 + p2a22 + r ( p1a12 + p2a22 ) = (1 + r ) ( p1a12 + p2a22 ) or in matrix form, ˜ a a12 ˛ 11 ˘ p2 ˛˝ ˇ °ˇ a21 a22 ˝˘ The problem now is to find the rate of profit and prices, which are consistent with the givens of the above-described economy. Clearly, the profit rate will correspond to the maximum eigenvalue of our hypothetical economy while equilibrium prices will correspond to the associated eigenvector. If we, therefore, write the above system in compact matrix form, we will have ˜ p ° 1

p2 ˛˝ = (1 + r )˜° p1

p (1+ r )−1 = pA where the eigenvalue ˜max = (1 + r )−1 and the eigenvector p corresponds to the vector of positive relative prices, which need to be fixed for their scale, as we show in the realistic numerical example that follows.

7.3 A Realistic numerical example and the estimation of direct prices Let us now take a representative numerical example based on the aggregated input–output data of the U.S. economy of the year 2014, the most recent input–output data available from WIOD (2016), meaningfully aggregated into five sectors and presented in Table 7.2.

Agriculture Manufac-turing Utilities FIRE Services Wages Gross operating surplus Total

815317.6 2022601.7 90093.8 580995.4 544680.0 986426.3 1175764.4 6215879.2

160349.9 139695.9 17554.1 77014.9 88066.5 152170.0 520510.7 1155362.0

54955.7 321277.4 19922.8 199948.7 124190.0 522102.0 471199.3 1713596

Manufacturing Utilities

Agriculture

Table 7.2 Aggregated into five industries input–output table, USA 2014

18700.2 589376.6 59301.6 908509.9 1307999.0 2213528.9 1978342.6 7075758.8

FIRE 38581.0 848977.2 349443.8 958569.6 3203842.6 5375267.2 4014932.5 14789614.0

Services

87340.9 1888539.9 270965.4 3346344.6 6300576.9

1155362 6215879.2 1713596 7075758.8 14789614

Final Demand Total

174 Linear model of production

Linear model of production 175

The matrix of input–output coefficients, A, is estimated by dividing the inputs of each sector of Table 7.2 by its total output. Thus, we get ° ˝ ˝ A =˝ ˝ ˝ ˛

0.1388 0.1209 0.0152 0.0667 0.0762

0.1312 0.3254 0.0145 0.0935 0.0876

0.0321 0.1875 0.0116 0.1167 0.0725

0.0026 0.0833 0.0084 0.1284 0.1849

0.0026 0.0574 0.0084 0.0648 0.2166

˙ ˇ ˇ ˇ ˇ ˇ ˆ

As expected, no column (or row, in our case) sum of the matrix A exceeds one which amounts that the economy produces surplus and thus it is capable of its reproduction and expansion. 7.3.1 Direct prices, prices of production and market prices If we symbolize the row vector of values of produced commodities by λ, the row vector of direct employment coefficients by, l, the indirect labor, that is, the labor contained in the inputs that are used in the current production of commodity j by ˜ j aij , then the direct and indirect labor requirements per unit of output are written in compact form as follows:

˜ = l + ˜ A which solves for ˜ = l [ I − A ]−1 Using the above input–output numerical example, we estimate the row vector of employment coefficients per unit of output, l, that is, the ratio of total employment (employees and self-employed) of sector j, Lj, times the wage rate, wj, of the sector j over the product of the economy’s average wage rate, w , times the output, xj, of the of the sector j. Thus, we get l=

w jL j = [0.0022     0.0027     0.0051    0.0053    0.0061] wx j

Where w = 59.3 is the economy’s average wage in thousands of U.S. dollars.1 The labor values, that is, the direct and indirect labor requirements per unit of output are estimated below:

˜ = l [ I − A ]−1 = [0.0053   0.0076    0.0086    0.0089    0.0094 ] The row vector λ gives the quantity of homogenized labor contained (directly and indirectly) in the output of each sector. The notion of value in Marx is, as we know, monetary. Thus, we have to transform the above quantities of direct and indirect labor time to direct prices (DP), which we symbolize by v j. For the estimation as well as for the comparison of DP with the market prices (MP), we consider that the MP of each unit of output of a sector is equal to 1 monetary unit (see Chapter 4). We assume that e is the row (summation) vector of ones and with its aid we stipulate the following

176 Linear model of production

normalization condition v˜ x = ˜ ex. The vector of unit labor values is transformed to DP as follows: ˙ 30950210 ˘ ˙ ex ˘ v = ˜ˇ = ˜ˇ  = [0.610    0.875   0.981  1.017  1.077] ˆ ˜x  ˆ 269965.35  where (ex )( ˜ x)−1 stands for the monetary expression of labor time. We observe that the vector of DP is extremely close to the vector of MP (which is equal to one). The proximity of DP to MP can be ascertained by the usual measures of deviations. The mean absolute deviation (MAD) with n = 5 is MAD = n −1 v − e = 0.1256 The mean absolute weighted deviation (MAWD) is 1155362.0  6215879.2  ˛xˆ MAWD = v − e ˙ ˘ = v − e  1713596.0  / 30950210 ˝ = 0.0813   ˜˛°˛ ˝ ex ˇ ex 7075758.8    14789614.0 ˜˛ ˛°˛˛ ˝ x

and the Steedman and Tomkins (1998) numéraire bias-free metric of deviation d-statistic is d = 2(1 − cos ˜ ) = 0.199 where ˜ is the estimated angle between the two vectors, v   and e, in comparison (see Section 4.3.1). 7.3.2 The Sraffian standard commodity For the needs of utilizing a convenient numéraire other than gross or net output, we refer to Sraffa’s standard commodity and standard ratios as the appropriate numéraire for our estimations provided that is scaled to the level of output. More specifically, we estimate the eigenvalue identified with the standard ratio of the vertically integrated input–output coefficients, which are the result of the assumption that wages are equal to zero and all value added is equivalent to profits. In other words, the maximum eigenvalue will be equal to the economy’s maximum rate of profit and the vector of output proportions will be the vector associated with the maximal eigenvalue. Thus, we have H = A [ I − A ]−1 The maximal eigenvalue that we get is ˜max = 0.934. Subsequently, we normalize the matrix H dividing it by the ˜max or what is the same by multiplying it by the standard ratio or the maximum rate of profit, R. The new estimated maximal eigenvalue is equal to one and the subdominant eigenvalues are scaled relative to the maximal eigenvalue. We find that the estimated subdominant eigenvalues are significantly less than one while the imaginary

Linear model of production 177

part in the last two eigenvalues is nearly three times smaller than the real part. In particular, we have 1.000, 0.132, 0.031, 0.009 − 0.003i, 0.009 + 0.003i The associated with the above maximal eigenvalue r.h.s. eigenvector, sc , of the matrix, H, is normalized appropriately so as to end up with the standard commodity ˘0.3691 ˘4693960    0.9219 11723611 ° vx ˙   ˘ 30950210  = 1110132  s = sc ˝ ˇ = 0.0873   vs ˛ c ˆ 0.4816  2.4339309  6124041      0.7673 9757541  which is used for our normalizations that follow.

7.4 Prices of production in circulating and fixed capital models I and II In this section, we estimate the prices of production (PP) defined as prices that incorporate the economy’s general rate of profit. The estimation of PP is carried out in both circulating and fixed capital models. The capital stock matrix used in the fixed capital model is estimated through two alternative ways. The first is the product of the column vector of investment shares times the row vector of capital–output ratio. The second way is based on the capital f lows matrix, which has been aggregated into five sectors. The so aggregated capital f low matrix is multiplied by the diagonal matrix of capital–output ratios. The resulting by the second method capital stock matrix is considered a better estimate, although the results of the two estimating methods are not so much different, as we show below. 7.4.1 Prices of production in a circulating capital model For the estimation of PP, we use the vector of real wage, that is, the basket of goods that workers spend their money wage on, that is w = pb where w is the money wage, p stands for the PP and b is the (nx1) vector of wage goods estimated by aggregating the column of the consumption expenditures to form five shares that are then multiplied by the economy-wide average wage: °0.4355 ˙ ˝9.4159  ˇ b = ˝1.3510   ˇ ˝ ˇ ˝16.684   ˇ ˝˛31.413 ˇˆ

178 Linear model of production

Alternatively, we could use the minimum instead of the average wage, but experience has shown that the average wage gives more stable results over the years for the same country and across countries. Having estimated the basket of wage goods normally consumed by workers with their money wage, we can proceed with the estimation of the nonnormalized prices of production, π, in the circulating capital model defined as follows: π = πA + πbl + rπ ( A + bl ) where bl is a new matrix that represents the quantity of commodity i, which is required for the consumption of workers in order to produce commodity j. The above relation after some manipulation gives the following eigenequation: πr −1 = π ( A + bl ) [ I − A − bl ]−1 Using the above realistic example of five sectors, we get the following maximal eigenvalue whose reciprocal gives us the rate of profit r = 1/2.7853774 = 0.359 and the normalized absolute eigenvalues are the following: 1.000, 0.132, 0.031, 0.009, 0.009. We observe, once again, that the subdominant eigenvalues are quite small relative to the maximal and the ratio of the subdominant to the dominant eigenvalue is 18.17%. A result that lends support to the view that the subdominant eigenvalues do not add or subtract much in the overall motion of the economy. In the last two of the subdominant eigenvalues, the imaginary parts (not shown) are three times lower than their real part and their module combined is nearly 2% of the maximal eigenvalue. The matrix H is as follows: ˘ 0.2044 0.2472 0.0950 0.03751 0.0315    0.2692 0.2301   0.3054 0.6524 0.4216 −1 0.0327 0.0480  H = ( A + bl ) [ I − A ] =  0.0349 0.0452 0.0356  0.2252 0.3394 0.3374 0.3435 0.2775   0.3474 0.4940 0.4565 0.5868 0.6220   The maximal eigenvalue associated with the unique positive left-hand-side (l.h.s.) eigenvector gives the vector of relative prices and in terms of our numerical example, we get the normalized standard commodity vector of PP ˝ vx ˇ ˝ 30950210 ˇ p = πˆ  = πˆ  = [0.687  1.060  0.970  0.963  0.972] ˙ πs ˘ ˙ 24579691 ˘ Once again, we invoke our usual statistics of deviation of PP from MP, and we get the following: The MAD is 0.0936 or 9.36%. The MAWD is 0.047 or 4.74%. The d-statistic is 0.158 or 15.8%.

Linear model of production 179

7.4.2 Prices of production in a fixed capital model I For the estimations of PP in a fixed capital stock model, we need to construct the matrix of capital stock per unit of output. For this purpose, we start with the estimation of the vector of capital stock per unit of output, k, whose data are available from WIOD at the 54-industry detail aggregated to the following five capital–output ratios or capital intensities. ° k = ˝ 2.105  ˛

0.584

1.513

0.838

2.63

˙ ˇˆ

We observe that there are wide differences in the estimated capital intensities as this can be judged by the standard deviation equal to 0.854. Hence, the average is equal to 1.535, while the coefficient of variation is equal to 0.557. In order to form the matrix of capital stock per unit of output, K, we need the investment or capital f lows matrix. The trouble with this matrix is that it is constructed sporadically and for different needs and classifications rendering extremely difficult its availability let alone its use which always requires adjustments. The lack of capital f lows matrix, we opine, may be surpassed by the investment shares of each sector, n, which are readily available in the usual input–output tables. Such a vector after its appropriate aggregation and estimates of its shares is displayed below: ° ˝ ˝ n =˝ ˝ ˝ ˛

0.0503  0.2905 0.2684   0.2109   0.1799

˙ ˇ ˇ ˇ ˇ ˇ ˆ

The multiplication of the two vectors n and k gives us the square matrix of capital stock coefficients per unit of output, K, that is ° ˝ ˝ K =˝ ˝ ˝ ˛

0.1060 0.6118 0.5654 0.4443 0.3789

0.0294 0.1697 0.1568 0.1233 0.1051

0.0761 0.4394 0.4060 0.3191 0.2721

0.0422 0.2433 0.2248 0.1767 0.1507

0.1325 0.7648 0.7067   0.5554 0.4737

˙ ˇ ˇ ˇ ˇ ˇ ˆ

The next step is to estimate the l.h.s. eigenvector corresponding to the non-normalized prices of production through the formation of the following eigenequation: or

π = πA + πbl + rπK πr −1 = πK [ I − A − bl ]−1

180 Linear model of production

The vertically integrated capital stock coefficients are the following: ˛ ˙ ˙ −1 H K1 = K[I − A − bl ] = ˙ ˙ ˙ ˝

0.2688 1.5516 1.4337 1.1267 0.9610

0.2643 1.5253 1.4095 1.1077 0.9448

0.2951 1.7029 1.5736 1.2366 1.0547

0.2722 1.5710 1.4517 1.1408 0.9730

0.3711 2.1416 1.9790 1.5552 1.3265

ˆ ˘ ˘ ˘ ˘ ˘ ˇ

Clearly, the columns of the matrix of capital coefficients, K, are very different and so is the sum of its columns that are the capital intensities of industries, eK = k. By contrast, the columns of the matrix of the vertically integrated capital coefficients, H K1, are quite similar for each industry and even closer to each other are the column sums, that is, the product of the row unit vector e times the matrix H K1 as indicated below: eH K1 = °˛ 5.342

5.252

5.863 

5.409   

7.373 ˝˙

The average vertically integrated capital intensity is equal to 5.848, as expected due to the process of vertical integration. The standard deviation is estimated at 0.885 and the coefficient of variation is equal to 0.151, that is, 3.68 times lower than that of the simple capital intensities; result indicative that the vertically integrated capital stock coefficients of the industries are characterized by much lower variability in comparison with the simple capital intensities. The rate of profit, that is, the reciprocal of the maximal eigenvalue is r=

1 = 17.14 5.8350108

The normalized eigenvalues, as a result of linear independence of the columns of H K1, are all zero except the maximal, which is equal to one. The prices of production according to this method of estimating the matrix of capital stock coefficients and after their normalization with the standard commodity are ˝ ex ˇ ˝ 30950210 ˇ p = πˆ  = πˆ  = [0.876    0.861    0.961   0.887  1.209] ˙ πs ˘ ˙ 24644288 ˘ Once again, we invoke the statistics of deviation of the PP from MP and we get The MAD is 0.1428 or 14.28%. The MAWD is 0.1560 or 15.60%. The d-static is 0.1500 or 15.00%. 7.4.3 Prices of production in a fixed capital model II The allocation of capital stock per unit of gross output over investment shares and the construction of a matrix of capital stock gives quite accurate

Linear model of production 181

results, when we compare them with those that we derive with more direct estimations of matrix K through an actual investment matrix. As a matter of fact, in the past, the BEA and the OECD had published investment or capital f lows matrices and from those available, we picked the one of the year 1990 published by the OECD which is of 34 industries. The appropriate aggregation of this matrix gave us the following 5×5 capital f lows matrix: ˜ 171 ˛ ˛ 17659 0 ˛ ˛ 4543 ˛ 116 °

0 87720 0 10299 0

914 49550 0 9484 0

0 163029 0 29551 0

18 35378 0 5384 0

˝ ˆ ˆ ˆ ˆ ˆ ˙

The matrix of investment shares is the following: ° ˝ ˝ N =˝ ˝ ˝ ˛

0.0076 0.7852 0 0.2020 0.0052

0 0.8949 0 0.1051 0

0.0152 0.8266 0 0.1582 0

0 0.8466 0 0.1534 0

0.0004 0.8675 0 0.1320 0

˙ ˇ ˇ ˇ ˇ ˇ ˆ

We observe that in the above aggregated matrix of five sectors, 11 of its 25 cells, that is, 44% are filled with zeros while four other cells are not too different from zero. Naturally, the allocation of capital stock per unit of gross output over the matrix of investment shares will give the matrix of capital stock coefficients that is as follows: ° ˝ ˝ K =˝ ˝ ˝ ˛

0.0160 1.6540 0 0.4255 0.0109

0 0.5230 0 0.0614 0

0.0231 1.2504 0 0.2393 0

0 0.7092 0 0.1285 0

0.0012 2.2842 0 0.3476 0

˙ ˇ ˇ ˇ ˇ ˇ ˆ

Clearly, the matrix of investment shares imposes its form on the resulting matrix of capital stock coefficients. In a comparison of the two capital stock matrices of models I and II, both have the same column norms, which by construction are equal to their respective capital–output ratios. The difference is in the allocation of the capital–output ratios over the industries. The ranks of the two matrices are equal to one, a result indicating that they will not have a very different maximal eigenvalue. Furthermore, the second and subdominant eigenvalues in the first matrix will be zeros while in the second matrix the second eigenvalue is expected to be very small and the rest, if not

182 Linear model of production

zero, nevertheless they will be in the neighborhood of zero. Such results will be stronger, much stronger, in the disaggregated matrices.

H K2

˙ ˇ ˇ = K [ I − A − bl ]−1 = ˇ ˇ ˇ ˆ

0.0224 4.4140 0 0.8919 0.0136

0.0083 4.5162 0 0.7237 0.0034

0.0290 4.9899 0 0.8420 0.0019

0.0052 4.6475 0 0.7549 0.0013

0.0068 6.3690 0 0.9963 0.0012

The matrices H K2 and H K1 contain different entries, but the sum of their columns is exactly equal to each other, eH K2 = eH K1, where e is the summation vector. The nominal rank of H K1is equal to 1, because of the linear dependence in its columns. By contrast, the nominal rank of the matrix H K2 is equal to 4 indicating that there are four linearly independent vectors; in reality, however, there is a kind of pseudo-independence and the effective rank of the matrix H K2 is in fact less than its nominal. The eigenvalues of the matrix H K2 are the following: 5.275, 0.0101+ 0.0218i, 0.0101 − 0.0228i, − 0.0002, 0 The rate of profit, that is, the reciprocal of the maximal eigenvalue is r=

1 = 18.96 5.275 

which is only slightly different from that of the fixed capital model I (r = 17.14). The estimated prices of production according to this method of estimating the matrix of capital stock coefficients and after their normalization with the standard commodity are ˝ ex ˇ ˝ 30950210 ˇ p = πˆ  = πˆ  = [0.907    0.891  1.0048   0.917  1.250] ˙ πs ˘ ˙ 34589626 ˘ We invoke the usual metrics of deviation of PP from MP and we get The MAD is 0.1021 or 10.21%. The MAWD is 0.1642 or 16.42%. The d-static is 0.1496 or 14.96%. These are quite similar to those derived in model I.

7.5 Prices of production/direct prices trajectories In this section, we study the price trajectories consequent upon changes in income distribution using our aggregated into five sectors input–output data of the U.S. economy of the year 2014. The price trajectories and, in general, changes in distribution are more transparent in models with a few sectors. We explore these changes used at the beginning a circulating capital model;

˘      

Linear model of production  183

subsequently, the model expands to include fixed capital estimated according to the two methods, whose details have been outlined in Chapter 4 and in Section 7.4. For the price trajectories, we invoke Equation 4.6 below: p = (1 − ρ ) v [ I − HRρ ]

−1

and we apply it in the case of the circulating capital model with H = ( A + bl ) [ I − A ]−1 and subsequently in the fixed capital models with H = K [ I − A ]−1 of course with appropriate adjustments in R. 7.5.1  PP/DP trajectories, circulating capital model As we can see from Figure 7.1, the movement of relative prices is monotonic and the same is true with the movement of capital–output ratio as shown in Figure 7.2 ruling out the case of Sraffian effect. We know that the circulating capital model is more likely to display switching in the price rate of profit (PRP) curves. In principle, we cannot rule completely out the change in rankings. However, we do know that crossings are more likely to occur when the number of industries increases, but even in this case, it has been shown empirically that the likelihood is very limited (see Chapter 4), exactly as suggested by theoretical studies (Pertz 1980; Mainwaring and Steedman 2000). The movement of the capital intensity, as measured by the circulating capital–­output ratio, induced by changes in the relative rate of profit is also 1.8 2

1.6 1.4

1

1.2 1

3

0.8 4 0.6 0.4

5 0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

1.1

Figure 7.1  Price trajectories and the relative rate of profit, circulating capital model, USA 2014.

184  Linear model of production

monotonic and does not display any empirically significant variations (Figure 7.2). In general, the reevaluation of capital stock in response to changes in income distribution is moderate and less pronounced than those of relative prices shown in Figure 7.2. 1.6

2

1.4 1

1.2 1

3

0.8 0.6 0.4

4 5 0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

1.1

Figure 7.2  Capital intensities and the relative rate of profit, circulating capital model, USA 2014.

This analysis in terms of a circulating capital model displayed linearities in the movement of relative prices and in capital intensities. As we have already pointed out, the results depend to a great extent, on the limited number of sectors; more industry-detailed input–output tables may give rise to non-monotonic movements in prices with extrema and crossings, as we have shown in Chapter 4. 7.5.2  PP/DP trajectories, fixed capital model I We repeat the same experiment using this time fixed capital model I in which we construct the matrix of capital stock coefficients by multiplying the row vector of capital stock per unit of output times the column vector of investment shares. In Figure 7.3, we display the paths of prices consequent upon changes in the relative rate of profit. As expected, the paths of relative prices are linear going either in the upward or downward direction. The capital–output ratios, as shown in Figure 7.4, remain the same regardless of the changes in the rate of profit, a result explained by the way in which the matrix of the capital stock coefficients matrix is constructed. The rank of such a matrix is equal to one and imposes its form when multiplied from the right by the Leontief inverse. In effect, the column norms of the matrix display little variability, a result that has to do with the process of vertical integration.

Linear model of production 185 2

1

1.8 1.6 1.4

5

1.2 2

1

3

0.8 0.6

4 0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

1.1

Figure 7.3 Price trajectories and the relative rate of profit, fixed capital model I, USA 2014. 5.5 1 4.5

3.5

5

2.5

2 3 4

1.5

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

1.1

Figure 7.4 Capital intensities and relative rate of profit, fixed capital model I, USA 2014.

7.5.3 PP/DP trajectories, fixed capital model II The above findings are contrasted to those derived from a fixed capital model in which we use the matrix of capital stock coefficients proper, that is, through the capital f low or investment matrix with the aid of which we construct new shares and the resulting matrix we multiply element by element by

186 Linear model of production

the capital–output vector. The so derived matrix of capital stock coefficients is then multiplied by the Leontief inverse also augmented by the matrix of workers consumption coefficients. The movement of relative prices is quite similar to that of the model I with some minimal differences lending support to the suggested treatment of the matrix capital stock employed in constructing the capital stock matrix in model I. The price trajectories are shown in Figure 7.5 while the capital intensities are shown in Figure 7.6. 2.2

1

2 1.8 1.6 5

1.4 1.2 1

2 3

0.8

4

0.6

0

0.1

0.2

0.3

0.4

0.5

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0.7

0.8

0.9

1

1.1

Figure 7.5 Price trajectories and the relative rate of profit, fixed capital model II, USA 2014. 6.1 1

5.6 5.1 4.6 4.1 3.6

5

3.1 2 3

2.6 2.1 1.6

4 0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

Figure 7.6 Capital intensities and relative rate of profit, fixed capital II.

1

1.1

Linear model of production 187

A cursory consideration of the price trajectories in the fixed capital models I and II shows that the ranking of prices is exactly the same. Furthermore, the differences in price trajectories are minimal, as we can see in industries 2 and 3, which almost coincide in the two models, whereas industries 1, 4 and 5 are in the same ranking and do not differ in any empirically significant way over the entire range of the relative rate of profit. Such price trajectories indicate that the movement of capital–output ratios does not show variability. In fact, in Figure 7.4, they are moving parallel to the horizontal axis and nearly parallel is their movement in the model II (Figure 7.6). Sectors 2 and 3 in both models have a capital–output ratio close to the standard ratio (in model I slightly higher and model II slightly lower) and so their trajectories are near one in both models. One could reasonably argue that the capital intensities of these industries are not far from the Sraffian standard product, in a sense, to the Ricardian practical invariable measure of value.

7.6 Wage rate of profit curves For the estimations of the WRP curves, we invoke from Chapter 5 the relevant relations and use them in both the circulating capital and the fixed capital models (in its two variants). In the case of circulating capital model, the WRP curves are derived using Equation 5.3, whose details are explained in Chapter 5: w=

ex

−1 l ˇ˘I − A − ˜ R ( A + bl ) x

The WRP curves of the fixed capital model are derived from Equation 5.4 (see Chapter 5), which is used in both of its variants. w=

ex −1 l [ I − A − ˜ RK ] x

Figure 7.7 displays the WRP curves utilizing the data of our aggregated into five sectors input–output table of the U.S. economy of the year 2014. The numeraire that we used was the gross output as we did in Chapter 5. The results are no different qualitatively using as numeraire the vector of workers’ consumption goods expenditures per unit of output and the same is true with the vector of employment coefficients. We observe in Figure 7.7, a very similar curvature for the circulating capital model, which is concave exactly as was found with the disaggregated data (2000, 2005, 2007 2010 and 2014) and also 2018 for the U.S. economy and in other countries and years as discussed in Chapter 5. Not surprisingly, the fixed capital models I and II displayed a convex shape with the fixed capital model I to display a much lighter curvature. Figure 7.8 in a panel of three graphs displays the WRP curves of our aggregated input–output table. The estimates of WRP curves were made for the circulating capital model according to Equation 5.5: −1 w = e x . / ˇ˘l ( I − A − ˜ R ( A + bl )) x 

Linear model of production

188 120 100

Fixed Capital Model I

80 60

Circulating Capital Model

40 Fixed Capital Model I 20 0

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Figure 7.7 WRP curves of circulating and fixed capital models, USA 2014.

where we end up with a dot (element-by-element) division of two vectors. Clearly, the numeraire here plays no role because it appears in both numerator and denominator in the five dot (element-by-element) divisions. The WRP curves in the cases of fixed capital models I and II are derived according to Equation 5.6: −1 w = e x . / ˆˇl ( I − A − ˜ RK ) x ˘

It is important to note that the above Formulae 5 and 6 from Chapter 5 essentially indicate the market price over the unit cost of production, whose difference between sectors has to do with the relative rate of profit. If ˜ = 0, we get the unit value or the same productivity of labor, which is equal to the maximum real wage. As ρ increases to its maximum, we have the sharing out of value added between workers and capitalists. From the WRP curves in Figure 7.8, we observe that in both the circulating and fixed capital models, we have near linearities. We also find that in the circulating capital model, the WRP curves may take any curvature convex or concave depending on its capital intensity relative to the average. The WRP curve in the fixed capital model I is more like a straight line with a light convexity. A result that we also found in our 65 and 49 (54) industry detail input–output models in Chapter 5. The fixed capital model II clearly displays a convex form, a result that is, also supported by the more disaggregated input–output data. Finally, the curvature of the fixed capital model II is higher than that of model I, a result that has to do with the construction

Linear model of production 189 200

Circula˜ng Capital Model

180 160 140

1

2

120

3

100 5

80

4

60 40 20 0

0

0.1

0.2

0.3

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200 180

Fixed Capital Model I

1

160 140

2

120

3

100 5

80 60

4

40 20 0

0

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Fixed Capital Model II

180 160

1

140

2

120 100 80

5

60

3 4

40 20 0

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Figure 7.8 WRP curves in circulating and fixed capital models I and II, USA 2014.

190 Linear model of production

methods of the matrices of fixed capital stock and the distribution of eigenvalues associated with them. In model I, the subdominant eigenvalues are zero. By contrast, in model II, the second, third and fourth eigenvalues are by far smaller than the maximal and the fifth eigenvalue is zero. The matrix of capital stock coefficients has a row filled with zeros and its nominal rank is therefore equal to four. This difference in distribution of eigenvalues is the reason that the curvature of WRP curves of model II is somewhat higher than that of model I.

7.7 The eigendecomposition and the approximations of price trajectories Following the discussion in Chapter 6 and in particular of Sections 6.3 and 6.4, we proceed with the eigen or spectral decomposition restricting ourselves to the circulating capital model starting with its linear approximation. The eigenvalues of the matrix

( A + bl) [ I − A ]−1 are 1.372,  0.352,  0.087,  0.025 − 0.008i, 

0.025 + 0.008i

We apply the first term from Equation 6.6 (for the details see Chapter 6) −1 −1 −1 ˜ ˜ H = y1x1ˆ x1ˆ y1 + 2 y 2 xˆ2 xˆ2 y 2 +  + n yn xˆn xˆn yn   ˜1 ˜1

(

)

(

)

(

)

which gives the linear approximation ˛ ˙ ˙ H1 = (y1x1° )−1 x1° y1 = ˙ ˙ ˙ ˝

0.062 0.216 0.025 0.182 0.329

0.104 0.360 0.042 0.304 0.549

0.077 0.267 0.031 0.225 0.406

0.069 0.240 0.028 0.202 0.365

0.065 0.226 0.026 0.190 0.344

ˆ ˘ ˘ ˘ ˘ ˘ ˇ

of matrix H ° ˝ ˝ H = ˝ ˝ ˝ ˛

0.204 0.305 0.035 0.225 0.347

0.247 0.655 0.045 0.339 0.494

0.095 0.422 0.036 0.337 0.456

0.038 0.269 0.033 0.343 0.587

0.032 0.231 0.048 0.277 0.622

˙ ˇ ˇ ˇ ˇ ˇ ˆ

Below we give the sum of columns of matrix H and its various approximations in order to get an idea of the proximity of these matrices. The sum of columns of H1 = 0.814 1.360 1.006 0.904 0.852 The sum of columns of H 2 = 0.805 1.332 0.999 0.914 0.864

Linear model of production 191

The sum of columns of H 3 = 0.813 1.327 0.993 0.915 0.866 The sum of columns of H = 1.117 1.781 1.346 1.270 1.210 The price trajectories are displayed in a panel of 5 graphs in Figure 7.9, where the dotted line is the actual price trajectory and the straight solid line stands for the linear approximation. Since our approximation is linear naturally for the extreme values of ρ (0, 1), the error will be zero. In our case (Figure 7.9), the maximum deviation between the estimated curve and its approximation is equal to 3.03% (industry 2) for ˜ = 0.50, which is almost exactly the estimated actual share of profit 1.6

1.3 1.25 1.2 1.15

A

1.1

r pp

ox

im

a˜

o

no

y1 str du n I f

1.45

1.3

Industry 1

Ap

1.15

pr

o

a xim

˜o

no

y2 str du n I f us Ind

2 try

1.05 1

0

0.2

0.4

0.6

0.8

1

1

1

0

0.2

0.998

Appro

xima˜

0.996

on of

Indus try 3

0.95

Ap

pr

0.994

0.6

0.8

1

im

a˜

on

of

0.8

1

Ind us try 4

3

0.99

ox

Ind us try 4

0.9

y str du In

0.992

0.988

0.4

1

0

0.2

0.4

0.6

0.8

1

0.85

0

0.2

0.4

0.6

1

0.95

Ap

0.9

pr

ox

0.85

im

a˜

on

of

In du str y5 Ind us try 5

0.8

0.75

0

0.2

0.4

0.6

0.8

1

Figure 7.9 Price trajectories and their linear approximation, circulating capital model, USA 2014.

192 Linear model of production

in national income assuming that the standard ratio is no different than the maximum rate of profit.

˜=

r 1/ 2.785 0.359 = = = 0.493 R 1/ 1.372 0.729

On the positive direction, the overall error of approximation is approximately 4.47% at ρ = 50%, and on the negative sign, the total average absolute deviation is 1.11%. Clearly, the linear approximation is pretty good, and there is no need to try a quadratic approximation because the gains in accuracy will be limited. Figure 7.10 lends overwhelming support to the linear approximation rendering superf luous the higher-order approximations. We observe that the neither the quadratic nor the cubic approximation improve the proximity in any significant way. In particular, the quadratic and cubic approximation gave the following matrices: H 2 = H1 +

0.369 ( y 2 x˛2 )−1 x˛2 y 2 2.785

˝ ˆ ˆ =ˆ ˆ ˆ ˙

0.163 0.416 0.037 0.277 0.439

0.082 0.234 0.024 0.173 0.293

0.093 0.281 0.029 0.218 0.377

0.047 0.219 0.030 0.212 0.406

0.04

0.039 0.201 0.029 0.202 0.394

ˇ      ˘

2

0.03

5

0.02

3

0.01 0

0

0.1

0.2

0.3

−0.01 −0.02

0.4

0.5

0.6

0.7

0.8

0.9

1

4 1

Figure 7.10 Deviation of actual and linear approximation price trajectories, USA 2014.

Linear model of production 193 1.3

1.6

Sector 1

Sector 2

1.5

1.24

linear approximation

linear approximation 1.4

1.18

Quadratic and cubic approximations

quadratic and cubic approximations

1.3

1.12

1.06

1

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

1.1

0.9

1

1

Sector 3

1.002

0.1

0.2

0.3

0.4

0.5

0.6

0.7

actual price trajectory

linear appro ximati on

0.92

0

0.1

0.2

0.9

quadratic approximation

actual price trajectory

0.99

0.88

0.3

0.4

0.5

0.6

0.7

0.8

0.9

0.86

1

0

0.1

0.2

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0.4

0.5

0.6

0.7

0.8

0.9

Sector 5

1

cubic and quadratic approximations 0.95

actual price trajectory

0.9

linear approximation 0.85

0.8

0.75

0

0.1

1

0.94

cubic approximation 0.994 0.992

0.9

Sector 4

0.96

0.996

0.8

cubic and quadratic approximation

0.98

0.998

0.988

0 1

linear appro ximati on

1

actual price trajectory

1.2

actual price trajectory

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Figure 7.11 Price trajectories and their linear, quadratic and cubic approximations.

1

Linear model of production

194

0.087 (y 3x˛3 )−1 x˛3 y 3 2.785 0.107 0.147 0.075 0.222 0.423 0.289 0.024 0.037 0.030 0.178 0.274 0.215 0.283 0.446 0.385

H3 = H2 + ˝ ˆ ˆ =ˆ ˆ ˆ ˙

0.051 0.218 0.030 0.212 0.404

0.044 0.196 0.029 0.203 0.391

ˇ      ˘

Furthermore, the fourth- and fifth-order approximations as they displayed conjugate figures did not improve the situation, when we tested their absolute values. In Figure 7.11, in the panel of five graphs, the dotted line stands for the actual price trajectory and the rest for its approximations. The linear approximation is represented by the solid straight line whereas the quadratic is represented by the dashed line. The cubic happens to coincide and to become nearly indistinguishable from the quadratic with the exception of industry 3, which at first sight is characterized by a phenomenally large curvature but on closer examination, we observe that its difference from one or the DP is only in the decimal point. We may say, therefore, that this is no different from the standard industry in that it is not subjected to changes in the face of variations in income distribution. Consequently, in the linear approximation of industry 3, the error of approximation is negligible; the maximum error is less than half of 1%! In Figure 7.12, we present the MADs in percentage form of the linear, quadratic and cubic approximations. Clearly, the linear approximation is a 2 % MAD linear approxima˜on

1.8 1.6 1.4 1.2

% MAD cubic approxima˜on

1 0.8 0.6 0.4

% MAD quadra˜c approxima˜on

0.2 0

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Figure 7.12 MADs of linear, quadratic and cubic approximations of price trajectories, USA 2014.

Linear model of production 195

stand-alone pretty good approximation that can be improved slightly by the application of a quadratic approximation; the cubic or higher-order approximations do not have much to add, and when the relative rate of profit increases, the rounding-off errors are magnified, thereby discouraging the use of higher-order approximations. In general, the aggregated input–output data give closer approximations between DP and PP, both of which are pretty close to MP. These findings are indicative that as the level of aggregation increases, the overlap between industries increases and become more similar to each other.

7.8 Hyper-basic industry The linear nature of the PRP and WRP curves encourages the research for the presence of a hyper-basic industry (or a few) embedded in the structure of the economy bearing most of the weight that gives rise to the quasi linearity of the curves. This is something very similar to the idea of onecommodity world or, what amounts to the same thing, the idea that of a single industry containing enough information to display trajectories that are similar to those that we would have derived had we had the information from all the sectors comprising the entire economy. It goes without saying, that a hyper-basic industry does not exist in reality, in the sense that it can be located in input–output data along with many other usual industries; however, such a hypothetical industry, if constructed may contain enough information, which is extracted, ideally, from the basic industries of the economy. The construction of such a hyper-basic industry bears some similarity to the pivotal role of the Sraffian standard industry that has been utilized in our analysis. From the above discussion, it follows that the matrix H1 although a linear approximation nevertheless is a pretty good approximation of matrix H, as this can be judged by an inspection of the graphs in Figures 7.11 and 7.12 as well as the statistics of deviation which are minimal. In the sense that both matrices give rise to price trajectories not very different from each other, we can decompose the matrix H1 into two matrices (factors). As in Chapter 6, we apply the Schur factorization (Meyer 2001, 508–509, see also Mariolis and Tsoulfidis 2016b, 2018) which gives the matrix of eigenvalues S, a diagonal matrix whose maximal eigenvalue is one and the right-hand eigenvector u1of matrix H1 associated with it. The column vector u1 of the matrix H1 will be uˆ1 = °˛ 0.5732

0.9580

0.7090      0.6367

0.6002 ˝˙

which when replaces the first column of the identity matrix, gives the new matrix U of dimensions 5×5. The matrix S can be now estimated as follows: SH1 =  U −1H1U

196 Linear model of production

The first row of the resulting matrix SH1 is the economically significant vector, which is displayed in Table 7.3. We observe that the first element of the so-estimated vector S is (approximately) equal to one and the rest on the first row are all positive or rather of the same sign. Finally, all the other elements of matrix S are in principle zero.2 Thus, with the aid of the Schur decomposition, we managed to transform the matrix H1 into the matrix S. Since matrix H1 gives rise to trajectories of prices quite similar to those of matrix H, it follows that matrix H1 is not very different from H. Therefore, the matrices H1 and H are quite similar to the matrix S. This similarity among the three matrices essentially amounts to the cells of the first row corresponding to a single industry of the matrix S. The composition of this so to speak hyper-basic industry, as the direct result of a similarity transformation is designed to encompass essentially the structural properties characterizing the entire economy. This approximation of the matrix H through the first term of its eigendecomposition, that is, the matrix H1 is further enhanced, the lower the spectral ratio. In our case, the eigenvalue gap is large enough ensuring a relative accurate approximation, as this is discussed in Chapter 6 and it is depicted in the graphs of Figures 6.10 and 6.11 and Table 6.3. In short, if the second eigenvalue tends to zero, the better will be the approximation. This is particularly pronounced in the fixed capital model. In the case of the circulating capital model, the application of Schur decomposition to the matrix HR gives the same matrix U1 as that obtained from the matrix H1, of course except by multiplication by a scalar; the latter does not affect the results when the first column of the matrix U1 replaces the first column of the identity matrix, which multiplied by the matrix H gives SH =  U −1HRU The first row of the matrix S gives us the economically significant vector displayed in Table 7.4. One wonders to what extent, if any, the two row vectors—displayed in Tables 7.3 and 7.4 derived from the rank 1 of matrix SH1 and the rank 5 of matrix SH  , respectively—relate close enough to each other. For this reason, we plot them in a scatter graph along with their linear regression line (Figure 7.13). The R-square of the two vectors is quite high at 92.7% Table 7.3 The matrix S H1 , Schur method 1.0000

0.3763

0.0439

0.3171

0.5732

‒5.5E-16 ‒2.0E-16 ‒3.6E-16 ‒4.7E-16

7.2E-10 ‒8.0E-10 1.4E-09 6.8E-10

‒9.8E-10 ‒9.0E-10 ‒2.8E-10 ‒2.0E-10

4.0E-09 4.2E-09 ‒7.3E-10 5.5E-11

‒1.1E-09 ‒1.1E-09 2.0E-10 ‒9.7E-11

Linear model of production 197 Table 7.4 The first row of the matrix SH 1.00

0.62

0.07

0.45

0.70

0.00 0.00 0.00 0.00

0.11 0.03 ‒0.05 ‒0.06

‒0.01 ‒0.01 0.00 0.01

‒0.03 0.04 0.07 0.03

-0.06 0.02 0.15 0.19

1 y = 1.001x - 0.1067 R² = 0.927

0.8

0.6

0.4

0.2

0 0.0

0.2

0.4

0.6

0.8

1.0

−0.2

Figure 7.13 First row of matrix SH vs. approximation SH1 .

indicating that the two estimated vectors move pretty much together. The MAD of the two estimated basic vectors is only 6.02% and the d-statistic is 21.23%. Of course, the number of observations is very small to claim unquestionable proximity; however, we know that the derived results are in line with those found for similar but larger dimension matrices in Chapter 6. Lastly and for the same purpose, we tried the singular value decomposition (SVD) whose results not surprisingly were approximately the same with those of Schur method. Let us start with the application of the SVD to the matrix H1 that gives us the following economically meaningful vector 0.3615 0.6041 0.4471 0.4015 0.3785 that replaces the first column of the identity matrix. Subsequently, we apply the similarity transformation SH1 =  U −1H1U and the matrix SH1 will be

198

Linear model of production

Table 7.5 The matrix SH1 , SVD method 1.0000

0.5967

0.0696

0.5029

0.9090

‒5.5E-10 ‒2.5E-10 ‒2.6E-10 ‒1.3E-10

3.9E-10 ‒9.5E-10 1.2E-09 6.1E-10

‒1.0E-09 ‒9.2E-10 ‒3.0E-10 ‒2.1E-10

3.7E-09 4.1E-09 ‒8.7E-10 ‒8.6E-12

‒1.6E-09 ‒1.4E-09 ‒3.6E-11 ‒2.1E-10

The nominal rank of the above matrix is equal to 5, but in fact is equal to one in that all the information of the matrix is contained in its first row that forms the economy’s hyper-basic industry. The results from the matrix HR gives the economically meaningful vector 0.3382 0.5561 0.4431 0.4435 0.4282 which as above, it replaces the first column of the identity matrix. Subsequently, we apply our usual similarity transformation and get Table 7.6 The first row of the matrix SH, SVD method 1.0000

0.6072

0.0693

0.4477

0.6906

0.0020 ‒0.0213 ‒0.0533 ‒0.0586

0.1030 0.0144 -0.0883 -0.1047

‒0.0082 ‒0.0066 ‒0.0087 0.0025

‒0.0208 0.0284 0.0324 ‒0.0051

‒0.0520 0.0008 0.0882 0.1224

The differences between the two methods of constructing the hyper-basic industry are minimal, and there is no particular reason to prefer one method over the other.

7.9 Summary and conclusions In this chapter, we carried out most of our estimations with the use of an aggregated into five sectors input–output table of the USA of the year 2014. We chose this particular input–output table of 54 industries because of the necessity for our estimations data. The purpose of this aggregation was to enable the reader to evaluate visually the arguments presented in Chapters 4–6 focusing on manageable dimensions matrices. We utilized two models: a circulating capital model and a fixed capital model in which the matrix of fixed capital has been estimated according to two methods. As expected, the fixed capital model, in both of its versions, gave more definitive results than the circulating capital that has been used extensively in similar studies. This by no means implies that the results in the

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two models are so much different; both models are very good approximations to reality and the fixed capital model is preferred to circulating. However, the lack of sufficient data of investment f lows matrices, as well as the vectors of capital stock and turnover times limits the applicability of fixed capital model. The circulating capital model is usually used precisely because its data requirements are not too demanding.

Notes 1 We assume that the differences in skills are taken care by the ratio of each sector’s wage over the economy’s average wage, assuming away all other factors (Botwinick 1993). 2 Practically, this might not be exactly the case and the other rows of S may contain cells containing near-zero numbers most likely due to rounding. In the subroutine of Gauss (or Matlab) that we utilized in our estimations, we found that the elements in the rows other than the first were trivial in size. Thus, although we know that the rank of H1 is equal to 1, the answer that we derive from Gauss or Matlab is 5.

8

Summing-up

8.1 Introduction In this book, we revisited the concept of capital and examined its treatment in economic theory. We argued that capital was of central importance in the economic theory, classical or neoclassical, and in fact, this is a contagious topic where the two major alternative approaches to economics meet and clash with each other. We insisted that the discussions about the theory of capital were not so much about its measurement per se (not that this did not matter) but rather its consistency with the requirements of the neoclassical theory. Two major debates on the theory of capital took place so far and a third one, characteristically different from the others, is underway. The first debate was a rather silent one and involved almost exclusively the pioneer neoclassical economists in the last quarter of the nineteenth century. In time, that the neoclassical theory was in its making compelling was the need for a consistent treatment of capital, and for that, it was imperative to circumvent its measuring in terms of money value. It goes without saying that the first neoclassical economists were fully cognizant of the issues at hand, that is, the measurement units of capital should be suitable for cost minimization and that must be independent of prices. The second capital theory debate is what came to be known as the Cambridge capital controversies (CCC), and it is the first essential debate because two competing approaches, the neoclassical and the classical debated fiercely with each other. The outcome of this debate is that economists from the Cambridge UK side prevailed, as this can be judged by the statements of the protagonists from both camps and, of course, from evaluating the arguments on both sides. The truth, however, is that the classical side of the debate kept discussing various aspects of capital theory, whereas the neoclassical demoted the significance of the discussions and continued using the usual production functions without wondering about the issues that were raised and remain unsettled. Lately, within the classical approach, there have been quite interesting developments dealing with core issues of the capital theory. Among these developments are the price rate of profit (PRP) trajectories and their determinants

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as well as their relationship with the wage rate of profit (WRP) curves. These recent developments are not only in theory but also they are tested using data from many economies over the years. Meanwhile, using input–output tables and network analyses, neoclassical economists grapple with questions having to do with the effects of internal or external shocks in the economy and the attainment of new equilibrium and its stability (Lucas 1977; Acemoglu et al. 2012). Thus, we can tell that although the two approaches move parallel to each other, we do not exclude the possibility of providing answers to the same core questions.

8.2 Capital theory debates in retrospect Two were the approaches to the measurement of capital within neoclassical economics in the last quarter of the nineteenth century. The first that measures capital as a scalar and that is the approach introduced by Jevons and Menger and later by Böhm-Bawerk. These economists used labor time to measure capital independently of prices and then by canceling out labor, what was left, it was pure calendar time as the measurement unit of capital. This “solution” was anything but satisfactory; in fact, this type of calendar time is what Ricardo utilized as waiting in production time (Principles I, ch. 1) to show that its effect on relative prices is very small. Similarly, Marx describes this kind of time as turnover time and uses it in the estimation of the total (circulating and variable) capital advanced. Marx, like Ricardo, argued that the effect of turnover time on prices and the rate of profit is minimal (Capital II, Part II). Thus, calendar time was already known to Ricardo and Marx but, unlike the neoclassical economists, considered it of secondary importance. Hence, the minor became the major in the neoclassical approach that “sacrificed” labor to keep only the time dimension. Léon Walras, also a pioneer of neoclassical approach and the originator of the second line of research, theorized capital as a vector of heterogeneous goods, each one of which deserved its reward. The idea was a sensible one, but it suffered from the lack of equalization of profit rates as his approach required (see Eatwell 1990, 2019; Garegnani 1990, 2012; Fratini 2019; Petri 2017, 2020 and the literature therein). Dissatisfaction with the proposed measurement units of capital and the compelling need for an alternative that circumvents the social implications of evaluating capital in terms of labor time, led neoclassical economists to drink the “bitter glass” of capital as a value magnitude. J.B. Clark and Wicksell tried hard to envisage a scalar presenting a value measure of capital; however, without being consistent with the neoclassical theory of value. In the 1930s, Hicks, Lindahl and others proposed alternative evaluations of capital, which, however, were in deviation of the long-period method of economic analysis. For some historians of economic thought (see Fratini 2019; Kurz 2020), this is the second round of capital theory debates with the first involving the pioneer neoclassical economists. In our view, this particular approach is not separate

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and, therefore, quite different from that of the first neoclassical economists because we are dealing with an internal to neoclassical discussion-extension and not with an altogether alternative approach and external critique. Capital and its estimation in a way consistent with the requirements of neoclassical economic theory continued to occupy center stage, albeit not through debates such as those in the 1960s. Indeed, this is the first and perhaps the only time that economists with competing theories of value exchanged views and leveled critiques regarding the conceptualization of capital and its internal consistency with the requirements of the neoclassical theory of value and distribution. The debate essentially sparked by Robinson’s (1953–1954) important article and continued in the Corfu conference 1958, where among the participants were included Piero Sraffa, Nicholas Kaldor and Austin Robinson from Cambridge UK side and Paul Samuelson, Robert Solow and Evsey Domar from Cambridge US side. In the list of participants, there were other major economists such as John Hicks (see Lutz and Hague 1961). The discussions and debates resumed after the publication of Sraffa’s (1960) book, whose main purpose was to redirect the economic theory to its old classical route. The book includes an implicit critique of the classical political economics approach, by arguing that the movement in relative prices no longer depends on the difference of an industry’s capital intensity relative to the economy’s average. The idea is that the industries’ capital intensities themselves are not fixed and their characterization may alternate from capital- to labor-intensive and back again as income distribution changes. These alternations depend on the non-linearity of relative price trajectories in the face of changes in income distribution. In other words, the non-linear relative price trajectories are imparted onto the movement in capital intensities, which change with income redistribution. A corollary of the above is that the WRP frontier, which is no different from the neoclassical economics relative price-factor frontier, will have, in general, a non-linear shape that gives rise to possibilities of reswitching depending on the number of curvatures and their importance. Consequently, one capital-intensive technique might be chosen as the cost-minimizing one at a low profit (interest) rate; the technique might be abandoned at an intermediate rate of profit and then selected again at a higher rate of profit. A result, contrary to the neoclassical expectation of scarcity prices, according to which labor- and not capitalintensive technique should be selected at a higher interest rate. Samuelson (1962), who knew well both the people and their arguments on both sides of the Atlantic, assumed the challenge to defend the neoclassical position. To this end, he introduced the concept of “surrogate production function” with the aid of which he could derive the main neoclassical propositions that the relative prices of factors of production ref lect their relative scarcities. More specifically, Samuelson showed that the derived WRP curves through his “surrogate production function” form a frontier corresponding to the well-known price-factor frontier, whose linearity gives rise to wellbehaved demand schedules for both capital and labor.

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The “surrogate production function” instead of resolving the challenges to neoclassical theory, it became the starting point of intense debates involving economists on both sides of the Atlantic or more precisely, neoclassical economists from the US side and economists in the Sraffian line of investigation mainly from the UK side.1 Levhari (1965) argued that the reswitching of techniques is not possible, which is equivalent to ascertaining the monotonic movement in both relative prices and WRP curves. Levhari’s (1965) article attracted the interest and stimulated further discussions in the UK Cambridge side. Sraffa’s students Pasinetti (1966) and Garegnani (1966) assumed the task of disproving Levhari’s (1965) argument showing the possibility of reswitching as a quite expected and, therefore, a quite normal result. The next year Levhari and Samuelson in their joint article (1966, p. 518) made the following sincere opening statement: “we wish to make it clear for the record that the non-reswitching theorem associated with us is definitely false. We are grateful to Dr. Pasinetti […]”. Furthermore, Samuelson (1966) not only admitted the possibility of reswitching of techniques but also showed its presence by utilizing a numerical example; in so doing, he recognized that there are issues with the neoclassical theorization of capital and that the reverse capital deepening cases cannot be ruled completely out. Notes Samuelson: Reswitching, whatever its empirical likelihood, does alert us to several vital possibilities. […] If all this causes headaches for those nostalgic for the old time parables of neoclassical writing, we must remind ourselves that scholars are not born to live an easy existence. We must respect and appraise, the facts of life. (Samuelson 1966, pp. 582–583) It is worth noting that such an acknowledgment does not imply Samuelson’s opposition to the use of production functions with capital including as an argument. This may be ascertained by his following bold statement: Until the laws of thermodynamics are repealed, I shall continue to relate outputs to inputs—i.e. to believe in production functions. Unless factors cease to have their rewards to be determined by bidding in quasicompetitive markets, I shall adhere to (generalized) neoclassical approximations in explaining their market remunerations. (Samuelson 1966, p. 444) Despite the declared victory of the UK Cambridge side of the debate regarding the fundamental inconsistency of the neoclassical theory of value, neither Samuelson nor other major neoclassical economists argued in favor of the Sraffian strand of the classical economics. Quite the contrary, neoclassical economists characterized the reverse capital-deepening effects and double-switching occurrences “paradoxical” results and, as such, they may

204 Summing-up

be added in the long list of paradoxes of neoclassical theory.2 Consequently, the neoclassical economists continued to utilize capital as a value magnitude and more than fully justifying Robinson’s (1953–1954) plea of the standard neoclassical practices regarding the units of measurement of capital, which are left hanging in the air in the usual production functions. The new generations of economists, by and large, have noticed the echo of something like a “capital debate” but the pressing needs for publications set-aside what is considered “counter-productive problematics” and continue using production functions without wondering about its measurement units and the related to this questions. Burmeister (2000), in evaluating the impact of the debate in the economic theory, argued that the Cambridge UK side hastened to pronounce the debate settled and, by doing so, some central issues that could be further and more deeply discussed were left aside. He continues, by arguing that the damage had been done, and Cambridge, UK, “declared victory”: Levhari was wrong, Samuelson was wrong, Solow was wrong, MIT was wrong and therefore neoclassical economics was wrong. As a result, there are some groups of economists who have abandoned neoclassical economics for their own refinements of classical economics. In the United States, on the other hand, mainstream economics goes on as if the controversy had never occurred. Macroeconomics textbooks discuss “capital” as if it were a well-defined concept—which it is not, except in a very special one-capital-good world (or under other unrealistically restrictive conditions). The problems of heterogeneous capital goods have also been ignored in the “rational expectations revolution” and in virtually all econometric work. (Burmeister 2000, p. 310). We doubt very much though that if the Cambridge UK side held a somewhat more modest stance would make any difference regarding the debate in the years that followed. From the above, it might be argued that the CCC was so “much ado about nothing”. On ref lection, however, we discover that the CCC contributed, one way or another, to the development of another strand of neoclassical theory based on a neo-Walrasian general or intertemporal equilibrium. In this strand of neoclassical theory, capital is no longer viewed as a factor of production but rather as an ordinary commodity, which like all the others is specified according to its place and delivery date. This strand of neoclassical theory returns to Walras’s notion of capital as a vector of physical goods, which need not be evaluated in money terms and express them in a scalar magnitude in line with the neoclassical tradition emanating from the time of J.B. Clark and Wicksell or even earlier. The trouble with this strand of the neoclassical theory is it dispenses with the traditional long-period method of analysis because there is no equalization of profit rates and so the existence

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of equilibrium is not in the discussions let alone its attainment and stability (Eatwell 2019; Fratini 2019). These theoretical developments were underway even before the climax of the CCC but, the weakness of the neoclassical economists to defend their major propositions paved the way for the development of the intertemporal strand of their theory, whose popularity increased quite rapidly. This was the first consequence of the particular treatment of capital not as a factor of production but as an ordinary good with a delivery date and location. The idea is that through the determination of a series of short-run equilibrium positions the long-run equilibrium path of the economy starts becoming visible. The trouble, here is that the classical notion of the attainment of equilibrium prices through the equalization of profit rates resulting from competition is spirited away. For example, in Ricardo, market and natural prices are equal (on average) in the long run and that the change in the market prices are explained by changes in their respective direct and indirect labor times (Tsoulfidis and Tsaliki 2019, ch. 3). Ricardo’s analysis is steadfast in the long-run setting and the short run is the stochastic part of attaining the long equilibrium. By contrast, the intertemporal general equilibrium is quite the opposite. The second consequence discussed also in Garegnani (1990, 2012) and later further elaborated by Eatwell (2019), Fratini (2019) and Petri (2020 and the literature cited therein) is that the saving and investment interaction may lead to multiple equilibrium positions and therefore instability. Reverse capital deepening and reswitching, therefore, may occur in this intertemporal general equilibrium neoclassical approach in the attainment of equilibrium between saving and investment.

8.3 Capital theory debates in prospect The significance of CCC is that by revealing the weak aspects of the neoclassical theory restored the confidence in the classical political economy approach. The latter, this time and after a long period of “rehabilitation” returns addressing effectively the few, mostly internal, critiques leveled against it, a great deal of which was anything but scientifically motivated. Over the years, the classical theory of value and distribution has proved that is capable of providing answers to the thorny questions regarding price changes induced by income redistribution and technological change. Furthermore, the classical theory of value not only has made theoretical advances but using input–output data and recent mathematical developments solve old problems and raise new questions that may lead to its further advancement. This knowledge and analytical tools were not available in the nineteenth century; today, more and more, the availability of data enables the classical theory of value to deal with modern problems and issues. The CCC was cast mainly in terms of theoretical debates and always without reference to actual data as this can be judged by the utilized numerical examples (see the Appendix of Chapter 3). The reswitching results, derived from these unrealistic numerical examples, were not noticed from both sides

206

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of the debate. Gradually, the neoclassical economists lost interest in the discussions for mainly two reasons: first, because they lost the debate, and second, the incomplete nature of the Sraffian strand of the classical theory was not appealing to them. On the other side, the Sraffians, by insisting that their critique was chief ly about the internal consistency of the neoclassical theory, considered superf luous any thoughts for empirical testing. After all, the logical critique is superior to the empirical one, and if successful, the testings and their results will not have to add anything quite new, let alone the problems with the data and the estimating techniques. There is no doubt that the Sraffians’ logical critique was formidable and shook the beliefs of major neoclassical economists; however, their insistence on the occasional instances of reswitching made them lose orientation and let other aspects of neoclassical economics like subjectivity, substitutability, causation in the marginal productivity theory of prices and neoclassical competition go intact. It is important to emphasize that the trouble with the application of the logical critique is that one may end up trapped into the assumptions of the theory under critique. It seems that most (fortunately not all) of Sraffians have fallen into such a trap by adopting (not all) in their analysis the concepts of perfect competition and substitutability, core ideas of neoclassical economics (Tsoulfidis 2015). Thus, it comes as no surprise that the first empirical results showing no or relatively few cases of reswitching in the WRP curves did not attract the attention that they deserved from both sides of the debate. The empirical research that started in the decade of the 1980s continued intensively in the more recent years and gave results overwhelmingly in support of the near linearity of both the PRP trajectories and WRP curves. These repeated occurrences of monotonicity in prices and the near linearity of both PRP and WRP curves rightfully gives them the status of empirical regularity. Hence, the reswitching and capital deepening, in reality, are not as general, as one would have expected after Sraffa (1960). The complex price feedback effects continue to exist, albeit their inf luence is limited to relatively few PP, and both theoretical and empirical research has shown that the alleged complexities of the price effects have been exaggerated. In recent years, we start distinguishing a third stage of the capital theory debates taking place, this time, exclusively among the followers of the general classical tradition. In this debate, old issues resurface, new ones are introduced and innovatively pursued. In the relevant literature, some economists in the classical tradition insist that empirical testings are not rightly contacted. For example, Kurz and Salvadori (1995) were from the first arguing that the question of reswitching is not appropriately tested; since the estimations of WRP curves are at a given time (year) without considering what happens with all the available techniques over time.3 This line of research, according to which we may select techniques from all the available, has been pursued by Zambelli et al. (2017), whose results show reverse capital deepening, which is the next worse thing (reswitching is the worst) to neoclassical economics. Han

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and Schefold (2006) and Schefold (2008, 2019, 2020) show that reswitching is more of a theoretical possibility. On a different line of research, Cogliano et al. (2018) also find near linearities in the WRP curves and the lack of reswitching in input–output data of the German economy. In a comparison of these researches, we observe that all of them adopt the neoclassical competition in either its perfect form (Han and Schefold 2006; Zambelli et al. 2017) or its monopolistic form (Cogliano et al. 2018). Hence, both approaches utilize one or the other aspect of the same unrealistic conception of competition according to which firms can freely pick the technique that suits them best from the book of blueprints. This conceptualization of “choice” of techniques has no contact to reality, where firms do not freely and without significant cost just pick techniques. A slight increase in wage does not mean that firms select a capital-intensive technique but simply their profits may fall. In effect, techniques change over long stretches of time, and at any given time, they are not responsive to changes in prices. In short, there is no substitutability between the factors of production, and Leontief ’s “cooking recipe” of fixed technology is well established in the empirical data. Consequently, the choice of techniques from the book of blueprints has no place in the broad classical approach and in general in any realistic approach. Furthermore, Sraffian economists are usually reluctant to use capital stock in their empirical analyses and, in so doing, in effect, what they estimate is the profit margin on cost. This is the reason why in Zambelli et al. (2017), the profit margin on cost almost exceeds the 250% level! The inclusion or not of fixed capital in the estimations exerts its effect on the curvature of the near-linear WRP curves from convex (fixed capital) to lightly concave (circulating capital). Furthermore, the employment of fixed capital allows the estimation of the maximum rate of profit, which in the classical tradition is theorized with a long-run falling tendency that depresses the actual rate of profit but, at the same time, increases productivity and reduces the unit cost of production and price. Thus, the near-linear WRP curve and a falling maximum rate of profit together preclude the case of more than one crossing provided that the productivity of labor is always on the rise (Carter 1980; Tsoulfidis and Tsaliki 2019, ch. 4). It goes without saying that one should not be indifferent between the kinds of capital; to the contrary, better estimates of capital stock should be more inclusive by utilizing the circulating capital and wages advanced whose estimation requires the respective turnover times. The fixed capital stock must also take into account the capacity utilization. The lack of required data on matrices of fixed capital stock as well as turnover times usually prevents the investigation to expand to more complete measures of capital advanced. However, the classical approach must become more innovative in the efforts to cope with the the lack of data on certain variables, which when available, experience shows, they work in favor of the classical view. Bródy’s (1997) conjecture about the stability properties of matrices having to do with the distribution of subdominant eigenvalues is what stimulated

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interest in testing the properties of the utilized technology matrices. The hitherto empirical research has repeatedly ascertained the exponential distribution of eigenvalues, where the subdominant eigenvalues are by far smaller than the dominant ones; this has become the currently key investigative issue in determining PP and the shape of their trajectories. There have been two major explanations of this skew distribution of eigenvalues and the resulting from these near linearities of PRP and WRP curves. The first claims that the particular distribution of eigenvalues is attributed to the randomness characteristics of the studied matrices. The second interpretation argues that the particular distribution has to do with the effective rank of the economic systems matrices, which is by far lower than that of their respective nominal rank. The randomness hypothesis states that as the dimensions of the matrix A or H = A [ I − A ]−1 R−1 increase the elements in each of their column become more like random variables and therefore the expected values of their mean tends to become the same. But the mean value of each column is the capital–output ratio and this is the explanation for the observed linearities in the trajectories of the PP and WRP curves. This should be looked as a tendency, which is strengthened by the size of matrices and forces subdominant eigenvalues to constellate near zero. However, in the empirical research presented in Chapters 4–7, the size of the utilized matrices is relatively small, and so the randomness hypothesis does not truly apply to them. Nevertheless, we observe quasi-linearities in PP, and the occurrence of non-linearities is more frequent as the size of matrices increase. Moreover, the random matrix hypothesis requires subdominant eigenvalues tending to zero as the size of the matrix increases. However, we present evidence that the distribution of eigenvalues is not consistent with the randomness hypothesis and Bródy’s conjecture as the second and third eigenvalues do not get smaller but larger relative to the maximal, as the size of the matrix increases (e.g., see Table 6.2). Hence, the empirical evidence regarding the size of matrices and the eigenvalue distribution repeatedly has shown that they are not consistent with the randomness hypothesis for realistic size matrices (see Mariolis and Tsoulfidis 2014, 2016a, 2016b, 2018; Shaikh and Nassif-Pirez 2018; Shaikh et al. 2020). Further tests that we ran (see Chapter 5) were not consistent with the randomness hypothesis. From the above, it follows that the observed linearities in PP and WRP curves must be explained from the internal structural properties of the system matrices and not from the alleged randomness properties. Furthermore, the assumption of randomness applied in a way very similar to the assumption of perfect competition in neoclassical economics. As competition is never perfect, so is with the randomness, an ideal situation that can be used as a yardstick to measure the extent that the linearity hypothesis holds. However, perfect competition is a concept derived not by observing how firms organize and, in reality, compete with each other in the markets, but rather as an imposed assumption to derive certain results. By way of the familiar fiction

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of the Walrasian auctioneer and the attainment of equilibrium prices, which is only possible by invoking the equally fictional assumption of perfect competition. Similar might be the assumption of randomness for it is not derived by observing how the matrices of technological coefficients are constructed and evolve over time but rather as an assumption that if true justifies the near linearities of PP and WRP curves. It is true that the particular distribution of eigenvalues is responsible for the observed linearities in PRP and WRP curves. In our view, the elements of the technological coefficients matrix, A, but more those of the fixed capital coefficients matrix, K, which is derived with the aid of the capital f low matrix, are anything but random. These matrices display specific repeated patterns that persist over the years, for the same country and across countries, as we have shown in our research. Starting with the matrix A, we know that the sum of its columns is less than one, if in the economy there is going to be any surplus. Furthermore, the main diagonal of the matrix A, more often than not, contains the largest element on each column and so the remaining elements are less, usually much less, than the diagonal ones. These salient features of the elements of matrix A are enhanced through its multiplication by the Leontief inverse and its normalization by the standard ratio, HR −1. Intuition and empirical observation ascertained these properties, which lend support to the view that the columns of matrix A are not as linearly independent as they appear to be. This effective dependency or, what is the same, “pseudo-independence” is maintained, if not strengthened, in the vertically integrated input–output coefficients HR −1. Hence, the presence of “pseudoindependence” makes us wonder about the “true” rank of the matrix and arrive at the concept of effective rank, that is, a more appropriate measure of the connectedness of the columns of the matrix A. The effective rank of a matrix is not measured in any unambiguous way, and in Chapters 6 and 7, we utilized an eigen or spectral decomposition to derive it in an objective and measurable way. The case of the fixed capital matrix derived from the capital f low table is much more straightforward since it contains many zero rows because with the exception of the investment goods-producing sectors, all the rest have zero rows. Hence, the resulting matrix of capital stock coefficients, naturally, will have too many zeros, and it can be thought of as a sparse matrix with not too strong interindustry connections. The post-multiplication of K matrix by the Leontief inverse will just pass its form on the resulting new matrix K [ I − A ]−1 R −1 whose effective rank will be very low, much lower, than that of the matrix A [ I − A ]−1 R −1. A low effective rank reinforces linearities, and this is the reason why the price trajectories in fixed capital are much more linear than those of circulating capital models. Consequently, the observed linearities in PRP trajectories and WRP curves are derived from the structural properties of the technological coefficients and capital stock matrices. The near linearities that we have derived

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do not imply random matrices but persistent systemicity contained in the already utilized matrices over time and across countries. A result rooted in the empirical regularities that withstand the time factor have to do chief ly with the technology and its evolution. To make a long story short, because random matrices give rise to a particular distribution of eigenvalues associated with linearities does not imply that the actual matrices are random.

8.4 Concluding remarks In the 1960s, the classical approach through the logical critique of the neoclassical theory of capital received recognition, but neoclassical economists (not all) returned to their business as usual. Nowadays, we are witnessing the broad classical approach making a comeback as a standalone theory, capable of answering old questions and raising new ones that can be answered in a much more satisfactory way than neoclassical theory can. From the above, it follows that there is another debate in the making concerning the capital theory, but this time, the debate is between economists following the general classical approach dealing with microeconomic questions in the main. In the current state of analysis, the major questions regarding the classical theory of value and distribution have been answered satisfactorily. For instance, the complex price feedback effects that may change the direction of PP are rare and occur, by and large, when PP are near to DP and their respective capital intensities are not far from the Sraffian standard ratio. In effect, the variability of PP with extrema is by far lower than that of monotonically moving PP. The WRP curves under these circumstances in the circulating capital model display some concave light curvature, while in the fixed capital model, the convex WRP curvature is found somewhat more pronounced. These are typical findings obtained in the relevant empirical studies, within the classical approach. While this is the case with the classical approach with respect to capital theory, the neoclassical theory still faces important problems that need solutions. Although reswitching was a formidable critique of the neoclassical economics in the 1960s, we find that the empirical research initially and then the theoretical led more and more to the further development of the classical approach. The neoclassical theory was also affected by the CCC and led to the development of its intertemporal variant. In the recent decades from the neoclassical theory we have another variant that uses the networks analysis, a parallel development to the classical approach and at some point there might be an exchange of analytical tools between them. Meanwhile, there are other issues that the neoclassical theory needs to find satisfactory answers, such as: – –

the extent to which the marginal productivity and factor payments are related to each other and their mono-causal relationship. the subjective nature of the theory for it is based on preferences and permeates costs.

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– –

the substitutability hypothesis in both consumption and production which is absolutely necessary for the neoclassical theory, however, its contact with reality is at least questionable. the conception of competition in its perfect or imperfect form, both of which are also disconnected from the reality.

At first sight, these issues do not appear to be important. However, when the analysis comes to major economic questions, such as economic growth, international trade and the like, the answers that neoclassical theory provides are far from being satisfactory. For instance, neoclassical growth theory argues for the convergence of the economies but instead, we observe more like divergence. Neoclassical international trade theory argues for free trade policies as the way for the less developed economies to reach the desired level of economic growth and prosperity; instead, we see the rich countries to become richer widening their gap from the poor ones. Of course, one could argue that some other countries, like China, are in their growth path; however, these countries have practiced anything but free trade policies. The neoclassical theory argues for stationary capital intensity and income shares. We have argued that, certainly, the capital intensity is rising (how else can someone explain the rising productivity of labor, if not, by the division of labor and rising capital intensity?) and this is what is found in our empirical research. Moreover, the income shares on which so many of the predictions of the production functions are based on, are not stationary; the profit share in the past few decades is in the rising and so are the incomes of the super-rich. Hence, the marginal productivity theory of income distribution appears like a fairy tale in explaining the rising inequalities. The classical approach, in recent years, has demonstrated vitality and potential to provide answers to the above questions. However, we must add that the classical approach, as it stands, needs to expand by addressing other key questions, such as those of the equilibrium level of output and employment. The latter, in the early 1980s, led to the simplistic idea to intermingle the classical theory of value and distribution with the Keynesian “effective demand”. However, such an idea did not prove fruitful, and it has been largely given away. The reason is that, unlike the Keynesian, the question of “effective demand” in the classical approach must be addressed according to its theory of capital accumulation. Hence, the classical and the Keynesian views have in common the causal relation running from investment to saving. However, unlike the Keynesian, the classical analysis treats saving and investment as interdependent variables, both motivated by profitability. As a consequence, the “effective demand” becomes cyclical, and structural emanating from within the process of capital accumulation (see Shaikh 2016; Chatzarakis and Tsaliki 2021). Hence, this treatment not only shows the boundaries of the theory of “effective demand” but also uses these boundaries for the further development of the theory of capital accumulation. In making capital accumulation

212 Summing-up

and the profit motive the center of analysis, issues such as economic growth, unemployment, income inequalities, international trade among others may find theoretically consistent answers.

Notes 1 We say “mainly” because heterodox economists are on both sides, and the same is true for the orthodox economists. 2 The paradoxes in neoclassical theory are listed in the following, of course, unofficial site https://en.wikipedia.org/wiki/List_of_paradoxes#Economics from which those on capital theory paradoxically are missing! 3 Kurz (2020) maintains his position pointing to the grave difficulties in carrying out fully and satisfactorily the testing procedures. Hagemann (2020) summarizes the views on the shape of the location of the WRP curves.

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Index

average period of production (APP) 22, 24, 25, 26, 28 auctioneer 209 best practice technique 122 Blaug, M. 16, 30, 213 Böhm-Bawerk, E. 4, 6, 8, 17, 24–26, 28, 30, 201, 213 book of blueprints 109, 121, 125, 126, 207 Bródy,A. 133–135, 139, 140, 151, 169, 207, 213 Bródy’s conjecture 132–136, 140, 151, 153, 159, 166, 203, 216 Burmeister, E. 204, 213 Cambridge capital controversies (CCC) xv, 1, 5, 6, 19, 31, 36, 60, 61, 93–96, 127, 143, 167 capacity utilization 130, 169, 171, 207 capital deepening 49, 123, 126, 131, 203, 205, 206, 214, 215, 218 capital flow table 62, 79, 90–92, 142, 177, 185, 209 capital goods 2, 4, 7, 10, 12–15, 18–21, 23, 24, 26–29, 31–34, 36–40, 47, 52, 54, 78, 90, 143, 204 capital intensity 1, 3, 13, 14, 35, 45, 47, 48, 53, 54, 62, 63, 68, 69, 71, 73, 77, 80, 84, 95, 106, 116, 125, 167, 180, 183, 188, 202, 211, 216, 220 capitalism 10, 14, 16, 19, 29, 89, 218–220 capital–labor ratio 2, 26, 40–43, 45, 47–49, 51–53, 62, 127, 131, 132 capital–output ratio ix, 66, 74–78, 84, 85, 87–89, 92, 108, 129, 142, 177, 179, 181–183, 184, 208 capital reversing 124, 216 capital stock 13, 24, 25, 29, 33, 48, 62, 64, 70, 78, 79, 88, 90, 91, 98, 100, 125, 126, 129, 130, 139, 141, 143, 159, 167, 169,

171, 177, 179, 180–182, 184–186, 190, 199, 207, 209 capital theories controversies see CCC capital theory xvii, 6, 8, 19, 31, 34, 36, 39, 53, 95, 123, 127, 131, 167, 171, 200, 210, 212–216 capital theory debates see CCC choice of technique 2, 4, 32, 35, 42, 56, 94, 103, 121, 123, 124, 136, 207 circulating capital model vi, vii, ix, x, xi, 14, 60, 62, 65, 67, 69, 70–78, 80–85, 98–125, 130, 131, 135, 139, 142–144, 156, 158, 161, 163, 166, 168, 169, 171, 177, 178, 182–184, 187, 188, 190, 191, 196, 198, 199, 207, 209, 210 Clark, J. B. 8, 16, 17, 27, 28, 32, 46, 131, 201, 204, 214 classical theory of value i, 1, 3, 5, 61, 88, 120, 205, 210, 211 coefficient of variation 70, 73, 82, 88, 132, 179, 180 Cogliano, J. 79, 92, 125, 126, 207, 214 column norm 181 competition classical (real) 121, 124, 127 competition monopolistic 207, 153 competition perfect 27, 36, 21–127, 153, 154, 206, 208, 209, 211 compound interest rate 24–26, 29, 56 consumer goods 20–24, 27, 31, 39, 40, 91, 143 Corfu conference 8, 202 corn model 12, 167 covariance coefficient 129 data of classical theory 20, 30 data of neoclassical theory 17, 18, 30 demand schedule for capital ix, 1, 2, 31, 35, 36, 44, 46, 50, 54, 61, 96, 127, 202 demand schedule for labor ix, 1, 36, 44, 45, 127, 202

222

Index

dimensionality reduction vi, 133, 140, 161 direct price (DP) vii, xv, 14, 15, 45, 62, 64–68, 70–80, 82, 84, 86–89, 92, 96, 99, 100, 105, 135, 144, 147, 167, 169, 171, 175, 176, 183–185, 194, 195, 210 division of labor 10, 29, 211 Dobb, M. 12, 214, 216 d-statistic 72, 81, 105, 107, 176, 178, 197

interest rate xiii, 1, 21–26, 29, 32, 33, 44, 54–58, 93, 127, 202 interest rate compound 24, 25, 29, 56 intertemporal general equilibrium 3, 7, 29, 36, 60, 204, 205, 210 invariable measure of value 13, 14, 77, 187 investment goods 90, 91, 209 isoquants ix, 35–38, 43–46, 60, 61, 96

Eatwell, J. 3, 12, 27, 33, 39, 60, 201, 205, 214, 215, 219 economic growth 39, 211 effective demand 62, 84, 211 effective rank vi, v, 84, 95, 96, 128, 142, 144, 149, 153, 159, 165, 167, 168, 169, 170, 182, 208, 209 eigendecomposition vi, vii, 5, 133, 140–144, 147, 149, 162, 167, 171, 190, 196 eigenspectrum x, 152 eigenvalues distribution x, xiii, 84, 136–138, 155, 170, 208, 215, 216, 219 eigenvectors 104, 134, 140, 148, 159, 160 employment coefficients 59, 61, 66, 92, 107, 108, 129, 175, 187 entropy index 168, 188 ‘envelope’ see wage-profit rate (WRP) frontier

Jevons, S. 4, 6, 16, 17, 21, 22, 26, 30, 33, 201, 215

factorization vi, x, xiii, 159–163, 167, 169, 195 factor price frontier 35, 50, 202 fixed capital model vi, vii, ix–xiii, 62, 65, 69, 70–84, 88, 89, 92, 94, 96–106, 109, 114–120, 129, 141–144, 147, 161, 163, 166–171, 177, 179, 180–189, 196, 199, 210 fundamental theorem of distribution 63 Garegnani, P. 2, 3, 27, 31, 33, 35, 36, 50, 52, 60, 97, 170, 201, 203, 205, 214 geometric mean 155, 157, 159, 170 Harcourt, G. 31, 36, 60, 214–215 Hicks, J. 8, 201, 202, 215 higher order goods 23 hyper-basic industry vi, vii, x, 5, 133, 161–167, 171, 195–198 input-output table x, xiii, 5, 16, 62, 72, 79, 83, 86, 89–95, 99, 101, 121, 123, 125, 136, 139–144, 152–155, 166, 171–174, 179, 213

Kaldor, N. 202 Keynes, J. M. 29, 30, 62, 211, 215, 220 Kuhn’s “protective belts” 122 Kurz, H. xvii, 25, 29, 30, 49, 121, 128, 201, 206, 212, 213, 215, 216, 219 labor commanded theory of value ix, 10, 11, 30 Leontief,W. 6, 16, 34, 90, 93–96, 101, 121, 128, 136, 139, 143, 169, 170, 173, 184, 186, 207, 209, 213, 215, 217 leontief inverse 90, 136, 143, 169, 170, 173, 184, 186, 209 leontief paradox 93, 217 Levhari, D. 35, 203, 204, 215 linear approximations x, xi, 141, 145, 147, 190–195 logical critique 34, 36, 206, 210 long-period method 15, 30, 36, 125, 201, 204, 205, 215 marginal efficiency of capital 29, 220 marginal productivity of capital 19, 21, 22, 27, 28, 29, 32–34, 46, 50, 94, 128, 131 marginal productivity of labor 27, 32 market prices (MP) xv, 7, 33, 67, 79, 88, 100, 105, 169, 175, 178, 180, 182, 195 Mariolis,T. 17, 70, 79, 84, 87, 91, 92, 95, 96, 100, 101, 132, 133, 141, 150, 153, 154, 155, 159, 170, 208 Marx, K. 4, 6, 7, 10, 14, 15, 19, 29, 30, 35, 45, 53, 54, 60–65, 68, 69, 77, 78, 83, 84, 88, 105, 116, 122, 124, 127, 166, 167, 172, 175, 201, 213, 214, 216, 217 matrix of depreciation 60, 62, 70, 90, 91, 97, 109, 135, 169, 171 matrix of technological coefficients 5, 59, 107, 108, 125, 142, 171, 172, 209 mean absolute deviation (MAD) x, xi, xv, 105, 107, 147–150, 162–165, 176, 178, 180, 182, 194, 197

Index mean absolute weighted deviation (MAWD) xv, 70, 176, 178, 180, 182 Menger, C. 4, 6, 16, 17, 23, 24, 30, 33, 201, 216 Mill, J.S. 4, 6, 13, 29, 216 money wage 10, 16, 34, 97, 108, 171, 177, 178 monetary expression of value 14, 54, 64, 66, 97, 135, 176 natural price 12, 14, 15, 36, 215 neoclassical theory of value 2, 4, 19, 20, 21, 29, 31, 35, 37, 39, 43–46, 52, 61, 116, 201–203 non-produced means of production 17, 18, 31, 37 normalization condition 97, 100, 105, 106, 176, 180, 182, 209 one-commodity world 2, 3, 35, 39, 41, 45, 46, 48–54, 60, 167, 195 output–capital ratio 48, 119, 219 Pasinetti, L. v, xiii, 23, 31, 35, 36, 51–59, 63, 65, 106, 203, 217 political economy 9, 30, 121, 128, 205, 214–220 price of production (PP) vii, ix, xv, 7, 14, 15, 20, 25, 64–68, 70–80, 82, 84–89, 92, 96, 97, 99, 105, 107, 108, 133, 135, 140–144, 147, 159, 167, 169, 171, 177–180, 182–185, 206–209 price rate of profit (PRP) trajectories or curves iv, vi, 4, 5, 36, 74, 84, 91, 96, 120, 122, 130, 132–135, 141–144, 147–150, 153–159, 166, 171, 183, 195, 200, 206, 208, 209 produced means of production v, 6, 7, 14, 18, 20, 31, 37–39, 53 production function 2, 8, 27, 31, 33–36, 38, 39, 45, 46, 54, 61, 131, 167, 200, 202–204, 211, 213–215, 217–219 production technique v, ix, 2–5, 20, 30–32, 35–37, 39, 41–45, 49–60, 79, 93, 94, 99, 101–103, 109, 113, 115–126, 131, 132, 202. 203, 206, 207, 213–217, 220 profit-wage ratio 5, 123 quadratic approximation 149, 192, 195 quantity of capital 8, 19, 20, 26, 28, 34, 214 randomness hypothesis vi, 129, 153, 154, 208

223

rate of profit vi, vii, ix–xi, xiii, xv, 3, 7, 8, 10–16, 20–22, 26–28, 30, 32–34, 36–46, 48–53, 57, 61, 64–66, 68, 73–78, 86, 88, 93, 96–98, 104, 110, 113, 114, 117, 119, 120, 123, 124–127, 131–132, 141, 147–149, 163, 166, 171, 173, 176–178, 180, 182, 188, 192, 195, 200–203, 207, 214 real wage xiii, 1, 2, 15, 16, 20, 33, 41–44, 48, 49, 53, 56, 61, 92, 98, 104, 110, 113, 118–120, 126–128, 177, 188 regulating capital 122, 220 regularities 5, 99, 125, 128, 153, 206 “regular” (“irregular”) behavior 85, 97, 98, 113, 114, 117, 119, 166, 216 relative prices 2, 10–13, 20, 34–36, 46, 61–63, 65–68, 74, 77, 78, 94, 131, 141, 144, 147, 166, 171, 173, 178, 183–186, 201–203, 214, 218 relative rate of profit ix, x, xi, 64, 68, 73, 76–78, 86–88, 98, 110, 113, 114, 117–119, 141, 147–149, 166, 183, 184–188, 195 relative scarcity 37, 45, 49, 50, 131 reswitching of techniques v, 2, 4, 31, 50–52, 59, 60, 99, 122, 123, 132, 203, 213 reverse capital deepening 123, 126, 203, 205, 206, 215, 218 Ricardian socialists 16 Ricardo, D. 1, 4, 6, 7, 12, 16, 21, 29, 30, 33, 34, 36, 53, 54, 60–65, 68, 69, 77, 78, 83, 84, 87, 88, 167, 201, 205, 213–217, 219 Robinson, J. 4, 8, 31, 34, 36, 37, 39, 131, 202, 204, 216, 218 Salvadori N 25, 30, 49, 121, 131, 206, 215, 218, 219 Samuelson, P. v, xiii, 2–4, 31, 35, 36, 39, 45, 46, 52–56, 60, 93, 94, 120, 127, 167, 170, 202–204, 215, 218 Schefold, B. 84, 95, 98, 113, 123–127, 129–131, 133, 153, 166, 169, 207, 215, 217, 218 schemes of reproduction 15, 53, 54, 166, 167, 172 Schur method xiii, 161, 163–165, 170, 196, 197 Schur factorization x, xiii, 5, 159–162, 195 Shaikh,A. xvii, 14, 19, 49, 66, 67, 75, 84, 87–89, 95, 96, 101, 106, 121, 127, 128, 130–133, 162–165, 167–169, 170, 197, 208 Shannon, C. 168, 219

224 Index Singular value decomposition (SVD) vi, xiii, xv, 133, 162–165, 167, 168, 170, 197, 198 Smith,A. 4, 6, 7, 9–12, 14–22, 29, 30, 214, 219 SocioEconomic Accounts (SEA) xv, 79, 91 Solow, R. xvii, 31, 36, 131, 132, 169, 202, 204, 213, 219 sparcity index 143 spectral decomposition 140, 190, 209 spectral flatness xv, 157, 159, 170 spectral gap 135, 169 spectral ratio vi, 4, 133–135, 161, 170, 196 Sraffa, P. 1–4, 7–8, 13, 16, 20, 31, 34–36, 62, 65, 68, 70, 78–84, 87–89, 92, 96, 106, 128, 139, 167, 176, 202, 203, 206, 213, 215–219 standard commodity vi, vii, 3, 13, 36, 66, 67, 77, 88, 94, 100, 104, 105, 108, 167, 176–178, 180, 182 standard ratio ix, 64, 69, 73–78, 82, 84–89, 92, 104, 106, 108, 110, 113, 119, 132, 167, 176, 187, 192, 209, 210 surrogate production function 2, 31, 46, 167, 170, 202, 203, 213, 217, 218 tableau économique 166, 167, 172 technological coefficients 5, 59, 107, 108, 125, 142, 171, 172, 209 technology 5, 13, 15, 18, 20, 27, 30, 33, 55, 59, 61, 88, 108–110, 113, 118, 126, 127, 133, 142, 172, 205, 210 Timmer, M. 61, 63, 78, 79, 82, 109, 136, 219 Torres-Gonzales, D. xvii, 130, 151, 219 transformation problem 14, 16, 19, 35, 53, 217, 218 Tsoulfidis, L. 13–16, 19, 29, 33, 45, 53, 60, 62, 63, 70, 79, 84, 87, 91, 92, 95–98,

100, 101, 104, 121, 122, 125, 127, 132, 133, 141, 150, 153, 154, 167, 169, 170, 205, 208, 215, 217, 220 Tsaliki, P. xvii, 13, 15, 19, 33, 45, 53, 63, 87, 91, 92, 95, 96, 101, 104, 121, 122, 125, 127, 133, 185, 205, 207, 211, 213, 217, 218, 220 Turnover time 13, 33, 65, 92, 130, 169, 199, 201, 207 unit labor values 66, 106, 110, 125, 176 utility 17, 23, 37, 46 vertical integration 66–69, 78, 79, 88, 89, 108, 109, 132, 135, 139, 141, 143, 154, 159, 166–169, 176, 180, 209 von Neumann, J. 124, 139, 215 wage rate of profit (WRP) curve vi, ix, x, xiii, xv, 1, 31, 34–36, 40, 45, 47–53, 56, 58, 59, 61–63, 67, 79, 84, 91, 93–105, 107–111, 113, 125–130, 132–135, 140, 141, 143, 147, 153, 159, 166, 171, 187–190, 195, 201–203, 206–210, 212 wage units 30 Walras, L. 4, 16, 17, 27, 28, 30, 33, 201, 204, 214, 215, 220 Wicksell, K. 4, 6, 17, 25, 28–30, 32, 33, 49, 69, 77, 78, 87, 88, 124, 201, 204, 217, 220 Wicksell effect 51, 68, 77, 87, 49, 124, 217 World Input-Output Database (WIOD) xv, 61, 63, 78, 79, 82, 91, 92, 101, 109, 118, 129, 135, 136, 155, 171, 173, 179 Yang, J. 130, 151, 220 Zambelli, S. 124, 125, 127, 131, 133, 206, 207, 220