Keynes, Sraffa and the Criticism of Neoclassical Theory: Essays in Honour of Heinz Kurz 1138231460, 9781138231467

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Keynes, Sraffa and the Criticism of Neoclassical Theory

Heinz Kurz is recognised internationally as a leading economic theorist and a foremost historian of economic thought. This book pays tribute to his outstanding contributions on the occasion of his 65th birthday by bringing together a unique collection of new essays by distinguished economists from around the world. Keynes, Sraffa and the Criticism of Neoclassical Theory comprises 24 essays, covering themes in Keynesian economic theory, in the development of the modern classical approach to economic theory, linear production models and the critique of neoclassical theory. The essays in this book will be an invaluable source of inspiration for economists interested in economic theory and in the evolution of economic thought. They will also be of interest to postgraduate and research students specialising in economic theory and in the history of economic thought. Neri Salvadori is Professor of Economics at the University of Pisa. Christian Gehrke is Professor of Economics at the University of Graz.

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Routledge studies in the history of economics

1. Economics as Literature Willie Henderson

9. The Economics of W. S. Jevons Sandra Peart

2. Socialism and Marginalism in Economics 1870–1930 Edited by Ian Steedman

10. Gandhi’s Economic Thought Ajit K. Dasgupta

3. Hayek’s Political Economy The socio-economics of order Steve Fleetwood 4. On the Origins of Classical Economics Distribution and value from William Petty to Adam Smith Tony Aspromourgos

11. Equilibrium and Economic Theory Edited by Giovanni Caravale 12. Austrian Economics in Debate Edited by Willem Keizer, Bert Tieben and Rudy van Zijp 13. Ancient Economic Thought Edited by B. B. Price

5. The Economics of Joan Robinson Edited by Maria Cristina Marcuzzo, Luigi Pasinetti and Alesandro Roncaglia

14. The Political Economy of Social Credit and Guild Socialism Frances Hutchinson and Brian Burkitt

6. The Evolutionist Economics of Léon Walras Albert Jolink

15. Economic Careers Economics and economists in Britain 1930–1970 Keith Tribe

7. Keynes and the ‘Classics’ A study in language, epistemology and mistaken identities Michel Verdon 8. The History of Game Theory, Vol 1 From the beginnings to 1945 Robert W. Dimand and Mary Ann Dimand

16. Understanding ‘Classical’ Economics Studies in long-period theory Heinz Kurz and Neri Salvadori 17. History of Environmental Economic Thought E. Kula

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18. Economic Thought in Communist and Post-Communist Europe Edited by Hans-Jürgen Wagener

28. The Canon in the History of Economics Critical essays Edited by Michalis Psalidopoulos

19. Studies in the History of French Political Economy From Bodin to Walras Edited by Gilbert Faccarello

29. Money and Growth Selected papers of Allyn Abbott Young Edited by Perry G. Mehrling and Roger J. Sandilands

20. The Economics of John Rae Edited by O. F. Hamouda, C. Lee and D. Mair 21. Keynes and the Neoclassical Synthesis Einsteinian versus Newtonian macroeconomics Teodoro Dario Togati 22. Historical Perspectives on Macroeconomics Sixty years after the ‘General Theory’ Edited by Philippe Fontaine and Albert Jolink 23. The Founding of Institutional Economics The leisure class and sovereignty Edited by Warren J. Samuels 24. Evolution of Austrian Economics From Menger to Lachmann Sandye Gloria 25. Marx’s Concept of Money: The God of Commodities Anitra Nelson 26. The Economics of James Steuart Edited by Ramón Tortajada 27. The Development of Economics in Europe since 1945 Edited by A. W. Bob Coats

30. The Social Economics of JeanBaptiste Say Markets and virtue Evelyn L. Forget 31. The Foundations of LaissezFaire The economics of Pierre de Boisguilbert Gilbert Faccarello 32. John Ruskin’s Political Economy Willie Henderson 33. Contributions to the History of Economic Thought Essays in honour of R. D. C. Black Edited by Antoin E. Murphy and Renee Prendergast 34. Towards an Unknown Marx A commentary on the manuscripts of 1861–63 Enrique Dussel 35. Economics and Interdisciplinary Exchange Edited by Guido Erreygers 36. Economics as the Art of Thought Essays in memory of G. L. S. Shackle Edited by Stephen F. Frowen and Peter Earl

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37. The Decline of Ricardian Economics Politics and economics in postRicardian theory Susan Pashkoff

46. Pareto, Economics and Society The mechanical analogy Michael McLure

38. Piero Sraffa His life, thought and cultural heritage Alessandro Roncaglia

47. The Cambridge Controversies in Capital Theory A study in the logic of theory development Jack Birner

39. Equilibrium and Disequilibrium in Economic Theory The Marshall–Walras divide Michel de Vroey

48. Economics Broadly Considered Essays in honour of Warren J. Samuels Edited by Steven G. Medema, Jeff Biddle and John B. Davis

40. The German Historical School The historical and ethical approach to economics Edited by Yuichi Shionoya

49. Physicians and Political Economy Six studies of the work of doctoreconomists Edited by Peter Groenewegen

41. Reflections on the Classical Canon in Economics Essays in honour of Samuel Hollander Edited by Sandra Peart and Evelyn Forget 42. Piero Sraffa’s Political Economy A centenary estimate Edited by Terenzio Cozzi and Roberto Marchionatti 43. The Contribution of Joseph Schumpeter to Economics Economic development and institutional change Richard Arena and Cecile Dangel

50. The Spread of Political Economy and the Professionalisation of Economists Economic societies in Europe, America and Japan in the nineteenth century Massimo Augello and Marco Guidi 51. Historians of Economics and Economic Thought The construction of disciplinary memory Steven G. Medema and Warren J. Samuels

44. On the Development of Longrun Neo-Classical Theory Tom Kompas

52. Competing Economic Theories Essays in memory of Giovanni Caravale Sergio Nisticò and Domenico Tosato

45. F. A. Hayek as a Political Economist Economic analysis and values Edited by Jack Birner, Pierre Garrouste and Thierry Aimar

53. Economic Thought and Policy in Less Developed Europe The nineteenth century Edited by Michalis Psalidopoulos and Maria-Eugenia Almedia Mata

54. Family Fictions and Family Facts Harriet Martineau, Adolphe Quetelet and the population question in England 1798–1859 Brian Cooper

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55. Eighteeth-Century Economics Peter Groenewegen 56. The Rise of Political Economy in the Scottish Enlightenment Edited by Tatsuya Sakamoto and Hideo Tanaka 57. Classics and Moderns in Economics, Volume I Essays on nineteenth and twentieth century economic thought Peter Groenewegen 58. Classics and Moderns in Economics, Volume II Essays on nineteenth and twentieth century economic thought Peter Groenewegen 59. Marshall’s Evolutionary Economics Tiziano Raffaelli 60. Money, Time and Rationality in Max Weber Austrian connections Stephen D. Parsons 61. Classical Macroeconomics Some modern variations and distortions James C. W. Ahiakpor 62. The Historical School of Economics in England and Japan Tamotsu Nishizawa

63. Classical Economics and Modern Theory Studies in long-period analysis Heinz D. Kurz and Neri Salvadori 64. A Bibliography of Female Economic Thought to 1940 Kirsten K. Madden, Janet A. Sietz and Michele Pujol 65. Economics, Economists and Expectations From microfoundations to macroeconomics Warren Young, Robert Leeson and William Darity Jnr. 66. The Political Economy of Public Finance in Britain, 1767–1873 Takuo Dome 67. Essays in the History of Economics Warren J. Samuels, Willie Henderson, Kirk D. Johnson and Marianne Johnson 68. History and Political Economy Essays in honour of P. D. Groenewegen Edited by Tony Aspromourgos and John Lodewijks 69. The Tradition of Free Trade Lars Magnusson 70. Evolution of the Market Process Austrian and Swedish economics Edited by Michel Bellet, Sandye Gloria-Palermo and Abdallah Zouache 71. Consumption as an Investment The fear of goods from Hesiod to Adam Smith Cosimo Perrotta

72. Jean-Baptiste Say and the Classical Canon in Economics The British connection in French classicism Samuel Hollander

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73. Knut Wicksell on Poverty No place is too exalted Knut Wicksell

81. Civil Happiness Economics and human flourishing in historical perspective Luigino Bruni 82. New Voices on Adam Smith Edited by Leonidas Montes and Eric Schliesser

74. Economists in Cambridge A study through their correspondence 1907–46 Edited by M. C. Marcuzzo and A. Rosselli

83. Making Chicago Price Theory Milton Friedman–George Stigler correspondence, 1945–57 Edited by J. Daniel Hammond and Claire H. Hammond

75. The Experiment in the History of Economics Edited by Philippe Fontaine and Robert Leonard

84. William Stanley Jevons and the Cutting Edge of Economics Bert Mosselmans

76. At the Origins of Mathematical Economics The Economics of A. N. Isnard (1748–1803) Richard van den Berg

85. A History of Econometrics in France From nature to models Philippe Le Gall

77. Money and Exchange Folktales and reality Sasan Fayazmanesh

86. Money and Markets A doctrinal approach Edited by Alberto Giacomin and Maria Cristina Marcuzzo

78. Economic Development and Social Change Historical roots and modern perspectives George Stathakis and Gianni Vaggi

87. Considerations on the Fundamental Principles of Pure Political Economy Vilfredo Pareto (Edited by Roberto Marchionatti and Fiorenzo Mornati)

79. Ethical Codes and Income Distribution A study of John Bates Clark and Thorstein Veblen Guglielmo Forges Davanzati

88. The Years of High Econometrics A short history of the generation that reinvented economics Francisco Louçã

80. Evaluating Adam Smith Creating the wealth of nations Willie Henderson

89. David Hume’s Political Economy Edited by Carl Wennerlind and Margaret Schabas

90. Interpreting Classical Economics Studies in long-period analysis Heinz D. Kurz and Neri Salvadori

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91. Keynes’s Vision Why the great depression did not return John Philip Jones 92. Monetary Theory in Retrospect The selected essays of Filippo Cesarano Filippo Cesarano 93. Keynes’s Theoretical Development From the tract to the general theory Toshiaki Hirai

100. Open Economics Economics in relation to other disciplines Edited by Richard Arena, Sheila Dow and Matthias Klaes 101. Rosa Luxemburg and the Critique of Political Economy Edited by Riccardo Bellofiore 102. Problems and Methods of Econometrics The Poincaré lectures of Ragnar Frisch 1933 Edited by Olav Bjerkholt and Ariane Dupont-Keiffer

94. Leading Contemporary Economists Economics at the cutting edge Edited by Steven Pressman

103. Criticisms of Classical Political Economy Menger, Austrian economics and the German historical school Gilles Campagnolo

95. The Science of Wealth Adam Smith and the framing of political economy Tony Aspromourgos

104. A History of Entrepreneurship Robert F. Hébert and Albert N. Link

96. Capital, Time and Transitional Dynamics Edited by Harald Hagemann and Roberto Scazzieri

105. Keynes on Monetary Policy, Finance and Uncertainty Liquidity preference theory and the global financial crisis Jorg Bibow

97. New Essays on Pareto’s Economic Theory Edited by Luigino Bruni and Aldo Montesano 98. Frank Knight and the Chicago School in American Economics Ross B. Emmett 99. A History of Economic Theory Essays in honour of Takashi Negishi Edited by Aiko Ikeo and Heinz D. Kurz

106. Kalecki’s Principle of Increasing Risk and Keynesian Economics Tracy Mott 107. Economic Theory and Economic Thought Essays in honour of Ian Steedman John Vint, J Stanley Metcalfe, Heinz D. Kurz, Neri Salvadori and Paul Samuelson

108. Political Economy, Public Policy and Monetary Economics Ludwig von Mises and the Austrian tradition Richard M. Ebeling

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109. Keynes and the British Humanist Tradition The moral purpose of the market David R. Andrews 110. Political Economy and Industrialism Banks in Saint-Simonian economic thought Gilles Jacoud 111. Studies in Social Economics Leon Walras (Translated by Jan van Daal and Donald Walker) 112. The Making of the Classical Theory of Economic Growth Anthony Brewer 113. The Origins of David Hume’s Economics Willie Henderson 114. Production, Distribution and Trade Edited by Adriano Birolo, Duncan Foley, Heinz D. Kurz, Bertram Schefold and Ian Steedman 115. The Essential Writings of Thorstein Veblen Edited by Charles Camic and Geoffrey Hodgson 116. Adam Smith and the Economy of the Passions Jan Horst Keppler

117. The Analysis of Linear Economic Systems Father Maurice Potron’s pioneering works Translated by Christian Bidard and Guido Erreygers 118. A Dynamic Approach to Economic Theory: Frisch Edited by Olav Bjerkholt and Duo Qin 119. Henry A. Abbati: Keynes’ Forgotten Precursor Serena Di Gaspare 120. Generations of Economists David Collard 121. Hayek, Mill and the Liberal Tradition Edited by Andrew Farrant 122. Marshall, Marshallians and Industrial Economics Edited by Tiziano Raffaelli 123. Austrian and German Economic Thought Kiichiro Yagi 124. The Evolution of Economic Theory Edited by Volker Caspari 125. Thomas Tooke and the Monetary Thought of Classical Economics Matthew Smith 126. Political Economy and Liberalism in France The contributions of Frédéric Bastiat Robert Leroux

127. Stalin’s Economist The economic contributions of Jenö Varga André Mommen

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128. E.E. Slutsky as Economist and Mathematician Crossing the limits of knowledge Vincent Barnett

129. Keynes, Sraffa and the Criticism of Neoclassical Theory Essays in honour of Heinz Kurz Edited by Neri Salvadori and Christian Gehrke

Keynes, Sraffa and the Criticism of Neoclassical Theory Downloaded by [Hacettepe University] at 12:42 20 April 2017

Essays in honour of Heinz Kurz Edited by Neri Salvadori and Christian Gehrke

First edition published 2011 by Routledge 2 Park Square, Milton Park, Abingdon, Oxon, OX14 4RN Simultaneously published in the USA and Canada by Routledge 711 Third Avenue, New York, NY 10017 Routledge is an imprint of the Taylor & Francis Group, an informa business

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© 2011 selection and editorial matter, Neri Salvadori and Christian Gehrke; individual chapters, the contributors The right of Neri Salvadori and Christian Gehrke to be identified as editors of this work has been asserted by them in accordance with the Copyright, Designs and Patent 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 Keynes, Sraffa and the criticism of neoclassical theory: essays in honour of Heinz Kurz / edited by Neri Salvadori and Christian Gehrke. p. cm. 1. Keynes, John Maynard, 1883-1946. 2. Sraffa, Piero. 3. Neoclassical school of economics. I. Salvadori, Neri. II. Gehrke, Christian. III. Kurz, Heinz-Dieter. HB99.7.K388155 2010 330.15ƍ7—dc22 2010045136 ISBN: 978-0-415-66450-9 (hbk) ISBN: 978-0-203-81772-8 (ebk) Typeset in Times New Roman by RefineCatch Limited, Bungay, Suffolk

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Contents

1

Contributors

xvi

Introduction

1

CHRISTIAN GEHRKE AND NERI SALVADORI

PART I

Keynes

5

2

7

Keynes’s Treatise on Money: the case for a rehabilitation JEAN CARTELIER

3

‘Animal Spirits’ in John Maynard Keynes’s General Theory of Employment, Interest, and Money: some short and sceptical remarks

21

INGO BARENS

4

Alternative routes of adjustment of saving to investment in the long period

33

MAN-SEOP PARK

PART II

Theory of growth 5

A simple model of path-dependent growth with some Keynesian features

57 59

AMIT BHADURI

6

Alternative models of growth and distribution with a simple formulation of endogenous technological change AMITAVA KRISHNA DUTT

67

xiv Contents

7 Labor heterogeneity, inequality and institutional change

84

PETER SKOTT

PART III

Sraffa 8 Sraffa and Trinity

99 101

JONATHAN SMITH

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9 Besicovitch, Sraffa and the existence of the Standard commodity

113

NERI SALVADORI

10 Sraffa and long-period general equilibrium

132

DUNCAN FOLEY

11 Surplus approach to value and distribution and structural economic dynamics: interpretation and uses of Sraffa’s analysis

146

ENRICO BELLINO AND ARIEL LUIS WIRKIERMAN

12 The production of customs by means of customs

170

ULRICH KRAUSE

13 A contribution to Weber’s theory of modern capitalism: amortization according to Sraffa as a rational substitution of missing markets

179

BERTRAM SCHEFOLD

PART IV

Linear models of production

199

14 Extensive rent and multiple equilibria

201

CHRISTIAN BIDARD

15 Why are there negative input–output coefficients? The ‘commodity technology assumption’ revisited

214

CHRISTIAN LAGER

16 Convergence of a short-run equilibrium to a long-run equilibrium in an emission permit scheme

223

EIJI B. HOSODA

17 Prices distorted from labour values TAKAO FUJIMOTO AND ALEJANDRO VALLE BAEZA

257

Contents xv PART V

Neoclassical theory and its critics

227

18 The stability of a competitive economy: a belated rejoinder

279

TAKASHI NEGISHI

19 A sonnet-divertimento for Heinz Kurz

286

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PAUL A. SAMUELSON AND ERKKO ETULA

20 Dominant firms, competition-deterring investment and antitrust policy

293

MIKA KATO AND WILLI SEMMLER

21 The market for saving in the theory of general intertemporal equilibrium

313

SERGIO PARRINELLO

22 Neoclassical theory: under-determined, over-determined or unable to move?

330

HARVEY GRAM

23 On/off inputs and their rentals

362

ARRIGO OPOCHER AND IAN STEEDMAN

24 Indivisible inputs and the probability of reswitching

373

IAN STEEDMAN

25 On the likelihood and relevance of reswitching and reverse capital deepening

380

FABIO PETRI

Heinz Kurz’s publications Index

419 435

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Contributors

Ingo Barens is Professor of Economics at the Technical University of Darmstadt, Germany. Enrico Bellino is Professor of Economics at the Catholic University of Milan, Italy. Amit Bhaduri is Professor of Economics at the University of Pavia, Italy, and Professor Emeritus at the Jawaharlal Nehru University, Delhi, India. Christian Bidard is Professor of Economics at the University of Paris OuestNanterre, France. Jean Cartelier is Professor of Economics at the University of Paris OuestNanterre, France. Amitava Krishna Dutt is Professor of Economics and Political Science at the University of Notre Dame, USA. Erkko Etula is an Economist in the Capital Markets Function of the Federal Reserve Bank of New York, USA. Duncan Foley is Professor of Economics at the New School University, New York, USA. Takao Fujimoto is Professor of Economics at Fukuoka University, Japan. Christian Gehrke is Professor of Economics at the University of Graz, Austria. Harvey Gram is Professor of Economics at Queens College, New York, and at the City University of New York, USA. Eiji B. Hosoda is Professor of Economics at Keio University, Japan. Mika Kato is Associate Professor of Economics at Howard University, USA. Ulrich Krause is Professor Emeritus of Mathematics at the University of Bremen, Germany. Christian Lager is Professor of Economics at the University of Graz, Austria.

Contributors xvii

Takashi Negishi is Professor Emeritus of Economics at the University of Tokyo, Japan. Arrigo Opocher is Professor of Economics at the University of Padua, Italy. Man-Seop Park is Professor of Economics at Korea University. Sergio Parrinello is Professor Emeritus of Economics at the University of Rome ‘La Sapienza’, Italy. Fabio Petri is Professor of Economics at the University of Siena, Italy.

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Neri Salvadori is Professor of Economics at the University of Pisa, Italy. Paul A. Samuelson was Institute Professor Emeritus at the Massachusetts Institute of Technology, Cambridge, USA. Bertram Schefold is Professor of Economics at the Johann-Wolfgang Goethe University, Frankfurt, Germany. Willi Semmler is Professor of Economics at the New School University, New York, USA. Peter Skott is Professor of Economics at the University of Massachusetts at Amherst, USA. Jonathan Smith is an Archivist in the Wren Library of Trinity College in Cambridge, UK. Ian Steedman is Professor Emeritus of Economics at the Manchester Metropolitan University in Manchester, UK. Alejandro Valle Baeza is Professor of Economics at the National Autonomous University of Mexico, Mexico. Ariel Luis Wirkierman is a PhD student at the Catholic University of Milan, Italy.

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Heinz D. Kurz

I 1PartIntroduction

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Christian Gehrke and Neri Salvadori Keynes

The essays collected in this volume honour the work of Heinz D. Kurz, who has contributed to the advancement of economic theory with scholarly dedication, good humour and untiring effort for over forty years. Early on in his career Kurz became deeply fascinated with Piero Sraffa’s work, which has provided a major source of inspiration for virtually all his research activities ever since. Kurz’s work focuses mainly on the historical reconstruction and modern re-formulation of the classical approach to economic theory, and his main research areas are the theory of production, value and distribution; the theory of capital; the theory of growth and technological change; the analysis of exhaustible and renewable resources; and, of course, the history of economic thought. Heinz D. Kurz was born on 29 March 1946 in Pfaffenhofen an der Ilm, a small Bavarian town just north of Munich. In 1948 his parents moved to Lohhof, a town even closer to Munich, where he later went to elementary school. While at boarding school, he spent a year at Eton College, UK, as an exchange pupil. He then studied for a Diploma in Economics at the University of Munich, from where he moved on to the University of Kiel in order to work on his doctoral dissertation, which he completed in 1975. The thesis, published in 1977 as Zur neoricardianischen Theorie des Allgemeinen Gleichgewichts der Produktion und Zirkulation (On the Neoricardian Theory of General Equilibrium in Production and Circulation) (Berlin, Duncker and Humblot), gained him recognition as a rigorous economic theorist and a foremost historian of economic thought. He became an Associate Professor of Economics at the University of Kiel from 1975 to 1979, and spent the academic year 1977–78 with a British Academy Visiting Scholarship at the University of Cambridge as a Visiting Fellow of Wolfson College. Shortly afterwards, in 1979, he was appointed as Professor of Economics at the University of Bremen on the strength of his scholarly publications, without having written a habilitation thesis (the then standard requirement for a professorship in Germany). In Bremen he collaborated closely with Harald Hagemann, his former colleague at the University of Kiel, and with Peter Kalmbach from the economics department, as well as with Ulrich Krause from the department of mathematics. For two almost consecutive periods, in 1981–82 and in 1983–85, he served as Dean of the Faculty LQDSDUWLFXODUO\GLI¿FXOWSHULRGZKLFKVDZWKHLQWURGXFWLRQRIDQHZFXUULFXOXP in economics and a major re-structuring of the faculty.

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2

Christian Gehrke and Neri Salvadori

In 1988, Heinz Kurz accepted a chair in the Department of Economics at the University of Graz, Austria, which he still holds. In the academic year 1990–91 he taught as the Theodor Heuss Professor at the Graduate Faculty of the New School for Social Research in New York, USA. In 1993, he initiated the founding of the Graz Schumpeter Society, which organizes and publishes the annual Graz 6FKXPSHWHU/HFWXUHVQRZLQWKHLU¿IWHHQWK\HDU6LQFHKHKDVDOVREHHQ Director of the Graz Schumpeter Centre, which organizes bi-annual summer schools as well as conferences and international research workshops. Besides his multifarious teaching and research activities in Graz, Kurz has held Visiting Professorships at over thirty universities from around the world. Moreover, he was one of the founders of the European Society for the History of Economic Thought and served as its Secretary General from 1999 to 2000 and as President from 2006 to 2008. He also chaired the ‘Committee for the history of economic thought’ in the association of economists from the German-speaking countries, the Verein für Socialpolitik, from 2006 to 2010. Kurz serves on the boards of several international journals and is a co-founder of The European Journal of the History of Economic Thought, of which he has been a managing editor since its foundation in 1993, and has also been a managing editor of Metroeconomica since 1998. He was appointed as the general editor of The Unpublished Writings of Piero Sraffa, which is to be published on behalf of Cambridge University Press and Trinity College, Cambridge. +HLQ].XU]LVQRWRQO\DYHU\SUROL¿FZULWHUEXWDOVRDPXFKVRXJKWDIWHU co-author and co-editor, who has written or edited some thirty books and published some 180 scholarly papers. A collection of his early essays was published as Capital, Distribution and Effective Demand (Cambridge, Polity Press, 1990). With his frequent co-author, Neri Salvadori, Kurz has written what he considers his most important work, Theory of Production: A Long-period Analysis (Cambridge, Cambridge University Press, 1995) and many joint papers; some of them were reprinted by Routledge in three edited collections of essays: Understanding ‘Classical’ Economics, Classical Economics and Modern Theory, and Interpreting Classical Economics (London, Routledge, 1998, 2003, and 2007). Kurz and Salvadori also co-edited the Elgar Companion to Classical Economics (Cheltenham, Edward Elgar, 1998, 2 vols) and The Legacy of Piero Sraffa (Cheltenham, Edward Elgar, 2003, 2 vols). Kurz also edited Critical Essays on Piero Sraffa’s Legacy in Economics (Cambridge, Cambridge University Press, 2000) and, jointly with Erik Dietzenbacher and Christian Lager, a collection of contributions to Input-output Analysis (Cheltenham, Edward Elgar, 3 vols, 1998). In German, a collection of Kurz’s essays was published as Ökonomisches Denken in klassischer Tradition (Marburg, Metropolis, 1998); there is also a small but well-selling book by him on Joseph A. Schumpeter. Ein Sozialökonom zwischen Marx und Walras (Marburg, Metropolis, 2004); and recently Kurz edited the collection Klassiker des ökonomischen Denkens (Munich, C. H. Beck, 2008 and 2009, 2 vols). For full references to the works mentioned, see the list of Heinz Kurz’s publications at the end of this volume.

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Introduction 3 The essays in this book develop three interconnected themes that run through PRGHUQWKHRU\DQG+HLQ].XU]¶VZRUN7KH¿UVWRIWKHVHWKHPHVFRQFHUQVWKH Keynesian idea that an economy, both in the short run and in the long run, is GHPDQGGULYHQ 7KH ¿UVW WKUHH HVVD\V H[SORUH WKLV WKHPH ZLWK DQ H[SOLFLW reference to Keynes’s contribution. Chapter 2, by Jean Cartelier, explores the transition within Keynes’s thought from the Treatise (disequilibrium dynamics) to the General Theory (static competitive equilibrium with involuntary unemployment). Chapter 3, by Ingo Barens, attempts to clarify the meaning, relevance and source of Keynes’s term ‘animal spirits’, also through an analysis of the structure of the General Theory, in order to understand the relevance of this notion for Keynes’s critique of orthodox economics. Chapter 4, by Man-Seop Park, analyses three alternative ways through which savings are generated to match investment in the long period: changes in income distribution, changes in the degree of capacity utilization, and changes in the size of productive capacity. Three further essays explore the same theme, but are more focused on the theory of growth. Chapter 5, by Amit Bhaduri, tries to capture some properties of path dependence through the evolving time structure of the capital stock in relation to variations in the state of aggregate demand and, in this way, to indicate how incorporating the Keynesian problem of effective demand affects the path dependence of the economy. Chapter 6, by Amitava Dutt, uses a simple framework within which alternative models (neoclassical new growth theory models, classical-Marxian and post-Keynesian models) of capital accumulation, distribution and endogenous technological change are represented and compared. Chapter 7, by Peter Skott, introduces labor heterogeneity, inequality and institutional change into the picture, but, differently from previous chapters, the aim is mainly applied. Another theme concerns Sraffa’s contribution to the understanding of price formation in the long period. Five essays explore this theme with an explicit reference to Sraffa’s work. Chapter 8, by Jonathan Smith, is a contribution to the biography of Sraffa and concerns his association with Trinity College over a period of forty-three years and the impact of such an association on Sraffa’s life and work. Chapter 9, by Neri Salvadori, explores the relationship between the proof of the existence of the Standard commodity contained in section 37 of Sraffa’s book and the proof supplied to Sraffa by Besicovitch on 21 September 1944, and postulates some reasons which led Sraffa to omit this proof in his book in favour of an incomplete argument. Chapter 10 is a general assessment of Sraffa’s 1960 book; its importance is also related to the author of the assessment, Duncan Foley. Chapter 11, by Enrico Bellino and Ariel Wirkierman, analyses and compares two different streams along which Sraffa’s analysis has been developed and extended. Chapter 12, by Ulrich Krause, moves from the circularity in the production structure of commodities of Production of Commodities by Means of Commodities to the circular structure of interacting individuals. Bertram Schefold, LQ&KDSWHUUHODWHV6UDIID¶VMRLQWSURGXFWLRQPHWKRGLQWKHWUHDWPHQWRI¿[HG capital to Max Weber’s theory of modern capitalism.

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4

Christian Gehrke and Neri Salvadori

Four further essays explore the same theme, but are more focused on linear models of production. Chapter 14, by Christian Bidard, is a contribution to extensive rent theory when more than one agricultural product exists. Chapter 15, by Christian Lager, helps to reduce the gap between applied input–output analysis DQG WKH 6UDI¿DQ HFRQRPLF OLWHUDWXUH E\ VXJJHVWLQJ ZD\V WR GHDO ZLWK MRLQW production in applied input–output models. Chapter 16, by Eiji Hosoda, uses a three-sectoral linear model to determine a formula for the supply of emission permits by which the government can meet the target amount of emission while keeping the economy in a steady-state equilibrium. Chapter 17, by Takao Fujimoto and Alejandro Valle Baeza, explores the relation between the labour value and the market price of a commodity and presents a new theory of the tendency of the rate RISUR¿WWRIDOOLQVSLWHRI2NLVKLR¶VWKHRUHP The third theme concerns Neoclassical theory and its criticism. Heinz Kurz has always been a critic of Neoclassical theory, but his rigour and competence attracted the interest of several eminent Neoclassical theorists like Paul A. Samuelson and Takashi Negishi, who on many occasions showed their friendship with him. In Chapter 18, Takashi Negishi provides a restatement of some of his results on the stability of a competitive economy. Chapter 19, by the late Paul Anthony Samuelson and Erkko Etula, introduces a model in which all margins are ‘spurious’ to help the understanding of the last half-century of ‘miracle’ developments outside of the United States. Chapter 20, by Mika Kato and Willi Semmler, provides a theoretical framework of the dynamics of competition between GRPLQDQW¿UPVDQGIULQJH¿UPVZKHUHGRPLQDQW¿UPVFDQEXLOGXSFRPSHWLWLRQ restricting capital. Chapters 21 and 22, by Sergio Parrinello and Harvey Gram, respectively, contribute to the recent debate on the consistency of the theory of general intertemporal equilibrium. Chapter 23, by Arrigo Opocher and Ian Steedman, shows by means of numerical examples that familiar results concerning marginal variations of inputs do not extend to the cases in which variations are not marginal, i.e. cases in which the quantity of an input may be either D or zero, where D is a positive number. Chapters 24 and 25, by Ian Steedman and Fabio Petri, respectively, contribute to the recent debate on the probability of reswitching.

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Part I

Keynes

2

Keynes’s Treatise on Money The case for a rehabilitation

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Jean Cartelier

The so-called ‘fundamental equations’ (in fact identities) are the core of Keynes’s Treatise on Money. 3ULFHV DUH GH¿QHG DQG GHWHUPLQHG  WKHUH DV WKH UDWLRV RI money expenditures to quantities of commodities brought to market. From these identities Keynes elicits three important propositions: 1 The price of consumption goods is independent from that of investment goods: ‘a fall in the price of consumption goods due to an excess of saving over investment does not in itself – if non accompanied by any change in the bearishness or bullishness of the public or in the volume of saving deposits . . . – require any opposite change in the price of new investment goods’ (Keynes (1930), p. 130).  3UR¿WV VHHP WR EH DQ LQH[KDXVWLEOH VRXUFH IRU HQWUHSUHQHXUV FRQVXPSWLRQ µSUR¿WVDVDVRXUFHRIFDSLWDOLQFUHPHQWIRUHQWUHSUHQHXUVDUHDZLGRZ¶VFUXVH which remains undepleted however much of them may be devoted to riotous living’ (Keynes (1930), p. 125).  7KLV µZLGRZ¶V FUXVH¶ DUJXPHQW PD\ EH H[WHQGHG ZLWKRXW GLI¿FXOW\ WR investment. Whatever the amount of saving, any increase of investment H[SHQVHV WULJJHUV DQ LGHQWLFDO LQFUHDVH LQ ZLQGIDOO SUR¿WV GH¿QHG DV WKH difference between investment and saving). The fact that these conclusions were rejected by Keynes’s followers and close IULHQGVUDLVHVDWOHDVWWZRLQWHUHVWLQJTXHVWLRQV7KH¿UVWRQHLVWRZKDWH[WHQW have discussions among the CircusLQÀXHQFHGWKHWUDQVLWLRQIURPWKHTreatise to General Theory? On that point we may follow Cristina Marcuzzo (2002). She VKRZVKRZ.DKQLQÀXHQFHG.H\QHVDQGOHGKLPWRLQWHJUDWH0DUVKDOOLDQHOHPHQWV (a short-term supply curve, for instance) into his own theory, which previously was almost free of them. Thanks to an analysis of their correspondence (partly unpublished) she evokes with great precision the different stages on the way to the effective demand theory. This point is not discussed here. The second question is: did Kahn’s critiques (and Keynes’s acceptance of them) lead to the abandonment of the monetary approach initiated by Wicksell? In this paper we will try to deal with the latter question.

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8 Jean Cartelier

We shall restrict our ambition to assessing the analytical relevance of Kahn’s arguments. The members of the CircusVHHPWRVKDUHWZRLQWXLWLRQV7KH¿UVWLV that prices and quantities should be distinguished. A (Marshallian) supply curve LVWKHµQDWXUDO¶GHYLFHWRRYHUFRPHWKHDSSDUHQW¿[LW\RITXDQWLWLHVLQ.H\QHV ‘fundamental equations’. The second is that taking into account the interdependence due to technique between the two sectors would invalidate the alleged independence of prices and the widow’s cruse paradox. Hicks (1967) will endorse both arguments 35 years later. Are these intuitions valid? Do they really lead to abandoning Keynes’s propositions above? Before answering these questions (in the negative), a brief reminder of the logic inherent in the TreatiseLVLQRUGHU¿UVWVHFWLRQ  It will provide the foundations for the two-sector model of the second section.

Fundamental equations’ logic: a recap Description of the economy The two sectors of the Treatise do not differ in the technique but in the destination of production. Commodities produced are homogeneous, since they are proportions of a unique global cost which is also the social income, denoted by E, which is the sum of all factors’ incomes (wages included) and of entrepreneurs’ QRUPDOSUR¿W1RWHWKDWZLQGIDOOSUR¿WVRUXQH[SHFWHGORVVHVDUHQRWSDUWRIE. If I' is the monetary cost of production of investment goods, E – I' is that of consumption goods. Besides expenditures forming the income E,1 entrepreneurs spend I on new investment goods. Income earners (only wage-earners are considered here for the sake of simplicity) spend E – S on consumption goods, saving, denoted by S, being taken as given. $QLPPHGLDWHFRQVHTXHQFHRIWKHVHGH¿QLWLRQVLVWKDWWKHVXPRIH[SHQGLWXUHV for the acquisition of goods, E – S + I, is not necessarily equal to the income or cost of production E. It is precisely the difference between these two monetary quantities (I – S) that Keynes calls ZLQGIDOOSUR¿WVRUORVVHV  respectively Qi and QIRUVHFWRULDODQGJOREDOZLQGIDOOSUR¿WV7KH\DUHQRWLQFOXGHGLQWKHVRFLDO income ELQFRQWUDVWZLWKQRUPDOSUR¿WV Keynes’s colleagues and friends apparently did not understand this approach. .H\QHVZDVOHGWRDEDQGRQLWODWHUZKLFKLVDSLW\6WUDQJHDVLWPD\DSSHDUDW¿UVW VLJKWWKHDERYHGHVFULSWLRQLVPHDQLQJIXO.H\QHVJLYHVDFOHDUMXVWL¿FDWLRQRILW LQWKHIROORZLQJVHQWHQFHDERXWZLQGIDOOSUR¿WV SUR¿WVRUORVVHV DUHDQHIIHFWRIWKHUHVWRIWKHVLWXDWLRQUDWKHUWKDQD FDXVH RI LW    SUR¿WV RU ORVVHV  KDYLQJ RQFH FRPH LQWR H[LVWHQFH become . . . a cause of what subsequently ensues; indeed, the mainspring of change in the existing economic system. This is the essential reason why it is useful to segregate them in our fundamental equation. (Keynes (1930), p. 126)

Keynes’s Treatise on Money 9 1RUPDOSUR¿WLVZLWKRXWDQ\DPELJXLW\HTXLOLEULXPSUR¿WWKDWLVSUR¿WVXFKWKDW H[SHFWHGDQGUHDOL]HGSUR¿WDUHHTXDO

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. . . that rate of remuneration which, if they [the entrepreneurs] were open to make new bargains with all the factors of production at the currently prevailing rates of earnings, would leave them under no motive either to increase or to decrease their scale of operations. (Keynes (1930), p. 112) It is of the utmost importance to note that a normal level of income, according to .H\QHVPDNHVVHQVHRQO\IRUWKHHQWUHSUHQHXUVSUR¿W DQGQRWIRUWKHIDFWRUVRI production (wages). Only entrepreneurs are supposed to react to a difference between expected and realized income. Assuming that ZLQGIDOO SUR¿WV are the mainspring of change amounts to saying that entrepreneurs alone are responsible for dynamic adjustments. There is nothing like a normal wage. Ascribing to wageearners any expectation about what their wage should be does not make sense in Keynes’s economics.2 E is not merely the social cost or income, it is also an equilibrium income. Keynes’s emphasis on situations of disequilibrium where Q z 0 is a distinctive feature of his Treatise. How is it possible that total expenditures on goods, E – S + I, could differ from income, E? This is equivalent to admitting that investment may not be limited by saving. Knut Wicksell had already answered this question many years before the Treatise. Wicksell opposed an economy in which investors and savers meet in loanable funds markets to another in which banks play a major role. In the former, WKDQNVWRWKHµODZRIVXSSO\DQGGHPDQG¶DJHQWV¿QGWKHLUZD\WRZDUGHTXLOLEULXP (‘a ball at the bottom of a bowl’ according to Wicksell). The market for loanable IXQGVHQVXUHVWKHHTXDOLW\EHWZHHQWKHUDWHRILQWHUHVWDQGWKHUDWHRISUR¿W,QWKH latter economy, the story is different: If to this [market for loanable funds] we add organized credit, and especially the activity of the banks, the connection between loan interest and interest on capital will become much less simple; indeed, it will then only exist at all by virtue of the connecting link of price movements. (Wicksell (1935), p. 194) Here, equilibrium is not stable, it is indifferent (‘a ball or cylinder on a plane’, ibid.). The reason for this change in the basic dynamic properties of the economy is that banks, according to Wicksell, do not face a capital constraint. Elasticity of WKHLU VXSSO\ RI PHDQV RI SD\PHQW LV YHU\ KLJK DQG LQ VRPH FDVHV LQ¿QLWH Expenditure on investment goods (and on factors of production) is made by entrepreneurs with the help of banks. They are not dependent on any condition of equilibrium. They are limited by entrepreneurs’ and/or banks’ willingness alone. Budgetary constraints are not the same in a monetary economy with banks as in one without them. They differ also from the constraints in usual models where

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10 Jean Cartelier

price determination is separate from payments and transactions. We must keep this in mind in order to understand why Kahn and Hicks rejected Keynes’s propositions mentioned above using the same misleading argumentation. In the economy of the Treatise, entrepreneurs, and only they, are capable of making WKH¿UVWPRYHFRQGLWLRQDOWRWKHEDQN¶VDJUHHPHQW 7KH\GHFLGHWKHOHYHORILQFRPH of the factors of production (here wage-earners). They decide also, along with banks, on the amount of investment independently of savers’ intentions and/or decisions. But these expenditures on investment goods are also receipts for entrepreneurs as a whole. Entrepreneurs’ total receipts are equal to the sum of wage-earners’ consumption expenditures plus investment (expenditures on themselves). It is quite possible for entrepreneurs to receive in return either more or less than their expenses. However, quite obviously, no violation of budgetary constraints is acceptable. $JHQWVZLWKPRQHWDU\VXUSOXVPXVW¿QDQFHLQVRPHZD\DJHQWVZLWKPRQHWDU\ GH¿FLWV$SRVVLELOLW\LVWKDWWKHIRUPHUDFTXLUHSHUSHWXDOERQGVVROGE\GH¿FLW entrepreneurs.3 If perpetual bonds were the only possibility, the market for bonds ZRXOG H[DFWO\ UHÀHFW WKH VLWXDWLRQ LQ WKH WZR PDUNHWV IRU JRRGV $ PRUH sophisticated description would take into consideration the possibility of hoarding. In that case banks could restore budgetary constraints by lending the deposits of H[FHVV DJHQWV WR GH¿FLW DJHQWV 7KH PDUNHW IRU ORDQV ZRXOG WKHQ UHÀHFW WKH situation in the other three markets. An even more complicated case would occur if banks were reluctant to lend all their deposits. A lender of last resort would enter the scene and decide to lend what is required in order to avoid generalized bankruptcy. In short, the economy is viable only if budgetary (here monetary) constraints are respected. In such an economy the global monetary constraint plays the role played by the Walras Law in standard models. In other words, the number of degrees of freedom is equal to the number of markets for goods plusWKHQXPEHURI¿QDQFLDOPDUNHWVminus one. In the simplest case, with only a market for perpetual bonds, this number is two. This simple fact is essential to understand one of the most controversial theses of the Treatise, the independence of the price of consumption goods, P, from that of investment goods, P'. Elements for a discussion Let us set down the matrix of payments associate to the Treatise’s description of the economy:4 Table 2.1 Consump. goods Consump. goods Invest. goods

Invest. goods

Factors

3UR¿W

Balances

0

I1

E – I'

I' – S

I' – S – I1

0

I2

I'

I – I'

I1 – I'

Factors

E–S

0

0

0

S

Total

E–S

I

E

I–S

0

Keynes’s Treatise on Money 11

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The Treatise’s ‘fundamental equations’ express the price of goods as the sum RIWZRWHUPVWKH¿UVWRQHLVWKHXQLWFRVWLQFOXGLQJQRUPDOSUR¿WE/O); the second RQHLVWKHZLQGIDOOSUR¿WRUORVV SHUXQLW ‘Fundamental equations’ are:5 P = E + I′ − S O R

(2.1)

P′ = E + I − I ′ O C

(2.2)

with R + C = O, R and C being respectively the ‘quantity’ of consumption and investment goods. Conditions of production and quantities R and C being given, P and P' differ only in that saving SLQÀXHQFHVWKHIRUPHUDQGQRWWKHODWWHUZKLOHLQYHVWPHQWI LQÀXHQFHVWKHODWWHUDQGQRWWKHIRUPHU7KHLQGHSHQGHQFHEHWZHHQERWKSULFHVLV another expression for the independence between investment and saving or for the SRVVLEOHH[LVWHQFHRIZLQGIDOOSUR¿WVRUORVVHV :LFNVHOO¶VFXPXODWLYHSURFHVVLV what ‘fundamental equations’ are about. The mere existence of a banking system PRGL¿HV WKH UHODWLRQ EHWZHHQ LQYHVWPHQW DQG VDYLQJ ,Q D VWDQGDUG PRGHO D difference between investment and saving is only virtual since I = S determines both the equilibrium rate of interest and the level of investment and saving. Disequilibrium is not real but only virtual as is the process of adjustment (tâtonnement). With banks, situations of disequilibrium actually exist. Starting from a position of equilibrium, ( I S Ÿ P P c ), an increase in investment expenditures (made real thanks to the banks) triggers an increase in investment goods price and leaves consumer goods prices unaffected. Kahn objects that Keynes’s conclusion holds only by virtue of an implicit DVVXPSWLRQWKDWµSUR¿WVDUHH[FOXVLYHO\GHYRWHGWREX\LQJLQYHVWPHQWJRRGVDQG losses exclusively made up by selling, or restricting the purchase of investment goods)’ (R. F. Kahn, letter to Keynes 5 April 1931, [JMK CW, p. 203]). In order to prove his assertion, Kahn assumes that a fraction kRIZLQGIDOOSUR¿WVQ1 + Q2 = Q, is spent on investment goods (1 – k being spent in consumption goods). Hicks (1967, pp. 196–197) will also make the same point. From the equation: Q1

k (Q1  Q2 )  I c  S

(2.3)

Kahn deduces that ‘if k = 0 . . ., Q1 is necessarily equal to [I' – S] and Q2 can have any value – in conformity to the exposition of the Treatise. . . . If k = 1 . . . Q2 is necessarily equal to [S – I'] and Q1 can have any value’ [JMK CW, p. 204]. Kahn’s argument is incorrect. Writing the equation above implies that we are at HTXLOLEULXP &RQVLGHU WKH PLVVLQJ HTXDWLRQ UHODWLYH WR ZLQGIDOO SUR¿WV LQ investment goods sector: Q2

(1  k )(Q1  Q2 )  I  I c

(2.4)

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12 Jean Cartelier

The sum of the two equations gives I { S, which is precisely what Keynes totally denies. For him, I = S D FRQGLWLRQ RI HTXLOLEULXP LV QRW VDWLV¿HG LQ general. $WWKHURRWRI.DKQ¶VLQWHUSUHWDWLRQLVWKHIDOVHYLHZWKDWZLQGIDOOSUR¿WVPD\ be spent on any type of goods, as is the case for income. The Treatise’s logic is RSSRVHGWRWKDWYLHZZLQGIDOOSUR¿WVRUORVVHV DUHWKHconsequence of effective entrepreneurs’ and wage-earners’ expenditures, not their cause. It is true that at HTXLOLEULXPFDXVDOLW\DQGZLQGIDOOSUR¿WVGLVDSSHDUDQGLQWHUGHSHQGHQFHVHHPV to prevail. But, in equilibrium as in disequilibrium, expenditures E and I are the primum movensRIWKHHFRQRP\$V.DOHFNLDI¿UPHGRQFHHQWUHSUHQHXUVFDQ decide to spend more but not to earn more. These expenditures are decided in view RIDQRUPDOSUR¿W(QWUHSUHQHXUVGRQRWFRQWURO S. Actual outcome, not under entrepreneurs’ control in a decentralized economy, is: consumption

investissement

income

profits

ES I  E Ÿ QG JJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJ

(2.5)

:LQGIDOOSUR¿WVDUHEURXJKWDERXWE\HQWUHSUHQHXUV¶DXWRQRPRXVH[SHQGLWXUHV 7KHUH LV QR VHQVH LQ WKLQNLQJ WKDW ZLQGIDOO SUR¿WV DUH ¿UVW REVHUYHG DQG then spent to a greater or lesser degree in a particular sector. Such a view is a total misinterpretation of Keynes. Kahn did not realize that, in Kalecki’s terms, entrepreneurs earn what they spend and wage-earners spend what they earn. However, it is worth following Kahn by introducing autonomous consumption expenditures, denoted by A, alongside investment ones. ‘Fundamental equations’ become: A  Ic  S (2.6) P E  R O Pc

E  I  Ic O C

(2.7)

ZKHUHLWLVFOHDUWKDWWRWDOZLQGIDOOSUR¿WVDUHA + I – S and that, S being given, prices P and P' depend on autonomous expenditures. But Keynes’s thesis that prices are independent of each other is not affected by Kahn’s suggestion. The widow’s cruse principle is a direct consequence of what has just been stated. Any increase in A (respectively in I ), ceteris paribus, generates an identical LQFUHDVHLQZLQGIDOOSUR¿WLQWKHFRQVXPHUJRRGVUHVSHFWLYHO\LQYHVWPHQWJRRGV  sector, with the other sector being unaffected. Nonetheless, other objections to Keynes have to be examined, particularly WKRVHUHODWLYHWR¿[HGTXDQWLWLHVDQGWRWKHLQWHUGHSHQGHQFHEHWZHHQVHFWRUV7KH ¿UVWUHTXLUHVWKHLQWURGXFWLRQRIDVXSSO\FXUYHLQRUGHUWRGLVWLQJXLVKTXDQWLW\DQG price effects. The second has at least two aspects: technical interdependence (input–output analysis) and relation between income and saving. A more explicit two-sector model than that of the Treatise will prove convenient to discuss these issues.

Keynes’s Treatise on Money 13

A simple two-sector model of a monetary economy The model

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Let us imagine a two-sector economy (two representative entrepreneurs) where LQSXW FRHI¿FLHQWV DUH FRQWLQXRXV DQG TXDVLFRQYH[ IXQFWLRQV RI TXDQWLWLHV RI production: [a11 (q1 ) … a12 (q1 ) … l1 (q1 )] Ÿ 1 unit of 1

(2.8)

[a21 (q2 ) … a22 (q2 ) … l2 (q2 )] Ÿ 1 unit of 2

(2.9)

where aij (qi ) and li (qi ) are respectively the quantities of commodity j and of labour used to produce one unit of commodity i when total production is qi. Following their basic instinct (or long-term expectations), entrepreneurs determine certain amounts of autonomous expenditures: let Dij be the autonomous expenditure of entrepreneur i to entrepreneur j. In addition, entrepreneurs have shortWHUPH[SHFWDWLRQVDERXWSULFHVDQGWKXVDERXWWKHLUQRUPDO SUR¿WV,I pie is the expected price of commodity i (for the sake of simplicity it is assumed that entrepreQHXUVKDYHLGHQWLFDOH[SHFWDWLRQV H[SHFWHGSUR¿WRIVHFWRUi depends on pie: Sie

qi ( pie (1  aii (qi ))  p ej aij (qi )  wli (qi ))

(2.10)

)RUDJLYHQH[SHFWHGSUR¿WWKHHTXDWLRQDERYHPD\EHLQWHUSUHWHGDVDVKRUWWHUP supply curve linking pie and qi , p ej and the technique being given.6 Quantities would be determined through the following system as functions of expected prices DQGQRUPDOSUR¿W ( p1e (1  a11 (q1 ))  p2e a12 (q1 )  wl1 (q1 ))q1 ( p2e (1  a22 (q2 ))  p1e a21 (q2 )  wl2 (q2 ))q2

S1e Se2

(2.11) (2.12)

That system along with autonomous expenditures determines entrepreneurs’ total expenditures. Thanks to banks, these expenditures are made real and irrevocable. Let us assume now that wage-earners spend a fraction Į of their income in sector 1 and a fraction ȕLQVHFWRU1RZDOOWKHH[SHQGLWXUHÀRZVRIWKHHFRQRP\DUH determined, as shown in Table 2.2. Entrepreneurs’ total balances are necessarily equal to those of wage-earners (with an opposite sign), whether the economy is in equilibrium or not. The sectors’ total receipts are given by: R1

Ford effect Interdependence Autonomous

       (2.13) e e q1 ( p1 a11 (q1 )  Dwl1 (q1 ))  q2 ( p1 a21 (q2 )  Dwl2 (q2 ))  D11  D21

R2

Ford effect Interdependence

     Autonomous   (2.14) e e q2 ( p2 a22 (q2 )  Ewl2 (q2 ))  q1 ( p2 a12 (q1 )  Ewl1 (q1 ))  D12  D22

14 Jean Cartelier Table 2.2

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1

2

Wageearners

Balance

1

P1e q2 a11 (q1) + D11

P2e q1 a11 (q1) + D12 wq1l1(q1)

P1e q2 a21 (q2) + Dw(q1l1(q1) + q2l2(q2)) – wq1l1(q1) + D21 – D12

2

P1e q2 a21 (q2) + D21 P2e q2 a22 (q2) + D22 wq2l2(q2)

P2e q1 a12 (q1) – P1e q2 a21 (q2) + Ew(q1l1(q1) + q2l2(q2)) – wq2l2(q2) + D12 – D21v

Wage- Dw(q1l1(q1) + earners q2l2(q2))

Ew(q1l1(q1) + q2l2(q2))

0

(1 – D – E)w (q1l1(q1) + q2l2(q2))

The Ford effect measures that fraction of receipts of entrepreneur i resulting directly or indirectly (through wage-earners) from his own expenditures. The LQWHUGHSHQGHQFHHIIHFWPHDVXUHVWKHLQÀXHQFHRIRQHVHFWRU¶VH[SHQGLWXUHVRQWKH receipts of another. From the total receipts of each sector and the quantities determined by supply curves it is easy to deduce market prices. It is exactly what Keynes’s ‘fundamental equations’ are designed for. Market prices are determined by the ratio of total receipts to the quantity supplied. By adopting such a rule, Keynes joined a long and venerable tradition, that of a monetary approach to a market economy (illustrated by some ‘mercantilists’: Cantillon, Steuart, Schumpeter, etc.).7 This tradition is still present today with strategic market games and the rule of ShapleyShubik which is just a direct application of Keynes’s ‘fundamental equations’ to a general framework of equilibrium. Two important points make that tradition interesting.  7KH¿UVWLVWKDWGLVHTXLOLEULXPLVHIIHFWLYHLQVWDQGDUGPRGHOVRQO\VLWXDWLRQV of equilibrium can be described since the conditions of equilibrium are part of the system of equations; Shapley-Shubik’s rule is not a condition of equilibrium; situations both of equilibrium and non-equilibrium are possible. 2 The second is that equilibrium or disequilibrium characterizes individuals (entrepreneurs) instead of markets, as is the case in standard models (where nobody but the auctioneer can observe markets). Dynamics spring directly from individuals (entrepreneurs) reacting to perceived disequilibria. Kahn did not realize how original and powerful Keynes’s ‘fundamental equations’ are, nor did Hicks 35 years later. Hicks interpreted the Treatise¶VDQDO\VLVDVD¿UVW VWHSLQD0DUVKDOOLDQWKHRU\ÀH[LEOHSULFHV¿[HGTXDQWLWLHV ZLWKWKHVHFRQG ÀH[LEOHTXDQWLWLHV¿[HGSULFHV FRPLQJODWHULQWKHGeneral Theory before a third VWHSÀH[LEOHZDJH VHH+LFNV SSII 7KHLQJHQXLW\DQGVKUHZGQHVV of Hicks’s overview of Keynes’s economics cannot be disputed. Nevertheless, one

Keynes’s Treatise on Money 15 of its aspects is to preserve theoreticians from an interrogation about standard approach foundations. Quantities are determined by supply curves and market prices by ShapleyShubik’s rule: p1

R1 q1

p1e a11 (q1 )  D wl1 (q1 ) 

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 p2

D11  D21 q1

p2e a22 (q2 )  E wl2 (q2 )  

D12  D22 q2

q2 e ( p1 a21 (q2 )  D wl2 (q2 )) q1

q1 e ( p2 a12 (q1 )  E wl1 (q1 )) q1

(2.15)

Due to interdependence between the two sectors, the market price of any given sector depends on what happens in the other. But this observation does not mean that there is only one degree of freedom in the system. Nothing prevents both money prices from moving in the same direction (which would be impossible in an economy with only one relative price). In a monetary economy, where an DFFRPPRGDWLQJEDQNLQJV\VWHPDOORZVHQWUHSUHQHXUVWR¿QDQFHWKHLUH[SHQGLWXUHV FRQVXPSWLRQLQYHVWPHQWSD\PHQWRIZDJHVDQGLQSXWV WKHÀRZRISD\PHQWVLV not necessarily paid out of income. It is in fact quite the contrary. Entrepreneurs’ expenditures are their source of income and not the other way round. As a consequence, prices, which are determined as ratios of money expenditures to quantities produced and brought to market, are independent from each other in the VHQVHSUHFLVHO\GH¿QHGE\.H\QHVLQWKHTreatise on Money: The reader will observe that the price level of consumption goods [here sector 1] is entirely independent of the price level of investment goods >KHUH VHFWRU @ *LYHQ WKH HI¿FLHQF\ ZDJHV DQG WKH GLIIHUHQFH EHWZHHQ the cost of new investment goods (as distinguished from their selling price) and the volume of saving, the price level of consumption goods is unequivocally determined, quite irrespective of the price level of investment goods. (Keynes (1930), p. 123) ,I µJLYHQ WKH HI¿FLHQF\ ZDJHV¶ LV LQWHUSUHWHG DV HTXLYDOHQW WR µJLYHQ WKH technique’, this quotation from Keynes perfectly corresponds to our model. The assertion that technical interdependence between sectors would ruin Keynes’s proposition about the independence of P and P' is incorrect. The fact that the distribution of autonomous expenditures between the sectors matters does not invalidate Keynes’s view. The model exhibits also the ‘widow’s cruse’ principle. Costs are determined by technique and expected prices, whereas receipts depend not only on the costs

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16 Jean Cartelier

but also on autonomous expenditures. Consequently, in spite of sectoral interdependence and separation between quantity and price effects, an increase in autonomous consumption (respectively investment) expenditures generates DQ HTXLYDOHQW LQFUHDVH LQ ZLQGIDOO SUR¿WV LQ WKH FRQVXPSWLRQ UHVSHFWLYHO\ investment) sector. What consideration of interdependence does, and what the TreatisePRGHOGRHVQRWDOORZLVWRUHYHDOWKHFRPSOH[LW\RIWKHLQÀXHQFHRI autonomous expenditures. But it is in no way an invalidation of Keynes’s basic propositions. In other words, if p1e and p2e both enter into equations p1 and p2 UHÀHFWLQJWKH technological interdependence between sectors, p1 does not enter into the determination of p2 (and vice-versa) as the equations above make clear. Market SULFHVDUHREYLRXVO\LQÀXHQFHGE\FRPPRQIDFWRUVQDPHO\ q1 and q2 but they are ultimately determined by different variables, D11  D12 on the one hand, and D21  D22 on the other. The argument may be recast in terms of Walras’s Law. Considering Keynes’s model of the Treatise exclusively at equilibrium means forgetting that a third market exists where positive balances (saving or supply of funds) face negative balances (excess of entrepreneurs’ expenditures over receipts). The fact that the global amount of both are identical (see Table 2.2) is due to Walras’s Law which holds in equilibrium and in disequilibrium. There are three markets and two degrees of freedom. Independence of investment goods price from consumer goods price follows. Does this proposition hold also in equilibrium (comparative statics)? No answer is to be found in the Treatise since the equilibrium level of E is not determined. Our model, more general, may help a little. A reformulation of the Treatise on Money In order to compare the model above with the one proposed by Keynes in the TreatiseWZRSRLQWVKDYHWREHFODUL¿HG 7KH¿UVWRQHFRQFHUQVWKHUHODWLRQVKLSEHWZHHQWKHQRUPDOSUR¿WLQWKHTreatise DQGWKHH[SHFWHGSUR¿WLQRXUPRGHO.H\QHVDGRSWVHTXDOLW\EHWZHHQVDYLQJ and investment as the condition of equilibrium. Investment D21 + D22 is a component of autonomous expenditures in our model, the other being autonomous consumption D12 + D12. Extending Keynes’s condition of equilibrium to all entrepreneurs’ autonomous expenditures amounts to restricting saving to wageHDUQHUVDQGWRDVVXPLQJ]HURQRUPDOSUR¿WZLWKRXWORVVRIJHQHUDOLW\OHWXV recall that there is no saving function in the Treatise). Consequently, the system becomes: ( p1e (1  a11 (q1 ))  p2e a12 (q1 )  wl1 (q1 ))q1 0 ( p2e (1  a22 (q2 ))  p1e a21 (q2 )  wl2 (q2 ))q2 0

(2.16)

The second point is relative to quantities. Keynes measures the production relatively to a given basis, as is done in social accounting. Taking the current period as a

Keynes’s Treatise on Money 17 reference, income E (according to the Treatise) is equal to w[q1l1(q1) + q2l2(q2)]. q1l1 (q1 ) The quantity of consumer goods R is thus R ; the wage w q1l1 (q1 )  q2l2 (q2 ) disappears in that expression. In the same way, the quantity of investment goods is:

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C

q2l2 (q2 ) . As in the Treatise: C  R q1l1 (q1 )  q2l2 (q2 )

O and

E O

W1 = w.

Now the model of the Treatise brings out our model above as a special case by adding the following assumptions: a11 (q1 ) a12 (q1 ) a21 (q2 ) a22 (q2 ) 0, E = 0, D11  D12 0. The model’s sector 1 produces only consumer goods while VHFWRUPD\EHLGHQWL¿HGDVSURGXFLQJRQO\LQYHVWPHQWJRRGV)LQDOO\GXHWRWKH absence in the Treatise of any assumption about the relation between income and saving, Dw(l1 (q1 )  l2 (q2 )) has to be replaced by w(l1 (q1 )  l2 (q2 ))  S with S exogenously given. Following the Treatise’s notations, E w(q1l1 (q1 )  q2l2 (q2 )), I c wq2l2 (q2 ) , PR w(q1l1 (q1 )  q2l2 (q2 ))  S , S { E  PR and I D21  D22 . It is easy to check that: q2 S wl2 (q2 )  q1 q1

p1

wl1 (q1 ) 

p2

D21  D22 q2

I C

(2.17)

Independence between D21  D22 , under the control of entrepreneurs, and S, under the control of wage-earners, ensures independence between p1 and p2, in the same manner that, in the 7UHDWLVHbetween P E  I c  S and P c E  I  I c O R O R S EHLQJ SUHVHQW RQO\ LQ WKH ¿UVW DQG I in the second. This degree of freedom disappears in equilibrium since equilibrium requires I = S. The Treatise’s comparative statics Due to the absence of a functional relation between income E and saving S in the Treatise, the level of equilibrium of E (at which I = S) is indeterminate. Consequently, no property of comparative statics can be shown within the model of the Treatise. Our model, which is more general, may remedy that indeterminacy. Unfortunately, the complete study of its comparative static properties is beyond our capacity. We shall limit ourselves to a simple case with the help of a numerical example. Let us assume: a11 (q1 ) a22 (q2 ) 0 ; a12 (q1 ) .1  .2q1; a21 (q2 ) .2  .4q2; l1 (q1 ) .5  .5q1; l2 (q2 ) .4  .6q2; qi ! 1; D = .8, E = 0, D11 D22 0, D12 .5 and D21 .6. Remember that Sie 0 and that wage is taken as the nominal unit w 1 . Relations between expected prices and quantities become:

18 Jean Cartelier

p1e p2e

.1 p2e  .2q1 p2e  .5  .5q1 .2 p1e  .4q2 p1e  .4  .6q2

(2.18)

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and the corresponding supply curves are:

q1

10.0 p1e  p2e  5.0 2.0 p2e  5.0

q2

5.0 p2e  p1e  2.0 2.0 p1e  3.0

(2.19)

Market prices are determined by the Shapley-Shubik rule where the pie s are replaced by the piʾs:

q1

.4q1  .4q12  .32q2  .48q22  .6 q1  .2q2  .4q22

q2

.5 q2  .1q1  .2q12

(2.20)

Reporting q1 and q2 , given by supply curves in Shapley-Shubik equations and using the condition of equilibrium pi pie , we obtain a system of two independent equations determining an equilibrium of prices (a unique real solution exists here). It is then easy to get the equilibrium value of the other variables. In Table 2.3 equilibrium values are calculated for two different levels of the autonomous investment expenditures. Table 2.3 D12 = .1

D12 = .5

D21 = .6

D21 = .6

Consumption goods price p1

1.4689

2.4769

Investment good price p2

1.0749

2.3742

Quantity of consumption goods q1

2.9041

3.2973

Quantity of investment goods q2

1.4893

2.0554

Income (E) w[l1 (q1) + l2 (q2)]

3.5

5.5

1.9061

1.7764

Variables

Real consumption of wage-earners

Dw ª¬l1  q1  l2  q2 º¼

.6 Real consumption of entrepreneurs 1 p1 a12(q2)q2 Real consumption of entrepreneurs 2 a21(q1)q1 Financial market monetary balance

D12 p2

p1 .40847

.24224

.58935

1.2788

.09303

.2106

1.3963 .7

1.8447 1.1

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Keynes’s Treatise on Money 19 The increase in autonomous expenditures leads to higher prices for both sectors in spite of the increase of quantities in both sectors. Interdependence between VHFWRUV KDV DQ LQÀXHQFH EXW GRHV QRW LQYDOLGDWH .H\QHV¶V SURSRVLWLRQV ZKLFK seem to be valid now in comparative statics also. The increase in quantities of consumption goods is associated with a decrease in entrepreneurs’ and wageearners’ consumption. This surprising result is due to the augmentation of the demand for consumption goods as input by the sector of investment goods. What is even more interesting is that levels of employment and real wages 1 ) and change, respectively 3.5 and 5.5 for the former and 0.6808 ( 1.4689 1 ) for the latter. Let us assume that the higher level of employment 0.4037 ( 2.4769 corresponds to full employment (in the absence of any supply of labour it is only a conjecture). As a consequence, the lower appears to be an involuntary unemployment equilibrium. In the model of the Treatise, and contrary to what Hicks thought, it is quite possible to prove the possible existence of an involuntary unemployment equilibrium: we need only to add to the Treatise a saving function ( S (1  D) E). Keynes perceived this point very soon after the publication of the Treatise, as is shown by his famous letter of 20 September 1931 adressed to Kahn: But if each level of aggregate output has an appropriate proportion of saving S to incomes attached to it, e. g. if f (O) or better suppose S = f1 E + E P E ⎛ Q ⎞ , then points of equilibrium output can be reached which fall short of f2 ⎜ ⎟ ⎝P⎠ Q maximum and zero. . . . If reachs zero before O reaches maximum, we O have ‘long-period unemployment’, i. e. an equilibrium position short of fullemployment. (JMK CW, p. 374)

( )

A decisive step toward the General Theory indeed!

Notes 1 E will become the factor cost in Chapter 6 of General Theory. 2 Keynes’s rejection of the so-called ‘second classical postulate’ in General Theory shows how important the difference between entrepreneurs and wage-earners is for him from an analytical point of view. 3 ‘These losses, which represent a failure to receive cash up to expectations from sales of current output, must be financed, and the non-receipt of the expected cash receipts must be somehow made good. The entrepreneurs can only make them good either by reducing their own bank deposits or by selling some of their other capital assets’ (Keynes (1930), p. 131). 4 Investment expenditures I are arbitrarily distributed between the two sectors with no loss of generality. 5 We have replaced for the sake of symmetry the second fundamental equation of the Treatise by that given in the text. It is easy to check that PR + P ′C = Π = E + I − S , O O O as in the Treatise.

20 Jean Cartelier 6 The supply curve may be calculated by maximization of expected profit. Keynes did not mention such a possibility in the Treatise. But adopting it would not change the principle of Keynes’s argument. 7 A brief look at that tradition may be found in Benetti and Cartelier (2000).

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References Benetti, Carlo and Cartelier, Jean (2000), ‘Money and Price Theory’, International Journal of Applied Economics and Econometrics, April–May. Hicks, John, R. (1967), Critical Essays in Monetary Theory, Oxford: Clarendon Press. Keynes, Johns M. (1930), A Treatise on Money, 2 vols, Volumes V and VI of The Collected Writings of John Maynard Keynes, London: Macmillan, 1971. 7KH*HQHUDO7KHRU\DQG$IWHU3DUW3UHSDUDWLRQ, Volume XIII of The Collected Writings of John Maynard Keynes, London: Macmillan, 1971. Marcuzzo, Maria Cristina (2002), ‘The Collaboration between J.M. Keynes and R.F. Kahn from the Treatise to the General Theory’ History of Political Economy 34: 2, pp. 421–47. Wicksell, Knut (1935), Lectures on Political Economy, London: Routledge & Sons.

3

‘Animal Spirits’ in John Maynard Keynes’s General Theory of Employment, Interest and Money Some short and sceptical remarks

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References $NHUORI*$DQG6KLOOHU5- Animal Spirits. How human psychology drives the economy, and why it matters for global capitalism3ULQFHWRQ3ULQFHWRQ8QLYHUVLW\ 3UHVV $NHUORI*$DQG6KLOOHU5- Animal Spirits. How human psychology drives the economy, and why it matters for global capitalism (with a new preface by the authors), QLQWKSULQWLQJ3ULQFHWRQ2[IRUG3ULQFHWRQ8QLYHUVLW\3UHVV %DUHQV ,   Geld und Unterbeschäftigung. John Maynard Keynes¶ Kritik der Selbstregulierungsvorstellung, Volkswirtschaftliche Schriften, Heft 368 %HUOLQ 'XQFNHU +XPEORW %DUHQV ,   µ)URP WKH ³%DQDQD 3DUDEOH´ WR WKH 3ULQFLSOH RI (IIHFWLYH 'HPDQG 6RPH UHIOHFWLRQV RQ WKH RULJLQ GHYHORSPHQW DQG VWUXFWXUH RI .H\QHV’ General 7KHRU\¶LQ:DONHU'$HG Perspectives on the History of Economic Thought Volume II: Twentieth-Century Economic Thought6HOHFWHG3DSHUVIURPWKH+LVWRU\ RI(FRQRPLFV6RFLHW\&RQIHUHQFHstHG$OGHUVKRW+DQWV(GZDUG(OJDUSS ± %HQQHWW05 µ7KHHDUO\KLVWRU\RIWKHV\QDSVH)URP3ODWRWR6KHUULQJWRQ¶Brain Research Bulletin9RO1RSS %HUNHOH\ *   µ$OFLSKURQ RU WKH 0LQXWH 3KLORVRSKHU LQ VHYHQ 'LDORJXHV¶ FRQWLQXHG LQThe Works of George Berkeley/RQGRQSS± &DUDEHOOL$ On Keynes’s Method/RQGRQ0DFPLOODQ &2('   The Compact Oxford English Dictionary 6HFRQG (GLWLRQ 2[IRUG &ODUHQGRQ3UHVV *HKUNH &   (GLWRULDO 1RWH RQ µ$Q $EVWUDFW RI D 7UHDWLVH RI +XPDQ 1DWXUH¶ SUHSDUHGIRUWKHIRUWKFRPLQJHGLWLRQRIThe Unpublished Writings of Piero Sraffa, *UD]PLPHR *O\QQ, µ7ZRPLOOHQQLDRIDQLPDOVSLULWV$QDQFLHQWWKHRU\DERXWQHUYHVSURYHG VXUSULVLQJO\KDUGWRGLVORGJH¶Nature9RO1RYHPEHUS +LOO*%1HG  Letters of David Hume to William Strahan2[IRUG&ODUHQGRQ 3UHVV +XPH' µ7KH/LIHRI'DYLG+XPH(VT¶LQ+LOO*%HG Letters of David Hume to William Strahan2[IRUG&ODUHQGRQ3UHVVSS[YLL±[O .H\QHV-0>@ µ$7UHDWLVHRQ0RQH\7KH3XUH7KHRU\RI0RQH\¶LQ 0RJJULGJH ' ( HG  The Collected Writings of John Maynard Keynes YRO9 /RQGRQ0DFPLOODQ .H\QHV-0>@ µ7KH*HQHUDO7KHRU\RI(PSOR\PHQW,QWHUHVWDQG0RQH\¶ LQ0RJJULGJH'(HG The Collected Writings of John Maynard KeynesYRO9,, /RQGRQ0DFPLOODQ .H\QHV-0D µ7KH*HQHUDO7KHRU\DQG$IWHU3DUW,3UHSDUDWLRQ¶LQ0RJJULGJH ' ( HG  The Collected Writings of John Maynard Keynes YRO ;,,, /RQGRQ 0DFPLOODQ .H\QHV-0E µ7KH*HQHUDO7KHRU\DQG$IWHU3DUW,,'HIHQFHDQG'HYHORSPHQW¶ LQ0RJJULGJH'(HG Collected Writings of John Maynard KeynesYRO;,9 /RQGRQ0DFPLOODQ

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Adjustment of saving to investment 39 being determined by the money rate of interest (Sraffa, 1960, p. 33). (Throughout the present chapter we will take the latter perspective on how the normal distribution of income is determined.)14 The uniformity of the industrial rates of SUR¿W LV DVVXPHG WR EH DFKLHYHG LQ WKH ORQJSHULRG HTXLOLEULXP WKURXJK IUHH movement of money capital in pursuit of higher returns. Then, for a particular WHFKQLTXHHPSOR\HGDVGRPLQDQWWKHUHZLOOLQHTXLOLEULXPEHDXQLTXHUHODWLRQVKLS DPRQJUHODWLYHSULFHVWKHQRUPDOUDWHRISUR¿WVDQGWKHUHDOZDJHUDWH 1RZWKHORQJSHULRGOHYHOVRIRXWSXWDQGRIHPSOR\PHQWDUHWKRVHZKLFKDUH FRQFRPLWDQWWRWKHORQJSHULRGFRQ¿JXUDWLRQRIWKHUDWHRISUR¿WVWKHUHDOZDJH rate and relative prices (Eatwell, 1983). The tendency towards the realization of WKHXQLIRUPUDWHRISUR¿WVLPSOLHVWKDWWKHVFDOHDQGFRPSRVLWLRQRIRXWSXWRQWKH one hand and the size and composition of productive capacity on the other tend to be adjusted one to the other. This full adjustment between output and productive FDSDFLW\LQWKHORQJSHULRGHTXLOLEULXPLQWXUQLPSRUWV±E\WKHYHU\GH¿QLWLRQRI long-period equilibrium±WKDWSURGXFWLYHFDSDFLW\ZLOOEHXWLOL]HGDWWKHQRUPDO OHYHOLQHTXLOLEULXP7KHVL]HRIDJJUHJDWHSURGXFWLYHFDSDFLW\GHSHQGVRQWKH level of aggregate net investment taken in the course of ‘gravitation’ towards HTXLOLEULXPWKHKLJKHUWKHOHYHORILQYHVWPHQWWKHODUJHUWKHVL]HRISURGXFWLYH capacity. A larger size of productive capacity implies a higher level of normal output (i.e. the level of output which would result if actual investment were such as to bring about the normal utilization of that productive capacity), and this higher level of normal output is the basis on which a higher level of saving will be generated FRUUHVSRQGLQJ WR D KLJKHU OHYHO RI LQYHVWPHQW %HWZHHQ WZR GLIIHUHQW IXOO\ adjusted positions,15 which result from different levels of aggregate investment EXWZLWKWKHVDPHWHFKQLTXHWKHVDPHQRUPDOUDWHRISUR¿WVDQGWKHVDPHUHODWLYH UDWHVRIJURZWKRIWKHFRPSRQHQWVRIFRQVXPSWLRQGHPDQGZHVKDOOREVHUYH±RQ WKHDVVXPSWLRQRIFRQVWDQWUHWXUQVWRVFDOH±WKHVDPHVHWRIUHODWLYHSULFHVWKH same real wage rate, the same composition of the stock of means of production and the same composition of output, but different sizes of the stock and output.16 It will be noted that what is autonomous is the volume of aggregate investment in value terms. The fact is, however, that the volume of aggregate investment in a certain period, let alone over a number of periods, is not a matter to be determined E\ D VLQJOH GHFLVLRQ E\ D VLQJOH SHUVRQ RU DXWKRULW\ LW LV PHUHO\ WKH VXP RI individual investments in that period or over a number of periods concerned. But when we are speaking of the volume of aggregate investment being autonomous, ZHDUHFRQFHLYLQJWKDWWKHUHLVLQ.H\QHV¶VWHUPVDJHQHUDOµVWDWHRIORQJWHUP expectations’ (Keynes, 1936, Ch. 12) in the economy as a whole; this depends on YDULRXV±HFRQRPLFSROLWLFDOLQVWLWXWLRQDOHWF±DVSHFWVRIWKHHFRQRP\VHH Eatwell, 1983; Park, 1994), and acts as a general incentive to invest, ‘regulating’ LQGLYLGXDOVKRUWSHULRGLQYHVWPHQWWKDWLVLWLVDXWRQRPRXVLQWKHSDUWLFXODUVHQVH that it is prior to saving. Where the economy settles in a stationary state, the size RISURGXFWLYHFDSDFLW\LQHTXLOLEULXPGHSHQGVRQWKHJHQHUDOVWDWHRIORQJWHUP H[SHFWDWLRQ LI WKH ODWWHU LV IRU WKH EHWWHU WKH IRUPHU ZLOO EH ODUJHU:KHUH WKH economy is growing, the average pace of accumulation will be faster if the state

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Adjustment of saving to investment 41 impacts on the real economy. As has been well noted, the latter conception is suggested by Sraffa (1960, p. 33). We are able to follow this suggestion by following Keynes closely by taking the value of investment as autonomous and thus by rejecting the Neo-Keynesian position of taking the rate of accumulation as autonomous. For in the long-period equilibrium, where productive capacity is utilized at the normal level, the exogeneity of the rate of accumulation precludes WKHSRVVLELOLW\RIWKHH[RJHQHLW\RIWKHUDWHRISUR¿WVRUWKHUHDOZDJHUDWH 17

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A conceptual framework for the alternative position This section will present a conceptual framework which would allow the third Keynesian route to be given analytic treatment. Throughout the argument, constant returns to scale are assumed. Let K0 stand for the size of the initial stock at period 0, v for the technical stock to output ratio, rIRUWKHQRUPDOUDWHRISUR¿WVDQGsr for the propensities to save RXW RI SUR¿WV VDYLQJ RXW RI ZDJHV LV DVVXPHG WR EH QLO 18 If the stock K0 is utilized at the normal level so as to produce the normal level of output (Y0*), then Y0* = v –1K0

(4.2)

and the level of net saving accruing from Y0*, which is called the normal level of net saving, will be S0* = sr Y0*

(4.3)

In equilibrium, the level of investment and that of saving coincide; the level of net investment (I0*) which corresponds to S0* in equilibrium will be called the normal level of net investment at period 0. Then the ratio, denoted by g*, which the normal level of net investment bears to the value of the stock will be g* ≡

I 0* = sr r K0

(4.4)

This rate, christened the normal rate of net accumulation, depends on the following:19 (a) the dominant technique in use, represented here by Q (b) the state of normal income distribution, which is determined in turn with reference to the dominant technique (Q and the labour to output ratio), once WKHQRUPDOUDWHRISUR¿WVLVGHWHUPLQHGE\WKHPRQH\UDWHVRILQWHUHVWDQG (c) saving behaviour ( sr ). All the variables listed above from (a) to (c), which we shall call the normal variables, are taken as given in the following arguments (it being thus implied that technical changes, among others, are exogenous). It immediately follows that the normal rate of net accumulation is determined prior to the size of the stock and the

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(1 + r )a + wl = 1



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Adjustment of saving to investment 47 to (4.13), equilibrium on the quantity side requires that output be equal to the sum of replacement, net investment and consumption; however, for our economy, gross investment is autonomously given. We are dealing with a growing economy (including a stationary one), and for this kind of economy, a volume of investment that is given at an absolute level makes little sense. This is where the system for GAG comes in to play the role. The system consisting of (4.13) and (4.14) describes a growing economy (if g* > 0) in which there is no constraint of effective demand: productive capacity is utilized at the normal level and labour employed at the full level continuously over time. Our theoretical intention is to analyse effects of the constraint of effective demand in a growing economy. Thus, the state of effective demand is best represented if analysis is carried out in terms of comparison with the GAG economy. For GAG to be possible, gross investment is required to be realized at the level of J * (1  g*) x * a . In contrast, in an economy where effective demand is a binding force, the state of effective demand is best represented by a ratio that gross investment in the economy bears to its counterpart in the GAG economy. Let us denote that ratio by z;26 that is, the level of gross investment in an EDC economy is zJ*. Then the counterpart of (4.13) for the EDC economy is zJ* + h = x

(4.13ƍ)

The value and the quantity sides are connected, precisely in the same way as in the GAG economy, through the saving behaviour of income groups: a fraction sr RISUR¿WVDQGQRIUDFWLRQRIZDJHVDUHVDYHGVRWKDWRQHKDV h

(1  sr )rxa  wxl

(4.15ƍ)

The system of equations for analysing an EDC economy in the long period consists of equations (4.12), (4.13ƍ) and (4.15ƍ). For a given value of z, one obtains the level of output as27 x

zJ * (1  sc r )a

zx*

(4.16)

One thus observes that higher investment (a higher value of z) leads to higher output (zx*). Productive equipment (zx*a) is utilized at the normal level to produce the normal level of gross saving ((1 + srr) zx*a) that matches the autonomously given, higher gross investment. The level of consumption is also higher (h = zh*). The employment of labour also moves in step with effective demand (zx*l).28 The principle of effective demand is in full operation. One will also note that a higher VDYLQJSURSHQVLW\RXWRISUR¿WVLVDVVRFLDWHGZLWKDORZHUOHYHORIRXWSXWDQGWKXV a lower level of saving – the paradox of saving. Despite differences in these YDULDEOHVWKHUDWHRISUR¿WVDQGWKHZDJHUDWHUHPDLQWKHVDPH As is seen in equation (4.16), the ratio that the exogenously given (gross) investment bears to the endogenously determined stock of means of production is

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Appendix :HVWDUWIURPSHULRG/HWXVKDYH Kt VWDQGIRUWKHVL]HRIWKHHTXLOLEULXPVWRFN FRUUHVSRQGLQJWRWKHORQJSHULRGOHYHORILQYHVWPHQW It DWSHULRGt! WKDWLVKt

Adjustment of saving to investment 49 is that part of the actual stock Kt at period t which would be utilized at the normal level when the investment at period t is It (1  g *)t I0. Then the degree of utilization at period t, where tLVVXI¿FLHQWO\ODUJHLV

K t Kt

ut {

(1  g *)t K 0 t 1

K0  ¦ I k

(4.A1)

k 0

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(1  g *)t K 0 t 1

K 0  ¦ Ik k 0

(1  g *)t §K · (1  g *)t  ¨ 0   1¸ © K0 ¹

and the annual proportionate rate of accumulation g t {

It Kt

(4.A2)

ut g*

Then

lim ut t of

g t 1 and lim t of

g*

(4.A3)

The ‘long-period average degree of utilization’ over W SHULRGVLVGH¿QHGDV W1

u (W) { W1 ¦ ut

(4.A4)

t 0

WKH¿UVWDSSUR[LPDWLRQRIZKLFKIRUDVXI¿FLHQWO\ODUJHIJLV u (W )

§ ln( K 0*/K 0 ) · 1  W 1 ¨ © ln(1  g*) ¸¹

(4.A5)

7KH VWDWH RI /7( IRU WKH EHWWHU LV UHSUHVHQWHG E\ D KLJKHU OHYHO RI I0 , which implies a higher level of K 0 . Thus as the state of LTE is for the better, the average degree of utilization u (W ) is higher. The long-period average rate of accumulation LVGH¿QHGDV W 1

g (W ) { W 1 ¦ g t

u (W ) g *

(4.A6)

t 0

Thus, the state of LTE for the better will be associated with a higher average rate of accumulation.

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Adjustment of saving to investment 51 12 Contrary to what has usually been conceived (e.g. Asimakopulos, 1985), the discussion between Keynes and Harrod in terms of the rates of growth on the occasion of the publication of the latter’s ‘An essay in dynamic theory’ does not indicate that Keynes would have considered the rate of accumulation to be autonomous. Referring to Harrod’s draft, Keynes argues that both the ‘warranted rate of growth’ and the ‘natural rate of growth’ are functions of the level of employment, and the ‘normal rate of growth’ is determined at the point where the former two rates of growth coincide; the level of employment corresponding to this normal rate of growth is called the ‘level of long-period employment’, probably less than full employment (Keynes, 1973, vol. 14, pp. 325–36). The level of employment being, as Keynes argues in the General Theory, dependent upon the level of investment, it follows that the various rates of growth referred to above are, for Keynes, not autonomous but functions of the level of investment. Also for Harrod, in his published article, the ‘actual rate of growth’ in a certain period is a function of the level of investment: the denominator of the actual rate of growth is ‘the value of the increment of capital per unit increment of output actually produced’ (Harrod, 1939, pp. 17–18). It would be redundant to say that his warranted rate of growth and the natural rate of growth are not autonomous in any sense which was given by Keynes to the level of investment. 13 The relation between the normal rate of profits and the real wage rate is affected, as Sraffa (1960) has argued, by political or monetary factors and by the technical conditions. As for the dependence of the normal degree of utilization on the distribution of income, see Kurz (1986). 14 On the relation between the rate of profits and the rate of interest in the light of Sraffa, see for example Panico (1980, 1988). 15 We speak of a particular ‘long-period equilibrium’ as corresponding to a given state of the dominant technique in use and the state of normal income distribution, regardless of the size and composition of the means of production. The term ‘fully-adjusted position’ is used to denote different positions of long-period equilibrium according to different sizes and compositions of the means of production. Vianello (1985) uses the term ‘fully-adjusted situations’ for the same purpose. 16 Kurz (1985) argues that an increase in the volume of aggregate investment in value terms accompanied by a different composition could be associated with a decrease in the volume of total output. This is because the respective productive processes produce output in different proportions: the case in question is possible if there is a relatively greater investment in the productive processes in which the unit production of output requires relatively greater productive equipment while there is a relatively smaller investment in the productive process in which the unit production requires relatively smaller productive equipment. 17 Thus, Robinson’s following remark is partly explained by her Neo-Keynesian hypothesis of the autonomy of the rate of accumulation: ‘. . . the suggestion that the rate of profits could be determined by monetary policy seems to be excessively fanciful’ (1979, p. 180). 18 These differential saving propensities are, however, not essential for our mechanism of adjustment: the constancy of the normal distribution of income makes redundant the role which is assigned to such differentiation of saving propensities. 19 In the case of the multi-sector model, it depends also on the composition of consumption demand (which affects the activity levels). 20 Changes in the state of LTE will either be induced by changes in the normal variables or reflect autonomous changes in the general ‘state of confidence’ (Keynes, 1936, Ch. 12). The present chapter does not touch upon the relation of the first type; it needs separate detailed studies. Thus, when we speak of different states of LTE, reference is being restricted to different states of confidence for the same normal variables. 21 This concept of the equilibrium stock for a given level of investment is derived by analogy with Harrod’s concept of the ‘justified’ investment, which is ‘that addition to capital goods in any period, which producers regard as ideally suited to the output

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Adjustment of saving to investment 53

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References Abraham-Frois, G. and Berrebi, E. 1979. Theory of Value, Prices and Accumulation: A Mathematical Integration of Marx, von Neumann and Sraffa, Cambridge: Cambridge University Press. Amadeo, E. J. 1985. Notes on capacity utilisation, distribution and accumulation, Contributions to Political Economy, vol. 5, pp. 83–94. Asimakopulos, A. 1985. Harrod on Harrod: the evolution of a ‘line of steady growth’, History of Political Economy, vol. 17, no. 4, pp. 619–35. Bardhan, P. 1995. The contributions of endogenous growth theory to the analysis of development problems: an assessment, in T. N. Srinivasan and J. Behrman (eds), Handbook of Development Economics, Vol. III, North Holland Publishing Co. Bharadwaj, K. and Schefold, B. (eds) 1990. Essays on Piero Sraffa. Critical Perspectives on the Revival of Classical Theory, London: Unwin Hyman. Ciccone, R. 1984. La teoria della distribuzione nell’analisi di Joan Robinson, Note economiche, no. 2, pp. 99–125. Ciccone, R. 1986. Accumulation and capacity utilization: some critical considerations on Joan Robinson’s theory of distribution, Political Economy – Studies in the Surplus Approach, vol. 2, no. 1, pp. 17–36. Committeri, M. 1986. Some comments on recent contributions on capital accumulation, income distribution and capacity utilization, Political Economy – Studies in the Surplus Approach, vol. 2, no. 2, pp. 161–86. Dutt, A. K. 1990. Growth, Distribution and Uneven Development, Cambridge: Cambridge University Press. Eatwell, J. 1983. The long-period theory of employment, Cambridge Journal of Economics, vol. 7, no. 3/4, pp. 269–85. Garegnani, P. 1962. Il problema della domanda effettiva nello sviluppo economico italiano, Rome: SVIMEZ. Gareganani, P. 1979. Notes on consumption, investment and effective demand: II, Cambridge Journal of Economics, vol. 3, no. 1, pp. 63–82. Garegnani, P. 1982a. Summary of the paper ‘Some notes for an analysis of accumulation’, paper presented at the conference Theories of Accumulation and the Control of the Economy, Udine. Garegnani, P. 1982b. Una polemica su Sraffa: a proposito di alcuni recenti interventi di Claudio Napoleoni, Rinascita, no. 25, 2 Juglio 1982, pp. 31–3. Garegnani, P. 1983. Two routes to effective demand: Comment on Kregel, in J. Kregel (ed), Distribution, Effective Demand and International Economic Relations, London: Macmillan. Garegnani, P. 1992. Some notes for an analysis of accumulation, in Halevi et al. (1992). Halevi, J., Laibman, D. and Nell, E. J. (eds) 1992. Beyond the Steady State: a Revival of Growth Theory, London: Macmillan. Harrod, R. 1939. An essay in dynamic theory, Economic Journal, vol. 49, no. 193, pp. 14–33. Kaldor, N. 1956. Alternative theories of distribution, Review of Economic Studies, vol. 23, no. 2, pp. 83–100. Kaldor, N. 1966. Marginal productivity and the macroeconomic theories of distribution: comment on Samuelson and Modigliani, Review of Economic Studies, vol. 33, pp. 309–19. Keynes, J. M. 1936. The General Theory of Employment, Interest and Money, as vol. VII of Keynes (1973 et seq.). Keynes, J. M. 1973. The General Theory and After: Part II, Defence and Development, as vol. XIV of Keynes (1973 et seq.).

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  Man-Seop Park .H\QHV-0et seqThe Collected Writings of John Maynard KeynesLQYROXPHV /RQGRQ0DFPLOODQ .XU]+'(IIHFWLYHGHPDQGLQDµ&ODVVLFDO¶PRGHORIYDOXHDQGGLVWULEXWLRQWKH PXOWLSOLHULQD6UDIILDQIUDPHZRUNThe Manchester SchoolYROQR± .XU]+'µ1RUPDO¶SRVLWLRQVDQGFDSLWDOXWLOL]DWLRQPolitical Economy – Studies in the Surplus ApproachYROQRSS± .XU] + '  $FFXPXODWLRQ GLVWULEXWLRQ DQG WKH µ.H\QHVLDQ +\SRWKHVLV¶ LQ %KDUDGZDMDQG6FKHIROG  .XU] + ' $FFXPXODWLRQ HIIHFWLYH GHPDQG DQG LQFRPH GLVWULEXWLRQ LQ +DOHYL et al.  .XU]+'*URZWKDQGGLVWULEXWLRQReview of Political EconomyYROQR SS± .XU] + ' DQG 6DOYDGRUL 1  Theory of Production. A Long Period Analysis &DPEULGJH&DPEULGJH8QLYHUVLW\3UHVV /DYRLH0Foundations of Post-Keynesian Economic Analysis$OGHUVKRW(GZDUG (OJDU /XFDV 5  2Q WKH PHFKDQLFV RI HFRQRPLF GHYHORSPHQW Journal of Monetary EconomicsYROSS± 3DOXPER$DQG7UH]]LQL$*URZWKZLWKRXWQRUPDOFDSDFLW\XWLOLVDWLRQEuropean Journal of the History of Economic ThoughtYROQRSS± 3DQLFR&0DU[¶VDQDO\VLVRIWKHUHODWLRQVKLSEHWZHHQWKHUDWHRILQWHUHVWDQGWKHUDWH RISURILWVCambridge Journal of EconomicsYROQRSS± 3DQLFR&Interest and Profit in the Theories of Value and Distribution/RQGRQ 0DFPLOODQ 3DUN06.H\QHV¶VWKHRU\RIHPSOR\PHQWVKRUWSHULRGDQDO\VLVLQWKHORQJSHULRG IUDPHZRUNContributions to Political EconomyYROSS± 3DUN06D1RUPDOYDOXHVDQGDYHUDJHYDOXHVMetroeconomicaYROQR SS± 3DUN 06 E$FFXPXODWLRQ FDSLWDO XWLOLVDWLRQ DQG GLVWULEXWLRQContributions to Political EconomyYROSS± 3DUN06$XWRQRPRXVGHPDQGDQGWKHZDUUDQWHGUDWHRIJURZWKContributions to Political EconomyYROSS± 3DVLQHWWL / /  5DWH RI SURILW DQG LQFRPH GLVWULEXWLRQ LQ UHODWLRQ WR WKH UDWH RI HFRQRPLF JURZWK DV LQ idem  Growth and Income Distribution. Essays in Economic Theory&DPEULGJH&DPEULGJH8QLYHUVLW\3UHVV 3DVLQHWWL//Lectures on the Theory of Production/RQGRQ0DFPLOODQ 5RELQVRQ-The Accumulation of Capital/RQGRQ0DFPLOODQ 5RELQVRQ-Essays in the Theory of Economic Growth/RQGRQ0DFPLOODQ 5RELQVRQ-*DUHJQDQLRQHIIHFWLYHGHPDQGCambridge Journal of EconomicsYRO QRSS± 5RPHU 3  *URZWK EDVHG RQ LQFUHDVLQJ UHWXUQV GXH WR VSHFLDOL]DWLRQ American Economic ReviewYROSS± 5RPHU3(QGRJHQRXVWHFKQRORJLFDOFKDQJHJournal of Political EconomyYRO SS6± 5RQFDJOLD$Sraffa and the Theory of Prices&KLFKHVWHU-RKQ:LOH\DQG6RQV 5RZWKRUQ5'HPDQGUHDOZDJHVDQGHFRQRPLFJURZWKThames Papers in Political Economy$XWXPQ5HSULQWHGLQ06DZ\HUHG Post-Keynesian Economics $OGHUVKRW(GZDUG(OJDU 6FKHIROG%1DFKZRUWHLQ6UDIID3Warenproduktion mittels Waren)UDQNIXUWD 06XKUNDPS9HUODJ

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Adjustment of saving to investment 55 Serrano, F. 1995. Long period effective demand and the Sraffian multiplier, Contributions to Political Economy, vol. 14, pp. 67–90. Setterfield, M. 2002. Demand-led Theories of Economic Growth, Cheltenham: Edward Elgar. Solow, R. 1956. A contribution to the theory of economic growth, Quarterly Journal of Economics, vol. 70, no. 1, pp. 65–94. Sraffa, P. 1960. Production of Commodities by Means of Commodities, Cambridge: Cambridge University Press. Steindl, J. 1952. Maturity and Stagnation in American Capitalism, Oxford: Basil Blackwell. Reprinted by Monthly Review Press, New York, 1976. Steindl, J. 1979. Stagnation theory and stagnation policy, Cambridge Journal of Economics, vol. 3, no. 1, pp. 1–14. Steindl, J. 1985. Distribution and growth, Political Economy – Studies in the Surplus Approach, vol. 1, no. 1, pp. 53–68. Trezzini, A. 1995. Capacity utilisation in the long run and the autonomous components of aggregate demand, Contributions to Political Economy, vol. 14, pp. 33–66. Trezzini, A. 1998. Capacity utilisation in the long run: some further considerations, Contributions to Political Economy, vol. 17, pp. 53–67. Vianello. F. 1985. The pace of accumulation, Political Economy – Studies in the Surplus Approach, vol. 1, no. 1, pp. 69–87.

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Part II

Theory of growth

5PartAIIsimple model of path-dependent growth with some Keynesian features

Theory of growth

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Notes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A simple model of path-dependent growth     6HH$UURZ¶VQHRFODVVLFDOJURZWKPRGHO OLQNLQJZRUNHUV¶OHDUQLQJE\GRLQJWR WKHH[LVWLQJVWRFNRIFDSLWDO%KDGXUL DUJXHVKRZSDWKGHSHQGHQWFDSLWDOVWRFN PLJKWDIIHFWWKHUDWHRIOHDUQLQJE\GRLQJDQGIXWXUHJURZWKSRWHQWLDO   +DUURG¶V   LQVWDELOLW\ ZKLFK ZDV PLVUHSUHVHQWHG DV D SUREOHP RI µIL[HG SUR SRUWLRQV¶E\6RORZ UHDSSHDUVKHUHZLWKIL[HGSURSRUWLRQVIRUHDFKYLQWDJH  ZKLOHVXEVWLWXWLRQZRUNVWKURXJKFKDQJLQJWKHDYHUDJHOLIHWLPHRIPDFKLQHV  ([WHQVLYH PDUJLQ GHWHUPLQLQJ WKH OLIH WLPH RI PDFKLQHV DQG WKH UDWH RI FDSDFLW\ XWLOL]DWLRQWKURXJKSURILWPD[LPL]DWLRQLVDQDORJRXVWRLQWHQVLYHPDUJLQGHWHUPLQLQJ WKHUDWLRRIFDSLWDOWRRXWSXWDWJLYHQIDFWRUSULFHVXQGHUSURILWPD[LPL]DWLRQ  7KLVLVVLPLODUWR0DOLQYDXG¶V GLVWLQFWLRQEHWZHHQSURILWPD[LPL]LQJµFODVVLFDO XQHPSOR\PHQW¶DQGGHPDQGFRQVWUDLQHGµ.H\QHVLDQXQHPSOR\PHQW¶  7KH LQIOXHQFH RI ORZHU FDSDFLW\ XWLOL]DWLRQ RQ LQYHVWPHQW LV FRQVLGHUHG E\ +DUURG   EXW KH OHDYHV RXW WKH HIIHFW RI FODVV GLVWULEXWLRQ HLWKHU RQ VDYLQJ RU RQ LQYHVWPHQWLQWURGXFHGLQ%KDGXULDQG0DUJOLQ   6KRUWHQLQJRIWKHOLIHWLPHRIDYHUDJHFDSLWDOJRRGVZRXOGWHQGWRUHGXFHHPSOR\PHQW DORQJZLWKFDSDFLW\DQGPLJKWSXWIXUWKHUGRZQZDUGSUHVVXUHRQWKHZDJHUDWH

References $UURZ.µ7KHHFRQRPLFLPSOLFDWLRQVRIOHDUQLQJE\GRLQJ¶Review of Economic Studies,± $UWKXU:%Increasing Returns and Path Dependence in the Economy$QQ$UERU 0LFKLJDQ8QLYHUVLW\3UHVV %DUUR5DQG6DODL0DUWLQ;Economic Growth, 1HZ x1T A11 + xT2 A 21 and xT2 = x1T A12 + xT2 A 22. Therefore, Besicovitch maintains, if u is a real number lower than 1, but so close to 1 that ux1T ! ux1T A11  xT2 A 21 still

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  Neri Salvadori holds, then by necessity xT ! uxT A  xT A . However, this is not necessarily WUXH,QGHHGLIPDWUL[A is decomposable and A 0, this is certainly false. It is reasonable to suppose that Besicovitch assumed that all commodities are basic DQGWKHUHIRUHPDWUL[A is indecomposable. Even in this case, however, the proof LVLQFRPSOHWHxT ALVVHPLSRVLWLYHEXWGRHVQRWQHHGWREHSRVLWLYH VLQFHZH PD\QHHGWRLWHUDWHWKHSURFHVVWREULQJKRPHWKHUHVXOW,QIDFWLIPDWUL[A is indecomposable, we are sure that xT t uxT A  xT A  and therefore the number of commodities with a positive surplus is increased and still no commodity has a negative surplus. Further, since at any iteration of the process the number of products with a positive surplus increases, the number of iterations needed to REWDLQDVXUSOXVLQDOOFRPPRGLWLHVLVFHUWDLQO\¿QLWHVLQFHLWLVORZHUWKDQn±h. 7KH¿UVWWKUHHWKHRUHPVRI¿OH'DUHLQWHQGHGWRVXSSRUWWZRIDFWV First, if there is a surplus of any type, industries may be proportioned in such a way as to get the surplus anywhere it is needed. Second, there is a surplus if, and only if, prices are positive and the wage rate is positive. The relationship with section RIWKHERRNE\6UDIID LVREYLRXV2QHRIWKHWZRVWHSVRIWKHDOJRULWKP LQWURGXFHGWKHUHFRQVLVWVH[DFWO\LQµDGMXVWLQJWKHSURSRUWLRQVRIWKHLQGXVWULHVRI the system in such a way that of each basic commodity a larger quantity is produced than is strictly necessary for replacement’. The fourth theorem concerns the H[LVWHQFH RI WKH 6WDQGDUG FRPPRGLW\ DQG ZLOO EH DQDO\VHG LQ WKH QH[WVHFWLRQ

Besicovitch’s proof The fourth theorem reads in plain English: If prices are positive, then there exist positive multipliers qa , . . ., qk such that the net output is proportional to the total of every kind of raw material. In modern notation the fourth Theorem states: p ! 0 w !   Ap  wl

p Ÿ x ! 0 R !     R xT A

xT 



The proof is similar to that provided by Sraffa, but is more detailed and closer to the description of an algorithm. It starts by assuming that there is a surplus with regard to all commodities. If there were a surplus only in some LQGXVWULHV WKHQ ZH FRXOG ¿QG D VWDUWLQJ SRLQW ZLWK D VXUSOXV LQ DOOindustries,

{

}

x 0 ∈ x > 0 xT l = E , xT [ I − A ] > 0T , since the assumption of Theorem 1 holds. Then the second step used by Sraffa is applied. That is, it is found that

O

O x

PD[ j

xT Ae j xT e j





so that xT >OI  A @ t 0T and xT >OI  A @ ! 0T . Then all the equations of commodities for which there is a surplus are multiplied by a common scalar lower than 1. Besicovitch thinks this is enough to obtain that all commodities are in VXUSOXVEXWWKLVGRHVQRWQHHGWREHWUXHVLQFHLQSXWFRHI¿FLHQWVDUHQRWDOOSRVLWLYH

Besicovitch and Sraffa   +RZHYHUVLQFHDOOFRPPRGLWLHVDUHDVVXPHGWREHEDVLFWKHLQSXWPDWUL[A is indecomposable and therefore we can get the desired result by iterating the same procedure, as seen above, in the analysis of the third theorem by Besicovitch. Let us consider the point in a more formal way. Let P ∈ ℜ and x  S be such that P xT t xTA DQG OHW XV GH¿QH WKH VHW RI indices

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⎧⎪ I Px = ⎨i ∈ {1, 2, ..., n} Pxi > ⎩⎪ IˆPx

­° ®i  ^   n` Pxi ¯°

n

∑x a j

j =1

n

¦x a j

ji

j 1

ji

⎫⎪ ⎬ ⎭⎪

½° ¾ ¿°





7KH DLP RI WKLV VWHS FRQVLVWV LQ ¿QGLQJ DQ LQWHQVLW\ YHFWRU Ix  VXFK WKDW I μφ( x) = {1, 2, ..., n} and, as a consequence, ˆI PIx ‡ . Besicovitch considers that this can be obtained if Ix is the function g ( P, x ) , where if i  ˆI Px  if i  I Px

°­ xi ® °¯ Kxi

gi  P, x



where Ș is a scalar lower than 1, but so close to 1 that P  Kxi !

¦

j ˆI P[

x j a ji  K¦ j I x j a ji P[

each i  IP[ 



That is, PD[  i I P[

¦

j ˆI P[

x j a ji

Pxi  ¦ j I x j a ji

 K  



P[

As mentioned above, this is not enough to obtain that , Pg  P, x ^     n` because some a ji may be nought. However, by construction, i  I Px implies that i ∈ I Pg ( P, x) and therefore I Pg  P, x ‹ I Px . On the other hand, I Pg P,x I Px if, and only if, a ji  , each i  I Px and each j  ˆI Px %XWWKHQPDWUL[A would be decomposable. This being impossible, we obtain that I Pg  P, x Š I Px . This is enough to say that WKHSURFHGXUHFDQEHLWHUDWHGIRUDQXPEHURIWLPHVORZHUWKDQWKH¿QLWH QXPEHU RIFRPPRGLWLHVDOVREHFDXVHLIlPx ^n}, then gȝ, x LVSURSRUWLRQDO to x +HQFHZHFDQGH¿QH h1  x

g O  x , x 

h j x

g O x , h j 1 x

I x



h n 1 x





j = 2, ..., n − 1

 

  Neri Salvadori There is one further aspect considered by Besicovitch. In a remark he argued that ‘we may keep one of our industries intact’ in order to avoid all multipliers becoming zero. With no loss of generality, assume that the industry in question is industry 1. Therefore function gȝ, x PXVWEHUHGH¿QHGDV

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gi  P, x

­ xi ° ° Kxi ° ®1 ° K xi ° °¯ xi

if i  ˆI Px and 1  ˆI Px if i  I and 1  ˆI Px

Px

if i  ˆI Px and 1  ˆI Px 



if i  I Px and 1  ˆI Px

Further, this function has the property that if lPx ^n}, then g  P , x x. As seen in the second section of this chapter, Sraffa followed a different, but equivalent, strategy to avoid all multipliers becoming zero. He kept the DPRXQWRIODERU¿[HG,IZHIROORZWKLVVWUDWHJ\WKHQIXQFWLRQgȝ, x must be UHGH¿QHGDV gi  P, x

°­ Txi ® °¯TKxi

if i  ˆI Px  if i  I Px



where T

¦

j ˆI Px

E  x j l j  K¦ j I x j l j



Px

Also this function has the property that if lPx ^n}, then gP, x  x. Also in Besicovitch’s proof there is a family of potential algorithms involved. ,QRUGHUWRKDYHDSURSHUDOJRULWKPZHPXVWKDYHDZD\WRGH¿QHKRZȘ is chosen. )RUH[DPSOHLIZHFKRVHȘ in the middle of the range in which it can vary, we would have

K

¦ j ˆI Px x j a ji 1 1  PD[    i I Px Pxi  ¦ j I x j a ji



Px

DQGLQJHQHUDODQ\SRVVLEOHFKRLFHFRXOGEHGH¿QHGDVDFKRLFHRIĮ < 1 in the H[SUHVVLRQ

K

D    D PD[  i I Px

¦

j ˆI Px

x j a ji

Pxi  ¦ j I x j a ji Px



Besicovitch and Sraffa   For each sequence {Įi`Įi ZHKDYHDGLIIHUHQWDOJRULWKPEXWZKDWHYHU sequence {Įi} is chosen, it is easily proved that the conditions stated by Salvadori  KROGDQGWKHUHIRUHDOOWKHSRWHQWLDODOJRULWKPVFRQVLGHUHGE\%HVLFRYLWFK converge to the desired result. In fact, for any given sequence {Įi} function Ix is continuous and can start from any point in S.

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Sraffa and Besicovitch Sraffa did not use the function Ix used by Besicovitch. He recognized that what is important is ‘adjusting the proportions of the industries of the system in such a way that of each basic commodity a larger quantity is produced than is strictly necessary for replacement’ and that at each step the desired result is closer, but he GLGQRWFRQVLGHUWKHIDFWWKDWWKHµLPDJLQDU\H[SHULPHQW¶PD\ZRUNWKURXJKDQ LQ¿QLWHQXPEHURIVWHSVZLWKRXWDSSURDFKLQJWKH6WDQGDUGFRPPRGLW\ Why did Sraffa not use the proof available to him and provided by Besicovitch LQ 6HSWHPEHU " $ VLPSOH DQVZHU FRXOG EH WKDW 6UDIID WKRXJKW WKDW WKH H[SRVLWLRQ RI WKH SURRI FRXOG EH VLPSOL¿HG DQG WKDW KH IDLOHG WR FDUU\ RXW WKH VLPSOL¿FDWLRQUHTXLUHG7KLVLVDSRVVLEOHLQWHUSUHWDWLRQ+RZHYHUWKHUHDUHRWKHU cases in which Sraffa made no use of an available proof by Besicovitch. For LQVWDQFHZKHQLQWKHV6UDIIDZDVIDFHGZLWKWKHQHHGWRGH¿QHEDVLFVDQG QRQEDVLFVLQMRLQWSURGXFWLRQKHFRQFHLYHGDGH¿QLWLRQLQWHUPVRIDWD[RQWKH SURGXFWLRQRIVLQJOHFRPPRGLWLHVDWLWKH $WD[RQWKHSURGXFWLRQRIDEDVLF FRPPRGLW\DIIHFWVDOOSULFHVDQGWKHZDJHUDWHIRUDJLYHQUDWHRISUR¿W ZKHUHDV DWD[RQWKHSURGXFWLRQRIDQRQEDVLFFRPPRGLW\DIIHFWVRQO\SULFHVRIVRPHQRQ EDVLFFRPPRGLWLHVLIWKHQXPHUDLUHLV¿[HGRQO\LQWHUPVRIEDVLFFRPPRGLWLHV  +HZDVFRQYLQFHGWRXVHWKHOLQHDUGHSHQGHQFHGH¿QLWLRQZH¿QGLQWKHERRN † E\%HVLFRYLWFKWKHZKROHVWRU\LVWROGE\.XU]DQG6DOYDGRUL 7KH WD[DUJXPHQWDSSHDUVLQWKHERRN† EXWLWLVDFRQVHTXHQFHQRWWKHGH¿QLWLRQ Interestingly, Besicovitch proved three months after Sraffa had accepted the GH¿QLWLRQLQWHUPVRIOLQHDUGHSHQGHQFHWKDW6UDIID¶VRULJLQDOLGHDZDVFRUUHFW DQGWKDWDFWXDOO\WKHGH¿QLWLRQFRXOGEHJLYHQLQWHUPVRIWKHWD[+RZHYHUWKH SURRILVH[WUHPHO\GHPDQGLQJLQWHUPVRIPDWKHPDWLFDOFDOFXODWLRQVVHH.XU]DQG 6DOYDGRUL   6UDIID PDGH QR PHQWLRQ RI WKLV SURRI E\ %HVLFRYLWFK LQ his book. %RWKWKHSURRIRIWKHH[LVWHQFHRIWKH6WDQGDUGFRPPRGLW\DQGWKHGLVWLQFWLRQ between basic and non-basic commodities recall the concluding remarks in Sraffa’s Preface to his book: ,WZLOOEHRQO\WRRREYLRXVWKDW,KDYHQRWDOZD\VIROORZHGWKHH[SHUWDGYLFH WKDWZDVJLYHQWRPH±SDUWLFXODUO\ZLWKUHJDUGWRWKHQRWDWLRQDGRSWHGZKLFK ,KDYHLQVLVWHGRQUHWDLQLQJDOWKRXJKDGPLWWHGO\RSHQWRREMHFWLRQLQVRPH UHVSHFWV DVEHLQJHDV\WRIROORZIRUWKHQRQPDWKHPDWLFDOUHDGHU 'HVSLWHKLVLQWHUHVWLQWKHH[LVWHQFHRIDSURRI6UDIIDZDVNHHQWRSURYLGHRQH only if it was ‘easy to follow for the non-mathematical reader’. He thought that

126

Neri Salvadori

the non-mathematical reader would understand his argument in section 37. If the PDWKHPDWLFDOUHDGHUZHUHWR¿QGLWLQFRPSOHWHWKHQVXFKDUHDGHUZRXOGDOVREH DEOHWR¿QGDFRPSOHWHSURRIZKLFK6UDIIDNQHZH[LVWHG

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Conclusion ,QWKLVFKDSWHU,KDYHH[SORUHGWKHUHODWLRQVKLSEHWZHHQWKHSURRIRIWKHH[LVWHQFH RI WKH 6WDQGDUG FRPPRGLW\ FRQWDLQHG LQ VHFWLRQ  RI 6UDIID¶V   ERRN DQG WKH SURRI VXSSOLHG WR 6UDIID E\ %HVLFRYLWFK RQ  6HSWHPEHU  DQG LQYHVWLJDWHGWKHFRPSOHWHQHVVDQGFRQVLVWHQF\RIVXFKDSURRI,DOVRSRVWXODWHG VRPH UHDVRQV ZKLFK OHG 6UDIID WR RPLW WKLV SURRI LQ KLV ERRN LQ IDYRXU RI DQ LQFRPSOHWHDUJXPHQW

Appendix A. An example Let ª  hº «k » ¬ ¼

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with 0@.XU]+'DQG6DOYDGRUL1 Theory of Production. A Long-Period Analysis. Cambridge University Press, Cambridge. [10] Lehrer, K. and Wagner, C. (1981). 5DWLRQDO &RQVHQVXV LQ 6FLHQFH DQG 6RFLHW\ $3KLORVRSKLFDODQG0DWKHPDWLFDO6WXG\'5HLGHO3XEOLVKLQJ&RPSDQ\'RUGUHFKW [11] Newman, M.E.J. (2010). 1HWZRUNV$Q,QWURGXFWLRQ. Oxford University Press, Oxford. [12] Penrose, L.S. (1952). 2QWKH2EMHFWLYH6WXG\RI&URZG%HKDYLRXU. H.K. Lewis & Co. Ltd., London. >@5HPDN5 .DQQGLH9RONVZLUWVFKDIWVOHKUHHLQHH[DNWH:LVVHQVFKDIWZHUGHQ" -DKUEFKHUIU1DWLRQDO|NRQRPLHXQG6WDWLVWLNSS± [14] Sraěa, P. (1960). Production of Commodities by Means of Commodities. Prelude to a &ULWLTXHRI(FRQRPLF7KHRU\Cambridge University Press, Cambridge. [15] Steedman, I. (1977). 0DU[DIWHU6UDěa. NLB, London. >@9LQW-0HWFDOIH-6.XU]+'6DOYDGRUL1DQG6DPXHOVRQ3$HGV   (FRQRPLF 7KHRU\ DQG (FRQRPLF 7KRXJKW (VVD\V LQ KRQRXU RI ,DQ 6WHHGPDQ Routledge, London.

13 A contribution to Weber’s theory of modern capitalism Amortization according to Sraffa as a rational substitution of missing markets

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Bertram Schefold Modern and other forms of capitalism Man has been aware of social and economic evolution for thousands of years. There are myths about the origin of man in probably all cultures, and they always involve comparisons between early and simple states of society and later, more developed ones. The philosophers of antiquity tried to rationalize this oral tradition; they discussed, for instance, whether the inventions were rather the result of accumulated practical experience1 or of rational philosophy, as Panaitios had asserted. Old is also the idea of political and social change. The Greeks distinguished different forms of government, and discussed the evolution from tribal rulers to kings of cities and realms, later to aristocratic rule, tyranny and eventually to democracy, with democracy itself undergoing well-observed institutional changes. Political change was associated with changes of the social forms, the emergence of new elites and the adoption of constitutions which legitimized political forms and social divisions. However, nobody before modern times seems to have associated this technical, political and social evolution with a corresponding change of economic forms, although the description of history always also involved the description of the change of some economic institutions, for instance concerning the introduction of coins, the policy of the monetary DXWKRULW\¿VFDODUUDQJHPHQWVUHGLVWULEXWLRQDQGWKHOLNH7KHUHZDVWKXVSDUWLDO awareness of what we mean when we say that the economic system was WUDQVIRUPHGEXWWKHFKDQJHRIWKHHFRQRPLFV\VWHPFRXOGEHLGHQWL¿HGRQO\ZKHQ the logic of the interdependence of a given economic system was understood, and this began only with classical thought. With the classical school, culminating in Marx, the idea of a ‘material’ (meaning an economic) determination of social and political life began to take hold, and with it the view of progress as a process of linear evolution. The older classical authors thought that free national markets and international free trade would ultimately prevail and displace the perceived arbitrariness of feudal institutions. Marx thought that competition would become imperfect and necessitate (and facilitate!) the transition to a socialization of property and production, but little room was left in this view for the roles of cultural forces and free and conscious choice in the shaping of development. Max Weber paradoxically accepted that modern capitalism represented a fateful outcome. The modern individual was born into a society in which it was the

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Bertram Schefold

inevitable task of most to lead disciplined professional lives to make a living and WR¿WLQWRDUDWLRQDOO\GHWHUPLQHGVWUXFWXUHQRWRQO\LQWKHVSKHUHRISURGXFWLRQ but also consumption. The paradox was that this rationality, which had emerged together with modern technology, modern law and modern attitudes to life, was not wholly derived from material factors but had been fostered by a religious movement, Calvinism. Those who had chosen to become Calvinists for religious conviction thereby helped to introduce habits which favoured a steady accumulation through frugality and saving and through regular production. A religious movement thus paved the way for secular change, and a spiritual departure had lasting material consequences. The Weber thesis has remained controversial to this day, but many objections have originated from misunderstandings. These began with the very concept of capitalism. ‘Modern’ capitalism meant the constellation which led to the emergence of industrial capitalism of the nineteenth century; it had spread in a fairly homogeneous form in western and central Europe. Weber accepted, indeed emphasized that various other forms of capitalism had existed in earlier periods and in other parts of the world – merchant capitalism had been present for thousands of years in many cultures. Especially Italian authors2 have argued for a long time that capitalism originated in Italy. Fanfani’s arguments are primarily those of an economic historian.3 Monumental is the work of Nuccio as a historian of economic thought;4 he believed to have shown that the capitalist spirit emerged among Italian catholic authors in the high Middle Ages. But the authors are at cross-purposes with Max Weber, since they do not take up his distinction between different forms of capitalism. He thought that modern capitalism had a number of VSHFL¿FFKDUDFWHULVWLFVHDFKZLWKLWVKLVWRU\VXFKDVWKHVHSDUDWLRQRIKRXVHKROG DQG ¿UP ZLWK WKHLU GLIIHUHQW UDWLRQDOLWLHV RI XWLOLW\ PD[LPL]DWLRQ DQG SUR¿W maximization – without the separation, the concerns of the family as the nucleus of the household would eventually endanger the working of the household as a ¿UP7KHFOHDULQVWLWXWLRQDOVHSDUDWLRQRIERWKUHTXLUHGDUDWLRQDOOHJDOIUDPHZRUN ZKLFKEHJDQWRHPHUJHEXWZKLFKZDVQRWVXI¿FLHQWO\GHYHORSHGLQWKH,WDOLDQ city-states of the Renaissance. This rationality was, according to Weber, not SUHVHQWLQVXI¿FLHQWO\GHYHORSHGIRUPLQPHUFDQWLOLVPHLWKHU,QKLVOHFWXUHVRQ HFRQRPLFKLVWRU\:HEHUDI¿UPVWKDWPHUFDQWLOLVPDQGWKHHDUO\IRUPVRIPRGHUQ capitalism have separate origins.5:HFDQQRWORFDWHWKHPKHUH6XI¿FHLWWRVD\ that, always according to Weber, industrial capitalism follows from modern rational capitalism, not the other way round. Modern capitalism is also characterized by free wage labour, for a rational allocation of labour is only possible on the basis of the wage contract. The different characteristics of modern capitalism are not reduced by Weber to one common cause; the multiplicity of causes means that the emergence of modern capitalism is to a certain extent to be regarded as an accident. If the rivalry of European nations had been subdued by a despotic force, the freedom of enterprises might have been suffocated. It is not easy to integrate Weber’s theory of modern capitalism with modern economic theory. Weber’s ideal types are similar to economic models in that they FRQVWUXFW DQ DUWL¿FLDO UHDOLW\ RIWHQ FRQWUDIDFWXDO LQ RUGHUWR KDYH D IUDPH RI

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reference which helps to order our observations of historical reality. Weber’s cautiousness and conscientiousness as a researcher induced him to use his ideal types mainly as negative characteristics. For instance, the capitalism of antiquity LVQRWPRGHUQDPRQJRWKHUWKLQJVEHFDXVHRIWKHLPSRUWDQFHRISROLWLFDOLQÀXHQFHV adventure capitalism is not modern because the conquerors, although capitalists LQVSLULWLQWKDWWKH\FHUWDLQO\ZHUHJUHHG\ZRXOGQRWEXLOGXSPRGHUQ¿UPVZKLFK accumulate steadily. Instead, they would retire to castles and get assimilated with the nobility as soon as they could. But if Weber shied away from a positive characterization of modern capitalism, he at the same time presented a wealth of hypotheses about it; extreme complexity and lack of closure go hand in hand in his approach. The same may be said of our actual economic theory, however. The multiplicity of models may be used to choose characterizations of different economic phases. The capitalism of the time of reconstruction after the Second World War with its high rates of industrial growth and its moderate growth cycles without spectacular ¿QDQFLDOFULVHVLVUDWKHUGLIIHUHQWIURPWKHSUHVHQWHUDRIFDSLWDOLVPLQZKLFKWKH VHUYLFHVHFWRUKDVEHFRPHWKHODUJHVWDQGLQZKLFK¿QDQFLDOPDUNHWVKDYHJURZQ tremendously, not least because the family and direct redistribution of the state have been replaced by pension and insurance schemes which compel most to VDIHJXDUGWKHLUOLIHDIWHUUHWLUHPHQWWKURXJKWKHEXLOGXSRI¿QDQFLDOIRUWXQHV7KH old project of the historical and institutional schools in England, Germany, the United States and elsewhere to construct different theories for different economic stages, for a while present as the project to construct different theories for ‘market’ and for ‘socialist’ economies, recurs today as the project to combine and recombine different economic models to explain or at least illustrate the economic functioning of different economic formations at different times and in different places in world history, but in particular in order to identify varieties of modern capitalism. The modern capitalism of the nineteenth century, which Weber had in view, was essentially one, but, for him, it had many roots. I here want to present an example of how Weber’s analysis may not just be repeated, but extended. It is based on my reading of Max Weber and on own older ZRUNRQ6UDIIDDQGWKHWKHRU\RI¿[HGFDSLWDO6 It is well known that Sraffa’s theory contains a model of the pricing of machines in the long period which implies a formula for depreciation and amortization. It is also well known that Max Weber and Sombart regarded book-keeping as one of the aspects of the rationalization of modern capitalism – the adventurer did not lose time with calculating the depreciation of his ships, nor did the Roman aristocrat have a clear LGHDRIWKHSUR¿WDELOLW\RIKLVHVWDWHV%XWPRGHUQPDQDJHUVW\SLFDOO\GRQRWWKURZ WKHPVHOYHVLQWRDGYHQWXUHVWKH\FDOFXODWH¿UVW±HYHQLIQRWDOZD\VFRUUHFWO\ 0RUHVSHFL¿FDOO\,ZDQWWRVKRZIROORZLQJ0D[:HEHUKRZWKHLQVWLWXWLRQV of slavery rendered rational capitalism in antiquity impossible and would have done so even if the slave owner had had a modern rationality. The ‘subjective’ factor (pre-modern rationality) and the ‘objective’ factor (the working of the institution ‘slavery’) each were an obstacle to the development in the direction of modern capitalism; Weber shows this for the objective factor, the case being

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obvious for the subjective one. The obstacles reinforced each other, but Weber was concerned to show that the ‘objective’ one would have been effective even in the absence of the ‘subjective’ one. Curiously, this will also shed light on the function of book-keeping and the calculation of amortization under rational capitalism, in that, again following Max Weber, these calculations can be interpreted as a necessary substitute for inevitably PLVVLQJPDUNHWV7KHJURZWKRIUDWLRQDOLW\LVUHÀHFWHGLQDQLQFUHDVLQJGHJUHHRI sophistication in the calculations. Hence, as an add-on, I shall discuss a contribution on amortisation by Chassot de Florencourt, who wrote an ingenious example of what Sraffa would call ‘handbooks of commercial arithmetic’ in the late eighteenth century. I hope that this is the sort of cocktail that will please my friend Heinz Kurz.

The challenge: capitalism in antiquity Max Weber has remained topical as an interpreter of the economic history of antiquity; the rediscovery of Weber as an ancient historian is primarily due to Moses I. Finley.7 Weber’s interest for slavery was not only a matter of economic history; it was also motivated by his travel to the United States in 1904. He had observed the effects of the civil war and felt that the discrimination of the blacks was scandalous.8 He extrapolated from Roman history with the transition from slavery to the colonate and dared to predict that the American South would have developed towards a system of land-lease to the slaves without the Civil War, but, since this could not be avoided, American society was divided into two castes. He returned to the problem of slavery often, not only in his writings on ancient history, but also in Economy and Society (Wirtschaft und Gesellschaft),9 as he had done already in his lectures on theoretical economics (Allgemeine [‘theoretische’] Nationalökonomie).10 There are several reasons for the recurrence of this problematic. The ancient sources on it are relatively numerous and inform us in different perspectives: Roman agrarian writers on the use of the slaves, the forensic orators on the differences in their juridical status, the historians on the LQÀXHQFHRIWKHLQVWLWXWLRQRIVODYHU\RQSROLWLFVDQGWKHSKLORVRSKHUVRQWKHPRUDO doubts, which were never suppressed completely in spite of the permanence of the LQVWLWXWLRQ0RUHVSHFL¿FDOO\:HEHUZDVLQWHUHVWHGLQDFRPSDULVRQEHWZHHQWKH modern capitalist enterprise and larger estates in antiquity, such as the Roman latifundia or the workshops of the slaves, the Greek ergasteria. Here, differences between the capitalism of antiquity and modern capitalism were to be demonstrated; it is astonishing how Weber was able to combine imagination and sober assessment, when he represented a situation. The following arguments are preeminent in his ‘Agrarverhältnisse im Altertum’11: 1 The reproduction of slavery is not regular, since it depends on war and conquest. Slave prices therefore vary and the rational calculation of their use is rendered GLI¿FXOW

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2 :HEHUDGGXFHVHYLGHQFHDFFRUGLQJWRZKLFKWKHUHZDVVRPHSUR¿WDEOHVODYH EUHHGLQJ+RZHYHUWKHPDOHVODYHVKDGWREHFRQ¿QHGWREDUUDFNVIRUWKHLU extensive use in agriculture. The female slaves were predominantly used in the households. Hence the cost of the breeding of slaves became too high, if it was not possible to use the women in the production of textiles. 3 The main use of slaves was in plantations, in nautical applications, the mines and in tax collection, and women were not useful in any of these applications.  :HEHUWKRXJKWWKDWLWZDVGLI¿FXOWEXWQHFHVVDU\WRPRWLYDWHWKHVODYHVH[FHSW for those who worked under the whip in plantations. He otherwise had little to say about motivation. 5 Hence he thought generally that the work of slaves remained rough and could QRWEHUH¿QHG 6 Slavery required more capital in comparison with the self-employed artisan or farmer and in comparison with wage labour, for the slaves had to be bought and needed to be maintained, even if they could temporarily not be used. This problem was particularly obvious regarding the equal distribution of work during the year in agriculture. The agricultural writers recommend to hire wage labourers in the season in which more labour is required. 7 The risks of illness and mortality are with the owner of the slave. They reinforce the uncertainty concerning the value of the slaves to be used. The agricultural writers therefore recommend to use wage labourers for dangerous work. 8 It is different with slaves in the households; then one renounces to the maximizaWLRQRISUR¿WVDQGVODYHVDQGWKHLUIDPLOLHVDUHPHPEHUVRIWKHKRXVHKROG 9 Weber considers the slave who works independently with his family as a source of rents, not as labour power.12 All this implies a microeconomic consequence: if a workshop, an ergasterium, is to function with slaves bought in the market, the slaves have to be cheap, and even WKHQDPRQRSRO\SULFHRIWKHSURGXFWPD\EHQHFHVVDU\IRUSUR¿WDELOLW\VLQFHWKH arguments which we have mentioned elevate the cost of production in comparison with those of independent artisans. The cost of the use of labour and capital can be calculated in the modern factory, by contrast. The cost of wage labour is regular and is associated with incentives, the cost of capital can be estimated by means of book-keeping; it permits the calculation of depreciation even if the factory and its machinery remain in function for a number of years without a change of ownership. Weber seems not to have made much of long-run prices in his teaching.13 He regarded it as characteristic for industrial capitalism that there are durable capital goods which can be used rationally on condition that prices are calculated by means of book-keeping which – as he would have said – express the chance that the equipment can be sold at these prices, if a sudden need arises, or that they represent a security of the corresponding value, if it is necessary to obtain a loan. For a Marshallian or for a classical economist, such a price is typically the normal SULFHRUWKHSULFHLQWKHORQJUXQ7KHSULFHVRIVODYHVE\FRQWUDVWÀXFWXDWHLQ

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FRQVHTXHQFHRIH[RJHQRXVLQÀXHQFHVZDUV DQGPDODGLHVDQGWKHSURGXFWLRQRI labour was less regular. Capital in the form of slaves was perhaps bought and sold even less frequently than capital in the form of machines in industrial capitalism, and the methods of book-keeping were unknown in antiquity. Hence, according to Weber, one did not really know what a slave was worth in the long run. Rational calculation in the workshops of antiquity could not develop according WR:HEHUEHFDXVHWKHVHSDUDWLRQRIKRXVHKROGDQG¿UPZDVODFNLQJ+HVHHPVWR assume constant returns to scale for the production in the ergasterium: if more slaves are employed, as in the manufacture of shields mentioned in Lysias,14 turnover rises, but no cheapening of production results, since each shield is SURGXFHGE\WKHVDPHKDQGLFUDIWPHWKRGDUH¿QHGGLYLVLRQRIODERXUGRHVQRW seem to take place. If slaves cannot be induced to work as well as a free artisan, the worse productivity must be compensated by cheaper subsistence. Weber WKRXJKWWKDWWKLVZDVQRWVXI¿FLHQWWRHQVXUHWKHFRPSHWLWLYHQHVVRIWKHHUJDVWHULXP hence he spoke of the necessity of a monopoly in order to ensure its economic survival. Although it is known that free workers and slaves occasionally worked side by side (see the inscription concerning the construction of the Erechtheion15), he did not assume real cooperation between them. He did not suppose that they used two different techniques either; he thought that the techniques were simple, craftsmanlike, without much use of capital. Weber concluded that manufacturing in antiquity was unstructured (‘amorph’). Weber’s theses are controversial. They do not explain the high quality of handicraft in antiquity, which remained a model for later generations and which has been admired for more than two thousand years. But we shall not question Weber’s assumptions further; instead we proceed to a representation of some aspects of his ideas in the form of a model.

A model of the valuation of machines and of slaves by means of the market, potentially aided by rational calculation The capital was mainly embodied in the slaves themselves. We compare the slave to a machine. To this end, we introduce a model of capitalist formation of prices. Fixed capital is to be represented for an analysis in the long run as a joint product according to Sraffa.16 The machine appears as a joint product in successive periods of production, but after each period, the machine is older by one period. In general, it will be used up more accordingly, and eventually it will not be worthwhile to XVHLWIXUWKHUEHFDXVHRIGLPLQLVKHGHI¿FLHQF\HYHQEHIRUHLWLVSK\VLFDOO\EURNHQ This joint production approach corresponds to Weber’s capital accounting (‘Kapitalsrechnung’), based on the comparison of the value of capital in the beginning and at the end of each economic period. The simplest possible example is as follows: corn is produced by means of corn, labour and a tractor, which can last for several years and is gradually used up, till it can not be used any more after T years. The tractor is produced by means of corn and labour. Wage labour is paid according to a wage rate w; corn capital is advanced. The price of the product must cover the cost of production, therefore

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of wage labour and of the use of capital, meaning the use of the machine or the VODYHLQFOXGLQJDQRUPDOSUR¿WSURSRUWLRQDOWRWKHDGYDQFHRIFDSLWDODFFRUGLQJ WRWKHUDWHRISUR¿Wr. Hence we obtain the following T + 1 equations: (1 + r )k0 p + wl0 = p0 ,

(13.0)

(1 + r )(k1 p + p0 ) + wl1 = p + p1,

(13.1)

(1 + r )(k2 p + p2 ) + wl2 = p,

(13.2)

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... (1 + r )(k2 p + p2 −1 ) + wlT = p .

(13.T)

Here kt , lt (t 1, . . ., T ) mean the amounts of corn and labour, which are used during T years in year t, aided by a tractor of price pt–1, in order to produce one unit of corn at price p and a tractor (which is one year older) at price pt. The tractor is broken at the end of year T; hence there is no price pT . By means of k0 and l0 a new tractor at price p0 LVSURGXFHGLQWKHSURFHVV¿UVWPHQWLRQHG  In order to calculate prices in this system, it is necessary, according to Sraffa,17 to multiply the equations (T), (T – 1), . . . , (1) successively by the factors 1, (1 + r), . . . , (1 + r)T – 1 and to add them all. As a result of this summation, the prices of the tractors of each age 1, . . ., T –1 are eliminated, as one can see easily; there remains DFRPELQHGHTXDWLRQLQZKLFKWKHFRHI¿FLHQWVRIFRUQDQGRIODERXULQSXWV kt , lt for t = 1, . . ., T and corn at its price p appear in a complicated polynomial in r: T

(1  r )T p0  ¦ (1  r )T  t >(1  r )kt p  wlt @ t 1

T

¦ (1  r )

T t

p.

(IG)

t 1

This so-called integrated equation (IG), in combination with (13.0), represents a reduced system, from which old machines have been eliminated.18 There remains the new machine, at price p0 . This is to be determined, together with the corn price p. Hence we have two equations, (13.0) and (IG), with four unknowns: two SULFHVWKHZDJHUDWHDQGWKHUDWHRISUR¿W$QXPpUDLUHKDVWREHGH¿QHGVLQFH the equations are homogeneous in prices and the wage rate. Once this is done, the wage rate and the two prices p0 and p can be calculated as functions of the rate of SUR¿WWKHSRO\QRPLDO,* OHDGVWRFRPSOLFDWHGHIIHFWVRISULFHVZLWKFKDQJHVRI distribution. It turns out that a higher real wage (a wage expressed in terms of a QXPpUDLUH LPSOLHVDORZHUUDWHRISUR¿WDQGYLFHYHUVD7KLVLVWKHVRFDOOHGZDJH FXUYH7KHZDJHLV]HURDWD¿QLWHPD[LPXPRIWKHUDWHRISUR¿WR. Most economists thought up to the middle of the nineteenth century that wages were determined by subsistence requirements of workers. Hence the real wage would be given, and WKHUDWHRISUR¿WZRXOGIROORZIURPWKHZDJHFXUYHDFFRUGLQJWRWKHPRGHO0RUH PRGHUQWKHRULHVRIGLVWULEXWLRQH[SODLQWKHUDWHRISUR¿WDFFRUGLQJWRWKHFRQGLWLRQV of accumulation or as determined by money rates of interest. Then the wage rate

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IROORZVIURPWKHZDJHFXUYHJLYHQWKHUDWHRISUR¿W:HQHHGQRWHQWHULQWRWKHVH extensions here, nor is it necessary to show how the model can be generalized to introduce machines not only in one sector but in several sectors of a multi-sectoral HFRQRP\7KHVLPSOHPRGHOVXI¿FHVIRURXUSXUSRVHV2QFHWKHSULFHVRI p0 and p have been determined, using (13.0) and (IG), the other prices can be calculated, using the remaining equations 1,. . ., (T  1) . 7KH HI¿FLHQF\ RI WKH PDFKLQH FDQ ZRUVHQ GXULQJ LWV OLIHWLPH OLNH PRWRU vehicles which require increasing maintenance work. If this tendency holds for all periods, it can be expressed by the assumption k1 d k2 d . . . d kT , l1 d l2 d . . . d lT . It is also conceivable that a machine works better, the longer it is used; then the inequalities are reversed k1 t k2 t . . . t kT , l1 t l2 t . . . t lT . One expects a superposition of both tendencies in reality. A machine can be XQGHUFRQVWUXFWLRQVRWKDWLWVSURGXFWLRQLVORZRU]HURLQWKHEHJLQQLQJ$SUR¿OH RIHI¿FLHQF\¿UVWULVLQJWKHQIDOOLQJLVWREHH[SHFWHGHVSHFLDOO\ZLWKODERXULQJ animals or in the historical case of slavery, which is here relevant. Prices then are not necessarily positive. It can be shown that corn and the new machine will receive positive prices in system (0  T ) in any case, if a surplus is produced, but DSRVLWLYHVXUSOXVLVQRWVXI¿FLHQWLQV\VWHP(0  T ) for positive prices of any old PDFKLQHIRULILWVHI¿FLHQF\IDOOVEHFDXVHPDLQWHQDQFHZRUNLQFUHDVHVLWFDQEH SUR¿WDEOHWRUHSODFHWKHROGPDFKLQHE\DQHZRQHEHIRUHWKHROGPDFKLQHLV broken, hence to truncate the lifetime of the machine. A machine which has EHFRPHLQHI¿FLHQWEHFDXVHRIWKHLQFUHDVHGFRVWRIPDLQWHQDQFHGRHVQRWDGG value to the product, but implies a deduction. In order to see this more precisely, we write the amortization of the machine by rearranging equation (13.t) in the following form: rpt 1  ( pt 1  pt )

p  (1  r )kt p  wlt .

(A)

Here we have on the right side the difference between the value of the produce, corn, and the direct costs of its production, that is corn advanced and the wage for labour. On the left-hand side we have the amortization, which results from the ¿QDQFLDOFKDUJHrpt–1, necessary because a machine of age t – 1 has to be advanced, and the change in the value of the machine, therefore the depreciation pt 1  pt . This depreciation usually is positive, because the machine falls in value as its HI¿FLHQF\GLPLQLVKHV7KHGHSUHFLDWLRQPD\EHFRPHVRODUJHWKDWDIXUWKHUXVHRI the machine inhibits production. This can be the case even if prices are equal labour values at r = 0; one then obtains negative values which indicate that the process of the use of the machine should be truncated, because it wastes labour in FRPSDULVRQZLWKWKHHI¿FLHQF\LQWKHXVHRIQHZPDFKLQHV0RUHJHQHUDOO\DWD JLYHQSRVLWLYHUDWHRISUR¿WDQGZLWKDGLPLQLVKLQJHI¿FLHQF\RIWKHPDFKLQHWKH diminution of the value will continue until the price of the aging machine has fallen below zero. Then it should not be used any more, but be replaced by an increased use of new machines. ,W LV RIWHQ VXI¿FLHQW WR PDNH WKH VLPSOH DVVXPSWLRQ WKDW WKH PDFKLQH KDV FRQVWDQWHI¿FLHQF\ZLWKk1 k2 . . . kT k , l1 l2 . . . lT l .

Weber’s theory of modern capitalism If then one has in addition r to that of a new machine is19

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pt p0

1

187

0 , the price of a t year old machine in relation

t . T

(L)

According to (L), depreciation is linear, if r = 0 and if prices are equal to labour values (which are obtained at r  ,IWKHUDWHRISUR¿WLVSRVLWLYHE\FRQWUDVW one obtains a progressive depreciation, for a reason which is intuitive: the ¿QDQFLDOFKDUJHLVUHODWLYHO\KLJKZLWKQHZPDFKLQHVEHFDXVHWKHPDFKLQHVDUH VWLOOYHU\YDOXDEOHVRWKDWGHSUHFLDWLRQPXVWEHORZVLQFHWKHVXPRI¿QDQFLDO FKDUJHDQGGHSUHFLDWLRQWKHUHIRUHWKHDPRUWL]DWLRQLVE\GH¿QLWLRQLQGHSHQGHQW RIDJHZLWKPDFKLQHVRIFRQVWDQWHI¿FLHQF\7KHULJKWKDQGVLGHRI$ LVE\ GH¿QLWLRQFRQVWDQWZLWKFRQVWDQWHI¿FLHQF\ 7KHORZHULQJRIWKHSULFHRIWKH machine pt–1LPSOLHVDORZHULQJRIWKH¿QDQFLDOFKDUJH+HQFHWKHGHSUHFLDWLRQ in (A) pt–1 – pt must increase with t2QH¿QGVLQPRGHUQHFRQRPLFSUDFWLFHWKDW linear and degressive depreciation prevail by comparison with progressive GHSUHFLDWLRQWKLVPD\EHMXVWL¿HGLQDZRUOGLQZKLFKWHFKQLFDOSURJUHVVSOD\V an important role.

Depreciation prior to Sraffa 7KH WKHRU\ RI DPRUWL]DWLRQ RI PDFKLQHV RI FKDQJLQJ HI¿FLHQF\ ZDV QRW known at the time of Max Weber. Marx would calculate using (L). However, handbooks of commercial arithmetic had already for at least a century contained formulas for the amortization of loans which were equivalent to the formula LPSOLHGKHUHIRUWKHDPRUWL]DWLRQRIDPDFKLQHRIFRQVWDQWHI¿FLHQF\7KHDQDORJ\ between the machine and the loan is clear from the following consideration: if someone buys equipment at price p0 for an amount of money X and uses that HTXLSPHQWDVVKRZQLQWKHHTXDWLRQ 7 ZLWKFRQVWDQWHI¿FLHQF\ he will in each period obtain a constant revenue, rendered by the right-hand side of (A); we assume that this revenue, measured in money, is equal to x. Revenue x is, as we can see from the left-hand side of (A), at the same time the amortization of the machine. We assume that each of these revenues, successively deposited in a bank, yields interest; the total of these revenues in period T is equal to T x + (1 + r ) x + . . . + (1 + r )T −1 x = ⎡⎣(1 + r ) − 1⎤⎦ (1/r ) x. If X is instead deposited in a bank from the beginning and yields interest at rate r, one obtains after T years (1 + r)T X. Both amounts must be equal, hence the yearly revenue or the amortization in money is x

r (1  r )T X (1  r )T  1

which tends to X/T for rĺWKHUHIRUHOLQHDUGHSUHFLDWLRQ

(F),

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The formula (F) had been known as an expression for the capital value of a series of yearly payments x at rate of interest r; but the connection with depreciation and amortization of equipment seems not to have been understood. Passow, in his textbook of 1912, was content with the vague statement: ‘Bei der Bemessung der Abschreibungssätze ist in erster Linie der Wunsch maßgebend, die insgesamt durch die Entwertung des betreffenden Gegenstandes eintretenden Kosten möglichst gleichmäßig auf die gesamte Gebrauchsdauer zu verteilen.’20 7KHGLVFUHSDQF\EHWZHHQWKHXVHRIIRUPXODVIRUDPRUWL]DWLRQLQWKH¿QDQFLDO sphere and in industry is curious, since the basis of the formula, the mathematical theory of compound interests, had been developed earlier and been applied in ¿QDQFLDOFDOFXODWLRQVZKLFKDOVRLQYROYHGUHDODVVHWV)RULQVWDQFHIRUPXOD) LV found in Chassot de Florencourt in 1781.21 Carl Chassot de Florencourt, of French descent, was employed in the German mining industry (dukedom of Braunschweig). His book contains numerous applications of mathematics to legal and economic problems, including population, probability theory and insurance, and it is interesting because it links problems typical for mercantilism (which in turn had medieval roots) with the then modern approach. An example is the following text: ‘Antichretischer Vertrag. . . . Er ist dann vorhanden, wenn der Schuldner dem Gläubiger, eine nutzbare Sache, auf eine gewisse Zeit übergiebt, sie gehörig zu nutzen, damit der Gläubiger, wegen des Verlustes am Nutzen seines Capitals, den der Schuldner zieht, entschädigt werde’ (p. 35). (English translation: Antichretic contract. . . . An antichretic contract exists, if the borrower delivers a useful thing to the lender for a certain time, so that the lender may use it to compensate the loss of the utility of his capital, which the borrower gets in his place.’) This was much discussed by scholastic and cameralist authors; examples are found, e.g., in Kaspar Klock.22 Today, the antichresis would be called a ‘swap’. The antichresis was discussed often in late scholasticism as one of the constructs in order to bypass the prohibition of usury. In fact, since the swap can take place without an explicit and open calculation of interest-taking on either side, usury could here be concealed perfectly. Florencourt, by contrast, being a self-conscious pioneer of enlightenment, calculates openly with interest and compound interest. He assumes that the yearly revenue derived from the useful thing in the hand of the lender is constant; the loss of interest to be accepted by the lender, because his capital X is in the hands of the borrower, will then be compensated exactly over T years at rate of interest r, if (F) holds, and this he proves, with some interesting generalizations (p. 36). Sraffa, when discussing (F), refers to the ‘handbooks of commercial arithmetic’ DVWH[WVLQZKLFKDPRUWL]DWLRQLVH[SODLQHGLQWKHFDVHRIFRQVWDQWHI¿FLHQF\DQG ZHPD\VXSSRVHXQGHUWKHDVVXPSWLRQRI¿QDQFLDOGHDOLQJV$SSDUHQWO\6UDIID primarily had two books in mind: the Doctrine of Interest and Annuities by Francis Baily (1808) and Arithmetical Books by de Morgan (1847).23 Baily’s book is much less varied and less sophisticated in the applications than the earlier Chassot GH)ORUHQFRXUW0DQ\QXPHULFDOH[DPSOHVUHQGHULWPRUHGLI¿FXOWWRLGHQWLI\WKH underlying theoretical ideas. The other book is a rich collection of references to books on commercial arithmetic, published in Great Britain and on the continent,

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but Chassot de Florencourt is missing. He deserves to be studied by historians of economic thought. His book should perhaps be described as a work on an intermediate level between the great works on calculus, probability theory and Newtonian physics which were published in the late eighteenth century by authors such as the mathematicians of the Bernoulli family and applied work on commercial arithmetic. However this may be, if we want to use Weber’s notions, we can state that the rationality of the capitalist process had not yet been developed to the point of a rational calculation of amortization and depreciation in a system of reproduction with a given technique such as Sraffa’s at the turn from the nineteenth to the WZHQWLHWKFHQWXU\QRWWRVSHDNRIIXUWKHUFRPSOLFDWLRQVVXFKDVWKHLQÀXHQFHRI technical progress on depreciation, although (F) had been known for more than one hundred years. The calculation of the value of capital or investment was of course far less developed in antiquity than in the ninteenth century, but not totally absent, as we can see from Columella, who made a rudimentary attempt to consider the effect of the rate of interest on the value of an investment (he compared the proceeds from investing in the build-up of a vineyard with a monetary investment) – an attempt which may be regarded as a beginning for a modern history of business administration.24 Schneider shows that formula (F) had already been used (in an older notation) in 1822 by von Oeynhausen in an industrial application, based on a consistent observation of Leibniz’s method to calculate compound interest.25 An increasing rationalization in the consideration of depreciation was furthered by the experiences of railway companies with the wear and tear of rails in the course of the nineteenth century;26 Marx knew this and discussed the problem at some length in vol. II of Das Kapital,27 with interesting insights on WKHYDULDELOLW\RIZHDUDQGWHDUEXWKHPDGHQRVLJQL¿FDQWDQDO\WLFDOSURJUHVVLQ the matter.

Depreciation: two formally equivalent approaches for two different institutional settings The theory of normal prices yields a norm of a kind. Imagine a world in which ¿QDO SURGXFWV ± WKH QHZ PDFKLQH DQG FRUQ ± DUH H[FKDQJHG UHJXODUO\ ROG machines, however, only spasmodically, because consumption takes place on a daily basis and new enterprises are built up often, while these and their equipment, once they have been installed, change hands only rarely. Markets for old machines are almost absent from this world. It follows that a formation of prices, compatible ZLWKXQLIRUPUDWHVRIZDJHVDQGSUR¿WVFDQWDNHSODFHRQO\LIWKHSULFHVRI¿QDO products result from a combination of the equations (13.0) and (IG), and an DQDO\VLVEDVHGRQ$ PXVWEHXVHG±LIHI¿FLHQF\LVFRQVWDQWLQDFRPELQDWLRQ with (F) – in order to calculate the book values of old machines. In practice, they are a substitute for prices, to be formed in markets which are not totally missing, but little used. The book values offer the ‘chance’, as Weber would have put it, to represent the ‘correct’ prices of old machines. Since this calculation, at least from a historical point of view, is not simple, especially because of the complicated

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form of (IG), we thus obtain a model for the increased rationality of modern capitalism. Our model also shows that this complicated calculation of depreciation is not necessary, if there is a regularly functioning and competitive market for old PDFKLQHV7KHWKHRUHWLFDOO\FRUUHFWSULFHVRI¿QLVKHGJRRGVREWDLQHLWKHUIURPWKH complete system of equations (13.0) – (13.T) (if the competitive process ensures WKHXQLIRUPLW\RIWKHUDWHRISUR¿W RUIURPHTXDWLRQ,* 7KHIRUPHUHTXDWLRQV only presuppose that prices are formed competitively in an anonymous market. The second procedure, the use of equation (IG), presupposes that all purchases and sales are discounted correctly during the lifetime of the machine. Strictly speaking, one only needs the price of the new machine for this; the prices of old machines, amortization and depreciation need not be known. However, in order to determine the optimal lifetime of the machine, it is necessary to calculate the formula (IG) for all technically possible durations of the machine. The HFRQRPLFDOO\HI¿FLHQWGXUDWLRQWKHQLVWKHRQHZKLFKSHUPLWVWRVHOOWKHSURGXFW DWWKHJLYHQUDWHRISUR¿WRULQWHUHVWPRVWFKHDSO\7KHSULFHVRIROGPDFKLQHVZLOO also be positive, given this particular duration. Functioning markets for old PDFKLQHVWKXVFDQUHSODFHWKHGLI¿FXOWERRNNHHSLQJDFFRUGLQJWR,* DQG$  But if the book-keeping is as inexact as our quotation from Passow (1912) suggests and if there is little trade in old equipment, prices cannot be correct and there is little hope that market prices will approach long-run prices.

Rationality and institutions Hence we arrive at last at the question which is crucial for the understanding of Weber’s theory of production under slavery: in how far is the difference of the form of production to be reduced to the lack of the formation of rationality in antiquity and in how far to other factors? In order to answer, we push Weber’s analysis further and analyse a hypothetical economy in which competitive capitalist slave owners operate latifundia with slaves and where there is also slave-breeding. We assume in this comparison that a slave takes the place of the machine. Apart from slaves there may be wage labour, as in the model in its present form (the wage labourers then are paid according to wage rate w), or one eliminates wage labour, which is formally done most simply by assuming w 7KHUDWHRISUR¿W WKHQLVDWLWVPD[LPXP:HVXSSRVHDVWKHW\SLFDOSUR¿OHIRUWKHHI¿FLHQF\RI slave labour, that the value of slaves will rise, at the given rate of interest, until they reach their maximum labour power – presumably between the age of twenty DQGWKLUW\DVIDUDVSK\VLFDOODERXULVFRQFHUQHG±ZKLOHHI¿FLHQF\ZLOOGLPLQLVK DIWHUZDUGV¿UVWVORZO\WKHQPRUHUDSLGO\,IWKHVODYHZDVWREHWUHDWHGH[DFWO\OLNH a machine, he would have to be taken out of the economic process as soon as the preservation of his labour power costs more at ruling prices than it delivers; a SHQVLRQZRXOGQRWH[LVWLIRQO\HFRQRPLFHI¿FLHQF\ZHUHWREHFRQVLGHUHG$Q extended trade of slaves of different ages, but otherwise equal capabilities, would WKHQGHWHUPLQHWKHLUYDOXHZKLFKZRXOG¿UVWLQFUHDVHWKHQGLPLQLVK DFFRUGLQJ

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WRWKHUHVXOWVRIWKHPRGHO7KHFKDUDFWHULVWLFSUR¿OHIRUWKHYDOXHRIVODYHVKDVLQ fact been observed in cliometric investigations of slave markets in the American southern states.28 The developed rationality of book-keeping would then not be necessary. In so far as prices would form according to the tendentially uniform rate RISUR¿WRQHZRXOGEHGHDOLQJZLWKDVLPSOHUIRUPRIFDSLWDOLVPLQFRPSHWLWLYH conditions. The lack of rational book-keeping thus is a lesser obstacle to capitalism based on slavery than Weber thought: if slaves at each age were homogeneous and if uniform prices of them resulted in competitive markets, they would be given by equations (13.0) – (13.T) and nobody needed to be aware of (IG). Marx might have said that these prices would be determined ‘behind the backs of the producers’. But Weber’s ideal type of the economy of antiquity assumes various LQHI¿FLHQFLHVKHXVHVDFKDUDFWHUL]DWLRQRIVODYHU\ZKLFKLVDSSOLFDEOHQRWRQO\ in the agricultural context, but also in ‘manufacturing’, with the peculiarities which have been mentioned, in particular the high need for capital and the necessity to maintain the slaves when they cannot be kept working, hence the use of slaves in manufacturing activities on the latifundia when it is not the time of WKHKDUYHVWZLWKWKHFRQVHTXHQWGLI¿FXOW\RIDODUJHGLVWDQFHWRWKHPDUNHWVZLWK WKHULVNVRIPRUWDOLW\ÀXFWXDWLRQVRISULFHVGXHWRZDUVZLWKFDOFXODWLRQUHQGHUHG GLI¿FXOW ZLWK HI¿FLHQW H[SORLWDWLRQ RQO\ RQ FRQGLWLRQ RI WKH XVH RI IRUFH DQG FRQ¿QHPHQWWRWKHEDUUDFNVZLWKDEDGVXFFHVVRIµEUHHGLQJ¶VRWKDWLWHYHQWXDOO\ seems quite natural that slavery evolves and develops into the colonate, according to Weber’s observations on the decline of (in particular Western) Rome in late antiquity. It follows that equations (13.1) – (13.T) cannot determine prices under the conditions which represent Weber’s view of the reality of slavery in antiquity. It is clear that, given the institutional obstacles mentioned by Weber, an extension of the rationality with a calculation of prices on the basis of (IG) would not lead to a reliable determination of normal prices either – they would come about only with the introduction of wage labour, where effort and recompense are FDOFXODEOH:HWKXV¿QGWKDWUDWLRQDOLW\SOD\VDVPDOOHUEXWWKHLQVWLWXWLRQDOJLYHQV no lesser, role in the explanation of the difference between the economic forms here compared than Weber thought. +LV DVVXPSWLRQV DERXW VODYHU\ DV DQ LQVWLWXWLRQ PXVW SHUKDSV EH TXDOL¿HG KRZHYHUIRUZH¿QGDFRQWUDVWZLWKPRGHUQUHVHDUFK5HFHQWHFRQRPLFKLVWRU\ GHQLHVWKDWVODYHU\LQWKH$PHULFDQVRXWKHUQVWDWHVZDVLQHI¿FLHQW,WFHUWDLQO\ZDV LPPRUDODQGDEMHFWEXWLWLVQRZDVVHUWHGQRWWRKDYHEHHQLQHI¿FLHQW,WRIWHQLV added that it does not save the human dignity of the slaves, if their work is FKDUDFWHUL]HGDVLQHI¿FLHQWWRFDOOWKHPOD]\DQGLQDSWLVVDLGWREHWKHMDUJRQRI the slaveholders. Wilfried Nippel believes that Weber is less cautious than Marx, regarding the comparison between Roman and American slavery, although both based their interpretations on the observation of the American conditions which today are interpreted in a different light.29 For Nippel joins the American UHVHDUFKHUVZKRGHQ\WKHWKHVLVRIWKHLQHI¿FLHQF\RIVODYHU\DQGVKRZWKDWD certain reproduction of slaves was possible, if the women were employed in cotton production. I am a historian of economic thought, not of economic and VRFLDOKLVWRU\,WKHUHIRUHGRQRWIHHOTXDOL¿HGWRMXGJHZKRZDVULJKWEXWLWVHHPV

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to me that slavery in antiquity was a highly differentiated phenomenon, probably much more so than early modern slavery. It makes a big difference whether one compares with the conditions in classical Greece or in imperial Rome. If slavery ZDVIDLUO\HI¿FLHQWWKH:HEHULDQTXHVWLRQZRXOGEHZK\WKLVZDVQRVXI¿FLHQW basis for the development of modern capitalist rationality. It did not develop in antiquity, but it did not originate in the ante-bellum South either, for it was introduced there from outside.

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The rationality of modern capitalism impossible in antiquity We recall that Weber wished to keep valuations out of science; what we have been considering are arguments that he would have regarded as objective. Because of the absence of frequent sales of slaves, because of their replacements via wars, because of the irregular character of their work in many applications so that their physical productivity remains uncertain, it is not possible to calculate their ‘prices’ correctly and modern book-keeping, even if it had existed, would have lacked the necessary foundations. Moreover, a sizeable number of slaves were integrated into the families after all, and here the intercourse between masters and slaves was more human – patriarchal slavery in the household is outside capitalist acquisition LQDQ\FDVH:HEHU¶VPDLQSRLQWLVWKDWWKHVHGLI¿FXOWLHVRIDVFHUWDLQLQJµFRUUHFW¶ prices would exist, even if the latifundia were administered with modern rationality or according to the rationality of an early modern plantation, because of differences in the institutions. The subsidiary point is that the institutional obstacles may have EHHQORZHUDWWLPHVEXWQHYHUORZHQRXJKIRUDVXI¿FLHQWOHQJWKRIWLPHWRSHUPLW the development of modern rationality. We now turn to the other side: modern capitalism. If an entrepreneur has to work with durable capital, as is the case during and after the Industrial Revolution, an at least approximately correct calculation of the value of capital becomes indispensable. Weber speaks of ‘Kapitalsrechnung’ and means not only the relationship between the values of old machines and depreciation, which we have considered, but also the valuation of investment, looking into the future.30 Where market prices have been stabilized in a modern economy to the extent that the market participants can reckon with (nearly) normal prices, the chance is offered to the industrial entrepreneur, following Schumpeter, to revolutionize the process of production by means of innovations and to obtain systematically and regularly, RQWKHEDVLVRIFDOFXODWLRQKLJKHUSUR¿WVWKDQFRPSHWLWRUVZLWKLPSURYHGPHWKRGV of production, until these have been imitated by others. Such chances existed to a much lesser extent prior to the formation of rational capitalism, but there were other possibilities – adventure capitalism for instance, as mentioned, or, in order WRUHPDLQZLWKWKH5RPDQVWKHSRVVLELOLW\WREHQH¿WIURPWKHUDSLGO\FKDQJLQJ conditions of war and peace in order to obtain contracts for armament and military supplies. Political daring, not sober calculation, possibly paired with an innovative idea, here led to riches. Hence it follows that Weber explains the capitalism of antiquity by means of a paradoxical procedure: he assumes a goal-oriented action, intended to be rational,

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on the part of the owners of the latifundia and shows that the action can in effect QRWEHUDWLRQDOEHFDXVHLWIRXQGHUVRQWKHURFNVRIWKHVSHFL¿FLQVWLWXWLRQVDQGWKH given technology. Eventually, the reader learns that the big fortunes were made by other ways. The acquisition by the exploitation of whole provinces is not accessible to the pure rational capitalist, for the conqueror needs other virtues. So far, the characteristics of the economy of antiquity are predominantly described in negative terms, and this is, as we saw in the beginning, characteristic of Weber. There was another rationality, not easily displaced by modern thinking. What was it? What were the Roman virtues and how were they connected with Roman empire-building and the Roman economy around the Mediterranean? Weber abstained from a description.

Weber and the theory of normal prices Our model suggests further conclusions regarding Weber’s theory. His explanation of slavery in antiquity is another instance of his peculiar approach, oriented towards a theory of action, to combine the Austrian theory of utility and classical views on distribution and accumulation (if one prefers: classical macroeconomics) in order to interpret this economic form in a manner which would hardly be possible on the basis of neoclassical theory. To begin with, a neoclassical would QRWVD\WKDWWKHÀXFWXDWLRQVRIWKHSULFHVRIDIDFWRUWKHVODYHV GXHWRZDUV prevented the formation of prices: market clearing prices could always be found in a sequence of temporary equilibria. Moreover, from the point of view of the Marshallian theory of normal prices, the ‘rational’ calculation would be regarded DVLQVLJQL¿FDQWVLQFH0DUVKDOOLDQHFRQRPLFVDVVXPHVFRPSHWLWLYHSULFHVVRWKDW the price always is given for the individual agent. Weber thought, in regard to the ¿UVWDUJXPHQWWKDWPDUNHWFOHDULQJSURFHVVHVGRQRWFRQVWLWXWHFDSLWDOLVPIRUWKDW UHTXLUHV V\VWHPDWLF HQWHUSULVH DQG LQGXVWULDO HQWHUSULVH FRXOG QRW ÀRXULVK LI temporary equilibria change rapidly. The second argument stands in contrast with his interpretation of what we call normal prices. Weber thought that prices were formed by rational agents in goal-oriented actions; his view of equilibrium price formation therefore differed from the neoclassical (Marshallian and Walrasian) interpretation and was decidedly Austrian: agents have expectations, seek information, form prices and act strategically with regard to rivals even in competitive processes. Such behaviour may be necessary in the context considered by Weber. If markets for old equipment exist only spasmodically and if normal prices have to be determined via the integrated equation (IG), it is questionable whether we would get a convergence of market prices to normal prices, if market prices were not themselves based on an approximately rational calculation of SULFHV)RUHYHQLQVLPSOHUPRGHOVZLWKRXW¿[HGFDSLWDOLIRQO\FLUFXODWLQJFDSLWDO is present, the convergence of market prices to normal prices cannot generally be proved.31 Weber’s ‘rational’ ‘Kapitalsrechnung’ therefore amounts to estimating the normal price of equipment directly, so that the question as to the convergence of market prices to normal prices can be dispensed with or has to be rephrased in a new way.

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Max Weber never published his view on economics in a coherent form. His IULHQGVOLNH(GJDU-DIIpKLVFRHGLWRURIWKHµ$UFKLYIU6R]LDOZLVVHQVFKDIWXQG Sozialpolitik’,32 reproached him for this omission, and others, like Schumpeter, came close to denying the extent of his knowledge of theoretical economics. Sociologists, anxious to reserve the great name for their own subject, have not done much to uncover Weber’s approach to theoretical economics, and the erroneous idea is widespread that he was no economist in the ordinary sense at all. Today, his lectures on theoretical economics have become accessible33 and the modern reconstruction of classical economics permits a better understanding of Weber’s curious eclecticism, which combined classical macroeconomics and 0DU[LDQ LQÀXHQFHV ZLWK 0HQJHU¶V WKHRU\ DV WKH SULPH H[DPSOH RI PRGHUQ rationality, as far as the theory of consumption under modern capitalism was concerned (consumption followed traditional patterns in other circumstances). To reconstruct Weber’s economics is an interesting and inspiring task, which we have here attempted in the case of a narrow application. I must renounce an attempt to explain Weber’s economics further, but I should ¿QDOO\OLNHWRORRNEDFNDWWKHDERYHPRGHOLQWKHRSSRVLWHSHUVSHFWLYHWKDWRI Sraffa. Sraffa represents an opposite case in that the analytical structure of his theory has been thoroughlyDQDO\VHGGXULQJWKHODVW\HDUVHVSHFLDOO\LQWKH¿UVW half of this period, but the question of its historical applicability has been left wide open. I have argued earlier that there is a historical dimension to the sequence of his models: not that labour values precede prices of production, but the formation RIWKHXQLIRUPUDWHRISUR¿WSUHFHGHVWKHLQWURGXFWLRQRIZDJHODERXU34 and his theory of joint production is hardly tenable without a historical element.35 Here ZHKDYHDQRWKHUH[DPSOHWKHWKHRU\RI¿[HGFDSLWDOLVZLWKDFHUWDLQKLVWRULFDO necessity, associated with the genesis of industrial capitalism, and, surprisingly, a link emerges with Max Weber.

Notes 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18

Seneca 2008, pp. 878–903. Fanfani 2005. Schefold forthcoming. Nuccio 2008. Weber 1924. Sraffa 1960. Bruhns 2006, pp. 151–83. Weber, Marianne 1926, p. 307 sq. and Radkau 2005, pp. 368–80. Weber 1980. Weber 2009. Weber 2006. Weber 2006, pp. 342–3. See e.g. Weber 2009, p. 144. Lysias 2004. Finley 1980, pp. 100–1. Sraffa 1960. Sraffa 1960, chapter X. Schefold 1989, pp. 145–78; 1997, pp. 211–91.

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19 Schefold 1989, p. 166. 20 Passow 1912, p. 367. ‘In the calculation of rates of depreciation the desire is decisive to distribute the costs deriving from the depreciation of the corresponding equipment as equally as possible over all periods during which this equipment is used’ (my translation). 21 Chassot de Florencourt 1781, pp. 35–7. 22 ‘An pactum Antichriseos emptioni Census adiectum usurarium sit’ (Klock 2009 [1651], book II, chapt. XIX, no. 52). 23 I owe this information to Heinz Kurz, to whom I am grateful for it. 24 Schneider 1981, pp. 86–87. 25 Schneider 1981, p. 337. 26 Littleton 1933, pp. 223–41. 27 Marx 1969, pp. 169–82. 28 Fogel and Engerman 1974. 29 Nippel 2005. 30 For the history of these concepts see: Schneider 1981 and Pollard 1963. According to Pollard it was ‘Sombart, who first placed accountancy practices at the centre of a theory of capitalist development’ (p. 75). Pollard shows how difficult it was to introduce a rational dealing with such concepts as capital, income, depreciation, even in the large industrial enterprises of the nineteenth century. 31 Caminati and Petri 1990.  -DIIpSS±KHUHSI 33 Weber 2009. 34 Schefold 1995. 35 Schefold 1989, 2004.

References Baily, Francis (1808) The doctrine of interest and annuities analytically investigated and explained, together with several useful tables connected with the subject. London: John Richardson. Bruhns, Hinnerk (2006) ‘Die verstehende Soziologie Max Webers im Kontext benachbarter Disziplinen. Max Webers Grundbegriffe ‘im Kontext seiner wirtschaftsgeschichtlichen Forschungen’, in: Lichtblau, Klaus (ed.) Max Webers „Grundbegriffe“. Kategorien der kultur- und sozialwissenschaftlichen Forschung. Wiesbaden: Verlag für Sozialwissenschaften 2006, pp. 151–83. Caminati, Mauro and Petri, Fabio (eds) (1990) ‘Convergence to Long-Period Position’s. Political Economy, Studies in the Surplus Approach, Special Issue, vol. 6, nos 1–2. Torino: Rosenberg & Sellier. Chassot de Florencourt, Carl (1781) Abhandlungen aus der juristischen und politischen Rechenkunst. Altenburg: Richterische Buchhandlung. Fanfani, Amintore (2005 [1944, 1934]) Cattolicesimo e protestantesimo nella formazione storica del capitalismo, a cura di Pietro Roggi. Milano: Vita e Pensiero. Finley, Moses I (1980) Ancient Slavery and Modern Ideology. London: Chatto & Windus. Fogel, Robert William and Engerman, Stanley L. (1974) Time on the Cross. Vol. 1: The Economics of American Negro Slavery. London: Wildwood House. -DIIp(GJDU µ'DVWKHRUHWLVFKH6\VWHPGHUNDSLWDOLVWLVFKHQ:LUWVFKDIWVRUGQXQJ¶ in: Archiv für Sozialwissenschaft und Sozialpolitik, 44. Bd., pp. 1–18. Klock, Kaspar (2009 [1651]) ‘Tractatus juridico-politico-polemico-historicus De Aerario. . . . Mit einer Einleitung hg. v. von Bertram Schefold’. Hildesheim: Olms

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2009. Reprint der Originalausgabe von 1651 in 2 Teilbänden [pp. V*–CXIII* vorn im ersten Teilband]. Historia Scientiarum (Wirtschaftswissenschaften). Ein Editionsprogramm der Fritz Thyssen Stiftung zur Geschichte der Wissenschaften in Deutschland. Littleton, Ananias Charles (1933) Accounting Evolution to 1900. New York: American Institute. Lysias (2004) Reden, Griechisch und deutsch. Eingeleitet, übersetzt und kommentiert von Ingeborg Huber, Bd. I. Darmstadt: Wissenschaftliche Buchgesellschaft 2004, Rede 12 (gegen Eratosthenes), pp. 146–81. Marx, Karl (1969 [1893, 1885]) Das Kapital, vol. II. Berlin: Dietz. Morgan, Augustus de (1847) Arithmetical books from the invention of printing to the present time, being brief notices of a large number of works drawn up from actual inspection. London: Taylor & Walton. Nippel, Wilfried (2005) ‘Marx, Weber und die Sklaverei’, in: Herrmann-Otto, Elisabeth (ed.) Unfreie Arbeits- und Lebensverhältnisse von der Antike bis zur Gegenwart. Hildesheim: Olms, pp. 317–56. Nuccio, Oskar (2008) La storia del pensiero economico italiano come storia della genesi dello spirito capitalistico. Rome: Luiss University Press. Passow, Richard (1912) ‘Bilanzwesen’, in: Wiese, Leopold von (ed.) Wirtschaft und Recht der Gegenwart, 2. Band. Tübingen: J.C.B. Mohr (Paul Siebeck), pp. 359–72. Pollard, Sydney (1963) ‘Capital accounting in the Industrial Revolution’, in: Yorkshire Bulletin of Economical and Social Research, vol. 15, no. 2, pp. 75–91. Radkau, Joachim (2005) Max Weber. Die Leidenschaft des Denkens. München: Hanser, pp. 368–80. Schefold, Bertram (1989 [1971) Mr. Sraffa on Joint Production and other Essays. London: Unwin & Hyman 1989, [now: Andover: Routledge]. See also: Schefold, Bertram (1971) Piero Sraffas Theorie der Kuppelproduktion, des Kapitals und der Rente. Dissertation, main part in English, entitled ‘Mr. Sraffa on Joint Production’. Basel: privately printed. Schefold, Bertram (1995) ‘Value and Price in a Historical Context’ (in English and Korean), Hanshin Journal of Economics, vol. 2, pp. 1–37 and 93–125. Seoul/Korea: Hanshin Institute for Economic Research. Schefold, Bertram (1997) Normal Prices, Technical Change and Accumulation. London: Macmillan. Schefold, Bertram (2004) ‘Joint production: triumph of economic over mathematical logic?’, in: Convegno internazionale Piero Sraffa (Roma, 11–12 Febbraio 2003) (Atti dei convegni lincei 200). Roma: Accademia nazionale dei lincei, pp. 303–31. Schefold, Bertram (forthcoming) ‘Amintore Fanfani e le tesi di Max Weber’, paper presented at the conference ‘Amintore Fanfani: formazione culturale, identità e responsabilità politica’, Università Cattolica del Sacro Cuore, Milan, 11–12 December 2008. To be published by the organisers of the conference. Schneider, Dieter (1981) Geschichte der betriebswirtschaftlichen Theorie. München, Wien: Oldenbourg. Seneca (2008) ‘Ad Lucilium epistolae’, XC, in: Stoa und Stoiker. Griechisch-lateinischdeutsch, ed. Rainer Nickel. Vol. II. Düsseldorf: Patmos. Sraffa, Piero (1960) Production of Commodities by Means of Commodities. Prelude to a critique of economic theory. Cambridge: Cambridge University Press. Weber, Marianne (1926) Max Weber. Ein Lebensbild. Tübingen: Mohr.

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Weber, Max (1924) Wirtschaftsgeschichte. Abriss der universalen Sozial- und WirtschaftsGeschichte aus den nachgelassenen Vorlesungen ed. S. Hellmann and M. Palyi. Munich and Leipzig: Duncker & Humblot. Weber, Max (1980 [1921/1922]) Wirtschaft und Gesellschaft. Grundriß der verstehenden Soziologie, besorgt von Johannes Winckelmann. Studienausgabe. Tübingen: J.C.B. Mohr (Paul Siebeck). Weber, Max (1992) Die Protestantische Ethik und der ‘Geist’ des Kapitalismus. Düsseldorf: Verlag Wirtschaft und Finanzen 1992. Klassiker der Nationalökonomie. Vademecum zu einem Klassiker der Geschichte ökonomischer Rationalität. Kommentar zur Faksimile-Ausgabe der 1905 erschienenen Erstdrucke von Weber, Max: Die Protestantische Ethik und der ‘Geist’ des Kapitalismus. Düsseldorf: Verlag Wirtschaft und Finanzen 1992. Klassiker der Nationalökonomie. Weber, Max (2006 [1908]) ‘Agrarverhältnisse im Altertum’ (3. Fassung), in: Max Weber Gesamtausgabe, Abt. I: Schriften und Reden, vol. 6: Zur Sozial- und Wirtschaftsgeschichte des Altertums. Schriften und Reden 1893–1908, ed. Jürgen Deininger. Tübingen: J.C.B. Mohr (Paul Siebeck), pp. 300–747. Weber, Max (2009) ‘Allgemeine (“theoretische”) Nationalökonomie. Vorlesungen 1894–1898’, in: Max Weber Gesamtausgabe, Abt. III: Vorlesungen und Vorlesungsnachschriften, vol. 1, ed. Wolfgang J. Mommsen et al. Tübingen: J.C.B. Mohr (Paul Siebeck).

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Part IV

Linear models of production

14 Extensive rent and multiple equilibria1

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Christian Bidard

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The origin of multiplicity ,QDFRUQ±LURQPRGHODORQJWHUPHTXLOLEULXPLVRIWKH5LFDUGLDQW\SHLIWKHSULFHV DUHGHWHUPLQHGE\DPDUJLQDOODQGDQGDQLURQPHWKRGRIWKHH[WHUQDOW\SHLIWKHUH DUHGHWHUPLQHGE\WZRLURQPHWKRGV([DPSOHKDVVKRZQWKHSRVVLEOHFRH[LVWHQFH RI WZR 5LFDUGLDQ HTXLOLEULD 7KH VHFWLRQ DERYH RQ WKH QRWLRQ RI DQ H[WHUQDO HTXLOLELUXPKDVH[SODLQHGWKDWUHJUHVVLRQLVDVRXUFHRIPXOWLSOLFLW\ZLWKRQH QRUPDODQGRQHH[WHUQDOHTXLOLEULXP7KHSUHVHQWVHFWLRQHVWDEOLVKHVDKLGGHQ FRQQHFWLRQ EHWZHHQ WKHVH WZR UHVXOWV DQG LGHQWL¿HV UHJUHVVLRQ DV WKH unique VRXUFHRIPXOWLSOLFLW\7RLOOXVWUDWHWKHSURSHUW\ZHUHWXUQWR([DPSOHIRUZKLFK WZR 5LFDUGLDQ HTXLOLEULD KDYH EHHQ IRXQG )XUWKHU LQYHVWLJDWLRQV VKRZ WKH H[LVWHQFHRIDWKLUGHTXLOLEULXPZKLFKLVRIWKHUHJUHVVLYHH[WHUQDOW\SH Example 4 7KHGDWDDUHWKRVHRI([DPSOH%HVLGHVWKHWZR5LFDUGLDQHTXLOLEULDLGHQWL¿HG DERYHDQRWKHUH[WHUQDOHTXLOLEULXPIRUZKLFKWKHSULFHYHFWRULVGH¿QHGE\WKH WZRLURQPHWKRGVDQGƍLV 0HWKRGTUFRUQWLURQODERXUDFUHVODQGĺTU corn 0HWKRGTUFRUQWLURQODERXUĺWLURQ 0HWKRGƍTUFRUQWLURQODERXUĺWLURQ 7KHSULFHVDUHp22ƍ  $WWKHVHSULFHVODQG\LHOGVDSRVLWLYHUHQWDQGLV IXOO\FXOWLYDWHGZKHUHDVFXOWLYDWLRQRQODQGƍZRXOGEHWRRFRVWO\0HWKRGLV RSHUDWHGIRUDORZOHYHORIGHPDQGDQGWKHQHZO\LQWURGXFHGPHWKRGƍ has a ORZHUUDWHRIWUDQVIRUPRIFRUQLQWRLURQWLURQSHUTXDUWHURIFRUQLQYHVWHG for method 2ƍLQVWHDGRIIRUPHWKRG 7KDWWKLUGHTXLOLEULXPLVWKHUHIRUH H[WHUQDODQGUHJUHVVLYH :HLQWHQGWRVKRZWKDWWKLVSURSHUW\LVJHQHUDOWKHFRH[LVWHQFHRIWZR5LFDUGLDQ VROXWLRQVLPSOLHVWKDWRIDWKLUGVROXWLRQZLWKUHJUHVVLYHH[WHUQDOUHQW7KHOHVVRQ LV WKDW PXOWLSOLFLW\ HYHQ ZKHQ LW WDNHV WKH IRUP RI WZR 5LFDUGLDQ HTXLOLEULD UHYHDOV WKH SUHVHQFH RI D UHJUHVVLYH H[WHUQDO UHQW EH\RQG WKH DSSHDUDQFHV regressive external rent is not aVRXUFHRIPXOWLSOLFLW\DVVKRZQLQWKHHDUOLHU VHFWLRQEXWWKHuniqueVRXUFHRIPXOWLSOLFLW\8QIRUWXQDWHO\WKHSURRIRIWKDW UHVXOWLVUDWKHULQYROYHG)RUWKHVDNHRIFRQYHQLHQFHWKHSURRILVUHOHJDWHGWRDQ DSSHQGL[ZKHUHLWZLOOOLHIRUHYHU /HWXVUDWKHUFRQVLGHUDODVWQXPHULFDOH[DPSOHWRLOOXVWUDWHDFRQVHTXHQFHRI WKDWUHVXOW$QRWHZRUWK\FKDUDFWHULVWLFRI([DPSOHLVWKDWWKHGHVFULSWLRQRI WKHHFRQRP\LVLQFRPSOHWHDVWKHGDWDUHODWLYHWRWKHSURGXFWLRQRIFRUQDUH ODFNLQJ

Extensive rent and multiple equilibria   Example 5 &RQVLGHUDFRUQ±LURQPRGHOZLWKH[WHQVLYHFXOWLYDWLRQDQGDFHUWDLQQXPEHURI ODQGV7KHUDWHRISUR¿WLVSHUFHQWDQGWKHWKUHHLURQPHWKRGVDUH

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0HWKRGFRUQLURQODERXU o WLURQ 0HWKRGƍFRUQLURQODERXU o WLURQ 0HWKRGƍƍFRUQLURQODERXU o WLURQ :H ORRN DW WKH SRVVLEOH H[LVWHQFH RI PXOWLSOH 5LFDUGLDQ HTXLOLEULD 6LQFH PXOWLSOLFLW\LPSOLHVWKHH[LVWHQFHRIDUHJUHVVLYHH[WHUQDOUHQWOHWXVLGHQWLI\WKH FRUUHVSRQGLQJHTXLOLEULXP:LWKWZRLURQPHWKRGVRSHUDWHGVLPXOWDQHRXVO\WKH RQO\WKUHHFDQGLGDWHVDUHPDGHRIWKHFRPELQDWLRQV" ƍ, 2ƍƍ RUƍ  7KH ¿UVW FRPELQDWLRQ LV H[FOXGHG EHFDXVH PHWKRG ƍ SD\V RYHUSUR¿WV DW LWV DVVRFLDWHGSULFHVDQGWKHVHFRQGEHFDXVHWKHZDJHZRXOGWKHQEHQHJDWLYHRQO\ WKHFRPELQDWLRQƍ FDQVXVWDLQDQH[WHUQDOHTXLOLEULXPZLWKODQG\LHOGLQJD SRVLWLYHUHQWDWSULFHVp22ƍ  +HUHHQWHUVDQHZDUJXPHQWFRQVLGHUDQ DUELWUDU\FRUQPHWKRGCZKLFKSD\VRYHUSUR¿WVDWSULFHVp22ƍWKHQWKHUHODWLYH cost of methods 2 and 2ƍZKHQPHWKRGCLVRSHUDWHGLVLQGHSHQGHQWRIWKHH[DFW FKDUDFWHUL]DWLRQRIC,IRQHPDNHVWKHFDOFXODWLRQIRUWKHPHWKRGCODERXU o FRUQLWWXUQVRXWWKDWPHWKRGLVFKHDSHUDQGLVRSHUDWHG¿UVW$QH[WHUQDOUHQW DSSHDUVZKHQPHWKRGƍLVLQWURGXFHG%XWVLQFHWKHUDWHRIWUDQVIRUPRIFRUQLQWR iron for method 2ƍTUFRUQJLYHVWLURQ LPSURYHVRQWKDWRIPHWKRG FRUQJLYHVWLURQ WKDWH[WHUQDOUHQWLVSURJUHVVLYH7KHRYHUDOOFRQFOXVLRQLV WKDWQRUHJUHVVLYHH[WHUQDOHTXLOLEULXPH[LVWVWKHUHIRUHWKH5LFDUGLDQHTXLOLEULXP LVXQLTXH5HPDUNDEO\HQRXJKWKDWUHVXOWIROORZVIURPWKHH[DPLQDWLRQRIWKH LURQPHWKRGVRQO\

Conclusion 7KLVVWXG\KDV¿UVWLGHQWL¿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¿FLHQWLURQPHWKRGPD\H[LVWDVZHOO5HJUHVVLRQLVD VRXUFHRIPXOWLSOLFLW\7KHPDLQUHVXOWRIRXUDQDO\VLVLVWKDWWKHFRH[LVWHQFHRI WZR5LFDUGLDQHTXLOLEULDLPSOLHVWKDWRIDWKLUGUHJUHVVLYHH[WHUQDOHTXLOLEULXP DQGWKHUHIRUHUHJUHVVLRQLVWKHXQLTXHVRXUFHRIPXOWLSOLFLW\$SDUDOOHOFDQEH GUDZQ ZLWK WKH WKHRU\ RI LQWHQVLYH UHQW SURSHU LQ WKDW FDVH PXOWLSOLFLW\ DOVR

210

Christian Bidard

comes IURP WKH LQVHUWLRQ RI D QHZ PHWKRG ZKLFK WXUQV RXW WR EH LQHI¿FLHQW (Bidard, 2010). The similarity suggests a research programme on more general land models.

Appendix

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Multiplicity requires regressive external rent Let there be an extensive cultivation model with an arbitrary number of corn methods and (for simplicity) two iron methods 2 and 2ƍ, a given positive rate of SUR¿Wr and a given demand vector d, and assume two Ricardian equilibria. The ¿UVWHTXLOLEULXPFDOOVIRUWKHPDUJLQDOODQGDQGWKHLURQPHWKRGWKHVHPHWKRGV GH¿QHWKHSULFHVWKHZDJHDQGWKHSRWHQWLDOUHQWVRQWKHRWKHUODQGVDQGGHWHUPLQH which lands are totally cultivated and which are left fallow. The same for the second equilibrium with the marginal land 1ƍ and the iron method 2ƍ. We intend to show the existence of a third equilibrium of the regressive external type. The proof LVLQWKUHHPDLQVWHSVZH¿UVWVWXG\WKHYDOXHVLGHDQGVKRZWKDWSRVLWLYHSULFHV are associated with the joint use of the iron methods 2 and 2ƍ (property P4 below), thus giving rise to external rent. Then the study of the quantity side shows that the external combination (2, 2ƍ) is regressive (property P8). Finally, we show that this combination can produce the required basket, therefore that (2, 2ƍ) is an external regressive equilibrium which coexists with the two Ricardian equilibria (property P9). )RUDJLYHQUDWHRISUR¿WOHW DQGƍ, 2ƍ GH¿QHWZR5LFDUGLDQHTXLOLEULD where 1 and 1ƍ are the respective marginal lands. The iron methods differ, otherwise the iron sector might be reduced to one method and the solution would be unique. Similarly, the marginal lands differ otherwise the associated cheaper iron method would be the same. The notation 3 ԰(1, 2ƍ ) means that the iron method 2ƍ is more costly at the positive equilibrium prices p12. 7KLV LV HTXLYDOHQW WR WKH IDFW WKDW PHWKRG  \LHOGV RYHUSUR¿WV DW SULFHV p12ƍ. 6\PPHWULFDOO\WKHVHFRQGHTXLOLEULXPLPSOLHVWKDW 3ƍ, 2ƍ) ԰(1ƍ, 2). If neither land 1 pays a positive rent at prices p1ƍ 2ƍ nor land 1ƍ at prices p12, an industrial system made of the methods (1, 1ƍ, 2, 2ƍ) (by ‘industrial system’, we mean that the land inputs are ignored because they are free) would admit both equilibria (1, 2) and (1ƍ, 2ƍ). This being excluded for industrial systems, we assume without loss of generality that land 1 yields a positive rent (or positive overSUR¿WV DWSULFHVp1ƍ2ƍ 3ƍ) ԰(1ƍ, 2ƍ).

Extensive rent and multiple equilibria   /HWODERXUEHFKRVHQDVQXPpUDLUH6LQFHWKHVHWƍ RIPHWKRGV\LHOGVPRUH WKDQWKHUXOLQJUDWHRISUR¿WDWWKHSRVLWLYHSULFHVpƍ2ƍWKHSULFHVpƍ DUHSRVLWLYH %\3 PHWKRG\LHOGVRYHUSUR¿WVDWSULFHVpƍ DQGE\3 LWSD\VH[WUDFRVWV DW SULFHV pƍ2ƍ 7KHUHIRUH PHWKRG  \LHOGV WKH UXOLQJ UDWH RI SUR¿W IRU VRPH DGHTXDWHO\FKRVHQFRQYH[FRPELQDWLRQOpƍ ±O) pƍ2ƍ 6LQFHPHWKRGƍ\LHOGV WKH UXOLQJ UDWH IRU DQ\ FRPELQDWLRQ RI pƍ and pƍ2ƍ  WKH SRVLWLYH YHFWRU WKHQ REWDLQHGLVWKHSULFHYHFWRUp22ƍ +HQFH

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Notes  :LWKDFNQRZOHGJHPHQWVWR*XLGR(UUH\JHUVIRUVWLPXODWLQJGLVFXVVLRQV  (TXDOLW\ LVJHQHULFDOO\QHFHVVDU\EXWQRWVXIILFLHQWDQGVKRXOGEHFRPSOHPHQWHG E\ DQ LQHTXDOLW\ VWDWLQJ WKDW WKH QXPEHU RI RSHUDWHG LQGXVWULDO PHWKRGV LV DW OHDVWHTXDOWRWKDWRILQGXVWULDOJRRGV7KHVHFRQGFRQGLWLRQH[FOXGHVWKHGHFRPSRVLWLRQ  

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References %LGDUG &   Prices, Reproduction, Scarcity &DPEULGJH &DPEULGJH 8QLYHUVLW\ 3UHVV %LGDUG& µ2Q6RPH3X]]OHVLQ5HQW7KHRU\¶LQ-9LQW-60HWFDOIH+'.XU] 16DOYDGRULDQG3$6DPXHOVRQHGVEconomic Theory and Economic Thought, /RQGRQ5RXWOHGJH '¶$JDWD$ µ7KH([LVWHQFHDQG8QLFLW\RI&RVW0LQLPL]LQJ6\VWHPVLQ,QWHQVLYH 5HQW7KHRU\¶MetroeconomicaYRO± (UUH\JHUV *   µ7HUUH UHQWH HW FKRL[ GH WHFKQLTXHV 8QH pWXGH VXU OD WKpRULH QpRULFDUGLHQQH¶PLPHR3K'WKHVLV8QLYHUVLW\RI3DULV;1DQWHUUH (UUH\JHUV* µ2QWKH8QLTXHQHVVRI6TXDUH&RVW0LQLPL]LQJ7HFKQLTXHV¶The Manchester SchoolYRO± .XU] + ' DQG 6DOYDGRUL 1   Theory of Production &DPEULGJH &DPEULGJH 8QLYHUVLW\3UHVV 5LFDUGR'>@ On the Principles of Political Economy and TaxationYRORI The Works and Correspondence of David Ricardo HG 3 6UDIID &DPEULGJH &DPEULGJH8QLYHUVLW\3UHVV 6DOYDGRUL 1   µ/DQG DQG &KRLFH RI 7HFKQLTXHV ZLWKLQ WKH 6UDIID )UDPHZRUN¶ Australian Economic PapersYRO± 6DXFLHU3 µ/HFKRL[GHWHFKQLTXHVHQVLWXDWLRQGHOLPLWDWLRQGHUHVVRXUFHV¶PLPHR 3K'WKHVLV8QLYHUVLW\RI3DULV,, 6UDIID 3   Production of Commodities by Means of Commodities &DPEULGJH &DPEULGJH8QLYHUVLW\3UHVV

15 Why are there negative input– output coefficients? The ‘commodity technology assumption’ revisited

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Christian Lager I came to know Heinz D. Kurz in spring 1988 when he became full professor in Graz, where I was working on my PhD. Because of the formal similarity to my RZQ ¿HOG RI UHVHDUFK ZKLFK ZDV DW WKDW WLPH LQSXW±RXWSXW DQDO\VLV EXW DOVR because of his sound arguments, Heinz needed little effort to convince me that the FODVVLFDODSSURDFKWRHFRQRPLFVSURYLGHVDULJRURXVWKHRUHWLFDOIUDPHIRULQSXW± output analysis. I dedicate this piece which may help to close the gap between DSSOLHGLQSXW±RXWSXWDQDO\VLVDQG6UDI¿DQHFRQRPLFOLWHUDWXUHWRWKHFestschrift in honour of my literate teacher and close friend Heinz. -RLQWSURGXFWLRQZDVDQGLV±LQDFWXDOIDFW±DQLPSRUWDQWLVVXH.XU] 6WHHGPDQRU.XU]DQG6DOYDGRULFKDSWHUVDQG 0RVWVWDWLVWLFDO RI¿FHVIUHTXHQWO\SXEOLVKLQSXW±RXWSXWVWDWLVWLFVZKLFKDUHEDVHGRQPDNHDQGXVH tables. There exist also some theoretical models of growth and distribution, in SDUWLFXODU PRGHOV EXLOW RQ WKH ZRUNV RI -RKQ YRQ 1HXPDQQ DQG 3LHUR 6UDIID which are based on input and output tables and therefore are able to account for joint production. Nevertheless, joint production is widely neglected. One will KDUGO\ ¿QG D WH[WERRN WKDW GRHV QRW WUHDW MRLQW SURGXFWLRQ DV DQ XQLPSRUWDQW complication rather than an interesting issue. The basic concepts of IO-analysis DUH DOVR OLPLWHG WR VLQJOH SURGXFWLRQ ,QSXW RXWSXW FRHI¿FLHQWV DUH GH¿QHG DV inputs of commodity i per unit of output of a single product j. In the case of joint production inputs refer to bundles of outputs and can by no means be allocated to a single product. But this is exactly the aim of so-called technology assumptions and other methods. 7KHUH DUH WZR EDVLF WHFKQRORJ\ DVVXPSWLRQV ± WKH FRPPRGLW\ WHFKQRORJ\ PRGHO &70  DQG WKH LQGXVWU\ WHFKQRORJ\ PRGHO ± VHYHUDO PL[HG RU K\EULG DVVXPSWLRQVDQGRWKHUPHWKRGV.RS-DQVHQDQGWHQ5DD DQGWHQ5DDDQG 5XHGD&DQWXFKH SRVWXODWHVRPHGHVLUDEOHDQGUHDVRQDEOHSURSHUWLHVDQG WHVWKRZZHOOVRPHPHWKRGVWRFRQVWUXFWFRPPRGLW\E\FRPPRGLW\FRHI¿FLHQWV SHUIRUP,WDSSHDUVWKDWRQO\WKH&70H[KLELWVDOOWKHGHVLUHGSURSHUWLHV ,Q WKLV FKDSWHU LW LV DUJXHG RQ SXUH WKHRUHWLFDO JURXQGV  WKDW µWHFKQRORJ\ DVVXPSWLRQV¶GH¿QHMRLQWSURGXFWLRQDZD\DQGDUHWKHUHIRUHQRWVXLWDEOHIRUWKDW case. If there are some activities which produce jointly more than one homogeneous SURGXFWLWLV±LQJHQHUDO±QRWSRVVLEOHWRGHULYH,2FRHI¿FLHQWVZKLFKUHIHUWRDQ output of one unit of a homogeneous product. The validity of the concepts of

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1HJDWLYHLQSXW±RXWSXWFRHI¿FLHQWV   ,2FRHI¿FLHQWVDQGRID/HRQWLHI,QYHUVHLVOLPLWHGWRVLQJOHSURGXFWLRQV\VWHPV and cannot easily be carried over to general joint production. The characteristics of joint production are studied and the very reason of QHJDWLYHLQWHQVLWLHVDQGQHJDWLYHTXDQWLWLHVLVGHWHFWHG,WLVGHPRQVWUDWHGWKDWD MRLQWSURGXFWLRQV\VWHPPD\QRWDOZD\VDGMXVWFRPSOHWHO\WRJLYHQ¿QDOGHPDQG and, therefore, excess production is possible. Following the track of von Neumann DQGDSSO\LQJWKHUXOHRIIUHHJRRGVIUHHGLVSRVDO WKHVROXWLRQVIRUTXDQWLWLHVDQG for prices can be determined by a linear programme. )LQDOO\DFODVVRIMRLQWSURGXFWLRQV\VWHPVLVGH¿QHGZKLFKH[KLELWVDOOWKH important characteristics of single production systems. Useful concepts developed by Neo-Ricardian scholars such as ‘adjustability’ or ‘all-productiveness’ are presented. /HW A, B  Rnun EHVHPLSRVLWLYHVTXDUHPDWULFHVRILQSXWVDQGRXWSXWVRIGL mension commodity by activity. Assume constant returns to scale, i.e. if  a k , b k is a feasible process characterized by the k-th column of the input and of the output matrix, then, for all intensities xk !  ,  a k xk , b k xk is also a feasible process. In other words, a proportional change in the inputs of an activity results into a proportional change in the outputs of that activity without changing the proportions of the outputs. Note, that constant returns to scale imply constant product mixFRHI¿FLHQWV+RZHYHUWKLVDVVXPSWLRQGRHVQRWSRVWXODWHFRPPRGLW\ technology! 7KH &70 DVVXPHV D VLQJOH SURGXFWLRQ SURFHVV IRU HDFK FRPPRGLW\ j, characterized by a vector c j  ZKRVH HOHPHQWV DUH WKH TXDQWLWLHV RI LQSXWV RI commodity iUHTXLUHGWRSURGXFHRQHXQLWRISURGXFWj. The inputs used by an activity k depend on the output structure of that activity. Hence each element aik of the input matrix A, which refer to inputs of an activity kUHÀHFWVDZHLJKWHG DYHUDJH RI WKH XQGHUO\LQJ FRPPRGLW\ E\ FRPPRGLW\ FRHI¿FLHQWV cij where the weights are determined by the outputs of process k. +HQFHWKH&70LVGH¿QHGE\ n

¦c b ij

jk

aik

œ CB

A 



j 1

If BLVUHJXODUWKHFRHI¿FLHQWVRIWKHXQREVHUYDEOHVLQJOHSURGXFWLRQSURFHVVHVFDQ be determined by

C

AB 1 



7KH&70DVVXPHVWKDWWKHUHLV±LQSULQFLSOH±DVLQJOHSURGXFWLRQV\VWHPZKLFK is not directly observable. Observable are ‘industries’ which produce some TXDQWLWLHV RI PRUH WKDQ RQH FRPPRGLW\ E\ FRPELQLQJ PRUH WKDQ RQH VLQJOH production process. Joint production is thus considered as a statistical problem FDXVHGE\DJJUHJDWLRQ7KLVDVVXPSWLRQPD\KDYHVRPHMXVWL¿FDWLRQLIDJJUHJDWH industries are concerned, which produce one characteristic primary product and, in addition, one or more secondary products which themselves are characteristic

  Christian Lager products of other industries. In this case, it is straightforward to calculate nonnegative matrices via a systematic search of possible sources of inaccuracy in the basic make and use tables. If, on the other hand, joint production proper is FRQFHUQHG WKH &7$ PLVVHV WKH SRLQW LV XQDFFHSWDEOH DQG DIWHU DOO LV QRW needed. Given input and output matrices, A and BDQGDYHFWRURIUHTXLUHGQHWSURGXFWV ¿QDO GHPDQG  GHQRWHG E\d7KH SUREOHP LV WR ¿QG D YHFWRU RI LQWHQVLWLHVx which supports the bundle d such that

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B  A x

d



7KHVWUDLJKWIRUZDUGVROXWLRQRI LV x

B  A

1

d 



Furthermore, we may calculate the vector of gross products by q

B  B  A d  1

Bx



1RWHWKDWWKHLQYHUVHPDWULFHVLQ DQG PD\EHZULWWHQDV

B  A

B 1  I  AB 1  1

1

 

and B B  A

1

 I  AB

1 1





1RWHWKDWWKHODWWHULQYHUVHPD\EHFRQVLGHUHGDVDJHQHUDOL]DWLRQRIWKH/HRQWLHI inverse. It is clear that the results for these solutions are the same as one would obtain E\XVLQJWKH&70+RZHYHUQRµWHFKQRORJ\DVVXPSWLRQV¶DUHUHTXLUHGQRUDQ\ PDWUL[ RI FRPPRGLW\ E\ FRPPRGLW\ LQSXW FRHI¿FLHQWV KDYH EHHQ FDOFXODWHG Furthermore, meaningful solutions may be obtained even if the output matrix is singular or if AB± contains negative elements. Though we have avoided the problem of a singular output matrix and QHJDWLYH FRPPRGLW\ E\ FRPPRGLW\ LQSXW FRHI¿FLHQWV WKHUH VWLOO UHPDLQV WKH SRVVLELOLW\ RI QHJDWLYH LQWHQVLWLHV DQG QHJDWLYH TXDQWLWLHV 7KH IROORZLQJ examples may be useful to detect the very reason for this problem of negative results. $VVXPHWKDWWKHUHDUHWZRSURFHVVHVE\ZKLFKWZRFRPPRGLWLHVZRRODQG PXWWRQ DUHMRLQWO\SURGXFHGE\PHDQVRIZRROPXWWRQDQGVRPHSULPDU\IDFWRUV §  · §  · /HW A ¨ ¸ and B ¨ ¸ be input and output matrices of dimension product ©  ¹ ©  ¹ by process. Neglecting the primary factors and considering a stationary economy,

1HJDWLYHLQSXW±RXWSXWFRHI¿FLHQWV  

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we may characterize the two processes also by the ‘netput’ matrix §  · B  A ¨ ¸ ZKHUHWKH¿UVWURZUHIHUVWRSURFHVVDQGWKHVHFRQGURZ ©   ¹ UHIHUVWRSURFHVV The set of producible net outputs is given by the set x 0  (B – A) §· § · x ¨ ¸ x  ¨ ¸ xDQGLVGHSLFWHGLQ)LJXUH ©  ¹ ©¹

Wool

Activity 1:

2 x1 –2 Set of producible net outputs

Mutton –1 x Activity 2: 2 4

Figure 15.1

Note that any non negative net product bundle of wool and mutton can be produced because the feasible set of net bundles contains the whole positive orthant. This observation is crucial for the non negativity of the solutions: i ii LLL iv

For all non negative bundles of net products d there exist a non negative vector of intensities x such that  B  A x d . §    · 1 The inverse  B  A ¨1 1 ¸!. ©  ¹ 1 §     · 7KHJHQHUDOL]HG/HRQWLHILQYHUVH  I  AB 1 ¨   ¸ ! . ©  ¹ Therefore, the bundle of gross outputs q all d t  .

 I  AB

1 1

d is non negative for

These nice features disappear, if we change the input matrix slightly such that LQVWHDGRIIRXUXQLWVRIZRROMXVWWZRXQLWVRIZRRODUHUHTXLUHGIRUWKHVHFRQG

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  Christian Lager process. While the output matrix remains unchanged, the input matrix is now § · §  · given by A ¨ ¸ . The new ‘netput’ matrix is B  A ¨ ¸ and the cone of ©  ¹ ©   ¹ SURGXFLEOHQHWRXWSXWVLVGHSLFWHGLQ)LJXUH In contrast to example 1, that set does not contain the whole positive orthant. Hence some proportions of net outputs are not producible. If more than four WLPHVWKHTXDQWLWLHVRIPXWWRQWKDQTXDQWLWLHVRIZRRODUHGHPDQGHGWKHQWKHUH H[LVWQRSRVLWLYHLQWHQVLWLHVE\ZKLFKWKHWZRSURFHVVHVFDQSURGXFHWKHUHTXLUHG bundle. §     · 1 The inverses B  A ¨© 1 ¸ exist but contain negative elements. 1   ¹ Hence solutions for any demanded bundle of net products can be calculated, but QHJDWLYHLQWHQVLWLHVIRUWKH¿UVWDFWLYLW\ZLOOUHVXOWLIWKHEXQGOHRIQHWSURGXFWV demanded is not an element of the producible set.

Wool

Process 1:

2 x 1 –2

Set of producible net outputs

Process 2:

1 x 2 4

x2 =12 d=

10 50 Mutton

x1 = –1

Figure 15.2

$VVXPH IRU LQVWDQFH WKDW  XQLWV RI ZRRO DQG  TXDQWLWLHV RI PXWWRQ DUH demanded. These proportions are not supported by the system. Even if only SURFHVVZKLFKVSHFLDOL]HVLQWKHSURGXFWLRQRIPXWWRQZLOOEHDFWLYDWHGWKHUH LVWRRPXFKZRRORUQRWHQRXJKPXWWRQ$VROXWLRQIRU FDQEHREWDLQHGEXW will result in a negative intensity for process 1 such that some wool will be ‘absorbed’ and additional mutton will be ‘produced’ by that process. Hence the § · §· §  · linear combination ¨ ¸ x  ¨ ¸ x ¨ ¸ has the solution x1 1 and x .    © ¹ © ¹ ©  ¹ Negative LQWHQVLWLHVDUHQRWDFFHSWDEOH+HQFHWKHPHWKRGSURSRVHGE\  will work for the case that the demanded bundle is producible but cannot be applied in general.

1HJDWLYHLQSXW±RXWSXWFRHI¿FLHQWV  

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Another route is suggested by von Neumann: allow for excess production, i.e. VXEVWLWXWHHTXDWLRQ E\WKHLQHTXDOLW\  B  A x t d , and, if necessary, invoke WKHUXOHRIIUHHJRRGVIUHHGLVSRVDO  ,QH[DPSOHWKLVDSSURDFKZLOOUHVXOWLQx1 DQGx 7KHGHPDQGIRU mutton is exactly met but wool is produced in excess and has to be disposed of by DIUHH GLVSRVDOSURFHVV)RUDPRUHFDUHIXODQGJHQHUDOGLVFXVVLRQRIWKHYRQ 1HXPDQQV\VWHPVZLWKIUHHRUFRVWO\GLVSRVDOVHH/DJHU  ,QRUGHUWRREWDLQVRPHPRUHULJRURXVDQGPRUHJHQHUDO¿QGLQJVZHPD\XWLOL]H WKH IROORZLQJ GH¿QLWLRQV DQG FRQFHSWV GHYHORSHG E\ VFKRODUV ZRUNLQJ LQ WKH 6UDI¿DQWUDGLWLRQ Definition 1 A system of production is strictly viable if it is possible to produce a positive net output. x t 

B  A x ! 

Note that in this case it is possible to produce any net output or more than that. +HQFH LW LV SRVVLEOH WR PDWFK DQ\ YHFWRU RI ¿QLVKHG SURGXFWV LI WKHUH LV IUHH disposal. Definition 2 A product i is said to be separately producible if it is possible to produce a net output consisting of one unit of that product but nothing else, i.e. xi t 

B  A x i

ei ,

where ei is a vector whose iWKHOHPHQWLVHTXDOWRDQGDOORWKHUHOHPHQWVDUH HTXDOWR]HUR Definition 3 A system of production is all-productive if all products are separately producible. All-productive systems have the following nice properties: x x

All-productive systems can produce any semi-positive net output, i.e. d t  x t  B  A x d. 1 If and only if the system is all-productive, then the inverse B  A is semii positive. Note that that the vector x LQGH¿QLWLRQLVWKHi-th column of the 1 inverse B  A . Therefore, an all-productive system has a semi-positive 1 QHWRXWSXWLQYHUVHE\GH¿QLWLRQDQGLI B  A t  , then all products are separately producible.

  Christian Lager x

x x

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x x

,I±EXWQRWRQO\LI±WKHV\VWHPLVDOOSURGXFWLYHWKHJHQHUDOL]HG/HRQWLHI 1 1 1 inverse is non-negative, i.e.  B  A t  Ÿ B  B  A  I  AB1 t  The proof for the ‘if statement’ follows from the fact that the product of two semi-positive matrices is non-negative. The evidence for the ‘not only if’ statement is provided by the example in the appendix. $OOSURGXFWLYHV\VWHPVKDYHDOZD\VVHPL SRVLWLYHVROXWLRQVIRULQWHQVLWLHV B  A 1 d x t . $OOSURGXFWLYH V\VWHPV KDYH DOZD\V VHPL SRVLWLYH VROXWLRQV IRU JURVV 1 outputs. Bx B B  A d q t . $OOSURGXFWLYHV\VWHPVKDYHDOZD\VVHPL SRVLWLYHVROXWLRQVIRUWRWDOFDSLWDO 1 UHTXLUHPHQWV Ax A B  A d k t . . $OOSURGXFWLYH V\VWHPV KDYH DOZD\V VHPL SRVLWLYH VROXWLRQV IRU ODERXU  1 values. l c B  A v t .

We may also explore the relation between semi-positive commodity by commodity LQSXWFRHI¿FLHQWVDQGDOOSURGXFWLYHV\VWHPV x

,I±EXWQRWRQO\LI±WKHFRPPRGLW\E\FRPPRGLW\FRHI¿FLHQWVDUHVHPL SRVLWLYHLWIROORZVWKDWWKHJHQHUDOL]HG/HRQWLHILQYHUVHLVVHPLSRVLWLYHLH 1 1 AB 1 t  Ÿ B B  A I  AB1 . The proof for the ‘if’ statement IROORZVIURPWKHIDFWWKDWWKHJHQHUDOL]HG/HRQWLHILQYHUVHLVHTXDOWRWKH sum of a Neumann series of convergent semi-positive matrices, i.e. f 1 1 1 t  I  AB ¦  AB . The evidence for the ‘not only if’ statement is t 

x x

provided by the example in the appendix. ,W IROORZV WKDW VHPLSRVLWLYH FRPPRGLW\ E\ FRPPRGLW\ FRHI¿FLHQWV JXD rantee semi-positive gross outputs, i.e. AB 1 t  Ÿ d t  q t   q = (I – AB±)± d. %XWVHPLSRVLWLYHFRPPRGLW\E\FRPPRGLW\FRHI¿FLHQWVdo not guarantee semi-positive intensity vectors.

Summary and conclusion In this chapter it is argued on pure theoretical grounds that a ‘technology DVVXPSWLRQ¶ RU DQ\ RWKHU PHWKRG WR HVWLPDWH FRPPRGLW\ E\ FRPPRGLW\ RU LQGXVWU\ E\ LQGXVWU\  LQSXW±RXWSXW PDWULFHV QHJOHFWV WKH YHU\ QDWXUH RI MRLQW SURGXFWLRQ ,Q SDUWLFXODU WKH FRPPRGLW\ WHFKQRORJ\ PRGHO GH¿QHV MRLQW production away by assuming that there are only single production processes and observed joint production is a result of aggregation of single production processes to industries. A straightforward joint production model is proposed which avoids FDOFXODWLQJFRPPRGLW\E\FRPPRGLW\LQSXW±RXWSXWPDWULFHV7KHYHU\UHDVRQIRU the still remaining problem of negative solutions is detected. The common feature of systems which may have negative solutions is that those systems do not support DQ\EXQGOHRIQHWSURGXFWV)LQDOO\DFODVVRIMRLQWSURGXFWLRQV\VWHPVLVLGHQWL¿HG which rules negative solutions out and exhibits some other nice features of single production systems.

1HJDWLYHLQSXW±RXWSXWFRHI¿FLHQWV  

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Appendix § · § · /HW A ¨ ¸ and B ¨ ¸.   © ¹ © ¹ § 8 1· The netput matrix  B  A ¨ ¸ shows clearly, that not all net bundles in ©   ¹ the positive orthant are producible, i.e. the system is not all productive. §     · 1 &RQVHTXHQWO\  B  A | ¨ ¸ t. ©    ¹ 1 1 1HYHUWKHOHVV WKH JHQHUDOL]HG /HRQWLHILQYHUVH B B  A I  AB 1





§    · |¨ t  , is positive. ©    ¸¹

Hence it is possible that there is a solution with negative intensities but gross outputs will be positive. 7KRXJK WKHUH LV D SRVLWLYH JHQHUDOL]HG /HRQWLHILQYHUVH WKH FRPPRGLW\ E\ FRPPRGLW\LQSXWPDWUL[FDOFXODWHGE\WKH&7$FRQWDLQVQHJDWLYHFRHI¿FLHQWV §    · C AB 1 | ¨ ¸. ©     ¹

Notes  7KLVURXWHKDVEHHQSURSRVHGE\6WHHQJH  1RWHWKDWDOVRVRPHQHJDWLYHDPRXQWVRIZRRORUPXWWRQDUHµSURGXFLEOH¶7KDWPHDQV WKDWWKHV\VWHPLVDOVRFDSDEOHWRGLVSRVHRIIVRPHH[LVWLQJTXDQWLWLHVRIZRRORUPXWWRQ  7KHFRQFHSWRIDQDOOSURGXFWLYHV\VWHPKDVEHHQLQWURGXFHGE\6FKHIROG  A similar concept, ‘the adjustment property’, has been presented by Bidard and (UUH\JHUV    1RWH WKDW VHPL SRVLWLYH SULFHV RI SURGXFWLRQ UHTXLUH WKDW WKH V\VWHP LV µr-all1 productive’ and therefore B  A   r t .

References %LGDUG&DQG*(UUH\JHUV µ6UDIIDDQG/HRQWLHIRQ-RLQW3URGXFWLRQ¶Review of Political Economy, 101RSS± .RS-DQVHQ3DQG7WHQ5DD µ7KH&KRLFHRI0RGHOLQWKH&RQVWUXFWLRQRI,QSXW 2XWSXW&RHIILFLHQW0DWULFHV¶International Economic Review, 31SS± .XU]+' µ&ODVVLFDODQG(DUO\1HRFODVVLFDO(FRQRPLVWVRQ-RLQW3URGXFWLRQ¶ Metroeconomica, 38SS± .XU] + ' DQG 1 6DOYDGRUL   Theory of Production. A Long-Period Analysis, &DPEULGJH&DPEULGJH8QLYHUVLW\3UHVV /DJHU& µ-RLQW3URGXFWLRQZLWK³5HVWULFWHG)UHH'LVSRVDO¶¶¶Metroeconomica, 52SS± 6DOYDGRUL1DQG,6WHHGPDQHGV  Joint Production of Commodities, Aldershot: Edward Elgar. 6FKHIROG %   µ0U6UDIIDRQ-RLQW3URGXFWLRQ¶3K' WKHVLV 8QLYHUVLW\ RI %DVHO mimeo.

  Christian Lager

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6FKHIROG %   Mr. Sraffa on Joint Production and other essays /RQGRQ 8QZLQ Hyman. 6WHHGPDQ, µ7KH(PSLULFDO,PSRUWDQFHRI-RLQW3URGXFWLRQ¶LQ&%LGDUGHG La Production Jointe: Nouveaux DebatesUHSULQWHGLQ6DOYDGRULDQG6WHHGPDQHGV  6WHHQJH$( µ7KH&RPPRGLW\7HFKQRORJ\5HYLVLWHG7KHRUHWLFDO%DVLVDQGDQ $SSOLFDWLRQWR(UURU/RFDWLRQLQWKH0DNH8VH)UDPHZRUN¶Economic Modelling, 7SS± WHQ5DD7DQG-05XHGD&DQWXFKH µ7KH&RQVWUXFWLRQRI,QSXW2XWSXW&RHIILFLHQW 0DWULFHVLQDQ$[LRPDWLF&RQWH[W6RPHIXUWKHUFRQVLGHUDWLRQV¶Economic Systems Research, 15 ±

16 Convergence of a short-run equilibrium to a long-run equilibrium in an emission permit scheme

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Eiji B. Hosoda*

Introduction Ronald Coase’s great insight on environmental policies is that once property rights are established for environment, environmental problems can be controlled in a decentralized fashion. This idea has led to the trade of emission permits for pollutants or environmentally unfriendly substances that has become very common nowadays. In emission permit trading, unlike Coase’s assumption of endogenous formation of a market, a market is designed by a government. As a result, the FUHDWLRQRIDQHZPDUNHWGRHVQRWDXWRPDWLFDOO\OHDGWRDQHI¿FLHQWDOORFDWLRQLQ particular because a government often fails to take into account the structural change of an economy that may occur in response to the creation of an emission permit market. If such a structural change is to occur, there is a large chance that the equilibrium targeted by a policy of tradable emission permits can never be achieved. 7KLV FKDSWHU GHPRQVWUDWHV WKDW HYHQ LI DQ HTXLOLEULXP WKDW VDWLV¿HV D JLYHQ emission target at every point of time is dynamically unstable, there is a practical and operational formula by which the government can determine an emission permit supply at each moment of time so as to achieve the target in the long run while guaranteeing the long-run stability of an equilibrium path. This study also REWDLQVFRQGLWLRQVXQGHUZKLFKDQHTXLOLEULXPSDWKWKDWVDWLV¿HVDJLYHQHPLVVLRQ target at every point of time is stable. In the case in which such a condition is not met, the emission permit supply formula is indispensable. For the purpose of the above study, utilization of a multisectoral disaggregated model is essential. This is because an impact made by the creation of a tradeable emission permit market and activation of an abatement sector implies a structural change of industries, and only by means of a multisectoral disaggregated model can ZHDQDO\VHWKHLQÀXHQFHRIWKHLQWURGXFWLRQRIWUDGDEOHHPLVVLRQSHUPLWVRQVWDELOLW\ conditions. It should be remembered that an analysis of effects of policy introduction or behavioural change on industrial structure has been conventionally considered as an important issue, as seen in Aoki (1977) and more recently Yano (1998). Another feature of our paper is that we adopt a linear descriptive model rather than a nonOLQHDURSWLPDOPRGHO,QDGGLWLRQWRFRQYHQLHQFHDVD¿UVWDSSUR[LPDWLRQWKHIRUPHU PRGHOPDNHVLWSRVVLEOHWRGHGXFHDQH[SOLFLWVROXWLRQSDWK'XHWRWKHYHU\QDWXUH of the model, we can demonstrate a practical and operational formula of emission

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224

Eiji B. Hosoda

permit supply by which the government can meet the target amount of emission while leading an economy to a steady-state equilibrium. Certainly, there have been seminal analytical works on dynamic aspects of tradable emission permits. For instance, Rubin (1996) has analysed the workings of tradable emission permits, taking banking and borrowing into account, and shown that a dynamic equilibrium path attains the least-cost solution. Nagurney DQG=KDQJ KDYHH[SORUHGWKHZRUNLQJRIDQDPELHQWSHUPLWVFKHPHLQD dynamic transportation model and deduced stability. Although their stance is similar to ours, there is a big difference between their model and ours; the former is based upon a tatonnement process but the latter is not. Grimaud (1999) has discussed the Aghion-Howitt type of endogenous growth model in a disaggregated version and successfully solved the implementation problems when there are some different types of distortions which are remedied by the introduction of emission permits. The disaggregation dealt with in the paper is, KRZHYHU GLIIHUHQW IURP RXUV WKH IRUPHU LPSOLHV H[LVWHQFH RI D QXPEHU RI intermediate goods, and the latter means that there are independent sectors of consumption goods and capital goods as well as an abatement sector. It is worth mentioning that this paper may be situated in the new research line of market quality developed by Yano (2008). Since structural stability is an important element of market quality, and since it is demonstrated in this paper that a proper supply rule of emission permits as institutional infrastructure is indispensable to maintain stability, the present chapter may be considered as natural application of the theory of market quality to environmental policies. 2XU SURFHGXUH WR VKRZ WKH PDLQ UHVXOWV LV DV IROORZV ,Q WKH QH[W VHFWLRQ ZH present the basic assumptions and fundamental model, and describe a short-run equilibrium and a steady-state equilibrium. In the third section, we will deduce stability conditions in a simple case where the target amount of emission permits is given and ¿[HGDWDQ\PRPHQWRIWLPH$OWKRXJKWKLVVXSSO\IRUPXODRISHUPLWVLVQRWÀH[LEOH the stability conditions obtained are considered as a good reference point for the following section. In section 4, we will demonstrate a supply formula of emission permits by which the government can lead an economy to a steady state where the target amount of emission is met. It will also be mentioned that the fundamental conclusions are applicable to the case in which emission permits are valid eternally.

The model Basic assumptions Our model is described as follows: we assume that there are three representative actors in an economy, i.e., workers, capitalists and the government. Since we assume that workers do not save but only consume, we identify them as consumers.1 In the same way, assuming that capitalists do not consume but only VDYHWRLQYHVWZHXVH¿UPVDQGFDSLWDOLVWVLQWHUFKDQJHDEO\ The detailed assumptions on the three actors are as follows. Workers are employed and paid by capitalists. They also receive the sales proceeds of emission permits from the government, since the government is assumed to be an abstract

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Equilibria in an emission permit scheme 225 entity to control the amount of emission permits and does not spend. Their income, therefore, consists of the wages and the sales proceeds of emission permits, and they spend all their income on consumption. Capitalists employ capital and workers, and organize production. They must also have a certain amount of emission permits, since production causes emission of pollutants, waste or whatever, which is regulated by the government. They earn SUR¿WVSD\LQJZDJHVWRZRUNHUVDQGHPLVVLRQFKDUJHVWRWKHJRYHUQPHQW7KH\ save all their income for investment or capital accumulation. The government’s role in this paper is very abstract as we have mentioned. The government issues emission permits every year, sells them to capitalists, and pays the proceeds of the sales to workers. Thus, the fundamental role of the government is just to control the amount of emission permits. Actually, how and to whom the sales proceeds are paid back does not matter in the following discussion. We assume, for a while, that emission permits are valid for a year, and must be renewed every year. The case where, once issued, emission permits remain valid IRU HYHU ZLOO EH EULHÀ\ GLVFXVVHG 7KH DOWHUQDWLYH DVVXPSWLRQ RQ GXUDWLRQ RI emission permits does not affect the essence of our argument. As for the production side, we assume the following three-sector model. The ¿UVWVHFWRUSURGXFHVDFDSLWDOJRRGDQGWKHVHFRQGVHFWRUDFRQVXPSWLRQJRRG Both sectors input a capital good and labour, emitting pollutants or waste. The third sector is a process which abates emission of pollutants or waste. This process inputs a capital good and labour but does not emit pollutants or waste. We assume that the charge for the abatement service is the same as the price of a unit of emission permits in equilibrium, since the difference is supposed to be adjusted LPPHGLDWHO\$OOWKHSURFHVVHVDUHVXEMHFWWRFRQVWDQWFRHI¿FLHQWWHFKQRORJ\ Notation and equations We adopt the following notation: p i (t) x j (t) r(t  w(t) p k (t) q(t) n ai li și bi K(t) B(t) L(t) b(t)

The price of the i-th commodity (a variable) at time t. The activity level of the j-th process (a variable) at time t. 7KHUDWHRISUR¿WDYDULDEOH DWWLPH t. The wage rate (a variable) at time t. The rental of capital at time t. The price of a unit of emission permits (a variable) at time t. The growth rate of labour supply (a given parameter). The input of the capital good per output in the i-th process (a constant). The labour input per output in the i-th process (a constant). The capital-labour ratio of the i-th process, namely ai /li. The emission rate per output in the i-th process (a constant). Capital stock at time t. The total amount of emission permits at time t. Labour supply at time (t). Per capita amount of emission permits at time t, namely B(t)/L(t).

226

Eiji B. Hosoda

If there is no room for misunderstanding, we sometimes abbreviate the time element t IURPPDWKHPDWLFDOH[SUHVVLRQV6LQFHZHDVVXPHWKDWWKHODERXUVXSSO\ increases at the constant rate n, the labour supply at the tWKSHULRGLVH[SUHVVHGDV L(t) = L0ent, where L0 is the initial amount of the labour supply. Now, the price system and the quantity system can be described as follows:

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Price system In a short-run equilibrium, each price equals unit cost of production, including purchase of emission permits. Thus, pi = pkai + wli + bi must holds for i = 1, 2. Furthermore, the price of emission permits q equals the abatement cost by arbitrage, and so, q = pk a3 + wl3 KROGV&RQVLGHULQJWKDWWKHUDWHRISUR¿WHTXDOV the capital rental divided by the price of capital good, we have r = pk /p1 or pk = rp1. Then, the following must hold: ⎧ p1 = rp1a1 + wl1 + qb1 ⎪ ⎨ p2 = rp1a2 + wl2 + qb2 ⎪ q = rp a + wl ⎩ 1 3 3 where (p1, p2, w, pk, q) • 0.

(16.1)

We adopt the consumption good as numerator, and set p2 = 1 in the following. The equations of (16.1) are the price-cost equations for the capital-good sector, the consumption-good sector and the abatement sector respectively. As we have mentioned, we assume that the price of emission permits equals the unit cost of abatement of the third sector by arbitrage. In other words, the difference between the emission right price and the abatement cost disappears immediately. It must be noted that all the price variables are determined irrespective of GHPDQGRQFHRQHRIWKHYDULDEOHVIRULQVWDQFHWKHUDWHRISUR¿Wr, is determined. Therefore, the short-run price formation is the same as the long-run one in the VHQVHWKDWSULFHVDUHGHWHUPLQHGRQFHWKHUDWHRISUR¿Wr, is determined by r = the growth rate of capital stock. This equality is obtained from the assumption on the behaviour of savings and consumption. Quantity system ⎧ x1 ⎪x ⎪⎪ 2 ⎨K ⎪L ⎪ ⎪⎩ B

= = = = =

rK wL a1 x1 l1 x1 b1 x1

+ + + +

Bq a2 x2 l2 x2 b2 x2

+ a3 x3 + l3 x3 − x3

(16.2)

where (x1, x2, x3) • 0. 7KH¿UVWDQGVHFRQGHTXDWLRQVRI PHDQWKHVXSSO\±GHPDQGHTXDOLW\IRU the capital good and the consumption good. Clearly, the third and fourth equations of (16.2) mean that both the capital stock and the labour are fully utilized. The

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Equilibria in an emission permit scheme 227 ¿IWKHTXDWLRQRI LPSOLHVWKDWWKHHPLVVLRQFRQVWUDLQWLVELQGLQJ2 In other words, supply of emission permits equals demand. 2QH UHPDUN LV QHFHVVDU\ KHUH ,Q WKH VXSSO\±GHPDQG EDODQFH H[SUHVVHG DERYHLH ZHDVVXPHWKDWDOOVXSSO\±GHPDQGUHODWLRQVKLSVDUHH[SUHVVHG DVHTXDOLWLHVLQVWHDGRILQHTXDOLWLHV7KLVPHDQVWKDWWKHUHLVQRH[FHVVVXSSO\IRU DQ\ JRRG RU HPLVVLRQ SHUPLWV7KH QR H[FHVV VXSSO\ FRQGLWLRQ LV SDUWLFXODUO\ important for emission permits, since it affects the target amount of emission permits, as shown later. Since x1LVQRWKLQJEXWFDSLWDODFFXPXODWLRQWKH¿UVWHTXDWLRQRI LPSOLHV < x1/K = K /K = rWKHJURZWKUDWHRIFDSLWDOVWRFNHTXDOVWKHUDWHRISUR¿W 0 and l1 L(t ) l2

Equilibria in an emission permit scheme 245 Dynamics '\QDPLFVLVH[SUHVVHGDVIROORZV ­ K ® ¯L

a1 x1 l1 x1

 a2 x2  l2 x2

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'H¿QLQJyi { xie±ntL0 and considering x1 into the following: Ay  nAy

K , we can transform the above equations

§ y1 · ¨© n ¸¹

⎛1 0 ⎞ 'H¿QLQJI1 as I1 ≡ ⎜ , we have ⎝ 0 0 ⎟⎠ y

§0 · (nI  A1 I1 ) y  A1 ¨ ¸ . ©n¹

/HWXVGH¿QH4 as 4 {±nI – A±I1).7KHQDSDUWLFXODUVROXWLRQLVH[SUHVVHGDV ⎛0 ⎞ y* = −( AΘ) −1 ⎜ ⎟ . ⎝n⎠ The general solution to a homogeneous part of the above differential equation is given as y(t) =Į1v1eO1tĮ2v2eO2t, where vi is an eigenvector corresponding to an eigenvalue Oi (i = 1, 2). The eigenvector and the eigenvalue are obtained by solving a characteristic equation of the differential equation as follows: det(ȜI – 4) = 0. Thus, we have O1 = − n +

1 l2 and O2 = − n detA

1 l2 is positive due to the feasibility condition If detA is positive, then − n + detA 1 n  . This means that the stability condition is detA < 0. Let us summarize the a1 UHVXOWVRIH[LVWHQFHRIHTXLOLEULXPDQGVWDELOLW\DVIROORZV

246

Eiji B. Hosoda

Proposition 6 (Shinkai and Uzawa) a1 K (t ) a2 holds, a short-run equilibrium with x1(t) > 0 and x2(t) > 0   l1 L(t ) l2 exits for t. (ii) If and only if detA < 0 holds, a short-run equilibrium, converges to a long-run equilibrium.

(i) If

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/HWXV¿QGHLJHQYHFWRUVv1 and v2. Solve the following for each Oi(i = 1, 2), 1 l2 and O2 ±n: where O1 = − n + detA (Oi I – 4) vi = 0. From this, we have ⎛ l2 ⎞ − v1 = ⎜ l1 ⎟ ⎜ ⎟ ⎝ 1 ⎠

⎛0⎞ and v2 = ⎜ ⎟ . ⎝1⎠

Consequently, we have obtained the general solution of the dynamic system in the following form: § l2 ·  §0· §0· y (t ) D 1 ¨ l1 ¸ e O1t  D 2 ¨ ¸ e O2t  ( A4) 1 ¨ ¸ . ¨ ¸ ©1¹ ©n¹ © 1 ¹

Appendix B A three-sectoral dynamics 1 In this section, let us prove Propositions 2 and 3. Proof of Proposition 2 We show the following: An equilibrium vector (x1(t), x2(t), x3(t)) is positive for any t if a†1/l†1 < k†0 < a†2/l†2 holds, where k† { (k0 + a3b0)/(1 + a3b0)· Proof. Let us consider (16.18) with (16.19). Namely, 1  l3b(t )

l1† y1 (t )  l2† y2 (t ),

where b(t) = b0 or b0e±nt. The left-hand side of (16.18) is constant or decreasing monotonicallv. Notice

y1 t

D 1

l2† H[S O1t  y1* l1†

(16.A.2)

Equilibria in an emission permit scheme 247 y2(t) = D1 H[SO1t) + D2H[SO2t) + y*2,

(16.A.3)

where O1 < O2 ±n < 0. Let us consider the following to cases: (i) Suppose Į1 > 0. Then, y1(t) d y *1 and y1(t) converges to y*1 monotonically. Since y1(t) increases and b(t) does not increase in (16.18), y2(t) decreases monotonically. Since y2(t) o y*2, we have y2(t) > 0 for all t. (ii) Suppose D1 < 0. Notice that y2(0) Į1Į2 + y*2 holds.

D (a) Suppose Į1 + Į2 > 0. Then, y2(0) > y*2 . And Į2!±Į1 > 0 and so  1 < 1

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D2

hold. If y2(t') < y* holds for some t´, then y2(t) = y*2 must hold for some t d t´. Then Į1H[SO1t) +Į2 H[SO2t) = 0 holds. This implies that −

α1 = exp (( λ2 − λ1 )t ) < 1 α2

holds. But (O2±O1)t t 0. A contradiction. Thus, y2(t) > y*2 for all t. (b) Suppose Į1 + Į2 < 0. Let us consider the following sub-cases. (b-1) If Į2 < 0, then y2(t) converges to y*2 monotonically. Thus, y2(t) > 0 for all t. (b-2) If Į2 > 0, then 0 < Į2±Į1 or equivalently |D2| < |D1|. Evaluating y 2 t D1O1 H[SO1t  D 2 O2 H[SO2 t at t = 0, we have y 2 (0) D 1 O1  D 2 O2 . Since O1 < O2 < 0 and 0 < D2±D1 hold, y 2 (0) = D1O1 + D2O2 > 0. By hypothesis y2 ±y*2 = D1 + D2 < 0 holds, we know that y2(t)¿UVWLQFUHDVHV Notice D1 H[SO1t) + D2 H[SO2t) = 0 holds for unique t. This means that D2H[SO2t  ±D1H[SO1t) holds, and thus we have t

§ D · ln ¨  1 ¸ /( O2  O1 ) ! 0. © D2 ¹

Hence, y2(t)¿UVWLQFUHDVHVDQGJRHVEH\RQG y*2 and turns to decrease, converging to y*2. Consequently, y2(t) cannot become non-positive. Therefore, (y1(t), y2(t)) or (x1(t), x2(t)) is positive. Positivitv of y3(t) (or x3(t)) is clear insofar as b0 < b*0KROGVŶ

248

Eiji B. Hosoda

Proof of Proposition 3 1RWLFHWKDWWKHVWDELOLW\FRQGLWLRQLVDOZD\VVDWLV¿HGLQFDVHLL DQGFDVHLLL  Thus a†1/l †1 < a2†/l2† KROGV LQ WKHVH FDVHV:KDW PDWWHUV LV WKH H[LVWHQFH FRQGLWLRQ LQWKHVKRUWUXQ7KHVHFRQGWKHLQHTXDOLW\RIWKHH[LVWHQFHFRQGLWLRQQDPHO\ (16.20) is a2† a2  b2 a3 k0  a3b0 { ! { k0† , l2  b2 l3 1  l3b0 l2†

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which is equivalent to (T 2  T 3) b0 ! (k0  T 2 )

b 1  ( k 0  T 3) 2 . l3 l2

Since T2 > T3 holds by hypothesis, the above is equivalent to b0 !

b2 ½ 1 ­ 1 ®( k 0  T 2 )  ( k0  T 3 ) ¾ . l3 l2 ¿ T 2  T3 ¯

6LPLODUO\WKH¿UVWLQHTXDOLW\RIWKHH[LVWHQFHFRQGLWLRQ LPSOLHV b0 !

b1 ½ 1 ­ 1 ®(k0  T1 )  (k0  T 3 ) ¾ . l3 l1 ¿ T1  T 3 ¯

Thus, the condition is equivalent to

b0 ! PD[ i

1,2

bi ½ 1 ­ 1 ®k0  T i  k0  T 3 ¾ l3 li ¿ Ti  T3 ¯

Appendix C A three-sectoral dynamics 2 ,Q WKLV $SSHQGL[ ZH FRQVLGHU PDWKHPDWLFDO GHGXFWLRQV LQ WKH VHFWLRQ RQ D stabilization policy. A short-run equilibrium $VKRUWUXQHTXLOLEULXPLQVHFWLRQRQDVWDELOL]DWLRQSROLF\LVH[SUHVVHGDV A† x(t )

§ a3 · § K (t ) · ¨© L(t ) ¸¹  B(t ) ¨© l ¸¹ , 3

Equilibria in an emission permit scheme 249 or A† y (t )

§ a3 · § k (t ) · ¨© 1 ¸¹  b(t ) ¨© l ¸¹ , 3

where b(t) = ȕ1{y1(t ±y*1} + ȕ2{y2(t ±y*2} + (E1y*1 + ȕ2y*2 + c0)e±nt.

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Let us consider a short-run equilibrium at t = 0. Ây (0)

§ a3 · § k0 · ¨© 1 ¸¹  c0 ¨© l ¸¹ . 3

If aˆ1 k0  c0 a3 aˆ2   1  c0 l3 lˆ1 lˆ2

(16.A.4)

LVVDWLV¿HG y LVSRVLWLYHDVZHKDYHVKRZQLQ$SSHQGL[$:HZLOOVKRZDV EHIRUHWKDWDVKRUWUXQHTXLOLEULXPDOZD\VH[LVWVIRUQRQQHJDWLYH t if the above LQHTXDOLW\LVVDWLV¿HG Dynamics Notice that (16.24) is fundamentally the same as (16.12). Indeed, the general solution to (16.24) is, as we have already shown, ­ °° y1 (t ) ® ° °¯ y2 (t )

§ · lˆ2 P1t * ¨ D 1 ˆ e  y1 ¸ l1 © ¹ (D 1e P1t  D 2 e P2t  y*2 ),

(16.A.5)

$VZHKDYHVKRZQLQWKHPDLQWH[Wμ1 and μ2 are solutions to the characteristic equation det{PI±±I1±nI)} = 0, and the eigenvectors corresponding to μ1 and μ2DUHH[SUHVVHGDV

v1

§ lˆ2 · ¨  ˆ ¸ and v 2 ¨ l1 ¸ ¨© 1 ¸¹

§0· ¨© 1 ¸¹ .

250

Eiji B. Hosoda

Notice that 1  l3b(t )

lˆ1 y1 (t )  lˆ2 y2 (t )

(16.A.6)

holds and this is equivalent to

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1  l3{( E1 y1*  E 2 y*2  c0 )e  nt  ( E1 y1*  E 2 y*2 )} lˆ1 y1 (t )  lˆ2 y2 (t ). Since (ȕ1y*1 + ȕ2y*2 + c0) is non-negative15, the left-hand side of the above inequality is a non-increasing function of t. Thus, completely the same argument DV$SSHQGL[%LVDSSOLFDEOHWRWKLVFDVHE\UHSODFLQJ Oi with μi. Therefore, if $  KROGV D VKRUWUXQ HTXLOLEULXP H[LVWV DQG FRQYHUJHV WR D ORQJUXQ equilibrium. It is easy to see that y3(t) (or x3(t)) is non-negative for small c0. For B(t) d b1x1(t) + b2x2(t) holds when c0LVVXI¿FLHQWO\VPDOO1RWLFH bi t ȕi.) The target amount of emission permits Proof of (16.26). From (16.A.5), we know that the following holds: lˆ1 y1 (t )  lˆ2 y2 (t ) lˆ1 y1*  lˆ2 y*2  lˆ2D 2 e P2t

(16.A.7)

On the other hand, from (16.A.6) we obtain lˆ1 y1 (t )  lˆ2 y2 (t ) 1  {( E1 y1*  E 2 y*2  c0 )e  nt  ( E1 y1*  E 2 y*2 )}l3

(16.A.8)

holds, taking t o f, we have lˆ1 y1*  lˆ2 y*2 1  ( E1 y1*  E 2 y*2 )l3 . Substituting this into (16.A.7), we have lˆ1 y1 (t )  lˆ2 y2 (t ) 1  ( E1 y1*  E 2 y*2 )l3  lˆ2D 2 e P2t .

(16.A.9)

It follows from (16.A.8) and (16.A.9), evaluated at t = 0 and from μ2 ±n that l3 ( E1 y1*  E 2 y*2  c0 ) lˆ2D 2 . Therefore,

D2

l3 ( E1 y1*  E 2 y*2  c0 ) lˆ 2

holds.

Equilibria in an emission permit scheme 251 2QWKHRWKHUKDQGIURP LQWKHVXEVHFWLRQRQ'\QDPLFVDQGFRQVLGHULQJ xi(t) = yient, we have

E1{ y1 (t )e nt  y1*e nt }L0  E 2 { y2 (t )e nt  y*2 e nt }L0 § lˆ · °½ °­ L0 ®D 1 ¨ E1 2 e( P1  n ) ¸ ¾  E 2D 2 L0 © lˆ1 ¹ ¿° ¯°

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o E 2D 2 L0 as t o 0, since μ1 < μ2 ±n holds. Hence, lim B (t ) t of

( E1 y1*  E 2 y*2  c0 ) L0  E 2D 2 L0 § l · L0 ¨1  E 2 3 ¸ ( E1 y1*  E 2 y*2  c0 ) lˆ ¹ © 2

L0

l2† ( E1 y1*  E 2 y*2  c0 ) Ŷ lˆ 2

The initial amount of emission permits Let us prove the following lemma: Lemma 1 2

Suppose that (16.25) or (16.27)LVVDWLV¿HGDVZHOODVȕ1 = b1 in cases (i) and (ii)7KHQWKHLQLWLDODPRXQWRIHPLVVLRQSHUPLWVLVQRQQHJDWLYHIRUDQ\ȕ2, if T†1• k0(= K0/L0) holds. Suppose that (16.25) or (16.27)LVVDWLV¿HGDVZHOODVȕ2 = b2 in cases (iii) and (iv)7KHQWKHLQLWLDODPRXQWRIHPLVVLRQSHUPLWVLVQRQQHJDWLYHIRUDQ\ȕ1, if T†2”N0 holds.

Proof. We assume here that c0 = 0 without loss of generality. Notice that the initial amount of emission permits B(  LV H[SUHVVHG DV B(0) = ȕ[(0), where ȕ{ (ȕ1ȕ2). Notice

E x(0)

§ K0 · © L0 ¸¹

E Â 1 ¨

§ lˆ 1 E¨ 2 det © lˆ1

 aˆ2 · § K 0 · ¸ ¨ ¸. aˆ1 ¹ © L0 ¹

Let E2 o±f withȕ1 = b1 orȕ1 o±f withȕ2 = b2. 6LQFH RU LVVDWLV¿HGE\K\SRWKHVLVZHNQRZWKDWx(0) is positive for ȕ1 = b1 with any negativeȕ2 or forȕ2 = b2 with any negativeȕ1. Furthermore, it is easy to calculate lim E x(0)

E 2 of

l1† L0 (T1†  k0 ) , l1l3 (T 3  T1 )

252

Eiji B. Hosoda

whereȕ1 = b1. Since T1 < k0 < E3KROGVLQFDVHVL ±LL ZHNQRZWKDW lim B (0)

E 2 of

lim E x(0) t 0

E 2 of

holds if T†1 • k0 LVVDWLV¿HG In the same way, we can easily show that, if T† ”N0 LVVDWLV¿HG, we obtain lim E x(0)

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E1 of

l2† L0 (k0  T 2† ) t 0, l2 l3 (T 2  T 3 )

LQFDVHVLLL ±LY ZKHUHȕ2 = b2, since T3 < k0 < T2 holds. Thus, we have shown that

lim B(0)

E1 of

lim E x(0) t 0.

E1 of

In any case, insofar as ȕ1 = b1KROGVLQFDVHVL ±LL RUȕ2 = b2 holds in cases LLL ±LY  B(0) does not go to non-positive value as ȕ2 or ȕ1 goes to minus LQ¿QLW\ 1H[WOHWXVFRQVLGHUKRZ B(0) changes corresponding to decrease in ȕ1 orȕ2: wB(0) wE 2

K ½ w ­ 1 § 0 · ® E  ¨ ¸¾ wE 2 ¯ © L0 ¹ ¿

§ K0 · w ­ 1 § K 0 · ½ (0, 1) Â 1 ¨ ¸  E ® Â ¨ ¸ ¾ . (16.A.10) L w E2 ¯ © 0¹ © L0 ¹ ¿

Calculating the second term in the right-hand side of (16.A.10), we have

w ­ 1 § K 0 · ½ ®  ¨ ¸¾ wE 2 ¯ © L0 ¹ ¿

­ ½ w lˆ2 ˆ w aˆ2 § w lˆ2 w aˆ2 · °  l1 ° aˆ1 ˆ § · ˆ l a  wE 2 wE 2 1 ¨ ° 2 2 ¸° § K0 · wE wE ® ¾ ¨ ˆ ¸ 2 2 2 det © l1 aˆ1 ¹ det ¨¨© 0 0 ¸¸¹° ¨© L0 ¸¹ ° °¯ °¿ aˆ1l3  lˆ1a3 1 § lˆ2  aˆ2 · § K 0 · 1 § a3 L0  l3 K 0 · ¨ ˆ ¸¨ ¸ ¨ ¸¹ 0 det det © l1 aˆ1 ¹ © L0 ¹ det © a1l3  l1a3 1 § a3 L0  l3 K 0 · x(0)  ¸¹ , 0 det det ¨©

where we have utilized â1l3± lˆ1 a3 = a1l3±l1a3 and x(0)

§ K0 · Â 1 ¨ ¸ © L0 ¹

1 § lˆ2 ¨ det © lˆ1

 aˆ2 · § K 0 · ¸ ¨ ¸. aˆ1 ¹ © L0 ¹

Equilibria in an emission permit scheme 253 Thus, we have

w B(0) wE 2

§ K0 · a l  l a §a L  l K · 1 (0, 1)  1 ¨ ¸  1 3 1 3 E x(0)  E ¨ 3 0 3 0 ¸. ˆ 0 ¹ det © L0 ¹ det A ©

Suppose that˜%(0)/˜ȕ2 = 0 holds. Then, we have

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l1l3 (T1  T 3 ) E x(0) detÂ

­ § K0 · § a L  l K ·½ 1  ®(0, 1)  1 ¨ ¸  E ¨ 3 0 3 0 ¸ ¾ , (16.A.11) L 0 ¹¿ © 0 ¹ det © ¯

where the right-hand side of (16.A.11) is calculated to be



· 1 † § a1† l1 L0 ¨ †  k0 ¸ . det © l1 ¹

where k0 { K 0/ L 0. 6LQFH WKH VWDELOLW\ FRQGLWLRQ LV VDWLV¿HG E\ K\SRWKHVLV ZH KDYH GHW < 0. Furthermore, T1 < T3KROGVLQFDVHVL ±LL +HQFH B(0) =ȕx(0) t 0 forȕ2 such that˜%/˜ȕ2 = 0. Coupling this result with the B(0) > 0 forȕ1 = b1 withȕ2 = 0 and limȕ2o±f B(0) t 0, we can tell that B (0) t 0 for any ȕ2 andȕ1 = b1 if a†1/l†1({ T†1) t k0 holds and the VWDELOLW\FRQGLWLRQLVVDWLV¿HG ,QH[DFWO\WKHVDPHZD\ZHFDQVKRZWKDW

w B(0) wE1

(0, 1) 

1 § lˆ2 ¨ det © lˆ1

 aˆ2 · § K 0 · l2 l3 (T 3  T 2 ) E x(0) ¸¨ ¸ det aˆ1 ¹ © L0 ¹

1 E 2 (l3 K 0  a3 L0 ). detÂ

Then, if˜%(0)/˜ȕ1 = 0 holds, we have l2 l3 (T 3  T 2 ) E x(0) detÂ

^

`



1 (lˆ2 K 0  aˆ2 L0 )  E 2 (l3 K 0  a3 L0 ) detÂ



a† · 1 † § l2 L0 ¨ k0  †2 ¸ . det l2 ¹ ©

Hence, if k0( { K0 /L0) t a†2/l†2 (T†2) holds, thenȕ[(0) t 0 holds, since det < 0 by hypothesis and T3 < k0 < T2LQFDVHVLLL ±LY &RXSOLQJWKLVUHVXOWZLWKWKHIDFW that B(0) > 0 forȕ2 = b2 withȕ1 = 0 and limȕ o±fB(0) t 0, we know that B(0) t 1 0 for any negativeȕ1 withȕ2 = b2, if T†2 d k0 holds and the stability condition is VDWLV¿HGŶ

254

Eiji B. Hosoda

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Notes * The author is very grateful to professors H-C. Chen, C. Gehrke, H. Kurz, C. Lager, N. Tarui, A. Maeda, Y. Maeda, and participants at the Graz seminar held on November 2009 for their helpful comments and criticism. 1 The assumptions on distribution and saving behaviour adopted in this paper do not essentially affect the results obtained here. 2 Of course, the emission constraint may be binding even when the abatement sector is QRWDFWLYDWHG%XWZHH[FOXGHWKLVFDVHVLQFHLWLVQRWHVVHQWLDOIRUWKHSXUSRVHRIWKLV paper. 3 Here, we are not interested in the case where the emission constraint is not binding, i.e. the case where B > b1x1 + b2x2 ±x3 holds in an equilibrium 4 It might be said that the constant amount of emission permits would not be compatible with a steady state, since a steady state implies that every variable grows at the same rate as the growth rate of labour supply. Yet, since the amount of per capita emission permits becomes negligible as time goes on in a growing economy, we may include that case in a steady state where b = 0. 5 This type can be seen in a waste management policy of a small town in Japan, where a certain number of tradeable waste bags are allocated to each family member. 6 For detailed argument on this type of differential equations, see Gandolfo, G. (1980). 7 For the system (16.1) and (16.2) to have a non-negative solution in a steady state, the growth rate nPXVWEHVPDOOHUWKDQWKHPD[LPXPUDWHZKLFKLVJLYHQDVa†1.   7KH XSSHU ERXQG ORVHV WKH PHDQLQJ LI ZH DOORZ D SRVVLELOLW\ RI H[FHVV VXSSO\ RI emission permits. 9 If we concentrate upon a neighbourhood of a long-run (steady-state) equilibrium, we do not have to be worried about this problem, since a short-run equilibrium always H[LVWV LQ WKH QHLJKERXUKRRG ([LVWHQFH RI D VKRUWUXQ HTXLOLEULXP LV KRZHYHU QRW guaranteed once it comes to an argument on a global stability. 10 We do not deal with the case where a target is the constant growth rate of emission permits, since the procedure is essentially the same. 0 11 Notice that Q † §¨ ·¸ when V = 0. ©n¹ 12 Here, we assume thea c0 = 0. 13 If B < 0 holds, this means that the government buy emission permits from capitalists. 14 See Shinkai, Y. (1960) and Uzawan, H. (1961) for details. 15 We have assume that E1 y1*  b2 y*2 t 0 LQWKHVXEVHFWLRQRQ'\QDPLFV

References $RNL0 µ'XDO6WDELOLW\LQD&DPEULGJHW\SH0RGHO¶ Review of Economic Studies, 9ROSS± $WNLQVRQ 6 (   µ0DUNHWDEOH 3ROOXWLRQ 3HUPLWV DQG $FLG 5DLQ ([WHUQDOLWLHV¶ Canadian Journal of Economics,9ROSS± Atkinson, S. E. and T, H. Tietenberg (1982) ‘The Empirical Properties of Two Classes of 'HVLJQ IRU 7UDQVIHUDEOH 'LVFKDUJH 3HUPLW 0DUNHWV¶ Journal of Environmental Economics and Management,9ROSS± Baumol, W. J. and W. E. Oates (1990) The Theory of Environmental Policy, second edition, Cambridge University Press, Cambridge. Coase, R. (1960) ‘The Problem of Social Cost’, Journal of Law and Economics, Vol. 3, SS±

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Equilibria in an emission permit scheme 255 )RVWHU9DQG5:+DKQ µ'HVLJQLQJ0RUH(IILFLHQW0DUNHWV/HVVRQVIURP/RV Angeles Smog Control’, Journal of Law and Economics,9RO;;;9,,,SS± Gandolfo, G. (1980) Economic Dynamics, 3rd edition, Springer-Verlag, Berlin. Grimaud, A. (1999) ‘Pollution Permits and Sustainable Growth in a Schumpeterian Model’, Journal of Environmental Economics and Management,9ROSS± Hahn, R. W. (1984) ‘Market Power and Transferable Property Rights’, Quarterly Journal of Economics,SS± Hahn, R. W. (1989) ‘Economic Prescriptions for Environmental Problems: How the Patient )ROORZHG WKH 'RFWRU¶V 2UGHU¶ Journal of Economic Perspectives, Vol. 3, No. 2, SS± +RVRGD( µ*URZWKDQG'LVWULEXWLRQXQGHUDQ(QYLURQPHQWDO5HVWULFWLRQ¶ The Manchester School9RO/;,,1RSS± +RVRGD( µ$Q(QYLURQPHQWDO5HVWULFWLRQDQG,QFRPH'LVWULEXWLRQLQD&DSLWDOLVW Economy’, Metroeconomica9RO1RSS± .UXSQLFN$ - : ( 2DWHV DQG ( 9 ' 9HUJ   µ2Q 0DUNHWDEOH$LU3ROOXWLRQ Permits: The Case for a System of Pollution Offsets’, Journal of Environmental Economics and Management9ROSS± McGartland, A. M. and W. E. Oates (1985) ‘Marketable Permits for the Prevention of (QYLURQPHQWDO'HWHULRUDWLRQ¶ Journal of Environmental Economics and Management,9ROSS± 0RQWJRPHU\ : '   µ0DUNHWV LQ /LFHQVHV DQG (IILFLHQW 3ROOXWLRQ &RQWURO 3UR grams’, Journal of Economic Theory,9RO1RSS± 1DJXUQH\$DQG'=KDQJ µ'\QDPLFVRID7UDQVSRUWDWLRQ3ROOXWLRQ3HUPLW6\VWHP with Stability Analysis and Computation’, Transportation Research Part D, Vol. 6, SS± 1DJXUQH\$DQG.'KDQGD µ$9DULDWLRQDO,QHTXDOLW\$SSURDFKIRU0DUNHWDEOH Pollution Permits’, Computational Economics,9ROSS± 1DJXUQH\$.'KDQGDDQG-6WUDQOXQG µ*HQHUDO0XOWLSURGXFW0XOWLSROOXWDQW Market Pollution Permit Model: A Variational Inequality Approach’, Energy Economics,9ROSS± Roberts, M. J. and M. Spence (1976) ‘Effluent Charges and Licenses under Uncertainty’, Journal of Public Economics,9ROSS± 5XELQ-' µ$0RGHORI,QWHUWHPSRUDO(PLVVLRQ7UDGLQJ%DQNLQJDQG%RUURZLQJ¶ Journal of Environmental Economics and Management,9ROSS± Shinkai, Y. (1960) ‘On Equilibrium Growth of Capital and Labour’, International Economic Review,9RO1RSS± 7LHWHQEHUJ7 µ7UDQVIHUDEOH'LVFKDUJH3HUPLWVDQGWKH&RQWURORI6WDWLRQDU\6RXUFH Air Pollution: A Survey and Synthesis’, Land Economics,9RO1RSS± Reprinted in Tietenberg (1995). 7LHWHQEHUJ7 µ7KH(PSLULFDO3URSHUWLHVRI7ZR&ODVVHVRI'HVLJQVIRU7UDQVIHUDEOH 'LVFKDUJH3HUPLW0DUNHWV¶ Journal of Environmental Economics and Management, 9ROSS±5HSULQWHGLQ7LHWHQEHUJ  Tietenberg, T. (1994) Economics and Environmental Policy, Edward Elgar, Cheltenham. Tietenberg, T. (1995) ‘Tradeable Permits for Pollution Control when Emission Location Matters: What have We Learned?’ Environmental and Resource Economics, Vol. 5, SS± Uzawa, H. (1961) ‘On a Two-Sector Model of Economic Growth’, Review of Economic Studies,9RO;;,;SS±

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Eiji B. Hosoda

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Van Egteren and M. Weber (1996) ‘Marketable Permits, Market Power, and Cheating’, Journal of Environmental Economics and Management,9ROSS± @ )XMLPRWR 7 +HUUHUR & 9LOODU $ µ$ 6HQVLWLYLW\ $QDO\VLV IRU /LQHDU 6\VWHPV Involving M-Matrices and Its Application to the Leontief Model’, Linear Algebra and Its Applications 64, 1985, pp. 85–91. >@)XMLPRWR7DQG5DQDGH5µ7HFKQLFDO&KDQJHVDQGWKH5DWHRI3UR¿WLQ0RGHOVZLWK Joint Production and Externalities’, Metroeconomica 49, 1998, pp. 129–38. >@)XMLPRWR7+HUUHUR&5DQDGH56LOYD-9LOODU$µ$&RPSOHWH&KDUDFWHUL]DWLRQRI Economies with the Nonsubstitution Property’, Economic Issues 8(2), 2003, pp. 63–70. KWWSZZZHFRQRPLFLVVXHVRUJDUFKLYHYROLQGH[KWPODFFHVVHG-DQXDU\  >@0DU[.Capital III.WKHRULJLQDO*HUPDQHGLWLRQLQKWWSZZZPDU[LVWVRUJ DUFKLYHPDU[LQGH[KWPDFFHVVHG-DQXDU\  >@0RULVKLPD0Equilibrium, Stability and Growth: a Multi-Sectoral Analysis, Oxford University Press, Oxford, 1964. >@0RULVKLPD0Marx’s Economics – A Dual Theory of Value and Growth, Cambridge University Press, Cambridge, 1973. >@0RULVKLPD0DQG6HWRQ)µ$JJUHJDWLRQLQ/HRQWLHI0DWULFHVDQGWKH/DERXU9DOXH Theory’, Econometrica 29, 1961, pp. 203–20. >@2NLVKLR1µ7HFKQLFDO&KDQJHDQGWKH5DWHRI3UR¿W¶Kobe University Economic Review 7, 1961, pp. 85–99. >@3DU\V:µ7KH'HYLDWLRQRI3ULFHVIURP/DERU9DOXHV¶American Economic Review 72, 1982, pp. 1208–12. >@6KDLNK$µ7KH7UDQVIRUPDWLRQIURP0DU[WR6UDIID¶LQRicardo, Marx and Sraffa, A. Freeman and E. Mandel (eds), Verso Books, London, 1984, pp. 43–84.

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Part V

Neoclassical theory and its critics

18 The stability of a competitive economy A belated rejoinder

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Takashi Negishi

I In September 1960, I read a paper on the stability of a competitive economy in Naples meeting of the Econometric Society (Bessiere, 1962). This was mainly a survey of researches carried on at Stanford University.1 Then, I was invited to contribute a properly revised and enlarged version of the paper to Econometrica. The paper was accepted to be published in Econometrica as one of the survey papers Econometrica was then publishing. Two years later, it was published as “Stability of a competitive economy, a survey article” (Negishi, 1962). Since then, this article has been widely read, and received many comments, among which the most detailed was the one from Professor Weintraub (Weintraub, 1991). This is a book on the economic dynamics and the stability analysis of equilibrium, written from the point of view of a new research program in the history of economic theory. It was my great honor as the author of the article to be commented on in detail by this book. This note is a much delayed partial rejoinder to comments from Professor Weintraub.2 It is by no means a critical review of Weintraub’s book itself, such is, of course, clearly beyond my power. What I wish to do is merely to clear up some minor points. The construction of this note is as follows. In section II below, it is made clear for what purpose the survey was written, and emphasized that it was clearly mentioned in the introductory section of the article. This is to insist that, at least, I am not guilty for what I had no intention to do, though I do not deny the consequential responsibility. In the following section III, the most important comment made by Weintraub is introduced. Sections IV and V are my rejoinder to it.

II Any survey article has to make clear from what point of view it intends to survey what kind of researches. Negishi (1962) stated, therefore, as follows in the abstract which is given before the text of the article is started. Some of the recent contributions to the problem of the stability of a competitive economy are surveyed. Emphasis is laid on the uses of economic laws such

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Takashi Negishi as Walras’ law, the homogeneity of demand functions with respect to prices, etc., in proving the stability, and on the development of models of nontâtonnement processes of adjusting in the market. (Negishi, 1962, p. 635)

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Furthermore, the Introduction of the article started as follows. The theory of the general competitive equilibrium, as developed by Leon Walras, has recently been reformulated in terms of fairly advanced mathematical methods. . . . However, it was not until the paper by Arrow and Hurwicz was published that the stability problem of a competitive economy was investigated systematically within the framework of general equilibrium analysis. (Negishi, 1962, p. 635) The problem is, therefore, in a perfectly competitive economy, composed of commodities of any number, say n commodities, whether a general equilibrium where demand and supply of all the commodities are equalized is stable. In other words, whether a general equilibrium can be established by changes in prices caused by the difference between demand and supply. In the consideration of this problem, furthermore, it is clearly stated that emphasis is laid (1) on the uses of economic laws such as Walras’ law, the homogeneity of demand functions with respect to prices, etc., in proving the stability, and (2) on the development of models of non-tâtonnement processes of adjusting in the market (Negishi, 1962, p. 635). The economic laws mentioned in (1) in the above can be established only when the economic models considered are closed ones, that is, they are completed systems. In other words, researches to be surveyed are those on the general equilibrium models, which include not only many but also all the commodities in the economy. Considerations on partial equilibrium models are not attempted, even though many (but not all) commodities are considered in such models. Of course, surveys from different points of view can be possible and fully worthwhile. For example, surveys from the point of view of the development of PDWKHPDWLFDOPHWKRGVDUHXWLOL]HG:HKDYHQRLQWHQWLRQWRGHQ\WKHVLJQL¿FDQFHRI such surveys. What we wish to emphasize is that such was not our aim in our survey on the developments of the stability of general equilibrium in Negishi (1962). For the sake of the completeness of the exposition, Walras’ law and the homogeneity of demand and supply functions with respect to prices are to be sketched out. Consider an economy of n commodities. Let p = (p1, . . ., pn) be a price vector, where pi is the price of the i-th commodity. The excess demand (the difference of demand and supply) for the i-th commodity Ei(p) is homogeneous of degree zero in p so that we have Ei(ȜS) = Ei(p) for any Ȝ > 0. This homogeneity implies that only the relative prices matter and that the real economy remains unchanged if the prices of all the commodities change proportionally. Summing up the individual budget FRQVWUDLQWVZHKDYHDQLGHQWLW\FDOOHG:DOUDV¶ODZLHȈi pi Ei = 0 for any p.

The stability of a competitive economy 281

III Among Weintraub (1991)’s comments on Negishi (1962), perhaps the most important and the most problematic one is the accusation that the latter ignores the contributions made by Bushaw and Clower (1954, 1957).3 In the below, we quote three paragraphs A, B and C from Weintraub(1991).

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A I submit that a potentially important paper, because it was not noted in the Negishi survey, came to be considered unimportant. It is well to be clear here: For a paper not to have been cited or otherwise mentioned in the Negishi VXUYH\PD\KDYHUHÀHFWHGDFRQVLGHUHGMXGJPHQWE\1HJLVKLWKDWWKHSDSHU ZDVQRWVXI¿FLHQWO\LPSRUWDQWRULQÀXHQWLDOLQWKHVHTXHQFHRIZRUNVWKDWKH wished to highlight. Such differences of opinion about the importance of a paper are hardly unusual in scholarly enterprises. Nevertheless, that judgment OHGWRDEHOLHIWKDWWKH/LDSXQRYPHWKRGVZHUH¿UVWLQWURGXFHGLQWRHFRQRPLFV literature by Arrow and Hurwicz in 1958, when in fact they had been introduced earlier.   6SHFL¿FDOO\,ZDQWWRGLVFXVVDSDSHUIURPEconometrica, which published most of the papers Negishi cited, titled ‘Price Determination in a Stock-Flow Economy’ (Bushaw and Clower, 1954). The import of the paper ZDVWKDWZKHUHDVÀRZDGMXVWPHQWPHFKDQLVPVFRXOGEHXQVWDEOHHYHQLQWKH linear case under not unreasonable assumptions, the addition of stockadjustment mechanisms could lead to stabilized systems (especially because stock mechanisms ‘average out’ past deviations). More important, however, is the fact that Clower and Bushaw explicitly used, cited, and discussed the second or indirect Liapunov technique for demonstrating asymptotic stability of the competitive equilibrium. (Weintraub, 1991, pp. 136–7) B [W]hat I wish to stress is that the Clower-Bushaw paper was lost in a way that the 1956 paper by Enthoven and Arrow, say, was not lost. Mathematically, the Enthoven-Arrow paper, titled ‘A Theorem on Expectations and the Stability of Equilibrium,’ was primitive compared with the Clower-Bushaw treatment, which used Malkin’s approach to stability via Liapunov functions. But the Enthoven-Arrow paper lives on in the Negishi survey and is in fact reprinted in the Arrow and Hurwicz (1977) collection of papers. Now, there are various reasons why one paper lives and another dies. On the basis of the fact Arrow and Hurwicz ‘solved’ the problem, it would seem likely that earlier papers by Arrow would get cited, and a paper by Clower would not. But in fact, Arrow and Hurwicz (1958), in their bibliography, and

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Takashi Negishi LQWKHWH[WLQWKHVHFWLRQRQWKHGH¿QLWLRQRIJOREDODV\PSWRWLFVWDELOLW\FLWHG the Bushow-Clower book (1957), which contained discussion on their paper, and did not cite the Enthoven-Arrow paper. There was no self-aggrandizement at work at all; Arrow and Hurwicz were scrupulously fair to their sources as they knew them. (Weintraub, 1991, p. 137)

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C I want to suggest that Negishi’s notice and mention of the Enthoven-Arrow SDSHUDQGKLVODFNRIQRWHRIWKH&ORZHU%XVKDZSDSHUVXI¿FHWRH[SODLQZK\ the former became a part of the ‘history,’ whereas the latter dropped from sight, even though Arrow and Hurwicz used and correctly cited the ClowerBushaw paper. Of course, the Clower-Bushaw paper was not exactly related to the sequence of papers that ended in the global stability theorem of Arrow et al. (1959). Yet the survey acted to identify a line of early work that served as precedent in some fashion or another. Negishi’s mention of Allais’s 1943 work created, later, a source for claiming that Allais had been a precursor in DVLJQL¿FDQWPDQQHUWRWKHODWHU/LDSXQRYEDVHGOLWHUDWXUH$QG1HJLVKL¶V failure to mention Clower-Bushaw precluded that work from likewise being WDNHQDVµVLJQL¿FDQW¶LQDSUHFXUVRUVHQVH (Weintraub, 1991, pp. 137–8)

IV It cannot be denied that a paper which is actually important may disappear from the consideration of the profession simply because it was not properly recognized in a survey. As is insisted in paragraph A quoted in section III, it cannot be denied that the profession’s recognition of the importance of the article, Bushaw and Clower (1954), might be delayed, simply because Negishi (1962) did not pay attention to it. +RZHYHUHYHQDVXUYH\DUWLFOHLVDVFLHQWL¿FDUWLFOH,QVXFKDQDUWLFOHZHKDYH to begin by limiting the scope of the investigation. As was mentioned in II above, Negishi (1962) clearly stated that emphasis is laid on the uses of economic laws such as Walras’ law, the homogeneity of demand functions with respect to prices, etc., in proving the stability (Negishi, 1962, p. 635). In other words, the development of mathematical methods utilized for the stability was beyond the scope of the survey. Naturally, Bushaw and Clower (1954) was not quoted in the survey, simply because Walras’ law and the homogeneity of demand functions are not utilized there in proving the stability. It was not mentioned, since it is beyond the scope of the survey. There was no other intention. Incidentally, in Negishi (1962), a mathematical method called Lyapunov’s method is noted only in footnote 10 on page 641. There is, of course, absolutely no statement that the Lyapunov methods

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The stability of a competitive economy 283 ZHUH¿UVWLQWURGXFHGLQWRWKHHFRQRPLFVOLWHUDWXUHE\$UURZDQG+XUZLF]LQ See, however, quotation A from Weintraub (1991) above. For Clower and Bushaw, it was unlucky that there happened to be no other survey papers which aim to be concerned mainly with the development of mathematical methods in economics. In fact, however, an article which is not concerned with the problems to be surveyed may sometimes be referred to in the negative sense. An example is the case of Enthoven and Arrow (1956) referred in quotation B in the above. This article shares a co-author with Arrow and Hurwicz (1958), which plays the leading role in our survey article, i.e., Negishi (1962). This article is, however, wanting in the recognition of the important role played by Walras’ law, which Negishi (1962) considers to be the most important contribution by Arrow and Hurwicz (1958). For Arrow, at least, this recognition did not exist in 1956. In this sense, the article by Enthoven and Arrow (1956) was referred to in Negishi (1962). If it remains in the history, being referred to in a negative sense, there is no help for it. It might have the devil’s luck. The case of Allais (1943), mentioned in quotation C, is the converse of the case of Bushaw and Clower (1954). The reason Negishi (1962) allocated a whole section to Allais is that the latter utilized Walras’ law skillfully in his demonstration of the stability. Furthermore, it is emphasized that Allais (1943) is a pioneering contribution from the early 1940s, and that it has been almost unknown to researchers in the English-speaking world. Of course, there is no mention of Lyapunov’s method.

V In quotation B in section III above, it is stated that Arrow and Hurwicz (1958), in their bibliography, and in the text in the section RQ WKH GH¿QLWLRQ RI JOREDO DV\PSWRWLF VWDELOLW\ FLWHG WKH %XVKRZ&ORZHU book (1957), which contained discussion on their paper. 7KLVVRXQGVYHU\VWUDQJH,QDVFLHQWL¿FSDSHUQRWDQH[SRVLWRU\SDSHUEXWDQ original paper), an original article by other authors should be quoted. In other words, an expository literature written later by the same author should not be quoted. In this case, therefore, Arrow and Hurwicz (1958) did not quote the Bushaw-Clower 1954 paper. They seem to quote another, different work of %XVKDZ DQG &ORZHU ZKLFK LV SUHVHQWHG IRU WKH ¿UVW WLPH LQ WKHLU ERRN LH Bushow-Clower (1957). As a matter of fact, the quotation by Arrow and Hurwicz (1958) from the book, Bushaw and Clower (1957), referred to in quotaion B in section III above, refers to footnote 44 attached to their Theorem 9. This theorem is concerned, however, not with the global stability in the case of the gross-substitutability, but with the local stability. Therefore, it has nothing to do with Lyapunov’s method.

284

Takashi Negishi Footnote 44. For the case m = 2, this result has already been established by Bushaw and Clower (1957, p. 81). The general case has been demonstrated independently by Hahn (1958) and Negishi (1958). (Arrow and Hurwicz, 1958, p. 546)

Unfortunately, however, this statement by Arrow and Hurwicz, which Weintraub called scrupulously fair, is an amusing blunder of masters due to their misreading of the literature or some other misapprehension. Bushaw and Clower (1957) requires

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a11aʊa12a21 > 0 (aij LV WKH SDUWLDO GLIIHUHQWLDO FRHI¿FLHQW RI H[FHVV GHPDQG IXQFWLRQ RI WKH ith commodity by the price of the jth commodity) as the independent condition for the stability, and they did not realize that it can be derived from the condition of the gross-substitutability (aij > 0, ij ) by the use of Walras’ law or the homogeneity of demand functions with respect to prices. Even in the case of three commodities, (i.e., m = 2), therefore, Bushaw and Clower (1957), unlike Arrow and Hurwicz (1958), Hahn (1958) and Negishi (1958), cannot be considered to have proven that WKHJURVVVXEVWLWXWDELOLW\DORQHFDQEHDVXI¿FLHQWFRQGLWLRQIRUWKHORFDOVWDELOLW\ Incidentally, a later reference by Arrow and Hahn (1971, pp. 167–8), to Bushaw and Clower (1957) has nothing to do with the stability condition. It is a problem of monopolistic competition. Furthermore, in the footnote on page 322 of the same book, which corresponds to the footnote 44 in Arrow and Hurwicz (1958), there exists no reference to Bushaw and Clower (1957) at all.4

Acknowledgment It is my great honor ad pleasure to contribute my paper to this Festschrift for Professor Heinz D. Kurz, with whom I have shared a common interest in the history of economics. I remember, with many thanks, our cooperation to organize sessions for the history of economics in International Economic Association world congresses in 1992 at Moscow and in 1995 at Tunis. Let me also takee this opportunity to express my sincere thanks for his kind efforts to edit A History of Economic Theory, Essays in honour of Takashi Negishi (Routledge, 2009).

Notes 1 Originally, this survey was planned by Frank H. Hahn, but for some reasons of his other works, I, as a joint researcher, replaced him to do it. I was on my way to Japan, after I had worked as a research associate at Stanford University, to complete my PhD work in University of Tokyo. 2 For my current view on the stability problem of a general equilibrium, see the Introduction to the first volume of my collected essays (Negishi, 1994, p. xv). 3 Weintraub’s views on other problems, for example, those on Allais and Lyapunov’s method, are generally pertinent. 4 Quotations are respectively from p. 180 and from p. 352 of Arrow and Hahn (1971).

The stability of a competitive economy 285

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References Allais, M. 1943, Traité d’ économie pure, Paris: Imprimerie Nationale. Arrow, K.J., and L. Hurwicz, 1958, “On the stability of the competitive equilibrium, I,” Econometrica, 26: 522–52. Arrow, K.J., H.D. Block and L. Hurwicz, 1959, “On the stability of the competitive equilibrium, II,” Econometrica, 27, 265–90. Arrow, K.J., and F.H. Hahn, 1971, General Competitive Analysis, San Francisco: HoldenDay. Arrow, K.J., and L. Hurwicz (eds), 1977. Studies in Resource Allocation Processes, Cambridge: Cambridge University Press. Bessiere, F., 1962, “Compte rendu du conger de Naples,” Econometrica, 30: 188–206. Bushaw, D., and R.W. Clower, 1954, “Price determination in a stock-flow economy,” Econometrica, 22: 328–43. Bushaw, D., and R.W. Clower, 1957, Introduction to Mathematical Economics, Homewood, IL: Richard D. Irwin. Enthoven, A.C., and K.J. Arrow, 1956, “A theorem on expectation and the stability of equilibrium,” Econometrica, 24, 288–93. Hahn, F.H., 1958, “Gross substitute and the dynamic stability of general equilibrium,” Econometrica, 26, 169–70. Negishi, T., 1958, “A Note on the stability of an economy where all goods are gross substitutes,” Econometrica, 26, 445–7. Negishi, T., 1962, “The stability of a competitive economy: a survey article,” Econometrica, 30, 635–69. Negishi, T., 1994, The General Equilibrium Theory, the collected essays of Takashi Negishi, Volume 1, Cheltenham: Edward Elgar. Weintraub, E.R., 1991, Stabilizing Dynamics, Cambridge: Cambridge University Press.

19 A sonnet-divertimento for Heinz Kurz

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Paul A. Samuelson and Erkko Etula

Editorial Note: This essay was originally submitted by Erkko Etula and the late Paul A. Samuelson to the Symposium “Production, Distribution, and Growth”, which was held in May 2006 at the University of Graz in honour of Heinz Kurz on the occasion of his sixtieth birthday. Professor Samuelson kindly agreed, in spring 2009, to the editors’ suggestion to arrange for its publication in some appropriate form. Dr Etula, in spring 2010, then kindly accepted our invitation to submit it to the present volume.

Richard Wagner, to celebrate his wife’s birthday, placed a small orchestra on the staircase to her bedchamber. That way Cosima Liszt-Wagner awoke to be the very ¿UVWDXGLHQFHWR³6LHJIULHG¶V,G\OO´ ,QOLNHWULEXWHWR+HLQ]3DXO6DPXHOVRQDQG(UNNR(WXODKDYHFRQFRFWHGD divertimento VFLHQWL¿F VRQQHW WR SLTXH KLV LQWHUHVW %HLQJ QRQPXVLFLDQV RXU QRWDWLRQVDUHQRWRIWKHWUHEOHDQGEDVVFOHIW\SH%XWWKHLUUXOHVDUHDVVWULFWDVWKH recipe for a Chopin piano gem. Time limitation necessitates only a token sampling QRZRIRXUIXWXUH¿QDORIIHULQJ We believe Herr Professor Kurz would like it. And what interests him has also peculiarly interested a young 1912 Graz professor of economics. And as well the present posy should be of interest to the likes of David R., Nassau S., Eugen von %%,UYLQJ))UDQN5DQG5REHUW06$OVRZHVKRXOGQRWIRUJHWWRPHQWLRQ Joan R. and Piero S. Robinson Crusoe is the protagonist of our little saga. Our mythical Crusoe is the Crusoe Homo Economicus not yet ever seen on land or sea. He seeks evenhandedly maximal consumptions in every t to t +1 interval of two heterogenous goods: à la 6UDIID&K, ZHGXEWKHP:KHDWDQG,URQ/LNH-60LOO &UXVRH also spends available real income 50–50 on C1(t  DQGC2(t   %\WULDODQGHUURU&UXVRHKDVGLVFRYHUHGDIHZGLIIHUHQWZD\VWRSURGXFHJURVV Q1 (t DQGDOVRJURVVQ2(t RXWRIKLV¿[HGDPRXQWRIXQLWGLUHFW/DERUL(t  ŁŁ L (t wheat + L (t iron,QLWLDOO\IRUDORQJWLPHSDVWWKHVHVROH/DERULQSXWV could produce only spartan diets of C1(t DQGC2(t %XWRYHUWLPHWKH ERXUJHRLV (QJOLVKPDQ FDVWDZD\ KDV OHDUQHG ¿YH DOWHUQDWLYH ZD\V WR SURGXFH Wheat’s Q1(t RXWRI/DERUt DQG:KHDWt DQG,URQt WKHVHGLIIHUHQWZD\V are labeled a, b, c, d, e.$QGE\FRLQFLGHQFH &UXVRHKDVDOVROHDUQHG¿YHGLIIHUHQW

A sonnet-divertimento for Heinz Kurz   Table 19.1 3URGXFWLRQVXEWHFKQLTXHVXVDEOH/LWHUDU\H[SODQDWLRQRIDW\SLFDOURZVXFK as cEHORZRI/DERU RI:KHDW RI,URQZLOOSURGXFHRQHSHULRG later 12 of Gross Wheat.

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Wheat sector

,URQVHFWRU

E1

E2

Subtechnique

Į0

Į1

Į2

a

1

0

0

1 4

0

b

1

1

0

2

0

c

1

1



12

0

d

1





19

1 10

0

e

1



9

20

1 10

0

A

1

0

0

0

1 4

B

1

0

1

0

2

C

1



1

0

12

D

1





0

19

1 10

E

1

9



0

20

1 10

ZD\VWRSURGXFH,URQ¶VQ2 (t RXWRI/DERUt DQG:KHDWt DQG,URQt LQSXWV these different ways are labeled A, B, C, D, E. Our Table 19.1 is the Rosetta Stone encapsulating all that is known about what economists will recognize is not a neoclassical production function of -% &ODUN:LFNVWHHG:LFNVHOO5DPVH\+LFNV'RXJODV6RORZ “marginal SURGXFWLYLWLHV>˜Q˜L@´PDLQVWUHDPW\SH,QVWHDG&UXVRHKDVVWXPEOHG onto the limited-substitutability technology of Leontief-Sraffa-Robinson-v. :LHVHU'DQW]LJ%HOOPDQ.XKQ7XFNHU¿QLWHPDWKW\SH3UDFWLFDO&UXVRHKDV zero awareness or interest in any scalar one-Kapital surrogates; nor in metaphysical “diminishing marginal utility” notions. What is objective and WHVWDEOHDERXWKLVGHPDQGWDVWHVLVWKDWIRUKLPDQG0LOO DOZD\VP1/P2 = C2/C1$QGZHUHKLV\HQIRU:KHDWWRULVHUHODWLYHWR,URQDQGWKHUHE\VKLIW demand to say P1/P2 = 2(C2/C1 WKDWZRXOGYLWDOO\DOWHUWKLVWDOHRIKLVULVH IURPUDJVWRULFKHV Table 19.1 has been singularly composed so that Crusoe’s optimal saving program FDQEHWUDQVSDUHQWO\VROYHGIRU,IDUHDGHUZHUHWRDOWHUDQ\RILWVQXPHU LFDOGDWDE\DQHSVLORQWKDWZRXOGOLWWOHFKDQJHRXUTXDOLWDWLYHQDUUDWLYHEXWLW ZRXOGFKDOOHQJHHYHQWKHJHQLXVRIORJLFLDQ)UDQN5DPVH\WRFRPSXWHTXLFNO\WKH new exact story. (Remark: This gives hope that what is singular here only slightly

  Paul A. Samuelson and Erkko Etula vitiates the generic story. Of course, future readers must test whether such hope is RQO\YDLQKRSH

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Crusoe’s exact dynamics The Rules of Our Game are simple but exacting. Axiom about Time Preference: Crusoe Homo Economicus states that he commits to live on thin air, with C1 (t Ł C2 (t ŁZKHQHYHUKLVVDFUL¿FHRIFXUUHQWCj can be rewarded by a greater future CjUHZDUGDQGVWXEERUQO\KHZLOOSHUVLVWLQDXVWHULW\IRUDQ\¿QLWHQXPEHURIT time periods needed to reach Mount Everests’ top Golden-Rule permanent C1g C2g C g . ,QQRWDWLRQWKDW&UXVRHZRXOGQRWXQGHUVWDQG5DPVH\OLNHKHLVDFWLQJDOZD\V so as to maximize the sum,1

{C1 (1) + C2 (1)} + {C1 (2) + C2 (2)} + . . .{C1 (t ) + C2 (t )} + . . . ∞

. . . = ∑ {C1 (t ) + C2 (t )}, 1

RXWWRLQ¿QLW\DQGVXEMHFWWR7DEOH¶VWHFKQLFDOFRQVWUDLQWV C j (t + 1) = Q j (t + 1) − K j (t + 1), j = 1, 2; etc. Niceties of proofs and explications are not needed at a birthday party itself. 7KHUHIRUHLWVXI¿FHVWRUHSRUWLQ7DEOHZKDWZLOOEHPDQGDWHGWRKDSSHQ intertemporally by Table 19.1 and the Rules of Our Game. Table 19.2 7LPHSUR¿OHRI&UXVRH¶VUDJVWRULFKHVSURJUDP Period (t) í

Subtechniques

K1t

L*

K 2t 0

Q1t 1

Q2t 1

C1t 1

C2t 1

.125

.125

.125

.125

.125

.125

aA

1

0

0

aA

1

0

0

0

1

abAB

1

.125

.125

    0

0

2

abAB

1

 

 

    0

0

3

bcBC

1

 

 

  0

0

 

bcBC

1





  0

0

5

bcBC

1





  0

0

0

6

bcBC

1





3.936

3.936

0

0

 

bcBC

1

3.936

3.936

5.295

5.295

0

0

 

cdCD

1

5.295

5.295

  0

0

9

cdCD

1



10

dD

1



 

    9.55

9.55

1.55g

1.55g

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A sonnet-divertimento for Heinz Kurz   8OWLPDWH*ROGHQ5XOH EOLVVLVVHHQWRWDNHDORQJWLPHWREHUHDFKHG:HFDOO attention to an important feature of thisVFHQDULR:LWK¿YHDOWHUQDWLYHRSWLRQVIRU HDFKRI:KHDWDQG,URQ&UXVRHFRXOGKDYHFKRVHQDYDVWYDULHW\RIPDQ\GLIIHUHQW combinations of activities: bcBC or bcDE or . . . . Note, however, that in every case he has ended up with the capital letter just matching the same little letter. This means that he ended always on the main diagonal in the (K1, K2 VSDFHZKHUH K1ŁK2. Why is this? Why did he not go from the origin in the (K1, K2 SODQHWR an off-diagonal point? That was because, by his own trial and error he found that this would have slowed his arrival at the Golden Age. How come Table 19.1 was VROXFN\DVWRSURGXFHVXFKDQHDV\¿QGLQJ"7KDWLVZKHUHRXURZQLQJHQXLW\ came in. Careful perusal of Table 19.1 will discern its “skew-symmetry.” Whatever can be true of lower case Wheat production can also be true here of upper case ,URQSURGXFWLRQ7KLVLVQRWDWDOODQLQVWDQFHRIcovert aggregate scalar capital contrivance. Rather our sole reason for choosing skew-symmetry is so that Crusoe would not have had to develop the ingenuity to solve two second-order difference HTXDWLRQVRI(XOHU/DJUDQJHFDOFXOXVRIYDULDWLRQVW\SH,QWKH/HRQWLHI6UDIID domain of limited substitutability and where discrete time intervals replace GLIIHUHQWLDOHTXDWLRQÀRZVRIYDULDEOHVPD[LPL]DWLRQLVVWUDLJKWIRUZDUG%XWVWLOO it is different from physicists’ historic maximizations.

'HWHUPLQDWH5LFDUGLDQZDJH±SUR¿WGLVWULEXWLYHVKDUHV Tables 19.1 and 19.2 together also resolve what Ricardo stated should be a principal preoccupation of political economy: How do supply-and-demand competitive markets determine what will be the respective distributive shares (“imputations”) of society’s total net harvest between the Wages of property-less ZRUNHUVDQGWKHVDIHXQLIRUP,QWHUHVWRU3UR¿W UHWXUQVWRRZQHUVRI:KHDWDQG Iron input assets? Crusoe, of course, is really a stand-in for a mob of Arrow-Debreu homogeneous workers and of alikeRZQHUVRI:KHDWDQG,URQKHWHURJHQHRXVFDSLWDOV2 And we ZLOO QH[W FRQ¿UP WKDW YHFWRUDO DEXQGDQFH RI KHWHURJHQHRXV FDSLWDOV ZLOO LI anything, raise real wage rates while depressing interest rates.3)RUDPXOWLSHUVRQ society, market mechanisms of supply and demand and Darwinian advantageseeking arbitrages accomplish unconsciously what Crusoe discovered by personal trial and error. Here is Table 19.3 on distributional pricing, as mandated by Tables 19.1 and 19.2, and displayed prior to any explanation. Why do the (W/Pj wages have to rise at the later stages of (K1, K2 YHFWRUDODEXQGDQFHV":K\PXVWWKHi uniform safe interest rate have to fall? Table 19.3’s successive (i, W/P1, W/P2 supply–demand determinants are the VROHRQHVWKDW DVVXUHIXOOHPSOR\PHQWXVHRIHDFKDQGHYHU\VFDUFHLQSXWDQG  DVVXUHWKDWZKHUHYHU:KHDWDQG,URQDUHHDFKEHLQJSURGXFHGWKHUHLVWDNLQJ place a full break-even of (L, K1, K2 WLPHt costs with time t + 1 sales revenues of :KHDWDQG,URQ

290

Paul A. Samuelson and Erkko Etula

Table 19.3 Distributive wage and interest levels at early and late stages

Distributive shares Stage

i indeterminate

aA

(W/P1

(W/P2





abA, abB, aAB, bAB, abAB







bcB, bcC, bBC, cBC, bcBC







cdC, cdD, cCD, dCD, cdCD



2.9

2.9

dD*ROGHQ$JH 

0

3.1

3.1

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Note: i of 3/ or 1/ or 1/PHDQVRURUSHUSHULRG

&UXVRH QHYHU FRXQWV DQG FRPSDUHV QXPEHUV RI HTXDWLRQV ZLWK QXPEHUV RI XQNQRZQVDQGQHYHUQHHGVWRUXOHRXWQHJDWLYHQXPEHUV%XW3LHURDQG(UNNRDQG 3DXO KDYH D GXW\ WR PRQLWRU WKDW MRE 1HZO\ HQODUJHG DFTXLVLWLRQV LQ WKH (K1, K2 YHFWRUDUHWKHUHDVRQZK\a and A had to become extinct and had to be replaced by c and C in the new bcBC stage. No magic is involved and no omissions can be permitted. 7KHVKDGHRI-RDQ5RELQVRQZLOOZDQWWRYHWWKLV,QYLVLEOH+DQGSUHWHQVLRQ 5HPDUN+LGGHQLQRXUVRQQHWLVGRXEOHVZLWFKLQJLWVHOI%XWQRWZLWKVWDQGLQJ LQWHUWHPSRUDO3DUHWRRSWLPDOLW\EHQLJQO\SUHYDLOV

Final remarks We have been careful to ensure that none of our implied “marginalisms” are “spurious marginalisms.” Heterogeneity of capitals has been explored: what holds for N   :KHDW DQG ,URQ FDSLWDOV SURSHUO\ JHQHUDOL]HG VKRXOG KROG IRU VD\ N = 10. Serious readers will want to go beyond our saga of zero systematic time preference. Armed with Table 19.1’s Rosetta Stone, any reader has the information QHHGHGWRGHGXFHWKHRSWLPDOVDYLQJWLPHSUR¿OHWKDWPD[LPL]HVVD\JHQHUDOL]HG Ramsey f

1

¦   G t 0

t

>ORJ C1 t    ORJ C2 t   @ G ! 

VXEMHFW WR RXU 5XOHV RI WKH *DPH FRQVWUDLQWV )RU WKLV WKH RSWLPDO SDWKV ZLOO GHSHQGRQįZLWK OLP^C1 t  C2 t ` ^C1ä  C2ä ` t of

where our previous Golden Rule C jg ! C äj for G > 0. ,QZRUGVKLJKHUįGLVFRXQWLQJIRUIXWXULW\ZLOOPRYHWKHXOWLPDWHDV\PSWRWH SRLQW RQ WKH PDLQ ƒ diagonal nearer to the origin. Neither classical nor contemporary mainstreamers will be surprised by this.

A sonnet-divertimento for Heinz Kurz

291

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2XUVDJDRUIDEOH KHOSVXQGHUVWDQGLQJLIXQGHUVWDQGLQJLVQHHGHGRIWKHODVW KDOIFHQWXU\¶V³PLUDFOH´GHYHORSPHQWVRXWVLGHRIWKH86,QRXUVDJD&UXVRH¶V VDYLQJSURJUDPVUDLVHGKLVOLYLQJVWDQGDUGVWZHOYHIROGLQ¿QLWHWLPH)ULGD\D similar human being on a similar island, if he became possessed of Crusoe’s Table 19.1 Rosetta Stone of knowledge, could hope to accomplish the same non-miracle. 6XEVWLWXWH$PHULFDIRU&UXVRHDQG-DSDQIRU)ULGD\WRUHDOL]HWKDWQRPLUDFOHV need be involved. ,IDQ\FOHYHUUHDGHUVGLVFRYHUDQ\KLGGHQPDUJLQDOLVPVLQRXUWDOH±KLGGHQLQ SODLQVLJKW±WKH\ZLOOQRWEH-%&ODUNPDUJLQDOLVPVQRUZLOOWKH\EH³VSXULRXV marginalisms.” Caveat Lector:5HDGHUV%HZDUH

Notes  2YHULQ¿QLWHWLPHVPDWKHPDWLFDOVHULHVFDQGLYHUJHWRLQ¿QLW\UDWKHUWKDQFRQYHUJHWR D¿QLWHOLPLW1/2 + 1/ + . . . + 1/2n . . . = 2 versus 1+ 1/2 + 1/3 + . . . + 1/n + . . . = f 5HDGHUV RI 5DPVH\   ZLOO SHUFHLYH WKDW ZKHQ &UXVRH ZRUNV HTXLYDOHQWO\ WR “minimize divergence from Golden-Rule maximal per-period C’s,” his optimal Cj (t  WLPHSUR¿OHZLOOFRQYHUJHIRUUHODWLYHO\ORZ7WLPHSHULRGV  /LNH6UDIID ZHKHUHLJQRUHZDJHIXQGDVVHWVRIUHQWLHUVZKRPD\KDYHDGYDQFHG at time t their wages prior to time t + 1 when the outputs produced can be sold. The fuller story is not hard to spell out and can be ignored here.  /LYLDWDQ6DPXHOVRQ DQG6DPXHOVRQ(WXOD KDYHFRQWULYHGMRLQWSURGXFW instances where i DQGW/Pj FDQULVHRUIDOOWRJHWKHUEHFDXVHRI:LFNVHOO(IIHFWVIRU P2/P1. Yet, zero deviations from Pareto optimality are implied for any competitive convex technology.  :KDWLVQRWJHQHULFKHUHLVWKHSURSHUW\WKDWIRUDOOį¶V¿QDODV\PSWRWLFHTXLOLEULXPLV on the K1 = K2 diagonal. Skew symmetry does have some minor costs.

References -%&ODUN 7KH'LVWULEXWLRQRI:HDOWK$7KHRU\RI:DJHV,QWHUHVWDQG3UR¿WV, The Macmillan Co., New York. 0 'REE   Theories of Value and Distribution since Adam Smith: Ideology and Economic Theory, Cambridge University Press, Cambridge. :/HRQWLHI The Structure of American Economy, 1919–1929, Harvard University Press, Cambridge, MA. 1/LYLDWDQDQG3$6DPXHOVRQ 1RWHVRQWXUQSLNHVVWDEOHDQGXQVWDEOHJournal of Economic Theory SS± -6 0LOO   Essays on Some Unsettled Questions of Political Economy, Parker, /RQGRQ5HSULQWHGLQ-05REVRQHG ± Collected Works of John Stuart Mill, University of Toronto Press, Toronto. % 2KOLQ   Interregional and International Trade, Harvard University Press, Cambridge, MA. )35DPVH\ $PDWKHPDWLFDOWKHRU\RIVDYLQJEconomic Journal SS± '5LFDUGR>@ The Works and Correspondence of David Ricardo, P. Sraffa, ed., with M.H. Dobb, On the Principles of Political Economy and Taxation, vol. 1. Cambridge University Press, Cambridge. -5RELQVRQ The Rate of Interest and Other Essays, Macmillan, London. -5RELQVRQ Accumulation of Capital, Macmillan, London.

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3$6DPXHOVRQ %ULHI¿UVWFULWLTXHRI6UDIID¶VXQ¿QLVKHGFULWLTXHRIPDUJLQDOLVP ,Q+*1XW]LQJHUHGRegulation, Competition and the Market Economy, Festschrift for Carl Christian von Weizs¨acker, Vandenhoeck and Ruprecht, G¨ottingen. 3$6DPXHOVRQDQG((WXOD &RPSOHWH:RUN8SRIWKH2QH6HFWRU6FDODU&DSLWDO 0RGHORI,QWHUHVW5DWHJapan and the World EconomySS± -$ 6FKXPSHWHU >@   Theorie der wirtschaftlichen Entwicklung, Leipzig: Duncker & Humblot. Translated as The Theory of Economic Development, Cambridge, MA: Harvard University Press. $6PLWK>@ An Inquiry into the Nature and Causes of the Wealth of Nations. ed. E. Cannan, New York: The Modern Library. 56RORZD $FRQWULEXWLRQWRWKHWKHRU\RIHFRQRPLFJURZWKQuarterly Journal of EconomicsSS± 56RORZE 7KHSURGXFWLRQIXQFWLRQDQGWKHWKHRU\RIFDSLWDOReview of Economic StudiesSS± 3 6UDIID   6XOOH UHOD]LRQL WUD FRVWL H TXDQWLWj SURGRWWD Annali di Economica ,, SS± 36UDIID 7KHODZVRIUHWXUQVXQGHUFRPSHWLWLYHFRQGLWLRQVEconomic Journal 36, pp. 535–50. 3 6UDIID   Production of Commodities by Means of Commodities: Prelude to a Critique of Economic Theory, Cambridge University Press, Cambridge. $ 0 ⎪ ⎨ xt = 0 if ⎪x < 0 ⎩ t

λt > 1 λt = 1 λt < 1

when Et > 0.

(20.13)

7KHUXOH VLPSO\VWDWHVWKDWWKHGRPLQDQW¿UPZDQWVWRLQYHVWGLVLQYHVW  ZKHQWKHPDUJLQDOEHQH¿WRILQFUHDVLQJDXQLWRIFRPSHWLWLRQUHVWULFWLQJFDSLWDO OtLVJUHDWHUOHVV WKDQLWVPDUJLQDOFRVWDQGWKHGRPLQDQW¿UPGRHVQRWKLQJ when they are same. When the non-negativity constraint is binding, . on the other hand, the dynamics of competition-restricting capital follow Et = E t = 0. Thus, from Eq. (20.1), the optimal competition-restricting investment follows xt

0

when Et

0.

(20.14)

7KHUHIRUHWKHGRPLQDQW¿UPZLOOQRWHQJDJHLQDQWLFRPSHWLWLYHDFWLYLW\ZKHQLW has no competition-restricting capital. Out-of-steady-state dynamics From Eq. (20.8) to Eq. (20.12), the market dynamics can be summarized by the law of motion of xt, xt

=

1 {(r + δ )(1 + ϕ ′ ) − p ′s ′st dt ϕ ′′

(20.15)

−( pt − C ′ )( s ′dt + st d ′p ′s ′ ) − rθ t + θt },

 and the law of motion of Et, Eq. (20.1). Note that Tt = T t = 0 when Et > 0. At a steady state, xt E t 0 holds. Three non-negative steady states, including a binding solution x* = E* = 0, possibly arise. Let’s call the three steady states E*l , E*m , and E*h in order of the steady-state size of the competitionrestricting capital. By conducting local stability analysis about each steady state, it turns out that the two steady states, E*l and E*h , are saddle stable points

298

Mika Kato and Willi Semmler

and the middle steady state, E*m , is an unstable point. We can interpret the lower stable steady state, E*l = 0, as a competitive-market environment where the GRPLQDQW¿UP¶VPDUNHWVKDUHLVQHJOLJLEO\VPDOOLH]HURLQRXUPRGHO7KH upper stable steady state, E*h > 0, on the other hand, can be interpreted as a highly dominated market. $QXPHULFDOH[DPSOH

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,QWKHIROORZLQJQXPHULFDOH[DPSOHZH¿QGPXOWLSOHVWHDG\VWDWHV We use the following simple functional forms: C (qt )

cqt ,

(20.16)

ϕ ( xt )

α xt2 ,

(20.17)

s ( Et ) =

Et ρ , χ ρ + Et ρ

(20.18)

p ( st ) = p c + ( p m − p c ) st ,

(20.19)

d ( pt ) = b − apt

(20.20)

and

where F, D,U> 1, x, a, b are all positive constants. :H¿UVWFRPSXWHWKHVWHDG\VWDWHV7KHSDUDPHWHUVDUHVHWDV Default example: r = .02,G = .15,U = 5, F = 30, F = .001, Į = .5, Sm = 8, S = 2, b = 10, a = .5. The default example has three steady states (Table 20.1). If the upper steady state, E*hLVDFKLHYHGWKHGRPLQDQW¿UPZLOOKDYHDKLJKPDUNHWVKDUHV* = 0.81, F

7DEOH Steady states (default) E*l

E*m

E*h

E – competition-restricting capital

0.00

19.61

39.85

x – investment level

0.00

2.94

5.98

V – market share

0.00

0.11

0.81

S – price level

2.00

2.64

6.83

d – market demand

9.00

8.68

6.58

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'RPLQDQW¿UPVDQGDQWLWUXVWSROLF\

299

in the long run. The market price is S (V*) = 6.83, which is greater than the competitive price but still lower than the monopolistic price. If the lower steady state, E*l LVDFKLHYHGRQWKHRWKHUKDQGWKHPDUNHWVKDUHRIWKHGRPLQDQW¿UPLV zero and the market price is the competitive price. This can be interpreted as a competitive market. The middle unstable steady state, Em*, will not be reached XQOHVVWKHGRPLQDQW¿UP¶VLQLWLDOFRPSHWLWLRQUHVWULFWLQJFDSLWDOFRLQFLGHVZLWKLW by accident. Yet, overall we want to remark here that the local analysis of computing the number of steady states and the local stability analysis do not necessarily help us WR¿QGWKHVWHDG\VWDWHWKDWLVDFWXDOO\UHDFKHGE\WKHPDUNHWIRUDJLYHQLQLWLDO condition. To obtain a complete picture of the global dynamics, we shall conduct DJOREDODQDO\VLVLQWKHQH[WVHFWLRQ+HUHZH¿UVWVNHWFKWKHIRXUSRVVLEOH longUXQ scenarios that can arise from our model: 1 +LJK FRQFHQWUDWLRQ: There are three steady states. Yet, for any initial FRPSHWLWLRQUHVWULFWLQJFDSLWDOWKHGRPLQDQW¿UPFKRRVHVWRHQJDJHLQDQWL competitive activities to reach the upper steady state, E *h. Therefore, in the long run, this scenario will entail a high concentration in the market; 2 6WDWHGHSHQGHQWHQYLURQPHQW: There are three steady states, yet there exists a WKUHVKROG VHSDUDWLQJ WZR GRPDLQV RI DWWUDFWLRQ 7KH GRPLQDQW ¿UP ZKLFK possesses the initial competition-restricting capital greater than this threshold chooses to engage in anti-competitive activities to reach the upper steady state, E *h. In this case, the market tends toward a high concentration in the long run. 7KHGRPLQDQW¿UPZKLFKSRVVHVVHVWKHLQLWLDOFRPSHWLWLRQUHVWULFWLQJFDSLWDO less than the threshold, on the other hand, chooses to reduce its competitionrestricting capital to move to the lower steady state, E l*. The market then tends toward competition in the long run; 3 &RPSHWLWLRQ: there are three steady states. Yet, for any initial competitionUHVWULFWLQJ FDSLWDO WKH GRPLQDQW ¿UP FKRRVHV WR UHGXFH WKH FRPSHWLWLRQ restricting capital to the lower steady state, E l*. The market will restore competition in the long run; and 4 &RPSHWLWLRQ (w. a solo attractor): there is a unique steady state at the lower steady state, E l*. For any initial competition-restricting capital, the dominant ¿UPUHGXFHVLWVFRPSHWLWLRQUHVWULFWLQJFDSLWDOWRWKHXQLTXHVWHDG\VWDWH7KH market will restore competition in the long run.

Global analysis $VDIRUHPHQWLRQHGWKHORFDODQDO\VLVGRHVQRWQHFHVVDULO\KHOSXVWR¿QGWKH steady state that is actually reached. To detect the long-run scenario, we require the global analysis. When multiple steady states emerge, traditional DSSURDFKHVVXFKDVWKHVWXG\XQGHUWDNHQE\3RQWU\DJLQ¶VPD[LPXPSULQFLSOH and the subsequent local stability analysis are not enough to obtain a complete

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picture of the global dynamics. To study the global dynamics, dy namic SURJUDPPLQJLVKHOSIXO,QWKLVDSSURDFKZH¿UVWFRPSXWHWKHSUHVHQWYDOXH RIWKH¿UPIRUHDFKFDQGLGDWHSDWKWRDGLIIHUHQWVWDEOHVWHDG\VWDWHVi, and WKHQZH¿QGWKHJOREDOYDOXHIXQFWLRQV = max Vi7KHGRPLQDQW¿UPIRUD given initial state, selects the path that yields the highest present value. The global value function allows us to detect the steady state which is actually reached. We here use the dynamic programming algorithm applied in Grüne and Semmler (2004)10 to solve the dynamic optimization problem numerically, to compute the global value function and the policy function, and to detect a threshold if it exists. The phase diagram for the default example is shown in Figure 20.2. It reveals the equilibrium candidates of the respective equations. There are two candidate paths; one goes towards the lower stable steady state, E*l, and the other goes towards the upper stable steady state, E*h. Figure 20.3 shows the global value function and the policy (optimal control) function computed by the GrüneSemmler algorithm. The arrows on the horizontal axis depicts the state-space dynamics, i.e. the direction of the motion of EtWKDWWKHGRPLQDQW¿UPDFWXDOO\ZLOO pursue. The global value function calculates the highest value that the dominant ¿UPFDQDFKLHYHIRUDJLYHQLQLWLDOFRQGLWLRQ,WLVREYLRXVWKDWWKHGHIDXOWH[DPSOH represents Scenario 1 (KLJKFRQFHQWUDWLRQ). In this case, the lower stable steady state, E*l ZLOO QHYHU EH UHDFKHG 7KH GRPLQDQW ¿UP UHJDUGOHVV RI LWV LQLWLDO competition-restricting capital, increases the capital to the upper steady state, Eh*. x 15 x=0

12.5 10

E=0

7.5 5 2.5

10

20

–2.5

)LJXUH High concentration (default)

30

40

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Loss of benefit $V D FRQVHTXHQFH RI WKH GRPLQDQW ¿UP¶V UHDOL]LQJ WKH XSSHU VWHDG\ VWDWH WKH market price exceeds the competitive price. Using the basic microeconomic theory, we can also calculate the welfare loss (see Figure 20.4) in this event. P

Total demand = d(p) Dominant firm’s share = sd(p)

p(s*) Welfare loss

p(0) = pc

s*d(p(s*))

(1–s*)d(p(s*))

)LJXUH Welfare loss from competition-restricting activities

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Mika Kato and Willi Semmler

The economic surplus under the competitive price, SF, is ∞

ES c = ∫ c d ( p )dp,

(20.21)

p

while the economic surplus under the upper-steady-state-level market price, S(V*), is

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ES * = ( p ( s*) − p c )d ( p ( s*)) + ∫

∞ p ( s *)

(20.22)

d ( p )dp.

7KHUHIRUH WKH ZHOIDUH ORVV GXH WR WKH GRPLQDQW ¿UP¶V FRPSHWLWLRQUHVWULFWLQJ activity is l

≡ ES c − ES * =

∫

p ( s *) pc

(20.23)

d ( p )dp − ( p ( s*) − p c )d ( p ( s*).

Table 20.2 reports the welfare loss of each steady state in the default case. As we NQRZWKDWWKHGRPLQDQW¿UPLQWKLVH[DPSOHZLOOPRYHWRWKHXSSHUVWHDG\VWDWH E*h WKHZHOIDUHORVVFDXVHGE\WKLVGRPLQDQW¿UPZKHQQRFRPSHWLWLRQSROLF\LV pursued by the regulatory agency, will be l = 5.84. 7DEOH Welfare loss (default)

l – welfare loss

E*l

E*m

E*h

0.00

1.10

5.84

Antitrust policy Based on the previous discussion, the regulatory agency’s intervention into the PDUNHW PD\ EH MXVWL¿HG XQGHU WKH IROORZLQJ FLUFXPVWDQFHV WKH PDUNHW DV LQ Scenario 1 (KLJKFRQFHQWUDWLRQ), as seen in the default example, and the market as in Scenario 2 (VWDWHGHSHQGHQWHQYLURQPHQW ZKHUHWKHGRPLQDQW¿UPFRPPDQGV competition-restricting capital beyond the threshold level. In this section, we study whether the regulatory agency can reduce or eliminate the possibility that WKH GRPLQDQW ¿UP XQGHU WKHVH FLUFXPVWDQFHV DFWXDOO\ DFKLHYHV D KLJK PDUNHW share in the long run. We view the parameter, G, in Eq. (20.1) as a policy instrument since the UHJXODWRU\ DJHQF\ FDQ FRXQWHUDFW WKH GRPLQDQW ¿UP¶V FRPSHWLWLRQUHVWULFWLQJ efforts. The regulatory agency can impose, for example, a shorter lifetime of patents, higher patent renewal fees, lower limits in political donations, more severe restrictions on political lobbying, stricter law enforcement to antitrust activities, etc. – all of those accelerate the depreciation of competition-restricting capital.

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Change in regulatory environment 7KH JOREDO DQDO\VLV LQ WKH SUHYLRXV VHFWLRQ ¿QGV WKDW WKH GHIDXOW H[DPSOH KDV Scenario 1 (KLJKFRQFHQWUDWLRQ 7KHGRPLQDQW¿UPLQ6FHQDULRWHQGVWRLQFUHDVH competition-restricting capital to the upper steady state, E*h, and this causes a welfare loss in the long run. The following numerical examples demonstrate that the regulatory policy may effectively work to alter such an undesirable scenario to a more preferable scenario such as Scenario 3 or 4 where a competitive state of the market is restored in the long run. First, we increase the parameter G, with the other parameters unchanged, from 0.15 (default) to 0.24. Table 20.3 (left) reports the new steady states. There are still three steady states. The new phase diagram (Figure 20.5) shows two candidate paths but the corresponding global analysis (Figure 20.6) reveals that WKH GRPLQDQW ¿UP FKRRVHV WKH SDWK WR WKH ORZHU VWHDG\ VWDWH E*l . This is Scenario 3 (FRPSHWLWLRQ). In this scenario, regardless of the initial market share, 7DEOH Steady states with tighter regulatory policyG= .24 (left) andG= .26 (right) E*l

E*m

E*h

E*l

E – competition-restricting capital

0.00

26.31

30.72

E

0.00

x – investment level

0.00

6.32

7.37

x

0.00

V – market share

0.00

0.34

0.53

V

0.00

S – price level

2.00

4.05

5.18

S

2.00

d – market demand

9.00

7.97

7.41

d

9.00

l – welfare loss

0.00

1.05

2.52

l

0.00

x 15 E=0

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x=0 10

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40

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. E=0

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20

30

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–2.5

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function (Figure 20.8) also suggest that this is Scenario 4 (FRPSHWLWLRQ with a sole DWWUDFWRU ,WFRQ¿UPVWKDWWKHUHJXODWRU\DJHQF\FDQDOVRPDNHFRPSHWLWLRQDVD sole attractor. It should be noted, however, that long-run results are also affected by some other factors. The regulatory agency, for example, may not need to use the regulatory instruments to the same extent to revive competition when the dominant ¿UPKDVWKHSRRUDELOLW\WRGHWHUIULQJH¿UPV:HPD\LQWHUSUHWF in our model as WKHGHWHUUHQFHHI¿FLHQF\RIWKHGRPLQDQW¿UP$JUHDWHUF leads to an downward shift of the V-shape market-share function in Eq. (20.18). Therefore, the dominant ¿UPZLWKDJUHDWHUFKDVWKHORZHUGHWHUUHQFHHI¿FLHQF\7KHIROORZLQJQXPHULFDO examples demonstrate this point. &RQVLGHUDOHVVHI¿FLHQWGRPLQDQW¿UP:H¿UVWLQFUHDVH F from 30 (default) to 40, again with the other parameters unchanged. The new example still has three steady states (Table 20.4, left) and therefore there are two candidate paths (Figure 20.9). Yet, the global study (Figure 20.10) detects Scenario 2 (VWDWHGHSHQGHQWHQYLURQPHQW). Hence there is a threshold level of competitionrestricting capital; above which the market gradually moves toward a high concentration and below which competition takes place in the long run. The long-run outcome in this scenario depends on the build-up of competitionUHVWULFWLQJFDSLWDOE\WKHGRPLQDQW¿UP:KLOHWKHUHLVVWLOOVRPHSRVVLELOLW\WKDW a high market share, E*h , is reached, the new domain of attraction about competition

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0.00

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0.00

x – investment level

0.00

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x

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0.00

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2.00

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2.00

d – market demand

9.00

8.38

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d

9.00

l – welfare loss

0.00

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l

0.00

is wider than the one in the default example. The regulatory agency in this scenario may want to intervene in the market further to alter this scenario to a safer scenario such as Scenario 3 or 4 where competition takes place anyway. Note that this can be achieved with less use of the regulatory instruments (i.e., changing Scenario 2 to 3 is easier than changing Scenario 1 to 3). Similarly, by setting even ORZHUGHWHUUHQFHHI¿FLHQF\F  ZH¿QGDXQLTXHVWHDG\VWDWH7DEOH ULJKW 7KHUHLVDVROHDWWUDFWRU)LJXUH DQGWKHGRPLQDQW¿UPJUDGXDOO\ reduces the competition-restricting capital (Figure 20.12). This is Scenario 4 (FRPSHWLWLRQwith a sole attractor). In this scenario, regulatory intervention is not necessary. x 15

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. x=0

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7.5

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2.5

. x=0

10

20

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50

60

E

308

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V.x

Value function

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0.128

0.014

Control variable x

–0.100 –2.000

2.400

6.800

11.200

15.600

E

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Conclusion We commenced our study of how to strengthen welfare-improving competition DPRQJ¿UPVE\LQWURGXFLQJRXUPRGHOZKHUHGRPLQDQW¿UPVDQGFRPSHWLWLYH IULQJH¿UPVFRPSHWHIRUDSULFHHODVWLFPDUNHWGHPDQGDQG\HWGRPLQDQW¿UPV have an ability to build up capital that restricts competition. :H¿UVWVKRZHGWKDWZLWKORFDOLQFUHDVLQJUHWXUQVLQVXFKFRPSHWLWLRQUHVWULFWLQJ DFWLYLWLHV E\ WKH GRPLQDQW ¿UP D PDUNHW FDQ WHQG WR FRPSHWLWLRQ RU KLJK FRQFHQWUDWLRQLQWKHORQJUXQ6SHFL¿FDOO\RXUPRGHOJLYHVULVHWRIRXUSRVVLEOH long-run scenarios; 1. high concentration, 2. state-dependent environment, 3. competition, and 4. competition with a sole attractor. Which of the scenarios HPHUJHV GHSHQGV RQ YDULRXV PDUNHWVSHFL¿F IDFWRUV DV ZHOO DV WKH UHJXODWRU\ environment. Furthermore, we presented a number of numerical examples and studied how an antitrust and competition policy can be designed to reduce the possible loss of HFRQRPLFEHQH¿WVGXHWRWKHEXLOGXSRIVXFKFRPSHWLWLRQUHVWULFWLQJFDSLWDO:H want to note that we have in mind here a competition policy that prevents restrictions to competition (as in the case of European competition policy) and not purely an antitrust policy that solely prevents industrial concentration (as in the case of the US antitrust policy). We consider the depreciation rate of the competition-restricting capital as a policy parameter. The regulatory agency can increase the depreciation UDWHWKURXJKFRXQWHUDFWLQJWKHGRPLQDQW¿UP¶VHIIRUWVRIEXLOGLQJLWXSIRUH[DPSOH

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309

by imposing shorter lifetime of a patent, restricting political lobbying, and exercising stricter law enforcement. Our numerical examples showed that such a regulatory policy can effectively enlarge the domain of attraction where competition takes place. Once the regulatory agency is successful in altering an undesirable scenario to a more preferable scenario, it does not have to intervene in the market persistently as competition will be restored in the long run. Finally, we want to note that the use of a dynamic programming approach is helpful to study the global dynamics and to detect a threshold and the superior or inferior domain of attraction. In this context, it is worth mentioning, although our VHWXSPD\OHQGLWVHOIWRDJDPHWKHRUHWLFDOPRGHOEHWZHHQWKHGRPLQDQW¿UP DQGWKHIULQJH¿UPVRUWKHGRPLQDQW¿UPDQGWKHUHJXODWRU\LQVWLWXWLRQ WKDWZH restricted ourselves to only one intertemporal pay-off function letting the comSHWLWLYHIULQJH¿UPVRUWKHUHJXODWRU\LQVWLWXWLRQRQO\SDUDPHWULFDOO\LQÀXHQFH the competition dynamics. A fully developed game theoretic model does not appear to be feasible at the moment since the numerical solution technique for a non-zero sum (Nash) differential game, with those nonlinearities used in our PRGHOKDVQRWEHHQWRRXUNQRZOHGJHVXI¿FLHQWO\GHYHORSHG\HW)XWXUHUHVHDUFK may take on this challenge.

Appendix: Numerical solution method :HKHUHEULHÀ\GHVFULEHWKHG\QDPLFSURJUDPPLQJDOJRULWKPDVDSSOLHGLQ*UQH and Semmler (2004) that enables us to numerically solve the dynamic model as proposed in the chapter. The feature of the dynamic programming algorithm is an adaptive discretization of the state space which leads to high numerical accuracy with moderate use of memory. 6XFK DV DOJRULWKP LV DSSOLHG WR GLVFRXQWHG LQ¿QLWH KRUL]RQ RSWLPDO FRQWURO problems of the type introduced in the text. In our model variants we have to numerically compute V(x) for ∞

V ( x) = max ∫ e − r f ( x, u )dt 0

u

s.t.x

g ( x, u )

where X represents the control variable and x a vector of state variables. ,QWKH¿UVWVWHSWKHFRQWLQXRXVWLPHRSWLPDOFRQWUROSUREOHPKDVWREHUHSODFHG E\D¿UVWRUGHUGLVFUHWHWLPHDSSUR[LPDWLRQJLYHQE\ ∞

Vh ( x) = max J h ( x, u ), J h ( x, u ) = h∑ (1 − Th)Uf ( xh (i ), ui ) j

(20.A.1)

i=0

where xXLVGH¿QHGE\WKHGLVFUHWHG\QDPLFV xh (0) = x, xh (i + 1) = xh (i ) + hg ( xi , ui )

(20.A.2)

310

Mika Kato and Willi Semmler

and h > 0 is the discretization time step. Note that j = (ji)i` here denotes a discrete 0 control sequence. The optimal value function is the unique solution of a discrete HamiltonJacobi-Bellman equation such as

Vh ( x) = max{hf ( x, uo ) + (1 − θh)Vh ( xh (1))} j

(20.A.3)

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where xh (1) denotes the discrete solution corresponding to the control and initial value x after one time step h. Abbreviating

Th (Vh )( x) = max{hf ( x, uo ) + (1 − θh)Vh ( xh (1))}, j

(20.A.4)

the second step of the algorithm now approximates the solution on grid * covering a compact subset of the state space, i.e. a compact interval [0, K] in our setup. Denoting the nodes of * by xi, i = 1,. . ., P, we are now looking for an approximation V *h satisfying VhΓ ( X i ) = Th (VhΓ )( X i )

(20.A.5)

for each node xi of the grid, where the value of V h* for points x which are not grid points (these are needed for the evaluation of Th) is determined by linear interpolation. We refer to the paper cited above for the description of iterative methods for the solution of (20.A.5). Note that an approximately optimal control law (in feedback form for the discrete dynamics) can be obtained from this approximation by taking the value j*(x) = j for j realizing the maximum in (20.A.3), where Vh is replaced by V h*. This procedure in particular allows the numerical computation of approximately optimal trajectories. ,Q RUGHU WR GLVWULEXWH WKH QRGHV RI WKH JULG HI¿FLHQWO\ ZH PDNH XVH RI a SRVWHULRUL error estimation. For each cell Cl of the grid * we compute K l := max | Th (VhΓ )(k ) − VhΓ (k ) | . k ∈cl

More precisely we approximate this value by evaluating the right-hand side in a number of test points. It can be shown that the error estimators Kl give upper and lower bounds for the real error (i.e. the difference between Vj and V *h) and hence VHUYHDVDQLQGLFDWRUIRUDSRVVLEOHORFDOUH¿QHPHQWRIWKHJULG*. It should be noted WKDWWKLVDGDSWLYHUH¿QHPHQWRIWKHJULGLVYHU\HIIHFWLYHIRUFRPSXWLQJVWHHSYDOXH functions and models with multiple equilibria, see Grüne and Semmler (2004).

Notes * We want to thank Herbert Dawid and Duncan Foley for helpful discussions, and Armon Rezai for help with research. We also thank participants at International Workshop on

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1 2

3 4 5

6 7 8 9

10

311

$JHQW%DVHG0RGHOVIRU(FRQRPLF3ROLF\'HVLJQ &RQIHUHQFHRQ(FRQRPLFV of Innovation and Industrial Dynamics (2006), Fifth International Workshop on Nonlinear Economic Dynamics (2007), and Ninth Workshop on Optimal Control, Dynamic Games and Nonlinear Dynamics (2007) for useful comments. See Baumol et al. (1982). The basic concept of limit pricing was established by Bain (1956), Modigliani (1958), Sylos-Labini (1962), and Gaskins (1971). The idea was then formalized by Spence (1977), Dixit (1979, 1980), and Milgrom and Roberts (1982). The Spence-Dixit model shows that an incumbency advantage lies in its capability of creating a large capacity and therefore charging a lower price to deter entry. In the Milgrom-Roberts model, the incumbent charges a low price under asymmetric information to signal low profitability of entry to potential entrants. See Tirole (1988) for further references. For details on the new paradigm of competition dynamics, see Audretsch et al. (2001). In particular the earlier German and now the European competition policy seem to focus on this point. When fringe firms or regulatory agencies can undermine the success of dominant firms by accelerating the latter’s obsolescence of the applied technology, then one can also view the dominant firms as playing a game against the fringe firms as in Judd and 3HWHUVHQ RUDJDLQVWWKHUHJXODWRU\DJHQFLHVDVIRUH[DPSOHGLVFXVVHGLQWKH robust control literature. This would result in a type of mini-max game. See Zhang and Semmler (2005) for a model in the tradition of the robust control literature. We want to note, however, that because of the possible complexity of such a differential game setup, which cannot be solved analytically, we restrict ourselves here to a modeling approach that involves only a single intertemporal pay-off function. As we will show further in the next section our model is more complex than the usual differential game models in industrial organization as for example reported in Dockner et al (2000, chs 9–12). See Baumol et al. (1982) for theory of contestable markets and Brock (1983) for an evaluation and criticism of this theory. Since ht is not allowed to exceed 0, then whenever ht = 0, we must forbid ht to increase. Thus, the problem has a state-space constraint. The same general idea can be found in Brock (1983) where the supply function of entrants is a reverse V-shape function of barriers-to-entry capital. It is not difficult to show in an n-firm Cournot oligopoly model that price is an increasing function of the concentration ratio. Many empirical studies show that there is a positive correlation of market share and rates of return. See, for example, Weiss (1963). For an extensive survey of earlier literature, see Semmler (1984). For a short description, see the Appendix.

References $XGUHWVFK'%%DXPRO:-DQG%XUNH$(µ&RPSHWLWLRQ3ROLF\LQ'\QDPLF Markets.’ ,QWHUQDWLRQDO-RXUQDORI,QGXVWULDO2UJDQL]DWLRQ 19: 613–34. Bain, J. S. 1956.%DUULHUVWR1HZ&RPSHWLWLRQ&DPEULGJH0$+DUYDUG8QLYHUVLW\3UHVV %DXPRO:-3DQ]DU-&DQG:LOOLJ5'&RQWHVWDEOH0DUNHWVDQGWKH7KHRU\RI ,QGXVWU\6WUXFWXUH New York: Harcourt Brace Jovanovich. %URFN:$µ3ULFLQJ3UHGDWLRQDQG(QWU\%DUULHUVLQ5HJXODWHG,QGXVWULHV¶,Q Evans, D. S. and Bornholz, R. (ed.) %UHDNLQJ 8S %HOO (VVD\V RQ ,QGXVWULDO 2UJDQL]DWLRQDQG5HJXODWLRQ New York: Elsevier Science Ltd. %URFN:$DQG'HFKHUW:'µ'\QDPLF5DPVH\3ULFLQJ¶-RXUQDORI,QWHUQDWLRQDO (FRQRPLFV, 26(3): 569–91.

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Dechert, D. W. and Nishimura, K. 1983. ‘Complete Characterization of Optimal Growth 3DWKVLQDQ$JJUHJDWH0RGHOZLWK1RQ&RQFDYH3URGXFWLRQ)XQFWLRQ¶-RXUQDORI (FRQRPLF7KHRU\ 31: 332–54. Dixit, A. 1979. ‘A Model of Duopoly Suggesting a Theory of Entry Barriers.’ %HOO-RXUQDO RI(FRQRPLFV 10: 20–32. Dixit, A. 1980. ‘The Role of Investment in Entry-Deterrence.’ (FRQRPLF-RXUQDO 90(357): 95–106. Dockner E. J., Jorgensen S., Long, N. V., and Sorger, G. 2000. 'LIIHUHQWLDO *DPHV LQ (FRQRPLFVDQG0DQDJHPHQW6FLHQFH&DPEULGJH&DPEULGJH8QLYHUVLW\3UHVV *DVNLQV':µ'\QDPLF/LPLW3ULFLQJ2SWLPDO3ULFLQJ8QGHU7KUHDWRI(QWU\¶ -RXUQDORI(FRQRPLF7KHRU\ 3: 306–22. *UQH / DQG 6HPPOHU :  µ8VLQJ '\QDPLF 3URJUDPPLQJ ZLWK$GDSWLYH *ULG 6FKHPHIRU2SWLPDO&RQWURO3UREOHPVLQ(FRQRPLFV¶-RXUQDORI(FRQRPLF'\QDPLFV and Control 28: 2427–56. -XGG . / DQG 3HWHUVHQ % &  µ'\QDPLF /LPLW 3ULFLQJ DQG ,QWHUQDO )LQDQFH¶ -RXUQDORI(FRQRPLF7KHRU\ 39: 368–99. 0LOJURP3DQG5REHUWV-µ/LPLW3ULFLQJDQG(QWU\8QGHU,QFRPSOHWH,QIRUPDWLRQ An Equilibrium Analysis.’ (FRQRPHWULFD 50: 443–59. Modigliani, F. 1958. ‘New Developments on the Oligopoly Front.’ -RXUQDORI3ROLWLFDO (FRQRP\ 66(3): 215–32. Semmler, W. 1984. &RPSHWLWLRQ 0RQRSRO\ DQG 'LIIHUHQWLDO 3URILW 5DWHV. New York: &ROXPELD8QLYHUVLW\3UHVV 6SHQFH$0µ(QWU\&DSDFLW\,QYHVWPHQWDQG2OLJRSROLVWLF3ULFLQJ¶%HOO-RXUQDO RI(FRQRPLFV 8: 534–44. 6WLJOLW]-(µ3ULFH5LJLGLWLHVDQG0DUNHW6WUXFWXUH¶$PHULFDQ(FRQRPLF5HYLHZ 74(2): 350–5. 6\ORV/DELQL32OLJRSRO\DQG7HFKQLFDO3URJUHVV Cambridge, MA: Harvard UniYHUVLW\3UHVV Tirole, Jean. 1988. 7KH7KHRU\RI,QGXVWULDO2UJDQL]DWLRQ&DPEULGJH0$0,73UHVV :HLVV/:µ$YHUDJH&RQFHQWUDWLRQ5DWLRVDQG,QGXVWULDO3HUIRUPDQFH¶-RXUQDORI ,QGXVWULDO(FRQRPLFV 11(3): 237–54. =KDQJ: DQG 6HPPOHU :  µ0RQHWDU\ 3ROLF\ 5XOHV XQGHU 0RGHO 8QFHUWDLQW\ Empirical Evidence, Adaptive Learning and Robust Control.’ 0DFURHFRQRPLF '\QDPLFV 9: 651–81.

21 The market for saving in the theory of general intertemporal equilibrium1

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Sergio Parrinello

To my friend and colleague, Heinz Kurz, who has successfully endeavoured to discover and disseminate the truth in economics, although he was aware of other truths like: ‘The disquieting fact remains that in economics propositions that have been proved wrong are still used by many of its practitioners.’ (Heinz D. Kurz and Neri Salvadori, Understanding Classical Economics, 1998) µ$QHZVFLHQWL¿FWUXWKGRHVQRWWULXPSKE\FRQYLQFLQJLWVRSSRQHQWVDQGPDNLQJ them see the light, but rather because its opponents eventually die, and a new generation grows up that is familiar with it.’ (Max Planck, $6FLHQWL¿F$XWRELRJUDSK\, 1949).

Introduction The recent debate on the theory of general intertemporal equilibrium among Garegnani (2000, 2003), Mandler (2005) and Parrinello (2005, 2008) with comment by Foley (2008), Petri (2004) and Schefold (2008) seems to be centred on two main issues. 1 It has been questioned whether a value aggregation of physical quantities is necessary in the theory of general intertemporal equilibrium; in particular whether an equilibrium condition between aggregate saving and aggregate investment for each period is as much determinant as an equilibrium condition in the market of a physical good or service. 2 It has been debated whether the possibility of reverse capital deepening and reswitching of techniques, which was proved for the aggregate version of the neoclassical theory of capital and distribution, can be a VSHFL¿F source of non meaningful intertemporal equilibria or, instead, can be neutralized by the same VXI¿FLHQW FRQGLWLRQV HJ WKH ZHDN D[LRP RI UHYHDOHG SUHIHUHQFHV RU WKH representative consumer) that for a long time have been adopted to prove the existence, uniqueness and stability of general equilibrium.

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The debate on these two issues will be resumed in the light of a recent contribution by Garegnani (2009). We shall use the abbreviation ‘Intertemporal’ instead of ‘model of general intertemporal economic equilibrium’; and the abbreviation ‘A-temporal’ instead of ‘model of general a-temporal economic equilibrium’. The adjective ‘a-temporal’ encompasses the terms ‘static’ or ‘oneperiod’, although we are aware of possible objections to the use of such expressions interchangeably. Clearly question 2 above arises only if the answer to question 1 is that aggregation is necessary. Ultimately the controversy seems to resolve itself into the acceptance or refusal of a syllogism of the following type. 1) Each Intertemporal can be formally converted into an A-temporal and shares the same properties, in terms of existence, uniqueness and stability of equilibrium which have been already proved for the static general equilibrium model. 2) Hahn-Garegnani’s model, which is taken as a prototype for the discussion at issue, is an Intertemporal. 3) Hence Hahn-Garegnani’s model is formally equivalent to an A-temporal and cannot exhibit properties which require the demonstration of special theorems. The premise (1) of the syllogism above is ambiguous because it presupposes that all A-temporals have a unique analytical structure and the same equilibrium properties. Schefold (2008) has convincingly demonstrated how an Intertemporal can be reduced to an A-temporal and that any solution to the former must be also a solution to the latter. However, the second part of premise (1) does not follow IURPWKH¿UVWRQH7KHRQHSHULRGRUVWDWLFPRGHOWRZKLFKDQIntertemporal is FRQYHUWHG SRVVHVVHV D VSHFLDO VWUXFWXUH ZKLFK UHÀHFWV WKDW RI LWV DQFHVWRU The existing theorems of stability, which have been proved for a standard A-temporal, may not be extended to the A-temporal corresponding to the Intertemporal. Garegnani (2000, 2003) has argued that non-meaningful equilibria can exist in the Intertemporal as a result of reverse capital deepening and reswitching of techniques. Elsewhere (Parrinello 2005, 2008), I have stressed that the theorems of tâtonnement stability cannot be extended from the A-temporal to the Intertemporal, because the adjustment mechanism is different in the two models. We should specify more carefully what is the structure of the Intertemporal, or its equivalent A-temporal, which opens the door to different equilibrium and disequilibrium properties related to the theory of capital. 7KH¿UVWWKUHHVHFWLRQVRIWKLVFKDSWHUVXPPDUL]HWKHHOHPHQWDU\QRWLRQVRI one-period versus multi-period budget constraints, of dependence among equilibrium conditions and of perfect substitutes, as an introduction to the main DUJXPHQW:HVKDOOUHLWHUDWHLQWKHIRXUWKDQG¿IWKVHFWLRQVWKDWDPDUNHWIRUVDYLQJ may and in a sense must exist in each period of the Intertemporal and intervenes in a special way in the determination of equilibrium. The next two sections specify the individual behaviour which underlies the markets for saving in the Intertemporal. We shall deal in the seventh section with the distinction among dated Walras laws (see Garegnani 2009). These seven sections should convince the reader that a market for saving must exist in each period of the Intertemporal

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if its structural form has to be reduced into an a-temporal form in which each agent is subjected to a unique budget constraint. More importantly, the properties of the market demand and supply functions of the corresponding A-temporal cannot be assumed as if the saving markets would not exist, but they must be derived from the properties of the demand and supply functions of the Intertemporal taken in its structural form. The penultimate section 8 presents a reformulation of the method of quasi-equilibrium used by Garegnani to describe some properties of the IQWHUWHPSRUDOE\IRFXVLQJRQWKHPDUNHWIRUVDYLQJRQO\7KH¿QDOVHFWLRQVXJJHVWV some lines for the development of the debate on question (2) presented above. The SUREOHPRIWKHFKRLFHRIWKHQXPHUDLUHZKLFKZDVEULHÀ\GLVFXVVHGLQDSUHYLRXV version of the present paper (see note 1), has been omitted here, because the author has dealt with the same problem more extensively and accurately elsewhere (Parrinello, 2010).

Alternative structures behind the budget constraints Dorfman, Samuelson and Solow (DOSSO, 1958) have warned their readers against a supposed equivalence between the intertemporal versus the atemporal theory (in their words: dynamic versus static models) of general equilibrium: But time does make a difference in economics – witness years of controversy over the theory of capital. To treat dynamic problems as nothing but special cases of static ones may simply rob us of the insights that a more direct theory might yield. After all, n commodoties at each of T dates are not simply nT separate commodities. There is a structure: sometimes it is useful to view them as T groups with date in common, sometimes as n groups with physical characteristic in common. (DOSSO, 1958, p. 265) It seems that this warning has been occasionally neglected. I will recall here three different structures of the budget constraints which can underlie the Walras law. Let us denote by T row-vectors the quantities of nT commodities which enter into the budget constraint(s) of a typical agent: x1 x2

x x

1,1

,. . . ., x1, n

2,1

,. . . ., x2, n

........................... xT

x

T ,1

,. . . ., xT , n

where each element xt, j of the xtYHFWRUVLVGH¿QHG xt , j { Zt , j  ct , j , the (positive or negative) difference between a given endowment Zt , j and a quantity ct , j of

316

Sergio Parrinello

commodity t, j to be chosen, t = 1, . . ., T; j = 1, . . ., n. Let us write the price column-vectors which are taken as given by the agent: p1 p2

p p

1,1

,. . . ., p1, n

2,1

,. . . ., p2, n

...........................

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pT

p

T ,1

,. . . ., pT , n

In the standard A-temporal the agent chooses the quantities x1 , x2 ,. . ., xT under the unique budget equation x1p1  x 2 p 2 ,. . ., xT pT

0

(21.1)

In the Intertemporal ZLWKRXWPDUNHWVIRUVDYLQJ, the agent chooses x1 , x2 ,. . ., xT under the budget constraints s1 º x 2 p 2 s2 »» ............... » » xT pT sT ¼ x1 p 1

(21.2)

where st, t = 1, . . ., T, is a slack scalar variable, subject to st t 0 , t = 1, 2,…, T. It means that in each period the agent cannot spend more than his current income. In this case an optimal choice x1 , x2 ,. . ., xT cannot be derived from a solution to a maximum utility problem under a unique budget constraint like (21.1). In the Intertemporal ZLWK PDUNHWV IRU VDYLQJ, the agent chooses not only x1 , x2 ,. . ., xT but also an optimal plan of savings or dissavings s1 s2 ,. . ., sT under the budget constraints º s1  x 2 p 2 s2 »» » ............... » sT 1  xT pT sT ¼ x1 p 1

s1

(21.3)

with sT 02. In this model an optimal plan x1 , x2 ,. . ., xT , coincides with that under the unique constraint (21.1) of the A-temporal, because (21.1) can be obtained from (21.3) through an ordered substitution of one equation for group t into the equation for t + 1, t = 1,. . ., T)ROORZLQJWKLVURXWHWR¿QG x1 , x2 ,. . ., xT , the equations (21.3) will serve to determine the optimal plan of saving s1 s2 ,. . ., sT by substitution of x1 , x2 ,. . ., xT into the same equations. Hence, equation (21.1), which appears in

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the a-temporal reduction of the Intertemporal, does not mean that the markets for aggregate saving do not exist. On the contrary, such markets must exist if we want to use the unique budget (21.1) instead of (21.3) without affecting an optimal choice x1 , x2 ,. . ., xT . Therefore an Intertemporal with complete markets possesses two structural features which are not shared by an A-temporal which is not the a-temporal equivalent of the Intertemporal: 1 A market for saving in value exists for each period. 2 The demand and supply of commodities in each period, although they are the result of a one-shot choice of the individual and can be calculated under a unique budget constraint, must satisfy distinct budget constraints for the different periods where dated savings intervene. We may even assume that all negotiations are settled at a point of time in the Intertemporal and we may adopt the device of dealing with n commodities available at periods 1, 2, . . . T as nT separate commodities. However, the delivery of a commodity of group t cannot be carried out if it is not yet available (see Garegnani, 2009) and this is the case if the delivery of the commodities of groups t – 1, . . .,1 is not carried out either. This is a peculiar feature which is not shared by the A-temporal in which the nT commodities are assumed to be ‘equally’ physical.

Different types of dependence among equilibrium conditions We need now to clarify what is meant by saying that an equilibrium condition depends on the others. A distinction already arises in the A-temporal with: n commodities, the corresponding market demand Di(.) and supply Oi(.) functions and prices Pi. For example, let us consider the following statements. 1 The equality between total intended expenditure and revenue n

¦ >D (.)  O (.) P @ i

i

0

i

(21.4)

i

depends on the equilibrium conditions Di (.) = Oi (.) i = 1. . . .n 2 Any equilibrium condition, say Dn (.) = On (.), depends on the others, Di (.) = Oi (.) i = 1. . . . n – 1, since the functions Di(.), Oi(.) satisfy the Walras law: n

¦ ª¬ D (.)  O (.) P º¼ { 0 i

i

i

(21.5)

i

In (21.5) the identity sign holds and means that the equality is valid also at GLVHTXLOLEULXP SULFHV ,QVWHDG HTXDWLRQ   LV VDWLV¿HG RQO\ DW HTXLOLEULXP prices, if identity (21.5) is not assumed. Therefore it is useful to distinguish the following types of dependence:

318 x x

Sergio Parrinello a ‘subordinate’ dependence of type (1) in which a condition is a linear combination of others and can be omitted. an ‘equivalent’ dependence of type (2) in which any equilibrium condition depends on the others because all market equilibrium conditions are assumed WRVDWLVI\WKH:DOUDVODZ,QWKLVFDVHQRVSHFL¿FFRQGLWLRQhas to be omitted instead of others.

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A pseudo market for saving in the Intertemporal Let us reconsider Hahn’s model in the simple case in which the techniques are given and all markets are cleared in equilibrium, with the possible exception of an excess supply on the labour market with a null salary. The economy lasts only two periods t = 0, 1, with two non storable goods a, b dated t = 0, 1. The production is FKDUDFWHUL]HGE\FRQVWDQWUHWXUQVWRVFDOHUHSUHVHQWHGE\¿[HGFRHI¿FLHQWV7KH endowments of goods a, b are given for t = 0 and all production is consumed at the end of period t = 1. The symbols used: x x x x x x

A0 , B0, L endowments of goods available at the beginning of period t = 0 and of labour in period t = 0; A1 , B1 quantities of goods produced in period t = 0 and available in period t = 1; (Pa0, Pb0, Pa1, Pb1, W) discounted nominal prices of goods and nominal wage rate; la, lb, aa, ba, ab, bbWHFKQLFDOFRHI¿FLHQWV Ca0, Cb0, Ca1, Cb1 quantities of goods consumed in period t = 0, 1; (.) denotes function.

Price equations under perfect competition Pa1

laW  aa Pa 0  ba Pb 0

(21.6)

Pb1

lbW  ab Pa 0  bb Pb 0

(21.7)

Equilibrium on the goods markets in period t = 0 A0

Ca 0 (˜)  aa A1  ab B1

(21.8)

B0

Cb 0 (˜)  ba A1  bb B1

(21.9)

Equilibrium on the labour market L t la A1  lb B1 with ( L  la A1  lb B1 )W

0

(21.10)

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Equilibrium on the goods markets in period t = 1 A1

Ca1 (˜)

(21.11)

B1

Cb1 (˜)

(21.12)

all variables being subjected to non negativity conditions. The unknowns of the model are the prices (Pa0, Pb0, Pa1, Pb1, W) and the quantities A1 , B1 . The written conditions are seven, but one of them is dependent in equivalent sense. Given a certain numeraire, e.g. assuming Pb1 = 1, in principle the system is determinate and a solution to equations (21.6) to (21.12) determines also the equilibrium rate of interest on the numeraire through the identity rb { Pb 0 /Pb1  1 . The Walras law can be written in the form:

[A

0

− Da 0 (.) ] Pa 0 + [ B0 − Db 0 (.) ] Pb 0 + ( L − LD (.) )W

+ ( A1 − Ca1 (.) ) Pa1 + ( B1 − Cb1 (.) ) Pb1 ≡ 0

(21.13)

with Da 0 (.) { Ca 0 (.)  I a 0 (.) , Db 0 (.) { Cb 0 (.)  I b 0 (.) , I a (.) { aa A1  ab B1 , I b (.) { ba A1  bb B1 . LD { la A1  lb B1 The identity (21.13) includes only the values of excess supplies on markets for SK\VLFDOJRRGV/HWXVSURYLVLRQDOO\GH¿QHWKHIXQFWLRQVIRUVDYLQJDQGLQYHVWPHQW by S0 ˜ { > A0  Ca 0 (˜) @ Pa 0  > B0  Cb 0 (˜) @ Pb 0

(21.14)

I 0 ˜ { I a 0 (.) Pa 0  I b 0 (.) Pb 0

(21.15)

The equality S0(.) = I0 GHSHQGVLQDVXERUGLQDWHVHQVH,WLVLPSOLFLWO\VDWLV¿HG E\ WKH HTXLOLEULXP VROXWLRQ DQG LV DV PXFK VXSHUÀXRXV DV WKH LGHQWLW\ n

¦ ª¬ D (.)  O (.) P ¼º { 0 in the A-temporal as mentioned in the second section i

i

i

i

(or its analogue identity in the Intertemporal). Such an interpretation dismisses the existence of a market for aggregate saving. We shall argue that the relevant saving PDUNHW HPHUJHV WKURXJK D GLIIHUHQW URXWH :H QHHG ¿UVW VRPH HOHPHQWDU\ explanation of the role of perfect substitutes in the theory of general equilibrium

When the aggregation of physical quantities becomes necessary Let us convert Hahn’s model into an A-temporal in which A0 , B0 , A1 , B1 L are distinct quantities of goods a0 , b0 , a1 , b1 and labour available in the same period.

320

Sergio Parrinello

Suppose that a0 , b0 are imperfect substitutes for consumption, whereas they are perfect substitutes as factors of production. This means that the marginal rate of technical substitution (MRTS) between the two inputs of a0 , b0 is given and constant. Let us assume MRTS = 1 by an appropriate choice of the units of PHDVXUH2QWKHEDVLVRIWKLVFRQYHQWLRQZHPD\¿QGLQHTXLOLEULXPWZRFDVHV x

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x

either Pa 0 Pb 0 , the price ratio is equal to MRTS and the distribution of a given expenditure between the two inputs is indifferent for the producer; or Pa 0 z Pb 0 , the price ratio is different from MRTS and the producer will use only one input (the cheapest).

If Pa 0

Pb 0, we replace the price equations III,1, III.2 of the original model with

3a1

Oa:  Na 30

(21.16)

3b1

Ob:  Nb 30

(21.17)

where Na, NbGHQRWHWKHLQSXWFRHI¿FLHQWVRIDJRRGPDGHRIWKHJRRGV a0 , b0 used LQLQGH¿QLWHSURSRUWLRQV7KHIXQFWLRQVRIDJJUHJDWHVDYLQJDQGLQYHVWPHQWDUH S (.)

 A0  Ca 0 (.)   B0  Cb 0 (.)

(21.18)

, (.)

Na $1  Nb %1

(21.19)

Notice that the aggregation of saving and investment in (21.14) and (21.15) is in value, whereas that in (21.18) and (21.19) is in pure physical units. In this case, a market for an aggregate variable replaces the markets of its components and S(.) = I(.)

(21.20)

is an equation which describes an equilibrium condition. The equation Pa 0 closes the model and the Walras 21.13 becomes:

Pb 0

P0 ª¬ A0  B0  Ca 0 (.)  Cb 0 (.)  I (.) º¼

 W  L  LD (.)   A1  Ca1 (.) Pa1   B1  Cb1 (.) Pb1 { 0

(21.21)

We observe that the equations A0 Ca 0 (.)  I a (.) ; B0 Cb 0 (.)  I b (.) are not equilibrium conditions like in the original Hahn model, where the supply of each initial endowment meets a demand for consumption and investment of the same commodity. Now the same equations serve only to determine the physical composition of the investment after the determination of the equilibrium prices. Suppose now that in equilibrium Pa 0 z Pb 0 instead of Pa 0 Pb 0. For example, let Pa 0  Pb 0 .

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In this case the producers will use only good a0 as a means of production and two distinct conditions of market equilibrium become determinant: A0

Ca 0 (˜)  I (.)

B0

Cb 0 (˜)

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and the price equations: 3a1

Oa:1  Na 3a 0

3b1

Ob:1  Nb 3a 0 .

We conclude that, if goods a0 , b0 are perfect substitutes for production and their relative price is equal to the respective marginal rate of technical substitution, the existence of an aggregate market for the two goods becomes a necessity also in the A-temporal. We could have assumed that goods a0 , b0 are perfect substitutes in consumption and imperfect substitutes for production and an analogous conclusion would have followed.

When value aggregation becomes necessary Let us go back to the Hahn’s original model where all goods are imperfect substitutes in consumption and in production. Its a-temporal form exhibits no aggregation. However goods a, b in the Intertemporal are perfect substitutes as carriers of purchasing power and a market for it is supposed to exist. In the AWHPSRUDOZHFDQQRW¿QGDFRQGLWLRQRIDXQLIRUPUDWHRIUHWXUQRQ capital invested or saved, although we can play with certain identities which resemble such a condition. We may start from the identity § Pa1 · § Pb 0 · § Pa 0 · Pb 0 ¨© P ¸¹ ˜ ¨© P ¸¹ ˜ ¨© P ¸¹ { P ; b1 a0 a1 b1

(21.22)

DQGGH¿QHWKHUHODWLYHSULFHV pa 0 { 1  ia {

Pa 0 P P , pa1 { a1 , pb 0 { b 0 { 1, Pb 0 Pb1 Pb 0

pb1 {

Pb1 { 1, Pb1

Pa 0 P , 1  ib { b 0 . Pa1 Pb1

and then write (21.22) in the form pa1 p 1  ia { b1 (1  ib ) pa 0 pb 0

(21.23)

322

Sergio Parrinello

We may even arbitrarily call ia , ib the own rates of interest on goods a0 , b0 . The form (21.23) remains an identity which is an expression of the law of a unique price (Jevons’ law) and it applies also if the markets are out of equilibrium. It belongs to the sphere of exchange, not to that of production processes. Now let us return to the Intertemporal and use the current prices pi, t i = a, b; t = 0, 1 WRGH¿QHWKHIDFWRUVRIUHWXUQRQFDSLWDOLQYHVWHGLQWKHWZRLQGXVWULHV

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1  ra {

aa pa 0

pa1 pb1 , 1  rb { ab pa 0  bb pb 0  lb w  ba pb 0  la w

pa 0 pb 0 , 1  rb { the aa pa 0  ba pb 0  la w ab pa 0  bb pb 0  lb w FRUUHVSRQGLQJIDFWRUVRISUR¿WFDOFXODWHGDWWKHSULFHVRIWKHFXUUHQWSHULRG7KH price equations (21.6) and (21.7) imply r a r b,QWHUPVRIWKHGH¿QLWLRQVDERYH we can write:

We define 1  ra {

pa1 1  ra pa 0

pb1 1  rb pb 0

(21.24)

By contrast with identity (21.23), equation (21.24) reformulates the equilibrium condition r a r b as an equality between the factors of return on each type of investement, multiplied by a factor of appreciation. In the corresponding A-temporal the same equation (21.24) can be obtained from the price equations (21.6) and (21.7), but must be interpreted as a relation between the relative prices of contemporary FRPPRGLWLHV,WVWLOOUHÀHFWVWKHDEVHQFHRISUR¿WVXQGHUSHUIHFWFRPSHWLWLRQEXWLW cannot mean equality between the rates of return on investment or saving. Instead in the Intertemporal the equation (21.24) extends itself to all rates of return – pure numbers per period of time – on capital invested or lent. In the presence of capital goods which are perfect substitutes as carriers of value, a market for aggregate saving must exist. This market can be called by different, but equivalent, names: market of saving, of credit or of future income. It is distinct from the pseudo market ZKLFK LQ WKH WKLUG VHFWLRQ ZDV GH¿QHG E\ WKH VDYLQJ IXQFWLRQ   DQG WKH investment function (21.15). We notice that, in the case of perfect substitutes in production or in consumption, some markets for separate goods merge themselves into an aggregate market. Instead, in each period of the Intertemporal, the market for saving adds to the markets of the individual components of the aggregate itself.

The wealth equation and the dated budget constraints at the level of individual agents Let us reconsider the individual choices. For the sake of the argument it is useful WRUHSODFHWKHGLVWLQFWLRQEHWZHHQKRXVHKROGVDQG¿UPVZLWKWKDWEHWZHHQ x x

m owners of the initial endowments which are only consumers and savers; n agents which possess only their labor force and act as consumers and producers.

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/HWXVEULHÀ\FDOOWKHPµVDYHUV¶DQGµSURGXFHUV¶UHVSHFWLYHO\)XUWKHUPRUHZH assume from now on that wages are paid post factum, i.e. at the end of the production period, in order to make easier the comparison of our formalization with that often used in the recent debate. The wealth equation of the saver and his dated budget constraints

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Let lower case letters denote the quantities of saver i corresponding to the market TXDQWLWLHVDOUHDG\GH¿QHG+HLVDVVXPHGWRPD[LPL]HKLVXWLOLW\IXQFWLRQ u i cai 0 , cbi 0 , cai 1 , cbi 1 subject to a0i Pa 0  b0i Pb 0 s1i  l iW

cai 0 Pa 0  cbi 0 Pb 0  s0i for t = 0

(21.25)

cai 1 Pa1  cbi 1 Pb1 for t = 1

Could s0i be different from s1i "2IFRXUVHDVLQGLFDWHGLQWKH¿UVWVHFWLRQ,QFRPH RU ZHDOWK ZKLFK DUH QRW FRQVXPHG DUH QRW D VXSSO\ RI VDYLQJ E\ GH¿QLWLRQ Nothing prevents one from calling ‘saving’ or ‘dissaving’ s0i { (a0i Pa 0  b0i Pb 0 )  (cai 0 Pa 0  cbi 0 Pb 0 ) on the basis of the plus or minus sign of this expression, but this value does not mean that a certain amount of purchasing power is moving over time. It is simply a slack variable. Instead, if we assume the existence of a market for saving, the equality s0i s1i must hold and the optimal consumption plan of the saver determined under the unique wealth constraint a0i Pa 0  b0i Pb 0  l iW

cai 0 Pa 0  cbi 0 Pb 0  cai 1 Pa1  cbi 1 Pb1

(21.26)

coincides with the choice under the two dated constraints (21.25). A solution to the maximum problem determines the demand for two consumption goods in the i i two periods, c a 0 , c b 0 , ca1 , cb1, a supply of labour li and, by substitution into (21.25)1 an optimal amount of saving s i which is the optimal common value of s0i , s1i. The wealth equation of the producer and his dated budget constraints The producer N maximizes his utility function u N caN0 , cbN0 , caN1 , cbN1 and chooses the optimal outputs a1N , b1N under the budget constraints: caN0 Pa 0  cbN0 Pb 0   aa a1N  ab b1N Pa 0  ba b1N  bb b1N Pb 0 · ¸ for t 0 ¸ N N N N N N N N ¸ l W  a1 Pa1  b1 Pb1 ca1 Pa1  cb1 Pb1  la a1  lb b1 W  s ¸ ¸¹ for t 1 sN

(21.27)

324

Sergio Parrinello

The market for saving brings about s0N s1N . The choice of caN0 , cbN0 , caN1 , cbN1 , that maximizes u N . PD[LPL]HVDOVRWKHSUR¿WVRIWKHSURGXFHU (a1N Pa1  b1N Pb1 ) 

3N

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ªla a1N  lbb1N W   aa a1N  abb1N Pa 0  ba a1N  bbb1N Pb 0 º . ¬ ¼

(21.28)

+HUHZHPHHWDFDVHRIEDGO\GH¿QHGGHPDQGDQGVXSSO\IXQFWLRQVGXHWRWKH assumptions of constant returns and perfect competition. Let us assume that the market prices are not arbitrarily given to the producers, but are equal to WKHUHVSHFWLYHXQLWFRVWVRISURGXFWLRQ7KLVLPSOLHVWKDWKLVKHUPD[LPXPSUR¿W is equal to zero and the corresponding demand for inputs and supply of outputs, combined with a demand for saving s N (the common value of s0N , s1N ), remain XQGHWHUPLQHG)RUWKHVDNHRIDUJXPHQWLWLVVXI¿FLHQWWRDVVXPHWKDWDQRSWLPDO choice caN0 , cbN0 , caN1 , cbN1 is associated with other market signals: a1N , b1N , s N .3 Since by assumption maximum 3 N 0 , the demand and supplies of the physical goods and labour service which are chosen under the unique budget constraint caN0 Pa 0  cbN0 Pb 0  caN1 Pa1  cbN1 Pb1

l NW

(21.29)

coincides with those under the separate constraints (21.27).

Walras laws and saving functions for each period /HWXVDJJUHJDWHWKHTXDQWLWLHVSURGXFHGLQWKH¿UVWSHULRGDQGFRQVXPHGLQWKH second one over the two types of agents:

¦a , N 1

Aa1

N

S (.) {

m

¦s

i

Bb1

¦b , C N 1

¦c

L a1

a1

N

L

 ¦ cbN1 , Cb1 N

¦c

L b1

L

 ¦ cbN1. N

GH¿QHVWKHPDUNHWVXSSO\IXQFWLRQRIVDYLQJZLWK s i denoting an

i

optimum value s i for saver i. I (.) {

n

¦s

N

GH¿QHVWKHPDUNHWGHPDQGIRUVDYLQJ

N

with analogous meaning of s N for producer N. Notice the difference between the functions S (.) , I (.) DQGWKHIXQFWLRQVRIVDYLQJDQGLQYHVWPHQWGH¿QHGLQWKH third section by S0 ˜ { > A0  Ca 0 (˜) @ Pa 0  > B0  Cb 0 (˜) @ Pb 0 ; I0 (˜) {Ia0Pa0 + Ib0Pb0. We have seen that S (.), I (.) are not to be reckoned among the determinants of equilibrium, whereas S (.), I (.) play the same role of the others equilibrium conditions on the markets of physical goods. Let us add the budget equations of period t = 0: for all savers a0i Pa 0  b0i Pb 0

cai 0 Pa 0  cbi 0 Pb 0  s i i = 1,…, m

7KHPDUNHWIRUVDYLQJ for all producers s N

325

caN0 Pa 0  cbN0 Pb 0   aa a1N  abb1N Pa 0  ba b1N  bbb1N

N Q and obtain the Walras law for period t = 0

>A

0

 Da 0 (.) @ Pa 0  > B0  Db 0 (.) @ Pb 0 { S (.)  I (.)

(21.28)

Similarly let us add the budget equations of period t = 1:

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for all savers s i  l iW

cai 1 Pa1  cbi 1 Pb1 i = 1,…, m

for all producers l NW  a1N Pa1  b1N Pb1

caN1 Pa1  cbN1 Pb1  la a1N  lbb1N W  s N

N Q and obtain the Walras law for period t = 1

>L  L

D

(.) @W  > S (.)  I (.) @  > A1  Ca1 (.) @ Pa1  > B1  Cb1 (.) @ Pb1 { 0 (21.29)

This model exhibits six equilibrium conditions associated with the markets of goods a, b in period t = 0, 1; with the labour market and with the market for saving, but only four relative prices – the relative price of goods a, b available at t = 0, 1 the real wage and the rate of interest on the numeraire. The existence of two Walras laws, which share the value of the same excess supply S(.) – I(.), allows one to eliminate a seeming overdetermination. Garegnani (2009, pp. 4,5) has pointed out that the existence of distinct budget constraints for each period and the corresponding dated Walras laws should be taken into account: two laws in our case. The dependence of one equilibrium condition for each period is of the equivalent type according to the distinction made in the second section. There is no reason to take the equation S(.) = I(.) as dependent, instead of an equilibrium condition on the markets of physical goods or on the labour market, except for the mathematical convenience to reduce the number of the budget constraints to only RQH LQ RUGHU WR ¿QG D VROXWLRQ WR WKH V\VWHP RI HTXDWLRQV E\ WKH PHWKRG RI substitution. Hence the properties of equilibrium (existence, uniqueness and stability) depend on the properties of the functions S(.), I(.) on the same foot as on those of the demand and supply functions of physical commodities. We observe that we cannot assert – as in the traditional A-temporal – that any condition of market equilibrium depends on all the other equilibrium conditions of the model as a consequence of a unique Walras law. Instead in the Intertemporal, a condition associated with one period depends on the conditions of the same period only. Furthermore, since the same excees supply of saving appears in the budget equations of two contiguous periods – and it sets the link between them – a disequilibrium on the market for saving of the model implies a disequilibrium on two other markets for physical commodities, one for each period. We shall exploit WKLV SURSHUW\ LQ WKH ¿QDO VHFWLRQ IRU D UHIRUPXODWLRQ RI WKH PHWKRG RI TXDVL

326

Sergio Parrinello

HTXLOLEULXPXVHGE\*DUHJQDQL +RZHYHUZH¿UVWQHHGWRFODULI\WKH different choice of the numeraire in the intertemporal versus a-temporal models.

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A revised method of quasi-equilibrium Let us take good b available in period t as distinct numeraires for t = 0, 1. The auctioneer adjusts the relative prices: Pa0/Pb0, Pa1/Pb1, W/Pb1, Pbo/Pb1 with the proviso that Pb0, Pb1 are strictly positive and of course Pb0/Pb0 = 1, Pb1/Pb1 = 1. Outside a full equilibrium, the auctioneer is supposed to increase (decrease) the relative price Pb0/Pb,1, i.e. the rate of interest rb { Pb 0 /Pb1  1 , in response to an excess demand (supply) of saving and to adjust Pa0/Pb0, Pa1/Pb1, W/Pb1 in order to keep in equilibrium all markets, except the market for saving and the two markets for good b (the numeraire) in period t = 0, 1. Since two dated Walras laws must hold, when the auctioneer calls a rate of interest rb at which also the market for saving is in equilibrium, the economy has reached a general intertemporal equilibrium. Therefore the method consists in the construction of an excess supply of aggregate saving E (.) { S (.)  I (.) where only one explicit independent variable appears: the rate of interest rb { Pb 0 /Pb1  1 . The model of quasi-equilibrium can be derived from the Hahn model described in the third section, supplemented with the dated Walras laws (21.28) and (21.29) of the seventh section. After such integration, we drop equations (21.9) and (21.12) of the original model and set rb { Pb 0 /Pb1 as a parameter. The complete system of equations of quasi-equilibrium follows: Price equations Pa1

laW  aa Pa 0  ba Pb 0

(21.30)

Pb1

lbW  ab Pa 0  bb Pb 0

(21.31)

Equilibrium on the market of good a in period t = 0 Ca 0 (˜)  aa A1  ab B1

A0

(21.32)

Equilibrium on the labour market L t la A1 (.)  lb B1 (.) with ( L  la A1 (.)  lb B1 (.))W

0

(21.33)

Walras law for t = 0 (corresponding to (21.28))

[A

0

− Ca 0 (.) − aa A1 − ab B1 ] Pa 0 + [ B0 − Cb 0 (.) − ba A1 − bb B1 ] Pb 0

≡ S (.) − I (.))

(21.34)

7KHPDUNHWIRUVDYLQJ

327

Equilibrium on the market of good a in period t = 1 A1

Ca1 (˜)

(21.35)

Walras law for period t = 1 (corresponding to (21.29))

[ S (⋅) − I (⋅)] + ⎡⎣ L − la A1

− lb B1 ⎤⎦ W

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+ [ A1 − Ca1 (⋅) ] Pa1 + [ B1 − Cb1 (⋅) ] Pb1 ≡ 0

(21.36)

Let us admit that (21.30), (21.31), (21.32), (21.33) and (21.35) can be solved and the unknowns written as single valued functions of the independent variable rb { Pb 0 /Pb1  1. The unknowns are the quantities A1 , B1 and the relative prices Pa0/ Pb0, Pa1/Pb1, W/Pb1 . By substitution of a solution into the function of the excess supply of saving E (.) { S (.)  I (.), selected from one of the two Walras laws, we derive the function E (rb ) { S (rb )  I (.rb ) . We observe that the separate functions S (rb ), I (.rb ) are not yet derived, but only their difference function E (rb ) and this PD\EHDVXI¿FLHQWUHVXOWIRUVRPHDQDO\WLFDOSXUSRVH,QVWHDGLIZHDUHLQWHUHVWHG in the properties of E (rb ) , we should go back to the sixth section and derive the functions S (rb ), I (.rb ) by splitting each identity (21.28), (21.29) and (21.30) into two identities which represent the aggregation of the budget constraints of each type of agent: the saver and the producer. The model of quasi-equilbrium can now be used, as in Garegnani (2000, 2003), to describe an economy with many markets by means of a market for aggregate saving. E (rb ) is represented by a single curve (see Figure 21.1) and each point of rb E(rb)

E

–1

Figure 21.1

328

Sergio Parrinello

the curve, which does not intersect the rb axis, describes a state of the economy with disequilibrium on three markets. Instead the points of intersection are points of full equilibrium, if all prices implicit in E (rb ) are non negative and therefore rb t 1 .

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What progress in the debate next? We do not pretend that our argument has settled the controversy around question (2) of the introduction: ‘whether the possibility of reverse capital deepening and reswitching of techniques, which was proved for the aggregate version of the neoclassical theory of capital and distribution, can be a VSHFL¿F source of non meaningful intertemporal equilibria’. However, it should be undisputed that the function I (.rb ), and therefore E (rb ), is the door through which those ‘perverse’ properties of an equilibrium can enter both into the Intertemporal and into its corresponding A-temporal. Some participants in the debate will be inclined to close that door; others to keep it open as much as possible. I believe that more SURJUHVVLQVXFKDWKHRUHWLFDOFRQWURYHUV\FDQEHDFKLHYHGLIVRPHGLI¿FXOWLHVDUH acknowledged on each side and eventually overcome. The ‘door’ should not be FORVHGE\DSSO\LQJWKHVWDQGDUGVXI¿FLHQWFRQGLWLRQVHJWKHDEVHQFHRILQFRPH effects) for meaningful equilibria – existence, uniqueness and stability – directly to the A-temporal associated with the Intertemporal. In my opinion it is a methodological infringement bypassing the structural form of the Intertemporal and attributing behavioural assumptions directly to its reduced A-temporal form. On the other side, anyone who has good reasons to leave the ‘door’ open should deal with the choice of techniques in the Intertemporal with initial capital stocks arbitrarily given; therefore outside a long-period analysis, which was at the centre of the debate during the 1960.4 Then, within a short-period context, the FRPSRVLWLRQRIWKHGHPDQGIRUFRQVXPSWLRQSOD\VDVSHFL¿FUROHDQGLQJHQHUDO WKH QXPEHU RI PHWKRGV RI SURGXFWLRQ ZKLFK EHORQJ WR D SUR¿W PD[LPL]LQJ technique, is not equal to the number of products, also in the absence of scarce natural resources. Finally and more importantly, some progress in this debate can be achieved by proving (Parrinello 2010) that the instability of an intertemporal equilibrium may derive from the presence of a market for saying, even in the absence of productive capital.

Notes 1 Revised version of the working paper, ‘The Market for Savings in the Theory of General Intertemporal Equilibrium’, Quaderno di Ricerca n. 8 2009, Università degli Studi Roma Tre; Centro di Ricerche e Documentazione ‘Piero Sraffa’. 2 We are assuming, here, as in Hahn’s model, that saving is null in the last period 3 The amounts a1N , b1N , s N might be determined after the determination of a general market equilibrium, by an assumption of symmetry among n producers: each is supposed to satisfy 1/n of the market demand for each output. 4 This point was raised by Schefold in an informal exchange and can be easily accepted.

7KHPDUNHWIRUVDYLQJ

329

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References DOSSO (Dorfman R., Samuelson P., Solow R.) (1958). Linear Programming and Economic Analysis. New York: McGraw-Hill. Foley, D. (2008). Comment to Parrinello (2008) ‘The stability of general intertemporal equilibrium: a note on Schefold’, Metroeconomica 59: 2, pp. 313–16. Garegnani, P. (2000, 2003). ‘Savings, investment and capital in a system of general intertemporal equilibrium’ in Kurz, H.D. (ed.) Critical Essays on Sraffa’s Legacy in Economics. Cambridge: Cambridge University Press, pp. 392–445. Second version (with two appendices and a mathematical note by M. Tucci), in Petri, F. and Hahn, F. (eds) (2003), General Equilibrium: Problems and Prosepects, London: Routledge, pp. 117–75. Garegnani, P. (2009). ‘On some missing equations in contemporary general equilibrium’. Centro di Ricerche e Documentazione ‘Piero Sraffa’ Quaderno di Ricerca n. 7, Università degli Studi di Roma Tre. Mandler, M. (2005). ‘Well-behaved production economies’, Metroeconomica, 56, 4, November, pp. 477–94. Parrinello, S. (2005). ‘Intertemporal competitive equilibrium, capital and the stability of tâtonnement pricing revisited’, Metroeconomica, 56, 4, November, pp. 514–31. Parrinello, S. (2008). ‘The stability of general intertemporal equilibrium: a note on Schefold’, Metroeconomica 59, 2, pp. 305–12. Parrinello, S. (2010). ‘Numeraire, savings and the instability of a competitive equilibrium’; Metroeconomica, first published online: 25 Nov. 2010 | DOI: 10.1111/j. 1467-999X.2010.04113.x. Petri, F. (2004). General Equilibrium, Capital and Macroeconomics: A Key to Recent Controversies in Equilibrium Theory, Cheltenham: Edward Elgar. Schefold, B. (2008) ‘Saving, investment and capital in a system of general intertemporal equilibrium – an extended comment on Garegnani with a note on Parrinello’, published in Chiodi, G., and Ditta, L. (eds), Sraffa or an Alternative Economics, Basingstoke: Palgrave Macmillan.

22 Neoclassical theory: underdetermined, over-determined, or unable to move?

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Harvey Gram

Introduction Complete stock-flow intertemporal equilibrium “reconciles” traditional neoclassical theory, in which the value of capital is a given parameter, with neoWalrasian general equilibrium theory, in which a vector of capital goods is a given parameter.1 As Garegnani (1976, 1990, 2005) has argued, however, “the associated ‘dynamic’ sequence of equilibria . . . appears to be of doubtful [theoretical and HPSLULFDO@ VLJQL¿FDQFH´ *DUHJQDQL   -XVW VXFK HTXLOLEULD KDYH EHFRPH central to mainstream neoclassical theory, especially in macroeconomics, despite the generally recognized instability of the associated saddle-path solutions to various models.2 Saddle-paths have the property that, in the face of any shock, prices must “jump” discretely onto a new “convergent” path even though they cannot be “approached” from any point off the equilibrium path. In this theory, prices therefore lack persistence in the face of shocks. The classical forces of competition are entirely emasculated (Eatwell, 1982), as is the process of production “as commonly understood” (Garegnani, 1990, p. 55). The analysis of a dynamic sequence of equilibria follows a discussion in the following two sections of certain preliminary questions concerning the number of degrees of freedom in a simple model which has recently received considerable attention in the ongoing debate in capital theory.

Are neoclassical models under-determined? Frank Hahn’s broadside against the “neo-Ricardians” (Hahn, 1982) has prompted interest in a three-input, two-output model. Notation is not standard and can be confusing when dates are used to distinguish prices for the same thing at different times.37KH¿UVWVWDWHPHQWRIWKHTXDQWLW\DQGSULFHUHODWLRQVIRUWKLVOLQHDUPRGHO4 avoids time altogether: a11 X 1  a12 X 2 d V1

p1 d w1a11  w2 a21  w3 a31

a21 X 1  a22 X 2 d V2

p2 d w1a12  w2 a22  w3 a32

(22.1)

a31 X 1  a32 X 2 d V3 The notation is taken from Samuelson (1953), a paper widely cited by trade theorists.5

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Neoclassical theory 331 Consider the three quantity relations as constraints in a primal linear programming problem with the objective: max p1 X 1  p2 X 2 where prices are given DQGTXDQWLWLHVDUHYDULDEOH([FHSWE\DÀXNHDWPRVWWZRTXDQWLW\UHODWLRQVDUH VDWLV¿HGDVequations at the point of maximum value, in which case at least one corresponding dual factor price is zero.6 Ranges for p1 /p2, in open intervals bounded by distinct slopes of three resource constraints, are associated with positive outputs of both goods when factor supplies allow for such full employment YDOXHPD[LPL]LQJRXWSXWV([FHSWLQJWKHÀXNHFDVHLQZKLFKDOOWKUHHUHVRXUFH constraints happen to meet in a point of positive outputs, just two non-zero factor prices are then determined. (Multiply the output price vector by the inverse of the two-by-two technique matrix associated with the two fully employed endowments.) The most interesting feature of the solution is the “reciprocity condition” (Samuelson, 1953), an application of the dual theorem of linear SURJUDPPLQJ ¿UVW HTXDWLRQ EHORZ  WKH HQYHORSH WKHRUHP VHFRQG DQG WKLUG equations), and Young’s Theorem (fourth equation). Value-maximizing output quantities and (dual) cost-minimizing input prices (or shadow values) are denoted by an asterisk. Y = G ( p1 , p2 ,V1 ,V2 ,V3 ) = p1 X 1* + p2 X 2* = w1*V1 + w*2 V2 + w3*V3 ∂G /∂p j = X *j ∂G /∂Vi = wi* ∂X *j /∂Vi = ∂wi* /∂p j

(22.2) →

p j X *j Vi ∂X *j w*V p j ∂wi* = i i * Y X j ∂Vi Y w* ∂p j i p

or ϕ j EVXi j = θ i Ewi j 7KH¿QDO³UHFLSURFLW\FRQGLWLRQ´LQHODVWLFLW\IRUPVKRZVWKDWDQHJDWLYHTXDQWLW\ elasticity EVXi j { Xˆ j Vˆi RU5\EF]\QVNL HIIHFWXVLQJDFLUFXPÀH[WRGHQRWH percentage change) implies and is implied by a negative price elasticity p Ewi j wˆ i pˆ j or Stolper-Samuelson (1941) effect; whereas if one such elasticity exceeds unity, the other may or may not exceed unity, depending of the size of the factor income share Ti in total cost and product output share M j in total value. Absent joint production, an increase in the price of one product, holding other prices constant, will always result in at least one “winner” among factors wˆ i pˆ j ! 1 (the price of the “winning factor” increases in terms of all goods because it increases in terms of the output price in question and all other output prices are constant); and at least one “loser” among factors wˆ i pˆ j  0 (the price of the “losing factor” decreases in terms of all prices). Apart from the simplest two-input, two-output case, where knowledge of relativeLQSXWLQWHQVLW\DORQHLVVXI¿FLHQWWRLGHQWLI\WKH³ZLQQHU´DQGWKH³ORVHU´ there is no obvious connection between cost shares and the identity of “winners” and “losers.” This is most easily shown for the case in which the number of inputs is equal to the number of outputs (with no joint production) so that the cost share

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332

Harvey Gram

matrix by industry in the differential form of the price equations T wˆ pˆ can be inverted to obtain wˆ T 1 pˆ .7 Chipman (1969) shows that “important” factors, as indicated by the size of the elements of ș, need not be associated with “winners” when the corresponding output price increases. Nevertheless, regardless of dimensionality and even in the presence of joint products, Samuelson’s GXDO ³UHFLSURFLW\ FRQGLWLRQ´ KROGV XQGHU FRQVWDQW UHWXUQV WR VFDOH -RQHV DQG Scheinkman, 1977). The dual solution for factor prices yields budget constraints for owners of factor endowments. Aggregate consumption choices may or may not correspond to YDOXHPD[LPL]LQJRXWSXWV,IQRWH[FHVVGHPDQGRUVXSSO\FDQEHVDWLV¿HGE\ LPSRUWVDQGH[SRUWV2WKHUZLVH¿[HGSRLQWWKHRUHPVHVWDEOLVKH[LVWHQFHRIDQ output price vector and an implied input price vector corresponding to a general equilibrium of supply and demand. When a vector of prices for inputs and outputs LV³FULHGRXW´E\D¿FWLWLRXV³DXFWLRQHHU´VXFKDYHFWRUPXVWEHfeasible – the SULFHUHODWLRQVPXVWEHVDWLV¿HG,IQRWXQERXQGHGQHWUHFHLSWVXQGHUFRQVWDQW returns to scale) will appear possible to individual producers unaware of the economy’s resource constraints. The fact that input prices are not completely free to vary once output prices are “announced” ensures that budget constraints are jointly feasible.8 In an intertemporal equilibrium of supply and demand, prices are further constrained (Burgstaller, 1994). In the neighborhood of supply and demand equilibrium, an increase in price for some particular output can be associated with excess demand. This “unstable” case is generally explained by income effects: the usual one associated with given income, the purchasing power of which moves in the opposite direction to the change in any price, affecting quantities demanded even when there is no substitution effect; and, in the present context, an effect associated with changing income as endowments change their value (Stolper-Samuelson effects). Fluke cases can arise. In one type, Mandler (2002) concludes that the three-input, two-output model is “underdetermined” when three linear quantity relations, written as equations, just happen to intersect in a single point in the two-dimensional output space where both outputs are positive. As long as the relative price ratio falls between the largest and smallest of the slopes of three such constraints, both outputs are produced at the point of maximum value. In the three-dimensional dual space of factor prices, two price equations intersect in a line (rather than a point) contained in the costminimizing resource plane. The meaning of “underdetermined” is that any point on this line yields cost minimizing factor prices, a “degenerate” case in the language of linear programming. Another type of indeterminacy can also occur. Suppose utility functions are VSHFL¿HG IRU WKH RZQHUV RI HQGRZPHQWV /HW HDFK PHPEHU RI JURXS$ KDYH LGHQWLFDOWDVWHVDQGRZQHUVKLSRIIDFWRUVDQG/HWHDFKPHPEHURIJURXS% have identical tastes and ownership of factors 2 and 3. Each maximizes a different Cobb-Douglas utility function, in which case expenditure shares are invariant to

Neoclassical theory 333 prices, and are given by the corresponding exponents (summing to unity) of two utility indexes: UA

D D A A E1 1

A A E2 2

UB

D D B B E1 1

B B E2 2

(22.3)

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Suppose the third factor input is in excess supply at all feasible outputs and so FRPPDQGV D ]HUR SULFH ([FOXVLYH RZQHUVKLS RI WKH ¿UVW WZR IDFWRUV DOORZV Stolper-Samuelson effects on income distribution to have their greatest impact. 7KH¿UVWRIWKHIROORZLQJHTXDWLRQVLVWKHNH\WRDÀXNHFDVH

E1A w1V1  E1B w2V2

p1 ( D1A  D1B )

p1 X 1

w1V1 + w2V2 = p1Q1 + p2 Q2

(22.4) (22.5)

Expressing input prices as functions of output prices for produced goods, and output quantities as functions of endowments of fully employed factors, allows WKH¿UVWHTXDWLRQWREHZULWWHQ B A E 1A wV 1 1 + E 1 w2V2 = E 1

+ E 1B

a22 p1 − a21 p2 V1 a11a22 − a12 a12

a V − a12V2 a11 p2 − a12 p1 V2 = p1 22 1 = p1 X 1 a11a12 − a12 a21 a11a22 − a21a12

(22.6)

Without loss of generality, assume a11a22  a12 a21 ! 0 so that good 1 is relatively intensive in the use of factor 1. With good 2 as numéraire: [ E1A a22V1  E1B a12V2  a22V1  a12V2 ] p1

E1A a21V1  E1B a11V2

(22.7)

The price p1 LVXQGH¿QHGLIbothWKHULJKWVLGHDQGWKHFRHI¿FLHQWRQWKHOHIWDUH zero. This yields two restrictions on parameters. In particular, if the ratio of group shares spent on each good is equal to the ratio of intercepts of the binding constraints on the corresponding axis, there results a continuum of equilibria for a21 /a22 d p1 /p2 d a11 /a12 , and in that sense, the model is under-determined. Mandler (2002, p. 208) notes how his type of under-determination can be dealt ZLWKE\DOORZLQJIRUIDFWRUVXEVWLWXWLRQ7KHVSHFL¿FIDFWRUVPRGHO-RQHV  LVDFDVHLQSRLQW$VIXOOHPSOR\PHQWUHDOORFDWLRQVRIWKHQRQVSHFL¿FIDFWRUWDNH place, a full employment concave-to-the-origin production possibilities curve is traced out. At every point on the curve, income distribution is different, but in this theory it is not income distribution that determines prices but the other way around. This is a hallmark of modern textbook neoclassical economics – the value of inputs is derived from the value of outputs. The Mandler-type of under-determination was regarded by Samuelson as “in some ways the most interesting analytical case” (Samuelson, 1953, p. 7), because

334

Harvey Gram

its resolution requires “the intensive relations . . . to be taken in conjunction with the scale or extensive relations . . . with the result: the equilibrium values of the w ’s will certainly depend upon the factor endowments” (ibid.). This remark and others in the section of his paper entitled “From factor prices to commodity prices: the classical case” indicate resistance to any suggestion that income distribution FDQEHVSHFL¿HGLQGHSHQGHQWO\RIDJHQHUDOHTXLOLEULXPRIVXSSO\DQGGHPDQG

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Are neoclassical models over-determined? A different interpretation of the three-input, two-output model is provided by Eatwell (1987), Eatwell and Milgate (1999) and Garegnani (2000). One input is non-produced “labor”9 and the other two inputs are dimensionally the same as period t  1 outputs, used up (net of period t  1 consumption) in period t production. Analysis by Garegnani (2000) has been commented on extensively by Schefold (2008), building on his earlier work (Schefold, 1990, 2000), and by others in Metroeconomica1RYHPEHU ,QVXI¿FLHQWO\HPSKDVL]HGLVWKHIDFW that, from a neoclassical perspective, the introduction of time requires a distinction WREHPDGHEHWZHHQVWRFNVDQGÀRZV,QFODVVLFDOWKHRU\DQDO\VLVRIVWRFNVDQG their associated prices generally arises only to the extent that land and other nonproduced natural resources are considered. This becomes evident in the classical DQDO\VLVRI¿[HGFDSLWDOZKHUHDOORXWSXWVDUHÀRZVLQFOXGLQJ³ROGPDFKLQHV´ (cf. Kurz and Salvadori, 2003). Eatwell and Milgate (1999) use the three-input, two-output model10 to elucidate the argument in Eatwell (1987) to the effect that the errors in Walrasian capital theory are clearly exposed in a quite general model that is at once true to Walras” assumptions and inherently overGHWHUPLQHG4XDQWLW\DQGSULFHUHODWLRQVDUH¿UVW written with notation appropriate to a reproduction model: aL1Q1t  aL 2 Q2t d L a11Q1t  a12 Q2t d Q1t 1  D1t 1 t 1 2

a21Q  a22 Q d Q t 1

t 2

t 1 2

D

P1t d aL1W t  a11 P1t 1  a21 P2t 1 P2t d aL 2W t  a12 P1t 1  a22 P2t 1

(22.8)

Commodities still have numbers, 1 and 2, rather than a and b (Garegnani, 2000); or w and b (Petri, 1999); or s and w (Eatwell and Milgate, 1999); and time is a superscript. Output quantities Q tj are available at the end of period t, when ZDJHVDUHSDLG&RHI¿FLHQWVGLVWLQJXLVKHGE\DQLQLWLDOVXEVFULSWGHQRWHLQSXWV of labor and commodities 1 and 2 per unit of each output. An important feature of the model concerns units of measurement and size restrictions placed on the elements of the input/output matrix, aij , i, j 1, 2. In allocation models, the only UHVWULFWLRQRQIDFWRUVHUYLFHÀRZVSHUXQLWRIRXWSXWLVWKDWHDFKVXFKFRHI¿FLHQW is non-negative and at least one in each process (aLj , a1 j , a2 j ) is strictly positive.11 ,QUHSURGXFWLRQPRGHOVQRQQHJDWLYHFRHI¿FLHQWVZLWKWKHVDPHVXEVFULSWV aii , are pure numbers restricted to the interval [0, 1] . Units of measurement of

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Neoclassical theory 335 QRQQHJDWLYHFRHI¿FLHQWV aij and a ji are reciprocal, allowing characteristic roots and YHFWRUV WR EH GH¿QHG DQG DUH UHVWULFWHG LQ VL]H WR VDWLVI\ WKH ZHOONQRZQ condition, which ensures reproducibility: 0 d a12 a21 d (1  a11 )(1  a22 ) d 0 . According to the quantity relations, outputs of one period Q tj 1 , net of consumption D tj 1 , are available for use in the next period. To describe Q tj 1 as ³DQ HQGRZPHQW´ WR EH FRQVXPHG RU XVHG XS DV D ÀRZ RI IDFWRU VHUYLFHV LV misleading: this vector is not a given parameter but a result of previous production. To insist nevertheless that any such vector ( Lt , Q1t 1 , Q2t 1 ) has the same analytical role in period t 1 as (V0t , V1t , V2t ) in an allocation model forces an interpretation on the price relations which leads to over-determination, as described by Eatwell and Milgate (1999). They note that the commodity output price on the left of each price relation is one period later than the commodity input price on the right side.12 ,QDFLUFXODWLQJFDSLWDOJRRGPRGHOWKHODWWHULQSXWSULFHFDQEHLGHQWL¿HGDVWKH “rental” rate for using up the commodity in question. If “rental” rates for each commodity as a proportion of its end-of-period price are forced into uniformity by classical competition (Eatwell, 1982), gross rates of return must satisfy: 1 r {

P1t 1 P1t

P2t 1 { 1 r P2t

(22.9)

Proportionality implies equal own-rates of interest on supply price (the denominators of each ratio). Imposing this restriction on the price equations (assuming both goods are produced) over-determines prices in the sense that, DOWKRXJKWKHHTXDWLRQVGHWHUPLQHSULFHVDVIXQFWLRQVRIWKHXQLIRUPUDWHRISUR¿W such “prices of production” are not generally consistent with supply and demand equilibrium, given arbitrary endowments of individual capital goods. Eatwell and Milgate (1999) argue that the degree of over-determination is equal to n  1 , where n is the number of reproducible inputs, from which it follows that onecapital-good models are inherently misleading to the extent that they hide the problem.13 Their argument runs in terms of excess demand functions, of which WKHUHDUH¿YHLQDWKUHHLQSXWWZRRXWSXWPRGHO2QHIRUWKHODERUPDUNHW LV GURSSHGE\:DOUDV´/DZDQGDQRWKHUWZRIRUHQGRIVHFRQGSHULRGRXWSXWVDUH VDWLV¿HG ZKHQ WKH SULFH UHODWLRQV DUH VDWLV¿HG DV HTXDWLRQV EHFDXVH with the SURSRUWLRQDOLW\ UHODWLRQ VDWLV¿HG (so that rates of return are equalized), there would be no excess production of either end-of-second-period output.14 The only markets left to be considered are for initial arbitrary endowments of commodities. At prices of production, there is no reason to suppose that such markets will clear, except by chance: what owners of endowments choose to consume rather than sell (receiving uniform rate-of-return-rents in the upcoming production period) must be just such a vector as to leave behind what is needed to produce market clearing end-of-second-period outputs. Exceptions to the over-determination that Eatwell and Milgate regard as generic are constructed by Schefold (2000, 2008) – not by choosing the endowment vector to ensure market clearing, but by choosing a utility function to generate just those

336

Harvey Gram

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demand vectors that make everything hang together. Schefold is entirely clear about what he is doing – the recipe for the utility function is provided. His purpose is to show that implausible increases in the wage and associated large (therefore implausible) changes in prices of production are necessary to explain how an economic system could become more labor-intensive in production when there is reswitching of techniques. Also analysed are the consequences of a shift away from current consumption in favor of accumulation to show again that the required shift in the distribution of income is counter-intuitive in the presence of reswitching.

Neoclassical intertemporal equilibrium has no degrees of freedom Discussion of over-determination of neoclassical general equilibrium models can be traced back to Wicksell, who grappled with the meaning of the supply of “capital” as a factor of production which can vary in form, as described more recently by George Stigler: “There is a different sense in which a factor may be held constant: its economic quantity (or value) can be held constant. We can hold the house-building tools at $2,000, say, but vary their form so that they are most appropriate to whatever quantity of labor we employ. With fewer men, we use fewer and more elaborate tools; with more men, we use more, but less elaborate, tools. Or conversely, if we are examining the marginal productivity of tools, we can hire fewer but abler workmen (with the same aggregate payroll) with fewer tools and more but less able workmen with many tools” (Stigler, 1987, p. 136, emphasis added).15 Walras took a different tack. In reconciling the enforcement E\FRPSHWLWLRQRIDXQLIRUPUDWHRISUR¿WRQWKHVXSSO\SULFHRIFDSLWDOZLWKWKH existence of an arbitrary endowment of capital goods, he obscured the problem by failing “to specify what determined the quantities of individual capital goods produced, requiring only that the total value of capital goods produced should equal the total value of savings” (Eatwell and Milgate, 1999, n. 3).16 The particular way in which Samuelson (1953) analysed over- and underdetermination turned on a seemingly arbitrary matter: the number of inputs and outputs in a model. Mandler’s description of the three-input, two-output model as under-determined is reconciled with the opposite description by Eatwell and Milgate simply by noticing that, in one case, three inputs (including two that are produced) are treated analytically as if they are all non-produced (the three-factor, two-product case that Samuelson thought most interesting) whereas, in the other case, only one input is a non-produced factor (for Samuelson, the less interesting classical one-factor, two-product case). Under-determination can be resolved for Mandler by adding an output, but over-determination is still a problem for Eatwell and Milgate if at least two inputs and outputs are the same commodities (a nonproduced input and a non-basic output accounting for the remaining third quantity and price relation in the three-by-three case). Their proportionality relation – a FRQVHTXHQFH RI WKH IRUFH RI FRPSHWLWLRQ ± FDQQRW JHQHUDOO\ EH VDWLV¿HG ZKHQ commodity endowments are given arbitrarily.17

Neoclassical theory 337 An issue raised by Eatwell and Milgate directs attention to an underlying debate about method. The proportionality relation central to their over-determination argument implies stationary relative prices: 1 r {

P1t 1 P1t

P2t 1 { 1 r P2t

o

P2t P1t

P2t 1 P1t 1

(22.10)

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In response to their own question – “to what extent is [over-determination] dependent on the assumption of a stationary price system” – they answer at some OHQJWKQRWLQJ¿UVWWKDW In a competitive economy the individual agent can know current prices, but typically (apart from the few commodities for which futures markets exist) cannot know future prices. . . . In the absence of such general equilibrium information the individual agent in a competitive economy makes decisions on the presumption that current prices will persist, i.e. that the price vector is stationary. It is therefore reasonable to characterise the stationary price system as a centre of gravitation, whilst recognising that through time that centre of gravitation will change. (Eatwell and Milgate, 1999, pp. 98–9) Granting that perfect foresight has been a central feature of neoclassical intertemporal equilibrium, and citing Malinvaud (1961), who laid the foundations for this approach in Malinvaud (1953), Eatwell and Milgate state a methodological position which echoes a central thesis of Garegnani (1976); namely, that the method of economic analysis followed by both the classics and the original QHRFODVVLFVWRRNDQDEUXSWDQGLQVXI¿FLHQWO\XQGHUVWRRGWXUQZLWKWKHZRUNRI /LQGDKO+LFNVDQG+D\HNLQWKHV Yet, whereas in [the model without produced means of production] some VHQVHPLJKWEHPDGHRIWKHLGHDRIÀXFWXDWLRQVDURXQGWKHHTXLOLEULXPSRLQW and of adjustment to the centre of gravitation through time, no such interpretation is possible in the intertemporal model because adjustment WKURXJKWLPHLVDOUHDG\LQFRUSRUDWHGLQWKHGH¿QLWLRQRIHTXLOLEULXP. In this model the concepts of short-run and long-run therefore have no meaning. All WKHLQIRUPDWLRQDERXWWKHHFRQRP\LVDOUHDG\LQFRUSRUDWHGLQWKHGH¿QLWLRQ RIHTXLOLEULXP7KHQRWLRQRIÀXFWXDWLRQDURXQGWKDWHTXLOLEULXPRURIWKH equilibrium as a centre of gravitation makes no sense. This is why the concepts of short-run and long-run have disappeared from general equilibrium DQDO\VLVWREHUHSODFHGE\FDWDORJXHVRIPRGHOV(DFKPRGHOLVVSHFL¿F . . . All generality if lost. (ibid., p. 99, emphasis in the original)

338

Harvey Gram

In taking on the First Welfare Theorem, they go further, arguing that the claim that a competitive economy is Pareto optimal:

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LVRQO\FUHGLEOHLIWKHLQWHUWHPSRUDOHTXLOLEULXPPRGHOLVGH¿QHGDVD complete description of the economy. . . . this is not a model of the economy, it is the economy . . . . There may be good reasons for making such a change in economic method. But none of the authors of intertemporal models appear to have put forward a case for change. (ibid., p. 102) André Burgstaller has put forward a case. The most important analytical argument in his 3URSHUW\DQG3ULFHV7RZDUGDXQL¿HGWKHRU\RIYDOXH (Burgstaller, 1994) makes clear an intimate connection between intertemporal equilibrium in an Arrow-Debreu model with full futures markets and perfect certainty and a far more widely used analytical framework (what Burgstaller calls quasi-Hamiltonian theory) that draws on the theory of optimal control.18 Among those who have written about Burgstaller’s book,19 Duncan Foley pays special attention to Burgstaller’s originality: %XUJVWDOOHU¶VULJRURXVGHYHORSPHQWRIWKHVWRFNÀRZDSSURDFKWRLQWHUWHPSRUDO economic analysis leads him to an original and provocative critique of the Arrow-Debreu model (see Arrow and Hahn (1971)). Burgstaller’s target here is not the Arrow-Debreu model as such, but the intertemporal equilibrium interpretation of it that arises from regarding the abstract commodities in the PRGHODVUHSUHVHQWLQJJRRGVIRUGHOLYHU\DWVSHFL¿FGDWHV,WLVVWULNLQJWKDW this interpretation requires no alteration in the assumptions of the ArrowDebreu model itself, from which we can conclude that the resulting theory of intertemporal economics has QRVXEVWDQWLYHVWUXFWXUHWKDWLVVSHFL¿FWRWKH passage of time. . . . Burgstaller’s complaint against the Arrow-Debreu equilibrium is . . . that its mode of presentation obscures the real economic forces at work in establishing the intertemporal price-quantity path . . . prices of titles to existing stocks of commodities . . . are implicit . . . and can be recovered by transforming the price of time-dated commodities in the appropriate way. . . . In Burgstaller’s view . . . speculators are the protagonists of the drama of intertemporal economics. . . . An intertemporal Arrow-Debreu equilibrium records the results of these speculative efforts in establishing a consistent system of prices, but hides the traces of the efforts themselves. (Foley, 2001, emphasis added) In light of Burgstaller’s emphasis on speculation, he could well have chosen as his main hero Irving Fisher, who captured so succinctly the distinctly neoclassical idea which provides the theoretical linchpin for all of Burgstaller’s arguments. The causal argument, Fisher wrote, “is . . . not from present to future but future to present” (Fisher, [1906] 1965, p. 328). This harkens back to a remarkable essay

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Neoclassical theory 339 of interpretation by Krishna Bharadwaj (1989) where Fisher is portrayed in a VXEWO\GLIIHUHQWOLJKWFRPSDUHGWRWKHRWKHURULJLQDOSUH+LFNV/LQGDKO+D\HN  QHRFODVVLFV,WZDVKHZKRZURWHWKDW³SULFHV¿QGWKHLUQRUPDOOHYHOEHFDXVH the sellers have been good speculators as to what prices would be” (Fisher, [1906] 1965, p. 188, emphasis added), thereby expressing future-to-present causation. In order to illustrate how neo-Walrasian intertemporal equilibrium theory avoids under-determination and over-determination, it is useful to have a model in hand, not very different from the three-input, two-output framework. To avoid Mandler under-determination, a non-basic sector is added to an optimal control problem with three outputs, all measured per unit of labor employed. The sum of discounted (positive time preference) non-basic output per unit of labor is the objective:20 f

max ³ y0 e  U t dt

(22.11)

0

subject to resource constraints, in which labor is normalized at unity and arbitrary capital good endowments k1 and k2 are measured per unit of labor. Non-negative RXWSXW ÀRZV RI FRQVXPSWLRQ JRRG y0 and capital goods y1 and y2 (gross investment) are, like stocks, measured per unit of labor: 1  a00 y0  a01 y1  a02 y2 t 0 k1  a10 y0  a11 y1  a12 y2 t 0 k2  a20 y0  a21 y1  a22 y2 t 0 yj t 0

j

(22.12)

0,1, 2

Full employment of labor and full utilization of stocks occurs at the point of intersection of quantity relations written as equations, assumed to yield positive output of all three goods.21 At each such point: § y0 · ¨y ¸ ¨ 1¸ © y2 ¹

§ a00 ¨a ¨ 10 © a20

a01 a11 a21

a02 · a12 ¸ ¸ a22 ¹

1

§1· §1· ¨ k ¸ {  M 1 ¨ k ¸ ¨ 1¸ ¨ 1¸ © k2 ¹ © k2 ¹

(22.13)

where M is the usual input/output matrix augmented by a row and column of FRHI¿FLHQWVIRUWKHQRQSURGXFHGLQSXWDQGWKHQRQEDVLFRXWSXW So-called state equations show how physical capital labor ratios change over time, thereby altering the full employment of labor (and full utilization of physical capital) composition of output. There are two such equations, linking changes in FDSLWDOVWRFNVWRRXWSXWÀRZVDQGWKHVWRFNVWKHPVHOYHVWKHJURZWKRIHPSOR\PHQW at rate n being incorporated into the state equations). The following are familiar from one good neoclassical models of capital accumulation (which explains the choice of notation).22

340

Harvey Gram

k1

y1  (n  G1 )k1

(22.13)

k2 = y2 − (n + G 2 )k2

(22.14)

Implicit in the primal maximization problem is a dual minimization problem. 9DULDEOHIDFWRUSULFHVGH¿QHGE\WKHZDJHDQGrental rates for the use of labor and capital, minimize total cost. Formally: f

min ³ ( w  R1k1  R2 k2 )e  U t dt

(22.15)

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0

subject to price relations where the price of the consumption good is normalized at unity: 1  a00 w  a10 R1  a20 R2 d 0 p1  a01 w  a11 R1  a21 R2 d 0

(22.16)

p2  a02 w  a12 R1  a22 R2 d 0 wt0

R1 t 0

R2 t 0

When all three goods are produced, price equations are solved for factor prices as functions of product prices, the usual neoclassical way of thinking about income distribution. As with physical capital labor ratios in the resource constraints, it is here assumed that output prices are such as to allow for an interior solution so that all factor prices are strictly positive: §w· ¨R ¸ ¨ 1¸ © R2 ¹

§ a00 ¨a ¨ 01 © a02

a10 a11 a12

a20 · a21 ¸ ¸ a22 ¹

1

§1· §1· ¨ p ¸ {  M T 1 ¨ p ¸ ¨ 1¸ ¨ 1¸ © p2 ¹ © p2 ¹

(22.17)

Dual state equations (co-state equations for the primal maximization problem) are:

p1 = (n + δ1 + ρ ) p1 − R1

(22.18)

p 2 = (n + δ 2 + ρ ) p2 − R2

(22.19)

'H¿QHUDWHVRIUHWXUQDVWKHUHQWDOUDWHIRUWKHVHUYLFHVRIHDFKNLQGRIFDSLWDO divided by its price as an asset, minus the rate of depreciation, plus the rate of capital gain (or loss, if negative). The relationship between rates of return and rental rates thus emerges: r1 {

R1 p  G1  1 p1 p1

and

r2 {

R2 p  G2  2 p2 p2

(22.20)

Neoclassical theory 341 Equate rates of return r1 n  U r2 to obtain the previous dual state (or primal costate) equations, often called equations of arbitrage (Burgstaller, 1994) linking dividends (or rental rates) and expected (and, in equilibrium, realized) capital gains or losses to a common rate of return. One sees immediately that if relative prices are constant, equal net rates of return require the proportionality relation central to the critique of Walrasian capital theory by Eatwell and Milgate (1999):

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d ( p1 /p2 ) dt

0Ÿ

p1 p1

Ÿ r1 {

p 2 p2 R1  p1G1 p1

R2  p2G 2 p2

r2

(22.21)

Before proceeding, it is useful to write the co-state equations of the primal optimal control problem in one further, equivalent way, again on the assumption that p1 /p2 is constant, in order to show that the demand price of each capital good is the discounted value of rents net of depreciation: p1

R1  p1G1 r

and

p2

R2  p2G 2 r

(22.22)

Co-state equations are thus equivalent to equality of “effective” rates of return,23 not on supply price, as in classical theory, but on demand price as in portfolio theory. Garegnani addresses the issue: :HKDYHTXDOL¿HGLQWZRZD\VWKHXQLIRUPUDWHRIUHWXUQHQVXULQJSHUVLVWHQF\ of the theoretical position: that it should be reckoned on the supply price of the capital goods and that when price changes over time are taken into consideration the uniformity in question should be the “effective” one. With UHVSHFWWRWKH¿UVWTXDOL¿FDWLRQWKHXQLIRUPUDWHon the demand price of the capital goods is tautologically achieved by arbitrage in any, however temporary, equilibrium, since the demand price of a capital good is the one that capitalizes future returns at the current rate of interest. It [demand price] will therefore fall below supply price whenever its [the capital good in question] rate of return over the supply price is below that of other capital goods: obviously, the uniformity achieved in this way in no way improves the persistence of the resulting equilibrium. (Garegnani, 2005, n. 7) The co-state equations of the primal optimal control problem are generally described as equations of arbitrage, equalizing yields on all assets, inclusive of capital gains and losses. The prices in question and their rates of change are asset SULFHVRU³GHPDQGSULFHV´LQ*DUHJQDQL¶VWHUPLQRORJ\6XSSO\SULFHVDUHÀRZ SULFHVIRUWKHRXWSXWVRISURGXFHGLQSXWV(TXDWLQJDVVHWSULFHVDQGÀRZSULFHVLW would appear, is precisely what Garegnani does not do,24 in sharp contradistinction to what is done in standard models of intertemporal equilibrium.25

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342

Harvey Gram

Adding non-zero rates of change of capital good asset prices may appear to SURYLGH VXI¿FLHQW GHJUHHV RI IUHHGRP WR UHVROYH (DWZHOO0LOJDWH RYHU determination. However, this raises the peculiar possibility of divergent paths along which “a capital good which decreases in price over time will have a rental which increases over time” (Petri, 1999, p. 34). Resolving one problem creates another, which must be carefully considered in view of the instability of dynamic price paths, long recognized by neoclassical theorists. For example, Solow writes about the “need for someone to take the long view . . . when the question at hand was the potential instability of the market for natural resources if it concentrates WRRKHDYLO\RQVSRWRUÀRZGHFLVLRQVDQGQRWHQRXJKRQfuture or stock decisions” (Solow, 1974, pp. 12–13, emphasis added). In the absence of natural resource constraints, the same Fisherian future-to-present problem arises in models with produced inputs: . . . arbitrage-induced own-rate relations . . . knit successive price ratios WRJHWKHULQVXFKDZD\WKDWRQO\VHTXHQFHVOHDGLQJWRHI¿FLHQWSURJUDPVFDQ DULVHORQJUXQDVVHWYDOXHPD[LPL]DWLRQDQGFXUUHQWSUR¿WPD[LPL]DWLRQ coincide . . . . The truly remarkable thing about the intertemporal invisible KDQGLVWKDWZKLOHLWUHVXOWVLQHI¿FLHQF\RYHUORQJSHULRGVRIWLPHLWUHTXLUHV only the most myopic vision on the part of market participants. . . . But for society as a whole, there is need for vision at a distance. (Dorfman, Samuelson, and Solow, p. 321, emphasis added) 1RZDGD\V³YLVLRQDWDGLVWDQFH´KDVFRPHWREHHPERGLHGIRUPDOO\DVD¿UVWRUGHU condition for the solution of dynamic optimization problems, namely, the “transversality condition”26 which, though central to the entire neo-Walrasian approach, seems almost slighted by some as a technical matter best left to pure mathematicians.27 In his Ely lecture, Solow makes but a single reference to this condition, while stating clearly that it is very much part of his thinking (Solow, 1974, p. 13). In an interview, Malinvaud does likewise, but almost in passing: “Overall, this work [on capital accumulation and resource allocation] contributed to make mathematical economists understand why they should pay attention to transversality FRQGLWLRQV´.UXHJHU $GGLQJYDULDEOHUDWHVRIFKDQJHRIDVVHWSULFHVVXI¿FLHQW to obtain degrees of freedom necessary to overcome Eatwell/Milgate overdetermination only succeeds by introducing inherent instability. The latter is then dealt with by imposing stabiliz-ing future-to-present “causation” via the transversality condition. The role of the transversality condition as a way of “hiding instability in plain sight”, while putting the theory into an analytical straitjacket, can now be illustrated. Returning to the model, solutions for non-zero full employment outputs and associated non-zero factor prices are substituted (see the Appendix) into the respective state equations for the primal/dual problems to obtain pairs of differential equations showing changes in physical capital labor ratios and commodity prices as functions of their levels. The role of dual Stolper-Samuelson and Rybczynski effects (the partial derivatives) emerges:

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Neoclassical theory 343 § k1 · ¨ ¸ ¨ ¸ ©¨ k2 ¹¸

§ wy1  ( n  G1 ) ¨ wk ¨ 1 wy 2 ¨ ¨© wk1

§ p 1 · ¨ ¸ ¨ ¸ © p 2 ¹

§ wR1  ( r  G1 ) ¨ wp 1 ¨ wR2 ¨ ¨© wp1

wy1 wk 2

· k ¸ § 1 · § K1 · ¸˜¨ ¸¨ ¸ wy 2 ¸ ¨ ¸ ¨ ¸  (n  G 2 ) ¸ © k2 ¹ ©¨ K 2 ¹¸ wk 2 ¹

(22.23)

wR1 wp2

· p ¸ § 1 · § P1 · ¨ ¸¨ ¸ ¸˜ wR1 ¸ ¨ ¸ ¨¨ ¸¸  (r  G 2 ) ¸ © p2 ¹ © P2 ¹ wp1 ¹

(22.24)

One immediately sees the source of Petri’s concern (Petri, 1999, p. 34). He considers a case in which a rental (his m is one of our Ri ) rises relative to the wRi R corresponding price so that ! i . Certainly, this may entail a falling price, pi wpi although both elements in the relevant row of the above price system matrix must be taken into account. In any case, the so-called “characteristic equations” for the primal state and costate equations (equivalently, dual costate and state equations) are easily formed: ª

§ wyi ·º  n  G i ¸ » Ok  ¹¼ ¬ i 1 © wk i 2

Ok2  « ¦ ¨

ª § wy1 · § wy · wy wy º  n  G1 ¸ ˜ ¨ 2  n  G 2 ¸  1 2 » «¨ ¹ © wk 2 ¹ wk2 wk1 ¼ ¬ © wk1 ª

(22.25)

0

§ wRi ·º  n  Gi  U ¸» Op  ¹¼ ¬ i 1 © wpi 2

O p2  « ¦ ¨

ª § wR1 · § wR · wR wR º  n  G1  U ¸ ˜ ¨ 2  n  G 2  U ¸  1 2 » «¨ ¹ © w p2 ¹ wp2 wp1 ¼ ¬ © wp1

0

(22.26)

By a standard formula, the sum of the characteristic roots in each case is minus WKHFRHI¿FLHQWRQWKHOLQHDUWHUPIURPZKLFKLWIROORZVWKDWWKHVXPRIDOOIRXU roots is a positive number equal to the number of produced inputs times the rate of time preference:

O p(1)  O p(2)

2 § wR · ¦ ¨ i  n  G i  U ¸ p w © ¹ i 1 i 2 § wy · ¦ ¨ i  n  G i ¸  2U k w ¹ i 1 © i

O p(1)  O p(2)  Ok(1)  Ok(2)

2U ! 0

 Ok(1)  Ok(2)  2 U

(22.27) (22.28)

344

Harvey Gram

The fact that not all roots can be negative is our First Result. If the roots of one system (or their common real parts) are negative, the roots of the other system (or their real parts) cannot both be negative. Thus, if one system can be described as a stable node (or focus), the other system is an unstable node (or focus) or a saddle path (real roots with opposite signs). Again, by a standard formula, the product of the roots is equal to the constant term in each characteristic equation. A Second Result (see Appendix) is obtained, written in various ways using the First Result:

O p(1) O p(2)

Ok(1) Ok(2)  U Ok(1)  Ok(2)  U 2

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Ÿ O p(1) O p(2)  Ok(1) Ok(2)

O p(1) O p(2)

U ª¬ U  Ok(1)  Ok(2) º¼

Ok(1) Ok(2)  U O p(1)  O p(2)  U 2

Ÿ O p(1) O p(2)  Ok(1) Ok(2)

(22.29)

U ª¬O p(1)  O p(2)  U º¼

'LDJUDPVLOOXVWUDWHWKHVHWZR5HVXOWV7KH¿UVWURRWIRUHDFKV\VWHPLVPHDVXUHG on the horizontal axis; the second, on the vertical axis. A point in the positive quadrant of Figure 22.1, where all roots are equal to U /2VDWLV¿HVERWK5HVXOWV Intercepts of congruent downward sloping lines measure O p(1)  O p(2) U and Ok(1)  Ok(2) U . The upper branch of a rectangular hyperbola (to be used presently) is tangent to both lines at the point where: O p(1) O p(2) Ok(1) Ok(2) U 2 /4. With all roots positive (including positive real parts of complex roots), the system is totally unstable: neither prices nor quantities can converge to an unstable node (or focus). The system cannot get into equilibrium – it must be there. rb E(rb)

E

–1

Figure 22.1

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Neoclassical theory 345 If, in this special case of the First Result where O p(1)  O p(2) U Ok(1)  Ok(2), the SRLQWGH¿QLQJWKHURRWVRIRQHV\VWHPLVDERYHRUEHORZ WKHPLGSRLQWRIeither congruent line, but still in the positive quadrant, the indicated hyperbola will be closer to the origin, intersecting the congruent lines in two points with coordinates equal to the positive roots of either system. Again, the system is totally unstable. If, on the other hand, the chosen point is in either quadrant where one root is positive and the other negative, so that O p(1) O p(2) Ok(1) Ok(2)  0 when O p(1)  O p(2) U Ok(1)  Ok(2) , the indicated hyperbolic curve lies in those quadrants, identifying roots corresponding to a saddle-path for each system. In the case of saddle-path solutions, the important implication is that if initial prices and/or initial quantities are not just right, the system generates divergent paths that lead to meaningless solutions (cf. Petri, 1999, p. 34). Setting initial prices and/or quantities in such a way as to place the system on a so-called “convergent saddle-path” LVDFFRPSOLVKHGE\LPSRVLQJWKH¿UVWRUGHU³WUDQVYHUVDOLW\´FRQGLWLRQIRUDVROXWLRQ to the corresponding optimal control problem. The theory has boxed itself into a FRUQHUZLWKRXWDQ\GHJUHHVRIIUHHGRPSULFHVDQGRUTXDQWLWLHVÀRZVQRWVWRFNV  must jump discretely to avoid divergent paths whenever a parameter shift alters the position of the “convergent saddle-path” (for examples, see Burgstaller, 1994). The saddle-path property of equilibrium paths represents theoretical progress for some (Begg, 1982), but a “theoretical crisis” for others (Garegnani, 1990, p. 54) insofar as it implies a lack of persistence of the theoretical position in the face of shocks. 2QHFDQJHQHUDOL]HIXO¿OOPHQWRIWKH)LUVW5HVXOWE\FKRRVLQJDQDUELWUDU\SRLQW to represent the roots of one system. As in Figure 22.1, the intercept of a line with slope 1 drawn through such a point measures the sum of the given roots by its intercept on either axis. Pick another arbitrary point for the roots of the other system and draw a second line through this point, also with a slope 1 . The First Result implies that the dashed line connecting U on each axis must then fall halfway between the lines so drawn in Figure 22.2.28 7KLV LOOXVWUDWHV D ORQJHVWDEOLVKHG UHVXOW -RUJHQVRQ   LI RQH V\VWHP LV stable, with both negative roots on the line that passes through the lower-left quadrant, then the other system is unstable. Its roots must fall within the strictly positive quadrant by the Second Result (see the Appendix) whenever the discount rate is positive ( U ! 0 ). As for saddle paths, if the roots of one system have opposite signs, those of the other may or may not also indicate a saddle path. Suppose that the line further from the origin is for the price system so that, by the Second Result, O p(1) O p(2)  Ok(1) Ok(2) U[(O p(1)  O p(2) )  U ] ! 0 . If O p(1) O p(2)  0 at a point on the upper line, Ok(1) Ok(2)  0 at a point on the lower line: in each system, the product of the roots is negative (double saddle paths). The converse does not follow because Ok(1) Ok(2)  0 does not require O p(1) O p(2)  0 -RUJHQVRQ GUHZ the following conclusion:29 If excess capacity is not admitted and the dual interpretation of the dynamic input-output system is retained, then . . . macroeconomic stability of the dynamic input-output system implies instability of the dual: causal

346

Harvey Gram

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, λ(2) λ(2) k p

ρ

ρ

ρ

λ(1) , λ(1) k p

Figure 22.2

determinacy of the prices requires causal indeterminacy of the outputs. The FRQFOXVLRQ LV WKDW H[FHVV FDSDFLW\ RU SRVLWLYH SUR¿W OHYHOV RU ERWK  LV QHFHVVDU\ DQG QRW PHUHO\ VXI¿FLHQW IRU WKH LQWHUSUHWDWLRQ RI WKH G\QDPLF input-output system and its dual as a model of an actual economy. -RUJHQVRQS “Causal indeterminacy” was once the general lesson to be drawn from any model that exhibited saddle-path instability. In rejecting such models, one maintains “the possibility of a correspondence between theory and observation” *DUHJQDQL  Q   1RZDGD\V WH[WERRNV DUH ¿OOHG ZLWK VXFK PRGHOV LQWHUSUHWHGLQH[DFWO\WKHZD\WKDW-RUJHQVRQWKRXJKWDEVXUGQDPHO\DVPRGHOV of an actual economy, whose predictions can be tested. On the contrary, neoWalrasian theory is seen by its critics no longer as a theory of prices but UDWKHU DV D GHVFULSWLRQ RI FRPSOHWH VWRFNÀRZ G\QDPLF HTXLOLEULXP ZLWK initial prices set in such a way as to avoid instability. The value of capital is therefore also set initially and, in certain models, its discounted value remains constant:30 One interesting sidelight before we leave the subject of intertemporal pricing: &RQVLGHUDQ\HI¿FLHQWFDSLWDOSURJUDPDQGLWVFRUUHVSRQGLQJSUR¿OHRISULFHV

Neoclassical theory 347

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and own-rates. At every point of time the value of the capital stock at current HI¿FLHQF\SULFHVGLVFRXQWHGEDFNWRWKHLQLWLDOWLPHLVDFRQVWDQWequal to its initial value. This law of conservation of discounted value of capital (or GLVFRXQWHG1HW1DWLRQDO3URGXFW UHÀHFWVDVGRWKHJUDQGODZVRIFRQVHUYDWLRQ of energy of physics, the maximizing nature of the path. (Dorfman, Samuelson, and Solow, 1958, p. 322) In the von Neumann-Samuelson model set forth in Dorfman, Samuelson, and Solow (1958), the given value-of-capital endowment of traditional neoclassical theory is thereby reconciled with an arbitrary capital goods endowment along an HI¿FLHQWFRPSHWLWLYHHTXLOLEULXPSDWKRIDFFXPXODWLRQ,WV$FKLOOHV+HHOLVLWV inherent instability, nowadays hidden in plain sight under the misleading name “convergent saddle-path” (Samuelson, 1972). It is therefore striking to note that the discussion of the three-input, two-output model was initiated by Frank Hahn (1982), for it is precisely the “Hahn problem” that has more than a passing UHVHPEODQFH WR -RUJHQVRQ¶V VWDELOLW\LQVWDELOLW\ UHVXOW31 As is made clear by Hahn,32 an intertemporal equilibrium of supply and demand requires asset prices, ZKLFKGHWHUPLQHÀRZSULFHVLQFRPSOHWHVWRFNÀRZHTXLOLEULXPWREHLQLWLDOO\VHW in a precise way to rule out divergent paths. Saddle-path solutions illustrate the theoretical crisis about which Garegnani (1990) has written so eloquently. He dates the beginning of the crisis to work by +LFNV/LQGDKODQG+D\HNLQWKHV:LWKWKHSXEOLFDWLRQRI3RQWU\DJLQHWDO (1962), this theory received a new lease on life and, in various manifestations, has come to dominate modern neoclassical economics. It is remarkable that some of WKHPRVWLQÀXHQWLDOHDUO\ZRUNLQ6KHOO DQG&DVVDQG6KHOO ZDV written at the time of the main debate on reswitching in The Quarterly Journal of Economics (November 1966) and at precisely the moment when Garegnani (1976) ODXQFKHG KLV PDLQ DUJXPHQW -XVW DV +DKQ KDV GLVPLVVHG WKH QHR5LFDUGLDQV without attempting to draw a connection between his own critique of conventional theory (the “Hahn problem”) and Garegnani’s critique of intertemporal equilibrium, so Samuelson (1999) draws a single lesson concerning the relationship between steady state consumption per capita and the rate of interest from the entire discussion of Sraffa (1960), not even mentioning his own deep misgivings about intertemporal theory. In a discussion of the required re-aiming behavior of speculators, he wrote: /HWXVWKRXJKLQGXOJHRXULPDJLQDWLRQVDQGFRQFHLYHRIIXWXUHVPDUNHWV :KDWGHWHUPLQHVWKHSUR¿OHRIIXWXUHVSULFHVWKDWZLOOEHVHOIIXO¿OOLQJ"(DFK SDWKFDOOVIRUDQGLVHYRNHGE\DGLIIHUHQWSUR¿OHRIIXWXUHVSULFHVRUGXDO values . . .). As Hahn suggests, we are left with the problem of indeterminacy RI WKH IXWXUHSULFH SUR¿OH    ,Q WKH ORQJ ORQJ IXWXUH DQ\ SDWK EXW WKH convergent one is going to frustrate somebody’s expectations and, crudely, is going to lead to bankruptcy for someone and to re-aiming for the system . . . . There will generally be a sharp-shooting speculator around, both to pick up the pieces after the debacle and – by foreseeing the debacle – to make money by doing some of the things that keep it from happening.

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348

Harvey Gram The image in my mind is that of a bicycle. The rider of the bicycle is the bulk of the market, a somewhat mystical concept to be sure – like its analogue, the well-informed speculator who gets his way in the end because his way is the correctly discerned way of the future; and those who think differently are bankrupted by their bets against (him and) the future. (It is easier to identify the well-informed speculator ex post than ex ante, and the image can easily dissolve into an empty tautology.) For a time, the less than omniscient market may chase down a false path . . . but when the system is led too far from the EDODQFHGJURZWK FRQ¿JXUDWLRQ VRPH HQWUHSUHQHXUV EHJLQ WR IRUHVHH WKH shoals ahead . . . and they act to push the system back toward the turnpike. . . . [P]ragmatically, this may be the best one can hope for in an uncertain world, where changes in technology and tastes are going on continuously, making it impossible to certify ex ante just what is the golden-age turnpike that the system should aim for even if it possessed a greater approximation to omniscience than it actually does. . . . One feels that the real world of tools, plant, and inventory contrasts with WKHSXUHO\¿QDQFLDOGUHDPZRUOGRILQGH¿QLWHJURXSVHOIIXO¿OOPHQW%XWFDQ WKLVIHHOLQJEHJLYHQGRFXPHQWDWLRQDQGSODXVLEOHH[SOLFDWLRQ"

7KHTXHVWLRQDULVHVIURPWKHWKHRU\7KHWKHRU\VSHFL¿HVFRPSOHWHVWRFNÀRZ equilibrium in, for the most part, continuous time models in which the production period collapses to an instant. Given stocks of resources resolve themselves continuously into new stocks held in “portfolios,” evaluated in such a way as to enforce uniform ex ante and ex post rates of return as anticipated and realized capital gains and losses exactly compensate for any differences in shadow (or actual) rents relative to prices for each individual asset. It is just such a straitjacket interpretation of the price system under competition that Burgstaller (1994) has revealed as implicit in an Arrow-Debreu model, with its complete set of forward PDUNHWV+HWKHQFDUHIXOO\HOXFLGDWHVWKHVLJQL¿FDQFHRIWKLVVWUXFWXUHRISULFHV within the context of a wide range of models, offering interpretations both novel and provocative. With no disequilibrium to respond to, the forces of competition within a general equilibrium of supply and demand have literally nothing to do and so the entire traditional method of analysis breaks down (Eatwell, 1982). Continuous equilibrium destroys the distinction between the short run and long run and so destroys the basis for macroeconomic policy. The transversality condition of optimal control has a great deal to do with this inability to understand a problem, which very much depends on an analytical distinction between the role of competitive forces in the short run and their long-run consequences – the classical methodology. For those engaged in the revival of classical economics, “. . . independence of the equilibria from future conditions had in fact become untenable in the face of the impermanence of the new equilibria imposed by the Walrasian capital vector: but that was the consequence of the change in the notion of capital and not its cause” (Garegnani, 2005, p. 429).

Neoclassical theory 349

Appendix 7KH LQSXWRXWSXW FRHI¿FLHQWV VXEPDWUL[ A { aij , i, j 1,... n (here n 2 ) ignores both non-produced factors and input requirements for non-basic sectors. 'H¿QHWKHDXJPHQWHGPDWUL[33

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M

§ a00 ¨a ¨ 10 © a20

a02 · a12 ¸ ¸ a22 ¹

a01 a11 a21

(22.A.1)

Denote by M ij the two-by-two determinants of sub-matrices formed by eliminating row i and column j where rows and columns are numbered i, j 0,1, 2 . For example:

M 21

a00 a10

a02 and M 22 a12

a00 a10

a01 a11

(22.A.2)

The full employment/utilization solution for outputs of produced inputs is given by the solution to the quantity relations, written as equations: § y0 · ¨y ¸ ¨ 1¸ © y2 ¹

§ a00 ¨a ¨ 10 © a20

a01 a11 a21

a02 · a12 ¸ ¸ a22 ¹

1

§1· ¨k ¸ ¨ 1¸ © k2 ¹

M 01 M 11 M k1  21 k2  M M M

y1



y2

M 02 M 12 M k1  22 k2  M M M

yield

(22.A.3)

Substitution into the state equations yields stock changes as functions of stock levels:

M 01 M 11 M  k1  21 k2  (n  G1 )k1 M M M

(22.A.4)

M M M k2 = y2 − (n + G 2 )k2 = 02 − 12 k1 + 22 k2 − (n + G 2 )k2 M M M

(22.A.5)

k1

y1  (n  G1 )k1



350

Harvey Gram

In matrix form:

§ k1 · ©¨ k ¹¸ 2

§ M 01 · ·  ¸ §k · ¨ M ¸ 1 ¸ ¸˜¨ ¸¨ M 22 ¸ © k2 ¹ ¨ M 02 ¸  (n  G 2 ) ¸ ¨© M ¸¹ M ¹

§ M 11  ( n  G1 ) ¨ M ¨ M ¨  12 ¨© M



M 21 M

(22.A.6)

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The steady state solution for the vector of per worker capital stocks is:

§ k1 · ¨© k ¸¹ 2

§ M 11  ( n  G1 ) ¨ M ¨ M ¨  12 ¨© M

· ¸ ¸ M 22 ¸  (n  G 2 ) ¸ M ¹ 

M 21 M

1

§ M 01 · ¨ M ¸ ¨ ¸ ¨ M 02 ¸ ¨©  M ¸¹

(22.A.7)

In similar fashion, the solution for factor prices is obtained on the assumption that all goods are produced so that the price relations can be written as equations: §w· ¨R ¸ ¨ 1¸ © R2 ¹

§ a00 ¨a ¨ 01 © a02

a10 a11 a12

a20 · a21 ¸ ¸ a22 ¹

1

§1· ¨p ¸ ¨ 1¸ © p2 ¹

M 10 M 11 M p1  12 p2  M M M

R1



R2

M 20 M 21 M p1  22 p2  M M M

yield

(22.A.8)

(22.A.9)

Replace factor prices in primal costate (dual state) equations with these solutions, DQGDVVXPHWKDWÀRZSULFHVLQWKHSULFHUHODWLRQVDUHequal to asset prices in the FRVWDWHHTXDWLRQVDNH\VWHSLQWKHVWRFNÀRZHTXLOLEULXPPRGHO6HH%XUJVWDOOHU (1994) for a model that allows for Tobin’s q . As before, a pair of differential equations follows: p1

( U  n  G1 ) p1  R1

p 2

( U  n  G 2 ) p2  R2 (r  G 2 ) p2 

(r  G1 ) p1 

M 10 M 11 M  p1  12 p2 (22.A.10) M M M

M 20 M 21 M  p1  22 p2 M M M

(22.A.11)

Neoclassical theory 351 In matrix form:

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§ p1 · ¨© p ¸¹ 2

M 12 § M 11 ·   ( U  n  G1 ) ¨ M ¸ M ¨ ¸˜ M 21 M 22 ¨ ¸   (U  n  G2 ) ¸ ¨© M M ¹

§ M 10 · § p1 · ¨ M ¸ ¨© p ¸¹  ¨¨ M ¸¸ 2 20 ¨©  M ¸¹

(22.A.12)

The steady state solution for asset prices is:

§ p1 · ¨© p ¸¹ 2

M 21 § M 11 ·   ( U  n  G1 ) ¨ M ¸ M ¨ ¸ M 12 M 22 ¨ ¸   (U  n  G2 ) ¸ ¨© M M ¹

1

§ M 10 ·  ¨ M ¸ ¨ ¸ ¨ M 20 ¸ ¨© M ¸¹

(22.A.13)

A striking feature of the differential equations for asset prices can be seen by comparing it with the solution to the primal state equations. Apart from the discount rate in the differential equations for asset prices, every element in the right side matrix is the same, but opposite in sign. On the quantity side, convergence requires examination of the roots of the characteristic equation:

Q  Ok I {

M 11  (n  G1 )  Ok M 

M 12 M



M 21 M

M 22  (n  G 2 )  Ok M (22.A.14)

§ wy1 · ¨© wk  n  G1 ¸¹  Ok 1

wy1 wk 2

wy2 wk1

§ wy2 · ¨© wk  n  G 2 ¸¹  Ok 2

0

352

Harvey Gram

1RWHKRZ5\EF]\QVNLHIIHFWVHQWHULQWRWKH¿UVWFKDUDFWHULVWLFHTXDWLRQ2QWKH price side, convergence requires examination of the roots of the characteristic equation:

 P  Op I {

M 11  ( U  n  G1 )  O p M

M 21 M

M 12 M



M 22  (U  n  G 2 )  Op M

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(22.A.15) 

wR1  n  G1  U  O p wp1 

wR1 w p2

 

wR2 wp1

wR2  n  G 2  U  Op wp2

0

Note how Stolper-Samuelson effects enter into the second characteristic equation.34 Characteristic equations for the primal state and costate equations are: ª

§ wyi ·º  n  G i ¸ » Ok ¹¼ ¬ i 1 © wk i 2

Ok2  « ¦ ¨

ª § wy · § wy · wy wy º  « ¨ 1  n  G1 ¸ ˜ ¨ 2  n  G 2 ¸  1 2 » ¹ © wk 2 ¹ wk2 wk1 ¼ ¬ © wk1 ª

0

(22.A.16)

§ wRi ·º  n  Gi  U ¸» Op ¹¼ ¬ i 1 © wpi 2

O p2  « ¦ ¨

ª § wR · § wR · wR wR º  « ¨ 1  n  G1  U ¸ ˜ ¨ 2  n  G 2  U ¸  1 2 » ¹ © w p2 ¹ wp2 wp1 ¼ ¬ © wp1

0 (22.A.17)

The sumRIWKHURRWVLQHDFKFDVHLVHTXDOWRPLQXVWKHFRHI¿FLHQWRQWKHOLQHDU term (minus the trace of P or Q ), whereas the productRIWKHURRWVLVWKH¿QDO constant term (the determinant of P or Q ). The discriminant or radical is the square of the difference in the diagonal elements plus the product of the offdiagonal elements (of P or Q ). By inspection, the only difference in the price system is the presence of the discount rate in each diagonal element of P compared to Q . Therefore, in the expression for the radical, which includes the difference in the diagonal elements (rather than their sum), the discount rate cancels out. The radical in each system is therefore the same: 2

ª § M 11 · §M ·º M M  (n  G1 ) ¸  ¨ 22  (n  G 2 ) ¸ »  4 21 12 «¨ M M M M © ¹ © ¹ ¬« ¼»

(22.A.18)

Neoclassical theory 353 The sum and product of the roots of the quantity system are: λk(1) + λk( 2 ) =

2

i =1

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Ok(1) Ok(2)

⎛ ∂yi

∑ ⎜⎝ ∂k

i

⎞ M + M 22 − n − δi ⎟ = 11 − δ1 − δ 2 − 2n M ⎠

ª § wy1 · § wy · wy wy º  n  G1 ¸ ˜ ¨ 2  n  G 2 ¸  1 2 » «¨ ¹ © wk 2 ¹ wk2 wk1 ¼ ¬ © wk1 M 11 M 22  M 21 M 12 M 11 (n  G 2 )  M 22 (n  G1 )  2 M M

(22.A.19)

(22.A.20)

 (n  G1 )(n  G 2 ) The roots on the price side are related to the roots on the quantity side, using reciprocity relations linking Rybczynski and Stolper-Samuelson effects. Two UHVXOWV DUH XVHG VXEVHTXHQWO\ 7KH ¿UVW LV REYLRXV E\ LQVSHFWLRQ WKH VHFRQG requires some algebra. First Result: O p(1)  O p(2)

2 § wR · ¦ ¨ i  n  G i  U ¸ p w © ¹ i 1 i

(22.A.21)

Ok(1) Ok(2)  U Ok(1)  Ok(2)  U 2

(22.A.22)

 Ok(1)  Ok(2)  2 U Second Result: O p(1) O p(2)

O p(1) O p(2)

ª § wR1 · § wR · wR wR º  n  G1  U ¸ ˜ ¨ 2  n  G 2  U ¸  1 2 » «¨ ¹ © w p2 ¹ wp2 wp1 ¼ ¬ © wp1 ª § wR1 · § wR · wR wR º  n  G1 ¸ ˜ ¨ 2  n  G 2 ¸  1 2 » «¨ ¹ © w p2 ¹ wp2 wp1 ¼ ¬ © wp1

(22.A.23)

§ wR · § wR ·  U ¨ 2  n  G 2 ¸  U ¨ 1  n  G1 ¸  U 2 © w p2 ¹ © wp1 ¹

Ok(1) Ok(2)  U  Ok(1)  Ok(2)  U 2

Consider steady state quantities. This point is a stable node if Ok(1)  0 and O  0, in which case, their sum is negative and their product is positive. The Results show that O p(1) ! 0 and O p(2) ! 0 , indicating an unstable node at the (2) k

point of steady state prices because O p(1) O p(2)

Ok(1) Ok(2)  U Ok(1)  Ok(2)  U 2 has

all positive terms on the right side when U ! 0 and so the left side is positive. Roots on the price side have the same sign, positive because their sum is positive

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354

Harvey Gram

by the First Result, given the premise that the roots on the quantity side are both negative. Consider steady state prices. This point is a stable node if O p(1)  0 and O p(2)  0, in which case, their sum is negative and their product is positive. The Results show that Ok(1) ! 0 and Ok(2) ! 0 , indicating an unstable node at the point of steady state quantities. From the First Result, O p(1)  O p(2)  0 implies Ok(1)  Ok(2) ! 2 U . The second two terms on the right of the Second Result sum to a negative number because the inequality just obtained implies: U Ok(1)  Ok(2)  U 2 ! 2 U 2  U 2 U 2 ! 0 and so  U Ok(1)  Ok(2)  U 2  0 . However, both sides of the Second Result are positive by the initial premise of a stable node on the SULFHVLGHDQGVRWKHSURGXFWRIWKHURRWVRQWKHTXDQWLW\VLGHPXVWEHVXI¿FLHQWO\ positive to offset the negative sum just obtained. In short, they must have the same sign, which is positive because, as noted, their sum is greater than 2 U ! 0. At any point on one of the lines in either of the diagrams in the text, the sum and product of the roots for one system are given, as represented graphically by the intercepts of the downward sloping line through the given point and the position relative to the origin of the relevant branch of a rectangular hyperbola asymptotic to the axes. Except in the isolated case of complex roots (for which line and hyperbolic curve are tangent), the curve will intersect the line in two SRLQWV7KLVPHDQVWKDWIRUDSRLQWGH¿QLQJJLYHQURRWVIRURQHV\VWHPWKHUHLVDQ equivalent alternative point with the same implications. This is not surprising in view of the fact the axes measure the roots in arbitrary order. That said, the implications for the other system are similar insofar as the diagram is concerned. To make this clear, suppose the quantity system roots are known so that Ok(1)  Ok(2) { K and Ok(1) Ok(2) { S. Substitute:

O

(1) p



 O p(2)  K

2U

S  U K  U

O p(1) O p(2)

O p(1) ¬ª 2 U  K  O p(1) ¼º

O

(1) 2 p

(22.A.24)

S  U K  U

 2 U  K O p(1)  S  U  K  U

0

This is a quadratic in one of the roots for the price system, indicating two answers compatible with the given roots for the quantity system. Pick one and substitute back to get the other. For example, let U .2 and suppose the roots to the quantity system are as follows, corresponding to an unstable node:

Ok(1)  Ok(2)

.3  .5

.8 { K and Ok(1) Ok(2)

O

  2 U  K O p(1)  S  U  K  U

O

 2.1 or  1.9

(1) 2 p

(1) p

.3 u .5

O

(1) 2 p

.15 { S (22.A.25)

 .4 O p(1)  .03

0

(22.A.26)

Neoclassical theory 355 The unstable node at steady state quantities is consistent with one of two pricesystem saddle paths:

O  O  K 2 U 2.1  O .4 1.9  O .4 (1) p

(2) p

o

(2) p

o

(2) p

o

O O O

(1) p

 O p(2)

(1) p

 O p(2)

(1) p

 O p(2)



.4  .8

2.1 1.9

.4 1.7

(22.A.27)

1.5

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Notes 1 In one such reconciliation (Samuelson, 1990; Kataoka and Hashimoto, 1995), the value capital/output ratio remains constant as prices and quantities of capital goods vary along an intertemporal equilibrium path. See Sato and Ramachandran (1990) for other such “laws of conservation.” 2 The ubiquity of saddle-path solutions is evident from a cursory glance at recent graduate level macroeconomic textbooks. See Caputo (2005) for a clear and exhaustive account of the underlying mathematical arguments. 3 Garegnani (2000) names his two commodities a0 , b0 , a1 , b1 where the subscript is the date of availability. Prices are denoted Pa , Pa , Pb , Pb , avoiding a possible confusion of powers with time superscripts. 4 Already, questions arise. Does the model assume constant returns to scale or are the per unit coefficients calculated at just one level of output for each commodity, using input TXDQWLWLHV WHFKQLFDOO\ UHTXLUHG IRU WKRVH SDUWLFXODU RXWSXWV" 'R VXFK WHFKQLFDO UH quirements correspond to a “best practice” technique or to an average of techniques DFWXDOO\LQXVH" 5 A special case of the three-input, two-output model is the specific-factors model -RQHV LQZKLFKWKHILUVWLQSXWLVWKHVHUYLFHRIXQGLIIHUHQWLDWHGODERUWKHVHFRQG is the service of land used only in “agriculture,” and the third is the service of a capital good used only in “manufacturing.” Two of six coefficients are then zero, as land is not used in “manufacturing” and capital is not used in “agriculture.” Diminishing returns to labor along a uniformly concave product curve in each sector (holding land constant in one case and capital constant in the other) fixes the signs of the derivatives of all coefficients with respect to the wage rate and rental rates for the two specific factor services. This is in line with the general neoclassical predilection to emphasize the role of technical substitution in all directions, excluding more general representations of choice of technique involving the use of heterogeneous capital goods, specific to SDUWLFXODUWHFKQLTXHVFI*DUHJQDQL -RQHV LVFRQYHQWLRQDOLQWKLVUHJDUG except for an isolated remark in another context where a related assumption arises concerning the mobility of capital goods between countries: “Some objection might be raised to the assumption that capital goods can be physically moved from one country WRDQRWKHU´-RQHVSS± 7KHVSHFLILFIDFWRUVPRGHO-RQHV KDVEHHQ transmogrified into the Heckscher-Ohlin trade model (with homogeneous labor and capital) by assuming that the two specific factors are, in the short run, temporarily located quantities of homogeneous capital which can be more efficiently allocated across sectors in the long run. “International mobility of capital” then comes to mean something almost impossible to imagine; namely, the costless instantaneous allocation of a physical stock of “machines” across all sectors and all countries in such a way as to equalize the value of marginal products everywhere through variation in sectoral and country-wide machine/labor ratios. 0

1

0

1

356

Harvey Gram

6 If only one quantity constraint is satisfied as an equation at the maximized value of output and both goods are produced, the relative price must equal the slope of the binding constraint and only one factor price is positive. Finally, one binding constraint and one positive output indicate a corner solution with one positive factor price. 7 The differential form of the price constraints follows from cost minimization under variable factor proportions which ensures that the first term in the following equation vanishes: m

¦ w (da i

ij

i 1

m

)  ¦ aij (dwi )

dp j

j

1,...n

o

i 1

m

¦ (w a Downloaded by [Hacettepe University] at 12:42 20 April 2017

i ij

/p j )(dwi /wi )

dp j /p j

i 1

8 Mandler (2005) lets excess demand functions depend on input prices, blurring the essential neoclassical idea that minimum cost income distribution is determined by output prices. He expresses uneasiness: “That firms just declare their production costs rests uneasily with prices being set by the market and (like adjustment costs) is ad hoc . . . . Informally, one may think of output prices responding very quickly to the possibility of unbounded profits.” 9 To regard labor as “non-produced” does not deny that workers must be sufficiently fed, clothed and sheltered and provided with that minimum of medical care required to allow for their physical reproduction. Nor does it deny that the educational system has, as one of its purposes, the reproduction of skills needed in the sphere of production. A classical model can recognize all this by specifying a vector of required “consumption,” sometimes subsumed in the input/output system itself (e.g. Boggio, 1992, pp. 284–7). 10 The reason so much attention has been given to this model is motivated by Hahn (1982), who used it to argue that “neo-Ricardians” have offered no fundamental criticism of mainstream theory. A similar model is analysed at length in Chapters 11 and 12 of Dorfman, Samuelson and Solow (1958), work that has not been cited in the discussion of Hahn (1982). See Petri (1999) for an extended response to Hahn (1982) in which it is argued that the “capital theory controversy” ended too quickly with the resolution of certain formal problems, leaving unanswered the deeper methodological questions set forth in Garegnani (1976). See also Garegnani (2005). 11 In standard input/output notation, labor coefficients are denoted a0 j . The process for a single non-basic consumption good is designated by (a00 , a10 , a20 ) and the technique matrix becomes square, of dimension n + 1. With more than one non-produced input and more than one non-basic, a notation based on Hicks (1965) could be adopted to replace a00 , a01 , a10 , a11 with respective matrices: E, b, D, a. Only the last is necessarily square. Direct and indirect requirements of non-produced inputs in non-basic sectors differ in the price and quantity relations when augmented by profit and growth factors (Gram, 1985) with consequent “perversities” for supply and demand relations. 12 The time superscript on the wage in the price relations indicates that wages are paid at the time the output is sold. Garegnani (2000) provides no time indicator; Petri (1999) assumes that wages are paid at the end of the period, but gives the wage the same date as the produced input prices, as opposed to the (later) price of output. 13 Such models are not uncommon. When the specific factors model in trade theory is H[WHQGHG-RQHV WRLQFRUSRUDWHDUDWHRIUHWXUQRQO\RQHLQSXWDQGRXWSXWDUH the same good, as is also true of the two-sector neoclassical growth model, set forth to UHYHDOLWVUHODWLRQVKLSWRWKHVWDQGDUGWZRVHFWRUPRGHOLQ-RQHV  14 Eatwell and Milgate (1999) appear to regard this point as obvious.

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Neoclassical theory 357 15 For Stigler, the problem is symmetrical: in one case the value of the tools is given; in the other, the value of the payroll. Such symmetry fails to recognize the difference between buying tools and buying (slave) workers. 16 For further discussion of the traditional neoclassical approach, see Garegnani (1960, 1990). 17 In passing, it may be noted that the number of dual price and quantity relations has both formal and substantive importance. Equality of the number of operated processes and the number of produced commodities in reproduction models has been discussed at length, especially in the context of joint production models where the most interesting problems arise. See various papers by Schefold (1997) and the references cited therein. It was the existence of negative prices in the context of joint production (Sraffa, 1960) that prompted an outpouring of analysis. See Pasinetti (1980) and the references cited therein. 18 Familiarity with the language of control theory – state equations, co-state equations, jump conditions, transversality conditions, etc – is nowadays taken for granted in graduate level economics programs. 19 See Gram (1996) and four articles together with a response from Burgstaller in Metroeconomica (May 2001). 20 Time invariant current period utility function is U ( y0 ) y0. See Bruno (1967, 1969), whom we follow closely. 21 Physical capital labor ratios are restricted to a certain range of values so that inequality constraints can be satisfied as equations. As long as prices are such as to generate a maximum value at the point of full employment, all three factor prices are determined. Mandler-type under-determination of factor prices is avoided. The theory can handle slack resource constraints and non-positive outputs. See Bruno (1967) for a detailed analysis. 22 If employment is constant, then n 0 ; and if capital goods are circulating intermediate products, rates of depreciation are G1 1 G 2 . Then ki yi  ki and the distinction between stock and flow is harder to see. 23 Garegnani (1990, pp. 54–5; 2000, pp. 392–3, n. 2) discusses effective rates of return that take into account changes in relative prices. 24 This is forthrightly stated (Garegnani, 2005, p. 422). See also the discussion of the demand for labor curve in Garegnani (2000, n. 15) where demand price is w and supply price is W. The example is offered as a way to understand “general equilibrium schedules of investment and saving” in an article where the “readers are . . . asked for some effort in entering a less familiar way of analysis . . . [and to] resist, if possible, the temptation to translate them too quickly into the language with which they are more familiar” Garegnani, 2000, p. 394. 25 See Burgstaller (1994) who, in one of the models in his book, does allow for a difference, thereby taking account of Tobin’s q-ratio, which may be interpreted as the ratio of demand price to supply price. 26 The easiest way to understand this condition is to consider the problem of minimizing the distance from a point to a boundary line. In a metric space characterized by the Euclidean norm, one first order condition eliminates all but straight lines from the point to the boundary line (any non-linear path is longer than some linear path); a second first order condition eliminates all but one of the linear paths, thereby determining the angle at which the minimum distance path crosses the boundary line: the “transverse.” Setting this angle correctly is therefore called the “transversality condition.” 27 The mathematical theory that gave a boost to neoclassical analysis of efficient programs of capital accumulation came to the attention of economists with the publication of Pontryagin et al. (1962), cited by Shell (1967). 28 The case in which both lines pass through the positive quadrant has been skipped over by imposing the condition that the sum of the roots on the upper line is greater than 2 U

358

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30

 32 33 34

Harvey Gram

rather than between U and 2 U (thereby ruling out a totally unstable system characterized by two unstable “nodes” or “foci”). -RUJHQVRQDOORZVIRUDXQLIRUPUDWHRISURILW+LVSULFHUHODWLRQVDUHZULWWHQKHUHZLWK a slight change of notation (the rate of profit is r in place of S and the rate of interest is iLQSODFHRI-RUJHQVRQ¶Vr, to avoid confusion): p = (1 + r) {pA + [i – ( p /p)] pB + wa0} where pA is the value of intermediate or circulating capital goods, [i  ( p /p)] pB represents interest charges minus capital gains (plus capital losses) on the value of durable capital good inputs, and wa0 is a vector of labor costs per unit of output. Matrix B LV D VWRFNIORZ PDWUL[ -RUJHQVRQ GURSV WKH LQWHUHVW UDWH EHFDXVH LW RQO\ DGGV D constant to the real part of each root of the price system. He identifies his price relations with those of Morishima (1958) and Solow (1959) and notes that Dorfman, Samuelson and Solow (1958) showed sufficiency rather than necessity of excess capacity or positive profit levels (or both) in order to establish a plausible connection between theory and reality. Aoki (1977) followed up with a model that allows for excess capacity (and full cost pricing). The “conservation of capital value” is a particular example of more general “conservation laws.” It may be noted that Bruno (1967, p. 192, n. 4), citing Dorfman, Samuelson and Solow (1958, p. 322), finds constant capital value over certain intervals of time only, in a fixed coefficients model with one basic and one non-basic good. Kataoka and Hashimoto (1995) generalize Samuelson (1990). 7KHUHLVQRUHIHUHQFHWR-RUJHQVRQ LQHLWKHU+DKQ RU+DKQ  Not exactly clear in Hahn (1987), where two of the arrows on the convergent saddlepath point away from rather that towards the steady state solution. This notation and the argument closely follow Michael Bruno (1967). Here, minors have all their subscripts reduced by one, compared to those found in Bruno. Bruno handles slack constraints and zero dual values in detail. Taking note of the role of Rybczynski and Stolper-Samuelson effects is an addition here made to the analysis in Bruno (1967), a paper not as readily available as Bruno (1969) where it is cited.

References Aoki, M. (1977) “Dual Stability in a Cambridge-Type Model,” Review of Economic Studies, Vol. 44, pp. 143–51. Arrow, K. and F. Hahn (1971) General Competitive Analysis, Holden-Day: San Francisco; Oliver and Boyd: Edinburgh. Begg, D. (1982) The Rational Expectations Revolution in Macroeconomics: Theories and Evidence-RKQV+RSNLQV8QLYHUVLW\3UHVV%DOWLPRUH Bharadwaj, K. (1989) “Sraffa’s Return to Classical Theory,” in K. Bharadwaj and B. Schefold, eds, (VVD\VRQ3LHUR6UDIID&ULWLFDO3HUVSHFWLYHVRQWKH5HYLYDORI&ODVVLFDO Theory8QZLQ+\PDQ/RQGRQSS± %RJJLR/ ³3URGXFWLRQ3ULFHVDQG'\QDPLF6WDELOLW\5HVXOWVDQG2SHQ4XHVWLRQV´ Manchester School, Vol. 60, pp. 264–94. Bruno, M. (1967) “Optimal Accumulation in Discrete Capital Models,” in Shell (1967). Bruno, M. (1969) “Fundamental Duality Relations in the Pure Theory of Capital and Growth,” Review of Economic Studies, Vol. 36, pp. 39–53. Burgstaller, A. (1994) Property and Prices: Toward a Unified Theory of Value, Cambridge University Press: Cambridge. Caputo, M. (2005) Foundations of Dynamic Analysis: Optimal Control Theory and Applications, Cambridge University Press: Cambridge.

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Neoclassical theory 359 Cass, D. and K. Shell (1976) eds, The Hamiltonian Approach to Dynamic Economics, Academic Press: New York. &KLSPDQ -   ³)DFWRU 3ULFH (TXDOL]DWLRQ DQG WKH 6WROSHU±6DPXHOVRQ 7KHRUHP´ International Economic Review, Vol. 10, pp. 399–406. Dorfman, R., P. Samuelson, and R. Solow (1958) Linear Programming and Economic Analysis, McGraw-Hill: New York. (DWZHOO- ³&RPSHWLWLRQ´LQ,%UDGOH\DQG0+RZDUGHGVClassical and Marxian Political Economy, St. Martin’s: New York, pp. 202–28. (DWZHOO- ³:DOUDV¶V7KHRU\RI&DSLWDO´LQ-(DWZHOO00LOJDWHDQG31HZPDQ eds, The New Palgrave: A Dictionary of Economics0DFPLOODQ/RQGRQ (DWZHOO - DQG 0 0LOJDWH   ³6RPH 'HILFLHQFLHV RI :DOUDVLDQ ,QWHUWHPSRUDO Equilibrium,” in Mongiovi, G. and F. Petri, eds., 9DOXH'LVWULEXWLRQDQG&DSLWDO Essays in honour of Pierangelo Garegnani5RXWOHGJH/RQGRQ Fisher, I. ([1906], 1965) The Nature of Capital and Income, A. M. Kelley: New York. Foley, D. (2001) “Notes on Burgstaller’s Property and Prices,” Metroeconomica, Vol. 52, pp. 138–48. Garegnani, P. (1960) Il Capitale nelle teorie della distribuzione, Giuffré: Milan. Garegnani, P. (1970) “Heterogeneous Capital, the Production Function and the Theory of Distribution,” Review of Economic Studies, Vol. 37, pp. 407–36. Garegnani, P. (1976) “On a Change in the Notion of Equilibrium in Recent Work on Value and Distribution,” in M. Brown, K. Sato, and P Zarembka, eds, Essays in Modern Capital Theory, North-Holland: Amsterdam. *DUHJQDQL3 ³4XDQWLW\RI&DSLWDO´LQ(DWZHOO-00LOJDWHDQG31HZPDQHGV Capital Theory, W. W. Norton: New York. Garegnani, P. (2000) “Savings, Investment and the Quantity of Capital,” in H. Kurz, ed., Critical Essays on Piero Sraffa’s Legacy in Economics, Cambridge University Press: Cambridge. Garegnani, P. (2005) “Capital and Intertemporal Equilibrium: A reply to Mandler,” Metroeconomica, Vol. 56, pp. 411–37. Gram, H. (1985) “Duality and Positive Profits,” Contributions to Political Economy, Vol. 4, pp. 61–71; reprinted in Ian Steedman, ed., Sraffian Economics, Vol. II, Edward Elgar: Aldershot, 1989, pp. 162–78. Gram, H. (1989) “Comment” on Krishna Bharadwaj, “Sraffa’s Return to Classical Theory,” in K. Bharadwaj and B. Schefold, eds, (VVD\VRQ3LHUR6UDIID&ULWLFDO3HUVSHFWLYHV on the Revival of Classical Theory8QZLQ+\PDQ/RQGRQSS± Gram, H. (1996) Review Burgstaller (1994), Journal of Economic Behavior and Organization, Vol. 31, pp. 445–54. Hahn, F. H. (1966) “Equilibrium Dynamics with Heterogeneous Capital Goods,” Quarterly Journal of Economics, Vol. 80, pp. 633–46. Hahn, F. H. (1982) “The neo-Ricardians,” Cambridge Journal of Economics, Vol. 6, pp. 353–74. +DKQ)+ ³+DKQ3UREOHP´LQ-(DWZHOO00LOJDWHDQG31HZPDQHGVThe New Palgrave: A Dictionary of Economics0DFPLOODQ/RQGRQ +LFNV-5 The Theory of Wages0DFPLOODQ/RQGRQ +LFNV-5 Value and Capital, Oxford University Press: Oxford. +LFNV-5 Capital and Growth, Oxford University Press: Oxford. Hotelling, H. (1931) “The Economics of Exhaustible Resources,” Journal of Political Economy, Vol. 39, pp. 137–75.

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-RQHV5: ³7KH6WUXFWXUHRI6LPSOH*HQHUDO(TXLOLEULXP0RGHOV´Journal of Political Economy, Vol. 73, pp. 557–72. -RQHV 5 :   ³,QWHUQDWLRQDO &DSLWDO 0RYHPHQWV DQG WKH 7KHRU\ RI 7DULIIV DQG Trade,” Quarterly Journal of Economics, Vol. 81, pp. 1–38. -RQHV 5 :   ³$ 7KUHH)DFWRU 0RGHO LQ 7KHRU\ 7UDGH DQG +LVWRU\´ LQ - 1 %KDJZDWL5$0XQGHOO5:-RQHVDQG-9DQHNHGV7UDGH%DODQFHRI3D\PHQWV and Growth: Paper in International Economics in Honor of Charles P. Kindleberger, North-Holland: Amsterdam. -RQHV5:DQG-6FKHLQNPDQ ³7KH5HOHYDQFHRIWKH7ZR6HFWRU3URGXFWLRQ Model in Trade Theory,” Journal of Political Economy, Vol. 80, pp. 909–35. -RUJHQVRQ' ³$'XDO6WDELOLW\7KHRUHP´Econometrica, Vol. 28, pp. 892–9. .DWDRND+DQG++DVKLPRWR ³1HZ&RQVHUYDWLRQ/DZVLQD1HRFODVVLFDOYRQ Neumann Model,” Journal of Mathematical Economics, Vol. 24, pp. 271–80. Krueger, A. (2003) “An Interview with Edmond Malinvaud,” Journal of Economic Perspectives, Vol. 17, pp. 181–98. Kurz, H. and N. Salvadori (2003) “Fund–Flow versus Flow–Flow in Production Theory: Reflections on Georgescu-Roegen’s Contribution,” Journal of Economic Behavior and Organization, Vol. 51, pp. 487–505. Malinvaud, E. (1953) “Capital Accumulation and Efficient Allocation of Resources,” Econometrica, Vol. 21, pp. 233–68. Malinvaud, E. (1961) “The Analogy between Atemporal and Intertemporal Theories of Resource Allocation,” Review of Economic Studies, Vol. 28, pp. 143–60. Mandler, M. (2002) “Classical and Neoclassical Indeterminacy in One-shot versus Ongoing Equilibria,” Metroeconomica, Vol. 53, pp. 203–22. Mandler, M. (2005) “Well-Behaved Production Economies,” Metroeconomica, Vol. 56, pp. 477–94. 0LOJDWH 0   ³2Q WKH 2ULJLQ RI WKH 1RWLRQ RI ǥ,QWHUWHPSRUDO (TXLOLEULXP¶´ Economica, Vol. 46, pp. 1–10. 0RULVKLPD 0   ³3ULFHV ,QWHUHVW DQG 3URILWV LQ D '\QDPLFDO /HRQWLHI 6\VWHP´ Econometrica, Vol. 26, pp. 358–80. 1HXPDQQ-YRQ>@ ³$0RGHORI*HQHUDO(FRQRPLF(TXLOLEULXP´Review of Economic Studies, Vol. 13, pp. 1–9. 3DVLQHWWL/ Essays on the Theory of Joint Production0DFPLOODQ/RQGRQ Petri, F. (1999) “Professor Hahn on the “neo-Ricardian” Criticism of Neoclassical Economics,” in G. Mongiovi and F. Petri, eds, 9DOXH'LVWULEXWLRQDQG&DSLWDO(VVD\V in Honour of Pierangelo Garegnani5RXWOHGJH/RQGRQ 3RQWU\DJLQ/69*%ROW\DQVNLL59*DPNUHOLG]HDQG()0LVKFKHQNR The Mathematical Theory of Optimal Processes, Interscience: New York. Ramsey, F. (1928) “A Mathematical Theory of Savings,” Economic Journal, Vol. 38, pp. 543–59. Rybczynski, T. (1955) “Factor Endowment and Relative Commodity Prices,” Economica, Vol. 22, pp. 336–41. Samuelson, P. (1953) “Prices of Factors and Goods in General Equilibrium,” Review of Economic Studies, Vol. 21, pp. 1–22. Samuelson, P. (1967) “Indeterminacy of Development in a Heterogeneous-Capital Model with Constant Saving Propensity,” in K. Shell, ed., Essays on the Theory of Optimal Economic Growth, MIT Press: Cambridge MA. Samuelson, P. (1972) “The General Saddlepoint Property of Optimal Control Motions,” Journal of Economic Theory, Vol. 5, pp. 102–20.

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Neoclassical theory 361 6DPXHOVRQ3 ³7ZR&RQVHUYDWLRQ/DZVLQ7KHRUHWLFDO(FRQRPLFV´LQ56DWRDQG R. V. Ramachandran, eds, Conservation Laws and Symmetry: Applications to Economics and Finance.OXZHU%RVWRQ'RUGUHFKW/RQGRQ 6DPXHOVRQ3 ³7KH6SHFLDO7KLQJ,/HDUQHGIURP6UDIID´LQ*0RQJLRYLDQG F. Petri, eds, 9DOXH 'LVWULEXWLRQ DQG &DSLWDO (VVD\V LQ +RQRXU RI 3LHUDQJHOR Garegnani5RXWOHGJH/RQGRQ 6FKHIROG %   ³-RLQW 3URGXFWLRQ ,QWHUWHPSRUDO 3UHIHUHQFHV DQG /RQJ 3HULRG Equilibrium, A Comment on Bidard,” 3ROLWLFDO (FRQRP\ 6WXGLHV LQ WKH 6XUSOXV Approach, Vol. 6, pp. 139–63. Schefold, B. (1997) 1RUPDO 3ULFHV 7HFKQLFDO &KDQJH DQG $FFXPXODWLRQ, Macmillan: /RQGRQ Schefold, B. (2000) “Paradoxes of Capital and Counterintuitive Changes of Distribution in an Intertemporal Equilibrium Model,” in H. Kurz, ed., Critical Essays on Piero Sraffa’s Legacy in Economics, Cambridge University Press: Cambridge. Schefold, B. (2008) “Savings, Investment and Capital in a System of General Intertemporal (TXLOLEULXP±DQ([WHQGHG&RPPHQW´LQ*&KLRGLDQG/'LWWDHGVPiero Sraffa or an Alternative Economics3DOJUDYH0DFPLOODQ/RQGRQ Shell, K. (1967) ed., Essays on the Theory of Optimal Growth, MIT Press: Cambridge MA. Solow, R. (1959) “Competitive Valuation in a Dynamic Input–Output System,” Econometrica, Vol. 27, pp. 30–53. Solow, R. (1974) “The Economics of Resources or the Resources of Economics,” American Economic Review, Vol. 64, No. 2, pp. 1–14. Sraffa, P. (1960) 3URGXFWLRQ RI &RPPRGLWLHV E\ 0HDQV RI &RPPRGLWLHV 3UHOXGH WR D Critique of Economic Theory, Cambridge University Press: Cambridge. Stigler, G. (1987) The Theory of Price, 4th edition, Macmillan: New York. Stolper, W. and P. Samuelson (1941) “Protection and Real Wages,” Review of Economic Studies, Vol. 9, No. 1, pp. 58–73.

23 On/off inputs and their rentals

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Arrigo Opocher and Ian Steedman

Introduction It is quite wrong to overemphasize the change, from one positive quantity to another positive quantity, in the use of inputs as their prices change. Often enough the quantity will change from positive to zero, or vice versa. This is no doubt recognized in the most abstract arguments based, say, on production sets, but is given far too little attention in general. For electricity generation, for example, the wind turbine is vital to a windfarm but a complete irrelevance in a nuclear reactor, or a hydroelectric scheme. What might be called ‘on/off’ produced input use should be studied just as carefully as continuous changes in input use, as input prices change. This chapter is a contribution to such a study. Our main purpose is to show, by means of a series of numerical examples, that there can be no presumption that the familiar ‘qualitative restrictions’, referred to the case of marginal variations, extend to the ‘on/off’ case. We shall also present a sketch of a more inclusive analysis.

On/off inputs in a small open economy /HWXV¿UVWFRQVLGHUDVPDOORSHQHFRQRP\LQZKLFKWKHUHLVMXVWRQHLQGXVWU\ IRUPHG E\ LGHQWLFDO ¿UPV ZLWK FRQVWDQW UHWXUQV WR VFDOH ZKLFK SURGXFH D FRQVXPSWLRQJRRGE\PHDQVRIKRPRJHQHRXVODERXUDQGDPDFKLQH(DFK¿UP can choose between two qualitatively different machines, each of which can be imported at internationally given prices. We initially ignore the fact that each machine might be employed at different ‘labour intensities’ and just assume that one machine (of either kind) is operated by one unit of labour. We also assume, throughout this chapter, that the interest rate is identically zero.1 Technological conditions are such that, using (one) machine 1, the output is 61 units of the consumption good; using (one) machine 2, the output is 50. Denoting by p1, p2 the prices of the two machines, in terms of the consumption good, and by w the real wage rate, the following inequalities hold under competitive conditions ”p1 + w

(23.1)

”p2 + w

(23.2)

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On/off inputs and their rentals 363 ZLWKDWOHDVWRQHVWULFWHTXDOLW\:KHQDPDFKLQHVSHFL¿FSURFHVVPDNHVORVVHV (strict inequality) it is not activated, and that machine is not used. The inequalities above determine the real wage and the machine in use as functions of the two given machine prices. Initially let the two prices be p1 = 50, p2 = 40. The real wage is 11, machine 1 is used, the process using machine 2 would makes losses and is not used. Now let both prices increase by 20%, to p1 = 60, p2 = 48. It is clear that now the real wage has dropped to 2, and machine 2 has replaced machine 1. This is, of course, an immediate consequence of the fact that the machine's share in the cost of production is lower in the case of machine 2, so that the use of this machine is relatively ‘advantaged’ by the worsening terms of trade. Yet this result is in some sense at odds with the conventional microeconomic wisdom: the price change has triggered an increase in the use of machine 2 from zero to positive, and in no way can we say that the price of machine 2 has fallen. On the contrary, it has increased both relative to the consumption good and relative to the wage, while it has remained constant relative to machine 1: using any sensible composite numéraire, we can safely say that there is a positive relationship between the use of machine 2 and its rental (while, asymmetrically, the use of machine 1 is always negatively related to its rental). $SUDFWLFDOVHQVHRIZKDWLVDW¿UVWVXUSULVLQJLQWKLVUHVSRQVHFDQEHFRQYH\HG by the case of a differential tariff levied on the two machines on the basis of the initial (seller’s) prices of 50 and 40. Assume the tariff on machine 1 is 20%, and that on machine 2 is 22%. At (buyer’s) prices of p1 = 60 p2 = 48.8, the more heavily taxed machine 2 comes into use, even though its (buyer’s) price has risen relative to all other prices. (The producers of machine 2 are advantaged, not hurt, by a heavier tariff!) The above cannot be claimed to depend on the absence of ‘substitutability’ between each machine and labour. For assume now that, in a different SOE still specialized in the consumption industry, one machine 1, together with 5 units of labour, can produce 150 units of the consumption good, while machine 2 can be smoothly substituted for labour in order to produce one unit of the consumption good in a way implicitly described by the Cobb-Douglas unit cost function c

10 0.6 0.4 w p2 248

Assuming again p2

(4 / 5) p1 , we have

150 ≤ p1 + 5w 10 0.6 ⎛ 4 ⎞ 1≤ w ⎜ p1 ⎟ ⎝5 ⎠ 248

(23.1a) 0.4

(23.2a)

with at least one strict equality. The industry must stay on the outer envelope of two curves in w, p1 space represented in Figure 23.1. We immediately see that not only does it still happen that a proportional increase in the two machine prices

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Arrigo Opocher and Ian Steedman

triggers a change from one machine to another, thus necessarily implying a positive input use/input rental relation for one machine, but also it does so ‘twice’ (at p1 = 34.472 and at p1 = 88.281) for which machine 2 ‘recurs’ after having been replaced by machine 1. (Smooth substitutability between machine 1 and labour would not rule out this possibility: to see this, just replace the RHS of equation D ZLWKD&(6FRVWIXQFWLRQKDYLQJDYDQLVKLQJO\VPDOOµı¶ Now we can express the use of each machine as a function of its rental, and the use of labour as a function of the real wage, assuming that the price ratio between the two machines remains constant.

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Wage

25 22.5 20 17.5 15 12.5 10 37.5

50

62.5

75

87.5 100 Price of machine 1

Figure 23.1

/HWWKHRXWSXWEH¿[HGDW7KHXVHRIPDFKLQHM1, is M1 = 0 for 0 < p1 < 34.472 and 88.281 < p1 M1 = 1 for 34.472 < p1 < 88.281 As far as machine 2 is concerned, things are slightly more complex, because, for the values of p1 which bring that machine into use, we must calculate the cost minimizing combination of that machine with labour. Keeping in mind that p2 = (4/5)p1, one PXVW¿UVWGLIIHUHQWLDWHWKHFRVWIXQFWLRQZLWKUHVSHFWWRp2, obtaining M2 = (75/31) (w/p2)0.6. It should be stressed, however, that differentiation alone is not enough to determine the cost minimizing amount of any machine, and it ‘works’ only when we know what machine is in use (more on this in the fourth section). It follows that M2 =

60 for 0 < p2 < 27.578 and 70.625 < p2 p2

M 2 = 0 for 27.578 < p2 < 70.625

On/off inputs and their rentals 365 7KHXVHRIODERXU¿QDOO\LVL = 5 when machine 1 is used, and L = (225/62) (p2/w)0.4 when machine 2 is in use. It follows that L=

90 for 0 < w < 12.344 and 23.105 < w w

L = 5 for 12.344 < w < 23.105 This is represented graphically in Figures 23.2 a, b and c. 1

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Machine 1

0.75

0.5

0.25

0 25

50

75

(a)

100 Rental 1

Figure 23.2a Use of machine 1 in relation to its rental Machine 2

7.5

5

2.5

0 25 (b)

Figure 23.2b Use of machine 2 in relation to its rental

50

75 Rental 2

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Arrigo Opocher and Ian Steedman

Labour 20

15

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10

5 12.5 (c)

25 Wage

Figure 23.2c Use of labour in relation to the real wage (axes NOT labelled from zero)

The reaction of an on/off input to a change in its rental (at a constant output) is E\GH¿QLWLRQGLVFRQWLQXRXV:HQRWLFHKRZHYHUWKDWWKHPHUHSUHVHQFHRIRQRII machines determines discontinuities also in the use of labour. It is also worth noting that, in this example, at no switch point is labour use positively related to the wage rate.

On/off inputs produced domestically Now let us assume that the machines are produced domestically (and only for domestic use). A further problem arises. When such inputs are out of use, they are not even produced and we do not know the long-run price and rental of something which does not exist as an economic entity. Since in this case the rental is unknown, it may be impossible to compare it with the rental when the same input is actually in use. Taking this point of view, one could not even assert the Generalized Substitution Theorem. Yet a more ‘constructive’ view is possible. In fact, there are restrictions on the long-run prices of potential outputs/inputs (and therefore on their rentals). When input i is out of use, a long-run equilibrium requires that both the potential users and the potential producers of i ¿QGLWSUR¿WDEOHWRGLVFDUGLW 2QHH[SHFWVWKHQWR¿QGLQWHUYDOVRISULFHVZKLFKµVXVWDLQ¶WKHFKRLFHRIneither producing nor usingDPDFKLQHRIDFHUWDLQNLQG$QDSSURSULDWHPRGL¿FDWLRQRI RXU¿UVWH[DPSOHVKRZVWKDWZHFDQVWLOOVD\ZKHWKHUDWDVZLWFKSRLQWWKHUHQWDO of each input rises or falls. Let us introduce a ‘basic’ machine, which is employed together with labour, in the production of either machine 1 or machine 2. One unit of labour and one ‘basic’ machine can produce 1.25 ‘basic’ machines. Moreover, let 1 ‘basic’ machine, 1 unit of labour and 5 units of land produce, alternatively, 1 unit of machine 1 or 1.25 units of machine 2. The consumption good is produced

On/off inputs and their rentals 367 DVLQWKH¿UVWH[DPSOH'HQRWLQJE\r the rent rate, and by p the price of the basic machine (always in terms of the consumption good), inequalities (23.1) and (23.2) must be supplemented by two further inequalities: p1”p + w + 5r

(23.3)

1.25 p2”p + w + 5r

(23.4)

Finally, since the ‘basic’ machine must always be produced, we have

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p = 4w

(23.5)

It will be clear that either equations (23.1) and (23.3) or equations (23.2) and (23.4) must hold as strict equalities, for otherwise the consumption good would not be produced. Now let r = 0. At w = 10.167, both the machine-1 industry and the consumption industry break even if p1 = 50.835. As far as machine 2 is concerned, a break-even price for one industry would induce losses in the other: no price can be determined. This does not mean, however, that we know nothing of p2. In fact, equations (23.2) and (23.4) set (rather narrow) upper and lower bounds, namely 40.668 and 39.833. 0RUHJHQHUDOO\HTXDWLRQV WR GH¿QHDUHDOZDJHUHDOUHQWIURQWLHU in which machine 1 is used at ‘low’ real rents. As the real rent increases (and the real wage falls), machine 2 is brought into use.2$WDVSHFL¿FSDLUr = 5, w = 6), both machines can be used and produced, and their rentals are fully determinate (p1 = 55, p2 = 44). At all other pairs, one rental is determined, while the other has upper and lower bounds, as in Figures 23.3 a and b.3 Rental 1

60

55

0 (a)

5

10 Land rent

Figure 23.3a The rental of machine 1 as a function of the real rent (vertical axis NOT labelled from zero)

368

Arrigo Opocher and Ian Steedman

Rental 2

50

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45

40 0 (b)

5

10 Land rent

Figure 23.3b The rental of machine 2 as a function of the real rent (vertical axis NOT labelled from zero)

An increasing real rent (and falling real wage) determines, at a critical point, the switch from machine 1 to machine 2, and we can unambiguously say that at the switch point both rentals increase. It follows at once that the relation between a machine’s use and its own rental is negative for machine 1 and positive for machine 2. It is also easy to see that this switch involves, again, a negative relation between labour use (in the consumption industry) and the real wage. The latter relation, however, is by no means a logical necessity. For let the production of a unit of machine 1 require, together with 1 basic machine, 1.4 units of labour (instead of 1) and 4 units of land (instead of 5). The production of machine 2 has become relatively more ‘land intensive’ and it is not surprising to ¿QGWKDWQRZLWLVXVHGDWµORZ¶UHDOUHQWV$VWKHUHDOUHQWLQFUHDVHVDQGWKHUHDO wage falls), crossing the critical point (r = 2.678, w = 7.857), machine 2 is replaced by machine 1. Now, in the consumption industry, labour use per unit of output is lower with machine 1 than it is with machine 2. It follows that in the neighbourhood of the switch point labour use is positively related to the real wage. It is worth stressing that labour use in the consumption industry falls in a situation in which the wage strictly falls relative to both machine 1 and machine 2; it also falls relative to the consumption good and land, whereas it remains constant in terms of the basic machine. It must also be remarked that this ‘unconventional’ reaction by no means GHSHQGVRQµ¿[HGFRHI¿FLHQWV¶)RUOHWODERXUEHVPRRWKO\VXEVWLWXWDEOHZLWKHDFK machine, according to the Cobb-Douglas cost functions c1

0.9 w0.15 p10.85

On/off inputs and their rentals 369 1.2 w0.2 p20.8

c2

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Let p1 = 5.4w + 4r; p2 = 4w + 4r.4 The ranking of c1, c2 crucially depends on the rent/wage ratio, z = r/w. In particular, c2 < c1LI”z”DQGc1 < c2 if z > 0.275. Now let min (c1, c2) = 1. We may obtain the relevant real wage in relation to z w=

1 for 0 ≤ z ≤ 0.275 1.2(4 + 4 z )0.8

w=

1 for z ≥ 0.275 0.9(5.4 + 4 z )0.85

Differentiating the relevant cost function with respect to w, and making some substitutions, we get l = 0.24(4 + 4z)0.8IRU”z” l = 0.135(5.4 + 4z)0.85 for z• As z increases from zero, and the real wage falls, labour use per unit of output starts increasing, because machine 2 is marginally substituted for labour. At z = 0.275, however, machine 2 is replaced by machine 1, and labour use drops, as shown in Figure 23.4. Further increases in z determine a sub stitution of machine 1 for labour, and labour use per unit of output increases again. Labour

0.9

0.85

0.8

0.75

0.7

0

0.25

0.5

1 0.75 Rent/wage ratio

Figure 23.4 Labour use per unit of output in relation to the rent/wage ratio (vertical axis NOT labelled from zero)

370

Arrigo Opocher and Ian Steedman

On/off inputs and conventional analysis The last example leads us to a theoretical explanation of the positive input use – input price relations discussed in this chapter. ,W VKRXOG EH ¿UVW UHPDUNHG WKDW QR H[DPSOH YLRODWHG WKH *HQHUDOL]HG 6XE stitution Theorem. In general, let xA, B EH WKH YHFWRU RI LQSXW XVH ZLWK D ¿[HG output) at input prices w A, B (by abuse of notation). The Theorem says that, on the basis of cost minimization alone,

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(xB – xA) (wB – wA ” ,QRXUH[DPSOHVWKHVHYHFWRUVKDYHWKUHHWHUPVWKH¿UVWUHODWLQJWRPDFKLQH the second to machine 2 and the third to labour. Let us denote by ‘A’ an initial situation, and by m1, m2, l the use of the three inputs when output is 1 (recall that FRQVWDQW UHWXUQV WR VFDOH KDYH EHHQ DVVXPHG WKURXJKRXW  ,Q WKH ¿UVW H[DPSOH of the second Section we have ( x B − x A ) = (− m1A , m2B , (l B − l A )), and (wB – wA) = (( p1B  p1A ), ( p2B  p2A ), (wB – wA)). Evaluating each term, the inner product is – (10/61) + (8/50) – (99/3050) = – 0.0364. It is left to the interested reader to generate initialDQG¿QDOVLWXDWLRQVXVLQJWKHVHFRQGH[DPSOH7KHWKLUG6HFWLRQ introduced a complication, because vector wi, i = A, B, has some undetermined, but bounded, terms. The undetermined terms have the same position in wi as the null terms in xi ,I LQ WKH LQLWLDO ¿QDO  VLWXDWLRQ PDFKLQH    LV XVHG DV above, then

{

(w B − w A ) = ⎡⎣( min p1B ≤ p1B ≤ max p1B ) − p1A ⎤⎦ ,

}

⎡ p2B − ( min p2A ≤ p2A ≤ max p2A )⎤ , ⎡⎣ wB − w A ⎤⎦ ⎣ ⎦ 1RZWKHWKHRUHPLVYHUL¿HGLI

 m1A min p1B  p1A  m2B  p2B  min p2A  l B  l A  wB  w A d 0 It is again left to the reader to substitute symbols with numbers and see that the 7KHRUHPLVLQIDFWYHUL¿HG All the examples have a common feature: there will never be two null addends on the left-hand side, and for this reason the Theorem is compatible with positive input use – input price relations. The problem with the Generalized Substitution 7KHRUHPLVWKDWLI¿UPVKDYH]HURQHWSUR¿WVERWKEHIRUHDQGDIWHUDFKDQJHDW least two (real) input prices must change5 and in this case the theorem is of no practical guidance. A particular version of the Theorem is commonly thought to be more useful, EXW XQIRUWXQDWHO\ LW H[FOXGHV E\ GH¿QLWLRQ WKH RQRII LQSXWV %\ 6KHSKDUG¶V Lemma, the cost function Jacobian, J, is the cost minimizing vector of input use.

On/off inputs and their rentals 371 If J > 0, then the cost function Hessian, H, determines the qualitative change in input use at any vector of input price change, according to the equation

dx = H × dw

(23.6)

%\QHJDWLYHVHPLGH¿QLWHGQHVVRIH, we have dw T ⋅ dx ≤ 0 , which proves the theorem for a particular case). In particular, dxi ¦ Hij wˆ j  wˆ i , i z j, where xi j ∂2c H ij = , and a ‘hat’ denotes a proportional rate of change. If all inputs are ∂wi ∂w j Hickian substitutes and the price of one of them falls relative to that of all others, then equation (23.6) tells us that its use (at constant output) increases. It may be tempting to infer that any input i such that xi > 0 reacts in this way, irrespective of whether there are on/off inputs or not. Our examples show that this inference would be wrong. Equation (23.6), then, offers no qualitative restriction in our case. In order to allow for on/off inputs, a representation of the cost function more general than usual is needed. Let technical knowledge in a certain industry allow for the alternative use of s ‘technologies’ each characterized by a technologyVSHFL¿FLQSXWDQGn common inputs. At a given output there are, so to speak, two choices to make, one concerning the technology and the other the input combination within that technology.6 Denoting by w and x the vectors of the n ‘common’ input prices and input uses, and by ( p1 , p2 , ", ps ) , ( m1 , m2 , ", ms ) the prices and quantities of technologyVSHFL¿FLQSXWVWKHFRVWIXQFWLRQFDQEHIRUPXODWHGDV

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c = min ⎡⎣c1 ( w , p1 ) , c2 ( w , p2 ) , ", cs ( w, ps )⎤⎦ , with ci ( w , pi ) = min x , mi

{(wx + p m ) f ( x, m ) = 0} , i

i

i

i

i = 1, 2, ", s

where fi (x, mi) is a transformation function associated to the given output level. Assuming for simplicity that all inputs are positive within each technologyVSHFL¿FFRVWIXQFWLRQWKH+HVVLDQDQGHTXDWLRQ LVZHOOGH¿QHGDWLQSXW prices w, p such that a single ci(w, pi) is the minimum. When, however, at switch points, more than one ci(w, pi) is minimum, we can make no use of the Hessian, and no simple qualitative restrictions seem to be available. The same is true, for discrete changes, when the industry shifts from one ‘technology’ to another.

Concluding remarks The presence of a ‘continuum of techniques’ should not be taken too literally, as meaning that on the isoquant any input can be marginally reduced by marginally expanding any of the others. At a given state of knowledge, it happens that different potential ‘technologies’ are characterized by qualitatively different inputs: a change in the long-run price system may determine a marginal change in input quantities

372

Arrigo Opocher and Ian Steedman

within the technology in use, and/or it may determine a shift from one technology to another. The familiar strict continuity hypothesis is no doubt convenient on analytical grounds, but it is relevant only if it generates results which apply also to more practical cases. We have shown in this chapter that the familiar results do not extend to on/off inputs. When such inputs are involved, as is often the case, it can happen that the use of an input rises (falls) when its rental rises (falls) relative to DOORWKHULQSXWUHQWDOV7KLVXQFRQYHQWLRQDOHIIHFWPD\FRQFHUQWHFKQRORJ\VSHFL¿F on/off inputs, as well as ‘common’ inputs.

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Notes 1 We want to make it clear that our results by no means depend on capital theoretic arguments. The reader aiming at more realism can verify that a positive (and variable) interest rate would not change the argument substantially. 2 The interested reader will verify that w

61 5  r for 0 d r d 5 6 6

w

10 

4 r for 5 d r d 12.5 5

3 Specifically, when 0 d r d 5 , we have p1

305 5  r 6 6

239 5 122 2  r d p2 d  r 6 6 3 3



:KHQ”r”ZHKDYH 51 

4 r d p1 d 50  r 5

p2

40 

4 r 5

4 When indeterminate, these are the minimum prices. For a rationale of using minimum prices, see next Section. 5 This fact is of course taken into account in long-run analyses of input use by the most DFFXUDWHDXWKRUV)RULQVWDQFH%ODQFKDUGDQG.DW] VWUHVVWKDWµDORQJUXQ labour demand curve [gives] the real wage consistent with other input prices and the condition that firms make a zero pure profit’. 6 Of course, from a mathematical point of view, there is only one choice, in n + s space.

Reference %ODQFKDUG2DQG.DW]/) µ:KDWZHNQRZDQGZKDWZHGRQRWNQRZDERXWWKH natural rate of unemployment’, The Journal of Economic Perspectives, 11 (1): 51–72.

24 Indivisible inputs and the probability of reswitching1

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Ian Steedman

‘Die ganzen Zahlen hat der liebe Gott gemacht; alles andere ist Menschenwerk.’ Leopold Kronecker (at the Deutsche Mathematiker Vereinigung, 1886)

Introduction We shall consider a stationary economy which produces a given net output of the consumption commodity, initially by means of a particular method. An alternative PHWKRGWKHQEHFRPHVDYDLODEOHDQGZHDVNZKLFKSDUDPHWHUVGH¿QLQJWKLVµQHZ¶ PHWKRGZLOOLPSO\WKDWRQHRIWKHWZRPHWKRGVZLOOEHSUHIHUUHGDWDOOUDWHVRISUR¿W which will imply that there will be a single switchpoint between the two methods, and which will imply double-switching between them. This will then allow us to discuss the probability of reswitching and, more importantly, to consider a technicality that can appear to stand in the way of such discussion. In the context of the particular model of production employed here, we shall reach the conclusion that the probability of reswitching is small – but it must be emphasized that we DWWDFK OHVV LPSRUWDQFH WR WKLV VSHFL¿F FRQFOXVLRQ WKDQ WR RXU GLVFXVVLRQ RI WKH WHFKQLFDOGLI¿FXOW\MXVWPHQWLRQHGDQGWRWKHSURSRVHGDSSURDFKWRRYHUFRPLQJLW

Two methods of production When using the incumbent, or ‘old’ method of production, our economy has two industries. In one of them, 400 units of a basic good and 1,000 units of labour are used to produce 500 units of the basic. In the other, 100 units of the basic and 200 units of labour are used to produce 600 units of the consumer good. With circulating capital and ex-postSD\PHQWRIZDJHVWKHFRUUHVSRQGLQJZDJHSUR¿W frontier is given by w=

3(1 − 4r ) (6 + r )

(24.1)

It is important to notice that, because the consumer good industry is the less labour-intensive one, the frontier (24.1) has the property (d2w/dr2)>0 for all nonnegative (r, w).

374

Ian Steedman

Suppose now that an alternative, ‘new’ method of production becomes available. This involves only one industry, in which K units of the consumer good and L units of labour can produce (K + 600) units of the consumer good (and hence the same QHWRXWSXWDVWKHµROG¶PHWKRG 7KHZDJHSUR¿WIURQWLHUIRUWKLVPHWKRGLVJLYHQE\

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⎛ 600 − Kr ⎞ w=⎜ ⎟⎠ ⎝ L

(24.2)

Restricting our attention always to the case of non-negative (r, w), we now ask how the two frontiers, (24.1) and (24.2), relate to each other. 7KHUHDUHLQHIIHFW¿YHGLIIHUHQWFDVHVWRFRQVLGHU7KHUHDGHUPD\ZLVKWR sketch the frontiers (24.1) and (24.2) to illustrate each case.) 1 If L < 1200 and K < 2400 there will be no switchpoint, because the new method will dominate the old one. Hence the corresponding area in Figure 24.1

Figure 24.1

Indivisible inputs and the probability of reswitching 375

2

3

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4

5

is labelled N, to indicate that the new method is always used as r rises from zero until w = 0. If L < 1200 and K > 2400 there will be a single switch. The corresponding area in Figure 24.1 is labelled NO, to show that there is a switch from the new to the old method as r rises. If L > 1200 and K < 2400 there will again be a single switch. But it will now be from the old method to the new one, as r rises, and hence the corresponding area in Figure 24.1 is labelled ON. If L is ‘much bigger’ than 1200 and K is ‘much bigger’ than 2400 there will be no switchpoint, because the old method will dominate the new one. The corresponding area in Figure 24.1 is thus labelled O. If L is ‘very little bigger’ than 1200 and K is ‘very little bigger’ than 2400 there will be double-switching, with the old method in use both at r = 0 and at w = 0 but with the new method in use for a certain range of (positive) r values. The corresponding area in Figure 24.1 is therefore labelled ONO.

7KXV IDU ZH KDYH VSHFL¿HG FDVHV   DQG   D OLWWOH YDJXHO\ ZKDW GR µPXFK bigger’ and ‘very little bigger’ mean here? The curve separating the regions O and ONO shows those values of (K, L) for which equating w in (24.1) and (24.2) leads to a quadratic equation in r that has a double root. (In geometric terms, these values of (K, L) make the straight line frontier (24.2) a tangent to the frontier   7KHFXUYHLQTXHVWLRQLVGH¿QHGE\ 6K

13L  600  5 L  L  1200

DQGKDVDQLQ¿QLWHVORSHDWL = 1200 and a zero slope at K = 2400. $V WKH UHDGHU PD\ DOUHDG\ KDYH QRWLFHG RXU ¿YH FDVHV DERYH H[FOXGH WKH possibility that (L – 1200) (K – 2400) = 0. That is because this possibility can lead to some ambiguity as to which region such cases ought to be assigned to. For example, if (L – 1200) = (K – 2400) = 0, is that case to be allocated to N (on the argument that the new frontier dominates the old one), or to ONO (on the argument that there are switchpoints at both r = 0 and w = 0)? We refuse to engage in logomachic disputation and proceed as follows. Since our argument tends to the ¿QGLQJWKDWGRXEOHVZLWFKLQJLVXQOLNHO\LWLVVHQVLEOHWRDOORFDWHGLVSXWDEOHFDVHV to ONO wherever possible and to neither NO nor ON whenever that can be avoided. On this understanding, then, Figure 24.1 allocates every possible (K, L) comELQDWLRQ WR RQH RI ¿YH UHJLRQV 7KH UHJLRQV 1 DQG 2 DUH RI QR LQWHUHVW IRU D discussion of switching and reswitching, while the remaining three regions are central to such a discussion.

The probability of reswitching: a dialogue Protagonist. While every relevant new method (K, L) can now be taken to lie in NO, or ONO, or ON, only those (K, L) lying in ONO give rise to reswitching.

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Ian Steedman

Using ‘ono’ to denote the area of the ONO region, etc., one can represent the probability of reswitching as

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⎛ ⎞ ono ⎜⎝ no + ono + on ⎟⎠ . 6LQFHµRQR¶LV¿QLWHZKLOHHDFKRIµQR¶DQGµRQ¶LVXQERXQGHGZHPD\FRQFOXGH that the probability of reswitching is vanishingly small. (This conclusion would, of course, change if one stipulated that extreme values of (K/L) are of no interest and that, e.g., 1 < (K/L    1RZ ERWK 12 DQG 21 ZLOO EH OLPLWHG WR ¿QLWH triangles and it can be shown that ⎛ ⎞ ⎛ 1 ⎞ ono ⎜⎝ no + ono + on ⎟⎠ < ⎜⎝ 385 ⎟⎠ . One could, of course, change the limits on (K/L) and/or consider non-uniform distributions of (K/L) within the assigned limits.) But whatever the details, Figure 24.1 embodies a method for assessing the probability of reswitching. Antagonist. No, no, your argument is quite unacceptable. Each of the areas you UHIHUWRLVDVHWZLWKLQ¿QLWHO\PDQ\PHPEHUV$QGZHKDYHNQRZQVLQFHWKH 1870s, thanks to Cantor and others, that in such cases a sub-set (like your ‘ono’) can have the same cardinality as the complete set (your ‘no+ono+on’). Your proposed probability measure is thus entirely spurious. Protagonist. It is far from self-evident that the Cantor-like arguments to which you refer are relevant to our problem. In supposing them to be relevant you are taking it for granted that LQSXWVDUHLQ¿QLWHO\GLYLVLEOH and that we thus need the continuum of real numbers to represent the alternative (K, L). Suppose, to the contrary, that for each of our inputs there is a smallest amount that ever needs to be considered and that all input quantities are represented by integer multiples of such smallest amounts. Now imagine, please, that Figure 24.1 is drawn on conventional graph paper and is thus covered in (a very large number of) small squares. Only points at the corners of those squares represent meaningful (K, L) combinations. So we can simply countWKHQXPEHURISRLQWVLQ212DQGWKDWQXPEHULV¿QLWH$QGPXFK smaller than 512 = 2601, in fact.) In NO and ON, however, the process of counting will never end, for there will EH LQ¿QLWHO\ PDQ\ FRUQHU SRLQWV ZLWK XQUHVWULFWHG 12 DQG 21  7KXV WKH probability of reswitching is indeed vanishingly small. (And incidentally, with very small squares, the number of relevant points would be roughly proportional to the area considered, so that our original (ono/no+ono+on) area proposal was a good approximation to a counting measure.) Antagonist. I recognize that input indivisibility certainly makes a big difference and I acknowledge that it was therefore a little hasty to dismiss your ‘relative

Indivisible inputs and the probability of reswitching 377 areas’ measure of probability out of hand, by reference to Cantor-like reasoning. I would draw it to your attention, however, that if one of the inputs were indivisible and the other one fully divisible then such reasoning would still be relevant. Thus you need both inputs to be indivisible in order to ward off the Cantor-like REMHFWLRQ6RDUHLQSXWVLQGLYLVLEOH"

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Input indivisibility . . . the factors of production, in the world as we know it, consist of indivisible units. (Robinson, 1948 [1933], p. 334) No one contends that inputs are in fact fully divisible. (Blaug, 1985 [1962], p. 456) . . . the divisibility axiom has to go. (Winter, 2008, p. 528) With respect to many inputs it is immediately apparent that, once they are carefully VSHFL¿HGWKHRQO\DPRXQWVLQZKLFKWKH\FDQEHXVHGDUH 3DUWLFXODU kinds of chisel and particular kinds of milling machine can only be used in quantities represented by integers; neither half a chisel, nor 11 2 chisels, nor yet 5S chisels are meaningful concepts. For such inputs, nearly all the non-negative real numbers are irrelevant as candidate representations of the amount of the input used. 7KHVDPHLVWUXHRIPDQ\µPRUH¿QHO\GLYLVLEOH¶NLQGVRILQSXW2IQRJDVHRXV liquid or metal input can one ever use less than one atom, or one molecule; nor can one use less than one electron of electricity, or one photon of light (and, by the way, in the context of nanotechnology it can be relevant to refer to input quantities this small). Whenever there is some such smallest meaningful unit of an input, all quantities of that input can be represented by integers; and no relevant quantity can then be represented by any other kind of real number. The same is true even when, for whatever practical reason, the minimum available input quantities are FRQVLGHUDEO\ODUJHUWKDQWKHSK\VLFDOPROHFXOHVHWFMXVWUHIHUUHGWR A worker is not divisible, of course. And the currently employed standard of time, based on the resonant frequency of the caesium 133 atom, gives over 9 billion cycles per second; that is a rather small unit of time (!) but it still is a discrete unit of time. Even if time is indeed a continuous variable, there can be reasons for treating it as indivisible. (cf. Frank, 1969, section 3.2.) In a given economy, in a given epoch, it is often not possible to employ labour of a particular kind for less than an hour, or less than forty hours, say. Even stopwatch-conscious lawyers do not charge by the microsecond; there is a much larger minimum unit of time on which they base their bills (for lawyers in Toronto, for example, that minimum unit is six minutes). These ‘social’ facts have nothing to do with the

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Ian Steedman

ultimate nature of time, of course, but they still mean that labour-time can in effect be indivisible for the capitalist choosing between (or among) alternative methods of production. (Actual buying and selling is done via money, not barter, and every currency has a smallest unit – be it the lira, the farthing, the cent, the won . . . Every amount of money can thus be expressed as an integer multiple of that smallest currency unit. Hence if every input quantity can be represented by the integers, so can every money payment for inputs.) There are good reasons, then, for saying that the continuum of all real numbers is not required for the representation of inputs, since these latter are – whether for physical, or practical or institutional reasons – indivisible. So if we adopt the sensible course of measuring the quantity of each input in terms of the smallest unit of that input, every input quantity will be represented by an integer. All other real numbers will, for this purpose, be completely irrelevant (be they rational, LUUDWLRQDODOJHEUDLFRUWUDQVFHQGHQWDO $Q\UHDGHULQFOLQHGWRUHMHFWWKLVFRQFOXVLRQ should feel obliged to offer convincing examples of truly divisible inputs. It need hardly be added that, with respect to our earlier argument about the probability of reswitching, it would be utterly beside the point to insist that at least RQHLQSXWLVµYHU\YHU\¿QHO\GLYLVLEOH¶7KDWLIWUXHZRXOGPHUHO\PHDQWKDWZH may have to use very large integers to represent input quantities; it would do nothing toward showing that we need all the real numbers for that representation. 7RSURYLGHWKHEDVLVIRUDQ\µ&DQWRUOLNH¶REMHFWLRQWRRXUSUREDELOLW\DUJXPHQW it would be necessary to maintain that at least one input is strictly divisible; and that claim may well be hard to sustain, not least if it is said to apply to every example of the choice of production methods.

Concluding remarks A particular, very simple model of production has been used as the vehicle for conveying the message that, in the presence of indivisible inputs, it may well be possible to examine the probability of reswitching, without resorting to Monte Carlo methods. As it happens, the model used can lead to the conclusion that the SUREDELOLW\LVKHUHYHU\VPDOO%XWLWSHUKDSVEHDUVUHSHWLWLRQWKDWWKLVVSHFL¿F conclusion has not been the main point of the chapter. (More interesting models of production may or may not yield a similar conclusion, after all.)2 Rather, the central thrust of the chapter has been that recognition of the indivisibility of inputs may permit the comparison of areas/volumes/hyper-volumes in input space as a way to assess the probability of reswitching. If this proposed procedure is DFFHSWHGZHVKDOOEHFRQWHQWZKDWHYHUVSHFL¿FFRQFOXVLRQVPD\EHDUULYHGDWE\ its use.

Notes 1 I would like to thank U. Krause, G. Lorimer, L. Mainwaring, S. Metcalfe, A. Opocher, 63DUULQHOOR 1 line ‘S’ is to the right of line ‘E’.7 It is the shaded area F in Figure 25.2a.

388

Fabio Petri



'



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Figures 25.2a and b

Having reached this pleasant graphical result, D’Ippolito proceeds in a way which appears marred by a logical slip. He argues (D’Ippolito, 1987, p. 17) that, for given r and v, the probability that the switch be ‘antineoclassical’ is given by the ratio between the surface of the area F of points satisfying both constraints, and the surface 1/ȡ2 of the whole square OCBQ, i.e. is given by ȡ2 times F(r, v) (if with F(r, v) one indicates the surface of F). 7KLVLVGLI¿FXOWWRDFFHSW/HWXVFRQFHGHWR'¶,SSROLWRIRUWKHVDNHRIDUJXPHQW WKHULJKWWROLPLWWKHLQTXLU\WRWKHFRHI¿FLHQWVa1, a2, and to assume, as he clearly does, that all points (a1, a2) compatible with the given r might occur with equal probability.8 But D’Ippolito forgets that he has assumed that the two techniques KDYHDVZLWFKSRLQWDWWKHJLYHQUDWHRISUR¿WVr, and that, since he has decided to call technique 2 the one which becomes dominant to the right of r, then (a1, a2) VDWLV¿HV E\DVVXPSWLRQWKHSRLQWVDERYHOLQHµ6¶DUHWKHUHIRUHRXWRIWKH question. So for given r and v, by assumption (a1, a2) is in the triangle ABC of Figure 25.2a if v < 1, or in the trapeze OCBH of Figure 25.2b if v > 1. Then the correct ratio – I will call it Z(r, v) – between area of ‘antineoclassical’ cases and area of possible cases, for a given r and a given v < 1, is the ratio between F(r, v), and the surface – I shall call it D(r, v) – of the triangle ABC of Figure 25.2a, while it is zero if Y•1. Under the assumption, which D’Ippolito explicitly makes (ibid., p. 18), that all values of v are equiprobable,9 one may therefore proceed to calculate the average probability Z*(r) that at a given r a switch be ‘antineoclassical’ under the assumption that v is random but < 1, by integrating Z(r, v) with respect to v from 0 to 1. Z*(r) is not the average probability Pme(r) that at a given r a switch be ‘antineoclassical’, because it is determined under the assumption that v < 1 so it leaves out the possibility that v > 1. But D’Ippolito says that the case v > 1 ‘would FRYHUWKHUHPDLQLQJRIFDVHV¶LELG., p. 16), so he appears to authorize us to assume that the probability that v < LV7KHQPme(r) is simply one half of Z*(r).

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Reswitching and reverse capital deepening 389 D’Ippolito, on the contrary, having said that the probability of an ‘antineoclassical’ switch, for a given r and a given v < 1 is given by U2F(r, v), simply goes on to integrate this probability over v from 0 to 1 in order to obtain his Pme(r), without mentioning any more the fact that there is also the case v > 1. ,KDYHEHHQXQDEOHWR¿QGDZD\WRPDNHKLVVHYHUDOVWDWHPHQWVFRQVLVWHQW10 His probabilities are reported in column 9 of Table 25.1, under the heading ‘D’Ippolito RULJLQDO¶7KH\FRQYHUJHWRDV r tends to +f. The calculation of the probability that a switch be ‘antineoclassical’ with WKHFRUUHFWLRQ,¿QGQHFHVVDU\LHPme(r) = Z*(r)/2, yields much higher values than the ones calculated by D’Ippolito; they are listed in column 10 of 7DEOHXQGHUWKHKHDGLQJµ'¶,SSROLWRFRUUHFWHG¶7KH\FRQYHUJHWRDV r tends to +f. Appendix A shows how to determine the surfaces, whose ratios determine the probability of an antineoclassical switch for given (r, v) according to D’Ippolito, and according to my correction. Perhaps D’Ippolito thought that he had the right to neglect to consider the cases v > 1 because in some way he was already taking the existence of those cases into account, by dividing F(r, v) by the whole surface of the square OBCQ instead of by the sole surface D(r, v).11 But then a more consistent estimation procedure would appear to be the following one. Let us drop the assumption that all values of v < 1 are equiprobable, by noticing that v < 1 and v > 1 are symmetrical and hence equiprobable only if no constraint RQWKHFRHI¿FLHQWVLVDGGHGVRWKDWDOOSRLQWVLQ2%&4PLJKWEHSLFNHGE\WKH random technique selection process; while here there is a constraint, and this is that at the switchpoint it is technique 2 which becomes dominant to the right of r. This constraint makes only the points in ABC eligible, i.e. a fraction of OCBQ which is the smaller, the smaller is v. If one then considers all values of a1, a2 as equally probable,12 one may conclude that the values of v are not all equiprobable, but are the more probable, the greater the portion of OCBQ that makes the switch possible. The natural assumption then is to assume that the probability of each value of v, or of each value of 1/v if v > 1,13 is proportional to the ratio between D(r, v) and the area of OCBQ. In other words, let us replace v in Z(r, v) with a variable y, 0 dy dGH¿QHGDV y = v if v d1 and y = 2 – (1/v) if v > 1. The density function of y, p(y), is assumed linear, going from 0 to 1 as y goes from 0 to 2, corresponding to the ratio D(r, y)/(1/U2) between the surface of ABC or of OCBH, and the surface of OCBQ. Then the average probability Pme(r WKDWDVZLWFKEHDQWLQHRFODVVLFDOLVWKHGH¿QLWH integral of Z(r, y)p(y) over y from 0 to 2, divided by 2; but since Z(r, y) = 0 for y > LWVXI¿FHVWRFDOFXODWHWKHGH¿QLWHLQWHJUDORIZ(r, v)p(v) over v from 0 to 1, and then divide by 2. Since for v < 1 it is D(r, v) = v/(2U2) (cf. Appendix A) then p(v) = v/2, so Pme(r  LV RQH KDOI RI WKH GH¿QLWH LQWHJUDO RI vZ(r, v) over v from 0 to 1. The resulting probabilities are listed in the last column of Table 25.1, under the heading ‘D’Ippolito reinterpreted’. I do not claim that this ‘reinterpretation’ has strong textual support.

390

Fabio Petri

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Part II. A different approach Now I explore a different method of estimating Pme(r). This method starts by assuming an initially given switchpoint C = (r*, w*) in the (r, w) plane in which one draws the w(r) curves of the different techniques. The form of a w(r) curve in this model depends on D, E, a. There are therefore three degrees of freedom. If we establish that the curve must pass through C and must have a given Rtr* (and > r* if w* > 0) and a given W tw* (and > w* if r* > 0), the three degrees of freedom are eliminated and the w(r) curve, i.e. the technique, is completely determined. The intuition behind what follows is that, by assuming an equal probability of all values of r between r* and R, and of w between w* and W, one must be able to determine the probability that two curves that switch at C have a second switch for a lower r so that the switch at C is antineoclassical. For brevity below I drop the asterisks. Not all quadruplets (w, r, R, W) correspond to an economically acceptable technique (D’Ippolito 1987, p. 34). For a given point C and a given a, the values of D and E must satisfy equations (25.3) and (25.5) where w, r and a are given; this can be re-written as w(1 + r)D + w[1 – (1 + r)a]ß = 1 – (1 + r)a

(25.15)

WD + W(1 – a)ß = 1 – a.

(25.16)

For a given W, this is a system of two linear equations in D, E. The solutions are ß=

w(1  r )(1  a )  W >1  (1  r )a @ wWr

(25.17)

D=

(W  w) >1  (1  r )a @ (1  a ) wWr

(25.18)

As long as wWr > 0, for ß to be non-negative the numerator on the right-hand side of (25.9) must be non-negative i.e. w(1 + r)(1 – a) t W[1 – (1 + r)a] which, since (1 – a)/a = R implies [1 – (1 + r)a]/(1 – a) = (R – r)/R, can be re-written as W d w(1 + r)

R { Wmax Rr

(25.19)

7KHULJKWKDQGVLGHRIWKLVLQHTXDOLW\GH¿QHVDQHZYDULDEOHWmax whose meaning ZLOOEHQRZFODUL¿HG([SUHVVLRQw/(R – r) is the absolute slope of the straight line connecting C with R (the point where the w(r) curve touches the horizontal axis),

Reswitching and reverse capital deepening 391

__ w(1 + r)R/(R – r)

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wR/(R – r)

C

r = 1.5

R

Figure 25.3

so wR/(R – r) is the value of w where this straight line crosses the vertical axis; see Figure 25.3. Therefore W cannot exceed a value Wmax(r, w, R) determined by the value of this point multiplied by (1 + r). This constraint W dWmax is the sole constraint besides W > w (assuming r > 0) and R > r: Dt0 does not pose a constraint because W – w t0 and [1 – (1 + r)a] (1 – a) = (R – r)/R t0; as to R, it can be chosen arbitrarily close to r by increasing a; and W can be chosen arbitrarily close to wWKURXJKWKHVXI¿FLHQWO\KLJKYDOXHV of D and E determined by equations (25.17) and (25.18). The w(r) curve will be concave (downwards) if W is below the point Wmax/(1 + r) = wR/(R – r) where the straight line through R and C crosses the vertical axis; it will be convex if W is in between this point and the point of ordinate Wmax. For given points C (with r > 0) and R, therefore, a given W satisfying the constraint w < W dWmax uniquely determines the w(r) curve. This means that all the possible w(r) curves passing through given points C = (r, w) and R can be generated by letting W vary in the interval (w, Wmax). It will be assumed that each value of W in this interval has the same probability. This does not appear to be a more arbitrary assumption than the analogous ones in D’Ippolito’s analysis. With CVWLOO¿[HGOHWXVQRZVXSSRVHWKDWQRDYDLODEOHWHFKQLTXHRIWKRVHZKRVH curve passes through C, has an associated RJUHDWHUWKDQDFHUWDLQ¿QLWHYDOXHRsup. ,¿QGVXFKDQDVVXPSWLRQZKLFKGRHVQRWSUHYHQWRQHIURP¿[LQJDYHU\KLJK Rsup, nor from admitting that technical progress increases Rsup) more reasonable than the assumption (implicit in D’Ippolito) that R can take any value, however great: the latter assumption would imply that one can get as near as one likes to producing with unassisted labour. Anyway my assumption is not necessary to the method proposed here; it can be seen as only an intermediate step to assuming Rsup = +f.

392

Fabio Petri

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Let us suppose that two (admissible) couplets (R, W) are repeatedly randomly selected, thus randomly selecting two w(r) curves through C and hence two techniques. The probability that the two values of R coincide is zero, so let us assume that they differ, and let us now call technique 1 the one associated with the lower R, and technique 2 the other one. Hence R1 < R2 by assumption. Note that WHFKQLTXHLVQRORQJHUGH¿QHGDVWKHRQHZKLFKLVGRPLQDQWWRWKHULJKWRIWKH switchpoint, but as the one with the higher R. The probability that W1 = W2 is analogously zero. A necessary condition for there to be a second intersection of the two w(r) curves either to the left or to the right of C, is that W1 < W2; see Figures 25.4a and 25.4b. w

w

C w1(r) w1(r)

C

O (a)

r

O

r

(b)

Figure 25.4

7KLVFRQGLWLRQLVDOVRQRWTXLWHVXI¿FLHQWEHFDXVHLWLVSRVVLEOHWKDWWKHWZRw(r) curves be tangent in C (see Figure 25.5c); but if W1 and W2 are, as we are assuming, continuous variables, the probability of this case is zero: once r, w, R2, W2 and R1 W2max but it is excluded that W2 > W1max. If we consider all values of Wi as equiprobable within its admissible interval, then the probability, that W1 < W2 conditional on W1 d W2max, is 1/2; while the probability that W1 < W2 conditional on W1 > W2max is zero; so the unconditional probability that W1 < W2 must be 1/2 times the ratio, to be indicated as Q, between (W2max – w) and (W1max – w). This ratio is given by: Q { (W2max – w)/(W1max – w)

(25.22)

= {[(1 + r)wR2/(R2 – r)] – w}/{[(1 + r)wR1/(R1 – r)] – w} = [(R1 – r)/(R2 – r)]/[(R1 + 1)/(R2 + 1)]. Q comes out not to depend on w; the probability, that W1 0

(25.34)

because the numerator is positive (0 < a < 1 if R > 0), and the denominator is positive because 0 < a(1 + r) < 1, owing to a = 1/(1 + R) and therefore a(1 + r) = (1 + r)/(1 + R) < 1. (Using a here instead of R has the same motivation as in equations (25.26) and following) W1^ is a function of a1, a2, r, w, W2: W1^ = W2·w(–a1 + a1·a2 + ra1a2 + 1 – a2 – ra2)/

(25.35)

(–rW2a2 + wa1a2 – wa2 + wra1a2 + a1rW2 – wa1 + w – wra1) Since w(r) is a hyperbola, as R1 increases continuously in the interval (r, R2) also W1^ increases continuously and goes through all values in the interval (w, W2). Thus, for a given R1, if W1 < W1^ then the slope in C of w1(r) is less (in absolute

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398

Fabio Petri

value) than the slope of w2(r), so the second switchpoint is to the right of C; if W1 > W1^, the second switchpoint is to the left of C. The probability that the switch in C be ‘antineoclassical’ depends therefore on the probability that W1 > W1^. We may assume that this probability is to 1 like the length of the interval (W1^, W2) is to the length of the interval (w, W2), and therefore that it is equal to (W2 – W1^)/(W2 – w). (For W2 and/or w tending to +f it might seem that the same problem arises, which earlier induced me to replace R with a; but this problem will not arise because W2 and wZLOOGLVDSSHDUIURPWKHIRUPXODVWKURXJKVLPSOL¿FDWLRQ For R1 tending to R2 this ratio tends to zero, and it tends to 1 for R1 tending to r. But how it varies within the interval (r, R2) is a complicated thing and for the calculation of the average value of (W2 – W1^)/(W2 – w), when both W2, R1 (or rather a1), and R2 (or rather a2) are considered random variables with a uniform probability distribution within the respective admissible intervals, Maple has been again indispensable. For given values of r, w, R2 (or rather a2), the values of W2 can vary in the interval (w, Wmax). It will be assumed that all values of W2 in this interval are equiprobable. The probability, that for given r, w, R2 and R1RQH¿QGVWKDWW1 > W1^LVJLYHQE\WKHGH¿QLWHLQWHJUDO W2 max

1 W2 max  w

³

w

W2  W1 ^ dW2 W2  w

(25.36)

where w is given, W2max = w(1 + r)R2/(R2 – r) = w(1 + r)(1 – a2)/(1 – a2 – ra2), and W1^ is given by equation (25.35) and is therefore a function of r, w, a1, a2 and W2. This probability is a function of r, w, a1 and a2. It is useful to reach this same probability in another way. Let x be a scalar, variable between 0 and 1, and, in the expression (W2 – W1^)/(W2 – w) let us replace W1^ with its value given by equation (25.27), and let us replace W2 with its expression in terms of w, Wmax and x, i.e. with: W2 = w + x[w(1 + r)(1 – a2)/(1 – a2 – ra2) – w].

(25.37)

(The expression inside the square brackets on the right-hand side of (25.37) is Wmax – w; so as x varies from 0 to 1, W2 varies from w to Wmax.) If we perform these two substitutions in the expression (W2 – W1^)/(W2 – w), w is eliminated and we obtain an expression for (W2 – W1^)/(W2 – w) which we shall indicate as PA: PA { (W2 – W1^)/(W2 – w) = r(a2 – a1 – a22 – ra22 + xra2 – xra1 + ra1·a2 + a1·a2)/(2ra2 – 2a1a2 – 2ra1a2 – 2ra22 – r2a22 + a22a1 + 2ra22a1 + r2a22a1 + xr2a2 – xr2a1 + a1 + 2a2 – 1 – a22).

(25.38)

Reswitching and reverse capital deepening 399 PA indicates the probability that, if two w(r) curves cross in C, this switchpoint is ‘antineoclassical’, when r, a1, a2 and W2 (i.e. x) are assigned. (As announced, W2 and w have disappeared.) PA is the basic function in what follows. Since x varies between 0 and 1, expression (25.36) becomes:

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PX {

³

1

0

PAdx

(25.39)

PX is the average probability that the switch in C is ‘antineoclassical’ if there is another switchpoint, when r, a1 and a2 are given while W2 is random between w and W2max. PX is independent of w, like PA; it is a function of r, a1 and a2. Again it can be calculated by approximation. Now let us consider only r and a2 as assigned and let us suppose all values of a1 in the interval (a2,1/(1 + r)) to be equiprobable; by integrating PX with respect to a1 on the interval (a2,1/(1 + r)) and dividing by [1/(1 + r)] – a2, we now determine the average probability that the switch in C is ‘antineoclassical’ if there is another switchpoint, when only r and a2 are assigned. Let this probability be indicated as PA1: PA1 {

1

1/ (1 r )

³

1  a2 1 r

PXda1

(25.40)

a2

Lastly, let a2inf { 1/(1 + Rsup) be the minimum possible value of a2 (and of a1), i.e. the one corresponding to the assigned Rsup. Let us integrate PA1 with respect to a2 on the interval (a2inf ,1/(1 + r)) and divide by the length of this interval; in this way we obtain the average probability P that the switch in C be the ‘antineoclassical’ one if there is another switchpoint, with only r and Rsup given: P(r, Rsup) {

1 1  a2inf 1 r

³

1/ (1 r ) a2 inf

PA1da2 where a2inf = (1 + Rsup) –1.

(25.41)

0DSOH95HOHDVH6WXGHQW9HUVLRQDJDLQGHWHUPLQHVZLWKRXWGLI¿FXOW\WKHYDOXHV of these integrals by approximating them with the method of rectangles (again I have used 40 rectangles). The basic function PA is very ‘regular’ so the approximations are certainly very good. The approximating function tends to 1 as r tends to Rsup. Table 25.3 shows the values of P for the same values of r and Rsup as Table 25.2 does for P*. We now have what we need: Table 25.1, columns 2 to 8 (under the heading ‘My method based on w(r) curves’), shows what we were looking for: the values of S(r, Rsup) = P*P which indicate the average probability that a switch in C (that is, at a certain value r RIWKHUDWHRISUR¿W LVµDQWLQHRFODVVLFDO¶IRUVHOHFWHGYDOXHV of Rsup. This probability is higher the higher is r for a given Rsup; it is on the

400

Fabio Petri

Table 25.3 P, i.e. probability, according to my method based on the w(r) curves, that, if two techniques switch twice, the switch at the given value of r is associated with reverse capital deepening

r

Rsup

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0.5

1

2

5

10

100

ĺ’

0.01

.1636

.1311

.1112

.0974

.0924

.0877

.0871

0.02

.2311

.1890

.1627

.1443

.1375

.1311

.1304

0.03

.2779

.2300

.1997

.1782

.1703

.1628

.1619

0.04

.3141

.2621

.2289

.2052

.1965

.1881

.1872

0.05

.3442

.2888

.2533

.2279

.2184

.2095

.2084

0.06

.3698

.3116

.2742

.2474

.2374

.2279

.2268

0.08

.4125

.3495

.3091

.2800

.2691

.2588

.2576

0.10

.4475

.3803

.3375

.3066

.2951

.2841

.2828

0.12

.4776

.4065

.3615

.3291

.3171

.3055

.3042

0.14

.5042

.4293

.3824

.3487

.3361

.3241

.3227

0.16

.5284

.4495

.4008

.3659

.3530

.3406

.3391

0.18

.5507

.4678

.4173

.3814

.3680

.3552

.3538

0.20

.5718

.4845

.4323

.3953

.3816

.3685

.3670

0.25

.6207

.5212

.4647

.4253

.4108

.3969

.3953

0.30

.6676

.5527

.4918

.4502

.4349

.4204

.4187

0.40

.7701

.6067

.5360

.4899

.4733

.4576

.4558

0.50

.6542

.5718

.5212

.5032

.4864

.4845

0.60

.6994

.6023

.5469

.5277

.5098

.5078

0.80

.7966

.6542

.5881

.5663

.5464

.5441

1.00

.6994

.6207

.5962

.5742

.5718

1.20

.7418

.6480

.6207

.5967

.5940

1.60

.8309

.6928

.6595

.6314

.6283

2.00

.7298

.6899

.6577

.6542

3.00

.8069

.7465

.7037

.6994

4.00

.8805

.7888

.7350

.7298

.8238

.7584

.7524

10

.8255

.8159

20

.8837

.8685

30

.9138

.8940

5.00

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Reswitching and reverse capital deepening 401 contrary lower the higher is Rsup for a given r. This shows that admitting no limit to Rsup tends to underestimate this probability relative to the – more plausible, it seems to me – cases in which the possible techniques can be presumed never to have an Rsup DERYH D FHUWDLQ ¿QLWH YDOXH $Q\ZD\ WKH OLPLWV WR ZKLFK WKH probability tends for Rsup tending to +f are also shown: the probability that a VZLWFKEHµDQWLQHRFODVVLFDO¶LVDERXWIRUr DERXWIRU r  'H¿QLWHO\QRWQHJOLJLEOH In conclusion, D’Ippolito’s 1987 results were deeply misleading. His proEDELOLWLHVDUHVLJQL¿FDQWO\ORZHUIRUWKHSODXVLEOHYDOXHVRIWKHUDWHRISUR¿WVWKDQ the probabilities determined in the three alternative ways analysed here. In particular the calculation method which follows D’Ippolito’s approach and statements most closely but corrects his logical slip (Table 25.1, next-to-last column: ‘D’Ippolito corrected’) arrives at probabilities enormously higher than his. The other two methods reach results less far from D’Ippolito’s, but they nonetheless arrive at much higher probabilities than D’Ippolito’s, especially for WKHORZYDOXHVRIWKHUDWHRISUR¿WVIRUZKLFK'¶,SSROLWRREWDLQVSDUWLFXODUO\VPDOO probabilities, e.g. for r WKH\HVWLPDWHSUREDELOLWLHVDURXQGDJDLQVWWKH RI'¶,SSROLWR :KDWDOVRHPHUJHVLVDVLJQL¿FDQWGHSHQGHQFHRIWKHUHVXOWVRQWKHDVVXPSWLRQV about the distribution of the probabilities of the parameters chosen to characterize techniques. It is unclear how one might decrease the arbitrariness of these assumptions. Therefore, as stressed by Salvadori (2000), there is a danger that, by changing them, one may obtain nearly any result.147KHVLJQL¿FDQFHDQGLQGHHGWKH legitimacy of exercises such as the one attempted here (or the one by Mainwaring and Steedman 2000) are therefore doubtful. Still, if one believes this kind of exercise to yield some useful information, then the message appears to be that the Samuelson-Garegnani model supplies no basis at all for believing that the likelihood of ‘antineoclassical’ switches can be considered negligible – rather the opposite.15

Part III. A restrictive assumption in Mainwaring and Steedman and other studies Leaving aside for the moment the doubts just raised about the relevance of the above type of exercise, a question implicitly posed by my results is why they are so different from the ones obtained by Mainwaring and Steedman (2000). They consider a two-sector model where both products are also capital goods and remain the same across changes of techniques, and try to estimate the ‘a priori’ probability of reswitching between two techniques (differing in the sole production method of commodity 2) which switch at least once. They again propose to measure this probability as the ratio of two areas representing possible and HTXLSUREDEOHYDOXHVRIFRHI¿FLHQWVWKLVUDWLRGHSHQGVRQPDQ\SDUDPHWHUVVR WKH\KDYHUHFRXUVHWRVLPXODWLRQV7KH\¿QGWKDWLIWKHWZRWHFKQLTXHVVZLWFKDWD given r0 the probability (that they call ȥ(r0)) of a second switch depends on the WHFKQLFDOFRHI¿FLHQWVVHOGRPJRLQJEHORZRIWHQEHLQJLQWKHUDQJHRIWR DQGULVLQJIRUFHUWDLQFRPELQDWLRQVRIYDOXHVRIWKHFRHI¿FLHQWVFRPELQDWLRQV ZKLFKKRZHYHUWKH\GHHPRIYHU\ORZSUREDELOLW\ DERYH$VHULHVRIFXUYHV

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402

Fabio Petri

showing how this probability varies with r0, R, and various assumptions about the WHFKQLFDOFRHI¿FLHQWVVXJJHVWVDWOHDVWWRPH0DLQZDULQJDQG6WHHGPDQGRQRW explicitly propose a recapitulatory single estimate) an average probability ȥ(r) in WKHUDQJHWRIRUSODXVLEOHYDOXHVRIr. From this, they conclude that ‘the SUREDELOLW\ RI IURQWLHU  UHVZLWFKLQJ DW DQ\ WZR UDWHV RI SUR¿W LQ D UHVWULFWHG domain is very small – typically less than one per cent’ (p. 345). An immediate observation on this conclusion is that there is no reason why, in order to assess the plausibility of the supply-and-demand approach to value and distribution, one should be interested in the likelihood of both switchpoints of two reswitching techniques appearing on the frontier of the wage curves. This is an irrelevant issue once the really relevant question is admitted to be: if there is a switch on the frontier, what is the probability that it be an antineoclassical one? This is the question relevant for the possibility of antineoclassical changes in technique as distribution changes, and is indeed the question asked by D’Ippolito, and by myself in Parts I and II. Therefore Mainwaring and Steedman should have rather concentrated on estimating the probability that if two techniques switch, they not only reswitch but do so at a lowerUDWHRISUR¿W16 neglecting the question whether either switch is dominated, or whether both switches will be inside a ‘restricted domain’. This probability is certainly lower than the average ȥ(r), which also considers second switches at a higherUDWHRISUR¿WEXWLIRQHFDQ assume as an indication the values I obtain for the Samuelson-Garegnani model for the probability P that, when two techniques that switch also reswitch, the other VZLWFKEHDWDORZHUUDWHRISUR¿WFI7DEOH WKHQIRUQRWYHU\KLJKYDOXHV of R17 one easily obtains that ȥ(r) should be no more than halved. So I would provisionally guess for the Mainwaring-Steedman model an average probability that a frontier switch be antineoclassical around 2 to 4 per cent instead of less than 1 per cent. Now an average probability of antineoclassical switches around 2 to 4 per cent (which means that in concrete instances their frequency could be even considerably KLJKHU  DSSHDUV VXI¿FLHQW WR DUJXH WKH µSRWHQWLDO JHQHUDOLW\¶ *DUHJQDQL  p. 72) of very antineoclassical behaviours of the value-capital/labour ratio. Still, the question remains why one obtains so much lower probabilities than for the Samuelson-Garegnani model. Further research on this issue would be necessary, but in all likelihood a main reason is that, as noted by Ciccone (1996, p. 45, fn. 8), the model assumes both goods to be common to both alternative techniques, so techniques that switch differ in the productive method of only one sector. On the contrary, in real economies it very seldom happens that different methods of production of a commodity do not require different intermediate goods or machines – so the cases in which at a switchpoint the capital goods directly and indirectly utilized in the production of the numéraire do not change should be considered exceptional. This is accepted, indeed made central, in the Samuelson-Garegnani model where a change of techniques always changes both methods because the capital good changes; it is on the contrary excluded in the Mainwaring-Steedman model, and the probability of reswitching is certainly reduced by this restrictive aspect of the model, because then reswitching can happen only if both w(r) curves

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Reswitching and reverse capital deepening 403 are concave or both are convex. Some indication that the reduction is very great can, I think, be obtained from a comparison with the Samuelson-Garegnani model, where this constraint on the shape of the w(r) curves of reswitching techniques does not hold. For the latter model I obtain very high values (cf. Table 25.2) for the probability ȝ that two techniques which switch also have a second switch, for example ȝ  IRUr DJDLQVWYDOXHVRIȥ(r ZKLFKVHOGRPJRDERYH for r LQWKH0DLQZDULQJ6WHHGPDQPRGHOFI)LJXUHVWRRIWKHLUSDSHU  True, in the Samuelson-Garegnani model there is only one capital good in each technique, and admitting two capital goods, both inputs to each other, which can both change with a change of technique, would require considering a three-goods model (at least one good, the consumption-numéraire good, cannot change across WHFKQLTXHV EXW,FDQVHHQRUHDVRQZK\WKLVPRGL¿FDWLRQVKRXOGVLJQL¿FDQWO\ restrict the shapes that w(r) curves of reswitching techniques can have, and WKHUHIRUHVKRXOGVLJQL¿FDQWO\UHGXFHWKHOLNHOLKRRGRIGRXEOHVZLWFKLQJUHODWLYHWR the Samuelson-Garegnani model. Therefore I consider my ȝ values more credible than Mainwaring and Steedman’s ȥ(r). The assumption that at switchpoints on the frontier the two techniques differ in the method of only one of the industries relevant for the determination of the w(r) curve is also made by D’Ippolito (1989) and by Han and Schefold (2006). The effect of this restrictive assumption on the likelihood of antineoclassical switches in models with many industries awaits study. If changes of technique change the capital goods utilized, more than one method (of the ones affecting the shape of the w(r) curve18) changes at a switchpoint. Accordingly, the set of possible shapes of w(r) curves passing through a given (r, w) point is considerably enlarged. Thus consider the model used by D’Ippolito (1989). He assumes a given (A1, l1) technique for an n-goods economy, which generates a certain w(r FXUYH+H¿[HV r = r*DQGFKRRVHVXQLWVIRUWKHVHYHUDOFRPPRGLWLHVVXFKWKDWKDYLQJ¿[HGWKH numéraire, w(r*) = 1 and p = (1, . . ., 1). Then by Montecarlo methods19 he UDQGRPO\ FKRRVHV GLIIHUHQW FRHI¿FLHQWV IRU RQH LQGXVWU\ VD\ WKH QXPpUDLUH industry 1, such that the new technique (A2, l2 WKDWGLIIHUVIURPWKH¿UVWRQHRQO\ IRUWKHPHWKRGRIWKH¿UVWLQGXVWU\ VZLWFKHVZLWKWKHROGRQHDWr*, that is, yields the same w = 1 and the same p = (1,. . .,1) for r = r*; this means that, with a1j, l1 UHVSHFWLYHO\WKHWHFKQLFDOFRHI¿FLHQWRILQSXWj and of labour in industry 1, the new FRHI¿FLHQWVRILQGXVWU\PXVWVDWLVI\ (a11 + a12 + . . . + a1n)(1 + r*) + l1 = 1. If n is great, if all industries affect the shape of the w(r) curve, and if ‘no sector of the economy is ‘‘large’’’ (Schefold 1997a, p. 279); that is, if changes in the WHFKQLFDOFRHI¿FLHQWVRIRQO\RQHVHFWRUFDQQRWJUHDWO\DIIHFWWKHw(r) curve, the new w(r) curve will be very close to the old one: for example the possible variation RI WKH FRHI¿FLHQWV RI LQGXVWU\  ZLOO QRW EH DEOH VLJQL¿FDQWO\ WR PRGLI\ WKH PD[LPXP ZDJH UDWH RU WKH PD[LPXP SUR¿W UDWH ,I RQ WKH FRQWUDU\ WKH QHZ technique uses different quasi-basic goods (that is, goods entering the numéraire basket or its direct or indirect means of production), then the number of quasi-

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basic industries that survive with an unchanged method at a switchpoint can be any number from n – 2 to zero, and prices at r* need remain equal to 1 only for the commodities which are quasi-basic in both techniques, possibly only one commodity;20 the new w(r) curve can differ from the old one much more GUDVWLFDOO\)RUH[DPSOHLWFDQKDYHLQÀHFWLRQSRLQWVLQFDVHVLQZKLFKWKHVHZRXOG be excluded by D’Ippolito’s conditions. The set of possible shapes of reswitching w(r) curves is then enlarged as well. It may be countered that this enlargement of the set of possible shapes of switching and of reswitching w(r) curves does not necessarily mean that reswitching becomes relatively more probable; however, the effect would seem to be precisely this one when contrasting the Mainwaring-Steedman model with the Samuelson-Garegnani model, so it seems legitimate to suspect that this might well be the general effect.

Part IV. On the relevance of the probability of reswitching But why is the probability of occurrence of reswitching considered important? As argued by many authors (e.g. by myself in Petri 2004, ch. 6), the simple possibility of reswitching21VXI¿FHVWRGHVWUR\WKHOHJLWLPDF\RIWKHFRQFHSWLRQRIFDSLWDODV a single factor of production of variable ‘form’; no possibility is thereby left of basing a general approach to value and distribution on that conception. This is particularly clear for the endowment of capital: the endowment of a non-existent factor does not exist; no possibility is thereby left of writing acceptable general equilibrium models with a given endowment of a single factor ‘capital’ when capital goods are in fact heterogeneous. But the problem with specifying the endowment of capital as a single quantity for economies with heterogenous capital goods had been perceived by the 1930s and was the main reason behind the shift from long-period general equilibrium models relying on capital as the value factor to the modern, neo-Walrasian versions. However, this shift was not accompanied by doubts about the working of capital-labour substitution in response to changes in the rate of interest (Petri 2004, pp. 156–60); as a result, the thesis remained dominant to this day of a negative elasticity of investment to the rate of interest. This characteristic of the investment function is fundamental for neoclassical macroeconomics, international economics, growth theory, etc. because it is necessary in order to argue that the rate of interest is capable of bringing investment into equality with full-employment savings, and thus to give some plausibility to the tendency toward a full-employment equilibrium which is fundamental for the neoclassical approach.22 Nowadays there are several attempts to derive this negative elasticity without relying on capital-labour substitution, but they are all YLWLDWHGE\JUDYHGH¿FLHQFLHV3HWULFK WKHUHIRUHLILWFRXOGEHFRQFOXGHG that it is impossible to rely on the traditional neoclassical conception of capitallabour substitution, the negative elasticity of investment with respect to the rate of interest would lose all theoretical credibility and therefore all credibility (it is generally admitted that it has very little empirical support), and then only very ignorant or very dogmatic economists could remain neoclassical.

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Reswitching and reverse capital deepening 405 Now, reverse capital deepening derived from reswitching has been used by the critics precisely to refute the neoclassical conception of capital-labour substitution and thus to criticize the traditional investment function. I view the insistence on a low probability of reswitching as an attempt to counter this criticism of the interest-elastic investment function. The implicit argument (I am not aware of it having been clearly spelled out in its entirety) would appear to be the following.23 If reverse capital deepening is highly improbable then, as Schefold (2010, p. 122) has put it: ‘each single small change of methods of production in different industries can only exert a small effect on the aggregates, and if the system is large and the changes are many, rare paradoxical [i.e. antineoclassical, F. P.] changes will, as it were, disappear in the noise of frequent transitions’; in other words a strong predominance of ‘neoclassical’ changes will imply a value-capital/labour ratio and hence an investment function almost everywhere negatively elastic with respect to the rate RISUR¿WUDWHRILQWHUHVW VXFKWKDWWKHSRUWLRQVZKHUHWKDWLVQRWWKHFDVHZLOOQRW EHDEOHWRGHWHUPLQHVLJQL¿FDQWLQGHWHUPLQDFLHVRIWKHHTXLOLEULXPLQWHUHVWUDWHRQ the investment-savings market. The idea seems to be representable as in Figure 25.5a, where the relatively small backward portions of the capital demand curve (and hence of the investment function derived from it – see below for a discussion RIKRZ DSSHDULQFDSDEOHRIFDXVLQJVLJQL¿FDQWLQGHWHUPLQDFLHVRIWKHOHYHORIWKH rate of interest that ensures equilibrium between full-employment savings and investment, so that one can speak of stability of the savings-investment market (as long at least as the savings function is not backward-bending, as one should assume if one wants to leave aside problems due to income effects in consumer choices). $ ¿UVW SUREOHP ZLWK WKLV DUJXPHQW WKDW HPHUJHV IURP WKH FRQVLGHUDWLRQV advanced in Parts I to III, is that the existing studies on the probability of r

r (savings) r1 r2

(investment)

O

K/L

O

K/L

Figure 25.5 The curves can be interpreted both as demand-for-capital and capitalsupply, or as investment demand and flow-supply of savings (in which case the L at the denominator on the abscissa is the flow of labour ‘freed’ by the gradual closure of the older plants).

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UHVZLWFKLQJGRQRWVXSSRUWDVXI¿FLHQWO\ORZSUREDELOLW\RIUHVZLWFKLQJ(YHQ conceding some meaningfulness to these studies (against the doubts raised above), as already declared, even as low an average probability of antineoclassical switches as on the frontier of 2 to 4 per cent – and I have pointed out reasons to consider these estimates too low, possibly vastly too low – would not bar the possibility, with nonnegligible probability, of instances in which the percentage of antineoclassical switches on the frontier were higher, even considerably higher, and as a result the demand-for-capital curve were utterly unable to support the thesis of a unique and stable equilibrium of the savings-investment market. Some historical realization of such instances should be expected to have happened, as I have observed elsewhere;24 and market economies have been able to function all the same, which strongly suggests that the forces determining distribution and employment are not the ones postulated by the neoclassical approach.25 But the argument has other weaknesses too, and I proceed to point them out, DOWKRXJKEULHÀ\IRUVSDFHUHDVRQV A second problem with the argument is that it requires that reverse capital deepening, rather than reswitching, be highly improbable. Now, reverse capital deepening can also be due to ‘price Wicksell effects’. Consider a single everywhere strictly concave w(r) curve: with no change in physical quantities, the value of capital per unit of labour in terms of the net product increases with the rate of SUR¿W7KHVDPHSRVLWLYHFRUUHODWLRQEHWZHHQUDWHRISUR¿WDQGFDSLWDOODERXUUDWLR can be observed even for intervals of values of r where the w(r) curve is convex, if it is concave in preceding intervals. The likelihood that a w(r) curve be concave or have concave sections would appear to be not less than that it be convex or with convex sections. A series of concave w(r) curves succeeding one another on the frontier may cause the value of capital per unit of labour to have many upwardsloping sections, or even to be entirely upward-sloping even in the absence of reswitching as demonstrated by Garegnani (1970) under an assumption of FRQWLQXRXV IURQWLHU VZLWFKLQJ LH LQ¿QLWH DOWHUQDWLYH WHFKQLTXHV )LJXUH  SURYLGHVDJUDSKLFDOH[DPSOHIRUDFDVHZLWKD¿QLWHQXPEHURIWHFKQLTXHV,QWKLV example (which is compatible with the assumptions of D’Ippolito’s model as well as with more complex models) the absence of reswitching guarantees that the change of the value of capital at switch points is not ‘antineoclassical’, nonetheless the overall behaviour of the function connecting the value of capital per unit of labour to the rate of interest is far from the one normally assumed in neoclassical analyses. The question thereby raised is whether reverse capital deepening caused by positive price Wicksell effects can be an additional cause of problems for neoclassical theory, besides the reverse capital deepening caused by reswitching. The answer is yes. Savings are generally admitted to depend not only on income but also on wealth. Therefore the supply of gross savings will be affected by price Wicksell effects too: in a given situation of capital stock adjusted to long-period technical choices and income distribution, and equilibrium between investment and savings, for small variations of r the change in the supply of savings can be GHFRPSRVHGLQWRWZRWKHHIIHFWRIWKHFKDQJHLQWKHUDWHRILQWHUHVWUDWHRISUR¿W 

Reswitching and reverse capital deepening 407

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w

r k

O

r

Figure 25.6

with an unchanged wealth of the owners of capital goods, and the effect of the change in wealth due to the change in the value of capital goods, with an unchanged rate of interest. If the second effect coincided with the price Wicksell effect upon the demand for capital, and assuming no ‘perverse’ income effects (no negative effect of changes in the rate of interest on the propensity to save), stability could be endangered only by antineoclassical switches of technique (Fratini 2009). However, there is no guarantee that a change in the value of capital goods with an unchanged rate of interest will induce an equal change in gross savings. Thus suppose that a rise in r from a situation of equality between investment and savings does not cause switches to a different technique nor changes in the amounts produced but raises the value of the existing vector of capital goods ZKLFKKDVQRWFKDQJHG E\EHFDXVHRISULFH:LFNVHOOHIIHFWVDQGWKDWDVD UHVXOWLQYHVWPHQWLQYDOXH LQFUHDVHVE\WKHUHLVQRJXDUDQWHHWKDWVDYLQJV will, because of the rise of rDQGRIWKHZHDOWKRIFRQVXPHUVULVHE\DQGLI VDYLQJV ULVH OHVV WKDQ  WKH HTXLOLEULXP EHWZHHQ VDYLQJV DQG LQYHVWPHQW LV unstable. Thus price Wicksell effects are an additional possible cause of problems for the stability of the savings-investment market in the neoclassical approach.

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A third problem with the argument is that it requires that the ‘neoclassical’ switches be able to guarantee a not only essentially decreasing, but also VXI¿FLHQWO\ elastic demand-for-value-capital function: as always with equilibria, a low elasticity of the demand function risks determining an implausible equilibrium value of the variable under discussion (e.g. a negative rate of interest, or so high a UDWH RI LQWHUHVW DQG KHQFH RI SUR¿W WKDW WKH FRUUHVSRQGLQJ UHDO ZDJH LV EHORZ subsistence) as well as implausible comparative statics (small shifts of the savings function or of the labour supply function, or of the investment function owing to technical progress, would be liable to cause enormous jumps of income distribution). Also, the lower the elasticity of the investment function ensured by ‘neoclassical’ switches, the more dangerous the occurrence of antineoclassical switches because these, even if infrequent and with modest effects on the demand for capital because LQÀXHQFLQJ RQO\ D VPDOO SDUW RI WKH HFRQRP\ GHFUHDVH DQ\ZD\ VRPHZKDW the overall elasticity of the function, with effects on the extent of possible indeterminacies of the equilibrium rate of interest potentially the greater, the lower this elasticity, cf. Figure 25.5b where, differently from Figure 25.5a, the distance between the two locally stable equilibrium rates of interest is not negligible. Unfortunately none of the authors who have attempted to determine the probability of antineoclassical switches for economies with many industries has gone on to analyse the elasticity of the value of capital per unit of labour with UHVSHFWWRWKHUDWHRISUR¿WUDWHRILQWHUHVW LQWKHPRGHOVWKH\H[DPLQH%XWWKHUH are reasons to suspect a very low elasticity even apart from antineoclassical VZLWFKHV7KHHPSLULFDOHQTXLULHVRI2FKRD 3HWURYLü 7VRX¿OGLV and Maniatis (2002) and others26 conclude that w(r) curves are nearly linear; Schefold (2010, p. 127) concurs, on the basis of the results of the empirical enquiry of Han and Schefold (2006); Bidard and Schatteman (2001) bring some analytical support to this view (cf. Schefold 2010, p. 122). This suggests that price Wicksell effects are not very relevant in the aggregate for observed techniques. This decreases the relevance of the problem discussed above but raises a new SUREOHP+DQDQG6FKHIROG ¿QGDYHU\VPDOOQXPEHURIVZLWFKSRLQWVWHQ on average, on the envelopes of the techniques with 33 different goods that they derive from empirical input – output tables; this result, if generalizable beyond the inevitable aggregation and not very great income variation connected with the use of input – output tables from similar advanced countries, means that a majority of industries experiences no change at all in optimal production methods as distribution changes within a realistic range.27 If then one accepts Schefold’s argument that generally a single switch in a many-goods economy will only have a very modest effect on the w(r) curve, the implication is that changes in technique induced by changes in distribution change the value-capital/labour ratio very little. On the other hand, if the near linearity of w(r) curves implies (as argued in the same papers) that the changes in relative prices induced by changes in distribution are generally small, substitution due to consumer choice (even assuming it is not ‘perverse’ owing to income effects) will be weak too. But then the investment function derived from the demand-for-capital function will be nearly vertical. The neoclassical approach is undermined anyway.

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Reswitching and reverse capital deepening 409 A fourth problem with the argument concerns the assumptions about the level of labour employment behind the derivation of the investment function. 7KH OHYHO RI WKH UDWH RI SUR¿W UDWH RI LQWHUHVW  GHWHUPLQHV WKH GHVLUHG YDOXH capital/labour ratio, which becomes a demand-for-value-capital function when one assumes a given (i.e. full) employment of labour.28 The concrete role of this function is to allow the derivation of the ÀRZ of gross investment implicit in the demand for the stock of capital, owing to the need to re-equip with the optimal capital goods the employed labour as capital goods are gradually used up; leaving aside for simplicity investment in circulating (intermediate) capital goods (anyway only the variations of their amounts are considered in current national income accounts), there results what in Petri (2004, p. 127) I have called the long-period investment function: at HDFKUDWHRISUR¿WLWPHDVXUHVWKHYDOXHRIFDSLWDOWREHHPSOR\HGLQQHZSODQWVDW the corresponding optimal value-capital/labour ratio, assuming normal (longSHULRG SULFHVDQGDJLYHQÀRZRIODERXUWREHHPSOR\HGLQQHZSODQWVEHFDXVH ‘freed’ by the gradual closure of existing plants as they reach the end of their economic life, in a situation of full labour employment and normal utilization of existing plants on average).29 This function, a reduced-scale copy of the demandfor-value-capital function,30 could be argued to give a good approximation to the DYHUDJHYDOXHRILQYHVWPHQWLQGXUDEOHFDSLWDORYHUVXI¿FLHQWO\ORQJSHULRGVLI capacity utilization was normal on average), even if most durable plants during that period were still not adapted to the new level of the interest rate because of even longer durability and were only earning residual quasi-rents: prices would anyway be determined by the newer plants, better adapted to current income distribution; the irregularities of the ‘freeing’ of labour owing to irregular age distribution of durable capital could be argued not to alter the fundamental terms of the question if one was LQWHUHVWHGLQDYHUDJHLQYHVWPHQWRYHUVXI¿FLHQWO\ORQJWLPHSHULRGV But the long-period investment function needs the full employment of labour. (VVHQWLDOO\ LW UHÀHFWV WKH FDSLWDOODERXU UDWLR in new plants, so investment is determined only if the denominator of the ratio is given. With unemployment, a JLYHQFDSLWDOODERXUUDWLRLQQHZSODQWVOHDYHVLQYHVWPHQWLQGHWHUPLQDWHWKHÀRZ RIODERXUHPSOR\HGLQQHZSODQWVFDQEHJUHDWHUWKDQWKHÀRZRIODERXUµIUHHG¶E\ the closure of the oldest plants, with a resulting gradual reduction of unemployment; or it can be smaller, with a resulting gradual increase of unemployment. Nowadays this is often forgotten, and one frequently meets textbooks where the interestelastic investment function used in the IS-LM model is derived from a decreasing marginal product of capital, forgetting that – even conceding the validity of the notion – a decreasing marginal product of capital requires a given employment of labour, which is not assumed in IS-LM analysis. If what is at issue is the validity of the neoclassical approach, the full employment of labour cannot be assumed. In fact, empirical evidence suggests the normal presence in capitalist economies of unemployment and of a capacity utilization vastly inferior to the technically maximum one; furthermore, the supply of labour hours can be varied even without changes in the number of employed labourers by varying the use of part time and overtime (over longer periods there is immigration and changes in the rate of participation that render labour supply largely determined by labour demand). The

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resulting considerable elasticity of production inevitably attributes a relevant role to aggregate demand in the determination of aggregate output; the resulting utilization of productive capacity will then be a main force affecting desired variations of productive capacity and hence investment. There has been a neoclassical approach to investment – now out of fashion – that tried to take account of this issue: the one of Jorgenson (1963) then popularized in the macroeconomics textbook of Dornbusch and Fischer (e.g. the 1987 edition). In this approach the rate of interest determines the desired K/L ratio but the denominator in this fraction is not determined by a full employment assumption: rather, industries determine their desired capital stock on the basis of the expected OHYHOVRIGHPDQG$WWKHDJJUHJDWHOHYHOWKHLQÀXHQFHRIWKHUDWHRILQWHUHVWRQ investment is then obtained as follows: the rate of interest selects the capitallabour proportion on the aggregate isoquant corresponding to the planned level and composition of aggregate output; the desired capital stock changes if either the rate of interest, or planned output (i.e. expected demand), or both, change; net investment is effected to adjust the actual capital stock to the desired capital stock. How the speed of adjustment is determined is left somewhat vague by Dornbusch and Fischer and is only econometrically estimated by Jorgenson; but what interests us now is that with this approach: a) for a given level of aggregate demand, the elasticity of the desired capital stock (and hence of investment31) to the rate of interest is less – possibly considerably less – than if the denominator of the K/L fraction were given: this is shown in Figure 25.7, where a change in distribution that changes the optimal K/L ratio from Į to ȕ causes an increase of desired capital from K1 to K3 LI ODERXU HPSOR\PHQW LV ¿[HG DW L1, from K1 to K2 if output is ¿[HGE LQYHVWPHQWUHOHYDQWO\GHSHQGVRQWKHYDULDWLRQVRIDJJUHJDWHGHPDQG i.e. on the acceleration principle.32 Thus the result of taking into account the presence of unemployment into an otherwise neoclassical approach to capital and investment is to have to admit SRVVLEOH VLJQL¿FDQW PXOWLSOLHUDFFHOHUDWRU LQWHUDFWLRQV DQG WR GHFUHDVH WKH interest-elasticity of the investment function derived from capital-labour K K/L = β K/L = α

K3 K2 K1

L1

Figure 25.7

L

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Reswitching and reverse capital deepening 411 substitution. Whatever (low) interest-elasticity of investment might be presumed still to exist when account is taken of this last observation and of the arguments of the preceding sections will then be in all likelihood swamped by the greater LQÀXHQFH RI DJJUHJDWH GHPDQG DQG LWV YDULDWLRQV ± D FRQFOXVLRQ FHUWDLQO\ QRW contradicted by the available empirical evidence.33 It seems clear then that the anti-Keynesian argument of a spontaneous tendency WRZDUGIXOOHPSOR\PHQWLIPRQH\ZDJHVDUHÀH[LEOHEDVHGRQWKHµ.H\QHVHIIHFW¶ is indefensible. There appears to be no reason to believe that decreases in money wages will have a stronger positive effect on investment via the (highly doubtful anyway) effect on and through the rate of interest, than a negative effect on investment due to the – initial, at least – decrease in real wages34 and hence in consumption expenditure. The consequent persistence of unemployment35 and the elasticity of production imply very different policy implications from the currently GRPLQDQWRQHV)RUH[DPSOHWKHÀH[LELOLW\RISURGXFWLRQLQUHVSRQVHWRFKDQJHV LQGHPDQGLPSOLHVWKDWWKHUHLVQRQHFHVVDU\LQÀXHQFHLQWKHVKRUWDVZHOODVLQ the long period, of changes in real wages on the demand for labour. In existing plants, where capital already has a given ‘form’, higher real wages will bring about little or no change in output per unit of labour: employment will depend on capacity utilization which will depend on aggregate demand. In new plants, the ÀH[LELOLW\RISURGXFWLRQRIFDSLWDOJRRGVLQGXVWULHVZLOOJHQHUDOO\SRVHQRSUREOHP ZLWKREWDLQLQJWKHLQSXWVUHTXLUHGE\WKHDGRSWLRQRIWKHQHZPRVWSUR¿WDEOH methods of production on the scale suggested by the expected level of aggregate demand, even if the latter is increasing considerably. Thus (apart from political reactions) there generally is no incompatibility between more employment and higher wages; all that is required is that the higher wages be accompanied by a stimulus to aggregate demand. This will be so even when it were the case that a higher wage implied a shift to more value-capital-intensive techniques and therefore required more savings: the increase in savings will be brought about by the increase in aggregate output.36 It can be concluded that the argument that the probability of reswitching is very low, disputable as it is, anyway would not rehabilitate traditional neoclassical analyses.

Appendix A I take from D’Ippolito (1987, p. 32) the following determination of the surface F(r, v). Refer to Figure 25.8, which is in fact the same as Figure 25.2a. F(r, v) is the difference between the areas of the triangles ABC and$ƍ%ƍ&. Since: OC = BC = 1/U OM = ME = 1 CM = r/U AC/CB = A'C/CB' = A'M/ME = $ƍ0 = v A'C = A'M – CM = v – r/U

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Fabio Petri 1 E

B

1 (1 + r)

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B'

A

A'

C

M

O

1 r / (1 + r) V

Figure 25.8

one obtains the areas of the two triangles as, respectively, ABC = (ACuCB)/2 = CB2v/2 = v/(2U) $ƍ%ƍ& = ($ƍ&u&%ƍ)/2 = $ƍ&2/(2v) = (v – r/U)2/(2v) as long as v tr/U, otherwise $ƍ%ƍ& = 0. Hence the area of F is F=

1 1 [v (Uv – r)2] if v tr/U; U 2 U2

F = v/(2U) if v < r/U. (v – r/U)2 Hence Z(r, v) = F(r, v)/D(r, v) = F(r, v)/ABC = 1 – Z(r, v) = 1. Therefore 1

Z*(r) = rȡ

³

r /U

[1 – (v – r/U)2/U2]dv.

( v  r /U )

U2

2

if vtr/U, otherwise

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Notes 1 This is admitted by several neoclassical economists. Two examples: ‘Does it not take time to establish equilibrium? By the time equilibrium would be established will we not have moved on to another “week” with new conditions, new expectations, etc.?’ (Bliss, 1975, p. 210). ‘In a real economy, however, trading, as well as production and consumption, goes on out of equilibrium. It follows that, in the course of convergence to equilibrium (assuming that occurs), endowments change. In turn this changes the set of equilibria. Put more succinctly, the set of equilibria is path dependent [. . .] [This path dependence] makes the calculation of equilibria corresponding to the initial state of the system essentially irrelevant’ (Fisher, 1983, p. 14); note how the reference to production implies that Fisher is referring not only to possible exchanges of endowments among consumers, a problem of secondary importance, but also to changes in the total endowment of each produced means of production. 2 That the definition of a state of rest of prices should include the full employment of labour becomes then highly questionable. The reason behind such an inclusion in the marginalist approach is the presumption that unless labour demand equals labour supply, the real wage will change. But this presumption is difficult to justify unless one can argue that the change in real wage needed to reach equality between labour demand and labour supply is not drastic. This requires both a downward-sloping and sufficiently elastic demand for labour, and some mechanism ensuring that if there is unemployment and wages decrease and this induces firms to hire more labour, the increased output will be sold without problems, that is, aggregate demand will increase in step with aggregate output (Say’s Law). Neither assumption is supported by modern GET owing both to income effects, and to the four difficulties of GET which render the theory silent on the behaviour of economies where adjustments are time-consuming. Whether wages will change in the presence of unemployment becomes then an open question: unless wage reductions considerably increase the demand for labour, a downward ‘stickiness’ of wages becomes both a necessity in order to avoid absurd conclusions such as wages falling to zero (Petri 2004, pp. 319–20), and a plausible outcome of social interactions on the basis of historical experience – an outcome that classical authors, for example, considered obvious. The assumption of indefinite wage decreases as long as there is unemployment could be considered by marginalist economists a natural premise to the definition of equilibrium only because their theory argued with some apparent plausibility that full employment could be reached by plausible wage decreases. Already with Keynes the problems with such an argument prompted the admission of social mechanisms rendering (money) wages ‘sticky’. 3 Representative examples are Hicks (1965, p. 156; 1973, p. 44); Eltis (1973, Ch. 5); Malinvaud (1986). For contrary views cf. e.g. Garegnani (1990, pp. 71–2); Ciccone (1996). 4 The widespread acceptance of the term ‘perverse’ to characterize phenomena that are simply in contradiction with the predictions of one’s preferred theory appears to betray a quasi-religious outrage at the emergence of phenomena questioning neoclassical certitudes, which has little to do with a correct scientific attitude. It is difficult to imagine a physicist calling ‘perverse’ the results of experiments contradicting accepted theories: in the natural sciences the reaction would more probably be one of excitement because unsuspected new aspects of reality would be emerging that could be expected, once understood, to permit a better mastery of the world. 5 ‘Not less than’, because the probability depends negatively on the assumed upper limit Rsup of the maximum rates of profit of the alternative techniques, and the values given in the text are the ones for Rsup ’ 6 The vertical intercept of the w(r) curve measures the value of net output per unit of labour, i.e., in our case, the physical production of consumption good per unit of total

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labour employment (the economy is assumed to be stationary). Let y1, y2 be these net outputs per unit of labour for technique 1 and 2, and assume y1 < y2. Then labour employment per unit of output, L = 1/y, is smaller with technique 2. Put net output equal to 1; since net output must equal net income i.e. y = 1 = wL + rK, if L is smaller in technique 2, then K must be greater. But the value of capital per unit of labour will also be greater at a switchpoint if the vertical intercept is greater, cf. equation (25.6). The meaning of v < 1 is that the capital per unit of product in the sole consumption good industry must be greater with technique 2 than with technique 1. It might on the contrary be argued that very low values of the coefficient a, implying very high values of the maximum rate of profits R, are less and less plausible the more they approach zero. It might also be argued that a cannot be very close to 1/(1 + r). He appears here to mean all values of v between 0 and 1, as made clear by the limits of integration in footnote 15, p. 18 of his article; if one were to interpret him literally then, since v can vary from 0 to + f, the probability that v will fall in any finite interval would be zero, i.e. the probability would be all concentrated at the value v = +f. The a priori symmetry of the possibilities v < 1 and v > 1 suggests instead to consider the two cases as equally probable for a random picking out of two techniques (giving or not rise to a switch: cf. section 3), i.e. to consider the probability that v' < v < v" < 1 with v' and v' assigned, equal to the probability that 1/v'< 1/v < 1/v". D’Ippolito appears to concur in this view (cf. below in the text). I am unable to accept Ciccone’s attempt (1996, pp. 51–4) to justify D’Ippolito’s procedure. Ciccone writes (p. 52, my translation): ‘Because of the symmetry between the conditions v < 1 and v > 1, the Pme(r) calculated for values of v included between 0 and 1 comes out to be in fact equal to the average probability obtainable for values of v included between 1 and + f’; but this is false, because if one follows D’Ippolito in calling technique 2 the one dominant to the right of r, then the second average probability is simply zero. It may be noticed that if one associated to each v < 1 the corresponding 1/v, the eligible portions of OCBQ would sum to exactly the area of OCBQ. To consider all points in OCBQ equally probable is clearly arbitrary in that they would not be equally probable if one decided e.g. that it is all admissible couples (D1, D2) that are equally probable. This arbitrariness is ineliminable from exercises of this kind. Cf. note 9 above on the need to replace v with 1/v when v > 1 in order to avoid having a zero probability of all finite intervals of values of v. Salvadori stresses in particular that a technique can be characterized in many different ways, each one based on different parameters, so that an assumption – arbitrary as it is anyway – of ‘equal probability’ of all values of these parameters within their acceptable range will generate different results for different characterizations of techniques. Indeed had I not replaced R with a in section 5, the numerical results would have been radically different. Laing (1991) attempts a similar comparison of areas, for an ‘Austrian’ model where the consumption good is produced by current labour, labour employed one period earlier, and labour employed two periods earlier; all he is able to argue is that ‘there is a much bigger volume [of possible coefficient values] where double-switching does not occur than where it does’ (Laing 1991, p. 187); how much bigger, he is unable to estimate numerically; the sole case in which he obtains a numerical estimate is the case which he deems the most favourable possible to RCD; the ratio of the two ‘volumes’ LQWKDWFDVHLVLQGLFDWLQJDSUREDELOLW\RI5&'ZKLFKPDNHVLWOLNHO\WKDWWKH range of probabilities for this model must be of the same far-from-negligible order of magnitudes as the ones found above for the Samuelson-Garegnani model. Laing’s conclusion that RCD and reswitching ‘are a possibility but are exceptional’ (p. 185) appears therefore totally unwarranted.

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Reswitching and reverse capital deepening 415 16 Mainwaring and Steedman prefer to speak of ‘double equi-profitability’ to mean that two w(r) curves cross twice, and, differently from the terminology adopted here, to restrict the term ‘reswitching’ to the cases of double equi-profitability where both switches are on the frontier. 17 From their Figures it would seem that Mainwaring and Steedman prefer to assume that R is not greater than 1. 18 It is always possible to ‘border’ each technique matrix with industries producing, as nonbasics, the capital goods only utilized in other techniques; then all techniques produce the same goods and one obtains that at a switchpoint the method of only one industry changes; but what is relevant for the change of the w(r) curve is whether there is a change of the goods (that can be called quasi-basic or also wage goods) entering the numéraire basket (but these cannot change) or directly or indirectly used in its production. 19 The chapter does not specify the procedure adopted to generate the random coefficients, and in particular the probability distribution assigned to them including the probability that a coefficient be zero; an announced longer paper explaining these issues has not been published owing to Professor D’Ippolito’s death; thus a thorough assessment of the plausibility of his 1989 results appears impossible. 20 The fact that D’Ippolito (1989) assumes that all prices remain equal to 1, that is, that all prices are the same at a switchpoint, shows that he is assuming that all quasi-basic goods are common to both techniques. The same assumption is stated by Han and Schefold (2006) on p. 741. 21 Actually, the simple possibility of the phenomenon that makes reswitching possible: the inversion of the movement of relative prices as the rate of profit rises, cf Sraffa 1960, p. 84, and Petri 2004, pp. 210–16. 22 On the Pigou or real-balance effect cf. Petri 2004 pp. 291–4; its weakness is admitted even by Patinkin. 23 In discussing this argument, I shall leave aside the arbitrariness in determining the valuecapital/labour ratio due to the arbitrariness in the choice of numéraire. Potestio (2010) has used this arbitrariness as a criticism of some presentations of the Sraffian critique of neoclassical capital theory, but the point of those presentations was that even leaving aside this problem as in certain cases it is possible to do (for example, by assuming a single consumption good which is then the natural numéraire because appropriate to measure the sacrifice – the potential consumption given up – connected with acts of saving) still powerful criticisms are possible. If the purpose is criticism, it is admissible to concede some ultimately illegitimate aspects to the argument to be criticized if that allows highlighting better the weakness of a more fundamental assumption. 24 ‘We have now had two centuries of capitalism, with all its technical changes and national peculiarities, so we have had very many “random” extractions of sets of alternative techniques. Thus even very low probabilities of reverse capital deepening would not make it unlikely that, at least in some countries and some historical periods, the capital-labour ratio schedule had upward-sloping sections, which should have resulted in at least some cases in phenomena that, to the contrary, have not been observed’ (Petri 2004, p. 254). 25 ‘However small the evaluated probability of the instances in which the principle of substitution does not operate, obviously prices and incomes would take shape, and would therefore have an explanation, also in those circumstances. One would thus be implicitly admitting the existence of a theory of distribution, alternative to the neoclassical one, and without any basis for excluding that this alternative theory, differently from the neoclassical one, may apply to the generality of cases’ (Ciccone 1996, p. 42, my translation). 26 These can be obtained from the references in Tsoufildis and Maniatis (2002). 27 Casual observation appears to suggest very little change in technology when firms move production plants from high-wage to low-wage countries, and this appears to

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confirm a predominant independence of modern optimal production methods from (realistic) changes in wages. Or a fully employed supply of labour which is a non-decreasing function of the real wage. If the rate of interest changes and normal prices with it, it is only in new plants that the new optimal production methods can be adopted. Already existing plants and other durable capital goods will go on being utilized as long as they earn non-negative quasirents, and there is little reason to assume that normal labour utilization in them will be relevantly affected by a higher real wage, given the little room for changes in production methods once the fixed plant is built. Thus, labour will be combined with capital goods adapted to new optimal technical choices only gradually, as existing plants gradually reach the end of their economic life and are replaced by new plants. Cf. Garegnani (1978–79, pp. 35, 64–5). The convenience of keeping existing plants in operation as long as quasi-rents are nonnegative still holds, so if output does not change investment will be again an opportune reduced-scale copy of the demand for capital. Dornbusch and Fischer do not place great stress on this implication of their approach, and do not recognize that it seriously questions the central role of the IS-LM model in their textbook: the relevant influence of variations of output on investment makes the IS curve a construction of little significance. Cf. Petri 2004, pp. 257–8. It is an interesting mental exercise to try to imagine the effect of teaching investment theory starting directly from the available empirical evidence, without any previous indoctrination with neoclassical notions of capital-labour substitution. Some time will be required for prices to decrease by the same percentage as money wages, even assuming such a price decrease to happen; and during this time real wages are reduced. This unemployment must be considered involuntary, because it does not depend on workers refusing real wage reductions: these would not get them a job anyway. Thus one might say, in neoclassical language, that owing to the adaptability of production to demand, relative factor proportions adapt to income distribution rather than the other way round.

References Bidard, C., and T. Schatteman, 2001, ‘The Spectrum of Random Matrices’, Economic Systems Research, 13: 3, 289–98. Bliss, C. J., 1975, ‘The reappraisal of Keynesian economics. An appraisal’, in M. Parkin and A. R. Nobay, eds, Current Economic Problems, Cambridge: Cambridge University Press. Bloise, G., and P. Reichlin, 2009, ‘An obtrusive remark on capital and comparative statics’, Metroeconomica 60: 1, 54–76. Ciccone, R., 1996, ‘Possibilità e probabilità di comportamento “perverso” del capitale’, Studi Economici 58: 1, 41–73. D’Ippolito, G., 1987, ‘Probabilità di perverso comportamento del capitale al variare del saggio di profitto. Il modello embrionale a due settori’, Note Economiche, 2, 5–37. D’Ippolito, G., 1989, ‘Delimitazione dell’area dei casi di comportamento perverso del capitale in un punto di mutamento della tecnica’, in Luigi Pasinetti, ed., Aspetti controversi della teoria del valore, Bologna: Il Mulino, 191–8. Dornbusch, R. and S. Fischer, 1987, Macroeconomics, 4th edn, New York: McGraw-Hill. Eltis, W., 1973, Growth and Distribution, London: Macmillan.

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Reswitching and reverse capital deepening 417 Fisher, F. M., 1983, Disequilibrium Foundations of Equilibrium Economics, Cambridge: Cambridge University Press. Fratini, S. M., 2009, ‘Real Wicksell effect, demand for capital and stability’, unpublished paper presented at Centro Sraffa, Rome. Garegnani, P., 1970, ‘Heterogeneous capital, the production function and the theory of distribution’, Review of Economic Studies, 407–36. Garegnani, P., 1978–79, ‘Notes on consumption, investment and effective demand. Part I’, Cambridge Journal of Economics, 2(4), 335–53, and ‘Notes on consumption, investment and effective demand. Part II’, Cambridge Journal of Economics, 3: 1, 63–82: as reprinted in John Eatwell and Murray Milgate, eds, Keynes’s Economics and the Theory of Value and Distribution, New York: Oxford University Press, 1983, pp. 21–69. Garegnani, P., 1990, ‘Quantity of capital’, in J. Eatwell, M. Milgate and P. Newman, eds, The New Palgrave: Capital Theory, London: Macmillan, pp. 1–78. Han, Z., and B. Schefold, 2006, ‘An empirical investigation of paradoxes: Reswitching and reverse capital deepening in capital theory’, Cambridge Journal of Economics 30: 5, 737–65. Hicks, J., 1965, Capital and Growth, Oxford: Clarendon Press. Hicks, J., 1973, Capital and Time, Oxford: Clarendon Press. Jorgenson, D. W., 1963, ‘Capital theory and investment behavior’, American Economic Review, 53(2), 247–59. Kurz, H. D., ed., 2000, Critical Essays on Piero Sraffa’s Legacy in Economics, Cambridge, Cambridge University Press. Laing, N. F., 1991, ‘The likelihood of capital-reversing and double-switching’, Bulletin of Economic Research, 43, 179–88. Mainwaring, L., and I. Steedman, 2000, ‘On the probability of re-switching and capital reversing in a two-sector Sraffian model’, in Kurz (2000), pp. 323–54. Malinvaud, E., 1986, ‘Pure profits as forced saving’, Scandinavian Journal of Economics 88: 1, 109–30. Ochoa, E., 1989, ‘Values, prices and wage-profit curves in the US economy’, Cambridge Journal of Economics, 13, 413–29. Petri, F., 2004, General Equilibrium, Capital and Macroeconomics. A Key to Recent Controversies in Equilibrium Theory, Cheltenham: Edward Elgar. 3HWURYLü 3  µ6KDSH RI D ZDJHSURILW FXUYH VRPH PHWKRGRORJ\ DQG HPSLULFDO evidence’, Metroeconomica, 42: 2 (June), 93–112. Potestio, P., 2010, ‘“Perverse cases” and the debate on neo-classical theory of distribution: recent contributions on an open issue’, in John Vint, J. Stanley Metcalfe, Heinz D. Kurz, Neri Salvadori and Paul A. Samuelson, eds, Economic Theory and Economic Thought, London: Routledge, pp. 138–60. Salvadori, N., 2000, ‘Comment on Mainwaring and Steedman’, in Kurz (2000), pp. 354–8. 6DOYDGRUL 1 DQG , 6WHHGPDQ  µ1R UHVZLWFKLQJ" 1R VZLWFKLQJ¶ Cambridge Journal of Economics, 12, 483–6. Samuelson, P. A., 1962, ‘Parable and realism in capital theory: the surrogate production function’, Review of Economic Studies, 29, 193–206. Schefold, B., 1976, ‘Relative prices as a function of the rate of profit: a mathematical note’, Zeitschrift für Nationalökonomie (Journal of Economics), 36, 21–48; repr. in Schefold 1997a, pp. 46–75. Schefold, B., 1997a, Normal Prices, Technical Change and Accumulation, London: Macmillan.

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Schefold, B., 1997b, ‘Classical Theory and Intertemporal Equilibrium’, in Schefold 1997a, pp. 425–501. Schefold, B., 2010, ‘Families of strongly curved and of nearly linear wage curves: a contribution to the debate about the surrogate production function’, in John Vint, J. S. Metcalfe, Heinz D. Kurz, N. Salvadori and Paul A. Samuelson, eds, Economic Theory and Economic Thought, London: Routledge, pp. 117–37. Sraffa, P., 1960, Production of Commodities by Means of Commodities, Cambridge: Cambridge University Press. Tsoufildis, L., and T. Maniatis, 2002, ‘Values, prices of production and market prices: some more evidence from the Greek economy’, Cambridge Journal of Economics, 26, 359–69.

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Heinz Kurz’s publications

Books Adriano Birolo, Duncan K. Foley, Heinz D. Kurz, Bertram Schefold and Ian Steedman (eds), 2010. Production, Distribution and Trade: Alternative Perspectives. Essays in Honour of Sergio Parrinello, London: Routledge. John Vint, J. Stanley Metcalfe, Heinz D. Kurz, Neri Salvadori and Paul A. Samuelson (eds), 2010. Economic Theory and Economic Thought. Essays in Honour of Ian Steedman. London: Routledge. Heinz D. Kurz (ed.), 2010. Studien zur Entwicklung der ökonomischen Theorie XXIV – Wechselseitige Einflüsse zwischen dem deutschen wirtschaftswissenschaftlichen Denken und dem anderer europäischer Sprachräume. Berlin: Duncker & Humblot. Horst Hanusch, Heinz D. Kurz and Christian Seidl (eds), 2010. Schumpeter for Our Century. Homo Oeconomicus vol. 27(1/2), München: Accedo. Aiko Ikeo and Heinz D. Kurz (eds), 2009. A History of Economic Theory. Essays in Honour of Takashi Negishi. London: Routledge. Heinz D. Kurz (ed.), 2009. Klassiker des ökonomischen Denkens, Band 2, München: C.H. Beck. Heinz D. Kurz (ed.), 2008. Critical Essays on Piero Sraffa’s Legacy in Economics. Paperback re-issue, Cambridge: Cambridge University Press. Heinz D. Kurz, Luigi Pasinetti and Neri Salvadori (eds), 2008. Piero Sraffa: The Man and the Scholar. Exploring His Unpublished Papers. Print on Demand Paperback, London: Routledge. Heinz D. Kurz, 2008. Shunpeta no mirai (The Future and Potential of Schumpeter) (in Japanese), Tokyo. Heinz D. Kurz (ed.), 2008. Klassiker des ökonomischen Denkens. Band 1, München: C.H. Beck. Heinz D. Kurz and Neri Salvadori, 2007. Interpreting Classical Economics. Studies in Long-period Analysis. London: Routledge. Heinz D. Kurz, Luigi Pasinetti and Neri Salvadori (eds), 2007. Piero Sraffa: The Man and the Scholar. Exploring His Unpublished Papers. London: Routledge. Heinz D. Kurz and Christian Gehrke (eds), 2006. David Ricardo: Über die Grundsätze der Politischen Ökonomie und der Besteuerung. 2nd revised edition, Marburg: Metropolis. Gerhard Huber, Hagen Krämer and Heinz D. Kurz (eds), 2005. Einkommensverteilung, technischer Fortschritt und struktureller Wandel. Festschrift für Peter Kalmbach. Marburg: Metropolis Verlag.

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420

Heinz Kurz’s publications

Heinz D. Kurz, 2004. Joseph A. Schumpeter. Ein Sozialökonom zwischen Marx und Walras. Marburg: Metropolis. (A Serbian translation was published in 2006 as Joseph Alois Schumpeter – Politicki Ekonomista Izmedu Marxa I Walrasa, Belgrad. Heinz D. Kurz and Neri Salvadori, 2003. Classical Economics and Modern Theory. Studies in Long-period Analysis. London: Routledge. Heinz D. Kurz and Neri Salvadori (eds), 2003. The Legacy of Piero Sraffa. Volumes I and II, Cheltenham: Edward Elgar. Stephan Boehm, Christian Gehrke, Heinz D. Kurz and Richard Sturn (eds), 2002. Is there Progress in Economics? Knowledge, Truth, and the History of Economic Thought. Cheltenham: Edward Elgar. Vitantonio Gioia and Heinz D. Kurz (eds), 2000. Science, Institutions and Economic Development. The Contribution of ‘German’ Economists and the reception in Italy (1860–1930). Milan: Giuffrè. Harald Hagemann and Heinz D. Kurz (eds), 1998. Political Economics in Retrospect. Essays in Memory of Adolph Lowe. Cheltenham: Edward Elgar. Heinz D. Kurz, Erik Dietzenbacher and Christian Lager (eds), 1998. Input–Output Analysis. Three volumes, Cheltenham: Edward Elgar. Heinz D. Kurz and Neri Salvadori (eds), 1998. The Elgar Companion to Classical Economics. Two volumes, Cheltenham: Edward Elgar. Heinz D. Kurz and Neri Salvadori, 1998. Understanding ‘Classical’ Economics. Studies in Long-period Theory. London and New York: Routledge. Heinz D. Kurz, 1998. Ökonomisches Denken in klassischer Tradition. Aufsätze zur Wirtschaftstheorie und Theoriegeschichte. Marburg: Metropolis. Heinz D. Kurz and Neri Salvadori, 1997. Theory of Production. A Long-Period Analysis. Paperback edition. Cambridge: Cambridge University Press. Lutz Beinsen and Heinz D. Kurz (eds), 1997. Ökonomie und Common Sense. Gunther Tichy zum 60. Geburtstag. Graz: Leykam. Heinz D. Kurz (ed.), 1997. Klaus H. Hennings. The Austrian Theory of Value and Capital. Studies in the Life and Work of Eugen von Böhm-Bawerk. Cheltenham: Edward Elgar. Heinz D. Kurz and Neri Salvadori, 1995. Theory of Production. A Long-Period Analysis. Cambridge: Cambridge University Press. (A Russian translation was published in 2004 as Teoria proisvodstva. Dolgosrotchny analys, Moskau: Financy y statistica. A Chinese translation was published by China Economic Publishing House in 2008.) Heinz D. Kurz (ed.), 1994. David Ricardo: Über die Grundsätze der politischen Ökonomie und der Besteuerung. Marburg: Metropolis. Heinz D. Kurz, 1993. United Germany and the New Europe. Cheltenham: Edward Elgar. Peter Kalmbach and Heinz D. Kurz, 1992. Chips und Jobs. Zu den Beschäftigungswirkungen des Einsatzes programmgesteuerter Arbeitsmittel. Marburg: Metropolis. Heinz D. Kurz, 1990. Capital, Distribution, and Effective Demand: Studies in the Classical Approach to Economic Theory. Cambridge: Polity Press. Heinz D. Kurz, 1990. Adam Smith (1723–1790). Ein Werk und seine Wirkungsgeschichte. Marburg: Metropolis. Heinz D. Kurz (ed.), 1989. Pierangelo Garegnani: Kapital, Einkommensverteilung und effektive Nachfrage: Beiträge zur Renaissance des klassischen Ansatzes in der Politischen Ökonomie. Marburg: Metropolis. Harald Hagemann and Heinz D. Kurz, 1984. Beschäftigung, Verteilung und Konjunktur. Zur Politischen Ökonomik der modernen Gesellschaft. Festschrift für Adolph Lowe. Bremen: Universität Bremen.

Heinz Kurz’s publications

421

Harald Hagemann, Heinz D. Kurz and Wolf Schäfer (eds), 1981. Die neue Makroökonomik. Marktungleichgewicht, Rationierung und Beschäftigung. Frankfurt and New York: Campus. Heinz D. Kurz, 1977. Zur neoricardianischen Theorie des Allgemeinen Gleichgewichts der Produktion und Zirkulation. Berlin: Duncker & Humblot.

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Articles +HLQ]'.XU]:KRLV*RLQJWR.LVV6OHHSLQJ%HDXW\"2QWKHǥ&ODVVLFDO¶$QDO\WLFDO Origins and Perspectives of Input-Output-Analysis, Review of Political Economy, vol. 23(1), pp. 25–47. Heinz D. Kurz, 2011. Two Masters – one Mind: Schumpeter zwischen Walras und Marx, in Karl Acham (ed.), Rechts-, Sozial- und Wirtschaftswissenschaften aus Graz. Zwischen empirischer Analyse und normativer Handlungsanweisung: wissenschaftsgeschichtliche Befunde aus drei Jahrhunderten. Wien: Böhlau, pp. 501–36. Heinz D. Kurz, 2010. Technical Progress, Capital Accumulation and Income Distribution in Classical Economics: Adam Smith, David Ricardo and Karl Marx, European Journal of the History of Economic Thought, vol. 17(5), pp. 1183–1222. +HLQ]'.XU]DQG&KULVWLDQ/DJHU(VNL.ODVLNøNWLVDWoÕODUÕQ.RQXPXQXQ