Demand, Equilibrium and Trade: Essays in Honour of Ivor F. Pearce

DEMAND, EQUILIBRIUM AND TRADE
This book presents thirteen essays by distinguished economists written to honour the work of Professor lvor F. Pearce. The chapters cover all aspects of economics, reflecting the wide-ranging contribution made by Ivor Pearce to the discipline of economics, but his major contributions are to demand theory, general-equilibrium theory, trade theory and capital theory. The book contains a number of important new works. W. M. Gorman re-examines the Le Chatelier principle from a general-equilibrium perspective, and shows that one can derive conclusions contrary to those of the principle. P. Simmons generalises a number of measures of comple mentarity, including that proposed by Pearce. lvor Pearce has explained the restrictive assumptions one needs to derive rigorously the familiar propositions of trade theory, but J. Peter Neary shows that, by aggrega tion, one can overcome some of these restrictions provided one is willing to ask a different set of questions. Christopher Bliss uses Keynes's insight ful interpretation of the famous Ramsey Rule in optimal growth to derive results in a wide range of growth models with considerable economy and elegance. Murray C. Kemp and his colleagues analyse the problems that might confront a market economy in managing the transition from an exhaustible to a renewable resource. The Pearce and Gabor analysis of money capital is developed by Donald W. Katzner in a general-equilibrium model and A. G. Schweinberger in a general-equilibrium model of inter national trade. Andre Gabor surveys recent developments in transfer pricing, while L. R. Klein and his colleagues use the project LINK models to test the purchasing power parity hypothesis. Some problems in using macroeconomic data and models are explored in chapters by Sir Richard Stone and by Alan Budd and Sean Holly. Finally, Victoria Chick pursues some ideas of Ivor Pearce on the inflationary consequence of expansionary fiscal policy and relates her analysis to more recent work of Pearce on institutional arrangements to secure sound money.
IVOR F. PEARCE
©Carmen Moll1984
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DEMAND, EQUILIBRIUM AND
Edited by
M MACMILLAN
Softcover reprint of the hardcover 1st edition 1984 978-0-333-33184-2
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First published 1984 by THE MACMILLAN PRESS LTD
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British Library Cataloguing in Publication Data Demand, equilibrium and trade. 1. Equilibrium (Economics) - Addresses, essays, lectures I. Ingham, A. II. Ulph, A. M. Ill. Pearce, lvor F. 330.15'43 HB145 ISBN 978-1-349-06360-4 ISBN 978-1-349-06358-1 (eBook) DOI 10.1007/978-1-349-06358-1
Contents
Preface vii The Contributors viii Bibliography of lvor F. Pearce ix Biographical Details of Ivor F. Pearce xii
Ivor Pearce as a Colleague D. C. Rowan xiii
Introduction A. Ingham and A. M. Ulph xvii
Le Chatelier and General Equilibrium W.M Gorman
2 A Complement to Pearce on Complements P. Simmons 19
3 The True Index S. N. Afriat 37
4 The Heckscher-Ohlin Model as an Aggregate J. Peter Neary 57
5 The Austrian Concept of Capital and the Pure Theory of International Trade A. G. Schweinberger 77
6 Notes on the Keynes-Ramsey Rule Christopher Bliss 93
19
vi Contents
7 On the Transition from an Exhaustible Resource Stock to an Inexhaustible Substitute Nguyen Manh Hung, Mu"ay C. Kemp and Ngo van Long 105
8 Capital and Walrasian Equilibrium Donald W. Katzner 123
9 On the Theory and Practice of Transfer Pricing Andre Gabor 149
10 Endogenous Exchange Rate in the Medium Term: A Weak Law of Purchasing Power Parity for the LINK System L. R. Klein, S. Fardoust and V. Filatov 171
11 Balancing the National Accounts: The Adjustment of Initial Estimates -A Neglected Stage in Measurement Sir Richard Stone 191
12 Short-term Models and Long-term Problems Alan Budd and Sean Holly 213
13 Monetary Increases and their Consequences: Streams, Backwaters and Floods Victoria Chick 237
Name Index 251
Subject Index 254
Preface
The essays in this volume were presented at a conference held at the University of Southampton in January 1982 to honour the work of Ivor Pearce, who retired from the Chair of Economic Theory, University of Southampton in September 1981. At the conference Ivor Pearce also delivered a lecture entitled 'The Time Is Not Yet Ripe'. This was the first of an annual series of lectures on economic policy sponsored by the Economics Department, University of Southampton.
We are grateful to many who made this conference possible. The British Academy, the British Council, BP, the Institute of Economic Affairs, Shell and Unilever all helped with financial assistance, without which the con ference would not have been possible. Of course, they do not necessarily share the views of the contributors. G. W. McKenzie and David F. Heathfield gave valuable assistance in the setting up and running of the conference. We would also like to thank Mrs Jan Gerrard, Mrs Julia Hepburn and Mrs Sarah Rollason for secretarial help in organising the conference and in preparing the manuscript.
A. I. A.M.U.
The Contributors
S. N. Afriat Christopher Bliss Alan Budd Victoria Chick S. Fardoust V. Filatov Andre Gabor W.M. Gorman Sean Holly Nguyen Manh Hung A. Ingham Donald W. Katzner Murray C. Kemp L. R. Klein Ngo van Long J. Peter Neary D. C. Rowan A. G. Schweinberger P. Simmons Sir Richard Stone A.M. Ulph
University of Ottawa Nuffield College, Oxford London Business School University College, London University of Pennsylvania University of Pennsylvania University of Nottingham Nuffield College, Oxford London Business School Universite Laval University of Southampton University of Massachusetts, Amherst University of New South Wales University of Pennsylvania Australian National University University College, Dublin University of Southampton University of East Anglia University of York King's College, Cambridge University of Southampton
viii
BOOKS
A Contribution to Demand Analysis (London: Oxford University Press, 1964).
International Trade (London: Macmillan, 1970). A Model of Output, Employment, Wages and Prices in the UK (Cambridge
University Press, 1976); with G. Anderson, C. Stromback, P. Trivedi. The Incredible Eurodollar (London: George Allen & Unwin, 1983); with
W. P. Hogan.
PAPERS
'A new approach to the theory of the flrm', Oxford Economic Papers, 4 (1952) 252-65.
'A note on Mr. Lerner's paper',Economica, 19 (1952) 16-18. 'The factor price equalisation myth', Review of Economic Studies, 19
(1952) 111-20. 'Total demand curves and general equilibrium', Review of Economic
Studies, 20 (1953) 216-27. 'Consumer's behaviour and the conditions for exchange stability - a note
on Mr. Spraos' paper',Economica, 22 (1955) 147-51. 'Total demand curves- a reply to Messrs. Ozga and Lancaster',Review of
Economic Studies, 23 (1955) 153-62.
ix
X Bibliography
'A study in price policy',Economica, 23 (1956) 114-27. 'Price policy with branded products', Review of Economic Studies, 24
(1957) 49-60. 'Demand analysis and the savings function', Economic Record, 34 (1958)
52-66. 'The place of money capital in the theory of production', Quarterly
Journal of Economics, 72 (1958) 537-57; with A. Gabor. 'A further note on factor commodity price relationships', Economic
Journal, 79 (1959) 725-32. 'The problem of the balance of payments', International Economic Review,
2 (1961) 1-28. 'A method of consumer demand analysis illustrated', Economica, 27
(1961) 371-94. 'An exact method of consumer demand analysis', Econometrica, 29
(1961) 499-516. 'The end of the golden age in Solovia', American Economic Review, 52
(1962) 1088-97. 'Community consumer demand theory', Australian Economic Papers, 1
(1962) 1-23. 'On the separability postulate', Economic Record, 41 (1965) 455-6. 'A framework for research on the real effects of capital movements',
Rivista Internazionale di Scienze Economiche (1965); with D. Rowan. 'More about factor price equalisation', International Economic Review,
8 (1967) 255-70. 'Rejoinder to Professor Samuelson', international Economic Review, 8
(1967) 296-9. 'Rejoinder to Professor McKenzie', International Economic Review, 8
{1967) 300-6. 'A look at the structure of optimal tariff rates', International Economic
Review, 11 (1970) 147-61; with D. Horwell. 'The Southampton econometric model of the UK and trading partners',
in The Econometric Study of the UK, Hilton and Heathfield (eds) (London: Macmillan, 1970) 29-52.
'Inflation in the Southampton econometric model', in The Current Infla tion, Johnson and Nobay (eds) (London: Macmillan, 1971) 38-42.
'Theory of wage differentials', Journal of International Economics, 1 (1971) 205-14.
'Some aspects of European monetary integration', in Issues in Monetary Economics, Johnson and Nobay (eds) (London: Oxford University Press, 1971) 75-97.
Bibliography xi
'On the uniqueness of competitive equilibrium (unbounded demand)', Econometrica, 41 (1973} 817-28; with J. Wise.
'On the uniqueness of competitive equilibrium (bounded demand)', Econometrica, 42 (1974) 921-32; with J. Wise.
'Matrices with dominating diagonal blocks' ,Journal of Economic Theory, 5 (1974) 159-70.
'Monopolistic competition and general equilibrium', Proceedings of the AUTE conference, in Cu"ent Economic Problems, Parkin and Nobay (eds} (Cambridge: Cambridge University Press, 1975) 93-110.
'A view of the Southampton econometric model', Proceedings of the SSRC Conference 1973, in Modelling the Economy, Renton (ed.) (London: Heinemann Educational Books, 1975) 83-123; with D. Heathfield.
'Resource conservation and the market mechanism', in Economics of Natural Resource Depletion, Pearce and Rose (eds) (London: Macmillan 1975) 191-203.
'Exact measures of welfare and the cost of living', Review of Economic Studies, 43 (1976) 465-8; with G. McKenzie.
'Stimulants to exertion ... a deficiency of excitements', in Catch '76, Seldon (ed.)(London: lEA, 1976} 113-21.
'Participation and the distribution of income', in The Economics of Co Determination, Heathfield (ed.) (London: Macmillan, 1977).
'Demand, Consumer Surplus and Sovereignty', in Modern Economic Thought, Weintraub (ed.)(Oxford: Blackwell, 1977) 217-45.
'Taxing the dole', in The State of Taxation, Seldon (ed.) (London: lEA, 1977) 91-107.
'Confrontation with Keynes', The Coming Confrontation, Seldon (ed.) (London: lEA, 1978) 91-115.
'A theory of money capital, general equilibrium and income distribution', in The Measurement of Capital, Patterson and Schott (eds) (London: Macmillan, 1979} 25-64.
'The incredible Eurodollar, a fable for our time', The Banker, 1980 (June) 35-48.
'Reforms required for the entrepreneur to serve public policy', in Prime Mover of Progress, Seldon ( ed.) (London: lEA, 1980) 129-43.
'Welfare economics- a synthesis', American Economic Review, 72 (1982} 660-82; with G. McKenzie.
'A tract on sound money: why and how', in The British Economy, Hawkins and McKenzie (eds) (London: Macmillan, 1982) 153-66; with D. Heathfield.
Bibliographical Details of I vor F. Pearce
Born 21 January 1916. Educated at Queen Elizabeth's Hospital, Bristol. Qualified as an accountant being elected to membership of the Institute of Management Accountants and worked in industry from 1932 to 1939. Served in HM Forces, 1940-6. Undergraduate at University of Bristol, 1946-9. Lecturer in Economics at University of Nottingham, 1949-56. Awarded PhD., 1954. Reader and then Professor of Economics at the Institute of Advanced Studies, Australian National University, 1956-61. Visiting Fellow of Nuffield College, Oxford, 1961-2. Professor of Eco nomic Theory at University of Southampton, 1962-81. Director of Research, Econometric Model Building Unit, University of Southampton, 1973-7. Visiting Professor at Wharton School, University of Pennsylvania, 1965 and 1969. University of Waterloo, 1971, University of California, 1972 and 1978, University of Massachusetts, 1976-7, University of Melbourne, 1978, University of Sydney, 1981, Visiting Fellow at Institute of Advanced Studies, Australian National University, 1972, and St Hilda's College, Melbourne, 1978. Elected Fellow of Econometric Society.
xii
D. C. ROWAN
The chapters in this volume were written to honour the work of Ivor Pearce, who, in September 1981, retired from the Chair of Economic Theory in the University of Southampton. Most of these contributions were delivered at a conference in Ivor's honour which was held in Southampton in January 1982. This introductory note is not an attempt to summarise or comment upon the contributions to the conference and this volume. Nor is it aimed at providing an appreciation -let alone an assessment - of Ivor's very considerable contribution to economics. This volume and the conference itself are the best indicators of the breadth, power and impact of Ivor's work. This note has the more modest purpose of complementing the scientific studies that follow with an appreciation of Ivor as a colleague. Inevitably, since Ivor and I have known each other so long, what I have written has a personal flavour. The reader, however, should not be misled by this. The views I express are shared by those who were and are his colleagues in the Department of Economics; and by many of his contemporaries in the faculty and the university.
Ivor and I first met when, getting on for thirty-six years ago, we were undergraduates together at the University of Bristol. Both of us had spent a considerable number of years in the Army and were, in our 'middle age', a little apprehensive about our ability to compete successfully with bright young rivals who had just left school and whose capacity for sustained concentration had not been impaired by Army life. As things turned out, our apprehensions were misplaced. Under the late Miles Fleming- a teacher to whom both of us owe an immense debt - we learned economics in the best of all possible ways: that is by continuous argument. In a small group, under Miles Fleming's patient but enthusiastic chairmanship, we consumed gallons of tea and debated most propositions of economic theory that we encountered. As a result we learned a great deal - and spent some two and half years in a way as pleasant as it was productive.
xili
xiv Ivor Pearce as a Colleague
During these discussions it became obvious that Ivor had an immense ability to stay with a problem until he was satisfied that he had correctly solved it. His mind was not only powerful. It was persistent. And to these two qualities he added a third: a willingness to dispute the assertions of the authorities and to treat all opinions, including his own, with a proper measure of scientific disrespect. Over the years his many colleagues, here, and elsewhere, have gained greatly from these qualities since they have encouraged him to devote much time and effort to their problems.
Perhaps I may be allowed a personal reminiscence to illustrate this point. In our final year, Ivor and I were friendly rivals since we both believed (rightly) that if we wished to enter academic life (as we did) we needed to obtain first-class honours and (wrongly) that the Department would not award more than one first. Nevertheless we still worked in co-operation, debating what we read and often seeking a more convincing analysis of problems than the literature seemed to offer. Not surprisingly, our reading programme was pretty intense and, during it, one of us, my memory does not now recall which, came upon Samuelson's {1948) treatment of the issue of factor-price equalisation, found it unconvincing and showed it to the other who found it the same.
From that occasion onwards Ivor became obsessed with the factor price issue and scarcely a day passed without his producing some new analysis of the conditions necessary and sufficient for equalisation to occur. Together we both spent more time on the problem than any optimal revision programme would have suggested. My interest, always less than Ivor's, waned relatively early. But Ivor's powerful and persistent mind continued with the problem after his appointment to a lectureship at Nottingham. I well remember my pleasure at seeing this persistence rewarded by his early and important paper in the Review of Economic Studies (1952).
Though Ivor and I did not become colleagues again in any formal sense until he took the chair in Economic Theory at Southampton in 1962, we nevertheless followed rather similar paths. I went to Melbourne in 1954 and, by 1956, was at the University of New South Wales when Ivor arrived in Sydney. He was, of course, then on his way to Canberra to join Trevor Swan at the Australian National University. I came to Southampton in 1960 and tried, successfully, to persuade Ivor to follow me. When he did, I believed that the university had appointed a brilliant and creative eco nomic theorist and a very worthy successor to his distinguished predecessor, the late Professor W. A. Armstrong. Moreover, because I had known him so long, I was confident that Ivor would not only make a major contribu tion to economics but also to economics at Southampton.
lvor Pearce as a Colleague XV
Economists' forecasts are typically objects of denigration. But both these forecasts have proved triumphantly correct. This volume -and the conference - are unassailable evidence for the first. But since Ivor's con tributions to the Department and its development are less well known, it seems essential to say something about the second.
The possession of a good and creative economic theorist is, at least in my view, a necessary, but not sufficient condition for building an effective department. When lvor arrived here, the Department, narrowly defined, had only two other members apart from myself. We urgently needed to expand our staff in order to offer an appropriate range and quality of teaching in economics. And we needed, no less urgently, to develop the teaching of econometrics. Unless both could be done, neither could be fully effective.
As everyone knows, the so-called 'Robbins Expansion' gave us the opportunity to do the former in the sense of providing us with vacancies.
Recruiting good staff - always a scarce factor - in the face of a general expansion of universities, raised quite other problems. And here there can be no doubt that lvor's reputation as a creative scholar was a significant element in attracting applicants of quality, just as his gentle and informal - but nevertheless searching -methods at interviews were of considerable help in the actual selection of new staff: a selection that, judged by the test of time, has had a considerable measure of success.
The establishment of econometrics became possible because of an appeal, rather reluctantly approved by the university, initiated by lvor and myself which, ultimately, received generous support from the Leverhulme Trust. In the early stages of this appeal results were scarcely promising. Most prospective donors were either more attracted to Oxbridge research projects or had been recently visited by representatives of importunate but prestigious colleges. It is doubtful whether we could have raised any signi ficant sum without Leverhulme's generosity. I think that it is more than probable that the favourable view taken by the Trust owed something to the fact that, as an honorary consultant, Ivor had, with Michael Greatorex, recently investigated some problems for Unilever's research department. Thus lvor contributed considerably to the establishment of the Depart ment of Econometrics (under Professor G. R. Fisher), and it is not too fanciful to argue that the success of this Department, in its turn, encour aged the Trust (in 1975) to establish a Department of Social Statistics (under Professor G. Kalton).
The Southampton Model project, which was financed by the SSRC from 1966 to 1976, owed a great deal to Ivor: for not only was the form of the model his conception but, again through SSRC finance, he acted
xvi lvor Pearce as a Colleague
as director of the project from 1973. For a number of reasons, of which the problems relating to data were perhaps the most important, the Southampton Model was less successful than had been hoped. It neverthe less produced a number of very worthwhile contributions and made pos sible a useful development of graduate work associated with it.
I have mentioned the Model - and the establishment of the Department of Econometrics - as two important developments in which Ivor played a considerable part. He also acted as Head of Department and demonstrated, in doing so, not only marked administrative ability but also that his frequently expressed view that Heads of Departments should be dicta torial rather than democratic had no operational implications. It is, of course, not possible for a Head of Department to consult all its members all of the time about all the issues that arise. But Ivor came close to doing so: exhibiting, in the process, monumental patience, as well as a remark able talent for listening.
Thus, in a number of ways, Ivor made administrative contributions to the development of the Department that could easily be underestimated but should not be. His main influence, however, has been felt in quite other ways. The foremost of these was, of course, through his command of economic theory. But this would have been far less valuable than, in fact, it has been- and will, I hope, continue to be -if he had not possessed an ability to communicate his ideas, at any level, in a way that was not only clear and stimulating but that was never, at any time, dismissive of the ideas of others. Over the twenty years of his tenure of the chair of Economic Theory members of staff, undergraduates and graduate students have all benefited from the generous way he has been prepared to give his time to discussing, analysing and clarifying their problems.
It may perhaps seem odd to stress this kind of activity: for surely this is precisely the way distinguished academics ought to behave? This is true; but, rather sadly, not all do. In this matter -as in so many others - I can, from more than thirty years' experience, assert that Ivor has given us all an example to follow, just as, thirty-six years or so ago, Ivor and I were given the same sort of example at Bristol by H. D. Dickinson and Miles Fleming.
For this example, and for his many other contributions, I offer my gratitude - secure in the knowledge that all his colleagues, past and present, join me in doing so_ And I am confident that I speak for the same wide and growing constituency when I conclude this note with every good wish for the future.
Introduction
A. INGHAM AND A. M. ULPH
The chapters presented in this volume, and the eminence of their authors, are a testament to the contribution of Ivor Pearce. Many of the authors directly acknowledge the influence of Ivcr's ideas on their own thinking. In the next few pages we shall show how the contributions are related to some of the themes that have recurred in Ivor's work. This will serve not only to introduce the chapters and illustrate the development of Ivor's ideas, but through the sheer diversity of the issues covered it will demon strate breadth of the contribution made to economic knowledge by lvor Pearce.
The volume starts with three essays in the area of demand analysis, and comparative statics. Gorman's chapter examines the Le Chatelier principle, an important feature of the Samuelsonian tradition in modern economics, the point at which lvor Pearce started. The Le Chatelier principle is widely used in both theoretical and applied economics yet it is only a partial result and Gorman shows that when considering the full economic system one obtains a general result that can be quite different and an anti Le Chatelier result. The concern with the interaction between economic agents and the pitfalls that lie where these interactions are ignored has been a concern of lvor Pearce throughout the whole of his career. A warning about obvious truths that are neither obvious nor true is an important but perhaps negative statement. However, as Gorman and Pearce himself have frequently shown, full and careful statements of the problem being considered lead to results and approaches that might not otherwise have been considered.
Ivor Pearce's concern with and use of econometrics stems from the fact that theory in itself can tell us very little about the workings of the economy. Yet theory has a vital role to play in empirical work. This point is illustrated by Simmons's chapter, which points out strongly the dangers of measurement without theory. In this chapter the danger lies of taking
xvii
xviii Introduction
demand systems and measuring complementarity without first thinking about what it is that complementarity might mean. lvor Pearce's import ant contribution to separability in the structure of utility and neutral want association was founded on using intuition to suggest appropriate restric tion and concepts, the full implications of which can be rigorously investi gated. This combination of intuition and rigour characterises all of Ivor's work. Afriat discusses one of the applications of this work. In present inflationary times probably no more important area could be thought of than the correct definition of a true cost of living index. Indeed such an index plays an important role in lvor's work on sound money as a means of controlling inflation.
The results of Simmons and Afriat relate to the individual consumer, and will not carry over to an economy with differing preferences. The problems attached to simplification that economists often use of assum ing a society of a large number of individuals with identical preferences have been pointed out both in lvor Pearce's work on community demand and on the use of total demand curves in models of monopolistic compe tition, where many models in popular use contain inherent contradictions.
The idea of studying the interactions between economic agents naturally leads to a study of international trade and the economic relationships between countries. Pearce's contribution to this area both through the factor price equalisation debate and his treatise has been immense. One of the most important aspects of this has been to caution against the indis criminate use of two-by-two international trade models and to provide results for more general models. Neary provides an important chapter that illuminates this approach. He takes a slightly different approach to the Pearcean one of asking if the results of two-by-two models hold in more general models, by asking what restrictions on the general model are needed for particular two-by-two results, in this case the Heckscher Ohlin theorem, to hold. Neary salvages the Heckscher-Ohlin, Rybczynski and Stolper-Samuelson theorems and factor price equalisation for the general model, but at considerable cost -equality of the numbers of goods and factors and no joint production or intermediate goods. Schweinberger considers the other main result in international trade theory - that of comparative advantage - in the context of an Austrian model of capital. Schweinberger illustrates well the advantages of the Pearce method of obtaining general results by carefully seeking out the condition for full general equilibrium. One problem with this formulation is that it is easy to come to the conclusion that everything depends on everything else, yet important and useful results can be derived and that comparative advantage is a general result obtainable in all general equilibrium models is one of great importance.
Introduction xix
Schweinberger's chapter neatly develops Ivor's work on international trade capital theory found in his treatise on international trade. lvor's writings on capital and growth started with his joint paper with Gabor in the Quarterly Journal of Economics, 1958. This work uses an Austrian approach and emphasises the importance of the time structure of produc tion, with the related concept of money capital. This approach is the basis for the treatment of capital in the general-equilibrium intertemporal model of Katzner's chapter. However, Ivor has been concerned not only with the perennial question of what is the appropriate concept of capital, but has also criticised the use of intertemporal welfare measures found in the work on 'golden rules' and optimal growth. This can be found in his paper 'The End of the Golden Age in Solovia', and in the context of models with exhaustible resources in his paper on 'Resource Conservation and the Market Mechanism'. In the latter context Ivor's criticisms may be partially offset by the existence of substitute inexhaustible technologies for the exhaustible resource, though even here, as Hung, Kemp and Long demon strate in their chapter, there are serious questions to be asked about the operation of market mechanisms. The final chapter in this grouping, by Bliss, still operates within the conventional optimal growth framework, but shows very elegantly how Keynes's intuitive interpretation of the famous Ramsey Rule for optimal growth can be exploited in a wide range of models to yield familiar results in a very neat way, an approach with which Ivor has considerable sympathy.
lvor has noted that his work with Gabor on money capital can also act as an accounting framework for firms, and this is pursued by Gabor in his chapter in a slightly different context - that of transfer pricing policies for firms. Gabor also takes up a theme that recurs in lvor's thinking about the way economists should approach the study of firms - the importance of looking at what firms actually do.
The final group of chapters are concerned with macroeconomic models both theoretical and empirical. The results that in general everything depeonds on everything else is a very powerful one for considering the restrictions that economic data can impose. Characteristically this is done by Pearce with great care in the definition of variables and concepts and their measurement. This causes great problems when confronted with imperfect data both in terms of incorrect concepts and inaccurate measure ment. The various chapters on the Southampton Econometric Model show this concern. Sir Richard Stone's chapter is therefore most apposite as it discusses the accurate measurement of appropriate concepts, and the unfortunate but essential idea of reconciliation.
The Southampton Model was first set out as the final chapter of Inter-
XX Introduction
national Trade and its purpose was to be able to answer important and interesting questions that could not be answered by theoretical arguments alone. The chapter by Klein and his associates reflects this interest with their attempts to test the purchasing power parity hypothesis using the LINK set of macroeconometric models. Budd and Holly argue that a number of the widely used models for forecasting the UK economy should not be used for other than rather short-term forecasts, since their speci fication of the operation of markets for traded goods is inappropriate for the medium and longer term. They argue that for these longer time horizons the Southampton Model, with its careful attention to inter national trade theory, is more applicable.
It is most appropriate that the ftnal chapter of the volume, by Victoria Chick, relates to the most recent work of Ivor Pearce. She looks at the possibly inflationary consequences of monetary growth used to finance a Keynesian expansion of aggregate demand, and concludes that economists have failed to reflect in their monetary theory the considerable changes that have taken place in the institutional structure of the fmancial system. While rejecting the relevance of the Keynesian framework, Ivor too has been concerned that the structure of the financial system, both domestic and international, has changed in ways that economists have not fully appreciated, and that it is now essential that the world's financial system should be based on the principle of preserving the real value of money. These themes, and his proposals for dealing with the problem of inflation, were spelt out in the Lecture on Economic Policy presented by lvor Pearce at the conference in January 1982 at which the contributions in this volume were presented.
1 Le Chatelier and General Equilibrium W. M. GORMAN
INTRODUCTION
I have recently been reading the papers presented at a conference on methodology sponsored by the Royal Economic Society in the home of lost causes and the presidential addresses that gave rise to it. Some of these1 are by old gentlemen like myself bewailing the conduct of the young, especially their use of mathematical and statistical arguments not heard in respectable households in the old days. As one these I have my own particular complaint: I was taught economics as a way of thinking about problems; it has become a body of theorems. Intuition and imagination have been sacrificed to precision and rigour, or so it seems to me.
Intuition and imagination are above all the qualities one enjoys in lvor Pearce's work, most particularly intuition about general equilibrium and its consequences. It is these qualities and in this context that I wish to celebrate. Ivor used them to particular effect in a famous debate with Paul Samuelson. I have chosen another Samuelsonian proposition, his Genera lised le Chatelier principle, as my text.
In discussing it I would like to praise George Duncan2 too, who intro duced me to economics as an engine of thought, and who, in particular, taught me to expect the results that I will attempt to prove, and that in one of the first lectures of the first term of my first year in Trinity College, Dublin. These results are that a firm, for instance, will increase its output by more in the long run following a given price increase, than in the short, because it has more ways of bringing it about; that the price change must have originated somewhere, possibly in a change in tastes or taxes; and that the rest of the economy will react to that initial change in such a way as to reduce its impact on the good in question, increasing
1
2 Le Chatelier and General Equilibrium
the production of substitutes, for instance, and reducing that of comple ments.
All that is rather woolly: are the substitutes in question substitutes in consumption or production? and what about cross-effects between these other goods? Such questions have to be answered if one is to be able to guess the order of magnitude of the effect in particular cases, for instance. Giving appropriate precision to such general notions is another of Ivor's gifts. It is also a skill in which modern economists score more highly than their forebears.
One final point before I get down to it. The sections that follow origi nated in lectures on 'doing economics economically' and were designed to inculcate the notion that time spent setting up a problem in appropriate terms can be well spent, and to illustrate some of the points to be con sidered at that stage. I have changed the arguments a good deal, especially in the next section but have kept to these general ideas. A good deal of what I will here say, therefore, will be old hat to the initiated; some, I fear, will be incomprehensible to the uninitiated. My apologies to both.
LE CHATELIER
Since we are interested in what happens when we change prices, they are the natural independent variables, and profit functions the natural mode of specification. From that point of view, the important fact is that the long-run profit function,
'if(p) ';;;!!: rr{p) (1.1)
the short, because whatever we can do in the short run, we can do in the long. We begin in long-run equilibrium, since the effects of past changes would otherwise be confounded with that of current, and assume it unique, since otherwise the long-run elasticity would be unbounded and there would be nothing to prove.3
At the initial prices p, then
1f (jf) = rr(jf) (1.2)
so that p = p minimises the difference 1f (p) - rr(p) between long- and short-run profits. In particular, therefore,
'if;(fi) = rr;(fi), each good i (1.3)
W.M. Gorman
both being equal, of course, to the common4 equilibrium output
Xt = 'iit(p) = tr;{P)
3
(1.4)
whose uniqueness guarantees differentiability. To examine the effect of changing prices we will have to differentiate again. Since they are convex, profit functions are almost everywhere tiwce differentiable. For simplicity, I will assume them so at fi .5 The second-order condition then yields
'I;'ii;;(fi)OtO;'~ 'I:,nt;W)Ot8;, each(} = (0 1, 02 ... ) ~ 0, (1.5)
the latter because profit functions are convex. In particular,
(1.6)
where the tildes once more denote long-run values. This is what we set out to prove.
Having found what we have been looking for, we should look to see whether there is not more to be found: that is, generalise. In doing so we may fmd something immediately useful, may discover 'why' our results hold, and failing either may find something useful to others in other contexts.
We used minimisation, convexity and, incidentally, differentiability. None depend on the axes. That suggests that we seek to interpret the general result ( 1.5) in terms of linear transformations, under which all three properties are invariant.
Linear transformations may be interpreted in terms of baskets of com modities, or, inverting, as yielding underlying characteristics, of which the actual commodities are themselves baskets. Take the first interpretation first. Let there be m commodities X = (X 1 , ••. , X m) packed in n linearly independent baskets Y = ( Y1 , ••• , Yn), of which Y1 contains a;; units of X; each i, j. Then a vector y = (y 1 , ••• , Yn) of basketfuls contains x1 = 'I;1-a1;Y; units of x1, each i, so that
x=Ay (1.7)
in the obvious notation. Moreover, the amount a11 of X1 in Y; is worth p1a11 , so that the total value of Y;'s contents is Q; = 'I;iptaif' and
q=ATp (1.8)
equally transparently, yielding the total expenditure
q.y =qry =prAy= pTx =p.x (1.9)
the total value of their contents, as one could have hoped. In general one need not have m = n, nor A nondegenerate, but that will be sufficient for
my purposes, so I will consider only that. I might not even do that were it not for the fact that the results will be useful in the next section also.
In this case we may write
B =A-t (1.10)
y =Bx,p =BTq (1.11)
in the place of ( 1. 7) and ( 1.8), so that the goods X may now be considered as baskets of basic characteristics Y = ( Y 1 , • •. , Yn), of which Xi contains bii units of Yi, each i, j, while its price Pi= ~iqibii• is just the value of the characteristics it contains. Of course some of the aii,bii will commonly be negative, selling short if you like, and some of the goods or characteristics may be free,6 because superabundant or valueless. Come to that, we have been measuring goods as net products, so that some of the xi will com monly be negative too. In terms of y and their prices the profit functions are
1i(p) = 'if(BTq) =: 'fi(q); rr(p) = tr(BTq) =: h(q) (1.12)
say, so that
in the obvious notation, as one would have hoped, and
(1.14)
Doing the same thing for the long-run profit function, and setting() k = bik
in ( 1.5) we see that
(1.15)
for composite commodites too. Indeed this is clearly the entire content of(1.5).
The content of our assumptions to date is then: increasing the price of any good increases its net supply and does so more in the long run than in the short.
Stop, look and listen: are further generalisations in order? For con firmed dualists the obvious questions to ask is: does the shadow price of a constraint perhaps fall less in the long run than the short, because the scarce resource can be used to better effect? Broadly, the answer is yes, but we need extra assumptions. While profit functions are necessarily convex in their price arguments, gross profit functions are not necessarily concave in the quantities, and at least local concavity is necessary for our
W.M. Gorman 5
proof, just as local convexity is for the second inequality in ( 1.5), ( 1.6) and (1.15).
Think of the constraints as representing quantities z = (z 1 , z 2 , ••• ) of very fixed inputs, Z, fixed that is in the long run as well as the short, and variable only in the very long run. 7 Write the gross profit functions as 1i(p,z), rr(p,z) with
1i(p, z) ?-1l(p, z) (1.16)
everywhere, and
1i (fi, Z) = rr(jj ,z) (1.17)
in the initial equilibrium, so that p = ji, z = z minimises the difference 1i (p, z) - rr(p, z) between long- and short-run profits, and thus8
(1.18)
=x; as in ( 1.3) and ( 1.4) while differentiation with respect to z8 yields
ns(ji,z)= rr8 (fi,Z) ( 1.19)
the shadow price of the s'th constraint, thus seen to be the same in the short and the long run. Finally, the second-order condition yields
"£'irst(p,z)8sfJt;;.. "Errst(ji,z)88 8t, each()
(1.20)
(1.21)
in the obvious notation, which would prove our result were 1i ss..;;; 0; while the corresponding general result would hold were
(1.22)
that is, if 1i (ji,.) were locally concave about z. Is it? There are two arguments leading to (1.22). For the first we note that
rr(fi, .) is the production function for the Hicks aggregate v = ji.x from the very fixed inputs Z. If the original technology for (X, Z) is convex, this production function is clearly globally concave, giving the desired result.9
For the second we assume that we are initially in very long-run compe titive equilibrium. If so ji, z minimise the loss of potential profit
p.x- r.z -1i (p,z) (1.23)
as in the usual proof of Shephard's lemma, yielding10 (1.18) and (1.19) as the first-order conditions and (1.20) by the second.
Of course (1.22) implies11 that Z are machines to be rented in the open market at the going prices r, which you might not be willing to require. If you are, however, you should be willing to consider composite machines, too, in the same manner as ordinary composite goods. (1.20) and (1.22) are then equivalent to the statement that the shadow price of each such com posite machine falls, when its number is increased, but falls less in the long run than the short.
Perhaps I have gone too far in avoiding the primal in these arguments. Should you find it so, I suggest you start from
in the first part of the discussion, and
S(z) 2_S(z)
(1.24)
(1.25)
in the second, in the obvious notation. I think you will find it worthwhile moving directly into the dual, using the profit and gross profit functions as I have been doing.
Perhaps I should add that the first argument goes straight through in the second model too, so that an increase in the price of a current good still increases its net supply, and by more in the long run than the short. It would be surprising indeed were a meaningful result to have been changed merely because we had mentioned a fixed parameter z explicitly.
LE CHATELIER IN EQUILIBRIUM
Setting up the Model
In the previous section I discussed the reaction of a particular sector to an exogenous price change. I called it a firm, but it might have been an industry, the productive sector as a whole or, for that matter, mutatis mutandis, a collection of households12 or of households and firms, or a country in a world economy ... The important thing is that they were price takers and the price change originated elsewhere.
In this, I close the model, including the sector where the change origi nated. It may be that consumers come to value a particular good more highly at the margin.13 Were the rest of the economy to be price takers, this would face them with a given price change and they would produce more of the good, net, than before. Were prices to stay at their new levels, they would increase production still more in the long run. But prices would not stay the same. To produce more of this good the firms would
W.M. Gorman 7
have to produce less of others net and that would face consumers with new supplies, from which new prices would emerge. I will contend that, properly construed, these interactions will go to reduce the long-run effect of the original change. 'Properly construed' means ignoring income effects. To make this possible, the change has to be only 'at the margin'. As it happens, the introduction of a small subsidy on the production of the good in question, whose costs are raised by appropriate lump-sum taxes, does the job just as well as an exogenous change in tastes or technologies.14
Since this is an oft analysed problem, the results would be more easily interpreted should we consider it.
Divide the economy, then, into two sectors: call them industry and households for defmiteness. The important thing is that the subsidy be paid to a firm whenever it sells the goods, z, to a household, or the initiat ing change be one in tastes rather than technology. Initially, I will assume the firms pocket the proceeds, adjusting their output to the new price.15
How should we set this up? In particular, in the primal or dual? With quantities or prices as independent variables?
The first stage is clearly to combine the firms into one sector, the households into another. That is best done in the dual. We just add the profit functions for the firms and the expenditure functions for the individual households, to get the corresponding functions for 'industry' and 'households' as a whole. Remember that we are going to start from an equilibrium and keep households at their equilibrium utility levels, so it is the expenditure functions at these levels that we use. Should we continue in the dual as we enter the main analysis, or switch to the primal? It hardly matters, as we will see. That is unusual: the fact that what different agents have in common are the prices they face commonly makes them the natural independent variables, and the dual formulation the simplest -just as in the aggregation we have been carrying out. In a two-sector model this is no longer an advantage. Just as both sectors face the same prices in equili brium, so one consumes what the other produces. The fact that quantities cannot be changed independently even in a single firm, because it is bound by its technology, while prices can, becomes an advantage when one good stands out from the rest in the problem itself.
It is to illustrate these facts that I will attack this problem in quantity space. In fact it would have been almost as natural and almost as easy to set it up in terms of the prices as independent variables.
Accordingly, I will represent the industry and households respectively by
z = -f(x)
z' = -g(x')
where -g(x') is the amount of the good Z whose supply we are investi gating the households require to secure the initia116 equilibrium utility levels ii, given they consume x' = (x~, . .. , x~) of X, while -f(x) is the amount industry can produce of Z, given it is producing x = (x 1 , ••• , xn) of X. Note that all goods are measured as industrial outputs and hence household inputs. Commonly, of course, many of the quantities will be negative.
Choose Z, which is in any case to be singled out, as numeraire. The supply and demand prices of X are then
(1.28)
That they should be simple derivatives, rather than ratio of derivatives, may seem a small thing. In fact we will need their derivatives in working out the effect of the subsidy; the formulae that results will be consider· ably simpler because of (1.28); that in turn will make it easier to see how to continue the analysis, and considerably easier to interpret the results. It is a trick to be remembered when one good is in any case to be treated differently from the rest, even when it is to be singled out for a price change as here- which hardly seems natural for a numeraire! In fact, what we do, of course, is to level a small ad valorem tax on the other goods. returning the proceeds, lump sum, to the households that paid them, rather than subsidising Z from lump sum taxes. What is normally a defect of the primal, that we cannot change any quantity independently of the others even in a single sector, has become an advantage.
Now consider our problem. Initially the prices are p, output and consumption both x, say. Immediately after the tax, prices are still p in the household sector, but p /c, say, in the industrial, where c > 1. Initially then
[j(x) = Ki(x) = Ji; (1.29)
(1.30)
and in the long-run equilibrium
Ki(x) = p; = p;/c = fi(x)/c (1.31)
a tilde again signifying 'long run' in the appropriate sense. To analyse these
W.M. Gorman 9
effects we will need to differentiate again, so that everything will depend on the gradients:
(1.32)
F= iftj(.x)], G = [gi;(x)] (1.33)
of the negative' 7 industrial production and household requirement func tions f( . ), g( . ) at x. Now it is always possible to reduce two symmetric matrices to diagonal form 18 by an appropriate proper linear transforma tion. Doing so will get rid of a complicated cross-relations between the X's that I mentioned at the end of the introduction, and effectively reduce the problem to the classical one of the incidence of an indirect tax in partial equilibrium, determined by the relative elasticities of supply and demand. All this, of course, in terms of the transformed or basic goods, Y, for which the household requirements and industrial production functions are locally additive. 19 This will be spelled out in the following subsection; in the subsection after that, I will return to the natural goods X and see what we can say about the behaviour of their prices in the light of what we will learn in the next subsection.
First let us prove that F and G can be diagonalised by a single trans formation, something known to mathematicians for a very long time, but not commonly used by economists.
I will assume the initial equilibrium x unique given ii, for the same reasons given above on p. 1. Since industry has maximised its profits ji.x- f(x) there, f( . ) is locally convex. For simplicity I will go further in assuming it locally strongly convex,20 that is
(1.34)
We may therefore reduce F to a strictly positive diagonal form by an orthogonal transformation, and to I by changing units. Do so, and apply the same transformation to G. Now reduce the transformed G to diagonal form by an orthogonal transformation, which of course leaves Fat I. Note that the households minimised the expenditure, p .x'-g(x'), required to achieve ii, at x. Hence g( . ) is locally concave and
I.JTGI.J ~ 0, each 1.J (1.35)
so that it is transformed to positive, though not necessarily strictly positive, diagonal form.
I suggest that this trick would be a useful one in many two-sector models in which the effect of small changes are to be examined.
The Model in Terms of Basic Goods: Exploiting Local Additivity
Put F and G into diagonal form, then, by the linear transformation
y =Bx,p =BTq (1.36)
z- -z =- {kilt(yi- .va +-!k(yj- .vY} +... (1.37)
z'- z =- {ki[;(y{- Y;) --!kai(yt- Y;)2 } +... (1.38)
which represent the industrial and household sectors respectively, where
a1 ~ 0, each i (1.39)
by (1.35) is the local convexity condition on tastes in the neighbourhood of the competitive equilibrium (Y, Z).
I will use this model to show that an exogenous change in tastes, leading to households fmding that one extra unit of Z, at the margin, does the work that
c>1 (1.40)
did before, will indeed lead to an increase in the output of Z, both immed iately21 and in the new equilibrium; that the latter will be the lesser, but an increase nevertheless, that, if we measure Y as industrial outputs, and their prices in terms of Z as numeraire, their prices will fall by
-d'if; = P;i/;dc = - Ptdqi > 0, each i (1.41)
in equilibrium, where the tilde once more stands for the 'long run' effect, conveniently interpreted, while the absence of a tilde from dq1 implies that it is the immediate impact effect of the small change inc from c = 1. Here
(I .42)
where
(1.43)
are the elasticity of supply and compensated elasticity of demand, respec tively, for Y1 at the initial equilibrium. This is the old formula for the inci dence of an indirect ad valorem tax at the rate 100dc per cent. In case it should have been dropped from the current curriculum I illustrate the argument in Figure 1.1, based on the fact that the slopes of the demand and supply curves at (Y, i[) are numerically proportional to the elasticities o1 and a1• Note that this is a partial equilibrium argument, neglecting inter-
W.M. Gorman 11
FIGURE 1.1
actions between the Y. It works here because we have effectively demo lished them by using basic goods and local additivity, a point to which I will turn in a moment.22
Since the price of Yi falls by 0 <Pi..;;;; 1 times as much in equilibrium as it did initially, so does its output. That is
(1.44)
which explains why the production of Z increases by less in equilibrium but nevertheless does increase.
I have been measuring all goods as industrial outputs. Trivially, there fore, each Yi is a substitute for Z both in industry and in the households, in the initial equilibrium.23 Had I measured the Y as inputs, but Z as an industrial output, each would have been a complement for it24 instead, a fact that should be borne in mind when interpreting the results below. In fact, the Y are substitutes for Z in both sectors, the -- Y complements, in an obvious notation. Nevertheless, the Y are unambiguous substitutes for Z and their prices have fallen, not risen as I had given you to expect in the Introduction. This is really a quibble, based on my insistence on keeping Z as numeraire. Had I allowed its price to rise to c > 1 as the logic of that statement required, qi would have become q{ = c{ft, where
dq{ /qi = aidc/(ai + oi) > 0 (1.45)
as claimed; how much greater depending on how much better a substitute, one might say25 , Yi is for Z in industry than in the households. It never increases as steeply as the price c of Z itself, however. Before introducing
the change in tastes explicitly, let me say a little about the locally additive formulation (1.37) and (1.38) in general, remembering that it is always available through transformations like ( 1.36).
First differentiate (1.37) and (1.38) with respect to Yt and yj to get
Yt - Yt = q; - lit
Yt- Yt = - (q{ - li;)/a;
(1.46)
(1.47)
to the first order, for the supply and demand equations. Note that the cross-elasticities between the Y vanish at the initial equilibrium, which is why the partial equilibrium analysis in Figure 1.1 worked. This is the key to the use of basic goods in general.
The supply and compensated demand elasticities
at= litay;/y;aqt = lidYt
( 1.48)
( 1.49)
(1.50)
by the local convexity condition (1.39). Since we have measured they as industrial outputs27
lit> 0, each i (1.51)
the common sign of a; and o1 is therefore that of y1, which may well be negative.
Now for the change in tastes. It replaces ( 1.38) by
c(z'- Z) =- p:;litCYi- Yt)- 1-LatCYi- Yt)2 } +... (1.52)
reflecting the fact that one extra unit of Z at the margin now does as much work as c > 1 did before.28 Note that the change is only at the margin; (Y, z) is still just sufficient to sustain the initial distribution ii of utility ,29
which is important because it enables me to neglect income effects. The demand price for Z might now be said to be c > 1 at the original equilibrium (Y, z), instead of the initial
c=1 ( 1.53)
and it is to this change that industry reacts. However, I will hold to Z as numeraire, and say that the demand price for y; becomes
Q; = li;/c (1.54)
initially instead, while the demand curve (1.47) for Y; becomes
W.M. Gorman
Yt- Y; =- c(qj- i[;)/a;, each i
to the first order. Faced by the prices q, industry3 0 produces
Yt = Yt + Q;- ift = Yt- q; + ii;jc
13
(1.55)
( 1.56)
units of Y;, each i, again to the first order, so that, using (1.46) too,
dy; = dq; = -q1dc < 0
as I claimed.
( 1.57)
(1.58)
So much for the impact effect. Next look at the new equilibrium where _, - _, - _, -q =q;y =y;z =z
With ( 1.46) and ( 1 .55) this yields
0 > d(f; = -p;i[;dc = p;dq1
= dy; = P;dY;
(1.59)
(1.60)
(1.61)
(1.62)
as claimed in, and just above (1.44). That is, the prices and production of the basic goods Y do indeed fall by less in the final equilibrium than immediately following the change in tastes, on our assumptions, but fall nevertheless and to a degree predicted by the old-fashioned, partial equi librium theory of tax incidence.
The equilibrium change in the output of Z
(1.63)
where
(1.64)
as a weighted average of the p;, each in (0, 1). The weights, q f, are pecu liar: they take that form because I reduced F to I rather than some other strictly positive diagonal matrix. In an earlier draft I presented {1.64) in two more easily interpreted forms. However (1.63-1.64) really speak for themselves, and space presses, so I have dropped them here. Broadly speak ing, what the formula tells us is that the output of Z increases in equi librium, though by less than it does initially ,32 a good deal less if the basic
goods are notably less good substitutes for Z on the demand than the supply side, only a little less if the opposite is true.
I leave the interpretation to you. One final word, though. It was the production of substitutes rather
than their prices that I now see I claimed would increase in the new equili brium. That was, compared with what it was immediately after the change, of course: that is
dYi- dyt =(pi- l)dyi =(I - Pt)?fidc > 0 = ai?fidcj(ai +81) (I.65)
by (I.6I), (I .52) and (I.62). QED
Natural Goods: Interpreting Their Price Changes
Remember that we have been measuring all goods, X, Y and Z itself, as industrial outputs initially, so that their prices are strictly positive33 then, and may therefore be normalised to unity. Equation (1.36)
Pt = '"E;b;iQ;, each i
(1.66)
(1.67)
Despite (I .66) the prices of the natural goods x in (I.36) are not weighted averages of the prices of the basic goods y they contain, since many of the b;i will often be negative. When that is strictly so, x1 contains b;1 < 0 units of Y;. or, equivalently
c;i: = -b;i > 0, when b;1 < 0 (I .68)
units of -Y; in the obvious terminology, introduced in the previous sub section, where it was mentioned that - Y;, as an industrial inpue4 and household output, is complementary to Z in both roles. Hence the 'c' in (1.68).
(1.69)
in the same strain, since x1 then contains s1;;;.. 0 units of Y;. which is a substitute for z.
Defme
(1.70)
W.M. Gorman 15
as the total value of the complements of, and substitutes for, Z in X1 res pectively at equilibrium prices. (1.66) then becomes
St = 1 + e1, each i (1.71)
reflecting the Hicksian dominance of substitution, or, more precisely the fact that the X, like Z are deftned as industrial outputs.
e1 and s1 are some sort of measures, then, of the importance of Xi as a complement of, as compared with a substitute for, Z. The other appro priate measure is presumably how close substitutes are the Yj such that
jeS(i) = { i :bit;;;. o} (1.72)
for Z on the average, and how strong complements the - Y1 such that
jeC(i) = { i : b11 < o} (1.73)
on the average too. Now (1.36) yields
dff1 = 'T.;bJid'iiJ = T.1b;iP;.de
= {xi- e1(p.1 - X1)} de, by (1.71)
where (1.74)
(1.75)
are the obvious averages to use, given that we take the strength of a relationship to mean its relative strength
(1.76)
on the demand as compared with the supply side, as we did in the previous subsection.35 Remember that 61 and o1 both take the sign of iJ and so may be considered measures of substitutability or complementarity on their respective sides, according as iJ is positive or negative; if you like, in our case as to whether jeS(i) or C(i).
I will not enter into any more detail on the matter here. Should you be interested, have a go.
A similar analysis may be applied to the changes dX using
Xt = 'T.jatiYi (1.77)
where A =B-1• Since (1.66)maybe writtenBTe =e, where e = (1 ,1, ... 1)T, e =ATe, too and hence
(1.78)
the direct analogue of the useful (1.66). Of course aii is here the amount of x; per unit of Yi rather than vice versa. These are slight complications. I leave the details to you again. The chapter is already overlong.
NOTES
1. As an old gentleman myself, I agree with a good deal of what they said. I wish we knew more about the other social sciences, for instance, were more deeply immersed in history, and more often knew the mathematics appropriate to our tasks. Given that we are not very bright, that means that I would like us to come from a greater variety of backgrounds. The modern American graduate school with its implicit assumption that there exists a platonic ideal training for economists, to which all actual programmes should approximate, is my bogeyman.
2. A man in many ways like Ivor, who might have become just as distin guished had he known more mathematics. He could not make head nor tail of the accelerator: but taught us about what have come to be known as Arrow-Debreu goods in one of his first lectures.
3. Not quite accurate, but not, I think, misleading. Consider it as an aside.
4. Note that this is a conclusion: the assumption was that short- and long-run profits were initially equal. This is more important in the discussion of the shadow prices of constraints below.
5. Otherwise some of the elasticities would not exist, of course. 6. And that either at isolated points, or, for instance, identically, because
nobody wants them, or they cost nothing to produce. This is particu larly so of characteristics. I have made extensive use of this fact elsewhere, but am going effectively to ignore it here, to keep an already complicated paper from becoming more complicated still.
7. This at the most in a convenient normalisation. Remember that x ;;.a, x 3 ;;oa3 and 3yx;;. 3ya are all equivalent statements, which do not transmit convexity or concavity. In general, then, there is no basic reason to assume either. It depends on the problem and its formulation.
8. Results again, not explicit assumptions. 9. Of course, the assumption is wildly over-sufficient in general, for given
p, that is. 10. Shephard's lemma is that x; = n;(p), for instance, and corresponds
precisely to Hicks's proposition that x; = 3mj3p;, the change in money income required to compensate for a small change in price. The simplest proof is to notice that p.x..;;; 1T(p) if x is feasible, while p.x = n(p), if it maximises profits at prices p. Hence p = p mini mises the waste n(p)- p.x, which yields x; = 7T;(P) and L1T;j(p)8;8i ;;. 0, given sufficient smoothness.
11. 'Implies' is too strong. 'Is suggested by the notion that' might be better.
W.M. Gorman 17
12. We would have had to ensure that the households remained equally 'well off' or, in a different formulation, that their 'marginal utilities of expenditure' were unaffected in an appropriate normalisation.
13. 'At the margin': to avoid income effects. 14. When it came to the point, I decided to model the change in tastes
after all. It is rather more revealing and I believe that interested readers should be able to model the subsidy themselves, once they have seen it.
15. An assumption that is possibly more acceptable when tastes change exogenously, facing each individual producer with stronger demand, with only his current production up for auction, than following a public subsidy.
16. That is not to say that the households remain as well off as before. They may be exuberant or miserable at the change: that is a matter of how we normalise utility before and after it.
Note, too, that only the first and second-order terms matter; the rest may change in quite a different manner.
17. Look at (1.26) and (1.27). Basically the minus sign is needed because I measure all goods as outputs.
18. The argument that follows turns on one of them being positive defi nite; but that is really beside the point. It is convenient because almost everybody knows about latent vectors. Even then, it would have been better to have applied it to P.G rather than F, since it is fg which is strictly convex at a unique equilibrium. However, it would not have affected the argument appreciably.
19. To the second order, which is all that often matters in comparative statics.
20. See note 19. 21. I assume a very short-run Walrasian market with supply predetermined,
so that it clears at the demand price. Note the absence of any explicit dynamics.
22. And which is the key to the value of this transformation both here and in other applications, I feel sure.
23. Think of the input - Y; in an obvious notation instead. It is used to produce Z in a fictitious plant and on its own. More will be needed to produce more Z and it will cost more. - Y; is a complement for Z.
24. Perhaps it is best to think of Z as leisure and -Z as homogeneous labour, the only input. People become fonder of leisure at the margin in my problem.
In general we may think of the economy made up of n separate sectors, each with its own households and firms. With full additivity, the households in sector j consume the Y; produced in its firms, and only this, though they may work in others. In general, some will have to, according to the distribution of profits.
25. In a distinctly lax use of words, whose only real justification is that Y; is related to Z and Z only, and as a substitute. Its elasticity may be thought to arise from this substitutability alone. It is always a prob lem how to get a unit free measure of substitutability.
26. Y; might vanish. The appeal of elasticities outweighs that considera tion. It is just a matter of words.
27. We have been maximising z + q.y. 28. Forgive the use of 'work'. 'Is just as good as' might be better. See
notes 12, 13 and 16 also. 29. Seenotesl2,13,16. 30. See notes 15 and 21. 31. dz = 'i:.ozfoy1.dy1 evaluated aty in (1.37). 3 2. On my definition of the impact effect as the result of market clearing
at the existing rate of production. Note the absence of an explicit dynamic. In practice people might look forward more percipiently; more probably in the case of a new publicly declared subsidy, than for changes in private tastes.
33. 'Strictly' positive is really an assumption. It is frequently convenient to assume that relatively few shared characteristics underlie the demand for a lot of goods. It leads to infinite elasticities of course and so would not be convenient here!
34. In an obvious notation. 35. Where it was made more acceptably by the analogy with the classical
theory of tax incidence. Better perhaps to have invented a name: 'relatively strongly related on the demand side' perhaps, with 'rela tively strong substitutes' and 'relatively strong complements' as the two cases? See note 25.
2 A Complement to Pearce on Complements
P. SIMMONS
There are almost as many measures of substitutability/complementarity as there are economists working in demand analysis - indeed, perhaps more since several eminent economists (Hicks, Samuelson, Pearce) have more than a single measure. There are cardinal measures based on the direct utility function (Auspitz and Leiben, Edgeworth-Pareto ); on the indirect utility function (McKenzie) and ordinal measures based on behaviour of the direct marginal rate of substitution (Hicks-Allen) and of the marginal rate of substitution of the expenditure function (Morishirna). There are measures based on properties of the compensated demand functions (Hicks-Allen, Pearce, Samuelson); on properties of the inverse compen sated demand functions (Hicks) and on the ordinary demand functions. Typically these measures all involve some common elements: they are defined locally in terms of partial derivatives of the relevant functions; they involve only pairs of goods but beyond those factors they have little in common. One of the reasons for such a plethora of measures appears to be that economists have not really decided what it is that they want to measure. At least five strands of argument can be found:
1. Substitution/complementarity relations should reflect the deviation from proportionality in the compensated demands for a pair of goods as prices vary.
2. Substitution/complementarity relations should reflect the deviation from proportionality in the uncompensated Marshallian demands for a pair of goods as prices vary. Pearce writes 'complementarity in con sumption should be reserved to describe goods which tend to be required
19
in fixed proportions in all circumstances of prices and income' (Pearce, 1964, p. 136).
3. Substitution/complementarity relations should reflect the deviation from proportionality of the compensated willingness to pay for a pair of goods as prices vary.
4. Substitution/complementarity relations should reflect the deviation from proportionality of the uncompensated willingness to pay for a pair of goods as prices vary.
5. Substitution/complementarity relations should reflect the movement in the relative budget shares of two goods as prices vary.
6. Substitution/complementarity measures should reflect some measure of the 'curvature' of consumers' preferences in dimensions of pairs of goods when preferences can be described in a variety of ways, for example, direct or indirect utility function, expenditure function, distance function.
We will refer to the flrst four notions as behavioural concepts of comple mentarity. This chapter argues that the measures that exist for those purposes are not wholly satisfactory either because they are not unit free, are not symmetric or unnecessarily restrict the price or quantity variations that are allowed. But using an approach of Pearce it is possible to find measures for each of the behavioural concepts that do not have these disadvantages and that are defined by a common operator with a striking interpretation: loosely it is the maximum observable variance in behaviour across the pair of commodities. This operator is also applicable to any group of commodities not just pairs.
THE EXISTING MEASURES
Let x1, p1 i = 1, ... , n refer to quantities and prices respectively; m to income, u to utility level. v(x) is the direct utility function; v(p,m) the indirect utility function; g(p, u) the expenditure function; d(x, u) the distance function and h(p, x) = g(p, u(x)) which might be interpreted as the Slutsky compensated expenditure function which is central to one of Samuelson's measures. No concern will be given to regularity condi tions; each of these functions is assumed to exist, be twice continuously differentiable and strictly monotonic in the appropriate variables and
direction (e.g. ag >O, ~ < 0) at all points for which we wish to defme apt apt
a measure of complementarity. Subscripts to functions denote partial
P. Simmons 21
(i) the compensated demand functions x; = xi(p, u) = g;(p, u);
(ii) the Marshallian demand function x; = ~i(p, m) = _ '!_;__(P ,!!!)_ ; V0 (p, m)
(iii) the compensated willingness to pay P; jm = qi(x, u) = q;(x' u) d(x, u)
(iv) the uncompensated willingness to pay Pi= tJ/(x, m) = A.v;(x).
Each of these functions is assumed to be continuously differentiable. Table 2.1 is designed to catalogue existing measures and their properties and to state the proposed measures. The existing measures are largely designed to answer different types of question. A relatively weak question would be to find some observable aspect of behaviour, a, in pairs of goods, F(i, j), such that if F(i, j) >a then i and j are substitutes while if F(i, j) < a then i and j are complements. A stronger question would be to use the measure F(i, j) to define an ordering of the degree of complementarity among pairs of goods so that i and j are more substitutable than k and I iff F(i, j) > F(k, 1). It is evident that to be useful for either purpose the classification procedure should be independent of the units of measure ment of prices or quantities. For the weaker approach where a is commonly taken as zero this requires that the sign of F(i, j) should be unit-free; for the stronger question an arbitrary variation in the units of measurement for different goods should generate a common monotonic transformation in F(.) for any pair of goods. The first necessity for a notion of comple mentarity/substitutability is that it should be unit-free in this sense. Of the measures in Table 2.1 all bar (e) have a unit-free sign while only (c) and (d) have a unit-free magnitude.
The second requirement for a measure follows from the view that complementarity refers to proportionality of some aspect of behaviour such as demands or willingness to pay for a pair of goods. The measure should thus reflect the degree of proportionality of the relevant aspect of behaviour under all possible variations of the independent variables and not just proportionality in the face of some restricted variations. Otherwise we could find circumstances under which the degree of com plementarity predicted by the measure was quite different from that observed.
A third requirement is symmetry of the measure between goods: the
T A
B L
E 2
P. Simmons 23
degree of complementarity between i and j should equal that between j and i. This is normally regarded as a very weak requirement.
Consider (a), (b) and (c); these all refer to movements of some measure of the compensated demands for goods i and j when the prices of these two goods vary. In a world where only those prices could vary and any price variation was automatically compensated this would tell us some thing about how consumption of the goods might vary. But of course neither of these conditions may hold; it is quite possible to construct cases in which two pairs of goods have identical degrees of complementarity according to any one of these measures but in which the consumption of the four goods move quite differently due to the variation in other prices and/or income. So to measure variations in consumption of goods we should at least take account of all price variations that are possible. Mea sures (a) and (b) also have a further drawback related to their dimension dependent nature; their magnitude but not their sign vary with the units of measurement of price and quantity of the different commodities. Measure (c) has both sign and magnitude unit free. So using (a) or (b) it would be possible for goods i and j to be more complementary than k and l with one set of units of quantity and price measurement but less complementary with another set of units. Finally it is evident that (c) is not symmetric between i and j although symmetry is a fairly minimal requirement for a property of a pair of goods to possess, indeed the requirements for symmetry are quite restrictive {Blackorby and Russell, 1981). These considerations suggest that (a), (b) and (c) do not perfectly fulfil the requirement of measuring the manner in which the consumption of a pair of goods may vary.
Consider the 'inverse demand' measures (f) and (g); Deaton {1979), and Deaton and Muellbauer (1980) appear to regard these as equivalent. However this does not appear to be correct. Direct calculation shows that2
i,j=l, ... ,n-1
i,j=l, ... ,n
where subscripts denote partial derivatives, in the first case at x and in the second at x/d(u, x).
Apart from the fact that the former is defined only fori, j =I= n while the latter is defined for all ij, these two expressions need not even agree in sign let alone in magnitude. One way of seeing this is to note Samuelson's point that the sign of the first expression can vary with the choice of the particular commodity that is to serve as the nth numeraire commodity (Samuelson, 1947). This is indeed a telling objection to the first criterion; it arises from requiring the variations in willingness to pay to be restricted to compensating variations in a circumstance in which there is no unique means of compensation. d;; itself seems to be quite a good measure; while not unit free its sign is independent of variations in units in the same way as measure (a). Again like (a) to which it is naturally dual, it suffers from failing to reflect all possible variations in the willingness to pay and con centrates only on those compensated variations arising from variations in the ith or jth quantity. Hence a variation in units of measurement for four commodities or a suitable variation in quantities other than those of the four goods can lead to a reversal of the degrees of complementarity of two pairs of goods. So we conclude that neither of the inverse measures is fully satisfactory in describing the degree of complementarity between a pair of commodities.
Neither (h) nor (i) are very closely related to consumption behaviour of goods i and j- it would be possible to have any variation of x;/x; with a given measure for either (h) or (i). Both were introduced as a means of overcoming the cardinal nature of (e) and presumably try to quantify the same notion of complementarity as underlies (e). The latter is based on what Samuelson refers to as introspective notions of how one's marginal utility behaves and both (h) and (i) seem to have this end in view. In fact we will argue below that (h) can be related to variation in willingness to pay and (i) to variation in the Marshallian demands.
ON PEARCE'S MEASURE OF COMPLEMENTARITY
From the above it is evident that measures of complementarity corres ponding to the behavioural concepts of proportionality of consumption of a pair of goods or proportionality of some measure of willingness to pay for a pair of goods are failing either because they are dimension dependent or because they do not take into account all possible varia tions in the relevant proportion. If we return to the rationale for Pearce's measure but adopt a slightly different application then we can deduce measures for either concept without those disadvantages.
P. Simmons 25
Pearce's approach was to consider x1 = g1(p, u) and two price displace ments
p' =(PI,··· •Pi-1 Pi+ t::Pi+!, · · · ,pn) II ( P = P1, · · · .P;-1 P; + EP;+1, · · · ,Pn)
with corresponding demands x'1 = g1(p', u); x"; = g1(p", u). His measure of complementarity between goods i and j was then inversely related to a measure of the angular distance between the two-dimensional vectors (x'; - x;, x'; - x;) and (x"1 - x1, x"; - x;). In Figure 2.1 l::i.ix refers to the former vector and t:/x to the latter vector, Pearce's measure is based on the angle 8. Apart from the difficulties of using 8 mentioned above, the particular formula used by Pearce to represent 8 has the difficulty that the denominator may vanish so that his measure becomes undefmed. He seems to state that this can occur only if there are two goods or if all substitution effects are zero.
x*
x,
FIGURE 2.1
In fact it appears that the denominator may vanish in more general cases than this. For example, suppose that g(p, u) =A (pj;, u) + b1p; + b;P; + P?P}-01 where Pi; is a vector of all prices except the ith and jth and A ( ·) is homogeneous of degree one in Pi;· Then the denominator of his expression vanishes but the compensated demands for i and j are only restricted to lie on a curve and are not proportional. Interestingly
enough in this example we might think of there being quite a lot of sub stitutability between i and j but very little between i and j and all other goods. The condition for the denominator of the Pearce measure to vanish is recognisable as the condition for the projection of the cost function on to the i.j plane to be a developable surface (Courant and Hilbert, 1962) that is, K;(p, u) = h(gi(p, u)) for some function h( o ). But then any cost function of the form
g(p, u) =A (pi;. u) + o:(p, u) Pt + w(o:(p, u)) P; + v(o:(p u))
where o:(p u) and w( •) satisfy 0 = Pt + w'(o:)p; + v'(o:) will generate the desired restriction with w( •) = h( • ). So in this sense there is quite a wide class of cases for which Pearce's measure is undefmed.
Nevertheless by going directly to what we want to measure - the ratio of compensated demands of i and j - we get a strong insight into the type of measure that will do. Select any measure of the ratio of consumption of the two goods variations in which will be unit free when prices vary. Then a lower bound to the degree of complementarity between the pair of goods is given by the variation of this measure taken in the direction of price change so that it is maximised. This yields the maximum local devia tion from proportionality of the two goods. For example, transform the compensated demand functions to
logxt = f (logpl, ... , logpn, u)
This has directional derivative
d1 - ~ alogx, . d' ti ( ) ogx1 -~ ---zkm rrec on z1, ... ,Zn
k alogpk
A measure of deviations from proportionality is given by the length of the vector (dlogx,, dlogx;)- "JJ.e where e is a two-dimensional vector of units. Selecting the square of the Euclidean norm the bound on complementarity is given by
max min II dlogx1 - "JJ., dlogx;- Xll 2 = (~ alogxt zf -X*) 2 Jl.zJI=I 1.>0 alogpk
+ (~ alogxi zf _ "A.* ) 2
P. Simmons 27
and substituting X* out of the objective function yields
min II dlog Xj- A, dlog X·- XII= -t (~(alogx~_- alog.xi_) zk) 2
i\.;;>o 1 3logpk 3logpk
Neglecting inessential constants our measure then becomes
llzll=l ~ -- - -- zk max ( (alogx1 3logxi) ) 2
3logpk 3logpk
Note that this is exactly the same measure as would have been achieved by starting from log x1/x; as an indicator of the ratio of consumption of the two goods and then considering normalised price variations that maximise the variation of this measure. Choosing the optimal zk leads to
3logx1jxi 3logpk
z;: = [~ logx1jxi 2] i k 3logpk
at the maximum and a measure for the degree of complementarity of i andj as
= ~ (gik - ~) 2 p'fc k gi gi
This is an upper bound for the local variation from proportionality of x1/x; and so represents a lower bound for the degree of complementarity. In geometrical terms it can be related to Pearce's measure. Let x* be the vector of compensated demands for i and j defined by
g_~ogx1fx~ x 1 = x; + ~ _a _x_1 _ __a_lo_,g::_P_,k'-c-~
k 3logpk [~ 3logx1jxi 2Ji k alogpk
= x; [1 + ~ ~!?gx~ x:] k dlogpk
F;} is represented by the length 8 being the maximal length projection of x* on x by a normalised percentage variation in prices. It is related to the difference in the angles a, 0 and is a measure of the greatest angular varia-
tion in x;/x; possible from price variations. Of course other measures are possible; for instance one might take the Radner definition of the angle a - (j. Then one would normalise x* and x to have unit length and consider the Euclidean length between the normalised x and x*. This is denoted 'Y in Figure 2.1. Similarly one could start from the beginning with another measure of proportionality such as the Radner angular distance, calculate the maximum distance variation from unit-norm price variations and treat these as a lower bound for the degree of complementarity. However it seems peculiarly intractable.
What sort of properties does our measure have? It is symmetric between pairs of goods and it obviously ranks pairs of goods in their degree of com plementarity in the same way as the possible observed variations in x;/x;. Of course a potential problem here is that goods i and j may have greater possible variation from proportionality than say k and I but the particular price variations involved are not observed. Hence we may actually observe a greater variation ink and I than in i and j. To cope with this one could adopt a partial ordering and require that i and j are more complementary than k and I only if the deviations from proportionality are smaller for all possible price changes. In the context of the proportionality measure log xdx; this would allow us to say that i andj are more complementary than k and l only if
I gis _ fu I < I gks _ gzs I g; g; gk gl
for all s. This is then likely to be a very partial ordering and so oflimited use.
There is not really any natural notion of independence or dividing line between substitutability and complementarity. Partly this is a reflection of using a measure that depends on xdx;. Indeed Pearce argues that inde pendence itself is not a meaningful idea and that substitutability is best regarded as the inverse of complementarity. One might say that goods are independent if the variations in their demands for different price variations is unrelated perhaps in the sense of having a low correlation.
The idea here is that movements in the consumption of one commodity impose no constraint on movements in the consumption of the other commodity. However this does not seem to lead to a great deal. Similarly, substitutability is merely the inverse of complementarity in this approach.
The measure does relate to others proposed. Consider the Morishima measure of complementarity:
a;; = ¥Ji P; - g;;Pj gi gi
P. Simmons 29
J J gl gk i
The proposed measure represents a measure of the distance between the vector of Morishima elasticities of substitution for the jth and for the ith goods

Demand, Equilibrium and Trade: Essays in Honour of Ivor F. Pearce

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