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Neoclassical General Equilibrium Theory as a Source of Powerful Concepts, Although Confronted with the Complexity of Market Economies

La théorie néoclassique de l’équilibre général, source de concepts puissants, mais confrontée à la complexité des économies de marché
Edmond Malinvaud
p. 3-14

Résumés

La théorie de l’équilibre général est un instrument d’analyse des économies de marché. La grande variété de concepts utilisés en théorie de l’équilibre général découle de la complexité des économies de marché. Dans cet article, je propose un aperçu de quelques étapes importantes du développement de la théorie de l’équilibre général. Je montre ensuite que les théoriciens qui utilisent des modèles d’équilibre général parviendront rarement à des conclusions sans référence à des observations quantifiées.

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Notes de l’auteur

The material of this article was presented at the Paris 2007 conference “General Equilibrium as Knowledge”. The author wishes to acknowledge the remarks by two anonymous referees. The editors thank Maxime Desmarais-Tremblay for technical assistance in the preparation of this article.

Texte intégral

1Three sentences will summarize the simple message of this article. First, the theory of general equilibrium was built in order to provide not a justification of capitalism but rather an instrument for the analysis of market economies. Second, indeed neoclassical general equilibrium theory is an essential part of our discipline because it provides a wide range of concepts, which are all essential for our reflections about such economies. Third, the multiplicity of concepts follows from the complexity of these economies, which is also responsible for the complexity of the theory. Hence, applications of the theory are challenging: definite conclusions would seldom emerge without precise reference to quantitative observations.

2My talk will have four parts. The first one will be focused on my maturing as an economist during the forties and fifties. The second one will be on Gérard Debreu’s contributions to general equilibrium theory (it seems to me that this particular selection provides an excellent observatory for the achievements reached by the axiomatic, purely mathematical, approach to the theory in question). In the third part I shall consider the more or less marginal topics on which I focused my attention within the same research program. The theme of the fourth part will be an illustration of the type of research that may put order in the more empirically oriented applications of equilibrium theory.

1. The Allais seminar and the Cowles Commission

3In my youth in the 1930s in the city of Limoges, I became familiar with industrial competition from abroad, unemployment, the struggles emerging from the working class and the presence of poverty. As I reached the age of 18 in 1941, awareness of such facts clashed with my discovery of the economic discipline, to which I was somewhat exposed in my occasional study of law, on the side of a main study of mathematics and sciences. Up to 1947, as the war was going on, I read a good deal of economics in such books as the Histoire des doctrines économiques by Gide and Rist (1944) or those dealing with business cycle facts or still with a wide range of theories. Although I had begun to also take courses in economics, my readings were very eclectic and my understanding similarly so.

4Meeting Maurice Allais at my age of 24 was a first great chance, because he immediately provided me with what was necessary to put order in my knowledge. He had just published two books in French on economic theory (Allais, 1943; 1947), which he had worked out himself in the relative isolation of the war. He was then starting to organize a seminar where young students would be working with him and helping him to be aware of the recent literature on economic theory. We were soon organized as a team of fewer than ten active members, inspired by the same research spirit as was found among young mathematicians, or other scientists in France at the same time. I shall just mention the two close to whom I would most work during the following years: Marcel Boiteux and Gérard Debreu.

5My second great chance was that Allais obtained for me a Rockefeller grant to work at the Cowles Commission in Chicago. From June 1950 to July 1951 I would share the life of an excellent international research group, which had been moreover revitalized in 1943, in order to work on a number of issues belonging in particular to the methodology of econometric inference and to general equilibrium theory. My visit was rather short, but it served to build tight links with many others, including by the way John Chipman as well as Gérard Debreu. We would often meet later in Europe or in the US. My duties in France would even permit me to accept two visiting professorships in Berkeley, for seven months each, one in 1961, the other in 1967.

2. Gérard Debreu’s contributions

6The neoclassical general equilibrium theory had developed since Léon Walras’ book (1874). An important addition to it was brought by Vilfredo Pareto, showing in 1909 that, in some sense, a competitive equilibrium was efficient, i.e. “Pareto efficient’’ as it was later labeled. Work on the theory in question was even particularly active during the 1930s and 1940s with such books as Hicks (1939), Value and Capital, Allais (1943), Samuelson (1947), Foundations of Economic Analysis, all tending to the emergence of a standard formalization with supplies and demands of producers and consumers. But, in one respect the mathematical systems that had been worked out were lacking in the rigor that pure mathematicians were then requiring: economists had stopped at checking that the number of variables to be explained was equal to the number of independent equations, which was notoriously insufficient to prove existence of an equilibrium. A few mathematicians meeting in Vienna, such as Wald and von Neumann, had explored ways to bypass the difficulty for quite particular models. Actually, it was the beginning of a period in which mathematical rigor was going to penetrate the theory of general equilibrium. As a graduate from Ecole Normale Supérieure, Gérard Debreu then realized such was going to be his job. I shall record the results in four points.

7Certainly, Debreu was not the only one to endorse that program. I could mention a list of others. But he was so typical and I knew him so well that it is convenient for this part of my presentation to restrict my attention to him. He contributed to many fundamental issues in general equilibrium theory: existence of equilibrium, robustness of the competitive equilibrium and specificity of excess demands. The unity of his project was to build a mathematics of economies

8The problem was to exhibit a set of fairly general sufficient hypotheses, which would imply this existence. The first result in this respect was published in 1954 in Econometrica jointly by two authors (Arrow and Debreu, 1954), with the title “Existence of an equilibrium for a competitive economy’’. The selection of the hypotheses was so persuasive that the phrase “Arrow-Debreu economy’’ was for long familiar. But Debreu was not fully satisfied: he was concerned with finding a better set of sufficient conditions and a more direct proof of existence of an equilibrium. He kept investigating the issue further with results contained in his 1959 book (Theory of value: An axiomatic analysis of economic equilibrium), and later complemented in the article “New concepts and techniques for equilibrium analysis’’ published in 1962 in the International Economic Review.

9For economists who know the complexities of market structures it is natural to wonder about the relevance of an equilibrium concept embodying the perfect competition hypothesis. Indeed, a substantial part of economic theory is validly devoted to imperfect competition. But intuition also suggests that, when the relevant markets are all large, the competitive general equilibrium may provide a sufficiently good approximation. Can that intuition be supported by a rigorous argument? Already in 1881 the Irish economist Edgeworth provided such an argument for a simple case, in which he showed that a market would tend to be competitive if the number of participants would increase indefinitely, each one tending to become negligible.

10This result was greatly generalized in 1963 by Debreu and Scarf proving “A limit theorem on the core of an economy’’. The concept of core was there borrowed from the theory of games, which was considered as making sense under imperfect competition. One could then speak of the robustness of the concept of a competitive equilibrium. All the more so as it turned out that similar limit theorems also applied to almost all concepts which could be substituted to that of core in order to characterize the outcome of imperfect competition (Aumann, 1987).

11How specific are the results of the classical theory of general equilibrium? In the late 1970s many students of general equilibrium theory were disappointed to learn that, taken alone, the theory in question was imposing only very few restrictions on the set of aggregate demand and supply functions of the various goods and services. The system to be solved in order to determine the values of all endogenous variables in the case of n goods or services was boiling down to n equations, each one stating that the net aggregate demand of a particular good or service was equal to zero at an equilibrium. Under the system of hypotheses that was most commonly posed, each net aggregate demand function had to be continuous and homogeneous in terms of prices. Moreover, the set of the n aggregate net demand functions had to satisfy the “Walras law’’, stating that the global value of the sum of all such net demands had to be identically equal to zero. But, as soon as these properties held, any specification was a valid case according to the theory. Debreu published a general proof of this result in 1974 (“Excess demand functions’’, Journal of Mathematical Economics, vol.1).

12Surprisingly, this mathematical result was taken as destructive by many economists, who were inclined to then argue that not only was the theory false, for such or such reason, but it was also void. This judgment clearly made no sense. The result meant only that, for meaningful applications of the theory, extra information besides perfect competition was also needed. The attitude in question was really revealing ignorance of the mathematical discipline and of the role of empirical information in applications of theories.

13From the outset, Debreu’s project was to build a mathematics of economies, which had to be rigorous and couched down in the concise language that was his distinctive mark. This lead him to define the concept of a measure space of economic agents, which allowed making sense of the idea that economic agents would be “neighboring’’. More generally Debreu wanted to explore meaningful categories such as “regular differentiable economies’’, which were appropriate for the study of a number of issues emerging in applied fields (e.g. his article in the 1976 American Economic Review Proceedings).

3. Approaches of mine, here and there

14When I left the Cowles Commission in the summer of 1951 I had a definite research project in mind, which had born there and would soon be brought to completion. Similarly, part of my time during my two visits in Berkeley was devoted to the search for particular Pareto efficient solutions to the planning problem. Research about ideal exchanges of risks on competitive market later led to the publication of two theoretical articles (Malinvaud, 1972; 1973). I will first present my work on capital accumulation and efficient allocation of resources (Malinvaud, 1953) and planning (Malinvaud, 1967). I then turn to the question of risk allocation and to elasticities of substitution.

15Economists working at Cowles in 1950-51 were well aware that general competitive equilibria could be applied to economies where markets for intertemporal trades were assumed to exist. In that case two goods with the same physical nature had to be considered as distinct “commodities’’ as soon they were meant to be available at different times. However, the number of commodities had to be infinite if the theory had to cover cases in which the time horizon was not assigned a given bound. Moreover, some of those economists who had been working on capital theory had particularly brought their attention to “stationary states’’, those in which the same technologies and needs, the same productions and consumptions, were assumed to occur period after period.

16I was then perceiving a double challenge: to explicitly extend the classical theory of the competitive general equilibrium to the case of an infinite horizon, and to show how stationary equilibria could appear as particular cases of general equilibria. That was actually a mathematical challenge in connection with two distinct branches of then modern mathematics. In particular an additional hypothesis was needed in order to exhibit a price system that was in conformity with those defined for economies with a bounded horizon. So, I describe my article as having aimed at consolidating the link between the general equilibrium theory of the allocation of resources and capital theory.

17More than most of my colleagues I was involved with the planning problem in market economies. My first published article, in 1950, had the title “L’expérience travailliste et la pensée économique anglaise’’. It was a survey of publications in England, in the immediate postwar years, about whether and how to plan the economy. The focus was notably on the book published by Abba Lerner in 1944 with the title The Economics of Control. Principles of Welfare Economics. My interest in the planning issues lasted throughout two decades, maintained in particular by my operational proximity to the Commissariat Général du Plan and my intellectual proximity to Pierre Massé.

18There already was a long tradition in economics about economic planning with its two aspects: (1) management of public utilities, since the time of Jules Dupuit (1844), (2) more generally the welfare economics of future trends in prices and productions. In the 1960s in France and a few other market economies, a standing problem in this respect was to know how to best organize the exchange of information where the economy was operated with a high degree of decentralization. I wrote then several articles on this subject; the best known probably being “Decentralized procedures for planning’’ (Malinvaud, 1967).

19A natural extension of the theory of general equilibrium beyond the intertemporal aspect was contemplated in the direction of risk taking. That was actually a multidimensional challenge, which greatly outgrew my own contribution. The latter aimed at investigating the impact of a remark made in particular by Kenneth Arrow. Many risks are individual in a modern economy, in the sense that each such risk bears on a person or a firm, which may be hit or not with a given probability. But many persons are exposed to similar risks with the same probabilities. These individual risks may then be taken as interchangeable by insurance companies. Given the laws of large numbers and perfect competition, the value of the insurance premium would then follow. This proposition was examined, at two distinct levels of rigor, in my “The allocation of individual risks in large markets’’ (Malinvaud, 1972), and “Markets for an exchange economy with individual risks’’ (Malinvaud, 1973).

20I consider these results as informative to some extent. But discussing the allocation of risks has other dimensions, which were early identified and follow from the fact that it is unrealistic, in market economies exposed to risks, to neglect the fact that information of agents are imperfect and known to be imperfect in different ways. So, a completely different approach had to be followed by younger economists.

21It would be a mistake to consider today neoclassical general equilibrium analysis as an outdated branch of economic theory. Actually this branch still serves for meaningful developments (see for instance Samuelson (2004)). As an illustration of that, I will present one branch of analysis that is also now frequently discussed: aggregate elasticities of substitution between factors of production. Those characteristics of equilibrium matter in two long standing economic theories dealing respectively with growth and income distribution. Now they may turn out to greatly matter also in the prospective analysis of the environment, where risk of overuse of natural resources may be the focus of an increasing concern. It so happens that I recently used a simple general equilibrium model in order to investigate the determinants of these aggregate elasticities (Malinvaud, 2006). Here is a brief presentation of the model and of some of its aggregate implications.

4. A general equilibrium model

4.1. The model

There are Image 100002010000000D000000194750CDF4BA75A2F6.png industries. Each industry Image 10000201000000A20000001980D939946480C27F.png produces output Image 100002010000001700000019F8D6281DA1518119.png of good Image 100002010000000D000000192188DBED1BEFBA70.png from inputs Image 10000201000000100000001997297B873AE4057C.png of Image 100002010000001300000019B6A4553C2253B9C0.png factors Image 100002010000008300000019064B5DE6E5BC524D.png with a constant-returns-to-scale technology. Production functions Image 100002010000001300000019EEADA4B0BE15F2D4.png are twice differentiable. Markets for goods and factors are assumed perfectly competitive. At equilibrium Image 1000020100000014000000197608864DE2908AB6.png is the price of factor Image 10000201000000070000001998F4615FA41BC809.png and Image 100002010000001500000019D266D482595317DA.png that of good Image 100002010000000D000000192188DBED1BEFBA70.png .

22The system of demand functions for goods is directly posed at the aggregate level and written as:

Image 100002010000028B00000035BDB138193C3361A6.png

with income r given by:

Image 100002010000028B0000005C3A7832D07808F24A.png

The functions Image 1000020100000013000000198448DD8C8A2D0E4A.png are differentiable, Image 1000020100000012000000190B0AF7B017E30F8D.png being the derivative with respect to ln Image 100002010000001500000019BD52C77CA4A3BE58.png and Image 1000020100000013000000193DF61A29D00529C2.png the derivative with respect to ln Image 100002010000000B00000019AB0453F279C4C8E7.png . In what follows the n-dimensional square matrix Image 1000020100000011000000194F69E328F6801B96.png is defined as having elements Image 1000020100000049000000196253EB9DCF89D1F8.png .

Market shares and cost shares in a reference equilibrium will be denoted by Image 10000201000000170000001940025D145985C4D1.png and Image 100002010000001400000019BC45C746514A3899.png :

Image 100002010000028B00000054BFBF094674873477.png

with the average cost shares:

Image 100002010000028B0000005974D1F0E0EC68E722.png

and we shall use the notation:

Image 100002010000028B0000003BF40A8ACE1F509CF5.png

The supply of factor Image 10000201000000070000001998F4615FA41BC809.png in the economy will be taken as exogenous, denoted by Image 1000020100000012000000199524CC1D2EF99E7A.png . Equilibrium will require:

Image 100002010000028B0000005DE0F7C3BB61F3618E.png

The purpose is a comparative static analysis of the infinitesimal changes such as Image 100002010000001A00000019501D56213B34BBF1.png or Image 100002010000001E00000019AFD8171C11571E6B.png affecting factor inputs Image 1000020100000011000000191E9A5F5B402E54CD.png or factor prices Image 1000020100000014000000192B7838F19E6F72B4.png .

23We focus here on relative changes, which we denote by capital letters:

Image 100002010000028B00000089736FCD7C5DAB52F2.png

The outcome of the analysis is the set of relations linking relative changes in the endogenous variables to possible relative changes Image 100002010000001600000019D353C58A534CD356.png in factor supplies. The aggregate elasticities of substitution are given by the system of the Image 100002010000001300000019AC5C13AD711A8F3E.png equations connecting the relative changes in factor prices Image 100002010000001600000019244D5B098EF9B8DD.png to the relative changes in factor supplies Image 100002010000001600000019D353C58A534CD356.png . It turns out that this system is conveniently written in matrix form as:

Image 100002010000028B0000003C29677532ADEB4108.png

24Our concern will now be to know the implications of this system, which must be first read and explained.

4.2. Some implications of the model: the two channels

The two column-matrices (i.e. vectors) (of relative changes) Image 1000020100000019000000197D3C22AEB81B5BDA.png and Image 100002010000001100000019CE959FBE4A41ECD2.png have m components (the number of factors). The other matrices are characteristics of the reference equilibrium, in comparison to which changes are measured. There are six such matrices: Image 1000020100000011000000194F69E328F6801B96.png was defined above and characterizes the price elasticities in the system of the demand functions for goods; Image 100002010000001100000019F73B813B677F9028.png is an Image 10000201000000280000001940CF74C14B90C354.png matrix with elements given by (5) and Image 100002010000001800000019ACB60ECDE3ED1AD5.png its transposed: they characterize the dispersion of cost shares across industries; Image 100002010000001D0000001900DC05BDB6887973.png is the diagonal matrix defined by the Image 100002010000000C00000019CACB98B17FF07BCC.png industry market shares given by (3a), Image 100002010000002100000019EB8C89D3B7E7C686.png is similarly the diagonal matrix defined by the average cost shares defined by (4). The matrix Image 100002010000001B0000001963E0A8158DC62163.png deserves a particular attention and plays a crucial role. It is defined by the weighted average :

Image 100002010000028B0000004366017E558E6A24C1.png

of matrices Image 10000201000000170000001930A598957C6E52B6.png , each one being a function of the first and second order elasticities of factor utilizations in its industry Image 100002010000000D00000019133B2671BD154BFF.png . For simplicity the formula leading to the computation of Image 10000201000000170000001930A598957C6E52B6.png will not be given here. Let me simply say that it defines a kind of pseudo-inverse of a matrix Image 1000020100000015000000199A329F4DC197337C.png involving first and second order derivatives of the production function Image 100002010000001400000019C437F890E449D0B5.png .

Actually, the form of the square bracket in the right-hand side of (8) illustrates the idea, often expressed in the literature, according to which there are two channels of adaptation of factor prices to changes in factor supplies. The first channel involves in each industry the direct substitution of relatively abundant factors to factors becoming scarcer (hence the role of matrices Image 10000201000000170000001930A598957C6E52B6.png ). The second channel is indirect and reflects two other adaptations: first, the relative prices of goods change; the price increases for goods that use much of the scarcer factors and decreases for those that use little of them; second, this change in prices induces substitutions between demands for goods. The importance of this second channel is reflected in the second term of the square bracket of formula (8), matrix Image 100002010000001100000019F73B813B677F9028.png giving the importance of changes in prices of goods, and matrix Image 1000020100000011000000194F69E328F6801B96.png that of substitutions between demands for different goods.

25From recognition of the existence of those two channels, inference was often drawn that their effects would add up and therefore elasticities of substitution would be larger at the aggregate level than on average at the industry level. But this intuition failed to be always right, far from that. The implications to be drawn from equation (8) are actually complex, as the equation itself is not transparent.

The most easy case occurs when the model is written for two factors only, for CES production functions with all the same elasticity of substitution Image 100002010000001800000019A34E2F818486548D.png (log-linearity would follow from Image 100002010000003F00000019352F14BF312A93CA.png ) and for a homothetic CES system of demands for goods (Image 100002010000001A00000019E5659E14CC9DC3AD.png being its elasticity of substitution). In that case (8) boils down to:

Image 100002010000028B00000082E2C97E219A93D443.png

(the same value of the variance holds when Image 100002010000002300000019B376DC82D0A4815A.png is replaced by Image 100002010000002700000019AFDEE3B151D0ACBF.png in the formula). The long square bracket gives the aggregate elasticity of substitution, Image 100002010000001600000019F0FAB12BBE8398DE.png say. It is equal to the industry elasticity Image 1000020100000019000000197597B5332E92149D.png if and only if Image 1000020100000041000000199A89861156D25F1F.png for all Image 100002010000000D00000019E52B65A1EAFA1FAB.png , or Image 100002010000004C00000019A16521345F0CE97D.png . Except for that, Image 100002010000004C00000019DF4C9CBE96640237.png if and only if Image 100002010000004E00000019AEA102D11E7D673A.png . The conjecture that Image 100002010000004E00000019AEA102D11E7D673A.png failed as soon as the elasticity of substitution between the demands for goods is smaller than the industry elasticity of substitution. Attached to Image 1000020100000019000000197597B5332E92149D.png the multiplier Image 1000020100000023000000192FB47A58D95E4976.png (a measure for the heterogeneity of cost shares between industries) exhibits the fact that aggregation attenuates the factor substitutability present within industries.

  • 0 What are the implications of the following changes with respect to the specification leading to (10 (...)

26The case in which formula (10) applies is very special. My article examines at some length many respects in which reality is more complex, so that the results derived from equation (8) are not as easily characterized0. It would, of course, be out of place to record here these results of a research taken as an example.

27The academic literature in economics now contains other similar applications of general equilibrium analysis. It seems to me that they should not be neglected in the history of economic theory because they may reveal a valuable evolution beyond the building period up to the end of 1970s.

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Bibliographie

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Notes

0 What are the implications of the following changes with respect to the specification leading to (10)? - if the number of factors is larger than 2, - if the elasticities of substitutions σh of the industry CES production functions differ, - if the production functions are more complex than CES, such as nested — CES, - if the income elasticities of the demands for goods differ from 1 and from one another.

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Edmond Malinvaud, « Neoclassical General Equilibrium Theory as a Source of Powerful Concepts, Although Confronted with the Complexity of Market Economies »Œconomia, 2-1 | 2012, 3-14.

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Edmond Malinvaud, « Neoclassical General Equilibrium Theory as a Source of Powerful Concepts, Although Confronted with the Complexity of Market Economies »Œconomia [En ligne], 2-1 | 2012, mis en ligne le 01 juin 2015, consulté le 29 mars 2024. URL : http://journals.openedition.org/oeconomia/1631 ; DOI : https://doi.org/10.4000/oeconomia.1631

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Edmond Malinvaud

CREST and Collège de France

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