The Neo-Walrasian Approach to General Economic Equilibrium

10.1.1. The conquest of the existence theorem

The rise of Nazism led to a diaspora of intellectuals. All the fervour of study and debate which had enlivened Berlin and Vienna in the 1920s ended in the following decade; and the move to the West began for the most important Mittel-European economists, apart from those like Schlesinger, who com­mitted suicide, or Remak, who died in a concentration camp.

The presence of these economists in the American intellectual circles of the 1940s and 1950s had many effects on the evolution of general-equilibrium theory. Even if the resumption of the American studies on this theory was stimulated, indirectly, more by Hicks’s Value and Capital than by the con­tributions of the members of the Viennese Kolloquium, it is still true that the work of Wald, von Neumann, and Morgenstern gave a sharp impulse to that resumption. With the fixed-point theorem, Wald and von Neumann had indicated the road to be taken to solve the existence problem. Moreover, von Neumann and Morgenstern’s 1944 book, the Theory of Games and Economic Behaviour, led (among other things) to the abandonment of traditional techniques of differential calculus and to a reorientation of mathematical economics towards the use of the techniques of convex analysis. An important contribution of that book was also the proof of the existence of solutions for a two-person zero-sum game—a demonstration later general­ized to an n-person game by John Nash in ‘Equilibrium Points in N-Person Games’ (1950).

A decisive stimulus to the resumption of the American studies on general equilibrium was given by two works by Samuelson, one of 1941, ‘The Stability of Equilibrium: Comparative Statics and Dynamics’ and one of 1947, Foundations of Economic Analysis.

In 1951s two articles were published which demonstrated the optimality properties of a competitive equilibrium: one by K. J. Arrow, ‘An Extension of the Basic Theorems of Classical Welfare Economics’ and one by Gerard Debreu, ‘The Coefficient of Resource Utilization’. In that period the two economists had begun to work together on the problem of the existence of solutions, and in 1952, at a meeting of the Econometric Society, they sub­mitted a key paper in which the longed-for peak was finally and compre­hensively conquered. The decisive instrument used in the conquest was Kakutani’s fixed-point theorem. The article was then published in 1954 with the title ‘Existence of an Equilibrium for a Competitive Economy’. We should also mention that, during that meeting of the Econometric Society, L. McKenzie presented a model of competitive equilibrium of international trade in which, under less general hypotheses, he solved the existence problem by means of mathematical techniques similar to those used by Arrow and Debreu.

The Walrasian ‘new testament’ was written in 1959. The author was Debreu, the title, simple and lapidary, Theory of Value. It is useful to present a brief summary of it here, if for no other reason than to show the beauty and the simplicity of the Walrasian dream come true.

The model assumes that the following data are known:

(1)the number of commodities, l;

(2)the number of producers, n;

(3)the number of consumers, m;

(4)the technology available to each producer;

(5) the physical constraints and psychological characteristics of each consumer, including tastes;

(6)the initial endowment of resources of each consumer;

(7) the share of the profits of each producer which belongs to each consumer.

The commodities are a set of goods and services specified in terms of their physical characteristics and place and time location. Thus, a good which is available today at two different locations is considered as two different commodities. The same is true for a good which is available on two different dates. Each commodity is given a price. The price vector is p = (p₁,...,p_l). The price is paid at the moment a deal is struck. All deals are struck at one place and at one time, even those on commodities to be delivered in the future. The prices of the latter are therefore ‘futures’ prices. This fact makes Debreu’s model one of inter-temporal equilibrium.

It is worth pointing out here that various attempts at formalizing a general inter-temporal equilibrium model had already been made by several eco­nomists, including Frisch and Tintner. One of particular interest was made in 1943 by Debreu’s teacher, Maurice Allais, of whom we must mention at least two important books: A la recherche d’une discipline economique: Premiere partie: L'Economie pure (1943) and Isconomie et interet (1947).

Let us now return to Debreu’s model. The technological constraints of producer j are represented by a production set, Yj, which contains all the technical combinations between inputs and outputs accessible to that pro­ducer. A ‘production plan’ is one of these combinations, and is expressed by the vector yj = (y_j1,..., y_jι), in which inputs are represented by negative elements and outputs by positive ones. The producer will choose a produc­tion plan so as to maximize profit π_j∙=py_j.

For each consumer, for example i, a consumption set, X_i, is defined, which contains all the combinations of commodities which the consumer can buy and sell. For some goods there are physical constraints. For example, it is impossible to sell more than a certain number of working hours per day. Furthermore, a preference ordering which expresses his tastes is defined for each consumer. Finally, given the resource endowment of consumer i, σ_i = (σ_i1,..., σ_i'ι) and his profit shares, θ_i = (θ_i1,..., θ_in), his wealth is defined, which is w_i = pσ_i + θ₁∙π, where π = (π_i,...

A state of the economy is an (m + n)-ple, (x, y) = (x₁,... x_m, y₁,... y_n) which includes all the plans of action of all the consumers and producers. Each element of the (m + n)~ple is a vector of l elements. In such a state of the economy, the net total demand is z = x — y — s. An equilibrium is an (m + n + 1)-ple, (x*, y*, p*) = (x₁*,... x_m*, y₁*,... y_n*, p*) such that: x* maximizes the satisfaction of all the consumers; y* maximizes the profits of all the producers; all and only the available resources are used, i.e. x* _ y* = s. The vector of equilibrium prices is p*.

Debreu proved that this vector exists under a series of hypotheses which are no less implausible than all those adopted before him and certainly more general. Here are some of the most important. Each consumption set must be convex, so that, if two consumption plans are in one set, this will also include all their linear and convex combinations. The consumers must be insatiable, in the sense that, for every chosen consumption plan, there will always be another which is preferred. An important assumption on the production side is that the total production set, Y = Y_j Y_j, is convex. This, together with the ‘inactivity hypothesis’ (i.e. 0 ∈ Y), excludes the possibility of increasing returns of scale: if two production plans are in the same pro­duction set, so are all their linear combinations. Moreover, the most recent research has allowed a little weakening of this hypothesis by admitting the possibility that single producers are able to benefit from ‘moderately’ increasing returns to scale.

There are other particularly irksome hypotheses concerning the existence of the markets and the forecasting ability of the economic agents. As con­tracts are stipulated in the present, for all commodities produced and delivered not only in the present but also in the future, futures markets must exist for the commodities available in all future periods—a hypothesis of which it is not even worth raising problems of realism. In fact, Debreu did not do so. Besides this, it is necessary to assume that economic agents are endowed with perfect foresight, as they know with precision, as consumers, the future evolution of their own preferences and, as producers, the future evolution of technology.

Debreu tried to avoid this peculiarity by introducing the notion of uncertainty, but he did so in a way which was no less odd, i.e. by attributing a further specification to goods, one relating to the ‘state of the world’. Thus, for example, a sack of corn available here and now would be a different commodity, not only from that available in another city now, or from that available there tomorrow, but also from that available here tomorrow in the case that there is an earthquake or any other act of God tonight. It is assumed that individuals are able to formulate plans of action with regard to all the commodities available anywhere, in all future periods, and in all possible states of the world. Besides this, it is obviously necessary to assume that there are ‘contingent’ markets for each possible state of the world, as all indi­viduals must be able to make deals in them. Many believe that this is a more reasonable way to account for behaviour with regard to future events—more reasonable than perfect foresight, for example.

Finally, we mention another set of unlikely hypotheses concerning, for example, the non-existence of externalities both in production (external economies or diseconomies of scale) and in consumption (any phenomenon of interdependence among consumers’ tastes and between production and consumption, fashions, hidden persuaders, etc.).

10.1.2. Defeat on the grounds of uniqueness and stability

Even though the goal of existence had been reached, the difficulties for neoclassical economists were not over. In fact, they were only just beginning, because it was also necessary to demonstrate that the equilibrium is in some way unique and stable. There are two reasons why uniqueness and stability are indispensable. One is, let us say, of a philosophical nature and is fundamental. We shall discuss it in the next section. The other, of a methodo­logical nature, will be dealt with straightaway.

The problem originates from the fact that a great deal of the reasoning with which the neoclassical economist explains the social meaning of the economic variables, prices, wages, etc., are the result of some comparative statics exercise. For example, in order to say that the relationship between the prices of two goods expresses their relative scarcity with respect to con­sumers’ tastes, one simply says that it is equal to their (marginal) rates of transformation and substitution. These marginal rates are defined in terms of ratios between ‘variations’ of the two goods. In reality these ‘variations’ are defined as the differences between the values that the variables assume in two different equilibrium states, even if they are interpreted as changes around a third intermediate equilibrium. If these exercises are to be correct, as had already been pointed out by Samuelson with the correspondence principle as early as the 1940s, the equilibrium around which they are undertaken must be stable and unique. If it were not so, even a very small change around the equilibrium would lead the economy far away from it, and the various rates of substitution, transformation, and the like would become meaningless. All the most important neoclassical arguments about the economic role of scarcity, the sovereignty of the consumer, the allocative efficiency of markets, etc., could no longer be sustained, for the simple reason that the concepts and the reasoning of comparative statics would no longer have any economic meaning.

Now, it is possible to obtain the desired results of stability and uniqueness, but the price in terms of the restrictiveness of the hypotheses is far too high. As early as 1936 Wald obtained results of uniqueness and stability by using some special hypotheses such as ‘diagonal dominance’ or ‘gross sub­stitutability’. Then in 1943 the global stability of the tatonnement process under hypotheses equivalent to those of gross substitutability was proved by Allais through applying Lyapunov’s second method. The gross sub­stitutability hypothesis implies that the aggregate excess demand of a com­modity decreases when its price increases, or the price of any other commodity decreases. Diagonal dominance implies that the aggregate excess demand of a commodity is more sensitive to a change in its price than to a change in the prices of all other commodities. The results of the most recent research on this argument are not very different from those obtained by Wald and Allais. In particular, it seems that the gross substitutability hypothesis is crucial to obtain stability. In fact, it was one of the hypotheses adopted by K. J. Arrow and L. Hurwicz in their article ‘On the Stability of Competitive Equilibrium I’ (1958). In this article cases were shown of economies characterized by a stable equilibrium under various special hypotheses—but this type of reasoning was not very elegant or general. The following year, however, a general theorem of global stability was obtained which still today remains a milestone in the evolution of stability theory. It is to be found in the article ‘On the Stability of Competitive Equilibrium II’ by

K. J. Arrow, H. D. Block, and L. Hurwicz. The most important of the hypotheses on which the theorem depends concerns the continuity of the excess demand functions, and, alas, gross substitutability.

The result was received most enthusiastically and raised confidence in the possibility of generalizing it by removing some of the most restrictive hypotheses on which it depended—so much so that in the following year another four or five significant articles on this subject were published. But among these there was one by Herbert Scarf which served immediately to dishearten the optimists. It was entitled ‘Some Examples of Global Instability of the Competitive Equilibrium’ (1960), and showed cases of rather simple economies in which equilibrium existed but was unstable. One important result obtained by Scarf was the demonstration that it is possible to obtain instability simply by introducing a complementarity hypothesis, and this was considered as a confirmation of the key role of the hypothesis of gross substitutability in obtaining stability. By 1964, when M. Morishima’s Equilibrium, Stability and Growth was published, the centrality of gross substitutability had become a more or less accepted fact.

Subsequently, there have been no great steps forward in this field of research. However, it is worth mentioning some small advances made by Arrow and Hahn in 1971 and by S. Smale in 1976. The trick consisted of the modification of Samuelson’s traditional tatonnement equation so as to obtain global stability without having to make very restrictive hypotheses about the excess demand functions. Unfortunately, however, it was necessary to introduce a few substitute hypotheses which were devoid of economic meaning.

In 1992 Saari provided a generalization of the instability result: a general equilibrium can be unstable even if its parts, i.e. the sub-sets of the economy, were stable. In other words, the stability properties of ‘small’ equilibrium systems do not carry over into bigger systems. The moral of the tale is now part of the official doctrine, and is that stability is not an intrinsic property of the general-equilibrium model.

Things are not very different in regard to the problem of uniqueness. Besides, the problems of stability and uniqueness are closely linked, in that, to the degree that it is possible that there are many equilibria, it is also possible that some of them are unstable. In fact, it had been clear right from the early 1950s, when Arrow and Debreu began working on the existence problem, that the general conditions used to demonstrate the existence of equilibrium were not sufficient also to guarantee uniqueness. And Debreu, more than anyone else, must have been aware of the analytical reasons for, and the theoretical implications of, this difficulty. This may explain both the rigorously axiomatic character he imposed on his research work (with the consequent refusal to listen to any criticism of irrelevance) and the almost snobbish absence from the Theory of Value of the usual neoclassical com­parative statics exercises.

The explanation why not much faith could be placed in the possibility of solving the problem of uniqueness was provided by H. F. Sonnenschein in ‘Market Excess Demand Functions’ (1972). This paper was followed by others by Sonnenschein, Debreu, Mantel, Kirman and Koch which con­firmed and consolidated the results.

In his 1972 article Sonnenschein finally demonstrated something that many people had suspected for some time: that the usual general hypotheses used to explain consumer behaviour, and from which the individual demand functions are derived, are not sufficient to place any significant restriction on the form of the aggregate demand functions. This showed that any hope of identifying general hypotheses on individual behaviour capable of generating aggregate excess-demand functions compatible with the uniqueness and stability of the equilibrium had to be given up.

To look at the problem from another angle: it is known that the results of uniqueness and stability can be obtained with some restrictive hypotheses about aggregate excess-demand functions; the problem is to know whether there is some set of particular assumptions about the behaviour of individuals which would justify such hypotheses. The answer is no. However restrictive the assumptions on individuals may be, the aggregate functions can assume practically any form. At most, it is possible to compel them to assume the properties of continuity and zero homogeneity and to obey Walras’s Law. These conditions are not sufficient to ensure the stability and the uniqueness of equilibrium. In the next section we will discuss the ‘philosophical’ consequences of this result.

There only remains to add something about an attempt made at that time to find, if not an escape route, at least a way to mitigate the seriousness of the problem. This attempt originated from a double hope: to be able to isolate the problem of stability from that of uniqueness, and then to be able to tame at least the latter. Such hopes were raised by the observation that, while with stability it was desirable to have global results, in the case of uniqueness a few local results might be sufficient. This was the road taken by Debreu with an article of 1970 and one of 1976, in which he introduced the notion of ‘regular economy’. This is one with a discrete set of equilibria, so that each equilib­rium has a neighbourhood in which it is unique. Regular economies exclude the most dangerous situation: that in which, near one equilibrium, there can be an infinite number of other equilibria—a situation which would make the state of equilibrium indeterminate. Moreover, the set of irregular economies that could possibly exist must be negligible. Finally, regular economies must be structurally stable, so that a small change in the parameters cannot gen­erate a catastrophic change in the equilibria.

By making use of Sard’s powerful theorem and by adopting two parti­cularly unlikely hypotheses, Debreu succeeded in demonstrating that regular economies have a set of equilibria which are not only discrete but also finite; that irregular economies make up a negligible set; and, finally, that the set of equilibria depends on their parameters, not only in a con­tinuous but also in a differentiable way. The two unlikely hypotheses on which these results depend are as follows: first, the individual demand functions must be differentiable; second, all the goods must be ‘desirable’, i.e. when their price approaches zero, the average excess demand for them tends to infinity. Now the hypothesis of differentiability seems rather brave. Economists are used to treating it as if it were normal, but this does not mean it is sensible; it only means that the economist’s education is generally successful in developing special gifts in the suspension of his critical fac­ulties. First, the differentiability hypothesis presupposes that individuals are able to formulate a precise demand, for example with regard to the vari­ation in the price of cars, not only for any number of cars but for any fraction of them; then, even worse, it implies that it is possible to determine the rate of variation of the demand for cars corresponding to every infin­itesimal variation in price. And even more ridiculous, if possible, is the hypothesis of desirability; which implies that, for example, when the price of water approaches zero, individuals will tend voluntarily to drown themselves or to try to hoard the seas. But the real problem of the theory of regular economies is not so much in the unrealistic hypotheses as in the fact that it does not help to solve the problem. In fact, once it has been proved that equilibria are not infinite, one still has to prove that they are dynamically stable. What can we do with the equilibria, even if they are ‘few’ in number, if they persist in moving the economy away from themselves?

10.1.3. Theendofaworld?

The general-equilibrium model has been under critical fire ever since it appeared on the scene; critics have never shown any signs of tiredness, and have been just as determined as supporters have been in ignoring the criti­cisms. The mass of critical literature has become so great that it is impossible to deal with it in one section of a book such as this. Here we shall restrict ourselves to presenting a schematic summary of the criticisms, and of responses to them, without drawing up a list of specific contributions or of specific authors.

The most widespread criticism is obviously that which calls for the need for realism, or for explanatory or forecasting power. This has also, perhaps, been the most heeded; but it has never proved decisive. Now, the fact that the general-equilibrium model is based on extreme hypotheses cannot be disputed—with its atomistic competition, the absence of externalities, the insatiability, the desirability, the differentiability, the futures and contingent markets for all the goods, and so on and so forth. What is its explanatory and forecasting power? What does it describe? What use is it? Why is it necessary to study it?

The first group of answers to these questions came from Anglo-American economists such as Arrow, Hahn, Townsend, and Roy Weintraub, to mention only those who have made the most recent contributions. Given their positivist backgrounds, these economists have been especially sensitive to this type of criticism. The counter-arguments they advanced can be basically reduced to four types. First, even though it is true that the general­equilibrium model in itself has no explanatory power, for the time being, there is no reason to despair; research moves forward, loosening and generalizing hypotheses, and this process can lead to an increase in the degree of realism, so that the ‘research programme’, of which that theory is the ‘hard core’, may turn out to be progressive in the end. Second, the general-equilibrium model already fulfils a fundamental heuristic function, as it is able to inspire a great deal of research and applied work in specific fields of economic theory in which it is possible to reach, and, in fact, have been reached, notable results in the field of forecasting. Third, the general-equilibrium model represents, vis-a-vis a great deal of research and many applications in specific fields, a general framework of theoretical reference, and acts as a deep structure capable of holding together theoretical pieces which are heterogeneous and independent of each other. Finally, the theory of general equilibrium can be used as a taxonomic instrument to classify different types of real economies by applying appropriate restrictions to them, such as the number and type of markets which are assumed to be open, the degree of competition, and the length of the time horizon within which it is assumed that decisions are taken.

Now all these arguments are rather weak, and for the same basic reason: that they are referring to something different from the model they wish to justify, rather than to the properties of the model itself. The first argument is only an appeal to the hope that the theory could, in future, become some­thing which it is not today. The second would be valid only if it could be demonstrated that those research projects and specific applications inspired by the general-equilibrium model have actually been helped by it and have not reached sound results, by chance, despite the model. The third should demonstrate that many of the results obtained in the field of specific applications could not be accommodated within a different general theor­etical system. Finally, the weakness of the fourth lies in the fact that the special worlds that can be obtained by rejecting some hypotheses of the general-equilibrium model—for example, the hypothesis of flexible prices so as to obtain non-Walrasian equilibria, or that of exchanges in equilibrium to obtain non-tatonnement processes—are, in fact, the result of a negation of that model, and it is hard to see how this can be considered as one of its virtues.

But there is another way of answering the question ‘What does the general-equilibrium model describe?’—one that resorts to a drastic answer: ‘Why should it describe anything?’ It is no accident that this is the path chosen, above all, by economists educated along the lines of the rationalist and conventionalist heritage of the homeland of Descartes and Poincare; we can cite the names of Debreu and Malinvaud, but not that of Allais, their master, who has always been more sensitive to the requirements of empirical research and practical applications. Debreu, more rigorously than the others, and by following an approach inspired by the epistemological stance of the ‘Bourbaki’ group, has set out the general-equilibrium model in terms of a strictly axiomatic theory. A ‘pure’ economic theory such as the Walrasian one, is an abstraction. As such, it has no need to justify its own hypotheses by induction, nor to test them by empirical research; and it is necessarily ‘irrealistic’. This is just as true for general-equilibrium theory as for any other. Is the post-Keynesian steady-state growth model more realistic? Or Sraffa’s standard commodity? Or Marx’s reproduction schemes, or Quesnay’s tableau economique? A pure theory is not an imitation, reflection, or description of reality, but a metaphor, or, in Samuelson’s well-chosen term, a parable. It is this attitude that has justified the neoclassical and all the other theoretical economists in continuing to work on theory by ignoring the problems of the realism of hypotheses.

From here, however, the debate moves onto new ground: that of relevance. Any theory, however pure, is never neutral in the types of problem it helps the economist to focus on or in the way in which it helps him to solve them. The general-equilibrium model has often been criticized as completely inadequate to tackle the really important problems: growth, change, the economic role of institutions, the behaviour of collective agents, etc. Today, while every neoclassical economist is ready to accept this criticism, none will consider it as decisive. The general-equilibrium model, its supporters main­tain, is not suitable to tackle these problems, which should be passed over to other sciences, such as sociology, history, and political science; but it is better suited than any other to tackle the problem of efficient allocation of scarce resources. Why should this problem be considered irrelevant? And who decides which problems are relevant? Is the fact that society has placed its best universities and richest research institutes at the neoclassical economists’ disposal devoid of all meaning?

Obviously, this immediately leads us on to a third battlefield—the ideo­logical one, in which it is mainly the Marxist critics who have distinguished themselves. It is true, they say, that the neoclassical theoretical system, of which the general-equilibrium model is the heart, reigns supreme in all the academic institutions of the capitalist world; but this demonstrates neither that it is a valid representation of that world nor that it is really useful in tackling important problems. Rather, it is only a representation in which the ruling classes recognize themselves. Is it not, perhaps, a model aiming to demonstrate the intrinsic tendency towards order and efficiency of a world made up of free, egoistic, and equal individuals? And to hide the fact that equality and liberty are only the formal attributes of the agents who meet on the market? One need only glance at the production system to realize that the individuals who are really free and equal are those who own the means necessary to avoid working and to exercise control over the work of others.

Inspiration for the general-equilibrium model can be traced back to Smith’s theory of the individualistic competitive equilibrium; a theory that has been greatly improved over the following two centuries, while keeping its substance intact. According to this view of the world, social order is the result of the interaction of a multiplicity of autonomous, self-interested, and rational individual agents. These enter into relationships among each other, not through the operation of the institutions, social groups, or other super­individual entities but by means only of the market. The fact that we are dealing with individual subjects is of key importance. In the neo-Walrasian theory they are called consumers and producers. And even producers, i.e. firms, are considered as individual decision-making agents, rather than organizations, as common sense would suggest. In this theory, in fact, the individuals who participate in the activity of the firm meet and take decisions before the activity begins, and they meet on the market. The entrepreneur buys goods and services; the workers and the savers sell them. After this, production starts. The resources, the goods, and the services bought by the entrepreneur enter into the activity of the firm, but not the individuals who have sold them to the firm. The decision-making subject of the firm remains an individual. On the market, goods are exchanged at values that are not influenced by any single agent, and depend only on the scarcity of the goods themselves in relation to consumers’ needs. These exchange relations, by ensuring maximum satisfaction to each agent, guarantee an efficient allocation of resources.

All things considered, it is fairly irrelevant that this impressive con­struction expresses a biased point of view. After all, why shouldn’t one be able to choose one’s point of view? In fact, ideological criticism has proved incapable of discouraging the general-equilibrium theorist from looking after his own subject in his own way. A truly effective criticism would be that of demonstrating the inconsistency of the construction itself, its inability to explain what it wants to explain, starting from its own premisses. It seems to us, however, that this criticism has never been raised, either by Marxists or by other heterodox economists.

It is a strange quirk of history, though, that it was the neoclassical eco­nomists themselves who finally produced the decisive criticism. And this was like reaching safe harbour after a tormented voyage which had lasted two centuries.

For the grand construction of the individualistic competitive equilibrium to be valid, it is necessary to demonstrate that the market is able actually to lead the economy towards a state of equilibrium. It must be only the market, and not any social institution or collective agent; otherwise, the essential individualistic premiss would be undermined. This is the fundamental problem, which Galiani and Smith thought they had ‘solved’ by assuming that the adjustment processes in disequilibrium are regulated by a ‘supreme’ or ‘invisible hand’; which Walras thought he had ‘solved’ by assuming that the tatonnement process is regulated by the ghost-like presence of an ‘auctioneer’, and which the modern followers of Smith and Walras have shown that it is impossible to solve.

In fact, the meaning of recent advances concerning the problem of stability and uniqueness is this: the behaviour of individuals is not sufficient to give the invisible hand the strength it needs to lead the market towards equilib­rium. In order to reach a stable equilibrium, it is necessary to advance strong hypotheses on the behaviour of certain aggregate variables; and the know­ledge of the criteria of individual behaviour alone is not enough in itself to justify any of these hypotheses. This simply means that an individualistic competitive economy is not granted, for it does not necessarily possess the strength to reach equilibrium, not even when regulated by the auctioneer’s ‘supreme hand’. Thus the ‘scientific’ basis of the theory of laissez-faire and orthodox economic doctrine is simply lacking. Awareness of the seriousness of this problem is fairly widespread today, at least among economic experts. As early as 1969, for example, John Hicks warned one of the present authors, his student at Oxford, not to be too enthusiastic about the proofs of exist­ence, as the theory would, in any case, run into the problem of stability. But it is true that many neoclassical economists tend to relegate the discussion of these problems to footnotes, to obiter dicta, and to verbal exchanges. Recently however, the crucial importance of this question has been brought to light by B. Ingrao and G. Israel in ‘General Economic Equilibrium: A History of Ineffectual Paradigmatic Shift’ (1985).

The reactions of neoclassical economists to this revelation took the form of feeling that a radical change in course was needed; but it is not clear which direction should be taken.

On the one hand, there has been an attempt to resort to modal logic and counter factuals. This was the road taken, for example, by Hahn in On the Notion of Equilibrium in Economics (1973). The general-equilibrium model does not describe actual reality, it is said, but only a possible ideal world. This does not make it less useful to economists: it could be used to teach them not to make hasty declarations with regard to the effectiveness of the invisible hand, and to understand the real world by observing its differences from the ideal world. For example, it would be possible to understand why per­manent unemployment exists in the real world simply by reflecting on the unlikely nature of the hypotheses which enable the general-equilibrium model to eliminate it.

This type of argument cannot be upheld, however, for two reasons. First, the conditions with which the existence of the general equilibrium is demonstrated are only sufficient and not necessary. This means that, if the proposition ‘If A, then B’ is true, the proposition ‘non-B because non-A’ is not necessarily true. The latter is the type of argument which should enlighten us about actual economic reality by telling us why it does not correspond to the possible ideal world of the general-equilibrium model. Well, this enlightenment is precluded. But there is an even more serious problem arising from the impossibility of demonstrating the stability of the individualistic equilibrium. The general-equilibrium model describes, not a possible world, albeit unreal, but, on the basis of its own hypotheses, an improbable world. It does not tell us that an individualistic equilibrium must be reached if the usual hypotheses on competition, convexity, and so forth are applied, but that it may not be reached, even by using these hypotheses, or, rather, precisely because of the most fundamental of them, those defining its indi­vidualistic nature. Therefore, the proposition itself, ‘If A, then B’, cannot be upheld; not because the world represented by A does not exist in reality but because its representation, A, is not warranted in theory.

A second position is assumed by authors like Grandmond and Hildebrand, responsible for the statistical approach to the stability of general equilibrium. On recognizing the futility of the search for general conditions of stability, these economists set out to discover the most likely conditions for it. And it was found that if individual preferences were adequately and foreseeably dispersed, then equilibrium might be stable. Needless to say, this is not a satisfactory way out of the issue, not least because it presupposes ad hoc assumptions on the utility functions of agents.

A third position is even more pessimistic. It has been put forward by Alan P. Kirman in a paper whose title tells us everything: ‘The Intrinsic Limits of Modern Economic Theory: The Emperor Has No Clothes’ (1989). Kirman believes that the only way to escape from the impasse following the collapse of the foundations of competitive-equilibrium theory is to abandon methodological individualism—which is tantamount to saying; ‘Let Samson die with all the Philistines’.

Methodological individualism in its weak versions is an epistemological criterion that serves to identify the subject and the research method of economic science. One of such versions is that of institutional individualism of J. Agassi, according to which only individuals have goals and interests, yet institutions and social aggregates affect and constrain individual behaviour. In this view, whatever the phenomenon studied by economics, it must be possible to define it as the result, not necessarily the sum, of a certain set of decisions or conducts by individual agents. This does not exclude the possibility of there being social phenomena which can be described in terms of collective behaviour, and of there being collective agents, social classes, institutions, etc. It only means that the economist who studies them must account for them in terms of the decisions and interactions of the individuals who have created them and are part of them. This position is shared by almost all non-neoclassical economic theorists, the only exceptions, perhaps, being some members of the German Historical School and a few extreme institutionalists.

The neo-Walrasian point of view, on the other hand, is based on a strong version of methodological individualism, one that should be more appro­priately defined as ‘ontological individualism’: only individual agents exist; their choices are not affected by any kind of externality; and their social relationships are not mediated by any institution other than a competitive market. This market is anything but an institution, which is why it must be governed by an invisible hand or an auctioneer. The hypothesis of no externalities is very important in this approach, for it makes it possible to obtain market demand and supply curves just by summing up the individual ones. Thus it turns out that the whole is precisely the summation of the parts, which is the most trivial (in this sense, the stronger) version of methodo­logical individualism. Well, it is this version that must be abandoned, and this is a problem not for the vast majority of economists, but only for the ‘orphans’ of Walras.

As Saari (1995, p. 228) made it clear, ‘The root of the difficulty... is that social sciences are based on aggregation procedures... The complexity of social sciences stems from the unlimited variety of individual preferences that define a domain of such enormous dimension that, once aggregated, they can generate all imaginable forms of pathological behaviour.’ The limit of methodological individualism, in the strong version, is the presumption that social interaction among individuals adds nothing new to what results from the behaviour of single individuals. Yet it is well known that groups of individuals can give rise to collective behavioural patterns that were not desired by any of their members, even though are determined by their individual decisions. A typical example is a group of opportunists who evade taxes to maximize their individual welfare. If everyone followed suit, there would obviously be a decline and reduction in public services, so that no individual will succeed in maximizing welfare.

10.1.4. Temporary equilibrium and money in general-equilibrium theory

The developments in general-equilibrium theory following the Arrow- Debreu-McKenzie model have not led to the formation of a new theoretical body of knowledge, but to the modification, and sometimes the elimination, of this and that postulate of the original model. In these developments the concept of temporary equilibrium has been particularly important, both from the point of view of the internal consistency of theory and in relation to the possibility of using it to study all those phenomena that are typical of a monetary economy, especially inflation and unemployment. The modern resumption of Hicks’s concept of temporary equilibrium comes from the work of K. Arrow and F. Hahn, General Competitive Analysis (1971) and J.-M. Grandmont, ‘Temporary General Equilibrium’ (1977).

The starting-point of the theory of temporary equilibrium has been the abandonment of the assumption that there is a complete system of markets, an assumption which is unattractive at both the empirical and the conceptual level. Already Roy Radner, in a work of 1968, had studied sequential economies in which transactions are made on any date, and in which the incompleteness of the markets makes it impossible to reduce economic activity to a single set of initial exchanges, as happens in inter-temporal equilibrium. Thus, instead of a timeless equilibrium, there is a ‘succession of temporary equilibria’. As we mentioned in section 8.2.4, at the basis of the Hicksian conception of temporary equilibrium was the device of the ‘week’, a period within which the economy reaches an equilibrium position. As the economic process occurs through time, and as there is only a limited number of futures markets, all the economic agents take decisions relating to a certain instant (the current ‘week’) subject to their plans and expectations about the future (successive ‘weeks’). In particular, they decide, for example, to save, by reducing today’s consumption, if they expect the prices of goods to fall in the future. Such conjectures may or may not be realized; if not, the agents will be forced to revise their plans according to the new data. In spite of this, the present decisions, already taken on the basis of incorrect expectations, once carried out, cannot be changed. In this way, future expectations, whether right or wrong, will influence the present equilibrium.

A temporary equilibrium, even though it is a general equilibrium in each ‘week’, changes through time as agents check their own expectations and revise their own plans. Grandmont used Hicks’s temporary-equilibrium scheme to introduce money into the general-equilibrium model. If goods are perishable and therefore cannot be carried from one period to another, individuals will be forced to ask for money to transfer their own savings through time. In this way, money carries out a reserve-of-value function: it allows individuals to transfer their own wealth from one period to another or, if necessary, from one place to another, or even from one state of nature to another. If individuals receive a certain quantity of money in each period, just like any other type of good, then money becomes, to all intents and purposes, part of the equilibrium scheme, without the possibility of separ­ating the economy into a ‘real part’ and a ‘monetary part’, as in the case of the traditional dichotomy. Thus, the amount of money present in the system will affect the determination of the prices of the various goods. In Grandmont’s model, inflation is not a purely monetary phenomenon, caused by a simple excess in the money supply, but is strictly linked to real phenomena and to the expectations of the agents.

The reserve-of-value function is not, however, the only one accomplished by liquidity. Historically, money developed as a means of exchange to facilitate the organization of the processes of decentralized exchange, processes in which there is no personification of the market such as that represented by the auctioneer.

If it makes no sense to introduce money into a model such as the Walrasian one, where, at each moment, it is certain that the exchanges will take place in equilibrium with the full satisfaction of all agents, it becomes, however, extremely important to use this instrument when it is assumed that exchanges take place in a series of physically separated ‘markets’ which are not in perfect communication. Thus, even considering the function of money as a means of exchange there are valid reasons to introduce it into the general-equilibrium model. As in the preceding case, however, it is necessary to modify the structure of the reference model, by abandoning, at least in part, the Walrasian world: for example, by admitting that exchanges among individuals can also occur outside equilibrium. This road has been followed by, among others, F. M. Fisher in Disequilibrium Foundations of Equilibrium Economics (1983).

There are many other recent examples of attempts to introduce money into more or less modified models of general equilibrium. Worth mentioning are those of F. Hahn (Equilibrium and Macroeconomics,1985, and Money, Growth and Stability, 1985). These attempts emphasize the function of money, either as a reserve of value or as a means of exchange (or even its role in speculative activity). In the present state of knowledge, however, this problem has still not found a definitive and fully satisfactory solution. The general-equilibrium model owes its strength and its weakness to the meta­phor of the auctioneer, a deus ex machina who carries out the co-ordination role necessary to make the plans of the single agents mutually compatible.

Money, once introduced, plays a co-ordinating role in the exchanges in which it partially replaces the role of the auctioneer and may conflict with it. The coexistence of a ‘real’ view, emphasizing the role of the auctioneer, and a ‘monetary’ view, stressing the role of money in the co-ordination of economic activities, is therefore an awkward one, if not contradictory. As long as this contradiction is not resolved in a theoretically satisfactory way, the introduction of money into the general-equilibrium model will be, to a certain degree, artificial. Considering the practical importance of issues such as inflation and unemployment, it is not surprising that the present state of the general-equilibrium model with money still gives rise to serious doubts and fiery debates.

10.2.