Two Visions of Equilibrium
The preceding discussion reveals a fundamental problem with general equilibrium theory. Are we talking about a state in which all agents’ choices are consistent or are we talking about the resting point of a dynamic system, which adjusts towards such a state?
Walras discussed both, and, in the end, seemed to settle for the mathematically more tractable, and less complex, vision which is the first.
The second brings up the problem of stability which turned out much later to be the bete noire of the theory. If we adopt the first approach then all that needs to be shown is that there are such states. There could, indeed, be many of them and we do not have to be concerned with the process by which they are achieved. This view has led to what Backhaus and Maks (2006) suggest is the caricature of what they describe as “the common university trained scholar’s” understanding of the Walrasian model, which is, “Walras developed the general economic equilibrium model, but did not care about uniqueness and stability of an equilibrium. It is a model with exchange and production only and it assumes an auctioneer who announces price vectors to establish the equilibrium” (Backhaus and Maks 2006: 1).Walras had an ambiguous view, although this has been the subject of vigorous debate (see, for example, Walker 1996, 1999; De Vroey 1999). On the one hand he pursued his “scientific” view, which consisted in showing that a solution to the system of excess demand functions existed. In this, he advanced the explanation that it was sufficient to count the number of equations and unknowns and that, since in his model the two were equal, the problem was solved. In fact it was Abraham Wald (1936) much later who showed that equilibrium did exist and for this it was necessary to use a fixed point theorem which was not proved till after Walras’s death. (The first fixed point theorem was proved by Brouwer 1912 and there were later generalizations by, for example, Kakutani 1941 which have been widely used in economics.) The alternative view, that of a system which would converge to an equilibrium state, invokes the problem of stability (see below).
However, before embarking on this subject, a number of remarks are in order. The idea that the essential problem was that of showing that equilibrium exists was far from being universally accepted, and yet what is often regarded as the apogee of General Equilibrium theory, the results of Arrow and Debreu (1954), do no more than this.
Furthermore, the vision of economic equilibrium as the solution of a set of simultaneous equations was criticized by a number of leading economic theorists and this is well illustrated by the observation of John Hicks (1939: 60) who remarked, disparagingly,
To some people (including no doubt Walras himself) the system of simultaneous equations determining a whole price-system seems to have vast significance. They derive intense satisfaction from the contemplation of such a system of subtly interrelated prices; and the further the analysis can be carried (in fact it can be carried a good way)... the better they are pleased, and the profounder the insight into the working of a competitive economic system they feel they get.
Nevertheless, the idea of finding a solution to a set of equations, which was, in large part, solved by Wald (1936), was in the spirit of the Bourbaki tradition and it is not surprising that it should have been Debreu who pushed Walrasian theory in this direction. He was following on from Maurice Allais, his mentor, who was clearly oriented in the direction of mathematical purity rather than realism. Allais said, “The fundamental Anglo-Saxon quality is satisfaction with the accumulation of facts. The need for clarity, for logical coherence and for synthesis is, for an Anglo-Saxon, only a minor need, if it is a need at all. For a Latin, and particularly a Frenchman, it is exactly the opposite” (Allais 1952: 58). However, in thinking in this way Allais was encouraging the distancing of economics from reality. Indeed this was exactly in the Bourbaki tradition for as Bourbaki (1949: 2) said:
Why do applications [of mathematics] ever succeed? Why is a certain amount of logical reasoning occasionally helpful in practical life? Why have some of the most intricate theories in mathematics become an indispensable tool to the modern physicist, to the engineer, and to the manufacturer of atom-bombs? Fortunately for us, the mathematician does not feel called upon to answer such questions.
Although to many economists this was not the appropriate route to follow, it is worth observing that considerable progress was made along the way in weakening the assumptions on individual characteristics necessary to achieve this result. Pareto (1906) showed that only an ordinal representation of preferences was necessary. A result of Shapley and Folkman could be used to show that the highly contestable assumption of the convexity of preferences was not necessary, if one was satisfied with a close approximation of the true equilibrium. (The lemma is stated and the original proof given in Starr 1969.) Incomplete preferences and other relaxations of the traditional axioms of rationality are discussed in Chipman et al. (1971).
A second and very important result of this line of thought was the demonstration by Vilfredo Pareto (1906), Walras’s successor at the University of Lausanne, that a competitive equilibrium is what is now referred to as a Pareto optimal state. This should be regarded essentially as an efficiency criterion. In such a state no individual can be made better off without making another worse off. This has widely, and often erroneously, been used to justify the market mechanism as a solution to society’s economic allocation problems. However, even before Pareto showed that a competitive equilibrium was efficient, in this sense Walras himself was at lengths to point out that there were no distributional considerations involved in defining such an equilibrium. Thus, ideas of equity and fairness nowhere figure in the standard general equilibrium framework. (This does not mean that such considerations cannot be included in the model as shown by Kolm 1968, Feldman and Kirman 1973 and Foley 1967. However, this literature is usually considered as an intriguing but unimportant side road.) Nevertheless, the importance attached to this result, which has come to be called the First Fundamental Theorem of Welfare Economics, can be measured by the disproportionate emphasis that has been given to the mathematical appendix to Pareto’s work (see Kirman 1998, 2013).
However, the very foundations of the analysis that led to this result are dependent on the notion of rational, optimizing agents, and Pareto in his later sociological work expressed a great deal of skepticism about this. He even went as far as to suggest that people spend a little part of their time taking non-rational decisions and the rest of their time rationalizing them!