Away from calculus? The rise and fall of general equilibrium theory
Regarding market models, it was Walras’s general equilibrium approach that experienced the most extraordinary developments during this time period. After being taken up by members of the so-called Vienna Colloquium in the 1930s, it blossomed in the US after World War II, and enjoyed particular support from the Cowles Commission in the 1940s and 1950s.
Its successes were largely due to the introduction into economics of topology, convex set theory and fixed-point theorems, with an entirely new way of thinking about mathematics that was based on axiomatization and originated in Bourbakist France (Weintraub 2002). A feeling of dissatisfaction with calculus was increasingly widespread, while, according to Nobel prize winner Gerard Debreu, reformulation of the theory in set-theoretical terms “forced a reexamination of several of the primitive concepts of the theory of general economic equilibrium. This was of great value” (1984: 269). From their Vienna experience, Oskar Morgenstern and John von Neumann also brought a view that the discipline needed “mathematical methods which diverge considerably from the techniques applied by older or by contemporary mathematical economists” (Morgenstern and von Neumann 1944: 1). Interestingly, they put the blame on the traditional linkages of calculus with physics: “the emphasis on mathematical methods seems to be shifted more towards combinatorics and set theory - and away from the algorithm of differential equations which dominate mathematical physics” (Morgenstern and von Neumann 1944: 45). Similarly, Debreu absorbed from Bourbakism the view that axiomatic mathematical structures must be seen as fully separated from any physical model, almost taking on a life of their own (Weintraub 2002).The new mathematics allowed for a sophisticated refinement of Walras’s approach, which goes beyond the simple count of the number of equations and number of unknowns, and may be named the Arrow-Debreu-McKenzie model after its main contributors.
One of its major achievements in the 1950s was a formal proof of existence of equilibrium, establishing that Walras’s notion of a set of prices that clear all markets is consistent. It showed that the system of simultaneous equations representing equality between supply and demand in all markets has a solution, with non-negative prices and quantities (Arrow and Debreu 1954; Debreu 1959; McKenzie 1959). Another success was the mathematical proof of the so-called two theorems of welfare economics. The first theorem states that a general equilibrium corresponds to a socially optimal allocation of resources, and the second states that, under some conditions, any socially optimal allocation of resources can be sustainable by a general equilibrium. These results amounted to rigorously establishing the desirable properties of the free market mechanism that earlier economists had put forward only intuitively. In sum, the new approach completely transformed general equilibrium theory, allowing to make it “rigorous, to generalize it, to simplify it, and to extend it in new directions” (Debreu 1984: 267).The new mathematics did not completely discard old-style tools, though. On the one hand, the assumption of continuity had always been preserved, only derivability being eliminated. A well-known criticism of this property is that it is just as unrealistic, approximate, and unable to tackle the question of indivisibilities, as in traditional models. Besides, continuous but not derivable functions are rare and ad hoc constructs, so that excluding them hardly leads to any gain in generality. On the other hand, and more importantly, Debreu himself felt the need to reintroduce differential calculus to rule out the possibility of a continuum of equilibria, which would have rendered the existence result void. In the 1970s, differentiable functions provided the “suitable conditions” (Debreu 1984: 271) to prove that the set of economies with a continuum of equilibriums is negligible.
Samuelson summarized well the failure of the project to completely replace the mathematical apparatus of economics: “I must agree that the Age of Debreu sees new and fruitful tools being used by mathematical economists... However, I do not honestly perceive any basic newness in the so-called non-physics mathematics... We benefit from the Debreu-von Neumann novelties and still operate in Isaac Newton’s style” (Samuelson 1989: 112).Still, the mix of set theory and calculus seemed to have the potential to provide the whole of economics with strong mathematical foundations. Problems started with attempts at proving two other properties of equilibrium, namely, stability and uniqueness. The former is meant to ensure that after an exogenous shock, the market mechanism can generate endogenous forces that bring it back to equilibrium; and uniqueness is needed to know where an adjustment process will drive the system after a shock. Yet proofs of stability and uniqueness under “tatonnement”, relying on systems of differential equations, could not be obtained under general conditions and required additional, excessively restrictive assumptions (Sonnenschein 1972; Mantel 1974; Debreu 1974).
Despite this ultimate failure, the project to rebuild economic theory on mathematical bases (admittedly, with varied methods and approaches) continued to be pursued for a long time, and the decline of general equilibrium theory as a major research programme only started in the 1990s after a few decisive attacks (Kirman 1989; Ingrao and Israel 1990). Still, general equilibrium theory long remained an important part of economists’ education (Mas-Colell et al. 1995) and the purely verbal style without any formalism, used by many of the forefathers of the discipline until the mid-twentieth century, has now virtually disappeared.