Kaldor on Debreu: The Critique of General Equilibrium Reconsidered

Abstract

This paper revisits Kaldor's methodological critique of orthodox economics. The main target of his critique was the theory of general equilibrium as expounded in the work of Debreu and others. Kaldor deemed this theory to be seriously flawed as an empirically adequate description of real-world economies. According to Kaldor, scientific progress was not possible in economics without a major act of demolition, by which he meant the destruction of the basic conceptual framework of the theory of general equilibrium. We extend Kaldor's critique by recourse to major developments in 20th century philosophy of mathematics, and then go on to demonstrate that Debreu's work, based as it is on Bourbakist formalism and in particular Cantorian set theory, is conceptually incompatible with Kaldor's requirements for an empirical science. This aspect of Kaldor's critique has not been explored, and as a consequence a major source of substantiating his critique has remained undeveloped.

Notes

¹This section draws in part on material in Section 1 of Boylan & O'Gorman (1997).

²While Kaldor's concern in his critical writings, particularly from the 1970s, was to dismantle the whole edifice of general equilibrium theory, it is interesting to note that at this time there emerged a series of papers that are conventionally interpreted as representing a major ‘internalist’ technical critique of the failure of general equilibrium theory to provide proofs of the uniqueness and stability based on general characterizations of preferences and technologies. These were the papers by Sonnenschein (1973), Mantel (1974) and Debreu (1974) – generally referred to as the SMD theorem – which inflicted what appeared to be a fatal wound to stability analysis in general equilibrium theory, certainly to any version of that theory which employed the Walrasian tâtonnement process and aggregate excess demand functions. These authors demonstrated that the only general properties possessed by the aggregate excess demand function (which is used to characterize the competitive equilibria) were those of continuity, homogeneity of degree zero, and the validity of Walras's law. Beyond that, as contained in the memorable phrase of Mas-Colell et al. (1995, p. 548), ‘anything goes.’ The SMD results showed, as Tohmé (2006, p. 214) succinctly summarized them, ‘that for every given system of equilibrium prices and its associated excess demands, an arbitrary economy can be defined, exhibiting the same aggregate behaviour and the same equilibria. That is, prices do not convey all the relevant information about the economy, since a “mock” one is able to generate the same aggregate demand.’ Kirman (2006, p. 257; emphasis in original) has recently noted, ‘The full force of the Sonnenschein, Mantel and Debreu (SMD) result is often not appreciated. Without stability or uniqueness, the intrinsic interest of economic analysis based on the general equilibrium model is extremely limited.’ While this sentiment would surely have found favour with Kaldor, it arguably falls far short of his more fundamental call for the ‘demolition’ of general equilibrium theory as a major inhibition to the development of economics as a science, and certainly as an empirical science. We do not pursue the implications of the SMD results here as this would take us too far from our central concerns in this paper.

³For a more detailed account of this Hilbertian, purely formalist reading of an axiomatic system, see Boylan & O'Gorman (2008).

⁴Debreu presupposed what is technically known as Brouwer's fixed point theorem in his proof of the existence of equilibrium. However, this theorem cannot be proved in non-Cantorian, computable mathematics. Moreover efforts by Scarf and others to render equilibrium constructable also fail. For more on this, see Boylan & O'Gorman (2008).

⁵We are currently completing an extended analysis of the Poincaré–Walras correspondence, with particular reference to the mathematization of economics as developed by Walras and its later development in the Neo-Walrasian programme.

⁶These developments in non-Cantorian mathematics and their relevance to economic methodology are discussed in more detail in Boylan & O'Gorman (forthcoming 2009).

⁷The phrase ‘theological economics’ was suggested to us by Carnap's characterization of Ramsey's work as ‘theological mathematics’ (Carnap, 1983, cited in Benacerraf & Putnam, 1983, p. 50).

⁸For Poincaré, the notion of what is in principle attainable in a finite number of steps requires the notion of potential infinity. Suppose we reject the relevance of the notion of potential infinity to the correct explication of the notion of what is in principle possible in a finite number of steps. On this supposition, there is some finite upper limit to the number of steps that are, in principle, possible. Call this upper limit . As Poincaré notes, in pure arithmetic we are not limited: there is nothing in principle wrong with the number etc. Thus, the Poincaré programme for pure mathematics is attempting to hold a middle ground between the too restrictive nature of a strict finitist approach to mathematics and the excesses of Cantorian actual infinity. This middle ground is the potentially infinite.

⁹Brown (2002) provides an interesting and readable introduction to these topics.

¹⁰As Poincaré notes, the number of steps ‘is greater than aleph zero’ (Poincaré, 1963, p. 67), the first of Cantor's transfinite numbers.

¹¹Velupillai, who is well aware of the efforts of Scarf and others to develop constructive general equilibrium, shows how the non-constructive Brouwer fixed point theorem is used in these efforts.