Some Consequences of the Unknottedness of the Walras Correspondence

Two basic properties concerning the dynamic behavior of competitive equilibria of exchange economies with complete markets are derived essentially from the fact that the Walras correspondence has no knots. I. One can view the work of Debreu, E. and H. Dierker, Balasko and others that introduced the notions of regular economy and regular equilibrium, and studied the local uniqueness and other more global properties of the Walras correspondence, as providing a formal mathematical framework for studying comparative statics properties of competitive equilibria. At the same time, P.A. Samuelson has pointed out repeatedly (see for instance Samuelson (1983)) that, in order to derive meaningful statements concerning the comparative statics properties of a given equilibrium, one also needs to take its stability properties into account. Samuelson also believed that, in many cases, perverse comparative statics properties are associated with unstable equilibria, and termed such a relationship, roughly, the correspondence principle. However, a basic problem with the correspondence principle and more importantly with the capability of doing meaningful comparative statics in Samuelson’s sense, is that it is not clear what dynamics one should use when evaluating the stability properties of a given competitive equilibrium. In this paper, we show that important information concerning the stability properties of competitive equilibria can be obtained directly from Dipartimento di Matematica, Università degli Studi di Pavia, 27100 Pavia, Italy. CORE, 34, Voie du Roman Pays, 1348 Louvain-la-Neuve, Belgium. We thank H. Polemarchakis and P. Siconolfi for useful comments. The first author is grateful to CORE for its generous hospitality, the second author gratefully acknowledges financial support from the European Commission, Grant ERBFMBICT972857. All errors are ours.