An Equilibrium Index Theory in an Economy with Convex Production Technologies

Abstract

Global analysis of economic equilibria for the purpose of counting the number of equilibria has been one of the important tasks of general equilibrium theories. Most of the analysis has relied on the mathematical tools borrowed from differential topology. In applying the tools economists have had to admit strong assumptions, two of which are differentiability and single-valuedness of excess demand fuctions (Dierker 1972; Nishimura 1978; Varian 1975; Yun 1981). If we look at the production sets even with the convexity assumption, we find it natural that there are kinks and flat pieces. Supply fuctions associated with these production sets are non-differentiable and multivalued. Kehoe(1983) and Mas-Colell (1985) concentrated on the problem of multivaluedness. They solved the problem by introducing a single-valued function whose fixed points are equivalent to the equilibria. Here, we solve both of the problems, non-differentiability and multivaluedness, by constructing a Lipschitz continuous function from which we develop an equilibrium index theory. We use generalized Jacobians for Lipschitz continuous functions defined by Clarke(1983) to compute the equilibrium indices.