GLOBAL ANALYSIS AND ECONOMICS IIA Extension of a theorem of Debreu

This note is a drastic revision and extension of the author’s paper (1972). We recall the following theorem of Debreu (1970) for a pure exchange economy : One is given C1 demand functions, fi, . . ,, f,, of m consumers, satisfying a certain boundary condition. Then for almost all initial allocations of commodities to the m consumers, there are only a finite number of price equilibria. Our goal is to remove the hypothesis of Cl demand functions, and in fact to remove the need for any well-defined demand functions. In their place, we study utility functions directly; our result is again finiteness and stability for the price equilibria. This is proved only for almost all initial allocations and almost all choices of utility functions. Our class of utility functions include those which define C’ demand functions. We use a different boundary condition than Debreu. No convexity or monotonicity assumptions are made on the utility functions. However, throughout, we do assume that they are of class C2 (have continuous second derivatives) and also that they have no critical points. Also we work with commodity bundles which may have some coordinates zero. In carrying out the above program we introduce an extension of the notion of price equilibria, which we hope to develop in future papers in the series. This note while written in the spirit of Smale (1972 and 1973) does not depend on these papers. Conversations with Debreu have been very helpful in writing this paper.