The Algebraic Geometry of Competitive Equilibrium

The classical tools of general equilibrium theory are convexity and general topology. One of Debreu’s lasting contributions has been to show how the tools of differential topology may serve to yield more refined information about equilibrium. In particular, Debreu (1970) showed that differential topology could provide a rigorous formalization of “counting equations and unknowns” to provide a satisfactory result on the determinacy of equilibrium. Debreu required that preferences be representable by C utility functions with non-vanishing gradients and that indifference surfaces have non-vanishing curvature. Following Debreu, a natural question to ask is: Are there other interesting classes of preferences (or demands) exhibiting regularity of behavior sufficient to guarantee the generic local determinacy of equilibrium prices? The first results in this direction were obtained by Rader (1972, 1973), who showed that generic finiteness of the equilibrium price set was a consequence of demand being differentiable almost everywhere and satisfying condition N : that the image of a null set is null. Rader also gave conditions on preferences sufficient to generate demand meeting these hypotheses; a further set of sufficient conditions was developed in a later paper (1979). The concave-utility requirement of Rader (1973) has only recently been relaxed to local concavifiability of the preference relation, by Pascoa and Werlang (1989). The work reported here extends the work of Debreu in a different direction. Its motivation comes from two sources. The first is a remark made by Rader at an NBER Conference in the 1970’s about the nature of demand when preferences are analytic. The second is the work of Blume and Zame (1989) and Schanuel, Simon and Zame (forthcoming) on the applications of algebraic geometry in non-cooperative game theory. When asked about the potential applications of the methods of those papers to general equilibrium theory, our response was to be skeptical because the required hypotheses seemed to be so strong. However, we later realized that these algebraic-geometric methods could in fact be extended to a much wider class of