Intersecting families of sets and the topology of cones in economics

Abstract

Two classical problems in economics, the existence of a market equilibrium and the existence of social choice functions, are formalized here by the properties of a family of cones associated with the economy. It was recently established that a necessary and sufficient condition for solving the former is the nonempty intersection of the family of cones, and one such condition for solving the latter is the acyclicity of the unions of its subfamilies. We show an unexpected but clear connection between the two problems by establishing a duality property of the homology groups of the nerve defined by the family of cones. In particular, we prove that the intersection of the family of cones is nonempty if and only if every subfamily has acyclic unions, thus identifying the two conditions that solve the two economic problems. In addition to their applications to economics, the results are shown to extend significantly several classical theorems, providing unified and simple proofs: Helly's theorem, Caratheodory's representation theorem, the Knaster-Kuratowski-Marzukiewicz theorem, Brouwer's fixed point theorem, and Leray's theorem on acyclic covers.