Existence of Competitive Equilibrium in Linear Exchange Economies: A Simple Proof Using Brouwer’s Fixed Point Theorem

Abstract

In this paper we show that for concave piecewise linear exchange economies every competitive equilibrium satisfies the property that the competitive allocation is a non-symmetric Nash bargaining solution with weights being the initial income of individual agents evaluated at the equilibrium price vector. We prove the existence of competitive equilibrium for concave piecewise linear exchange economies by obtaining a non-symmetric Nash bargaining solution with weights being defined appropriately. In a later section we provide a simpler proof of the same result using the Brouwer’s fixed point theorem, when all utility functions are linear. In both cases the proofs pivotal step is the concave maximization problem due to Eisenberg and Gale and minor variations of it. We also provide a proof of the same results for economies where agents’ utility functions are concave, continuously differentiable and homogeneous. In this case the main argument revolves around the concave maximization problem due to Eisenberg. Unlike previous results, we do not require all initial endowments to lie on a fixed ray through the origin. 1. Introduction : Eisenberg and Gale (1959) showed that for linear exchange economies with individual initial endowment bundles lying on a fixed ray through the origin, the proof of existence of a competitive equilibrium reduces to finding a solution for a concave programming problem. In particular they showed that the corresponding competitive allocation is a nonsymmetric Nash bargaining solution (see Kalai (1977)) with weights being given by the share of each individual in the aggregate initial endowment. Moore (2007) refers to the non-symmetric Nash bargaining solution as a solution to the Cobb-Douglas-Eisenberg (CDE) aggregator function. In this paper we first show that for concave piecewise linear exchange economies every competitive equilibrium satisfies the property that a competitive allocation is a non-symmetric