Market Equilibria in Fisher’s Model for Concave Economic Environments

Abstract

Eisenberg and Gale [1959] showed that market equilibrium in Fisher’s model with linear utility functions could be obtained by maximizing a concave objective function subject to linear constraints. Eisenberg [1961] generalized this result to concave and linearly homogeneous functions. Eisenberg’s result can be used to establish that market equilibrium in economic environments where utility functions are homogeneous of arbitrary degree can be obtained by maximizing a concave objective function subject to linear constraints. This note provides an independent proof of this latter result in a setting slightly more general than the one considered by Fisher. We allow for a technology to convert resources that an economy may be initially endowed with into consumable goods. While shadow price of the resources is a byproduct of our analysis, we are able to obtain the existence of equilibrium prices of the consumable goods by solving a concave-linear optimization problem, whose objective function is similar to the ones considered by Eisenberg and Gale [1959] and Eisenberg [1961].