General equilibrium and the theory of directed graphs

Abstract

The classical Arrow-Debreu existence theorem for a general equilibrium requires an assumption of positive endowments of all commodities by all consumers. This paper presents new results for relaxing this assumption for a class of economic models in which utility functions and production possibility sets are limited to certain special but useful forms, including the commonly used Cobb-Douglas and constant elasticity of substitution (CES) forms. For a given model, a directed graph is constructed which relates the sources of commodities and profits to users of commodities and recipients of profits, and it is proved that the existence of a competitive equilibrium is related to the strong connectedness of the graph. This allows equilibrium existence testing for economic models of arbitrary size by efficient computational algorithms.

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