Approximate Solutions of Walrasian Equilibrium Inequalities with Bounded Marginal Utilities of Income

Abstract

Recently Cherchye et al. (2011) reformulated the Walrasian equilibrium inequalities, introduced by Brown and Matzkin (1996), as an integer programming problem and proved that solving the Walrasian equilibrium inequalities is NP-hard. Brown and Shannon (2002) derived an equivalent system of equilibrium inequalities, i.e., the dual Walrasian equilibrium inequalities. That is, the Walrasian equilibrium inequalities are solvable if the dual Walrasian equilibrium inequalities are solvable.

We show that solving the dual Walrsian equilibrium inequalities is equivalent to solving a NP-hard minimization problem. Approximation theorems are polynomial time algorithms for computing approximate solutions of NP-hard minimization problems. The primary contribution of this paper is an approximation theorem for the equivalent NP-hard minimization problem. In this theorem, we derive explicit bounds, where the degree of approximation is determined by observable market data.

Keywords: Algorithmic game theory, Computable general equilibrium theory, Refutable theories of value

JEL Classification: B41, C68, D46