Computational Complexity of the Walrasian Equilibrium Inequalities

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. Following Brown and Shannon (2000), we reformulate the Walrasian equilibrium inequalities as the Hicksian equilibrium inequalities. Brown and Shannon proved that the Walrasian equilibrium inequalities are solvable iff the Hicksian equilibrium inequalities are solvable. We show that solving the Hicksian equilibrium inequalities is equivalent to solving an NP-hard minimization problem. Approximation theorems are polynomial time algorithms for computing approximate solutions of NP-hard minimization problems. The contribution of this paper is an approximation theorem for the NP-hard minimization, over indirect utility functions of consumers, of the maximum distance, over observations, between social endowments and aggregate Marshallian demands. In this theorem, we propose a polynomial time algorithm for computing an approximate solution to the Walrasian equilibrium inequalities, where explicit bounds on the degree of approximation are determined by observable market data.