Approximate Solutions of Walrasian and Gorman Polar Form 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 dual Walrasian equilibrium inequalities. Brown and Shannon proved that the Walrasian equilibrium inequalities are solvable iff the dual Walrasian equilibrium inequalities are solvable. We show that solving the dual Walrasian 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 propose a polynomial time algorithm for computing an approximate solution to the dual Walrasian equilibrium inequalities, where the marginal utilities of income are uniformly bounded. We derive explicit bounds on the degree of approximation from observable market data. The second contribution is the derivation of the Gorman polar form equilibrium inequalities for an exchange economy, where each consumer is endowed with an indirect utility function in Gorman polar form. If the marginal utilities of income are uniformly bounded then we prove a similar approximation theorem for the Gorman polar form equilibrium inequalities.