Tractable General Equilibrium

TL;DR

This paper introduces a new class of algorithms called mirror extragradient algorithms for solving Walrasian economies, ensuring polynomial-time convergence in many cases, including complex and pathological economies, by framing the problem as a variational inequality.

Contribution

The paper develops mirror extragradient algorithms for Walrasian economies, providing convergence guarantees via variational inequalities and addressing previous convergence issues in pathological cases.

Findings

Algorithms converge rapidly in large economies with diverse utility functions.

Mirror extragradient process successfully handles the Scarf economy, a known pathological case.

Experimental results confirm fast convergence without failure across various economy models.

Abstract

We study Walrasian economies (or general equilibrium models) and their solution concept, the Walrasian equilibrium. A key challenge in this domain is identifying price-adjustment processes that converge to equilibrium. One such process, t\^atonnement, is an auction-like algorithm first proposed in 1874 by L\'eon Walras. While continuous-time variants of t\^atonnement are known to converge to equilibrium in economies satisfying the Weak Axiom of Revealed Preferences (WARP), the process fails to converge in a pathological Walrasian economy known as the Scarf economy. To address these issues, we analyze Walrasian economies using variational inequalities (VIs), an optimization framework. We introduce the class of mirror extragradient algorithms, which, under suitable Lipschitz-continuity-like assumptions, converge to a solution of any VI satisfying the Minty condition in polynomial time. We show that the set of Walrasian equilibria of any balanced economy-which includes among others Arrow-Debreu economies-corresponds to the solution set of an associated VI that satisfies the Minty condition but is generally discontinuous. Applying the mirror extragradient algorithm to this VI we obtain a class of t\^atonnement-like processes, which we call the mirror extrat\^atonnement process. While our VI formulation is generally discontinuous, it is Lipschitz-continuous in variationally stable Walrasian economies with bounded elasticity-including those satisfying WARP and the Scarf economy-thus establishing the polynomial-time convergence of mirror extrat\^atonnement in these economies. We validate our approach through experiments on large Arrow-Debreu economies with Cobb-Douglas, Leontief, and CES consumers, as well as the Scarf economy, demonstrating fast convergence in all cases without failure.