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Tractable General Equilibrium

Denizalp Goktas, Amy Greenwald

TL;DR

The paper tackles the computational challenge of stabilizing prices to Walrasian equilibria in general equilibrium models, including problematic Scarf economies. It introduces a tractable variational-inequality framework and a mirror extragradient method, linking equilibria to strong solutions of VI problems on the unit box and leveraging the Minty condition with pathwise Bregman-continuity to guarantee polynomial-time convergence. The mirror extratâtonnement process emerges as a natural, tâtonnement-like price-adjustment rule that converges in balanced economies and specifically in the Scarf economy, with strong theoretical guarantees and empirical validation on large-scale Arrow-Debreu-style economies across multiple utility forms. Overall, the work shifts the perspective on computational intractability in general equilibrium from a hardness barrier to a discontinuity-driven challenge, offering practical algorithms for computing Walrasian equilibria in a broad class of economies.

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, tâtonnement, is an auction-like algorithm first proposed in 1874 by Léon Walras. While continuous-time variants of tâtonnement 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 tâtonnement-like processes, which we call the mirror extratâtonnement 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 extratâtonnement 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.

Tractable General Equilibrium

TL;DR

The paper tackles the computational challenge of stabilizing prices to Walrasian equilibria in general equilibrium models, including problematic Scarf economies. It introduces a tractable variational-inequality framework and a mirror extragradient method, linking equilibria to strong solutions of VI problems on the unit box and leveraging the Minty condition with pathwise Bregman-continuity to guarantee polynomial-time convergence. The mirror extratâtonnement process emerges as a natural, tâtonnement-like price-adjustment rule that converges in balanced economies and specifically in the Scarf economy, with strong theoretical guarantees and empirical validation on large-scale Arrow-Debreu-style economies across multiple utility forms. Overall, the work shifts the perspective on computational intractability in general equilibrium from a hardness barrier to a discontinuity-driven challenge, offering practical algorithms for computing Walrasian equilibria in a broad class of economies.

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, tâtonnement, is an auction-like algorithm first proposed in 1874 by Léon Walras. While continuous-time variants of tâtonnement 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 tâtonnement-like processes, which we call the mirror extratâtonnement 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 extratâtonnement 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.

Paper Structure

This paper contains 43 sections, 33 theorems, 102 equations, 3 figures, 2 tables, 2 algorithms.

Table of Contents

  1. Introduction
  2. A Tractable Variational Inequality Framework for Walrasian Economies
  3. Technical Contributions
  4. Variational Inequalities
  5. Walrasian Economies
  6. Preliminaries
  7. Notation.
  8. Functions.
  9. Correspondences.
  10. Bregman divergence
  11. Variational Inequalities
  12. Solution Concepts and Properties
  13. First-Order Methods
  14. Mirror Gradient Algorithm
  15. Global Convergence of the Mirror Extragradient Algorithm
  16. ...and 28 more sections

Key Result

Theorem 3.1

Let $(\set, \vioper)$ be a VI satisfying the Minty condition with $\set$ non-empty, compact, and convex; and $h_{}$ a $1$-strongly-convex and $\kappa$-Lipschitz-smooth kernel function. Consider the mirror extragradient algorithm (alg:VI_mirror_extragrad) run with the VI $(\set, \vioper)$, the kernel

Figures (3)

  • Figure 1: Phase Portraits of Tâtonnement, and Extratâtonnement for the Scarf Economy, and Results of Experiments 1-7.
  • Figure : Mirror Extragradient Algorithm
  • Figure : Mirror Extratâtonnement Process

Theorems & Definitions (63)

  • Remark 1: Solution Set Properties
  • Theorem 3.1: Mirror Extragradient Method Convergence
  • Theorem 3.2: Mirror Extragradient Method Local Convergence
  • Theorem 4.1: Walrasian economies as Complementarity Problems
  • Theorem 4.2: Balanced economies as VIs
  • Lemma 1: $\varepsilon$-strong solution $\implies$ $\varepsilon$-Walrasian equilibrium
  • Theorem 4.3: Competitive economies as VIs
  • Theorem 4.4: Existence of Walrasian Equilibrium
  • Lemma 2: Balanced Economies are Variationally Stable on the Unit Box
  • Remark 2: Mirror Extratâtonnement is a Natural Price-Adjustment Process
  • ...and 53 more