A Consumer-Theoretic Characterization of Fisher Market Equilibria

TL;DR

This paper introduces a new convex programming approach to analyze Fisher market equilibria with complex utility functions, linking consumer theory with market dynamics and suggesting convergence properties of t ext{"a}tonnement.

Contribution

It develops a convex program based on expenditure functions for CCH utilities, connecting equilibrium prices with market excess demand and analyzing t ext{"a}tonnement convergence.

Findings

Convex program characterizes Fisher market equilibria with CCH utilities.

Subdifferential of the dual equals negative excess demand, linking to t ext{"a}tonnement.

Experimental results indicate t ext{"a}tonnement may converge at rate O((1+E)/t^2).

Abstract

In this paper, we bring consumer theory to bear in the analysis of Fisher markets whose buyers have arbitrary continuous, concave, homogeneous (CCH) utility functions representing locally non-satiated preferences. The main tools we use are the dual concepts of expenditure minimization and indirect utility maximization. First, we use expenditure functions to construct a new convex program whose dual, like the dual of the Eisenberg-Gale program, characterizes the equilibrium prices of CCH Fisher markets. We then prove that the subdifferential of the dual of our convex program is equal to the negative excess demand in the associated market, which makes generalized gradient descent equivalent to computing equilibrium prices via t\^atonnement. Finally, we run a series of experiments which suggest that t\^atonnement may converge at a rate of $O\left(\frac{(1+E)}{t^2}\right)$ in CCH Fisher markets that comprise buyers with elasticity of demand bounded by $E$. Our novel characterization of equilibrium prices may provide a path to proving the convergence of t\^atonnement in Fisher markets beyond those in which buyers utilities exhibit constant elasticity of substitution.