Tâtonnement in Homothetic Fisher Markets

Abstract

A prevalent theme in the economics and computation literature is to identify natural price-adjustment processes by which sellers and buyers in a market can discover equilibrium prices. An example of such a process is tâtonnement, an auction-like algorithm first proposed in 1874 by French economist Walras in which sellers adjust prices based on the Marshallian demands of buyers, i.e., budget-constrained utility-maximizing demands. A dual concept in consumer theory is a buyer's Hicksian demand, i.e., consumptions that minimize expenditure while achieving a desired utility level. In this paper, we identify the maximum of the absolute value of the elasticity of the Hicksian demand, i.e., the maximum percentage change in the Hicksian demand of any good w.r.t. the change in the price of some other good, as an economic parameter sufficient to capture and explain a range of convergent and non-convergent tâtonnement behaviors in a broad class of markets. In particular, we prove the convergence of tâtonnement at a rate of O((1+ε2)/T), in homothetic Fisher markets with bounded price elasticity of Hicksian demand, i.e., Fisher markets in which consumers have preferences represented by homogeneous utility functions and the price elasticity of their Hicksian demand is bounded, where ε is the maximum absolute value of the price elasticity of Hicksian demand across all buyers. Our result not only generalizes known convergence results for CES Fisher markets, but extends them to mixed nested CES markets and Fisher markets with continuous, possibly non-concave, homogeneous utility functions. Our convergence rate covers the full spectrum of nested CES utilities, including Leontief and linear utilities, unifying previously existing disparate convergence and non-convergence results. In particular, for ε = 0, i.e., Leontief markets, we recover the best-known convergence rate of O(1/T), and as ε → ∞, e.g., linear Fisher markets, we obtain non-convergent behavior, as expected.