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Tâtonnement in Homothetic Fisher Markets

Denizalp Goktas, Jiayi Zhao, Amy Greenwald

TL;DR

The paper addresses the convergence of price-adjustment dynamics (tâtonnement) in Fisher markets with homothetic utilities. It introduces the maximum Hicksian price elasticity across buyers, $\varepsilon$, as the key economic parameter that governs convergence and proves that entropic tâtonnement converges at rate $O((1+\varepsilon^2)/T)$, unifying and extending results from Leontief to linear and nested CES markets, even when utilities are non-concave. The analysis leverages a dual convex program (Eisenberg–Gale) and shows that the associated potential is KL-smooth, enabling a mirror-descent-based argument. This provides a broad computational perspective on market equilibria, linking economic assumptions to rigorous convergence guarantees. The work suggests future directions for exploring negative cross-price Hicksian elasticities and potential tighter bounds beyond the current $O((1+\varepsilon^2)/T)$ rate.

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. A dual concept in consumer theory is a buyer's Hicksian demand. In this paper, we identify the maximum of the absolute value of the elasticity of the Hicksian demand, 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 $O((1+\varepsilon^2)/T)$, 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 $\varepsilon \geq 0$ 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 $\varepsilon = 0$, i.e., Leontief markets, we recover the best-known convergence rate of $O(1/T)$, and as $\varepsilon \to \infty$, e.g., linear Fisher markets, we obtain non-convergent behavior, as expected.

Tâtonnement in Homothetic Fisher Markets

TL;DR

The paper addresses the convergence of price-adjustment dynamics (tâtonnement) in Fisher markets with homothetic utilities. It introduces the maximum Hicksian price elasticity across buyers, , as the key economic parameter that governs convergence and proves that entropic tâtonnement converges at rate , unifying and extending results from Leontief to linear and nested CES markets, even when utilities are non-concave. The analysis leverages a dual convex program (Eisenberg–Gale) and shows that the associated potential is KL-smooth, enabling a mirror-descent-based argument. This provides a broad computational perspective on market equilibria, linking economic assumptions to rigorous convergence guarantees. The work suggests future directions for exploring negative cross-price Hicksian elasticities and potential tighter bounds beyond the current rate.

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. A dual concept in consumer theory is a buyer's Hicksian demand. In this paper, we identify the maximum of the absolute value of the elasticity of the Hicksian demand, 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 , 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 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 , i.e., Leontief markets, we recover the best-known convergence rate of , and as , e.g., linear Fisher markets, we obtain non-convergent behavior, as expected.
Paper Structure (20 sections, 21 theorems, 72 equations, 1 figure, 1 table)

This paper contains 20 sections, 21 theorems, 72 equations, 1 figure, 1 table.

Table of Contents

  1. Introduction
  2. Interpretation via Pseudo-Game Theory
  3. Technical Contributions
  4. Related Work
  5. Preliminaries
  6. Notation.
  7. Mathematical Preliminaries
  8. Mirror Descent
  9. Consumer Theory
  10. Fisher Markets
  11. Homothetic Fisher Markets
  12. Market Parameters
  13. Convergence Bounds for Entropic Tâtonnement
  14. Base case: $t = 0$.
  15. Inductive hypothesis.
  16. ...and 5 more sections

Key Result

Theorem 2.1

Let $\{{\boldsymbol x}^{t}\}_{t}$ be the iterates generated by mirror descent with Bregman divergence $\divergence[h]$. Suppose $f$ and $h$ are convex, and for all $t \in \mathbb N$ and for some $\gamma > 0$, it holds that $f(\bm{x}^{(t+1)}) \leq \lapprox[f][\bm{x}^{(t+1)}][\bm{x}^{(t)}] + \gamma \d

Figures (1)

  • Figure 1: A summary of known results in Fisher markets.

Theorems & Definitions (35)

  • Theorem 2.1: grad-prop-response,fisher-tatonnement
  • Definition 2.2: Competitive Equilibrium
  • Theorem 2.3: goktas2022consumer
  • Theorem 2.4: goktas2022consumer
  • Lemma 3.0
  • Lemma 3.1
  • proof : Proof of \ref{['lemma:expend_change']}
  • Lemma 4.0
  • proof : Proof of \ref{['ineq-devanur']}
  • Lemma 4.0: Bounded Indirect Utility for Homothetic Fisher Markets
  • ...and 25 more