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A Consumer-Theoretic Characterization of Fisher Market Equilibria

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  • Denizalp Goktas
  • Enrique Areyan Viqueira
  • Amy Greenwald

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.

Suggested Citation

  • Denizalp Goktas & Enrique Areyan Viqueira & Amy Greenwald, 2021. "A Consumer-Theoretic Characterization of Fisher Market Equilibria," Papers 2107.08153, arXiv.org, revised Jan 2022.
  • Handle: RePEc:arx:papers:2107.08153
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    References listed on IDEAS

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    1. William C. Brainard & Herbert E. Scarf, 2005. "How to Compute Equilibrium Prices in 1891," American Journal of Economics and Sociology, Wiley Blackwell, vol. 64(1), pages 57-83, January.
    2. Xavier Vives, 1987. "Small Income Effects: A Marshallian Theory of Consumer Surplus and Downward Sloping Demand," The Review of Economic Studies, Review of Economic Studies Ltd, vol. 54(1), pages 87-103.
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    Cited by:

    1. Denizalp Goktas & Amy Greenwald, 2022. "Gradient Descent Ascent in Min-Max Stackelberg Games," Papers 2208.09690, arXiv.org.
    2. Denizalp Goktas & Jiayi Zhao & Amy Greenwald, 2023. "T\^atonnement in Homothetic Fisher Markets," Papers 2306.04890, arXiv.org.

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