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Approximate Solutions of Walrasian and Gorman Polar Form Equilibrium Inequalities

In: Affective Decision Making Under Uncertainty

Author

Listed:
  • Donald J. Brown

    (Yale University)

Abstract

Recently Cherchye et al. (2011) reformulated the Walrasian equilibrium inequalities, introduced by Brown and Matzkin (1996), as an integer programming problem and proved that solving the Walrasian equilibrium inequalities is NP-hard. Following Brown and Shannon (2000), we reformulate the Walrasian equilibrium inequalities as the dual Walrasian equilibrium inequalities. Brown and Shannon proved that the Walrasian equilibrium inequalities are solvable iff the dual Walrasian equilibrium inequalities are solvable. We show that solving the dual Walrasian equilibrium inequalities is equivalent to solving a NP-hard minimization problem. Approximation theorems are polynomial time algorithms for computing approximate solutions of NP-hard minimization problems. The primary contribution of this paper is an approximation theorem for the equivalent NP-hard minimization problem. In this theorem, we propose a polynomial time algorithm for computing an approximate solution to the dual Walrasian equilibrium inequalities, where the marginal utilities of income are uniformly bounded. We derive explicit bounds on the degree of approximation from observable market data. The second contribution is the derivation of the Gorman polar form equilibrium inequalities for an exchange economy, where each consumer is endowed with an indirect utility function in Gorman polar form. If the marginal utilities of income are uniformly bounded then we prove a similar approximation theorem for the Gorman polar form equilibrium inequalities.

Suggested Citation

  • Donald J. Brown, 2020. "Approximate Solutions of Walrasian and Gorman Polar Form Equilibrium Inequalities," Lecture Notes in Economics and Mathematical Systems, in: Affective Decision Making Under Uncertainty, pages 41-54, Springer.
  • Handle: RePEc:spr:lnechp:978-3-030-59512-8_4
    DOI: 10.1007/978-3-030-59512-8_4
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    More about this item

    Keywords

    Algorithmic game theory; Computable general equilibrium theory; Refutable theories of value;
    All these keywords.

    JEL classification:

    • B41 - Schools of Economic Thought and Methodology - - Economic Methodology - - - Economic Methodology
    • C68 - Mathematical and Quantitative Methods - - Mathematical Methods; Programming Models; Mathematical and Simulation Modeling - - - Computable General Equilibrium Models
    • D58 - Microeconomics - - General Equilibrium and Disequilibrium - - - Computable and Other Applied General Equilibrium Models

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