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Approximate Solutions of the Walrasian Equilibrium Inequalities with Bounded Marginal Utilities of Income

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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. Brown and Shannon (2002) derived an equivalent system of equilibrium inequalities, i.e., the dual Walrasian equilibrium inequalities. That is, 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 derive explicit bounds, where the degree of approximation is determined by observable market data.

Suggested Citation

  • Donald J. Brown, 2014. "Approximate Solutions of the Walrasian Equilibrium Inequalities with Bounded Marginal Utilities of Income," Cowles Foundation Discussion Papers 1955, Cowles Foundation for Research in Economics, Yale University.
    • Handle: RePEc:cwl:cwldpp:1955
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    File URL: https://cowles.yale.edu/sites/default/files/files/pub/d19/d1955.pdf
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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. Donald J. Brown & Chris Shannon, 2000. "Uniqueness, Stability, and Comparative Statics in Rationalizable Walrasian Markets," Econometrica, Econometric Society, vol. 68(6), pages 1529-1540, November.
    3. Donald J. Brown & Caterina Calsamiglia, 2003. "The Strong Law of Demand," Cowles Foundation Discussion Papers 1399, Cowles Foundation for Research in Economics, Yale University.
    4. Donald J. Brown & Rosa L. Matzkin, 2008. "Testable Restrictions on the Equilibrium Manifold," Lecture Notes in Economics and Mathematical Systems, in: Computational Aspects of General Equilibrium Theory, pages 11-25, Springer.
    5. Donald Brown & Felix Kubler, 2008. "Computational Aspects of General Equilibrium Theory," Lecture Notes in Economics and Mathematical Systems, Springer, number 978-3-540-76591-2, October.
    6. Donald J. Brown & Ravi Kannan, 2008. "Two Algorithms for Solving the Walrasian Equilibrium Inequalities," Lecture Notes in Economics and Mathematical Systems, in: Computational Aspects of General Equilibrium Theory, pages 69-77, Springer.
    Full references (including those not matched with items on IDEAS)

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    More about this item

    Keywords

    Rationalizable Walrasian markets; NP-hard minimization problems; Approximation theorems;
    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
    • D46 - Microeconomics - - Market Structure, Pricing, and Design - - - Value Theory

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