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Competitive equilibria in semi-algebraic economies

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  • Kubler, Felix
  • Schmedders, Karl

Abstract

This paper develops a method to compute the equilibrium correspondence for exchange economies with semi-algebraic preferences. Given a class of semi-algebraic exchange economies parameterized by individual endowments and possibly other exogenous variables such as preference parameters or asset payoffs, there exists a semi-algebraic correspondence that maps parameters to positive numbers such that for generic parameters each competitive equilibrium can be associated with an element of the correspondence and each endogenous variable (i.e. prices and consumptions) is a rational function of that value of the correspondence and the parameters. This correspondence can be characterized as zeros of a univariate polynomial equation that satisfy additional polynomial inequalities. This polynomial as well as the rational functions that determine equilibrium can be computed using versions of Buchberger's algorithm which is part of most computer algebra systems. The computation is exact whenever the input data (i.e. preference parameters etc.) are rational. Therefore, the result provides theoretical foundations for a systematic analysis of multiplicity in applied general equilibrium.

Suggested Citation

  • Kubler, Felix & Schmedders, Karl, 2010. "Competitive equilibria in semi-algebraic economies," Journal of Economic Theory, Elsevier, vol. 145(1), pages 301-330, January.
  • Handle: RePEc:eee:jetheo:v:145:y:2010:i:1:p:301-330
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    References listed on IDEAS

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    1. Mas-Colell,Andreu, 1990. "The Theory of General Economic Equilibrium," Cambridge Books, Cambridge University Press, number 9780521388702, Fall.
    2. Cass, David, 2006. "Musings on the Cass trick," Journal of Mathematical Economics, Elsevier, vol. 42(4-5), pages 374-383, August.
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    7. Blume, Lawrence E & Zame, William R, 1994. "The Algebraic Geometry of Perfect and Sequential Equilibrium," Econometrica, Econometric Society, vol. 62(4), pages 783-794, July.
    8. Kubler, Felix & Schmedders, Karl, 2000. "Computing Equilibria in Stochastic Finance Economies," Computational Economics, Springer;Society for Computational Economics, vol. 15(1-2), pages 145-172, April.
    9. Mas-Colell, Andreu, 1977. "On the equilibrium price set of an exchange economy," Journal of Mathematical Economics, Elsevier, vol. 4(2), pages 117-126, August.
    10. Kam-Chau Wong & Marcel K. Richter, 1999. "Non-computability of competitive equilibrium," Economic Theory, Springer;Society for the Advancement of Economic Theory (SAET), vol. 14(1), pages 1-27.
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    12. Gjerstad, S., 1996. "Multiple Equilibria in Exchange Economies with Homothetic, Nearly Identical Preferences," Papers 288, Minnesota - Center for Economic Research.
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    15. Felix Kubler, 2007. "Approximate Generalizations and Computational Experiments," Econometrica, Econometric Society, vol. 75(4), pages 967-992, July.
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    Cited by:

    1. Tao Zha & Juan F. Rubio-Ramirez & Daniel F. Waggoner & Andrew T. Foerster, 2010. "Perturbation Methods for Markov-Switching Models," 2010 Meeting Papers 239, Society for Economic Dynamics.
    2. Andrew Foerster & Juan F. Rubio‐Ramírez & Daniel F. Waggoner & Tao Zha, 2016. "Perturbation methods for Markov‐switching dynamic stochastic general equilibrium models," Quantitative Economics, Econometric Society, vol. 7(2), pages 637-669, July.
    3. Arias-R., Omar Fdo., 2014. "A condition for determinacy of optimal strategies in zero-sum convex polynomial games," MPRA Paper 57099, University Library of Munich, Germany.
    4. repec:eee:apmaco:v:321:y:2018:i:c:p:614-632 is not listed on IDEAS
    5. repec:eee:dyncon:v:84:y:2017:i:c:p:77-90 is not listed on IDEAS
    6. Toda, Alexis Akira, 2017. "Huggett economies with multiple stationary equilibria," Journal of Economic Dynamics and Control, Elsevier, vol. 84(C), pages 77-90.
    7. Soares, Helena & Sequeira, Tiago Neves & Marques, Pedro Macias & Gomes, Orlando & Ferreira-Lopes, Alexandra, 2018. "Social infrastructure and the preservation of physical capital: Equilibria and transitional dynamics," Applied Mathematics and Computation, Elsevier, vol. 321(C), pages 614-632.
    8. Orrego, Fabrizio, 2011. "Demografía y precios de activos," Revista Estudios Económicos, Banco Central de Reserva del Perú, issue 22, pages 83-101.
    9. Ian Ayres & Colin Rowat & Nasser Zakariya, 2011. "Optimal voting rules for two-member tenure committees," Social Choice and Welfare, Springer;The Society for Social Choice and Welfare, vol. 36(2), pages 323-354, February.
    10. Li, Xiaoliang & Wang, Dongming, 2014. "Computing equilibria of semi-algebraic economies using triangular decomposition and real solution classification," Journal of Mathematical Economics, Elsevier, vol. 54(C), pages 48-58.
    11. Arias-R., Omar Fdo., 2014. "On the pseudo-equilibrium manifold in semi-algebraic economies with real financial assets," MPRA Paper 54297, University Library of Munich, Germany.

    More about this item

    Keywords

    Semi-algebraic preferences Equilibrium correspondence Polynomial equations Grobner bases Equilibrium multiplicity;

    JEL classification:

    • D50 - Microeconomics - - General Equilibrium and Disequilibrium - - - General
    • C63 - Mathematical and Quantitative Methods - - Mathematical Methods; Programming Models; Mathematical and Simulation Modeling - - - Computational Techniques

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