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Competitive Equilibria in Semi-Algebraic Economies

Author

Listed:
  • Felix Kuber

    () (Department of Economics, University of Pennsylvania)

  • Karl Schmedders

    () (Kellogg – MEDS, Northwestern University)

Abstract

This paper examines the equilibrium correspondence in Arrow-Debreu exchange economies with semi-algebraic preferences. We show that a generic semi-algebraic exchange economy gives rise to a square system of polynomial equations with finitely many solutions. The competitive equilibria form a subset of the solution set and can be identified by verifying finitely many polynomial inequalities. We apply methods from computational algebraic geometry to obtain an equivalent polynomial system of equations that essentially reduces the computation of all equilibria to finding all roots of a univariate polynomial. This polynomial can be used to determine an upper bound on the number of equilibria and to approximate all equilibria numerically. We illustrate our results and computational method with several examples. In particular, we show that in economies with two commodities and two agents with CES utility, the number of competitive equilibria is never larger than three and that multiplicity of equilibria is rare in that it only occurs for a very small fraction of individual endowments and preference parameters.

Suggested Citation

  • Felix Kuber & Karl Schmedders, 2007. "Competitive Equilibria in Semi-Algebraic Economies," PIER Working Paper Archive 07-013, Penn Institute for Economic Research, Department of Economics, University of Pennsylvania.
  • Handle: RePEc:pen:papers:07-013
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    File URL: http://economics.sas.upenn.edu/system/files/working-papers/07-013.pdf
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    References listed on IDEAS

    as
    1. Mas-Colell,Andreu, 1990. "The Theory of General Economic Equilibrium," Cambridge Books, Cambridge University Press, number 9780521388702, December.
    2. Cass, David, 2006. "Musings on the Cass trick," Journal of Mathematical Economics, Elsevier, vol. 42(4-5), pages 374-383, August.
    3. P. Jean-Jacques Herings & Felix Kubler, 2002. "Computing Equilibria in Finance Economies," Mathematics of Operations Research, INFORMS, vol. 27(4), pages 637-646, November.
    4. Anderson, Robert M. & Raimondo, Roberto C., 2007. "Incomplete markets with no Hart points," Theoretical Economics, Econometric Society, vol. 2(2), June.
    5. Smale, S., 1974. "Global analysis and economics IIA : Extension of a theorem of Debreu," Journal of Mathematical Economics, Elsevier, vol. 1(1), pages 1-14, March.
    6. Chiappori, Pierre-Andre & Rochet, Jean-Charles, 1987. "Revealed Preferences and Differentiable Demand: Notes and Comments," Econometrica, Econometric Society, vol. 55(3), pages 687-691, May.
    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.
    11. Brown, Donald J & DeMarzo, Peter M & Eaves, B Curtis, 1996. "Computing Equilibria When Asset Markets Are Incomplete," Econometrica, Econometric Society, vol. 64(1), pages 1-27, January.
    12. Gjerstad, S., 1996. "Multiple Equilibria in Exchange Economies with Homothetic, Nearly Identical Preferences," Papers 288, Minnesota - Center for Economic Research.
    13. Balasko, Yves, 1979. "Economies with a finite but large number of equilibria," Journal of Mathematical Economics, Elsevier, vol. 6(2), pages 145-147, July.
    14. Brown, Donald J & Matzkin, Rosa L, 1996. "Testable Restrictions on the Equilibrium Manifold," Econometrica, Econometric Society, vol. 64(6), pages 1249-1262, November.
    15. Felix Kubler, 2007. "Approximate Generalizations and Computational Experiments," Econometrica, Econometric Society, vol. 75(4), pages 967-992, July.
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    Citations

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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

    computable general equilibrium; semi-algebraic economy; Groebner bases;

    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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