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Competitive Equilibria in Semi-Algebraic Economies

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  • 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.

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Bibliographic Info

Paper provided by Penn Institute for Economic Research, Department of Economics, University of Pennsylvania in its series PIER Working Paper Archive with number 07-013.

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Length: 37 pages
Date of creation: 22 Mar 2007
Date of revision:
Handle: RePEc:pen:papers:07-013

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

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References

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  1. David Cass, 2006. "Musings on the Cass Trick," PIER Working Paper Archive 06-011, Penn Institute for Economic Research, Department of Economics, University of Pennsylvania.
  2. 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.
  3. Brown, Donald J & Matzkin, Rosa L, 1996. "Testable Restrictions on the Equilibrium Manifold," Econometrica, Econometric Society, vol. 64(6), pages 1249-62, November.
  4. Lawrence E. Blume & William R. Zame, 1993. "The Algebraic Geometry of Perfect and Sequential Equilibrium," Game Theory and Information 9309001, EconWPA.
  5. Balasko, Yves, 1979. "Economies with a finite but large number of equilibria," Journal of Mathematical Economics, Elsevier, vol. 6(2), pages 145-147, July.
  6. Herings,P. Jean-Jacques & Kubler,Felix, 2002. "Computing Equilibria in Finance Economies," Research Memorandum 010, Maastricht University, Maastricht Research School of Economics of Technology and Organization (METEOR).
  7. Kam-Chau Wong & Marcel K. Richter, 1999. "Non-computability of competitive equilibrium," Economic Theory, Springer, vol. 14(1), pages 1-27.
  8. 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.
  9. Kubler, Felix & Schmedders, Karl, 2000. "Computing Equilibria in Stochastic Finance Economies," Computational Economics, Society for Computational Economics, vol. 15(1-2), pages 145-72, April.
  10. Felix Kubler, 2007. "Approximate Generalizations and Computational Experiments," Econometrica, Econometric Society, vol. 75(4), pages 967-992, 07.
  11. Gjerstad, S., 1996. "Multiple Equilibria in Exchange Economies with Homothetic, Nearly Identical Preferences," Papers 288, Minnesota - Center for Economic Research.
  12. Anderson, Robert M. & Raimondo, Roberto C., 2007. "Incomplete markets with no Hart points," Theoretical Economics, Econometric Society, vol. 2(2), June.
  13. Chiappori, Pierre-Andre & Rochet, Jean-Charles, 1987. "Revealed Preferences and Differentiable Demand: Notes and Comments," Econometrica, Econometric Society, vol. 55(3), pages 687-91, May.
  14. 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.
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Cited by:
  1. Ian Ayres & Colin Rowat & Nasser Zakariya, 2011. "Optimal voting rules for two-member tenure committees," Social Choice and Welfare, Springer, vol. 36(2), pages 323-354, February.
  2. Helena Soares & Tiago Neves Sequeira & Pedro Macias Marques & Orlando Gomes & Alexandra Ferreira-Lopes, 2012. "Social Infrastructure and the Preservation of Physical Capital: Equilibria and Transitional Dynamics," Working Papers Series 2 12-04, ISCTE-IUL, Business Research Unit (BRU-IUL).
  3. Orrego, Fabrizio, 2011. "Demografía y precios de activos," Revista Estudios Económicos, Banco Central de Reserva del Perú, issue 22, pages 83-101.
  4. 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.
  5. 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.

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