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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.

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Bibliographic Info

Article provided by Elsevier in its journal Journal of Economic Theory.

Volume (Year): 145 (2010)
Issue (Month): 1 (January)
Pages: 301-330

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Handle: RePEc:eee:jetheo:v:145:y:2010:i:1:p:301-330

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Web page: http://www.elsevier.com/locate/inca/622869

Related research

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

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References

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  1. Blume, Lawrence E & Zame, William R, 1994. "The Algebraic Geometry of Perfect and Sequential Equilibrium," Econometrica, Econometric Society, vol. 62(4), pages 783-94, July.
  2. 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.
  3. 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.
  4. Felix Kubler, 2007. "Approximate Generalizations and Computational Experiments," Econometrica, Econometric Society, vol. 75(4), pages 967-992, 07.
  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. 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.
  7. 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.
  8. P.J.J. Herings & F. Kubler, 2001. "Computing Equilibria in Finance Economies," GE, Growth, Math methods 0205003, EconWPA.
  9. Gjerstad, S., 1996. "Multiple Equilibria in Exchange Economies with Homothetic, Nearly Identical Preferences," Papers 288, Minnesota - Center for Economic Research.
  10. Kam-Chau Wong & Marcel K. Richter, 1999. "Non-computability of competitive equilibrium," Economic Theory, Springer, vol. 14(1), pages 1-27.
  11. Anderson, Robert M. & Raimondo, Roberto C., 2007. "Incomplete markets with no Hart points," Theoretical Economics, Econometric Society, vol. 2(2), June.
  12. Brown, Donald J & Matzkin, Rosa L, 1996. "Testable Restrictions on the Equilibrium Manifold," Econometrica, Econometric Society, vol. 64(6), pages 1249-62, November.
  13. 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.
  14. 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.
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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. 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.
  4. Orrego, Fabrizio, 2011. "Demografía y precios de activos," Revista Estudios Económicos, Banco Central de Reserva del Perú, issue 22, pages 83-101.

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