Competitive Equilibria in Semi-Algebraic Economies
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.
|Date of creation:||22 Mar 2007|
|Date of revision:|
|Contact details of provider:|| Postal: 3718 Locust Walk, Philadelphia, PA 19104|
Web page: http://economics.sas.upenn.edu/pier
More information through EDIRC
Please report citation or reference errors to , or , if you are the registered author of the cited work, log in to your RePEc Author Service profile, click on "citations" and make appropriate adjustments.:
- Gjerstad, S., 1996. "Multiple Equilibria in Exchange Economies with Homothetic, Nearly Identical Preferences," Papers 288, Minnesota - Center for Economic Research.
- Kubler, Felix & Schmedders, Karl, 2000. "Computing Equilibria in Stochastic Finance Economies," Computational Economics, Springer;Society for Computational Economics, vol. 15(1-2), pages 145-72, April.
- P.J.J. Herings & F. Kubler, 2001.
"Computing Equilibria in Finance Economies,"
GE, Growth, Math methods
- 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).
- Herings P. Jean-Jacques & Kubler Felix, 2000. "Computing Equilibria in Finance Economies," Research Memorandum 010, Maastricht University, Maastricht Research School of Economics of Technology and Organization (METEOR).
- Mas-Colell,Andreu, 1990.
"The Theory of General Economic Equilibrium,"
Cambridge University Press, number 9780521388702, November.
- Brown, Donald J & Matzkin, Rosa L, 1996.
"Testable Restrictions on the Equilibrium Manifold,"
Econometric Society, vol. 64(6), pages 1249-62, November.
- 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.
- Cass, David, 2006.
"Musings on the Cass trick,"
Journal of Mathematical Economics,
Elsevier, vol. 42(4-5), pages 374-383, August.
- Anderson, Robert M. & Raimondo, Roberto C., 2007. "Incomplete markets with no Hart points," Theoretical Economics, Econometric Society, vol. 2(2), June.
- Blume, Lawrence E & Zame, William R, 1994.
"The Algebraic Geometry of Perfect and Sequential Equilibrium,"
Econometric Society, vol. 62(4), pages 783-94, July.
- Lawrence E. Blume & William R. Zame, 1993. "The Algebraic Geometry of Perfect and Sequential Equilibrium," Game Theory and Information 9309001, EconWPA.
- 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.
- 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.
- Balasko, Yves, 1979. "Economies with a finite but large number of equilibria," Journal of Mathematical Economics, Elsevier, vol. 6(2), pages 145-147, July.
- 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.
- Felix Kubler, 2007. "Approximate Generalizations and Computational Experiments," Econometrica, Econometric Society, vol. 75(4), pages 967-992, 07.
- 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.
When requesting a correction, please mention this item's handle: RePEc:pen:papers:07-013. See general information about how to correct material in RePEc.
For technical questions regarding this item, or to correct its authors, title, abstract, bibliographic or download information, contact: (Dolly Guarini)
If references are entirely missing, you can add them using this form.