Competitive Equilibria in Semi-Algebraic Economies

37 Pages Posted: 28 Mar 2007

See all articles by Felix Kubler

Felix Kubler

University of Zurich; Swiss Finance Institute

Karl Schmedders

University of Zurich

Date Written: March 22, 2007

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.

Keywords: Computable general equilibrium, semi-algebraic economy, Groebner bases

JEL Classification: D50, C63

Suggested Citation

Kubler, Felix E. and Schmedders, Karl, Competitive Equilibria in Semi-Algebraic Economies (March 22, 2007). PIER Working Paper No. 07-013. Available at SSRN: https://ssrn.com/abstract=976890 or http://dx.doi.org/10.2139/ssrn.976890

Felix E. Kubler

University of Zurich ( email )

Rämistrasse 71
Zürich, CH-8006
Switzerland

Swiss Finance Institute

c/o University of Geneva
40, Bd du Pont-d'Arve
CH-1211 Geneva 4
Switzerland

Karl Schmedders (Contact Author)

University of Zurich ( email )

Moussonstrasse 15
Zürich, CH-8044
Switzerland
+41 (0)44 634 3770 (Phone)

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