Constructing Pure-Exchange Economies with Many Equilibria

Abstract: We develop a restart algorithm based on Scarf’s (1973) algorithm for computing approximate Brouwer fixed points. We use the algorithm to compute all of the equilibria of a general equilibrium pure-exchange model with four consumers, four goods, and 15 equilibria. The mathematical result that motivates the algorithm is a fixed-point index theorem that provides a sufficient condition for uniqueness of equilibrium and a necessary condition for multiplicity of equilibria. Examining the structure of the model with 15 equilibria provides us with a method for constructing higher dimensional models with even more equilibria. For example, using our method, we can construct a pure-exchange economy with eight consumers and eight goods that has (at least) 255 equilibria.

Keywords: Uniqueness of equilibrium; Multiplicity of equilibrium; Computation of equilibrium;

JEL Classification: C63; D51; C62; C60;

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Bibliographic Information

Provider: Federal Reserve Bank of Minneapolis

Part of Series: Staff Report

Publication Date: 2021-12-13

Number: 631