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Constructing Pure-Exchange Economies with Many Equilibria

Staff Report 631 | Revised December 31, 2021

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Pascal Gauthier Nordic Investment Bank
Timothy J. Kehoe Consultant, University of Minnesota, and National Bureau of Economic Research
Erwan Quintin University of Wisconsin-Madison
Constructing Pure-Exchange Economies with Many Equilibria


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.

Published in: _Economic Theory_ (Vol. 73, Iss. 2-3, April 2022, pp. 541-564),