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Constructing Pure-Exchange Economies with Many Equilibria

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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.

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  • Pascal Gauthier & Timothy J. Kehoe & Erwan Quintin, 2021. "Constructing Pure-Exchange Economies with Many Equilibria," Staff Report 631, Federal Reserve Bank of Minneapolis.
    • Handle: RePEc:fip:fedmsr:93490
    DOI: 10.21034/sr.631
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    1. 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.
    2. Herbert E. Scarf, 2008. "Neighborhood Systems for Production Sets with Indivisibilities," Palgrave Macmillan Books, in: Zaifu Yang (ed.), Herbert Scarf’s Contributions to Economics, Game Theory and Operations Research, chapter 5, pages 105-130, Palgrave Macmillan.
    3. Felix Kubler & Karl Schmedders, 2008. "Approximate Versus Exact Equilibria in Dynamic Economies," Lecture Notes in Economics and Mathematical Systems, in: Computational Aspects of General Equilibrium Theory, pages 135-163, Springer.
    4. Kenneth J. Arrow & Timothy J. Kehoe, 1994. "Distinguished Fellow: Herbert Scarf's Contributions to Economics," Journal of Economic Perspectives, American Economic Association, vol. 8(4), pages 161-181, Fall.
    5. Varian, Hal R, 1975. "A Third Remark on the Number of Equilibria of an Economy," Econometrica, Econometric Society, vol. 43(5-6), pages 985-986, Sept.-Nov.
    6. Kehoe, Timothy J, 1980. "An Index Theorem for General Equilibrium Models with Production," Econometrica, Econometric Society, vol. 48(5), pages 1211-1232, July.
    7. Chipman, John S., 1974. "Homothetic preferences and aggregation," Journal of Economic Theory, Elsevier, vol. 8(1), pages 26-38, May.
    8. Barnett,William A. & Cornet,Bernard & D'Aspremont,Claude & Gabszewicz,Jean & Mas-Colell,Andreu (ed.), 1991. "Equilibrium Theory and Applications," Cambridge Books, Cambridge University Press, number 9780521392198, October.
    9. Dierker, Egbert, 1972. "Two Remarks on the Number of Equilibria of an Economy," Econometrica, Econometric Society, vol. 40(5), pages 951-953, September.
    10. E. Eisenberg, 1961. "Aggregation of Utility Functions," Management Science, INFORMS, vol. 7(4), pages 337-350, July.
    11. Debreu, Gerard, 1970. "Economies with a Finite Set of Equilibria," Econometrica, Econometric Society, vol. 38(3), pages 387-392, May.
    12. Kehoe, Timothy J., 1991. "Computation and multiplicity of equilibria," Handbook of Mathematical Economics, in: W. Hildenbrand & H. Sonnenschein (ed.), Handbook of Mathematical Economics, edition 1, volume 4, chapter 38, pages 2049-2144, Elsevier.
    13. Herbert E. Scarf, 1967. "The Approximation of Fixed Points of a Continuous Mapping," Cowles Foundation Discussion Papers 216R, Cowles Foundation for Research in Economics, Yale University.
    14. B. Curtis Eaves & Herbert Scarf, 1976. "The Solution of Systems of Piecewise Linear Equations," Mathematics of Operations Research, INFORMS, vol. 1(1), pages 1-27, February.
    15. Timothy J. Kehoe, 1985. "Multiplicity of Equilibria and Comparative Statics," The Quarterly Journal of Economics, President and Fellows of Harvard College, vol. 100(1), pages 119-147.
    16. W. Hildenbrand & H. Sonnenschein (ed.), 1991. "Handbook of Mathematical Economics," Handbook of Mathematical Economics, Elsevier, edition 1, volume 4, number 4.
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    Cited by:

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    More about this item

    Keywords

    Uniqueness of equilibrium; Multiplicity of equilibrium; Computation of equilibrium;
    All these keywords.

    JEL classification:

    • C63 - Mathematical and Quantitative Methods - - Mathematical Methods; Programming Models; Mathematical and Simulation Modeling - - - Computational Techniques
    • D51 - Microeconomics - - General Equilibrium and Disequilibrium - - - Exchange and Production Economies
    • C62 - Mathematical and Quantitative Methods - - Mathematical Methods; Programming Models; Mathematical and Simulation Modeling - - - Existence and Stability Conditions of Equilibrium
    • C60 - Mathematical and Quantitative Methods - - Mathematical Methods; Programming Models; Mathematical and Simulation Modeling - - - General

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