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Global Invertibility of Excess Demand Functions

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  • Enrique Covarrubias

Abstract

In this paper we provide necessary and sufficient conditions for the excess demand function of a pure exchange economy to be globally invertible so that there is a unique equilibrium. Indeed, we show that an excess demand function is globally invertible if and only if its Jacobian never vanishes and it is a proper map. Our result includes as special cases many partial results found in the literature that imply global uniqueness including Gale-Nikaido conditions and properties related to stability of equilibria. Furthermore, by showing that the condition is necessary, we are implicitly finding the weakest possible condition

Suggested Citation

  • Enrique Covarrubias, 2013. "Global Invertibility of Excess Demand Functions," Working Papers 2013-21, Banco de México.
  • Handle: RePEc:bdm:wpaper:2013-21
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    References listed on IDEAS

    as
    1. Mas-Colell,Andreu, 1990. "The Theory of General Economic Equilibrium," Cambridge Books, Cambridge University Press, number 9780521388702, March.
    2. Covarrubias, Enrique, 2013. "The number of equilibria of smooth infinite economies," Journal of Mathematical Economics, Elsevier, vol. 49(4), pages 263-265.
    3. Wagstaff, Peter, 1975. "A Uniqueness Theorem," International Economic Review, Department of Economics, University of Pennsylvania and Osaka University Institute of Social and Economic Research Association, vol. 16(2), pages 521-524, June.
    4. Benhabib, Jess & Nishimura, Kazuo, 1979. "On the Uniqueness of Steady States in an Economy with Heterogeneous Capital Goods," International Economic Review, Department of Economics, University of Pennsylvania and Osaka University Institute of Social and Economic Research Association, vol. 20(1), pages 59-82, February.
    5. Nishimura, Zazuo, 1979. "On the uniqueness theorems by Arrow and Hahn," Journal of Economic Theory, Elsevier, vol. 21(2), pages 348-352, October.
    6. Debreu, Gerard, 1984. "Economic Theory in the Mathematical Mode," American Economic Review, American Economic Association, vol. 74(3), pages 267-278, June.
    7. Anjan Mukherji, 1997. "On the uniqueness of competitive equilibrium," Economic Theory, Springer;Society for the Advancement of Economic Theory (SAET), vol. 10(3), pages 509-520.
    8. Pearce, Ivor F & Wise, J, 1974. "On the Uniqueness of Competitive Equilibrium: Part II, Bounded Demand," Econometrica, Econometric Society, vol. 42(5), pages 921-932, September.
    9. Mukherji, Anjan, 1995. "A Locally Stable Adjustment Process," Econometrica, Econometric Society, vol. 63(2), pages 441-448, March.
    10. Pearce, I F & Wise, J, 1973. "On the Uniqueness of Competitive Equilibrium: Part I, Unbounded Demand," Econometrica, Econometric Society, vol. 41(5), pages 817-828, September.
    11. 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.
    12. Dierker, Egbert, 1972. "Two Remarks on the Number of Equilibria of an Economy," Econometrica, Econometric Society, vol. 40(5), pages 951-953, September.
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    More about this item

    Keywords

    Excess demand function; invertibility; uniqueness; Jacobian; proper map;

    JEL classification:

    • C62 - Mathematical and Quantitative Methods - - Mathematical Methods; Programming Models; Mathematical and Simulation Modeling - - - Existence and Stability Conditions of Equilibrium
    • C65 - Mathematical and Quantitative Methods - - Mathematical Methods; Programming Models; Mathematical and Simulation Modeling - - - Miscellaneous Mathematical Tools
    • D51 - Microeconomics - - General Equilibrium and Disequilibrium - - - Exchange and Production Economies
    • D52 - Microeconomics - - General Equilibrium and Disequilibrium - - - Incomplete Markets
    • D58 - Microeconomics - - General Equilibrium and Disequilibrium - - - Computable and Other Applied General Equilibrium Models

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