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Quadratic Concavity and Determinacy of Equilibrium

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  • Chris Shannon

    (University of California, Berkeley)

  • William R. Zame

    (University of California, Los Angeles)

Abstract

One of the central features of classical models of competitive markets is the generic determinacy of competitive equilibria. For smooth economies with a finite number of commodities and a finite number of consumers, almost all initial endowments admit only a finite number of competitive equilibria, and these equilibria vary (locally) smoothly with endowments; thus equilibrium comparative statics are locally determinate. This paper establishes parallel results for economies with finitely many consumers and infinitely many commodities. The most important new condition we introduce, quadratic concavity, rules out preferences in which goods are perfect substitutes globally, locally, or asymptotically. Our framework is sufficiently general to encompass many of the models that have proved important in the study of continuous-time trading in financial markets, trading over an infinite time horizon, and trading of finely differentiated commodities.
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Suggested Citation

  • Chris Shannon & William R. Zame, 2000. "Quadratic Concavity and Determinacy of Equilibrium," GE, Growth, Math methods 9912001, EconWPA.
    • Handle: RePEc:wpa:wuwpge:9912001
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    References listed on IDEAS

    as
    1. Debreu, Gerard, 1970. "Economies with a Finite Set of Equilibria," Econometrica, Econometric Society, vol. 38(3), pages 387-392, May.
    2. Balasko, Yves, 1997. "Equilibrium analysis of the infinite horizon model with smooth discounted utility functions," Journal of Economic Dynamics and Control, Elsevier, vol. 21(4-5), pages 783-829, May.
    3. Anderson Robert M. & Zame William R., 2001. "Genericity with Infinitely Many Parameters," The B.E. Journal of Theoretical Economics, De Gruyter, vol. 1(1), pages 1-64, February.
    4. Kehoe, Timothy J. & Levine, David K. & Mas-Colell, Andreu & Zame, William R., 1989. "Determinacy of equilibrium in large-scale economies," Journal of Mathematical Economics, Elsevier, vol. 18(3), pages 231-262, June.
    5. Timothy J. Kehoe & David K. Levine & Andreu Mas-Colell & William Zame, 1989. "Determinacy of Equilibrium in Large Square Economies," Levine's Working Paper Archive 46, David K. Levine.
    6. Chichilnisky, Graciela & Zhou, Yuqing, 1998. "Smooth infinite economies," Journal of Mathematical Economics, Elsevier, vol. 29(1), pages 27-42, January.
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    Citations

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    Cited by:

    1. Martins-da-Rocha, V. Filipe & Riedel, Frank, 2010. "On equilibrium prices in continuous time," Journal of Economic Theory, Elsevier, vol. 145(3), pages 1086-1112, May.
    2. Covarrubias, Enrique, 2013. "The number of equilibria of smooth infinite economies," Journal of Mathematical Economics, Elsevier, vol. 49(4), pages 263-265.
    3. Riedel, Frank, 2005. "Generic determinacy of equilibria with local substitution," Journal of Mathematical Economics, Elsevier, vol. 41(4-5), pages 603-616, August.
    4. John Geanakoplos, 2008. "Overlapping Generations Models of General Equilibrium," Cowles Foundation Discussion Papers 1663, Cowles Foundation for Research in Economics, Yale University.
    5. Accinelli, E. & Covarrubias, E., 2014. "An extension of the Sard–Smale Theorem to convex domains with an empty interior," Journal of Mathematical Economics, Elsevier, vol. 55(C), pages 123-128.
    6. Herrendorf, Berthold & Valentinyi, Akos, 2006. "On the stability of the two-sector neoclassical growth model with externalities," Journal of Economic Dynamics and Control, Elsevier, vol. 30(8), pages 1339-1361, August.
    7. Berthold Herrendorf & Akos Valentinyi, 2002. "Neoclassical Growth Model with Externalities," IEHAS Discussion Papers 0203, Institute of Economics, Centre for Economic and Regional Studies, Hungarian Academy of Sciences.
    8. Covarrubias Enrique, 2010. "Regular Infinite Economies," The B.E. Journal of Theoretical Economics, De Gruyter, vol. 10(1), pages 1-21, July.
    9. John Geanakoplos, 2008. "Overlapping Generations Models of General Equilibrium," Levine's Working Paper Archive 122247000000002225, David K. Levine.
    10. Bloise, Gaetano & Reichlin, Pietro, 2011. "Asset prices, debt constraints and inefficiency," Journal of Economic Theory, Elsevier, vol. 146(4), pages 1520-1546, July.
    11. Molina-Abraldes, Antonio & Pintos-Clapés, Juan, 2008. "Pareto optimality in continuous-time OLG economies," Journal of Mathematical Economics, Elsevier, vol. 44(9-10), pages 933-950, September.
    12. Mertens, Jean-François & Rubinchik, Anna, 2014. "Essential properties of Lp,q spaces (the amalgams) and the implicit function theorem for equilibrium analysis in continuous time," Journal of Mathematical Economics, Elsevier, vol. 50(C), pages 187-196.
    13. Donald J. Brown & Ravi Kannan, 2003. "Indeterminacy, Nonparametric Calibration and Counterfactual Equilibria," Cowles Foundation Discussion Papers 1426, Cowles Foundation for Research in Economics, Yale University.
    14. DEMICHELIS, Stefano & POLEMARCHAKIS, Heracles, 2000. "Life-span and the determinacy of equilibrium in economies of overlapping generations," CORE Discussion Papers 2000034, Université catholique de Louvain, Center for Operations Research and Econometrics (CORE).
    15. Rubinchik, Anna, 2015. "The chase of a multi-armed economist for the elusive social discount rate," Working Papers WP2015/8, University of Haifa, Department of Economics, revised 18 Nov 2015.
    16. Accinelli, Elvio, 2013. "The equilibrium set of infinite dimensional Walrasian economies and the natural projection," Journal of Mathematical Economics, Elsevier, vol. 49(6), pages 435-440.
    17. H. Polemarchakis & S. Demichelis, 2002. "Frequency of Trade and the Determinancy of Equilibrium Paths: Logarithmic Economies of Overlapping Generations Under Certainty," Working Papers 2002-16, Brown University, Department of Economics.
    18. Eric Bond & Kazumichi Iwasa & Kazuo Nishimura, 2011. "A dynamic two country Heckscher–Ohlin model with non-homothetic preferences," Economic Theory, Springer;Society for the Advancement of Economic Theory (SAET), vol. 48(1), pages 171-204, September.
    19. Accinelli, Elvio & Covarrubias, Enrique, 2014. "Smooth economic analysis for general spaces of commodities," MPRA Paper 53222, University Library of Munich, Germany.

    More about this item

    JEL classification:

    • C62 - Mathematical and Quantitative Methods - - Mathematical Methods; Programming Models; Mathematical and Simulation Modeling - - - Existence and Stability Conditions of Equilibrium
    • D41 - Microeconomics - - Market Structure, Pricing, and Design - - - Perfect Competition
    • 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

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