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Arbitrage, Equilibrium, and Nonsatiation

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  • Nizar Allouch

    ()
    (Queen Mary, University of London)

  • Cuong Le Van

    ()
    (University of Paris 1)

  • Frank H. Page, Jr.

    ()
    (University of Alabama)

Abstract

In his seminal paper on arbitrage and competitive equilibrium in unbounded exchange economies, Werner (Econometrica, 1987) proved the existence of a competitive equilibrium, under a price no-arbitrage condition, without assuming either local or global nonsatiation. Werner's existence result contrasts sharply with classical existence results for bounded exchange economies which require, at minimum, global nonsatiation at rational allocations. Why do unbounded exchange economies admit existence without local or global nonsatiation? This question is the focus of our paper. We make two main contributions to the theory of arbitrage and competitive equilibrium. First, we show that, in general, in unbounded exchange economies (for example, asset exchange economies allowing short sales), even if some agents' preferences are satiated, the absence of arbitrage is sufficient for the existence of competitive equilibria, as long as each agent who is satiated has a nonempty set of useful net trades - that is, as long as agents' preferences satisfy weak nonsatiation. Second, we provide a new approach to proving existence in unbounded exchange economies. The key step in our new approach is to transform the original economy to an economy satisfying global nonsatiation such that all equilibria of the transformed economy are equilibria of the original economy. What our approach makes clear is that it is precisely the condition of weak nonsatiation - a condition considerably weaker than local or global nonsatiation - that makes possible this transformation. Moreover, as we show via examples, without weak nonsatiation, existence fails.

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Bibliographic Info

Paper provided by Queen Mary, University of London, School of Economics and Finance in its series Working Papers with number 512.

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Date of creation: Apr 2004
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Handle: RePEc:qmw:qmwecw:wp512

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Keywords: Arbitrage; Asset market equilibrium; Nonsatiation; Recession cones;

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  1. Dana, Rose-Anne & Le Van, Cuong & Magnien, François, 1996. "On the different notions of arbitrage and existence of equilibrium," CEPREMAP Working Papers (Couverture Orange) 9616, CEPREMAP.
  2. Page, F.H.Jr. & Wooders, M.H. & Monteiro, P.K., 1999. "Inconsequential Arbitrage," The Warwick Economics Research Paper Series (TWERPS) 561, University of Warwick, Department of Economics.
  3. Florenzano, Monigue & Le Van, Cuong, 1986. "A note on the Gale-Nikaido-Debreu lemma and the existence of general equilibrium," Economics Letters, Elsevier, vol. 22(2-3), pages 107-110.
  4. Kim, Chongmin, 1998. "Stochastic Dominance, Pareto Optimality, and Equilibrium Asset Pricing," Review of Economic Studies, Wiley Blackwell, Wiley Blackwell, vol. 65(2), pages 341-56, April.
  5. Bergstrom, Theodore C., 1976. "How to discard `free disposability' - at no cost," Journal of Mathematical Economics, Elsevier, vol. 3(2), pages 131-134, July.
  6. Hammond, Peter J., 1983. "Overlapping expectations and Hart's conditions for equilibrium in a securities model," Journal of Economic Theory, Elsevier, vol. 31(1), pages 170-175, October.
  7. Milne, Frank, 1976. "Default Risk in a General Equilibrium Asset Economy with Incomplete Markets," International Economic Review, Department of Economics, University of Pennsylvania and Osaka University Institute of Social and Economic Research Association, vol. 17(3), pages 613-25, October.
  8. Monique Florenzano, 2008. "General equilibrium," Documents de travail du Centre d'Economie de la Sorbonne, Université Panthéon-Sorbonne (Paris 1), Centre d'Economie de la Sorbonne b08005, Université Panthéon-Sorbonne (Paris 1), Centre d'Economie de la Sorbonne.
  9. Allouch, Nizar & Le Van, Cuong & Page, Jr. Frank H., 2001. "The geometry of arbitrage and the existence of competitive equilibrium," The Warwick Economics Research Paper Series (TWERPS) 598, University of Warwick, Department of Economics.
  10. Grandmont, Jean-Michel, 1993. "Temporary general equilibrium theory," Handbook of Mathematical Economics, Elsevier, in: K. J. Arrow & M.D. Intriligator (ed.), Handbook of Mathematical Economics, edition 4, volume 2, chapter 19, pages 879-922 Elsevier.
  11. Allouch, N., 2000. "Equilibrium and Bounded Arbitrage," Papiers d'Economie Mathématique et Applications, Université Panthéon-Sorbonne (Paris 1) 2000.78, Université Panthéon-Sorbonne (Paris 1).
  12. Page Jr., Frank H. & Wooders, Myrna Holtz, 1996. "A necessary and sufficient condition for the compactness of individually rational and feasible outcomes and the existence of an equilibrium," Economics Letters, Elsevier, vol. 52(2), pages 153-162, August.
  13. Allouch, Nizar, 2002. "An equilibrium existence result with short selling," Journal of Mathematical Economics, Elsevier, vol. 37(2), pages 81-94, April.
  14. Geistdoerfer-Florenzano, Monique, 1982. "The gale-nikaido-debreu lemma and the existence of transitive equilibrium with or without the free-disposal assumption," Journal of Mathematical Economics, Elsevier, vol. 9(1-2), pages 113-134, January.
  15. Milne, Frank, 1980. "Short-Selling, Default Risk and the Existence of Equilibrium in a Securities Model," International Economic Review, Department of Economics, University of Pennsylvania and Osaka University Institute of Social and Economic Research Association, vol. 21(2), pages 255-67, June.
  16. Gale, D. & Mas-Colell, A., 1975. "An equilibrium existence theorem for a general model without ordered preferences," Journal of Mathematical Economics, Elsevier, vol. 2(1), pages 9-15, March.
  17. Grandmont, Jean-Michel, 1972. "Continuity properties of a von Neumann-Morgenstern utility," Journal of Economic Theory, Elsevier, vol. 4(1), pages 45-57, February.
  18. Nielsen, Lars Tyge, 1989. "Asset Market Equilibrium with Short-Selling," Review of Economic Studies, Wiley Blackwell, Wiley Blackwell, vol. 56(3), pages 467-73, July.
  19. Green, Jerry R, 1973. "Temporary General Equilibrium in a Sequential Trading Model with Spot and Futures Transactions," Econometrica, Econometric Society, Econometric Society, vol. 41(6), pages 1103-23, November.
  20. Hart, Oliver D., 1974. "On the existence of equilibrium in a securities model," Journal of Economic Theory, Elsevier, vol. 9(3), pages 293-311, November.
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Cited by:
  1. Allouch, Nizar & Le Van, Cuong & Page, Frank Jr., 2002. "The geometry of arbitrage and the existence of competitive equilibrium," Journal of Mathematical Economics, Elsevier, vol. 38(4), pages 373-391, December.
  2. repec:hal:journl:halshs-00086096 is not listed on IDEAS
  3. Nizar Allouch, 2000. "Edgeworth and Walras Equilibria of an Arbitrage-Free Exchange Economy," Econometric Society World Congress 2000 Contributed Papers 1901, Econometric Society.

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