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Arbitrage and equilibrium in unbounded exchange economies with satiation

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  • Nizar Allouch

    (Queen Mary, University of London - QMUL (Londres, Angleterre) - Queen Mary, University of London - QMUL (Londres, Angleterre))

  • Cuong Le Van

    () (CES - Centre d'économie de la Sorbonne - CNRS : UMR8174 - Université Paris I - Panthéon-Sorbonne)

  • Frank H. Page

    (UA - University of Alabama - 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. First, we show that in unbounded exchange economies, even if some agents' preferences are satiated, the absence of arbitrage is suffcient 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 weaknonsatiation. 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.

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Bibliographic Info

Paper provided by HAL in its series Université Paris1 Panthéon-Sorbonne (Post-Print and Working Papers) with number halshs-00096040.

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Date of creation: Sep 2006
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Publication status: Published, Journal of Mathematical Economics, 2006, 42, 06, 661-674
Handle: RePEc:hal:cesptp:halshs-00096040

Note: View the original document on HAL open archive server: http://halshs.archives-ouvertes.fr/halshs-00096040
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Related research

Keywords: Arbitrage; Asset Market Equilibrium; Nonsatiation; Recession Cones.;

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References

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  1. 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.
  2. Nizar Allouch, 2000. "Edgeworth and Walras Equilibria of an Arbitrage-Free Exchange Economy," Econometric Society World Congress 2000 Contributed Papers 1901, Econometric Society.
  3. Monique Florenzano, 2008. "General equilibrium," Documents de travail du Centre d'Economie de la Sorbonne b08005, Université Panthéon-Sorbonne (Paris 1), Centre d'Economie de la Sorbonne.
  4. Grandmont, Jean-Michel, 1977. "Temporary General Equilibrium Theory," Econometrica, Econometric Society, vol. 45(3), pages 535-72, April.
  5. Nielsen, Lars Tyge, 1989. "Asset Market Equilibrium with Short-Selling," Review of Economic Studies, Wiley Blackwell, vol. 56(3), pages 467-73, July.
  6. 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.
  7. 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.
  8. Dana, R.-A. & Le Van, C. & Magnien, F., 1999. "On the Different Notions of Arbitrage and Existence of Equilibrium," Papiers d'Economie Mathématique et Applications 1999.34, Université Panthéon-Sorbonne (Paris 1).
  9. PageJr., Frank H. & Wooders, Myrna H. & Monteiro, Paulo K., 2000. "Inconsequential arbitrage," Journal of Mathematical Economics, Elsevier, vol. 34(4), pages 439-469, December.
  10. 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.
  11. 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.
  12. Green, Jerry R, 1973. "Temporary General Equilibrium in a Sequential Trading Model with Spot and Futures Transactions," Econometrica, Econometric Society, vol. 41(6), pages 1103-23, November.
  13. 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.
  14. Bergstrom, Theodore C., 1976. "How to discard `free disposability' - at no cost," Journal of Mathematical Economics, Elsevier, vol. 3(2), pages 131-134, July.
  15. Werner, Jan, 1987. "Arbitrage and the Existence of Competitive Equilibrium," Econometrica, Econometric Society, vol. 55(6), pages 1403-18, November.
  16. Grandmont, Jean-Michel, 1972. "Continuity properties of a von Neumann-Morgenstern utility," Journal of Economic Theory, Elsevier, vol. 4(1), pages 45-57, February.
  17. 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.
  18. 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.
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Cited by:
  1. De Giorgi, Enrico & Hens, Thorsten & Rieger, Marc Oliver, 2010. "Financial market equilibria with cumulative prospect theory," Journal of Mathematical Economics, Elsevier, vol. 46(5), pages 633-651, September.
  2. Martins-da-Rocha, V. Filipe & Monteiro, Paulo K., 2009. "Unbounded exchange economies with satiation: How far can we go?," Journal of Mathematical Economics, Elsevier, vol. 45(7-8), pages 465-478, July.
  3. Won, Dong Chul & Yannelis, Nicholas C., 2008. "Equilibrium theory with unbounded consumption sets and non-ordered preferences: Part I. Non-satiation," Journal of Mathematical Economics, Elsevier, vol. 44(11), pages 1266-1283, December.
  4. Won, Dong Chul & Yannelis, Nicholas C., 2011. "Equilibrium theory with satiable and non-ordered preferences," Journal of Mathematical Economics, Elsevier, vol. 47(2), pages 245-250, March.

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