Economic Theory

, Volume 29, Issue 3, pp 549–564 | Cite as

Supporting weakly Pareto optimal allocations in infinite dimensional nonconvex economies

  • Monique Florenzano
  • Pascal Gourdel
  • Alejandro Jofré
Research Article


In this paper, we prove a new version of the Second Welfare Theorem for economies with a finite number of agents and an infinite number of commodities, when the preference correspondences are not convex-valued and/or when the total production set is not convex. For this kind of nonconvex economies, a recent result, obtained by one of the authors, introduces conditions which, when applied to the convex case, give for Banach commodity spaces the well-known result of decentralization by continuous prices of Pareto-optimal allocations under an interiority condition. In this paper, in order to prove a different version of the Second Welfare Theorem, we reinforce the conditions on the commodity space, assumed here to be a Banach lattice, and introduce a nonconvex version of the properness assumptions on preferences and the total production set. Applied to the convex case, our result becomes the usual Second Welfare Theorem when properness assumptions replace the interiority condition. The proof uses a Hahn-Banach Theorem generalization by Borwein and Jofré (in Joper Res Appl Math 48:169–180, 1997) which allows to separate nonconvex sets in general Banach spaces


Second welfare theorem Nonconvex economies Banach spaces Subdifferential Banach lattices Properness assumptions 

JEL Classification Numbers

D51 D6 


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  1. Aliprantis C.D., Brown D.J., Burkinshaw O. Existence and optimality of competitive equilibria. Berlin Heidelberg New York: Springer 1990Google Scholar
  2. Allouch M., Florenzano M. (2004). Edgeworth and Walras equilibria of an arbitrage-free exchange economy. Econ Theory 23:353–370CrossRefGoogle Scholar
  3. Anderson R.M. (1988). The second welfare theorem with nonconvex preferences. Econometrica 56:361–382CrossRefGoogle Scholar
  4. Arrow K.J., Hahn F.H. (1971). General competitive analysis. Holden-Day, San FranciscoGoogle Scholar
  5. Bonnisseau J.M., Cornet B. (1988). Valuation equilibrium and Pareto optimum in non-convex economies. J Math Econo 17:293–308CrossRefGoogle Scholar
  6. Borwein J.M., Jofré A. (1997). A non-convex separation property in Banach Spaces. J Oper Res Appl Math 48:169–180CrossRefGoogle Scholar
  7. Debreu G. Valuation equilibrium and Pareto optimum. In: Proceedings of the national academy of sciences, vol. 40, pp. 588–592 (1954)Google Scholar
  8. Florenzano M. (1978). Eléments maximaux des préordres partiels sur les ensembles compacts. R.A.I.R.O. Recherche opérationnelle 12:277–283Google Scholar
  9. Florenzano, M., Gourdel, P., Jofré, A.: Supporting weakly Pareto optimal allocations in infinite dimensional nonconvex economies. Cahiers de la MSE, série bleue (CERMSEM) 2002.129Google Scholar
  10. Guesnerie R. (1975). Pareto optimality in non-convex economies. Econometrica 43:1–29CrossRefGoogle Scholar
  11. Ioffe A. (1984). Approximate subdifferentials and applications 1. The finite dimensional theory. Trans Am Math Soc 281(1):389–416Google Scholar
  12. Ioffe A. (1986). Approximate subdifferentials and applications 2. Functions on locally convex spaces. Mathematica 33:111–128Google Scholar
  13. Ioffe A. (1989). Approximate subdifferentials and applications 3. The metric theory. Mathematica 36:1–38Google Scholar
  14. Ioffe A. (2000). Codirectional compactness, metric regularity and subdifferential calculus. In: Théra M., (ed.) Constructive, experimental, and nonlinear analysis. Canadian Mathematical Society 27:123–163Google Scholar
  15. Jofré, A.: A second-welfare theorem in nonconvex economies. In: Théra, M. (ed.) Constructive, experimental, and nonlinear analysis, vol. 27, pp. 175–184. Canadian Mathematical Society (2000)Google Scholar
  16. Jofré A., Rivera Cayupi J.R. The second-welfare theorem with public goods in non-convex non-transitive economies with externalities. Universidad de Chile, Mimeo 2002Google Scholar
  17. Khan M.A. (1991). Ioffe’s normal cone and the foundations of welfare economics: the infinite dimensional theory. J Math Anal Appl 161(1):284–298CrossRefGoogle Scholar
  18. Khan M.A., Vohra R. (1988). Pareto optimal allocations of non-convex economies in locally convex spaces. Nonlinear Anal 12:943–950CrossRefGoogle Scholar
  19. Lefebvre I. Application de la théorie du point fixe à une approche directe de la non-vacuité du cœur d’une économie. Thèse de Doctorat Université de Paris 1 (2000)Google Scholar
  20. Mas-Colell A. (1986a). The price equilibrium existence problem in topological vector lattices. Econometrica 54:1039–1055CrossRefGoogle Scholar
  21. Mas-Colell A. (1986b). Valuation equilibria and Pareto optimum revisited. In: Hildenbrand W., Mas-Colell A (eds). Contributions to mathematical economics, In honor of Gérard Debreu. North-Holland, Amsterdam, pp. 317–331Google Scholar
  22. Mas-Colell A., Zame W. (1991). Equilibrium theory in in infinite dimensional spaces. In: Hildenbrand W., Sonnenschein H (eds). Handbook of mathematical economics, vol. IV,. North-Holland, Amsterdam, pp. 1835–1898Google Scholar
  23. Mordukhovich B.S. (2000). An abstract extremal principle with applications to welfare economics. J Math Anal Appl 251:187–216CrossRefGoogle Scholar
  24. Mordukhovich B.S. (2001). The extremal principle and its applications to optimization and economics. In: Rubinov A., Glover B (eds). Optimization and related topics. Kluwer, Dordrecht, pp. 343-369Google Scholar
  25. Mordukhovich B.S., Shao Y. (1996). Nonsmooth sequential analysis in Asplund spaces. Trans Am Math Soc 348:1235–1280CrossRefGoogle Scholar
  26. Podczeck H. (1996). Equilibria in vector lattices without ordered preferences or uniform properness. J Math Econ 25:465–485CrossRefGoogle Scholar
  27. Richard S.F. (1986). Competitive equilibria in Riesz spaces Mimeographed. Carnegie Mellon University, GSIA, PittsburghGoogle Scholar
  28. Rockafellar R.T. (1980). Generalized directional derivatives and subgradients of nonconvex functions. Can J Math XXXII(2):257–280Google Scholar
  29. Starr R.M. (1969). Quasi-equilibria in markets with non-convex preferences. Econometrica 37(1):25–38CrossRefGoogle Scholar
  30. Tourky R. (1998). A new approach to the limit theorem on the core of an economy in vector lattices. J Econ Theory 78:321–328CrossRefGoogle Scholar
  31. Tourky R. (1999). The limit theorem on the core of a production economy with unordered preferences. Econ Theory 14(1):219–226Google Scholar

Copyright information

© Springer-Verlag 2005

Authors and Affiliations

  • Monique Florenzano
    • 1
  • Pascal Gourdel
    • 2
  • Alejandro Jofré
    • 3
  1. 1.CNRS–CERMSEMUMR CNRS 8095, Université Paris 1Paris Cedex 13France
  2. 2.CERMSEMUMR CNRS 8095,Université Paris 1Paris Cedex 13France
  3. 3.Centro de Modelamiento MatematicoUMR CNRS 2071, Universidad de ChileCorreo 3Chile

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