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Positivity

, Volume 9, Issue 3, pp 541–568 | Cite as

Nonlinear Prices in Nonconvex Economies with Classical Pareto and Strong Pareto Optimal Allocations

  • Boris S. MordukhovichEmail author
Article

Abstract

The paper is devoted to applications of modern tools of variational analysis to equilibrium models of welfare economics involving generally nonconvex economies with infinite-dimensional commodity spaces. The main results relate to the so-called generalized/extended second welfare theorem ensuring an equilibrium price support at Pareto optimal allocations. Based on advanced tools of variational analysis and generalized differentiation, we establish refined results of this type with the novel usage of nonlinear prices at the three types to optimal allocations: weak Pareto, Pareto, and strong Pareto. We pay a special attention to strong Pareto optimal allocations in economies with ordering commodity spaces showing that enhanced results for them do not require, in contrast to the classical types of weak Pareto and Pareto optimality, any net demand qualification conditions.

Keywords

generalized differentiation Pareto and strong Pareto optimality price equilibrium variational analysis welfare economics 

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References

  1. 1.
    Abramovich, Y.A., Aliprantis, C.D., Burkinshaw, O. 1992Positive operators on Krein spacesActa Appl. Math.27122CrossRefGoogle Scholar
  2. 2.
    Aliprantis, C.D., Florenzano, M. and Tourky, R.: Linear and non-linear price decentralization, J. Econ. Theory, to appear.Google Scholar
  3. 3.
    Aliprantis, C.D., Monteiro, P.K., Tourky, R. 2004Non-marketed options, non-existence of equilibria, and non-linear pricesJ. Econ. Theory114345357CrossRefGoogle Scholar
  4. 4.
    Aliprantis, C.D., Tourky, R., Yannelis, N.C. 2001A theory of value with non-linear prices. Equilibrium analysis beyond vector latticesJ. Econ. Theory1002272CrossRefGoogle Scholar
  5. 5.
    Arrow, K.J.: An extension of the basic theorems of classical welfare economics, in: Proceedings of the Second Berkeley Symposium on Mathematical Statistics and Probability, University of California, Berkeley, California, 1951, pp. 507–532.Google Scholar
  6. 6.
    Borwein, J.M., Jofré, A. 1998A nonconvex separation property in Banach spacesMath. Meth. Oper. Res.48169179CrossRefGoogle Scholar
  7. 7.
    Borwein, J.M. and Zhu, Q.J.: Techniques of variational analysis: an introduction, Springer, New York, to appear.Google Scholar
  8. 8.
    Cornet, B. 1990Marginal cost pricing and Pareto optimalityChampsaur, P. eds. Essays in Honor of Edmond MalinvaudMIT PressCambridge, Massachusetts1453Vol. 1Google Scholar
  9. 9.
    Debreu, G. 1951The coefficient of resource utilizationEconometrica19273292Google Scholar
  10. 10.
    Fabian, M., Mordukhovich, B.S. 1998Smooth variational principles and characterizations of Asplund spacesSet-Valued Anal.6381406CrossRefGoogle Scholar
  11. 11.
    Fabian, M., Mordukhovich, B.S. 2003Sequential normal compactness versus topological normal compactness in variational analysisNonlinear Anal.5410571067CrossRefGoogle Scholar
  12. 12.
    Flam, S.D.: Inf-convolution, nonconvex separation, games, and welfare economics, preprint, 2004.Google Scholar
  13. 13.
    Florenzano, M., Gourdel, P. and Jofré, A.: Supporting weakly Pareto optimal allocations in infinite dimensional nonconvex economies, preprint, 2003.Google Scholar
  14. 14.
    Ioffe, A.D.: Codirectional compactness, metric regularity and subdifferential calculus, in: Constructive, Experimental and Nonlinear Analysis, M. Théra (ed.), Canad. Math. Soc. Conf. Proc. 27 (2000), 123–164.Google Scholar
  15. 15.
    Jofré, A.: A second-welfare theorem in nonconvex economics, in: Constructive, Experimental and Nonlinear Analysis, M. Théra (ed.), Canad. Math. Soc. Conf. Proc. 27 (2000), 175–184.Google Scholar
  16. 16.
    Jofré, A. and Rivera J.: A nonconvex separation property and some applications, Math. Progr., Ser. A, to appear.Google Scholar
  17. 17.
    Jofré, A., Rockafellar, R.T. and Wets R.J-B.: A variational inequality scheme for determining an economic equilibrium of classical or extended type, in: Variational Analysis and Applications, F. Giannessi and A. Maugeri (ed.), Kluwer, Dordrecht, to appear.Google Scholar
  18. 18.
    Khan, M.A. 1991Ioffe’s normal cone and the foundations of welfare economics: The infinite dimensional theoryJ. Math. Anal. Appl.161284298CrossRefGoogle Scholar
  19. 19.
    Khan, M.A. 1999The Mordukhovich normal cone and the foundations of welfare economicsJ. Public Economic Theory1309338CrossRefGoogle Scholar
  20. 20.
    Kruger, A.Y., Mordukhovich, B.S. 1980Extremal points and the Euler equation in nonsmooth optimizationDokl. Akad. Nauk BSSR24684687Google Scholar
  21. 21.
    Malcolm, G.G., Mordukhovich, B.S. 2001Pareto optimality in nonconvex economies with infinite-dimensional commodity spacesJ. Global Optim.20323346CrossRefGoogle Scholar
  22. 22.
    Mas-Colell, A. 1986The price equilibrium existence problem in topological vector latticesEconometrica5410391053Google Scholar
  23. 23.
    Mordukhovich, B.S. 1976Maximum principle in problems of time optimal control with nonsmooth constraintsJ. Appl. Math. Mech.40960969CrossRefGoogle Scholar
  24. 24.
    Mordukhovich, B.S. 1980Metric approximations and necessary optimality conditions for general classes of nonsmooth extremal problemsSoviet Math. Dokl.22526530Google Scholar
  25. 25.
    Mordukhovich, B.S. 2000Abstract extremal principle with applications to welfare economicsJ. Math. Anal. Appl.251187216CrossRefGoogle Scholar
  26. 26.
    Mordukhovich, B.S. 2001The extremal principle and its applications to optimization and economicsRubinov, A.Glover, B. eds. Optimization and Related TopicsKluwerDordrecht343369Google Scholar
  27. 27.
    Mordukhovich, B.S. 2005Variational Analysis and Generalized Differentiation, Vol. I: Basic Theory, Vol. II: ApplicationsSpringerBerlinGoogle Scholar
  28. 28.
    Mordukhovich, B.S., Shao, Y. 1996Nonsmooth sequential analysis in Asplund spacesTrans. Amer. Math. Soc.34812351280CrossRefGoogle Scholar
  29. 29.
    Phelps, R.R. 1993Convex Functions, Monotone Operators and Differentiability2SpringerBerlinGoogle Scholar
  30. 30.
    Rockafellar, R.T., Wets, R.J-B. 1998Variational AnalysisSpringerBerlinGoogle Scholar
  31. 31.
    Samuelson P.A.: The pure theory of public expenditures, Review Econ. Stat. 36, 387–389.Google Scholar
  32. 32.
    Zhu, Q.J. 2004Nonconvex separation theorem for multifunctions, subdifferential calculus and applicationsSet-Valued Anal.12275290CrossRefGoogle Scholar

Copyright information

© Springer 2005

Authors and Affiliations

  1. 1.Department of MathematicsWayne State UniversityDetroitUSA

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