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Some remarks on the Walras equilibrium problem in Lebesgue spaces

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Abstract

We introduce a parametric variational inequality in order to model the time dependent Walras economic equilibrium and discuss its relation with an integral formulation in the spaces (L ∞, L 1). The role of monotonicity is analysed and, as a classical example, we study the Walras problem using the Cobb–Douglas functions in this new functional setting.

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References

  1. Appel J., Zabresko P.: Nonlinear Superposition Operators. Cambridge University Press, Cambridge (1990)

    Google Scholar 

  2. Aubin J.P., Frankowska H.: Set Valued Analysis. Birkhäuser, Boston (1990)

    MATH  Google Scholar 

  3. Arrow K.J., Debrew G.: Existence of an equilibrium for a competitive economy. Econometrica 22, 265–290 (1954)

    Article  MATH  MathSciNet  Google Scholar 

  4. Arrow, K.J., Hahn, F.H.: General Competitive Analysis. In: Bliss, C.J. Intriligator, M.D. (eds.) Advanced textbooks in economics (1991)

  5. Border K.C.: Fixed Point Theorems with Applications to Economics and Game Theory. Cambridge University Press, Cambridge (1985)

    Book  MATH  Google Scholar 

  6. Causa, A., Raciti, F.: Lipschitz continuity results for a class of variational inequalities and applications: a geometrical approach. J. Optim. Theory Appl. (to appear) doi:10.1007/s10957-009-9622-4

  7. Caruso A.O., Khan A.A., Raciti F.: Continuity results for a class of variational inequalities with applications to time-dependent network problems. Numer. Funct. Anal. Optim. 30(11–12), 1272–1288 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  8. Dafermos S.: Traffic equilibrium and variational inequalities. Transp. Sci. 14, 42–54 (1980)

    Article  MathSciNet  Google Scholar 

  9. Dafermos S.: Exchange price equilibria and variational inequalities. Math. Program. 46, 391–402 (1990)

    Article  MATH  MathSciNet  Google Scholar 

  10. Daniele, P., Maugeri, A., Oettli, W.: Time-dependent traffic equilibria. J. Optim. Theory Appl. 103(3), December (1999)

  11. Daniele, P.: Dynamic Networks and Evolutionary Variational Inequalities, pp. 543–555. E. Elagar Publishing Limited, Cheltenham (2006)

  12. Donato M.B., Milasi M., Vitanza C.: Dynamic Walrasian price equilibrium problem: evolutionary variational approach with sensitivity analysis. Optim. Lett. 2, 113–126 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  13. Falsaperla P., Raciti F.: Improved, non-iterative algorithm for the calculation of the equilibrium in the traffic network problem. J. Optim. Theory Appl. 133, 401–411 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  14. Facchinei F., Pang J.-S.: Finite-Dimensional Variational Inequalities and Complementarity Problems, Two Volumes. Springer, New York (2003)

    Google Scholar 

  15. Friesz T.L., Bernstein D., Smith T.E., Tobin R.L., Wie B.-W.: A variational inequality formulation of the dynamic network user equilibrium problem. Oper. Res. 41(1), 179–191 (1993)

    Article  MATH  MathSciNet  Google Scholar 

  16. Giannessi, F., Maugeri, A. (eds): Variational Inequalities and Network Equilibrium Problems, Erice 1994. Plenum Press, New York (1995)

    Google Scholar 

  17. Gwinner, J., Raciti, F.: Random equilibrium problems on networks, Math. Comput. Modelling, 43 (2006)

  18. Gwinner J., Raciti F.: On a class of random variational inequalities on random sets. Numer. Funct. Anal. Optim. 2(5–6), 618–636 (2006)

    MathSciNet  Google Scholar 

  19. Hartman P., Stampacchia G.: On some nonlinear elliptic differential functional equations. Acta Math. 115, 271–310 (1966)

    Article  MATH  MathSciNet  Google Scholar 

  20. Jofré, A., Rockafellar, R.T., Wets, R.J.B.: Variational Inequalities and economic equilibrium. Math. Oper. Res. 32 (2007)

  21. Kinderlehrer D., Stampacchia G.: An Introduction to Variational Inequalities and their Applications. Academic Press, New York (1980)

    MATH  Google Scholar 

  22. Lions J.L., Stampacchia G.: Variational Inequalities. Comm. Pure Appl. Math. 20, 493–519 (1967)

    Article  MATH  MathSciNet  Google Scholar 

  23. Maugeri A., Raciti F.: On existence theorems for monotone and nonmonotone variational inequalities. J. Convex Anal. 16, 809–911 (2009)

    MathSciNet  Google Scholar 

  24. Maugeri A., Raciti F.: On general infinite dimensional complementarity problems. Optim. Lett. 2(1), 71–90 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  25. Nagurney A.: Network Economics: A Variational Inequality Approach. Kluwer, Dordrecht (1993)

    MATH  Google Scholar 

  26. Raciti F.: Bicriterion weight varying spatial price networks. J. Optim. Theory Appl. 122(2), 387–403 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  27. Rouhani, B.D., Khan, A.A.: On the embedding of variational inequalities. In: Proceedings of the American Mathematical Society, vol. 131, no. 12, pp. 3861–3871 (2003)

  28. Stampacchia G.: Formes bilineaires sur les ensembles convexes. Compt. Rend. Acad. Sci. Paris 258, 4413–4416 (1964)

    MATH  MathSciNet  Google Scholar 

  29. Stampacchia, G.: Variational Inequalities. In: Theory and Applications of Monotone Operators. Proc. N.A.T.O. Adv. Study Inst., Venice 1968, pp. 101–192. Oderisi, Gubbio, Italy (1968)

  30. Walras L.: Elements d’Economie Politique Pure. Corbaz, Lausanne, Switzerland (1874)

    Google Scholar 

  31. Wardrop, J.G.: Some theoretical aspects of road traffic research. In: Proceedings of the Institute of Civil Engineers, Part II, pp. 325–378 (1952)

  32. Yen N.D.: Lipschitz continuity of solutions of variational inequalities with a parametric polyhedral constraint. Math. Oper. Res. 20, 695–708 (1995)

    Article  MATH  MathSciNet  Google Scholar 

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  1. Dipartimento di Matematica e Informatica, Università di Catania, Viale A. Doria 6, 95125, Catania, Italy

    A. Causa & F. Raciti

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  1. A. Causa
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  2. F. Raciti
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Correspondence to F. Raciti.

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Causa, A., Raciti, F. Some remarks on the Walras equilibrium problem in Lebesgue spaces. Optim Lett 5, 99–112 (2011). https://doi.org/10.1007/s11590-010-0193-y

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