Advertisement

Set-Valued Analysis

, Volume 2, Issue 1–2, pp 159–173 | Cite as

On variational stability in competitive economies

  • Sjur Didrik FlÅm
Article

Abstract

We explore the variational stability of supply, demand and equilibria in perfectly competitive economies. The appropriate and, in fact, minimal limit notion is furnished by the Kuratowski-Painlevé concept of set convergence together with its functional analogues epi and hypo convergence. When technologies and preferences converge is such manners we show, subject to compactness assumptions, that observable features of approximate economies cluster to those of the limiting economy. Such findings are important in applied economic analysis.

Mathematics Subject Classification (1991)

90A11 

Key words

Epi and hypo convergence sub and super-differentials Hotelling's and Shepard's lemmata expenditure and benefit functions Hicksian and excess demand indirect utility competitive equilibria 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. Attouch, H. (1977): Convergences de fonctions convexes, des sous-différentiels et semi-groupes associés,Comptes Rend. Acad. Sci. Paris 284, 539–542.Google Scholar
  2. Attouch, H. (1984):Variational Convergence for Functions and Operators, Pitman, Boston.Google Scholar
  3. Attouch, H., Azé, D., and Beer, G. (1991): On some inverse stability problems for the epigraphical sum,Nonlinear Analysis 16 (3), 241–254.Google Scholar
  4. Attouch, H. and Beer, G. (1991):On the Convergence of Subdifferentials of Convex Functions, WP, Montpellier.Google Scholar
  5. Attouch, H. and Wets, R.J-B. (1986): Isometries for the Legendre-Fenchel transform,Trans. Amer. Math. Soc. 296, 33–60.Google Scholar
  6. Attouch, H. and Wets, R.J-B. (1991): Quantitative stability of variational systems: I The epigraphical distance,Trans. Amer. Math. Soc. 328, 695–730.Google Scholar
  7. Aubin, J.P. and Ekeland, I. (1984):Applied Nonlinear Analysis, Wiley, New York.Google Scholar
  8. Aumann, R.J. (1966): Existence of a competitive equilibrium in markets with a continuum of traders,Econometrica 34, 1–17.Google Scholar
  9. Azé, D. and Volle, M. (1990): A stability result in quasi-convex programming,J. Optim. Theory Appl. 67 (1), 175–184.Google Scholar
  10. Beer, G. (1991):The Slice Topology: A Viable Alternative to Mosco-Convergence in Non-reflexive Spaces, WP, California State University, LA.Google Scholar
  11. Bewley, T.F. (1972): Existence of equilibria in economies with infinitely many commodities,J. Econ. Theory 4, 514–540.Google Scholar
  12. Cavazzuti, E. and Pacchiarotti, N. (1986): Convergence of Nash equilibria,Boll. U.M.I. 6 (5B), 247–266.Google Scholar
  13. Diewert, W.E. (1982): Duality approaches to microeconomic theory, in Arrowet al. (eds.),Handbook of Mathematical Economics, North-Holland, Amsterdam, vol. II Chapt. 14.Google Scholar
  14. Debreu, G. (1959):The Theory of Value, Wiley, New York.Google Scholar
  15. Debreu, G. (1967): Integration of correspondences,Proc. Fifth Berkeley Symp. Math. Stat. Prob., Univ. of California Press, Berkeley, vol. II, Part I, pp. 351–372.Google Scholar
  16. Kreps, D.M. (1990):Microeconomic Theory, Harvester Wheatsheaf, New York.Google Scholar
  17. Lucchetti, R. and Patrone, F. (1986): Closure and upper semicontinuity results in mathematical programming, Nash and economic equilibria,Optimization 17, 619–628.Google Scholar
  18. Lucchetti, R. (1985): Stability in Pareto problems, Univ. degli Studi di Milano, Dip. di Matematica, 14.Google Scholar
  19. Lucchetti, R., Papageorgiou, N.S., and Patrone, F. (1987): Convergence and approximation results for measurable multifunctions,Proc. Amer. Math. Soc. 100, 551–556.Google Scholar
  20. Luenberger, D.G. (1992): Benefit functions and duality,J. Math. Economics 21, 461–481.Google Scholar
  21. Mosco, V. (1971): On the continuity of the Young-Fenchel transform,J. Math. Anal. Appl. 35, 518–535.Google Scholar
  22. Rockafellar, R.T. (1970):Convex Analysis, Princeton University Press, New Jersey.Google Scholar
  23. Varian, H. (1992):Microeconomic Analysis, 3 edn., Norton, New York.Google Scholar
  24. Volle, M. (1984): Convergence en nivaux et en epigraphs,C.R. Acad. Sci. Paris 299, 295–298.Google Scholar
  25. Wijsman, R. (1966): Convergence of sequences of convex sets, cones and functions II,Trans. Amer. Math. Soc. 123, 32–45.Google Scholar
  26. Yannelis, N.C. (1985): On a market equilibrium theorem with an infinite number of commodities,J. Math. Anal. Appl. 108, 595–599.Google Scholar

Copyright information

© Kluwer Academic Publishers 1994

Authors and Affiliations

  • Sjur Didrik FlÅm
    • 1
  1. 1.Economics DepartmentBergen UniversityNorway

Personalised recommendations