Skip to main content

Variational analysis and mathematical economics 2: Nonsmooth regular economies

  • Conference paper
Advances in Mathematical Economics

Part of the book series: Advances in Mathematical Economics ((MATHECON,volume 14))

Abstract

We introduce and study a quantitative concept of regularity (in the spirit of variational analysis) for exchange economies with set-valued demand correspondences and prove that in case the latter are semi-algebraic, every economy with the exception of a set of smaller dimension are regular. We also discuss the determinacy problem for such economies.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 89.00
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 119.00
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD 169.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Similar content being viewed by others

References

  1. Aubin, J.P., Ekeland, I.: Applied Nonlinear Analysis. J. Wiley and Sons, New York (1984)

    Google Scholar 

  2. Blume, L.E., Zame W.R.: The algebraic geometry of competitive equilibrium. In: Neuefeind, W. (ed.) General Equilibrium and International Trade. In Memoraum Trout Rader, pp. 53–66. Springer, New York (1993)

    Google Scholar 

  3. Bochnak, J., Coste, M., Roy, M.F.: Real Algebraic Geometry, E.M.G. vol. 36. Springer, Berlin (1998)

    Google Scholar 

  4. Borwein, J.M., Zhuang, D.M.: Verifiable necessary and sufficient conditions for openness and regularity of set-valued and single-valued maps. J. Math. Anal. Appl. 134, 441–459 (1988)

    Article  Google Scholar 

  5. Coste, M.: An Introduction to O-Minimal Geometry. Inst. Rech. Math., Univ. de Rennes. http://name.math.univ-rennes1.fr/michel.coste/polyens/OMIN.pdf (1999)

  6. Debreu, G.: Economies with a finite set of equilibria. Econometrica 38, 387–392 (1970)

    Article  Google Scholar 

  7. Debreu, G.: Smooth preferences. Econometrica 40, 603–615 (1972)

    Article  Google Scholar 

  8. Dierker, E.: Regular economies. In: Arrow, K.J., Intriligator, M.D. (eds.) Handbook of Mathematical Economics, vol. 3, pp. 795–830. North-Holland, Amsterdam (1982)

    Google Scholar 

  9. Dmitruk, A.V., Milyutin, A.A., Osmolovskii, N.P.: Ljusternik’s theorem and the theory of extrema. Russ. Math. Surv. 35(6), 11–51 (1980)

    Article  Google Scholar 

  10. Dontchev, A.L.: The Graves theorem revisited. In: Mordukhovich, B.S., Sussmann, H.J. (eds.) Nonlinear Analysis and Geometric Methods in Deterministic Optimal Control, pp. 59–81. Springer, New York (1996)

    Google Scholar 

  11. Dontchev, A.L., Rockafellar, R.T.: Implicit Function and Solution Mapping. Springer, Berlin (2009)

    Book  Google Scholar 

  12. van den Dries, L.: Tame Topology and O-Minimal Structures. Cambridge University Press, Cambridge (1998)

    Book  Google Scholar 

  13. van den Dries, L., Miller, C.: Geometric categories and o-minimal structures. Duke Math. J. 84, 497–540 (1996)

    Article  Google Scholar 

  14. Drusvyatskiy, V., Lewis, A.S.: Semi-algebraic functions have small subdifferentials. (2010, preprint)

    Google Scholar 

  15. Graves, L.M.: Some mapping theorems. Duke Math. J. 17, 111–114 (1950)

    Article  Google Scholar 

  16. Guillemin, V., Pollack, A.: Differential Topology. Prentice Hall, Englewood Cliffs, NJ (1976)

    Google Scholar 

  17. Ioffe, A.D.: Non-smooth analysis: differential calculus of non-differentiable mappings. Trans. Am. Math. Soc. 255, 1–55 (1981)

    Article  Google Scholar 

  18. Ioffe, A.D.: Metric regularity and subdifferential calculus. Uspehi Mat. Nauk 55(3), 103–162 (2000) (in Russian), English translation: Russ. Math. Surv. 55(3), 501–558 (2000)

    Google Scholar 

  19. Ioffe, A.D.: Critical values of set-valued mappings with stratifiable graphs. Extensions of Sard and Smale-Sard theorems. Proc. Am. Math. Soc. 136, 3111–3119 (2008)

    Article  Google Scholar 

  20. Ioffe, A.D.: Variational analysis and mathematical economics 1. Subdifferential calculus and the second theorem of welfare economics. Adv. Math. Econ. 12, 71–95 (2009)

    Google Scholar 

  21. Ioffe, A.D., Tihomirov, V.M.: Theory of Extremal Problems. Nauka, Moscow (1974) (in Russian); English translation: North Holland (1979)

    Google Scholar 

  22. Kubler, F., Schmedders, K.: Competitive equilibria in semi-algebraic economies. J. Econ. Theory 145, 301–330 (2010)

    Article  Google Scholar 

  23. Łojasiewicz, S.: Ensembles Semi-Analytiques. IHES Lecture Notes (1965)

    Google Scholar 

  24. Mas-Colell, A.: The Theory of General Economic Equilibrium. A Differentiable Approach. Cambridge University Press, Cambridge (1985)

    Google Scholar 

  25. Mordukhovich, B.S.: Variational Analysis and Generalized Differentiation. Springer, Berlin (2005)

    Google Scholar 

  26. Penot, J.-P.: Metric regularity, openness and Lipschitzean behaviour of multifunctions. Nonlinear Anal. 13, 629–643 (1989)

    Article  Google Scholar 

  27. Robinson, S.M.: Stability theory for system of inequalities. Part II: differentiable nonlinear systems. SIAM J. Numer. Anal. 13, 497–513 (1976)

    Google Scholar 

  28. Rockafellar, R.T., Wets, R.J.B.: Variational Analysis. Springer, Berlin (1998)

    Book  Google Scholar 

  29. Smale, S.: Global analysis and economics. J. Math. Econ. 1, 1–14 (1974)

    Article  Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to A. D. Ioffe .

Editor information

Editors and Affiliations

Rights and permissions

Reprints and permissions

Copyright information

© 2011 Springer

About this paper

Cite this paper

Ioffe, A.D. (2011). Variational analysis and mathematical economics 2: Nonsmooth regular economies. In: Kusuoka, S., Maruyama, T. (eds) Advances in Mathematical Economics. Advances in Mathematical Economics, vol 14. Springer, Tokyo. https://doi.org/10.1007/978-4-431-53883-7_2

Download citation

Publish with us

Policies and ethics