Skip to main content
SpringerLink
Log in

Variational Analysis for the Consumer Theory

  • Published:
Journal of Optimization Theory and Applications Aims and scope Submit manuscript

Abstract

We look for an interpretation of the demand correspondence in the consumer theory as a generalized derivative of the inverse utility function. We test the main concepts of nonsmooth analysis for such an objective. The proofs only use classical methods in optimization such as penalization and optimality conditions.

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

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Diewert, W.E.: Applications of duality theory. In: Intriligator, M.D., Kendrick, D.A. (eds.) Frontiers of Quantitative Economics, vol II, pp. 106–171. North-Holland, Amsterdam (1974)

    Google Scholar 

  2. Penot, J.-P.: The bearing of duality on microeconomics. Adv. Math. Econ. 7, 113–139 (2005)

    Article  MathSciNet  Google Scholar 

  3. Borwein, J.M., Zhu, Q.J.: Techniques of Variational Analysis. CMS Books in Maths. Springer, New York (2005)

    MATH  Google Scholar 

  4. Clarke, F.H.: Optimization and Nonsmooth Analysis. SIAM, Philadelphia (1990)

    Book  MATH  Google Scholar 

  5. Mordukhovich, B.: Variational Analysis and Generalized Differentiation. I Basic Theory. Grundlehren der Mathematischen Wisssenschaften, vol. 330. Springer, Berlin (2006)

    Google Scholar 

  6. Penot, J.-P.: Calculus Without Derivatives. Graduate Texts in Mathematics, vol. 266. Springer, New York (2013)

    Book  MATH  Google Scholar 

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

    Book  MATH  Google Scholar 

  8. Shirotzek, W.: Nonsmooth Analysis. Springer, Berlin (2007)

    Book  Google Scholar 

  9. Crouzeix, J.-P.: Duality between direct and indirect utility functions. Differentiability properties. J. Math. Econ. 12, 149–165 (1983)

    Article  MathSciNet  MATH  Google Scholar 

  10. Crouzeix, J.-P.: La Convexité Généralisée en Économie Mathématique. ESAIM Proc. 13, 31–40 (2003)

    MathSciNet  MATH  Google Scholar 

  11. Intriligator, M.D.: Mathematical Optimization and Economic Theory. Prentice-Hall, Englewood Cliffs (1971)

    Google Scholar 

  12. Martínez-Legaz, J.-E.: Duality between direct and indirect utility functions under minimal hypothesis. J. Math. Econ. 20, 199–209 (1991)

    Article  MATH  Google Scholar 

  13. Martínez-Legaz, J.-E.: Generalized convex duality and its economic applications. In: Hadjisavvas, N., Komlósi, S., Schaible, S. (eds.) Handbook of Generalized Convexity and Generalized Monotonicity. Nonconvex Optimization and Its Applications, vol. 76, pp. 237–292. Springer, New York (2005)

    Chapter  Google Scholar 

  14. Penot, J.-P.: Some properties of the demand correspondence in the consumer theory. Submitted

  15. Penot, J.-P.: What is quasiconvex analysis? Optimization 47, 35–110 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  16. Greenberg, H.P., Pierskalla, W.P.: Quasiconjugate function and surrogate duality. Cah. Cent. d’Etude Rech. Oper. 15, 437–448 (1973)

    MathSciNet  MATH  Google Scholar 

  17. Aussel, D., Daniilidis, A.: Normal characterization of the main classes of quasiconvex functions. Set-Valued Anal. 8, 219–236 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  18. Aussel, D., Daniilidis, A.: Normal Cones to Sublevel Sets: An Axiomatic Approach. Applications in Quasiconvexity and Pseudoconvexity. Generalized Convexity and Generalized Monotonicity, Karlovassi, 1999. Lecture Notes in Econom. and Math. Systems, vol. 502, pp. 88–101. Springer, Berlin (2001)

    Book  Google Scholar 

  19. Aussel, D., Hadjisavvas, N.: Adjusted sublevel sets, normal operator, and quasi-convex programming. SIAM J. Optim. 16(2), 358–367 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  20. Borde, J., Crouzeix, J.-P.: Continuity of the normal cones to the level sets of quasiconvex functions. J. Optim. Theory Appl. 66(3), 415–429 (1990)

    Article  MathSciNet  MATH  Google Scholar 

  21. Penot, J.-P., Zălinescu, C.: Harmonic sum and duality. J. Convex Anal. 7(1), 95–113 (2000)

    MathSciNet  MATH  Google Scholar 

  22. Ngai, H.V., Théra, M.: Error bounds and implicit multifunction theorem in smooth Banach spaces and applications to optimization. Set-Valued Anal. 12(1–2), 195–223 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  23. Plastria, F.: Lower subdifferentiable functions and their minimization by cutting plane. J. Optim. Theory Appl. 46(1), 37–54 (1985)

    Article  MathSciNet  MATH  Google Scholar 

  24. Moreau, J.-J.: Inf-convolution, sous-additivité, convexité des fonctions numériques. J. Math. Pures Appl. 49, 109–154 (1970)

    MathSciNet  MATH  Google Scholar 

  25. Penot, J.-P.: Unilateral Analysis and Duality. In: Audet, C., Hansen, P., Savard, G. (eds.) Essays and Surveys in Global Optimization, pp. 1–37. Springer, New York (2005)

    Chapter  Google Scholar 

  26. Roy, R.: De l’Utilité. Hermann, Paris (1942)

    MATH  Google Scholar 

  27. Penot, J.-P.: On regularity conditions in mathematical programming. Math. Program. Stud. 19, 167–199 (1982)

    Article  MathSciNet  MATH  Google Scholar 

  28. Zălinescu, C.: Convex Analysis in General Vector Spaces. World Scientific, Singapore (2002)

    Book  MATH  Google Scholar 

  29. Borwein, J.M., Lewis, A.S.: Convex Analysis and Nonlinear Optimization. Theory and Examples. CMS Books in Maths. Springer, New York (2000)

    Book  MATH  Google Scholar 

Download references

Acknowledgements

We are grateful to Jean-Pierre Crouzeix and Nicolas Hadjisavvas for their incentives to study the consumer theory and for stimulating discussions about that topic.

Author information

Authors and Affiliations

  1. Laboratoire Jacques-Louis Lions, Université Pierre et Marie Curie, 4 place Jussieu, 75252, Paris Cedex 05, France

    Jean-Paul Penot

Authors
  1. Jean-Paul Penot
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Jean-Paul Penot.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Penot, JP. Variational Analysis for the Consumer Theory. J Optim Theory Appl 159, 769–794 (2013). https://doi.org/10.1007/s10957-013-0289-5

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10957-013-0289-5

Keywords

Search

Navigation