Skip to main content
SpringerLink
Log in

A method in demand analysis connected with the Monge—Kantorovich problem

  • Chapter
Advances in Mathematical Economics

Part of the book series: Advances in Mathematical Economics ((MATHECON,volume 7))

Abstract

A method in demand analysis based on the Monge—Kantorovich duality is developed. We characterize (insatiate) demand functions that are rationalized, in different meanings, by concave utility functions with some additional properties such as upper semi-continuity, continuity, non-decrease, strict concavity, positive homogeneity and so on. The characterizations are some kinds of abstract cyclic monotonicity strengthening revealed preference axioms, and also they may be considered as an extension of the Afriat—Varian theory to an arbitrary (infinite) set of ‘observed data’. Particular attention is paid to the case of smooth functions.

Supported in part by Russian Foundation for Humanitarian Sciences (project 03-02-00027). A part of the material of this paper was presented at the international conference “Kantorovich memorial. Mathematics and economics: old problems and new approaches”, St.-Petersburg, January, 8–13, 2004.

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

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 84.99
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 109.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD 109.99
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

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. Afriat, S.N.: The construction of utility functions from expenditure data. International Economic Review 8, 67–77 (1967)

    MATH  Google Scholar 

  2. Afriat, S.N.: On a system of inequalities on demand analysis: an extension of the classical method. International Economic Review 14, 460–472 (1973)

    MATH  MathSciNet  Google Scholar 

  3. Carlier, G., Ekeland, I., Levin, V.L., Shananin, A.A.: A system of inequalities arising in mathematical economics and connected with the Monge—Kantorovich problem. Ceremade—UMR 7534—Université Paris Dauphine, No 0213, 3 Mai 2002

    Google Scholar 

  4. Chipman, J.S., Hurwicz, L., Richter, M., Sonnenschein, H., eds.: Preferences, Utility and Demand. Harcourt, Brace, Jovanovich, New York 1971

    Google Scholar 

  5. Fenchel, W.: Über konvexe Funktionen mit vorgeschriebenen Niveaumannigfaltigkeiten. Math. Z. 63, 496–506 (1956)

    Article  MATH  MathSciNet  Google Scholar 

  6. Houthakker, H.S.: Revealed preference and the utility function. Economica 17, 159–174 (1950)

    Article  MathSciNet  Google Scholar 

  7. Houthakker, H.S.: The present state of consumption theory. Econometrica 29, 704–740 (1961)

    MATH  MathSciNet  Google Scholar 

  8. Kannai, Y.: Concavifiability and constructions of concave utility functions. Journal of Mathematical Economics 4, 1–56 (1977)

    Article  MATH  MathSciNet  Google Scholar 

  9. Kantorovich, L.V.: On mass transfer. Dokl. Akad. Nauk SSSR 37(7–8), 199–201 (1942) (Russian)

    Google Scholar 

  10. Kantorovich, L.V., Akilov, G.P.: Functional Analysis, 3rd ed. Moscow: Nauka 1984 (Russian)

    Google Scholar 

  11. Kantorovich, L.V., Rubinshtein, G.S.: On a function space and some extremal problems. Dokl. Akad. Nauk SSSR 115, 1058–1061 (1957) (Russian)

    MathSciNet  Google Scholar 

  12. Kantorovich, L.V., Rubinshtein, G.S.: On a space of countably additive functions. Vestnik Leningrad Univ. 13(7), 52–59 (1958) (Russian)

    MathSciNet  Google Scholar 

  13. Levin, V.L.: A formula for optimal value of Monge-Kantorovich problem with a smooth cost function and characterization of cyclically monotone mappings. Matem. Sbornik 181,N12, 1694–1709 (1990) (Russian); English translation in Math. USSR Sbornik 71, No 2, 533–548 (1992)

    MATH  Google Scholar 

  14. Levin, V.L.: General Monge—Kantorovich problem and its applications in measure theory and mathematical economics. In: Functional Analysis, Optimization, and Mathematical Economics. A Collection of Papers Dedicated to the Memory of L.V. Kantorovich (L.J. Leifman, ed.). pp. 141–176 N.Y. Oxford: Oxford Univ. Press 1990

    Google Scholar 

  15. Levin, V.L.: Some applications of set-valued mappings in mathematical economics. Journal of Mathematical Economics 20, 69–87 (1991)

    Article  MATH  MathSciNet  Google Scholar 

  16. Levin, V.L.: A superlinear multifunction arising in connection with mass transfer problems. Set-Valued Analysis 4,N1, 41–65 (1996)

    Article  MATH  MathSciNet  Google Scholar 

  17. Levin V.L.: Duality for a non-topological version of the mass transportation problem. In: Distributions with Fixed Marginals and Related Topics (L. Rüschendorf, B. Schweizer, M.D. Taylor eds.). IMS Lecture Notes Monogr. Ser. 28, pp.175–186 Inst. Math. Statist., Hayward 1996

    Google Scholar 

  18. Levin, V.L.: Topics in the duality theory for mass transfer problems. In: Distributions with given marginals and moment problems (V. Beneš, J. Štěpan eds.). pp.243–252 Kluwer Academic Publishers 1997

    Google Scholar 

  19. Levin, V.L.: On duality theory for non-topological variants of the mass transfer problem. Matem. Sbornik 188,N4, 95–126 (1997) (Russian); English translation in Sbornik: Mathematics 188, N4, 571–602 (1997)

    MATH  Google Scholar 

  20. Levin, V.L.: Reduced cost functions and their applications. Journal of Mathem. Economics 28, 155–186 (1997)

    Article  MATH  Google Scholar 

  21. Levin, V.L.: Abstract cyclical monotonicity and Monge solutions for the general Monge—Kantorovich problem. Set-Valued Analysis 7, 7–32 (1999)

    Article  MathSciNet  Google Scholar 

  22. Levin, V.L.: A method in utility theory connected with the Monge—Kantorovich problem. Working paper WP/2000/089, Moscow, CEMI Russian Academy of Sciences, 2000

    Google Scholar 

  23. Levin, V.L.: The Monge-Kantorovich problems and stochastic preference relations. Advances in Math. Economics 3, 97–124 (2001)

    MATH  Google Scholar 

  24. Levin, V.L.: Optimal solutions of the Monge problem. Advances in Mathematical Economics 6, 85–122 (2004)

    MATH  Google Scholar 

  25. Levin, V.L., Milyutin, A.A.: The problem of mass transfer with a discontinuous cost function and a mass statement of the duality problem for convex extremal problems. Russian Math. Surveys 34(3), 1–78 (1979)

    Article  MathSciNet  Google Scholar 

  26. Monge, G.: Mémoire sur la théorie des déblais et de remblais. Histoire de l’Académie Royale des Sciences de Paris, avec les Mémoires de Mathématique et de Physique pour la même année, 666–704 (1781)

    Google Scholar 

  27. Pospelova, L.Ya., Shananin, A.A.: The value of nonrationality for consumer behavior and generalized nonparametric method. Mathematical Modeling 10, No.4, 105–116 (1998) (Russian)

    Google Scholar 

  28. Rachev, S.T., Rüschendorf, L.: Mass Transportation Problems. Volume 1: Theory, Volume 2: Applications. Springer-Verlag 1998

    Google Scholar 

  29. Richter, M.: Revealed preference theory. Econometrica 34, 635–645 (1966)

    MATH  Google Scholar 

  30. Rockafellar, R.T.: Convex Analysis. Princeton University Press, Princeton, 1970

    Google Scholar 

  31. Samuelson, P.A.: Foundations of Economic Analysis. Harvard University Press 1947

    Google Scholar 

  32. Samuelson, P.A.: Consumption theory in terms of revealed preference. Economica 15, 243–253 (1948)

    Article  Google Scholar 

  33. Uzawa, H.: Preference and rational choice in the theory of consumption. In: Mathematical Methods in the Social Sciences (K.J. Arrow, S. Karlin, P. Suppes eds.). Stanford University Press 1959

    Google Scholar 

  34. Varian, H.R.: The nonparametric approach to demand analysis. Econometrica 50, 945–973 (1982)

    MATH  MathSciNet  Google Scholar 

  35. Varian, H.R.: Non-parametric tests of consumer behaviour. The Review of Economic Studies V(1), No.160, 99–110 (1983)

    Article  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Central Economics and Mathematics Institute of Russian Academy of Sciences, 47 Nakhimovskii Prospect, 117418, Moscow, Russia

    Vladimir L. Levin

Authors
  1. Vladimir L. Levin
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

Editors and Affiliations

  1. Graduate School of Mathematical Sciences, University of Tokyo, 3-8-1 Komaba, Meguro-ku, Tokyo, 153-0041, Japan

    Shigeo Kusuoka (Professor) (Professor)

  2. Graduate Faculty of Economics, Hitotsubashi University, Kunitachi, Tokyo, 186-8601, Japan

    Akira Yamazaki (Professor) (Professor)

Rights and permissions

Reprints and permissions

Copyright information

© 2005 Springer-Verlag

About this chapter

Cite this chapter

Levin, V.L. (2005). A method in demand analysis connected with the Monge—Kantorovich problem. In: Kusuoka, S., Yamazaki, A. (eds) Advances in Mathematical Economics. Advances in Mathematical Economics, vol 7. Springer, Tokyo. https://doi.org/10.1007/4-431-27233-X_3

Download citation

Publish with us

Policies and ethics

Search

Navigation