On the Stability and Solution Sensitivity of a Consumer Problem

Abstract

Various stability properties and a result on solution sensitivity of a consumer problem are obtained in this paper. Focusing on some nice features of the budget map, we are able to establish the continuity and the locally Lipschitz continuity of the indirect utility function, as well as the Lipschitz–Hölder continuity of the demand map under a minimal set of assumptions. The recent work of Penot (J Nonlinear Convex Anal 15:1071–1085, 2014) is our starting point, while an implicit function theorem of Borwein (J Optim Theory Appl 48:9–52, 1986) and a theorem of Yen (Appl Math Optim 31:245–255, 1995) on solution sensitivity of parametric variational inequalities are the main tools in our proofs.

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

References

  1. 1.

    Takayama, A.: Mathematical Economics. The Dryden Press, Hinsdale (1974)

    Google Scholar 

  2. 2.

    Intriligator, M.D.: Mathematical Optimization and Economic Theory, 2nd edn. SIAM, Philadelphia (2002)

    Google Scholar 

  3. 3.

    Chiang, A.C., Wainwright, K.: Fundamental Methods of Mathematical Economics, 4th edn. McGraw-Hill, New York (2005)

    Google Scholar 

  4. 4.

    Nicholson, W., Snyder, C.: Microeconomic Theory: Basic Principles and Extension, 7th edn. South-Western, Cengage Learning, Boston (2012)

    Google Scholar 

  5. 5.

    Rasmussen, S.: Production Economics: The Basic Theory of Production. Optimization, 2nd edn. Springer, Berlin (2013)

    Google Scholar 

  6. 6.

    Jarrahi, F., Abdul-Kader, W.: Performance evaluation of a multi-product production line: an approximation method. Appl. Math. Model. 39, 3619–3636 (2015)

    Article  MathSciNet  Google Scholar 

  7. 7.

    Keskin, G.A., Omurca, S.I., Aydin, N., Ekinci, E.: A comparative study of production-inventory model for determining effective production quantity and safety stock level. Appl. Math. Model. 39, 6359–6374 (2015)

    Article  MathSciNet  Google Scholar 

  8. 8.

    Miyagishima, K.: Implementability and equity in production economies with unequal skills. Rev. Econ. Des. 19, 247–257 (2015)

    MATH  MathSciNet  Google Scholar 

  9. 9.

    Penot, J.-P.: Variational analysis for the consumer theory. J. Optim. Theory Appl. 159, 769–794 (2013)

    Article  MATH  MathSciNet  Google Scholar 

  10. 10.

    Penot, J.-P.: Some properties of the demand correspondence in the consumer theory. J. Nonlinear Convex Anal. 15, 1071–1085 (2014)

    MATH  MathSciNet  Google Scholar 

  11. 11.

    Tsao, Y.-C.: A piecewise nonlinear optimization for a production-inventory model under maintenance, variable setup costs, and trade credits. Ann. Oper. Res. 233, 465–481 (2015)

    Article  MATH  MathSciNet  Google Scholar 

  12. 12.

    Ane, B.K., Tarasyev, A.M., Watanabe, C.: Construction of nonlinear stabilizer for trajectories of economic growth. J. Optim. Theory Appl. 134, 303–320 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  13. 13.

    Aseev, S.M., Kryazhimskii, A.V.: The Pontryagin maximum principle and problems of optimal economic growth, translation in Mat. Inst. Steklova. 257 (2007); translation in Proc. Steklov Inst. Math. 257 (2007) (in Russian)

  14. 14.

    Krasovskii, A.A., Tarasyev, A.M.: Construction of nonlinear regulators in economic growth models. Proc. Steklov Inst. Math. 268(suppl. 1), S143–S154 (2010)

    Article  MATH  MathSciNet  Google Scholar 

  15. 15.

    Hadjisavvas, N., Penot, J.-P.: Revisiting the problem of integrability in utility theory. Optimization 64, 2495–2509 (2015)

    Article  MATH  MathSciNet  Google Scholar 

  16. 16.

    Beggs, A.: Sensitivity analysis of boundary equilibria. Econ. Theory. doi:10.1007/s00199-016-0987-y (2016)

  17. 17.

    Diewert, W.E.: Applications of duality theory. In: Frontiers of Quantitative Economics, vol. II, pp. 106–206. North-Holland, Amsterdam (1974)

  18. 18.

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

    Article  MATH  MathSciNet  Google Scholar 

  19. 19.

    Martínez-Legaz, J.-E., Santos, M.S.: Duality between direct and indirect preferences. Econ. Theory 3, 335–351 (1993)

    Article  MATH  MathSciNet  Google Scholar 

  20. 20.

    Borwein, J.M.: Stability and regular points of inequality systems. J. Optim. Theory Appl. 48, 9–52 (1986)

    Article  MATH  MathSciNet  Google Scholar 

  21. 21.

    Yen, N.D.: Hölder continuity of solutions to a parametric variational inequality. Appl. Math. Optim. 31, 245–255 (1995)

    Article  MATH  MathSciNet  Google Scholar 

  22. 22.

    Bonnans, J.F., Shapiro, A.: Perturbation Analysis of Optimization Problems. Springer, New York (2000)

    Google Scholar 

  23. 23.

    Mordukhovich, B.S.: Variational Analysis and Generalized Differentiation (Vol. I. Basic Theory, Vol. II. Applications). Springer, Berlin (2006)

    Google Scholar 

  24. 24.

    Aubin, J.-P.: Lipschitz behavior of solutions to convex minimization problems. Math. Oper. Res. 9, 87–111 (1984)

    Article  MATH  MathSciNet  Google Scholar 

  25. 25.

    Yen, N.D.: Implicit function theorems for set-valued maps. Acta Math. Vietnam. 12, 17–28 (1987)

    MATH  MathSciNet  Google Scholar 

  26. 26.

    Clarke, F.H.: Optimization and Nonsmooth Analysis, 2nd edn. McGraw-Hill, SIAM, Philadelphia (2002)

    Google Scholar 

  27. 27.

    Aubin, J.P., Frankowska, H.: Set-Valued Analysis. Birkhäuser, Boston (2009). (reprint of the 1990 edition)

    Google Scholar 

  28. 28.

    Gfrerer, H., Mordukhovich, B.S.: Robinson stability of parametric constraint systems via variational analysis. Preprint, July 26 (2016)

  29. 29.

    Rudin, W.: Functional Analysis, 2nd edn. McGraw-Hill Inc, New York (1991)

    Google Scholar 

  30. 30.

    Brezis, H.: Functional Analysi Sobolev Spaces and Partial Differential Equations. Springer, New York (2011)

    Google Scholar 

  31. 31.

    Jeyakumar, V., Yen, N.D.: Solution stability of nonsmooth continuous systems with applications to cone-constrained optimization. SIAM J. Optim. 14, 1106–1127 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  32. 32.

    Huyen, D.T.K., Yen, N.D.: Coderivatives and the solution map of a linear constraint system. SIAM J. Optim. 26, 986–1007 (2016)

    Article  MATH  MathSciNet  Google Scholar 

  33. 33.

    Yen, N.D.: Stability of the solution set of perturbed nonsmooth inequality systems and application. J. Optim. Theory Appl. 93, 199–225 (1997)

    Article  MATH  MathSciNet  Google Scholar 

  34. 34.

    Vasilev, F.P.: Numerical Methods for Solving Extremal Problems, 2nd edn. Nauka, Moscow (1988). (in Russian)

    Google Scholar 

  35. 35.

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

    Google Scholar 

  36. 36.

    Hadjisavvas, N., Komlósi, S., Schaible, S. (eds.): Handbook of Generalized Convexity and Generalized Monotonicity. Springer, New York (2005)

    Google Scholar 

  37. 37.

    Yao, J.-C.: Variational inequalities with generalized monotone operators. Math. Oper. Res. 19, 691–705 (1994)

    Article  MATH  MathSciNet  Google Scholar 

  38. 38.

    Yao, J.-C.: Multi-valued variational inequalities with K-pseudomonotone operators. J. Optim. Theory Appl. 80, 63–74 (1994)

    Article  MATH  MathSciNet  Google Scholar 

  39. 39.

    Yao, J.-C., Chadli, O.: Pseudomonotone complementarity problems and variational in equalities. In: Hadjisavvas, N., Komlósi, S., Schaible, S. (eds.) Handbook of Generalized Convexity and Generalized Monotonicity, pp. 501–558. Springer, Berlin (2005)

    Google Scholar 

  40. 40.

    Allen, W.B., Doherty, N.A., Weigelt, K., Mansfield, E.: Managerial Economics. Theory, Applications, and Cases, 6th edn. W. W. Norton and Company, New York (2005)

    Google Scholar 

Download references

Acknowledgements

This work was supported by National Foundation for Science & Technology Development (Vietnam) under grant number 101.01-2014.37 and the Grant MOST 105-2221-E-039-009-MY3 (Taiwan). The authors would like to thank the anonymous referees, the Associate Editor, Professor Jean-Paul Penot, and Professor Hoang Xuan Phu for helpful comments and suggestions on the first version of the present paper.

Author information

Affiliations

Authors

Corresponding author

Correspondence to Nguyen Dong Yen.

Additional information

Communicated by Vladimir Veliov.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Huong, V.T., Yao, J. & Yen, N.D. On the Stability and Solution Sensitivity of a Consumer Problem. J Optim Theory Appl 175, 567–589 (2017). https://doi.org/10.1007/s10957-017-1164-6

Download citation

Keywords

  • Consumer problem
  • Producer problem
  • Budget map
  • Indirect utility function
  • Demand map
  • Continuity
  • Lipschitz–Hölder continuity

Mathematics Subject Classification

  • 91B16
  • 91B42
  • 91B38
  • 46N10
  • 49J53