Advertisement

Journal of Optimization Theory and Applications

, Volume 175, Issue 2, pp 567–589 | Cite as

On the Stability and Solution Sensitivity of a Consumer Problem

  • Vu Thi Huong
  • Jen-Chih Yao
  • Nguyen Dong YenEmail author
Article
  • 257 Downloads

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.

Keywords

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

Mathematics Subject Classification

91B16 91B42 91B38 46N10 49J53 

Notes

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.

References

  1. 1.
    Takayama, A.: Mathematical Economics. The Dryden Press, Hinsdale (1974)zbMATHGoogle Scholar
  2. 2.
    Intriligator, M.D.: Mathematical Optimization and Economic Theory, 2nd edn. SIAM, Philadelphia (2002)CrossRefzbMATHGoogle 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)CrossRefGoogle 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)CrossRefMathSciNetGoogle 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)CrossRefMathSciNetGoogle Scholar
  8. 8.
    Miyagishima, K.: Implementability and equity in production economies with unequal skills. Rev. Econ. Des. 19, 247–257 (2015)zbMATHMathSciNetGoogle Scholar
  9. 9.
    Penot, J.-P.: Variational analysis for the consumer theory. J. Optim. Theory Appl. 159, 769–794 (2013)CrossRefzbMATHMathSciNetGoogle Scholar
  10. 10.
    Penot, J.-P.: Some properties of the demand correspondence in the consumer theory. J. Nonlinear Convex Anal. 15, 1071–1085 (2014)zbMATHMathSciNetGoogle 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)CrossRefzbMATHMathSciNetGoogle 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)CrossRefzbMATHMathSciNetGoogle 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) Google Scholar
  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)CrossRefzbMATHMathSciNetGoogle Scholar
  15. 15.
    Hadjisavvas, N., Penot, J.-P.: Revisiting the problem of integrability in utility theory. Optimization 64, 2495–2509 (2015)CrossRefzbMATHMathSciNetGoogle 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)Google Scholar
  18. 18.
    Crouzeix, J.-P.: Duality between direct and indirect utility functions. Differentiability properties. J. Math. Econ. 12, 149–165 (1983)CrossRefzbMATHMathSciNetGoogle Scholar
  19. 19.
    Martínez-Legaz, J.-E., Santos, M.S.: Duality between direct and indirect preferences. Econ. Theory 3, 335–351 (1993)CrossRefzbMATHMathSciNetGoogle Scholar
  20. 20.
    Borwein, J.M.: Stability and regular points of inequality systems. J. Optim. Theory Appl. 48, 9–52 (1986)CrossRefzbMATHMathSciNetGoogle Scholar
  21. 21.
    Yen, N.D.: Hölder continuity of solutions to a parametric variational inequality. Appl. Math. Optim. 31, 245–255 (1995)CrossRefzbMATHMathSciNetGoogle Scholar
  22. 22.
    Bonnans, J.F., Shapiro, A.: Perturbation Analysis of Optimization Problems. Springer, New York (2000)CrossRefzbMATHGoogle 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)CrossRefzbMATHMathSciNetGoogle Scholar
  25. 25.
    Yen, N.D.: Implicit function theorems for set-valued maps. Acta Math. Vietnam. 12, 17–28 (1987)zbMATHMathSciNetGoogle Scholar
  26. 26.
    Clarke, F.H.: Optimization and Nonsmooth Analysis, 2nd edn. McGraw-Hill, SIAM, Philadelphia (2002)zbMATHGoogle Scholar
  27. 27.
    Aubin, J.P., Frankowska, H.: Set-Valued Analysis. Birkhäuser, Boston (2009). (reprint of the 1990 edition) CrossRefzbMATHGoogle Scholar
  28. 28.
    Gfrerer, H., Mordukhovich, B.S.: Robinson stability of parametric constraint systems via variational analysis. Preprint, July 26 (2016)Google Scholar
  29. 29.
    Rudin, W.: Functional Analysis, 2nd edn. McGraw-Hill Inc, New York (1991)zbMATHGoogle Scholar
  30. 30.
    Brezis, H.: Functional Analysi Sobolev Spaces and Partial Differential Equations. Springer, New York (2011)zbMATHGoogle 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)CrossRefzbMATHMathSciNetGoogle 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)CrossRefzbMATHMathSciNetGoogle 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)CrossRefzbMATHMathSciNetGoogle 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)zbMATHGoogle Scholar
  36. 36.
    Hadjisavvas, N., Komlósi, S., Schaible, S. (eds.): Handbook of Generalized Convexity and Generalized Monotonicity. Springer, New York (2005)zbMATHGoogle Scholar
  37. 37.
    Yao, J.-C.: Variational inequalities with generalized monotone operators. Math. Oper. Res. 19, 691–705 (1994)CrossRefzbMATHMathSciNetGoogle Scholar
  38. 38.
    Yao, J.-C.: Multi-valued variational inequalities with K-pseudomonotone operators. J. Optim. Theory Appl. 80, 63–74 (1994)CrossRefzbMATHMathSciNetGoogle 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)CrossRefGoogle 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

Copyright information

© Springer Science+Business Media, LLC 2017

Authors and Affiliations

  1. 1.Graduate Training Center, Institute of MathematicsVietnam Academy of Science and TechnologyHanoiVietnam
  2. 2.Center for General EducationChina Medical UniversityTaichungTaiwan
  3. 3.Institute of MathematicsVietnam Academy of Science and TechnologyHanoiVietnam

Personalised recommendations