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On the Stability and Solution Sensitivity of a Consumer Problem

Author

Listed:
  • Vu Thi Huong

    (Vietnam Academy of Science and Technology)

  • Jen-Chih Yao

    (China Medical University)

  • Nguyen Dong Yen

    (Vietnam Academy of Science and Technology)

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.

Suggested Citation

  • Vu Thi Huong & Jen-Chih Yao & Nguyen Dong Yen, 2017. "On the Stability and Solution Sensitivity of a Consumer Problem," Journal of Optimization Theory and Applications, Springer, vol. 175(2), pages 567-589, November.
    • Handle: RePEc:spr:joptap:v:175:y:2017:i:2:d:10.1007_s10957-017-1164-6
    DOI: 10.1007/s10957-017-1164-6
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    References listed on IDEAS

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    1. Jean-Paul Penot, 2013. "Variational Analysis for the Consumer Theory," Journal of Optimization Theory and Applications, Springer, vol. 159(3), pages 769-794, December.
    2. Jean-Pierre Aubin, 1984. "Lipschitz Behavior of Solutions to Convex Minimization Problems," Mathematics of Operations Research, INFORMS, vol. 9(1), pages 87-111, February.
    3. Jen-Chih Yao, 1994. "Variational Inequalities with Generalized Monotone Operators," Mathematics of Operations Research, INFORMS, vol. 19(3), pages 691-705, August.
    4. Martinez-Legaz, Juan-Enrique & Santos, Manuel S, 1993. "Duality between Direct and Indirect Preferences," Economic Theory, Springer;Society for the Advancement of Economic Theory (SAET), vol. 3(2), pages 335-351, April.
    5. B. K. Ane & A. M. Tarasyev & C. Watanabe, 2007. "Construction of Nonlinear Stabilizer for Trajectories of Economic Growth," Journal of Optimization Theory and Applications, Springer, vol. 134(2), pages 303-320, August.
    6. N. D. Yen, 1997. "Stability of the Solution Set of Perturbed Nonsmooth Inequality Systems and Application," Journal of Optimization Theory and Applications, Springer, vol. 93(1), pages 199-225, April.
    7. Crouzeix, J. -P., 1983. "Duality between direct and indirect utility functions : Differentiability properties," Journal of Mathematical Economics, Elsevier, vol. 12(2), pages 149-165, October.
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    Full references (including those not matched with items on IDEAS)

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