On the Stability and Solution Sensitivity of a Consumer Problem


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.

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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.

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Correspondence to Nguyen Dong Yen.

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Communicated by Vladimir Veliov.

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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

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

Mathematics Subject Classification

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