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


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.


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

Mathematics Subject Classification

91B16 91B42 91B38 46N10 49J53 



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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© 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

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