IDEAS home Printed from https://ideas.repec.org/p/arx/papers/2112.06815.html
   My bibliography  Save this paper

Envelope theorem and discontinuous optimisation: the case of positioning choice problems

Author

Listed:
  • Jean-Gabriel Lauzier

Abstract

This article examines differentiability properties of the value function of positioning choice problems, a class of optimisation problems in finite-dimensional Euclidean spaces. We show that positioning choice problems' value function is always almost everywhere differentiable even when the objective function is discontinuous. To obtain this result we first show that the Dini superdifferential is always well-defined for the maxima of positioning choice problems. This last property allows to state first-order necessary conditions in terms of Dini supergradients. We then prove our main result, which is an ad-hoc envelope theorem for positioning choice problems. Lastly, after discussing necessity of some key assumptions, we conjecture that similar theorems might hold in other spaces as well.

Suggested Citation

  • Jean-Gabriel Lauzier, 2021. "Envelope theorem and discontinuous optimisation: the case of positioning choice problems," Papers 2112.06815, arXiv.org.
    • Handle: RePEc:arx:papers:2112.06815
    as

    Download full text from publisher

    File URL: http://arxiv.org/pdf/2112.06815
    File Function: Latest version
    Download Restriction: no
    ---><---

    References listed on IDEAS

    as
    1. Jean-Gabriel Lauzier, 2021. "Ex-post moral hazard and manipulation-proof contracts," Papers 2112.06811, arXiv.org.
    2. Milgrom, Paul & Shannon, Chris, 1994. "Monotone Comparative Statics," Econometrica, Econometric Society, vol. 62(1), pages 157-180, January.
    3. John K.-H Quah, 2007. "The Comparative Statics of Constrained Optimization Problems," Econometrica, Econometric Society, vol. 75(2), pages 401-431, March.
    Full references (including those not matched with items on IDEAS)

    Citations

    Citations are extracted by the CitEc Project, subscribe to its RSS feed for this item.
    as


    Cited by:

    1. Jean-Gabriel Lauzier, 2021. "Ex-post moral hazard and manipulation-proof contracts," Papers 2112.06811, arXiv.org.

    Most related items

    These are the items that most often cite the same works as this one and are cited by the same works as this one.
    1. Uttiya Paul & Tarun Sabarwal, 2023. "Directional monotone comparative statics in function spaces," Economic Theory Bulletin, Springer;Society for the Advancement of Economic Theory (SAET), vol. 11(1), pages 153-169, April.
    2. Nikolai Kukushkin, 2015. "The single crossing conditions for incomplete preferences," International Journal of Game Theory, Springer;Game Theory Society, vol. 44(1), pages 225-251, February.
    3. Roy, Sunanda & Sabarwal, Tarun, 2012. "Characterizing stability properties in games with strategic substitutes," Games and Economic Behavior, Elsevier, vol. 75(1), pages 337-353.
    4. repec:kan:wpaper:201412 is not listed on IDEAS
    5. Finn Christensen, 2019. "Comparative statics and heterogeneity," Economic Theory, Springer;Society for the Advancement of Economic Theory (SAET), vol. 67(3), pages 665-702, April.
    6. John Quah, 2016. "Supermodular Correspondences," Economics Series Working Papers 795, University of Oxford, Department of Economics.
    7. Kukushkin, Nikolai S., 2013. "Approximate Nash equilibrium under the single crossing conditions," MPRA Paper 44320, University Library of Munich, Germany.
    8. Shuoxun Zhang & Tarun Sabarwal & Li Gan, 2015. "Strategic Or Nonstrategic: The Role Of Financial Benefit In Bankruptcy," Economic Inquiry, Western Economic Association International, vol. 53(2), pages 1004-1018, April.
    9. Roy, Sunanda & Sabarwal, Tarun, 2010. "Monotone comparative statics for games with strategic substitutes," Journal of Mathematical Economics, Elsevier, vol. 46(5), pages 793-806, September.
    10. Koji Shirai, 2013. "Welfare variations and the comparative statics of demand," Economic Theory, Springer;Society for the Advancement of Economic Theory (SAET), vol. 53(2), pages 315-333, June.
    11. Andrés Carvajal, 2010. "The testable implications of competitive equilibrium in economies with externalities," Economic Theory, Springer;Society for the Advancement of Economic Theory (SAET), vol. 45(1), pages 349-378, October.
    12. Nikolai Kukushkin, 2013. "Monotone comparative statics: changes in preferences versus changes in the feasible set," Economic Theory, Springer;Society for the Advancement of Economic Theory (SAET), vol. 52(3), pages 1039-1060, April.
    13. Anne-Christine Barthel & Tarun Sabarwal, 2018. "Directional monotone comparative statics," Economic Theory, Springer;Society for the Advancement of Economic Theory (SAET), vol. 66(3), pages 557-591, October.
    14. Kukushkin, Nikolai S., 2018. "Better response dynamics and Nash equilibrium in discontinuous games," Journal of Mathematical Economics, Elsevier, vol. 74(C), pages 68-78.
    15. Wang, Jianli & Li, Jingyuan, 2016. "Lattice-based monotone comparative statics on saving with Selden/Kreps–Porteus preferences," Journal of Mathematical Economics, Elsevier, vol. 65(C), pages 132-138.
    16. Andrew J. Monaco & Tarun Sabarwal, 2016. "Games with strategic complements and substitutes," Economic Theory, Springer;Society for the Advancement of Economic Theory (SAET), vol. 62(1), pages 65-91, June.
    17. Bar Light, 2021. "Stochastic Comparative Statics in Markov Decision Processes," Mathematics of Operations Research, INFORMS, vol. 46(2), pages 797-810, May.
    18. Koji Shirai, 2008. "A generalization of monotone comparative statics," Economics Bulletin, AccessEcon, vol. 3(39), pages 1-9.
    19. repec:kan:wpaper:201502 is not listed on IDEAS
    20. Łukasz Balbus & Paweł Dziewulski & Kevin Reffett & Łukasz Woźny, 2019. "A qualitative theory of large games with strategic complementarities," Economic Theory, Springer;Society for the Advancement of Economic Theory (SAET), vol. 67(3), pages 497-523, April.
    21. Kukushkin, Nikolai S., 2011. "Monotone comparative statics: Changes in preferences vs changes in the feasible set," MPRA Paper 31612, University Library of Munich, Germany.
    22. Elena Antoniadou & Leonard J Mirman & Richard Ruble, 2014. "Lattices and Lotteries," Mathematics of Operations Research, INFORMS, vol. 39(2), pages 445-463, May.

    More about this item

    NEP fields

    This paper has been announced in the following NEP Reports:

    Statistics

    Access and download statistics

    Corrections

    All material on this site has been provided by the respective publishers and authors. You can help correct errors and omissions. When requesting a correction, please mention this item's handle: RePEc:arx:papers:2112.06815. See general information about how to correct material in RePEc.

    If you have authored this item and are not yet registered with RePEc, we encourage you to do it here. This allows to link your profile to this item. It also allows you to accept potential citations to this item that we are uncertain about.

    If CitEc recognized a bibliographic reference but did not link an item in RePEc to it, you can help with this form .

    If you know of missing items citing this one, you can help us creating those links by adding the relevant references in the same way as above, for each refering item. If you are a registered author of this item, you may also want to check the "citations" tab in your RePEc Author Service profile, as there may be some citations waiting for confirmation.

    For technical questions regarding this item, or to correct its authors, title, abstract, bibliographic or download information, contact: arXiv administrators (email available below). General contact details of provider: http://arxiv.org/ .

    Please note that corrections may take a couple of weeks to filter through the various RePEc services.

    IDEAS is a RePEc service. RePEc uses bibliographic data supplied by the respective publishers.