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On the (non-)differentiability of the optimal value function when the optimal solution is unique

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  • Oyama, Daisuke
  • Takenawa, Tomoyuki

Abstract

We present examples of a parameterized optimization problem, with a continuous objective function differentiable with respect to the parameter, that admits a unique optimal solution, but whose optimal value function is not differentiable. We also show independence of Danskin’s and Milgrom and Segal’s envelope theorems.

Suggested Citation

  • Oyama, Daisuke & Takenawa, Tomoyuki, 2018. "On the (non-)differentiability of the optimal value function when the optimal solution is unique," Journal of Mathematical Economics, Elsevier, vol. 76(C), pages 21-32.
    • Handle: RePEc:eee:mateco:v:76:y:2018:i:c:p:21-32
    DOI: 10.1016/j.jmateco.2018.02.004
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    References listed on IDEAS

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    1. Paul Milgrom & Ilya Segal, 2002. "Envelope Theorems for Arbitrary Choice Sets," Econometrica, Econometric Society, vol. 70(2), pages 583-601, March.
    2. HALKIN, Hubert, 1974. "Implicit functions and optimization problems without continuous differentiability of the data," LIDAM Reprints CORE 184, Université catholique de Louvain, Center for Operations Research and Econometrics (CORE).
    3. Benveniste, L M & Scheinkman, J A, 1979. "On the Differentiability of the Value Function in Dynamic Models of Economics," Econometrica, Econometric Society, vol. 47(3), pages 727-732, May.
    4. Mas-Colell, Andreu & Whinston, Michael D. & Green, Jerry R., 1995. "Microeconomic Theory," OUP Catalogue, Oxford University Press, number 9780195102680, Decembrie.
    5. William Hogan, 1973. "Directional Derivatives for Extremal-Value Functions with Applications to the Completely Convex Case," Operations Research, INFORMS, vol. 21(1), pages 188-209, February.
    6. Charalambos D. Aliprantis & Kim C. Border, 2006. "Infinite Dimensional Analysis," Springer Books, Springer, edition 0, number 978-3-540-29587-7, November.
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    Cited by:

    1. Marimon, Ramon & Werner, Jan, 2021. "The envelope theorem, Euler and Bellman equations, without differentiability," Journal of Economic Theory, Elsevier, vol. 196(C).
    2. Guusje Delsing & Michel Mandjes & Peter Spreij & Erik Winands, 2021. "On Capital Allocation for a Risk Measure Derived from Ruin Theory," Papers 2103.16264, arXiv.org.
    3. Sunil Dutta & Stefan Reichelstein, 2021. "Capacity Rights and Full-Cost Transfer Pricing," Management Science, INFORMS, vol. 67(2), pages 1303-1325, February.
    4. Ludvig Sinander, 2019. "The converse envelope theorem," Papers 1909.11219, arXiv.org, revised Jun 2022.
    5. Sunil Dutta & Stefan J. Reichelstein, 2019. "Capacity Rights and Full Cost Transfer Pricing," CESifo Working Paper Series 7968, CESifo.

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