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A nonsmooth approach to envelope theorems

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  • Morand, Olivier
  • Reffett, Kevin
  • Tarafdar, Suchismita

Abstract

We develop a nonsmooth approach to envelope theorems applicable to a broad class of parameterized constrained nonlinear optimization problems that arise typically in economic applications with nonconvexities and/or nonsmooth objectives. Our methods emphasize the role of the Strict Mangasarian–Fromovitz Constraint Qualification (SMFCQ), and include envelope theorems for both the convex and nonconvex case, allow for noninterior solutions as well as equality and inequality constraints. We give new sufficient conditions for the value function to be directionally differentiable, as well as continuously differentiable. We apply our results to stochastic growth models with Markov shocks and constrained lattice programming problems.

Suggested Citation

  • Morand, Olivier & Reffett, Kevin & Tarafdar, Suchismita, 2015. "A nonsmooth approach to envelope theorems," Journal of Mathematical Economics, Elsevier, vol. 61(C), pages 157-165.
    • Handle: RePEc:eee:mateco:v:61:y:2015:i:c:p:157-165
    DOI: 10.1016/j.jmateco.2015.09.001
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    References listed on IDEAS

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    1. John K.-H Quah, 2007. "The Comparative Statics of Constrained Optimization Problems," Econometrica, Econometric Society, vol. 75(2), pages 401-431, March.
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    4. Rincón-Zapatero, Juan Pablo & Santos, Manuel S., 2009. "Differentiability of the value function without interiority assumptions," Journal of Economic Theory, Elsevier, vol. 144(5), pages 1948-1964, September.
    5. 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.
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    7. Hopenhayn, Hugo A & Prescott, Edward C, 1992. "Stochastic Monotonicity and Stationary Distributions for Dynamic Economies," Econometrica, Econometric Society, vol. 60(6), pages 1387-1406, November.
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    Cited by:

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    3. Kitagawa, Toru & Montiel Olea, José Luis & Payne, Jonathan & Velez, Amilcar, 2020. "Posterior distribution of nondifferentiable functions," Journal of Econometrics, Elsevier, vol. 217(1), pages 161-175.
    4. Ludvig Sinander, 2019. "The converse envelope theorem," Papers 1909.11219, arXiv.org, revised Jun 2022.
    5. Lehrer, Ehud & Light, Bar, 2018. "The effect of interest rates on consumption in an income fluctuation problem," Journal of Economic Dynamics and Control, Elsevier, vol. 94(C), pages 63-71.
    6. Paweł Dziewulski, 2011. "On Time-to-Build Economies with Multiple-Stage Investments," Gospodarka Narodowa. The Polish Journal of Economics, Warsaw School of Economics, issue 9, pages 23-49.

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