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Differentiability of the Value Function without Interiority Assumptions

Author

Listed:
  • Manuel Santos

    (Department of Economics, University of Miami)

  • Juan Pablo Rincon-Zapatero

    (Department of Economics, Universidad Carlos III de Madrid)

Abstract

This paper studies first–order differentiability properties of the value function in concave dynamic programs. Motivated by economic considerations, we dispense with commonly imposed interiority assumptions. We suppose that the correspondence of feasible choices varies with the vector of state variables, and we allow the optimal solution to belong to the boundary of this correspondence. Under minimal assumptions we prove that the value function is continuously differentiable. We then discuss this result in the context of some economic models, and focus on some examples in which our assumptions are not met and the value function is not differentiable.

Suggested Citation

  • Manuel Santos & Juan Pablo Rincon-Zapatero, 2007. "Differentiability of the Value Function without Interiority Assumptions," Working Papers 0704, University of Miami, Department of Economics.
    • Handle: RePEc:mia:wpaper:0704
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    Cited by:

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    7. Zhigang Feng & Jianjun Miao & Adrian Peralta‐Alva & Manuel S. Santos, 2014. "Numerical Simulation Of Nonoptimal Dynamic Equilibrium Models," International Economic Review, Department of Economics, University of Pennsylvania and Osaka University Institute of Social and Economic Research Association, vol. 55(1), pages 83-110, February.
    8. Strulovici, Bruno & Szydlowski, Martin, 2015. "On the smoothness of value functions and the existence of optimal strategies in diffusion models," Journal of Economic Theory, Elsevier, vol. 159(PB), pages 1016-1055.
    9. Juan Pablo Rincón-Zapatero, 2020. "Differentiability of the value function and Euler equation in non-concave discrete-time stochastic dynamic programming," Economic Theory Bulletin, Springer;Society for the Advancement of Economic Theory (SAET), vol. 8(1), pages 79-88, April.
    10. Strulovici, Bruno & Szydlowski, Martin, 2012. "On the Smoothness of Value Functions," MPRA Paper 36326, University Library of Munich, Germany, revised 31 Jan 2012.
    11. 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.
    12. Robert Kirkby Author-Email: robertkirkby@gmail.com|, 2017. "Convergence of Discretized Value Function Iteration," Computational Economics, Springer;Society for Computational Economics, vol. 49(1), pages 117-153, January.
    13. Rocheteau, Guillaume & Weill, Pierre-Olivier & Wong, Tsz-Nga, 2021. "An heterogeneous-agent New-Monetarist model with an application to unemployment," Journal of Monetary Economics, Elsevier, vol. 117(C), pages 64-90.
    14. Pontus Rendahl, 2015. "Inequality Constraints and Euler Equation‐based Solution Methods," Economic Journal, Royal Economic Society, vol. 125(585), pages 1110-1135, June.
    15. Jaime McGovern & Olivier Morand & Kevin Reffett, 2013. "Computing minimal state space recursive equilibrium in OLG models with stochastic production," Economic Theory, Springer;Society for the Advancement of Economic Theory (SAET), vol. 54(3), pages 623-674, November.
    16. Cao, Dan, 2020. "Recursive equilibrium in Krusell and Smith (1998)," Journal of Economic Theory, Elsevier, vol. 186(C).
    17. Andrew Clausen & Carlo Strub, 2012. "Envelope theorems for non-smooth and non-concave optimization," ECON - Working Papers 062, Department of Economics - University of Zurich.
    18. Rohit Lamba & Ilia Krasikov, 2017. "A Theory of Dynamic Contracting with Financial Constraints," 2017 Meeting Papers 1544, Society for Economic Dynamics.
    19. Dziewulski, Paweł, . "On time-to-build economies with multiple stage investments," Gospodarka Narodowa-The Polish Journal of Economics, Szkoła Główna Handlowa w Warszawie / SGH Warsaw School of Economics, vol. 2011(9).
    20. F. García Castaño & M. Melguizo Padial, 2015. "A natural extension of the classical envelope theorem in vector differential programming," Journal of Global Optimization, Springer, vol. 63(4), pages 757-775, December.
    21. Morand, Olivier & Reffett, Kevin & Tarafdar, Suchismita, 2015. "A nonsmooth approach to envelope theorems," Journal of Mathematical Economics, Elsevier, vol. 61(C), pages 157-165.

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    JEL classification:

    • C61 - Mathematical and Quantitative Methods - - Mathematical Methods; Programming Models; Mathematical and Simulation Modeling - - - Optimization Techniques; Programming Models; Dynamic Analysis
    • C63 - Mathematical and Quantitative Methods - - Mathematical Methods; Programming Models; Mathematical and Simulation Modeling - - - Computational Techniques
    • P51 - Political Economy and Comparative Economic Systems - - Comparative Economic Systems - - - Comparative Analysis of Economic Systems

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