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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 focuss 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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    References listed on IDEAS

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    Citations

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    Cited by:

    1. Sickles, Robin C. & Williams, Jenny, 2008. "Turning from crime: A dynamic perspective," Journal of Econometrics, Elsevier, vol. 145(1-2), pages 158-173, July.
    2. 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.
    3. Robert Kirkby, 2017. "Convergence of Discretized Value Function Iteration," Computational Economics, Springer;Society for Computational Economics, vol. 49(1), pages 117-153, January.
    4. Bruno Strulovici & Martin Szydlowski, 2012. "On the Smoothness of Value Functions," Discussion Papers 1542, Northwestern University, Center for Mathematical Studies in Economics and Management Science.
    5. Andrew Clausen & Carlo Strub, 2012. "Envelope theorems for non-smooth and non-concave optimization," ECON - Working Papers 062, Department of Economics - University of Zurich.
    6. Pontus Rendahl, 2015. "Inequality Constraints and Euler Equation‐based Solution Methods," Economic Journal, Royal Economic Society, vol. 125(585), pages 1110-1135, June.
    7. Morand, Olivier & Reffett, Kevin & Tarafdar, Suchismita, 2015. "A nonsmooth approach to envelope theorems," Journal of Mathematical Economics, Elsevier, vol. 61(C), pages 157-165.
    8. 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.
    9. repec:eee:ecolet:v:163:y:2018:i:c:p:10-12 is not listed on IDEAS
    10. repec:spr:joptap:v:176:y:2018:i:3:d:10.1007_s10957-018-1241-5 is not listed on IDEAS
    11. 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, pages 83-110, February.
    12. 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.
    13. Rohit Lamba & Ilia Krasikov, 2017. "A Theory of Dynamic Contracting with Financial Constraints," 2017 Meeting Papers 1544, Society for Economic Dynamics.

    More about this item

    Keywords

    Constrained optimization; value and policy functions; differentiability; envelope theorem; shadow price;

    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 - Economic Systems - - Comparative Economic Systems - - - Comparative Analysis of Economic Systems

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