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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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    1. Jonathan Thomas & Tim Worrall, 1994. "Foreign Direct Investment and the Risk of Expropriation," The Review of Economic Studies, Review of Economic Studies Ltd, vol. 61(1), pages 81-108.
    2. Manuel S. Santos & Michael Woodford, 1997. "Rational Asset Pricing Bubbles," Econometrica, Econometric Society, vol. 65(1), pages 19-58, January.
    3. Le Van, Cuong & Morhaim, Lisa, 2002. "Optimal Growth Models with Bounded or Unbounded Returns: A Unifying Approach," Journal of Economic Theory, Elsevier, vol. 105(1), pages 158-187, July.
    4. Christiano, Lawrence J. & Fisher, Jonas D. M., 2000. "Algorithms for solving dynamic models with occasionally binding constraints," Journal of Economic Dynamics and Control, Elsevier, vol. 24(8), pages 1179-1232, July.
    5. Amir, Rabah & Mirman, Leonard J & Perkins, William R, 1991. "One-Sector Nonclassical Optimal Growth: Optimality Conditions and Comparative Dynamics," International Economic Review, Department of Economics, University of Pennsylvania and Osaka University Institute of Social and Economic Research Association, vol. 32(3), pages 625-644, August.
    6. Christopher Phelan & Ennio Stacchetti, 2001. "Sequential Equilibria in a Ramsey Tax Model," Econometrica, Econometric Society, vol. 69(6), pages 1491-1518, November.
    7. Timothy J. Kehoe & David K. Levine, 1993. "Debt-Constrained Asset Markets," The Review of Economic Studies, Review of Economic Studies Ltd, vol. 60(4), pages 865-888.
    8. K. Askri & C. Le Van, 1998. "Differentiability of the Value Function of Nonclassical Optimal Growth Models," Journal of Optimization Theory and Applications, Springer, vol. 97(3), pages 591-604, June.
    9. Jonathan Thomas & Tim Worrall, 1988. "Self-Enforcing Wage Contracts," The Review of Economic Studies, Review of Economic Studies Ltd, vol. 55(4), pages 541-554.
    10. Paul Milgrom & Ilya Segal, 2002. "Envelope Theorems for Arbitrary Choice Sets," Econometrica, Econometric Society, vol. 70(2), pages 583-601, March.
    11. Mirman, Leonard J. & Zilcha, Itzhak, 1975. "On optimal growth under uncertainty," Journal of Economic Theory, Elsevier, vol. 11(3), pages 329-339, December.
    12. Kydland, Finn E. & Prescott, Edward C., 1980. "Dynamic optimal taxation, rational expectations and optimal control," Journal of Economic Dynamics and Control, Elsevier, vol. 2(1), pages 79-91, May.
    13. Fernando Alvarez & Urban J. Jermann, 2000. "Efficiency, Equilibrium, and Asset Pricing with Risk of Default," Econometrica, Econometric Society, vol. 68(4), pages 775-798, July.
    14. Sargent, Thomas J., 1980. ""Tobin's q" and the rate of investment in general equilibrium," Carnegie-Rochester Conference Series on Public Policy, Elsevier, vol. 12(1), pages 107-154, January.
    15. Koeppl Thorsten V., 2006. "Differentiability of the Efficient Frontier when Commitment to Risk Sharing is Limited," The B.E. Journal of Macroeconomics, De Gruyter, vol. 6(1), pages 1-6, April.
    16. Hopenhayn, Hugo A & Nicolini, Juan Pablo, 1997. "Optimal Unemployment Insurance," Journal of Political Economy, University of Chicago Press, vol. 105(2), pages 412-438, April.
    17. Lucas, Robert E, Jr, 1980. "Equilibrium in a Pure Currency Economy," Economic Inquiry, Western Economic Association International, vol. 18(2), pages 203-220, April.
    18. Montrucchio, Luigi & Privileggi, Fabio, 2001. "On Fragility of Bubbles in Equilibrium Asset Pricing Models of Lucas-Type," Journal of Economic Theory, Elsevier, vol. 101(1), pages 158-188, November.
    19. 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.
    20. Benveniste, L. M. & Scheinkman, J. A., 1982. "Duality theory for dynamic optimization models of economics: The continuous time case," Journal of Economic Theory, Elsevier, vol. 27(1), pages 1-19, June.
    21. Lucas, Robert E, Jr, 1978. "Asset Prices in an Exchange Economy," Econometrica, Econometric Society, vol. 46(6), pages 1429-1445, November.
    22. Cass, David, 2006. "Competitive equilibrium with incomplete financial markets," Journal of Mathematical Economics, Elsevier, vol. 42(4-5), pages 384-405, August.
    23. Martin L. Weitzman, 1973. "Duality Theory for Infinite Horizon Convex Models," Management Science, INFORMS, vol. 19(7), pages 783-789, March.
    24. Narayana R. Kocherlakota, 1996. "Implications of Efficient Risk Sharing without Commitment," The Review of Economic Studies, Review of Economic Studies Ltd, vol. 63(4), pages 595-609.
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    Cited by:

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    2. Andrew Clausen & Carlo Strub, 2012. "Envelope theorems for non-smooth and non-concave optimization," ECON - Working Papers 062, Department of Economics - University of Zurich.
    3. Pierri, Damian Rene & Reffett, Kevin, 2021. "Memory, multiple equilibria and emerging market crises," UC3M Working papers. Economics 32871, Universidad Carlos III de Madrid. Departamento de Economía.
    4. Olivier Morand & Kevin Reffett & Suchismita Tarafdar, 2018. "Generalized Envelope Theorems: Applications to Dynamic Programming," Journal of Optimization Theory and Applications, Springer, vol. 176(3), pages 650-687, March.
    5. 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.
    6. 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.
    7. Strulovici, Bruno & Szydlowski, Martin, 2012. "On the Smoothness of Value Functions," MPRA Paper 36326, University Library of Munich, Germany, revised 31 Jan 2012.
    8. 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.
    9. Pontus Rendahl, 2015. "Inequality Constraints and Euler Equation‐based Solution Methods," Economic Journal, Royal Economic Society, vol. 125(585), pages 1110-1135, June.
    10. 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.
    11. Rohit Lamba & Ilia Krasikov, 2017. "A Theory of Dynamic Contracting with Financial Constraints," 2017 Meeting Papers 1544, Society for Economic Dynamics.
    12. Sickles, Robin C. & Williams, Jenny, 2008. "Turning from crime: A dynamic perspective," Journal of Econometrics, Elsevier, vol. 145(1-2), pages 158-173, July.
    13. Tessa Bold, 2009. "Implications of Endogenous Group Formation for Efficient Risk‐Sharing," Economic Journal, Royal Economic Society, vol. 119(536), pages 562-591, March.
    14. Rincón-Zapatero, Juan Pablo & Zhao, Yanyun, 2018. "Envelope theorem in dynamic economic models with recursive utility," Economics Letters, Elsevier, vol. 163(C), pages 10-12.
    15. 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.
    16. 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.
    17. 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.
    18. Cao, Dan, 2020. "Recursive equilibrium in Krusell and Smith (1998)," Journal of Economic Theory, Elsevier, vol. 186(C).
    19. 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.
    20. 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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    More about this item

    Keywords

    Constrained optimization; value and policy functions; differentiability; envelope theorem; shadow price;
    All these keywords.

    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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