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Necessary and Sufficient Conditions for a Solution of the Bellman Equation to be the Value Function: A General Principle

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Listed:
  • Takashi Kamihigashi

    (RIEB, Kobe University - Kobe University)

  • Cuong Le Van

    (CES - Centre d'économie de la Sorbonne - UP1 - Université Paris 1 Panthéon-Sorbonne - CNRS - Centre National de la Recherche Scientifique, PSE - Paris School of Economics - UP1 - Université Paris 1 Panthéon-Sorbonne - ENS-PSL - École normale supérieure - Paris - PSL - Université Paris Sciences et Lettres - EHESS - École des hautes études en sciences sociales - ENPC - École des Ponts ParisTech - CNRS - Centre National de la Recherche Scientifique - INRAE - Institut National de Recherche pour l’Agriculture, l’Alimentation et l’Environnement, IPAG - Business School, VCREME - Van Xuan Center of Research in Economics, Management and Environment)

Abstract

In this paper, we give Necessary and Sufficient Conditions for a Solution of the Belman Equation to be the Value Function. This result is a general principle. It requires no structure beyond the common framework of discrete-time stationary optimization problems with time-additive returns. In particular, the state space X is an arbitrary set.

Suggested Citation

  • Takashi Kamihigashi & Cuong Le Van, 2015. "Necessary and Sufficient Conditions for a Solution of the Bellman Equation to be the Value Function: A General Principle," Post-Print halshs-01159177, HAL.
    • Handle: RePEc:hal:journl:halshs-01159177
    Note: View the original document on HAL open archive server: https://shs.hal.science/halshs-01159177
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    References listed on IDEAS

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    1. repec:dau:papers:123456789/13605 is not listed on IDEAS
    2. 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.
    3. David Gale, 1967. "On Optimal Development in a Multi-Sector Economy," The Review of Economic Studies, Review of Economic Studies Ltd, vol. 34(1), pages 1-18.
    4. Takashi Kamihigashi, 2008. "On the principle of optimality for nonstationary deterministic dynamic programming," International Journal of Economic Theory, The International Society for Economic Theory, vol. 4(4), pages 519-525, December.
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    7. Takashi Kamihigashi, 2011. "Existence and Uniqueness of a Fixed Point for the Bellman Operator in Deterministic Dynamic Programming," Discussion Paper Series DP2011-23, Research Institute for Economics & Business Administration, Kobe University.
    8. Cuong Le Van & Rose-Anne Dana, 2003. "Dynamic Programming in Economics," Post-Print halshs-00119098, HAL.
    9. V. Filipe Martins-da-Rocha & Yiannis Vailakis, 2010. "Existence and Uniqueness of a Fixed Point for Local Contractions," Econometrica, Econometric Society, vol. 78(3), pages 1127-1141, May.
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    More about this item

    Keywords

    dynamic programming; Bellman equation; value function; fixed point;
    All these keywords.

    JEL classification:

    • C61 - Mathematical and Quantitative Methods - - Mathematical Methods; Programming Models; Mathematical and Simulation Modeling - - - Optimization Techniques; Programming Models; Dynamic Analysis

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