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Infnite-Horizon Deterministic Dynamic Programming in Discrete Time: A Monotone Convergence Principle

Author

Listed:
  • Takashi Kamihigashi

    (Research Institute for Economics & Business Administration (RIEB), Kobe University, Japan)

  • Masayuki Yao

    (Graduate School of Economics, Keio University)

Abstract

We consider infinite-horizon deterministic dynamic programming problems in discrete time. We show that the value function is always a fixed point of a modified version of the Bellman operator. We also show that value iteration monotonically converges to the value function if the initial function is dominated by the value function, is mapped upward by the modified Bellman operator, and satisfies a transversality-like condition. These results require no assumption except for the general framework of infinite-horizon deterministic dynamic programming.

Suggested Citation

  • Takashi Kamihigashi & Masayuki Yao, 2015. "Infnite-Horizon Deterministic Dynamic Programming in Discrete Time: A Monotone Convergence Principle," Discussion Paper Series DP2015-32, Research Institute for Economics & Business Administration, Kobe University.
    • Handle: RePEc:kob:dpaper:dp2015-32
    as

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    File URL: https://www.rieb.kobe-u.ac.jp/academic/ra/dp/English/DP2015-32.pdf
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    References listed on IDEAS

    as
    1. Takashi Kamihigashi, 2014. "Elementary results on solutions to the bellman equation of dynamic programming: existence, uniqueness, and convergence," Economic Theory, Springer;Society for the Advancement of Economic Theory (SAET), vol. 56(2), pages 251-273, June.
    2. Le Van, Cuong & Vailakis, Yiannis, 2005. "Recursive utility and optimal growth with bounded or unbounded returns," Journal of Economic Theory, Elsevier, vol. 123(2), pages 187-209, August.
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    4. Takashi Kamihigashi & Kevin Reffett & Masayuki Yao, 2014. "An Application of Kleene's Fixed Point Theorem to Dynamic Programming: A Note," Working Papers 2014-398, Department of Research, Ipag Business School.
    5. Takashi Kamihigashi, 2014. "An order-theoretic approach to dynamic programming: an exposition," Economic Theory Bulletin, Springer;Society for the Advancement of Economic Theory (SAET), vol. 2(1), pages 13-21, April.
    6. 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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    8. Alvarez, Fernando & Stokey, Nancy L., 1998. "Dynamic Programming with Homogeneous Functions," Journal of Economic Theory, Elsevier, vol. 82(1), pages 167-189, September.
    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.
    Full references (including those not matched with items on IDEAS)

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    More about this item

    Keywords

    Dynamic programming; Bellman operator; Fixed point; Value iteration;
    All these keywords.

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