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On the Principle of Optimality for Nonstationary Deterministic Dynamic Programming

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  • Takashi Kamihigashi

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

Abstract

This note studies a general nonstationary infinite-horizon optimization problem in discrete time. We allow the state space in each period to be an arbitrary set, and the return function in each period to be unbounded. We do not require discounting, and do not require the constraint correspondence in each period to be nonempty-valued. The objective function is defined as the limit superior or inferior of the finite sums of return functions. We show that the sequence of time-indexed value functions satisfies the Bellman equation if and only if its right-hand side is well defined, i.e., it does not involve -∞+∞.

Suggested Citation

  • Takashi Kamihigashi, 2007. "On the Principle of Optimality for Nonstationary Deterministic Dynamic Programming," Discussion Paper Series 200, Research Institute for Economics & Business Administration, Kobe University.
    • Handle: RePEc:kob:dpaper:200
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    File URL: https://www.rieb.kobe-u.ac.jp/academic/ra/dp/English/dp200.pdf
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    References listed on IDEAS

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    Cited by:

    1. Agnieszka Wiszniewska-Matyszkiel & Rajani Singh, 2020. "When Inaccuracies in Value Functions Do Not Propagate on Optima and Equilibria," Mathematics, MDPI, vol. 8(7), pages 1-25, July.
    2. Takashi Kamihigashi & Kevin Reffett & Masayuki Yao, 2014. "An Application of Kleene's Fixed Point Theorem to Dynamic Programming: A Note," Discussion Paper Series DP2014-24, Research Institute for Economics & Business Administration, Kobe University, revised Jul 2014.
    3. Ronaldo Carpio & Takashi Kamihigashi, 2019. "Fast Value Iteration: An Application of Legendre-Fenchel Duality to a Class of Deterministic Dynamic Programming Problems in Discrete Time," Discussion Paper Series DP2019-24, Research Institute for Economics & Business Administration, Kobe University.
    4. 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.
    5. 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.
    6. 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.
    7. Ronaldo Carpio & Takashi Kamihigashi, 2015. "Fast Bellman Iteration: An Application of Legendre-Fenchel Duality to Infinite-Horizon Dynamic Programming in Discrete Time," Discussion Paper Series DP2015-11, Research Institute for Economics & Business Administration, Kobe University.
    8. 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.
    9. Masayuki Yao, 2016. "Recursive Utility and the Solution to the Bellman Equation," Discussion Paper Series DP2016-08, Research Institute for Economics & Business Administration, Kobe University.
    10. 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.
    11. Robert A. Becker, 2012. "Optimal growth with heterogeneous agents and the twisted turnpike: An example," International Journal of Economic Theory, The International Society for Economic Theory, vol. 8(1), pages 27-47, March.
    12. Takashi Kamihigashi, 2013. "Ergodic chaos and aggregate stability: A deterministic discrete-choice model of wealth distribution dynamics," International Journal of Economic Theory, The International Society for Economic Theory, vol. 9(1), pages 45-56, March.
    13. Ronaldo Carpio & Takashi Kamihigashi, 2016. "Fast Bellman Iteration: An Application of Legendre-Fenchel Duality to Deterministic Dynamic Programming in Discrete Time," Discussion Paper Series DP2016-04, Research Institute for Economics & Business Administration, Kobe University.
    14. Takashi Kamihigashi & Masayuki Yao, 2016. "Infinite-Horizon Deterministic Dynamic Programming in Discrete Time: A Monotone Convergence Principle and a Penalty Method," Discussion Paper Series DP2016-05, Research Institute for Economics & Business Administration, Kobe University, revised May 2016.
    15. Robert A. Becker, 2012. "Optimal growth with heterogeneous agents and the twisted turnpike: An example," International Journal of Economic Theory, The International Society for Economic Theory, vol. 8(1), pages 27-47, March.
    16. Takashi Kamihigashi & Masayuki Yao, 2015. "Deterministic Dynamic Programming in Discrete Time: A Monotone Convergence Principle," Discussion Paper Series DP2015-15, Research Institute for Economics & Business Administration, Kobe University.
    17. Luis A. Alcala, 2016. "On the time consistency of collective preferences," Papers 1607.02688, arXiv.org, revised Jul 2018.

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    More about this item

    Keywords

    Bellman equation; Dynamic programming; Principle of optimality; Value function;
    All these keywords.

    JEL classification:

    • C61 - Mathematical and Quantitative Methods - - Mathematical Methods; Programming Models; Mathematical and Simulation Modeling - - - Optimization Techniques; Programming Models; Dynamic Analysis
    • O41 - Economic Development, Innovation, Technological Change, and Growth - - Economic Growth and Aggregate Productivity - - - One, Two, and Multisector Growth Models

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