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An Order-Theoretic Approach to Dynamic Programming: An Exposition

Author

Listed:
  • Takashi Kamihigashi

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

Abstract
In this note, we discuss an order-theoretic approach to dynamic programming. In particular, we explain how order-theoretic fixed point theorems can be used to establish the existence of a fixed point of the Bellman operator, as well as why they are not sufficient to characterize the value function. By doing this, we present the logic behind the simple yet useful result recently obtained by Kamihigashi (2013) based on this order-theoretic approach.

Suggested Citation

  • Takashi Kamihigashi, 2013. "An Order-Theoretic Approach to Dynamic Programming: An Exposition," Discussion Paper Series DP2013-29, Research Institute for Economics & Business Administration, Kobe University, revised Nov 2013.
    • Handle: RePEc:kob:dpaper:dp2013-29
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    File URL: https://www.rieb.kobe-u.ac.jp/academic/ra/dp/English/DP2013-29.pdf
    File Function: Revised version, 2013
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    References listed on IDEAS

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    1. Larry Karp & Jiangfeng Zhang, 2016. "Taxes Versus Quantities for a Stock Pollutant with Endogenous Abatement Costs and Asymmetric Information," Studies in Economic Theory, in: Graciela Chichilnisky & Armon Rezai (ed.), The Economics of the Global Environment, pages 493-533, Springer.
    2. Yann Algan & Edouard Challe & Xavier Ragot, 2011. "Incomplete markets and the output–inflation tradeoff," Economic Theory, Springer;Society for the Advancement of Economic Theory (SAET), vol. 46(1), pages 55-84, January.
    3. 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.
    4. 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.
    5. 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.
    6. Mihaela Schaar & Jie Xu & William Zame, 2013. "Efficient online exchange via fiat money," Economic Theory, Springer;Society for the Advancement of Economic Theory (SAET), vol. 54(2), pages 211-248, October.
    7. Yann Algan & Edouard Challe & Xavier Ragot, 2011. "Incomplete markets and the output-inflation tradeoff," Post-Print halshs-00654652, HAL.
    8. Santanu Roy & Itzhak Zilcha, 2012. "Stochastic growth with short-run prediction of shocks," Economic Theory, Springer;Society for the Advancement of Economic Theory (SAET), vol. 51(3), pages 539-580, November.
    9. 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.
    10. Prajit K. Dutta & Roy Radner, 2016. "Capital Growth in a Global Warming Model: Will China and India Sign a Climate Treaty?," Studies in Economic Theory, in: Graciela Chichilnisky & Armon Rezai (ed.), The Economics of the Global Environment, pages 277-310, Springer.
    11. Catarina Reis, 2013. "Taxation without commitment," Economic Theory, Springer;Society for the Advancement of Economic Theory (SAET), vol. 52(2), pages 565-588, March.
    12. Janusz Matkowski & Andrzej Nowak, 2011. "On discounted dynamic programming with unbounded returns," Economic Theory, Springer;Society for the Advancement of Economic Theory (SAET), vol. 46(3), pages 455-474, April.
    13. Francis Bloch & Nicolas Houy, 2012. "Optimal assignment of durable objects to successive agents," Economic Theory, Springer;Society for the Advancement of Economic Theory (SAET), vol. 51(1), pages 13-33, September.
    14. Gastón Llanes & Stefano Trento, 2012. "Patent policy, patent pools, and the accumulation of claims in sequential innovation," Economic Theory, Springer;Society for the Advancement of Economic Theory (SAET), vol. 50(3), pages 703-725, August.
    15. Juan Rincón-Zapatero & Carlos Rodríguez-Palmero, 2007. "Recursive utility with unbounded aggregators," Economic Theory, Springer;Society for the Advancement of Economic Theory (SAET), vol. 33(2), pages 381-391, November.
    16. Helios Herrera & César Martinelli, 2013. "Oligarchy, democracy, and state capacity," Economic Theory, Springer;Society for the Advancement of Economic Theory (SAET), vol. 52(1), pages 165-186, January.
    17. Juan Pablo RincÛn-Zapatero & Carlos RodrÌguez-Palmero, 2003. "Existence and Uniqueness of Solutions to the Bellman Equation in the Unbounded Case," Econometrica, Econometric Society, vol. 71(5), pages 1519-1555, September.
    18. repec:dau:papers:123456789/4077 is not listed on IDEAS
    19. repec:hal:spmain:info:hdl:2441/58mqn6l8gc96n8lujj5560pkuf is not listed on IDEAS
    20. Jorge DurÂn, 2000. "On dynamic programming with unbounded returns," Economic Theory, Springer;Society for the Advancement of Economic Theory (SAET), vol. 15(2), pages 339-352.
    21. Yann Algan & Edouard Challe & Xavier Ragot, 2011. "Incomplete markets and the output-inflation tradeoff," PSE-Ecole d'économie de Paris (Postprint) halshs-00654652, HAL.
    22. 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.
    23. Aditya Goenka & Lin Liu, 2012. "Infectious diseases and endogenous fluctuations," Economic Theory, Springer;Society for the Advancement of Economic Theory (SAET), vol. 50(1), pages 125-149, May.
    24. Charalambos D. Aliprantis & Kim C. Border, 2006. "Infinite Dimensional Analysis," Springer Books, Springer, edition 0, number 978-3-540-29587-7, February.
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    Citations

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    Cited by:

    1. 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.
    2. 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.
    3. 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.
    4. 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.
    5. 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.
    6. 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.
    7. 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.

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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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