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On Discounted Dynamic Programming with Unbounded Returns

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  • Matkowski, Janusz
  • Nowak, Andrzej S.

Abstract

In this paper, we apply the idea of $k$-local contraction of \cite{zec, zet} to study discounted stochastic dynamic programming models with unbounded returns. Our main results concern the existence of a unique solution to the Bellman equation and are applied to the theory of stochastic optimal growth. Also a discussion of some subtle issues concerning k-local and global contractions is included.

Suggested Citation

  • Matkowski, Janusz & Nowak, Andrzej S., 2008. "On Discounted Dynamic Programming with Unbounded Returns," MPRA Paper 12215, University Library of Munich, Germany.
  • Handle: RePEc:pra:mprapa:12215
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    References listed on IDEAS

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    1. 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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    8. 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.
    9. 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.
    10. Boud, John III, 1990. "Recursive utility and the Ramsey problem," Journal of Economic Theory, Elsevier, vol. 50(2), pages 326-345, April.
    11. 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.
    12. Nowak, Andrzej S. & Szajowski, Krzysztof, 1998. "Nonzero-sum Stochastic Games," MPRA Paper 19995, University Library of Munich, Germany, revised 1999.
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    Cited by:

    1. V. Filipe Martins-da-Rocha & Yiannis Vailakis, 2013. "Fixed point for local contractions: Applications to recursive utility," International Journal of Economic Theory, The International Society for Economic Theory, vol. 9(1), pages 23-33, March.
    2. Philippe Bich & Jean-Pierre Drugeon & Lisa Morhaim, 2017. "On Temporal Aggregators and Dynamic Programming," Université Paris1 Panthéon-Sorbonne (Post-Print and Working Papers) halshs-01437496, HAL.
    3. Anna Jaśkiewicz & Andrzej Nowak, 2011. "Stochastic Games with Unbounded Payoffs: Applications to Robust Control in Economics," Dynamic Games and Applications, Springer, vol. 1(2), pages 253-279, June.
    4. Lukasz Balbus & Kevin Reffett & Lukasz Wozny, 2016. "On uniqueness of time-consistent Markov policies for quasi-hyperbolic consumers under uncertainty," Working Papers 2016-020, Warsaw School of Economics, Collegium of Economic Analysis.
    5. Philippe Bich & Jean-Pierre Drugeon & Lisa Morhaim, 2015. "On Aggregators and Dynamic Programming," Documents de travail du Centre d'Economie de la Sorbonne 15053, Université Panthéon-Sorbonne (Paris 1), Centre d'Economie de la Sorbonne.
    6. repec:spr:etbull:v:5:y:2017:i:2:d:10.1007_s40505-016-0112-1 is not listed on IDEAS
    7. 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.
    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. 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.
    10. Matthias Messner & Nicola Pavoni & Christopher Sleet, "undated". "Contractive Dual Methods for Incentive Problems," GSIA Working Papers 2012-E26, Carnegie Mellon University, Tepper School of Business.
    11. Philippe Bich & Jean-Pierre Drugeon & Lisa Morhaim, 2015. "On Aggregators and Dynamic Programming," Université Paris1 Panthéon-Sorbonne (Post-Print and Working Papers) halshs-01169552, HAL.
    12. 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.
    13. repec:eee:jetheo:v:173:y:2018:i:c:p:181-200 is not listed on IDEAS
    14. Łukasz Balbus & Kevin Reffett & Łukasz Woźny, 2015. "Time consistent Markov policies in dynamic economies with quasi-hyperbolic consumers," International Journal of Game Theory, Springer;Game Theory Society, vol. 44(1), pages 83-112, February.
    15. Nicole Bauerle & Anna Ja'skiewicz, 2015. "Stochastic Optimal Growth Model with Risk Sensitive Preferences," Papers 1509.05638, arXiv.org.
    16. Tapan Mitra & Gerhard Sorger, 2014. "Extinction in common property resource models: an analytically tractable example," Economic Theory, Springer;Society for the Advancement of Economic Theory (SAET), vol. 57(1), pages 41-57, September.
    17. repec:hal:journl:halshs-01169552 is not listed on IDEAS
    18. 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.
    19. repec:eee:jetheo:v:172:y:2017:i:c:p:1-25 is not listed on IDEAS
    20. 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.
    21. João Brogueira & Fabian Schütze, 2017. "Existence and uniqueness of equilibrium in Lucas’ asset pricing model when utility is unbounded," Economic Theory Bulletin, Springer;Society for the Advancement of Economic Theory (SAET), vol. 5(2), pages 179-190, October.

    More about this item

    Keywords

    Stochastic dynamic programming; Bellman functional equation; contraction mapping; stochastic optimal growth;

    JEL classification:

    • D90 - Microeconomics - - Micro-Based Behavioral Economics - - - General
    • D91 - Microeconomics - - Micro-Based Behavioral Economics - - - Role and Effects of Psychological, Emotional, Social, and Cognitive Factors on Decision Making
    • C61 - Mathematical and Quantitative Methods - - Mathematical Methods; Programming Models; Mathematical and Simulation Modeling - - - Optimization Techniques; Programming Models; Dynamic Analysis

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