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Unbounded Markov Dynamic Programming with Weighted Supremum Norm Perov Contractions

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  • Alexis Akira Toda

Abstract

This paper shows the usefulness of the Perov contraction theorem, which is a generalization of the classical Banach contraction theorem, for solving Markov dynamic programming problems. When the reward function is unbounded, combining an appropriate weighted supremum norm with the Perov contraction theorem yields a unique fixed point of the Bellman operator under weaker conditions than existing approaches. An application to the optimal savings problem shows that the average growth rate condition derived from the spectral radius of a certain nonnegative matrix is sufficient and almost necessary for obtaining a solution.

Suggested Citation

  • Alexis Akira Toda, 2023. "Unbounded Markov Dynamic Programming with Weighted Supremum Norm Perov Contractions," Papers 2310.04593, arXiv.org.
  • Handle: RePEc:arx:papers:2310.04593
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    References listed on IDEAS

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    1. Toda, Alexis Akira, 2019. "Wealth distribution with random discount factors," Journal of Monetary Economics, Elsevier, vol. 104(C), pages 101-113.
    2. Jaroslav Borovička & John Stachurski, 2020. "Necessary and Sufficient Conditions for Existence and Uniqueness of Recursive Utilities," Journal of Finance, American Finance Association, vol. 75(3), pages 1457-1493, June.
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    5. 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.
    6. Steven A. Lippman, 1975. "On Dynamic Programming with Unbounded Rewards," Management Science, INFORMS, vol. 21(11), pages 1225-1233, July.
    7. Ma, Qingyin & Toda, Alexis Akira, 2022. "Asymptotic linearity of consumption functions and computational efficiency," Journal of Mathematical Economics, Elsevier, vol. 98(C).
    8. John Stachurski, 2009. "Economic Dynamics: Theory and Computation," MIT Press Books, The MIT Press, edition 1, volume 1, number 0262012774, December.
    9. Alexis Akira Toda, 2021. "Perov's Contraction Principle and Dynamic Programming with Stochastic Discounting," Papers 2103.14173, arXiv.org, revised Sep 2021.
    10. Gaetano Bloise & Cuong Le Van & Yiannis Vailakis, 2024. "Do not Blame Bellman: It Is Koopmans' Fault," Econometrica, Econometric Society, vol. 92(1), pages 111-140, January.
    11. Li, Huiyu & Stachurski, John, 2014. "Solving the income fluctuation problem with unbounded rewards," Journal of Economic Dynamics and Control, Elsevier, vol. 45(C), pages 353-365.
    12. Stachurski, John & Zhang, Junnan, 2021. "Dynamic programming with state-dependent discounting," Journal of Economic Theory, Elsevier, vol. 192(C).
    13. Ma, Qingyin & Toda, Alexis Akira, 2021. "A theory of the saving rate of the rich," Journal of Economic Theory, Elsevier, vol. 192(C).
    14. Schechtman, Jack, 1976. "An income fluctuation problem," Journal of Economic Theory, Elsevier, vol. 12(2), pages 218-241, April.
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    More about this item

    JEL classification:

    • C61 - Mathematical and Quantitative Methods - - Mathematical Methods; Programming Models; Mathematical and Simulation Modeling - - - Optimization Techniques; Programming Models; Dynamic Analysis
    • C65 - Mathematical and Quantitative Methods - - Mathematical Methods; Programming Models; Mathematical and Simulation Modeling - - - Miscellaneous Mathematical Tools
    • D14 - Microeconomics - - Household Behavior - - - Household Saving; Personal Finance
    • E21 - Macroeconomics and Monetary Economics - - Consumption, Saving, Production, Employment, and Investment - - - Consumption; Saving; Wealth

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