IDEAS home Printed from https://ideas.repec.org/p/arx/papers/2310.04593.html
   My bibliography  Save this paper

Unbounded Markov Dynamic Programming with Weighted Supremum Norm Perov Contractions

Author

Listed:
  • 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
    as

    Download full text from publisher

    File URL: http://arxiv.org/pdf/2310.04593
    File Function: Latest version
    Download Restriction: no
    ---><---

    References listed on IDEAS

    as
    1. Ma, Qingyin & Toda, Alexis Akira, 2022. "Asymptotic linearity of consumption functions and computational efficiency," Journal of Mathematical Economics, Elsevier, vol. 98(C).
    2. John Stachurski, 2009. "Economic Dynamics: Theory and Computation," MIT Press Books, The MIT Press, edition 1, volume 1, number 0262012774, December.
    3. Alexis Akira Toda, 2021. "Perov's Contraction Principle and Dynamic Programming with Stochastic Discounting," Papers 2103.14173, arXiv.org, revised Sep 2021.
    4. Ma, Qingyin & Toda, Alexis Akira, 2021. "A theory of the saving rate of the rich," Journal of Economic Theory, Elsevier, vol. 192(C).
    5. Schechtman, Jack, 1976. "An income fluctuation problem," Journal of Economic Theory, Elsevier, vol. 12(2), pages 218-241, April.
    Full references (including those not matched with items on IDEAS)

    Most related items

    These are the items that most often cite the same works as this one and are cited by the same works as this one.
    1. Alexis Akira Toda, 2024. "Unbounded Markov dynamic programming with weighted supremum norm Perov contractions," Economic Theory Bulletin, Springer;Society for the Advancement of Economic Theory (SAET), vol. 12(2), pages 141-156, December.
    2. Toda, Alexis Akira & Walsh, Kieran James, 2024. "Recent advances on uniqueness of competitive equilibrium," Journal of Mathematical Economics, Elsevier, vol. 113(C).
    3. Ma, Qingyin & Stachurski, John & Toda, Alexis Akira, 2020. "The income fluctuation problem and the evolution of wealth," Journal of Economic Theory, Elsevier, vol. 187(C).
    4. Lee, Byoungchan, 2023. "Wealth Inequality and Endogenous Growth," Journal of Monetary Economics, Elsevier, vol. 133(C), pages 132-148.
    5. Thomas J. Sargent & John Stachurski, 2024. "Dynamic Programming: Finite States," Papers 2401.10473, arXiv.org.
    6. Qingyin Ma & John Stachurski & Alexis Akira Toda, 2018. "The Income Fluctuation Problem with Capital Income Risk: Optimality and Stability," Papers 1812.01320, arXiv.org.
    7. Yang, C.C. & Zhao, Xueya & Zhu, Shenghao, 2023. "Tax progressivity and the Pareto tail of income distributions," Economics Letters, Elsevier, vol. 231(C).
    8. Karine Gobert & Michel Poitevin, 2006. "Non-commitment and savings in dynamic risk-sharing contracts," Economic Theory, Springer;Society for the Advancement of Economic Theory (SAET), vol. 28(2), pages 357-372, June.
    9. Michael Magill & Martine Quinzii, "undated". "Equity, Bonds, Growth And Inflation In A Quadratic Infinite Horizon Economy," Department of Economics 98-08, California Davis - Department of Economics.
    10. Dubois, Pierre & Magnac, Thierry, 2024. "Optimal intertemporal curative drug expenses: The case of hepatitis C in France," Journal of Health Economics, Elsevier, vol. 94(C).
    11. Lilia Maliar & Serguei Maliar & John B. Taylor & Inna Tsener, 2020. "A tractable framework for analyzing a class of nonstationary Markov models," Quantitative Economics, Econometric Society, vol. 11(4), pages 1289-1323, November.
    12. Mark Huggett, 2004. "Precautionary Wealth Accumulation," The Review of Economic Studies, Review of Economic Studies Ltd, vol. 71(3), pages 769-781.
    13. repec:cvs:starer:8415 is not listed on IDEAS
    14. Bloise, Gaetano & Vailakis, Yiannis, 2018. "Convex dynamic programming with (bounded) recursive utility," Journal of Economic Theory, Elsevier, vol. 173(C), pages 118-141.
    15. Thomas H. Jørgensen, 2016. "Euler equation estimation: Children and credit constraints," Quantitative Economics, Econometric Society, vol. 7(3), pages 935-968, November.
    16. Benhabib, Jess & Bisin, Alberto & Zhu, Shenghao, 2015. "The wealth distribution in Bewley economies with capital income risk," Journal of Economic Theory, Elsevier, vol. 159(PA), pages 489-515.
    17. 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.
    18. Shenghao Zhu, 2020. "Existence Of Stationary Equilibrium In An Incomplete‐Market Model With Endogenous Labor Supply," International Economic Review, Department of Economics, University of Pennsylvania and Osaka University Institute of Social and Economic Research Association, vol. 61(3), pages 1115-1138, August.
    19. van Wieringen, Wessel N. & Stam, Koen A. & Peeters, Carel F.W. & van de Wiel, Mark A., 2020. "Updating of the Gaussian graphical model through targeted penalized estimation," Journal of Multivariate Analysis, Elsevier, vol. 178(C).
    20. Xavier Boutin, 2015. "Mergers and the Dynamics of Innovation," Working Papers ECARES ECARES 2015-15, ULB -- Universite Libre de Bruxelles.
    21. Kamihigashi, Takashi & Stachurski, John, 2012. "An order-theoretic mixing condition for monotone Markov chains," Statistics & Probability Letters, Elsevier, vol. 82(2), pages 262-267.

    More about this item

    Statistics

    Access and download statistics

    Corrections

    All material on this site has been provided by the respective publishers and authors. You can help correct errors and omissions. When requesting a correction, please mention this item's handle: RePEc:arx:papers:2310.04593. See general information about how to correct material in RePEc.

    If you have authored this item and are not yet registered with RePEc, we encourage you to do it here. This allows to link your profile to this item. It also allows you to accept potential citations to this item that we are uncertain about.

    If CitEc recognized a bibliographic reference but did not link an item in RePEc to it, you can help with this form .

    If you know of missing items citing this one, you can help us creating those links by adding the relevant references in the same way as above, for each refering item. If you are a registered author of this item, you may also want to check the "citations" tab in your RePEc Author Service profile, as there may be some citations waiting for confirmation.

    For technical questions regarding this item, or to correct its authors, title, abstract, bibliographic or download information, contact: arXiv administrators (email available below). General contact details of provider: http://arxiv.org/ .

    Please note that corrections may take a couple of weeks to filter through the various RePEc services.

    IDEAS is a RePEc service. RePEc uses bibliographic data supplied by the respective publishers.