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Dynamic Programming with State-Dependent Discounting

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  • John Stachurski
  • Junnan Zhang

Abstract

This paper extends the core results of discrete time infinite horizon dynamic programming theory to the case of state-dependent discounting. The traditional constant-discount condition requires that the discount factor of the controller is strictly less than one. Here we replace the constant factor with a discount factor process and require, in essence, that the process is strictly less than one on average in the long run. We prove that, under this condition, the standard optimality results can be recovered, including Bellman's principle of optimality, convergence of value function iteration and convergence of policy function iteration. We also show that the condition cannot be weakened in many standard settings. The dynamic programming framework considered in the paper is general enough to contain features such as recursive preferences. Several applications are discussed.

Suggested Citation

  • John Stachurski & Junnan Zhang, 2019. "Dynamic Programming with State-Dependent Discounting," Papers 1908.08800, arXiv.org, revised Sep 2019.
  • Handle: RePEc:arx:papers:1908.08800
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    File URL: http://arxiv.org/pdf/1908.08800
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    References listed on IDEAS

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    3. Higashi, Youichiro & Hyogo, Kazuya & Takeoka, Norio, 2009. "Subjective random discounting and intertemporal choice," Journal of Economic Theory, Elsevier, vol. 144(3), pages 1015-1053, May.
    4. Youichiro Higashi & Kazuya Hyogo & Norio Takeoka & Hiroyuki Tanaka, 2017. "Comparative impatience under random discounting," Economic Theory, Springer;Society for the Advancement of Economic Theory (SAET), vol. 63(3), pages 621-651, March.
    5. Mehra, Rajnish & Sah, Raaj, 2002. "Mood fluctuations, projection bias, and volatility of equity prices," Journal of Economic Dynamics and Control, Elsevier, vol. 26(5), pages 869-887, May.
    6. Timothy S. Hills & Taisuke Nakata, 2018. "Fiscal Multipliers at the Zero Lower Bound: The Role of Policy Inertia," Journal of Money, Credit and Banking, Blackwell Publishing, vol. 50(1), pages 155-172, February.
    7. Boud, John III, 1990. "Recursive utility and the Ramsey problem," Journal of Economic Theory, Elsevier, vol. 50(2), pages 326-345, April.
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    Blog mentions

    As found by EconAcademics.org, the blog aggregator for Economics research:
    1. Dynamic Programming with State-Dependent Discounting
      by Christian Zimmermann in NEP-DGE blog on 2019-09-25 22:14:42

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