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Failure of Smooth Pasting Principle and Nonexistence of Equilibrium Stopping Rules under Time-Inconsistency

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  • Ken Seng Tan
  • Wei Wei
  • Xun Yu Zhou

Abstract

This paper considers a time-inconsistent stopping problem in which the inconsistency arises from non-constant time preference rates. We show that the smooth pasting principle, the main approach that has been used to construct explicit solutions for conventional time-consistent optimal stopping problems, may fail under time-inconsistency. Specifically, we prove that the smooth pasting principle solves a time-inconsistent problem within the intra-personal game theoretic framework if and only if a certain inequality on the model primitives is satisfied. We show that the violation of this inequality can happen even for very simple non-exponential discount functions. Moreover, we demonstrate that the stopping problem does not admit any intra-personal equilibrium whenever the smooth pasting principle fails. The "negative" results in this paper caution blindly extending the classical approaches for time-consistent stopping problems to their time-inconsistent counterparts.

Suggested Citation

  • Ken Seng Tan & Wei Wei & Xun Yu Zhou, 2018. "Failure of Smooth Pasting Principle and Nonexistence of Equilibrium Stopping Rules under Time-Inconsistency," Papers 1807.01785, arXiv.org, revised Sep 2019.
    • Handle: RePEc:arx:papers:1807.01785
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    References listed on IDEAS

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    1. Grenadier, Steven R. & Wang, Neng, 2007. "Investment under uncertainty and time-inconsistent preferences," Journal of Financial Economics, Elsevier, vol. 84(1), pages 2-39, April.
    2. Karp, Larry, 2007. "Non-constant discounting in continuous time," Journal of Economic Theory, Elsevier, vol. 132(1), pages 557-568, January.
    3. repec:dau:papers:123456789/11473 is not listed on IDEAS
    4. George Loewenstein & Drazen Prelec, 1992. "Anomalies in Intertemporal Choice: Evidence and an Interpretation," The Quarterly Journal of Economics, President and Fellows of Harvard College, vol. 107(2), pages 573-597.
    5. Hsiaw, Alice, 2013. "Goal-setting and self-control," Journal of Economic Theory, Elsevier, vol. 148(2), pages 601-626.
    6. Paul A. Samuelson, 1937. "A Note on Measurement of Utility," The Review of Economic Studies, Review of Economic Studies Ltd, vol. 4(2), pages 155-161.
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    Cited by:

    1. Schürhoff, Norman & Dimopoulos, Theodosios Sakis, 2021. "Self-inflicted debt crises," CEPR Discussion Papers 15781, C.E.P.R. Discussion Papers.
    2. Marcel Nutz & Yuchong Zhang, 2019. "Conditional Optimal Stopping: A Time-Inconsistent Optimization," Papers 1901.05802, arXiv.org, revised Oct 2019.

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