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

23 Pages Posted: 19 Jul 2018

See all articles by Ken Seng Tan

Ken Seng Tan

University of Waterloo

Wei Wei

University of Waterloo - Department of Statistics and Actuarial Science

Xun Yu Zhou

Columbia University - Department of Industrial Engineering and Operations Research (IEOR)

Date Written: June 26, 2018

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.

Keywords: time inconsistency, equilibrium stopping rule, intra-personal game, smooth pasting, nonexistence

JEL Classification: D81, D90, G11

Suggested Citation

Tan, Ken Seng and Wei, Wei and Zhou, Xunyu, Failure of Smooth Pasting Principle and Nonexistence of Equilibrium Stopping Rules under Time-Inconsistency (June 26, 2018). Available at SSRN: https://ssrn.com/abstract=3203774 or http://dx.doi.org/10.2139/ssrn.3203774

Ken Seng Tan

University of Waterloo ( email )

Waterloo, Ontario N2L 3G1
Canada

Wei Wei (Contact Author)

University of Waterloo - Department of Statistics and Actuarial Science ( email )

200 University Avenue West
Waterloo, Ontario N2L 3G1
Croatia

Xunyu Zhou

Columbia University - Department of Industrial Engineering and Operations Research (IEOR) ( email )

331 S.W. Mudd Building
500 West 120th Street
New York, NY 10027
United States

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