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

This paper considers time-inconsistent stopping problems in which the inconsistency arises from a class of nonexponential discount functions called weighted discount functions. We show that the smooth pasting (SP) principle, the main approach that is used to construct explicit solutions for classical time-consistent optimal stopping problems, may fail under time-inconsistency. Specifically, a mere change of the discount function from exponential to nonexponential (everything else being the same) will cause the SP approach to fail. In general, we prove that SP solves a time-inconsistent problem, within the intrapersonal game theoretic framework with a general nonlinear cost functional and a geometric Brownian motion, if and only if certain inequalities on the model primitives are satisfied. In the special case of a real options problem, we show that while these inequalities hold trivially for the exponential discount function, they may not hold even for very simple nonexponential discount functions. Moreover, we show that the real options problem actually does not admit any equilibrium whenever the SP approach fails. The negative results in this paper caution one from blindly extending the classical approach for time-consistent stopping problems to their time-inconsistent counterparts.