  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.
