Renegotiation and Dynamic Inconsistency: Contracting with Non-Exponential Discounting

Abstract

This paper studies a continuous-time, nite-horizon contracting problem with renegotiation
and dynamic inconsistency arising from non-exponential discounting. The
problem is formulated as a dynamic game played among the agent, the principal and
their respective future "selves", each with their own discount function. We identify
the principal optimal renegotiation-proof contract as a Markov Perfect Equilibrium
(MPE) of the game, prove such a MPE exists, and characterize the optimal contract
via an extended Hamilton-Jacobi-Bellman system. We solve the optimal contract in
closed form when the discount functions of the selves are related by time di erence,
a property that is satis ed by common forms of non-exponential discounting such as
quasi-hyperbolic discounting and anticipatory utility. In particular, quasi-hyperbolic
discounting leads to a U-shaped action path and anticipatory utility leads to a humshaped
path, both are qualitatively di erent from the monotonic action path that
would arise under exponential discounting.