Renegotiation and Dynamic Inconsistency: Contracting with Non-Exponential Discounting

This paper studies a continuous-time, finite-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 difference, a property that is satisfied 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 different from the monotonic action path that would arise under exponential discounting.