Contracting with Non-Exponential Discounting: Moral Hazard and Dynamic Inconsistency

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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.