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"Contracting with Non-Exponential Discounting: Moral Hazard and Dynamic Inconsistency" by Can Urgun, Doruk Cetemen, and Felix Zhiyu Feng
We develop a framework for dynamic moral hazard problems with dynamic inconsistencies resulting from general, non-exponential discount functions. We derive the
principal-optimal contract as a Markov Perfect Equilibrium of the game played between the agent’s and the principal’s future selves. Such contract exists even when both contracting parties have dynamically inconsistent discount functions, and can be characterized via a system of differential equations rather than the classical Hamilton Jacobi-Bellman equation. We demonstrate the applicability of our framework by solving two examples in closed form: one with quasi-hyperbolic discounting, and one with anticipatory utility.