Repeated moral hazard with a time-inconsistent agent

I consider a repeated principal-agent model with moral hazard, in which the agent has βδ-preferences, which are widely used to capture time-inconsistency. I analyze the case where the agent is sophisticated in the sense that he is fully aware of his inconsistent discounting. I characterize the optimal wage scheme for such an agent and compare it to time-consistent benchmarks. The marginal cost of rewarding the agent for high output today exceeds the marginal benefit of delaying these rewards until tomorrow. In this sense, the principal does not smooth the agent's rewards over time. When facing a sophisticated agent, it is optimal for the principal to reward the good performance more and punish the bad performance more in the early period, relative to the optimal wage scheme for a time-consistent agent.