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Punish Underperformance with Resting Optimal Dynamic Contracts in the Presence of Switching Cost

This paper studies a dynamic principal-agent setting in which the principal needs to dynamically schedule an agent to work or rest. When the agent is motivated to work, the arrival rate of a Poisson process increases, which increases the principal's payoff. Resting, on the other hand, serves as a threat to the agent by delaying future payments. A key feature of our setting is a switching cost whenever the agent stops resting and starts working. We formulate the problem as an optimal control model with switching, and fully characterize the optimal control policies under different parameter settings. Our analysis shows that when the switching cost is not too high, the optimal contract demonstrates a generalized control-band structure, and may involve randomly switching the agent to rest. The length of each resting episode, on the other hand, is fixed. Overall, the optimal contract is easy to describe, compute, and implement.