Optimal finite horizon contract with limited commitment

We study a finite horizon optimal contracting problem with limited commitment. A risk-neutral principal enters into an insurance contract with a risk-averse agent who receives a stochastic income stream and cannot commit to keeping the contract. We consider a general concave utility function and a general process. We use the dual approach and the Lagrangian method to solve our optimization problem by transforming the dual problem into an infinite series of optimal stopping problems. We derive the optimal contract by representing the optimal intermediate and terminal payments from the principal to the agent in a closed-form. We show that the contract begins with a low level of payment to the agent and ratchets up the payment if the stochastic income of the agent rises above a pre-specified threshold level. In particular, if the agent’s income follows a geometric Brownian motion, the threshold level is a deterministic decreasing function of time. We also show that the final payment depends on the history of the agent’s income and the sale value of the production facility at the final time.