Moral hazard and optimal insurance contract with a continuum effort

In the research works, moral hazard is usually represented in two natural states (accident and no accident). In this case, the determination of the optimal contract could be made graphically. The mathematicaing is become more complicated when we consider infinite natural states and efforts under the monotone likelihood ratio property. In fact, the particular form of incentive constraints introduces a non-convex problem. In this paper, under technical conditions we show that the non-convex problem has a solution which is a new result and we determine the optimal contract such that the optimal wealth of the insured is a non-increasing function of the loss.