Pareto efficient insurance contracts when the insurer's cost function is discontinuous

We consider the problem of efficient insurance contracts when the cost structure includes a fixed cost per claim. We prove existence of efficient insurance contracts and that the indemnity function in such contracts is non-decreasing in the damage. We further show that either there is no insurance, or the indemnity is positive for all losses, or efficient insurance contracts have a unique jump. We study variants of the model and provide a generalization to the case of non expected utilities. Our results are then applied to Townsend's model of deterministic auditing.