Optimal insurance design under limited liability

We study optimal demand for insurance in a classical expected utility setting where the insured party has limited liability and has access to three different types of progressively more restrictive contracts. At one end, with no restrictions on the indemnity schedule, it is optimal to fully insure certain losses while leaving others uninsured. At the other end, if indemnity schedules and retained losses are assumed to be increasing functions of the underlying loss, the optimal insurance policies are shown to be capped deductibles. For the intermediate case when the indemnity schedule is only an increasing function of the loss, we find that optimal contracts exhibit a richer structure beyond the capped policies suggested in earlier literature. Our study extends and provides a unifying perspective on the existing literature on optimal insurance under limited liability.