Multiple losses, ex-ante moral hazard, and the non-optimality of the standard insurance contract

Under certain conditions the optimal insurance policy will offer full coverage above a deductible, as Arrow and others have shown long time ago. Interestingly, the same design of insurance policies applies in case of a single loss and ex-ante moral hazard. However, many insurance policies provide coverage against a variety of losses and the possibilities for the insured to affect the probabilities of each possible loss might be substantially different. The optimal design of a insurance contract providing coverage against different losses therefore should generally differ from the standard form under moral hazard. The paper concentrates on the conditions under which the standard insurance contract holds under moral hazard and more than one loss. It gives some evidence that many insurance contracts should be split up. The main result is, that the relative changes of probabilities due to precautious activities are decisive. On the other hand, under moral hazard it is rarely ever optimal to combine two losses in one insurance contract prescribing only a single deductible for both losses if both losses can occur simultaneously.