Optimal insurance under moral hazard in loss reduction

This study investigates the optimal insurance when moral hazard exists in loss reduction. We identify that the optimal insurance is full insurance up to a limit and partial insurance above that limit. In case of partial insurance, the indemnity schedule for prudent individual is convex, linear, or concave in loss, depending on the shapes of the utility and loss distribution. The optimal insurance may include a deductible for large losses only when the indemnity schedule is convex. It may also include a fixed reimbursement when the schedule is convex or concave. When the loss distribution belongs to the one dimensional exponential family with canonical form, the indemnity schedule is concave under IARA and CARA, whereas it can be concave or convex under DARA. •The optimal indemnity can be convex, linear or concave in loss under prudence.•The optimal insurance may contain a deductible when indemnity is convex.•The optimal insurance may include a fixed reimbursement when indemnity is convex or concave.