'Optimal Insurance with Belief Heterogeneity'
We re-examine the problem of demand for insurance indemnification when the insured and the insurer disagree about the likelihoods associated with the realizations of the insurable loss. For ease of comparison with the classical literature, we adopt the original setting of Arrow, but allow for divergence in beliefs between the insurer and the insured. We do not impose the no sabotage condition on admissible indemnities. Instead, we impose a state-verification cost that the insurer can incur in order to verify the loss severity, which rules out ex post moral hazard issues that could otherwise arise from possible misreporting of the loss by the insured. Moreover, unlike the existing literature, we do not impose conditions on the type or level of disagreement about probabilities, and we allow in particular for singularity between the beliefs, that is, disagreement about zero-probability events. We characterize the optimal indemnity for any type or level of belief heterogeneity, and we show that it has a simple two-part structure: full insurance on an event to which the insurer assigns zero probability, and a variable deductible on the complement of this event. We then introduce a belief singularity metric and show that there exists a maximal level of belief singularity; and we characterize the optimal indemnity at this limit. Finally, we show that for any subjective belief that the insured can have, there exists another subjective belief that exhibits the same level of singularity with respect to the insurer's subjective belief, such that the corresponding optimal indemnity is full insurance on a uniquely and endogenously determined event to which the insurer assigns zero probability, and a linear deductible on the complement of this event.