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This paper studies the design of Pareto-optimal insurance under the heterogeneous beliefs of the insured and insurer. To accommodate a wide range of belief heterogeneity, we allow the likelihood ratio function to be non-monotone. To prevent the  ex post  moral hazard issue, the incentive compatibility condition is exogenously imposed to restrict the indemnity function. An implicit characterization of the optimal indemnity function is presented first by using the calculus of variations. Based on the point-wise maximizer to the problem, we partition the domain of loss into disjoint pieces and derive the parametric form of the optimal indemnity function over each piece through its implicit characterization. The main result of this paper generalizes those in the literature and provides insights for related problems.

Pareto-optimal insurance under heterogeneous beliefs and incentive compatibility