Optimal Insurance Without the Nonnegativity Constraint on Indemnities: Ambiguity and Belief Heterogeneity

In Arrow's (1971) classical problem of optimal insurance design, a linear deductible schedule is optimal for an Expected-Utility (EU) maximizing decision maker (DM), if the premium depends on the indemnity's actuarial value, if the DM and the insurer share the same probabilistic beliefs about the realizations of the random loss, and under the classical constraints that, in each state of the world, the indemnity is nonnegative and does not exceed the value of the loss. Raviv (1979) re-examined Arrow's problem and concluded that the presence of a deductible is due to both the nonnegativity constraint on the indemnity and the variability in the cost of insurance. In an effort to test this statement, Gollier (1987) relaxes the nonnegativity constraint and argues that the existence of a deductible is only due to the variability in the cost of insurance. In this paper, we test whether the intuition behind Gollier's result still holds under more general preferences for the DM and the insurer. We consider a setting of ambiguity (one-sided and then two-sided) and a setting of belief heterogeneity. We drop the nonnegativity constraint and assume no cost (or a fixed cost) to the insurer, and we derive closed-form analytical solutions to the problems that we formulate. In particular, we show that an optimal indemnity (resp., retention) no longer includes a deductible provision (resp., is not equal to the realized loss). Moreover, the optimal indemnity (resp., retention) can be negative (resp., higher than the realized loss) for small values of the loss, or in case of no loss. This is in line with the intuition behind Gollier's finding in the case of belief homogeneity and no ambiguity.