Arrow’s theorem of the deductible: moral hazard and stop-loss in health insurance

We show that the logic of Arrow’s theorem of the deductible, i.e. that it is optimal to focus insurance coverage on the states with largest expenditures, remains at work in a model with ex post moral hazard. The optimal insurance contract takes the form of a system of “implicit deductibles”, i.e. it results in the same indemnities as a contract with 100% coverage above a variable deductible positively related to the elasticity of medical expenditures with respect to the coverage rate. In a model with an explicit stop-loss arrangement, this stop-loss takes the form of a deductible, i.e. there is no reimbursement for expenses below the stop-loss amount. One motivation to have some co-insurance below the deductible arises if regular health care expenditures in a situation of standard health have a negative effect on the probability of getting into a state with large medical expenses.