Insurance contracts with multiple levels of deductibles

Insurance contracts with multiple levels of deductibles (henceforth MLD contracts) are very common. The major goal of this paper is to suggest theoretical foundations for the prevalence of this type of contracts when agents have different, not necessarily concave, utility functions that are private information. We show that an important property of such contracts is that they reduce risk in the sense of Bickel&Lehmann dispersion, see [2] and [3], which is necessary and sufficient to ensure that more risk averse agents are willing to pay more for insurance that offers more protection; the celebrated Rotschild&Stiglitz [12] risk concept lacks this property. Agents whose utility functions belong to the star-shape class, see [6] (that includes concave as a special case), can be separated by their Arrow-Pratt risk measure, regardless whether contracts are fair or subfair. To achieve separation under concavity, contracts must be increasigly subfair. We also show that contrary to our (initial) conjecture all parties can benefit from having contracts with more deductible levels; what matters is how competitive insurance markets are, or how tightly they are regulated.