Optimal insurance and generalized deductibles

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Abstract
This report is intended as a contribution to the theory of demand for insurance. In many circumstances, it appears that, given a range of alternative possible insurance policies, the insured would prefer a policy offering complete coverage beyond a deductible. In an earlier paper (Arrow [1]; reprinted in Arrow [3], pp. 212-216), this argument was developed for the case where the risk being insured against was, effectively, loss of income. Recently, Ehrlich and Becker [4] have extended these results considerably, as well as analyzing other responses of the insured to the price of insurance, responses beyond the scope of this study. For some other related work, see Pashigian, Schkade, and Menefee [8], Smith [12], and Gould [6]. However, income is not the only uncertainty, especially in the context of health insurance, and only under special and unrealistic circumstances can it be held that the other uncertainties have income equivalents. Put loosely, the marginal utility of income will in general depend not only on the amount of income but also on the state of the individual or, more generally, on the state of the world.

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