A general model of insurance under adverse selection

This paper considers optimal insurance schemes in a principal-agent multi-dimensional environment in which two types of risk averse agents differ in both risk and attitude to risk. Risk corresponds to any pair of distribution functions (not necessarily ordered by any of the usual dominance relations) and attitudes to risk are represented by any pair of non-decreasing and concave utility functions (not necessarily ordered by risk aversion). Results obtained in one-dimensional models that considered these effects separately and under more restricted conditions, are preserved in the more general set-up, but some of the questions we study can only be posed in the more general framework. The main results obtained for optimal insurance schemes are: (i) Insurance schemes preserve the order of certainty equivalents; consequently, the latter constitute a one-dimensional representation of types. (ii) Agents with the lower certainty equivalent are assigned full insurance. Partial insurance assigned to the others may entail randomization. (iii) Partially insured positions are an increasing function of the ratios of the probabilities that the two types assign to the uninsured positions. Most of these properties are preserved when, due to competition or other reasons, the insured certainty equivalents can not be set below pre-determined levels.