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Individual Risk and Mutual Insurance

David Cass
Graciela Chichilnisky
o-Mou Wu

Abstract

This paper examines how, in the presence of individual risk, economic efficiency can be achieved without an unrealistically large number of contingent claims. Market uncertainty is specified in such a way that general types of individual risk and collective risk are properly accounted for and so that, specifically, market clearing is always satisfied ex post as well as ex ante. We show that consistency of beliefs and optimality of allocation can be guaranteed with an appropriate array of pure Arrow securities to spread collective risk and mutual insurance policies to pool individual risk. When there is individual risk common to like groups of individuals, pooling risk by means of mutual insurance permits substantial economizing on market transactions, as compared to those required if dealing instead with the full complement of pure Arrow securities. We show that if there are $N$ households (consisting of $H$ types), each facing the possibility of being in $S$ individual states together with $T$ collective states, then ensuring Pareto optimally requires only $H(S - 1)T$ independent mutual insurance policies plus $T$ pure Arrow securities. Our results also help to clarify the question of which missing markets may affect allocational efficiency.