The Incentive Implications of Incomplete Insurance: The Multiplicative Case

Abstract

Suppose that a fair insurance policy is available to risk-averse economic agents contingent on only "observable" variables. The risk-averse agents will purchase incomplete insurance to maximize their expected utility. The first order condition is typically an equality of expected "marginal" utility. whether or not weighted by the level of income, conditional on observable states. It is interesting to know how the "levels" of utility are ranked among these states given the first order conditions. We will show that the ranking of the "level" of expected utility depends on the degree of risk aversion. Suppose an implicit contract model with severance payments. Workers are laid off with a fixed known probability. When a worker is laid off, he is paid severance payments and released for a search of new employment. Whether he is reemployed or self-employed or how much he is earning is unverifiable by the original employer, so that severance payments cannot be contingent on income after layoff. In this sense, severance payments are incomplete insurance. Suppose that income after layoff is proportional to severance payments with the proportion being stochastic. The first order condition is given as an equality of an expected marginal utility of income after layoff weighted by that income to a marginal utility of wages for a retained worker weighted by the amount of severance payments. For example, when the relative risk aversion is more than one and constant, and the mean of proportion of yield on severance payments is less than one, the utility of the retained is larger than the expected utility of the laid off. Other cases are worked out, too. This claim is proved using the following theorem. Provided that the relative risk aversion is greater than one, the "level" of utility of sure income is greater (less) than the expected utility, if the utility function is of increasing (decreasing, respectively) relative risk aversion. The result is reverse for the case that the degree of risk aversion is less than one. All assertions are rigorously proved.