Center for Economic Research, Department of Economics, University of Minnesota
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
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.
Ito, T. and Machina, M., (1983), "The Incentive Implications of Incomplete Insurance: The Multiplicative Case", Discussion Paper No. 175, Center for Economic Research, Department of Economics, University of Minnesota.
Ito, Takatoshi; Machina, Mark.
The Incentive Implications of Incomplete Insurance: The Multiplicative Case.
Center for Economic Research, Department of Economics, University of Minnesota.
Retrieved from the University of Minnesota Digital Conservancy,
Content distributed via the University of Minnesota's Digital Conservancy may be subject to additional license and use restrictions applied by the depositor.