Introduction

There have been numerous contributions to economic theory of uncertainty and insurance. Despite well-developed theory in risk aversion in a complete market framework, economic behavior in an incomplete market case has not been investigated in sufficient depth. It was only recently that Ross (1981) showed that a basic theorem of Pratt (1964) about the relationship between the degree of risk aversion and the size of risk premium does not necessarily hold without a stronger condition in incomplete market situations.

Suppose that a risk-averse agent faces different amounts of income for different states of nature. It is well known that optimal risk sharing between a risk-averse agent and a risk-neutral agent (insurance company) results in equalization of net income (income after premium and coverage) across different states of nature, given that all states of nature are verifiable, that is, that markets are complete. Suppose now that a subset of states of nature are not verifiable individually, but only as a group of states, by an insurance (risk-neutral) agent. If uncertainty is additive to insurance coverage, then an optimal insurance policy is such that the expected marginal utility of net income over a group of unverifiable states of nature is equal to the marginal utility of the net (sure) income at a verifiable state of nature. If uncertainty is multiplicative, then an optimal insurance policy would equalize the expected marginal utility weighted by the multiplicative stochastic yield to the marginal utility of a sure income at a verifiable state.