(Adapted from Chapter 3 in Kreps (1990).) Imagine a greedy, risk-averse, expected utility maximizing consumer whose end-of-period income level is subject to some uncertainty. The income will be Y with probability p¯ and Y < Y with probability 1 − p¯.

Think of  = Y − Y as some loss the consumer might incur due to an accident. An insurance company is willing to insure against this loss by paying  to the consumer if she sustains the loss. In return, the company wants an upfront premium of δ. The consumer may choose partial coverage in the sense that if she pays a premium of aδ, she will receive a if she sustains the loss. Let u denote the von Neumann–Morgenstern utility function of the consumer. Assume for simplicity that the premium is paid at the end of the period.

(a) Show that the first-order condition for the choice of a is p¯δu
(Y − aδ) = (1 − p¯)( − δ)u
(Y − (1 − a) − aδ).

(b) Show that if the insurance is actuarially fair in the sense that the expected payout (1 − p¯) equals the premium δ, then the consumer will purchase full insurance, that is a = 1 is optimal.

(c) Show that if the insurance is actuarially unfair, meaning (1 − p¯) < δ, then the consumer will purchase partial insurance, that is the optimal a is less than 1.