A consumer who enjoys risk has an initial wealth W and a utility function U(W). He faces a probability P of having an accident. In the event of an accident, he loses an amount L.

Assume that insurance is available at a price x per dollar of insurance.

a) Show that the optimal choice of this consumer is given by

(1-P)/P)*((U'(W_g)/(U'(W_b)) = (1-x)/x

where W_g = ax and W_b = W-ax-L+a

b) Show that, if x = P , this consumer is fully insured.

c) Show that, if x > P, this consumer partially insures.