Problem on utility funtion probability

Suppose that your utility, U, is a function only of wealth, Y, and that U(Y) is as drawn below. In this graph, note that U(Y) increases linearly between points a and b.

Suppose further that you do not know whether or not you will be sick, but you do know that the probability of becoming sick is p (while the probability of staying healthy is 1-p). If you do get sick, your wealth will be Y_s = 0. If you do not get sick, your wealth will be Y_h > 0.

1940_utility function.jpg

(1) Write an expression for expected income, EI, and an expression for expected utility without insurance.

(2) Assume that a < EI < b. Draw, on the graph above, a line showing expected utility without insurance. Also draw a line showing expected utility with actuarially fair full insurance.

(3) Consider an actuarially fair partial insurance contract that offers a if you are sick and b if you are healthy. Would your utility with such a contract be greater or less than your utility with an actuarially fair full insurance contract? Briefly, explain.

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