Consider an individual whose utility function over income I is U(I), where U is increasing smoothly in I and is concave (in other words, our basic assumptions throughout this chapter). Let I_S = 0 be this person’s income if he is sick, let I_H > 0 be his income if he is healthy, let p be his probability of being sick, let E[I] be expected income, and let E[U] be his expected utility when he has no insurance.

a. Write down algebraic expressions for both E[I] and E[U] in terms of the other parameters of the model.

b. Consider a full insurance product that guarantees this individual E[I]. Create a diagram in U–I space. Draw the individual’s utility curve and the lines representing I_S, I_H, and E[I]. Then draw and label a line segment that corresponds to the utility gain, ΔU, from buying this insurance product. Draw and label another line segment, M, which corresponds to the consumer surplus from the purchase of insurance (that is, the monetary value of the utility gain from buying insurance).

c. Derive an algebraic expression for M. [Hint: you may assume that U is invertible and its inverse is U−1].

d. Draw a graph plotting how M changes as p (the probability of being sick) varies between 0 and 1. Describe intuitively why this graph has the shape that it does.