13 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 IS = 0 be this person's income if he is sick, let IH > 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.

1. Consider a full insurance product that guarantees this individual E[I]. Make adiagram in U-I space. Draw the individual's utility curve and the lines representing IS, IH, and E[I]. Then draw and label a line segment that corresponds to the utility gain, U, 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).

1. Derive an algebraic expression for M. [Hint: you may assume that U is invertible and its inverse is U1].

1. Draw a graph plotting how M changes as p (the probability of being sick) varies between 0 and 1. [Hint: draw a coordinate plane with p on the x-axis and M on the y-axis.] Describe intuitively why this graph has the shape that it does.