Question

2. Consider an individual whose utility function over income I is U(I), where U is increas- ing smoothly in I and is concave (in other words, our basic assumptions throughout this chapter). Let Is be this person's income if he is sick, let 1н be his income if he is healthy, let p be his probability of being sick, let EI be expected income, and let ElU] be his expected utility when he has no insurance. Assum e 0 < 1s < IH . (8 points (a) Write down expressions for both EI and EU] in terms of the other parameters of the model (b) Consider a full insurance product that guarantees this individual El/]. Create a diagram in U- I space. Draw the individual's utility curve and the lines repre- senting Is, I, and ElI]. 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) (c) Derive an algebraic expression for M. To simplify the calculation, assume 1s = 0 and U(Is)0. 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. Hint: draw a coordinate plane with M on the x-axis and M on the y-axis. Describe intuitively why this graph has the shape that it does