7.8 In Equation 7.30 we showed that the amount an individual is willing to pay to avoid a fair gamble (h) is given by p ¼ 0.5E(h2

)r(W ), where r(W ) is the measure of absolute risk aversion at this person’s initial level of wealth. In this problem we look at the size of this payment as a function of the size of the risk faced and this person’s level of wealth.

a. Consider a fair gamble (v) of winning or losing $1. For this gamble, what is E(v 2

)?

b. Now consider varying the gamble in part

(a) by multiplying each prize by a positive constant k. Let h ¼ kv. What is the value of E(h2

)?

c. Suppose this person has a logarithmic utility function U(W) ¼ ln W. What is a general expression for r(W )?

d. Compute the risk premium ( p) for k ¼ 0.5, 1, and 2 and for W ¼ 10 and 100. What do you conclude by comparing the six values?