Question: Show that the linear-plus-exponential utility function in Problem I has decreasing risk aversion.
Problem I: Repeat Problem II with U(x) = 0:0003x 8:48e-x/2775. A utility function of this form is called linear-plus-exponential, because it contains both linear (0.0003x) and exponential terms. It has a number of interesting and useful properties, including the fact that it switches only once among any pair of lotteries (such as those in Problem II) as wealth increases (see Bell 1995a, b).
Problem II: A bettor with utility function U(x) = ln(x), where x is total wealth, has a choice between the following two alternatives:
A Win $10,000 with probability 0:2
Win $1,000 with probability 0:8
B Win $3,000 with probability 0:9
Lose $2,000 with probability 0:1
a. If the bettor currently has $2,500, should he choose A or B?
b. Repeat a, assuming the bettor has $5,000.
c. Repeat a, assuming the bettor has $10,000.
d. Do you think that this pattern of choices between A and B is reasonable? Why or why not?