8.8 For the constant relative risk aversion utility function (Equation 8.62) we showed that the degree of risk aversion is measured by (1 — R). In Chapter 3 we showed that the elasticity of substitution for the same function is given by 1/(1 — R). Hence, the measures are reciprocals of each other. Using this result, discuss the following questions:

a. Why is risk aversion related to an individual's willingness to substitute wealth between states of the world? What phenomenon is being captured by both concepts?

b. How would you interpret the polar cases R = 1 and R— — °° in both the risk-aversion and substitution frameworks?

c. A rise in the price of contingent claims in "bad" times (Pb) will induce substitution and income effects into the demands for Wg and Wh. If the individual has a fixed budget to devote to these two goods, how will choices among them be affected? Why might Wg rise or fall depending on the degree of risk aversion exhibited by the individual?

d. Suppose that empirical data suggest an individual requires an average return of 0.5 per cent if he or she is to be tempted to invest in an investment that has a 50-50 chance of gaining or losing 5 percent. That is, this person gets the same utility from Wo as from an even bet on 1.055 Wo and 0.955 Wo.
i. What value of R is consistent with this behavior?
ii. How much average return would this person require to accept a 50-50 chance of gaining or losing 10 percent?
Note: This part requires solving nonlinear equations, so approximate solutions will suffice.
The comparison of the risk/reward trade-off illustrates what is called the "equity premium puzzle," in that risky investments seem to actually earn much more than is consistent with the degree of risk-aversion suggested by other data. See N. R. Kocherlakota, "The Equity Premium: It's Still a Puzzle" Journal of Economic Literature (March 1996):
42-71.