Question

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 \(R=-\infty\) in both the risk-aversion and substitution frameworks? c. A rise in the price of contingent claims in "bad" times \(\left(P_{b}\right)\) will induce substitution and income effects into the demands for \(W_{g}\) and \(W_{b} .\) If the individual has a fixed budget to devote to these two goods, how will choices among them be affected? Why might \(W_{g}\) 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 percent 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 \(W_{0}\) as from an even bet on \(1.055 W_{0}\) and \(0.955 W_{0}\) 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 ): \(42-71\)

Short answer

Answer: The constant relative risk aversion utility function is related to risk aversion and the willingness to substitute wealth between different states of the world as it connects uncertainty and individual preferences. As risk aversion increases, an individual becomes less willing to take risks and prefers more stable outcomes. The elasticity of substitution reflects how easily an individual can adjust their preferences for different wealth states. These two factors combined help determine the individual's behavior under uncertainty and their willingness to take risks and allocate their wealth accordingly.

Step-by-step solution

Underlying connection

Risk aversion and the willingness to substitute wealth between different states of the world are both connected to uncertainty and how an individual copes with it. As risk aversion increases, an individual becomes less willing to take risks and would prefer a more stable outcome. The elasticity of substitution reflects how easily an individual can adjust their preferences for different wealth states. #b. Polar cases interpretations#

Case 1: R = 1

When R = 1, this individual has a "logarithmic" utility function, meaning that they are risk-averse, but their degree of risk aversion doesn't change as wealth changes. They have a fixed preference towards risks. In terms of substitution, their elasticity is 1, showing that they are relatively flexible in substituting wealth between different states.

Case 2: R = -∞

When R = -∞, the individual is extremely risk-averse, meaning that they are not willing to take any risks even for a small potential gain. In terms of substitution, their elasticity of substitution is 0, suggesting that they are not able to substitute wealth between different states of the world at all. #c. Effects of a rise in Pb#

Substitution and income effects

When the price of contingent claims in bad times (Pb) rises, the individual faces substitution and income effects. The substitution effect pushes the individual to substitute away from the more expensive good (Wb) and increase their demand for the cheaper good (Wg). The income effect, however, depends on the individual's degree of risk aversion.

How Wg might rise or fall

Depending on the individual's degree of risk aversion, they may react differently to the rise in Pb. A very risk-averse person might choose to allocate more of their fixed budget to Wg to avoid potential losses in the bad state. On the other hand, a less risk-averse person might prefer to maintain or even increase their holdings of Wb, hoping for a higher return in good times. #d. Value of R and required average return#

Part i: Finding R

We need to find a value of R that makes the individual indifferent between keeping W0 and taking an even bet on 1.055W0 and 0.955W0. First, set up the utility function with constant relative risk aversion: Utility(W) = W^(1-R) / (1-R) Then calculate the expected utility of the bet: Expected utility = 0.5 * Utility(1.055W0) + 0.5 * Utility(0.955W0) Next, set the above expected utility equal to the utility of keeping W0 and solve for R: Utility(W0) = Expected utility

Part ii: Finding required average return

In this part, we want to find the average return the individual would require to accept a 50-50 chance of gaining or losing 10 percent. We first set up the expected utility equation for this investment: Expected utility = 0.5 * Utility(1.1W0) + 0.5 * Utility(0.9W0) Now, we need to make this expected utility equal to the utility of keeping W0 times some required average return (1+required_average_return), and solve for the required average return: Utility(W0 * (1 + required_average_return)) = Expected utility After finding the required average return, the comparison between the risk/reward trade-off will show how the equity premium puzzle exists in real-world investments.