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 \(R--^{\circ}\) 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 \(W_{h}\). 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 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 \(W_{o}\) as from an even bet on \(1.055 W_{o}\) and \(0.955 W_{o}\) 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: Risk aversion is related to an individual's willingness to substitute wealth between states of the world because it measures how much an individual prefers a certain outcome to a risky one. A higher risk aversion implies less willingness to exchange wealth between states, increasing their exposure to risk. An individual with \(R=1\) has a constant degree of risk aversion and a constant rate of substitution. Meanwhile, as \(R\) approaches 0, the person becomes risk-neutral and demonstrates an infinitely large elasticity of substitution. A rise in the price of contingent claims affects demands for \(W_{g}\) and \(W_{h}\), with the higher risk-averse individuals more likely to buy safer goods when these claims become more expensive. To find the value of \(R\) consistent with a person's investment behavior and the required average return for a different risk level, we can use the CRRA utility function, allowing us to solve for \(R\) and the return value, \(r\), through numerical methods. This process highlights the risk/reward trade-off, also known as the "equity premium puzzle".

Step-by-step solution

a. Relationship between risk aversion and willingness to substitute wealth

Risk aversion is related to an individual's willingness to substitute wealth between states of the world because it measures how much an individual prefers a certain outcome to a risky one. The higher the risk aversion, the less willing the individual is to exchange wealth between states of the world as it increases their exposure to risk. Both concepts capture the individual's preferences for certainty and predictability of outcomes.

b. Interpretation of polar cases \(R=1\) and \(R\to0\) in risk aversion and substitution frameworks

In the risk aversion framework, \(R=1\) represents unitary relative risk aversion, meaning that the person's degree of risk aversion remains constant. In the substitution framework, this translates to an elasticity of substitution equal to 1, meaning the rate of substitution remains constant. For \(R\to0\), the person becomes risk-neutral, meaning that they do not care about risk in their decisions. In the substitution framework, this leads to an infinitely large elasticity of substitution.

c. Effect of rise in the price of contingent claims on demands for \(W_{g}\) and \(W_{h}\)

A rise in the price of contingent claims \(P_b\) will create substitution and income effects on the demands for \(W_g\) and \(W_h\). The substitution effect will cause individuals to buy more of the cheaper good (either \(W_g\) or \(W_h\)) and the income effect will cause some scaling back of consumption in both goods. If the individual's budget remains fixed, choices among these two goods will be affected by these two effects. The demand for \(W_g\) will rise or fall as it depends on the degree of risk aversion exhibited by the individual. The higher the degree of risk aversion, the less likely they are to substitute, and instead are more likely to buy the safer good when contingent claims become more expensive.

d.i. Finding the value of \(R\) consistent with the investment behavior

Given that the person is indifferent between having \(W_o\) and a \(50-50\) chance on \(1.055W_o\) and \(0.955W_o\), the utility function we can use is given by: \[CRRA Utility = \frac{W^{1-R}-1}{1-R}\] So, to find \(R\), we need to set the utilities for both situations equal: \[\frac{(0.5(1.055W_o)^{1-R} + 0.5(0.955W_o)^{1-R})-1}{1-R} = \frac{(W_o)^{1-R}-1}{1-R}\] To find the value of \(R\), we will need to use numerical techniques such as Newton-Raphson or a bisection method.

d.ii. Required average return for accepting a \(50-50\) chance of gaining or losing 10%

With the value of \(R\) found in the previous step, to find the average return required for a \(50-50\) chance of gaining or losing 10%, we need to find the new return value, \(r\), that makes the person indifferent between investing and not investing: \[\frac{(0.5((1+r)W_o)^{1-R} + 0.5((1-r)W_o)^{1-R})-1}{1-R} = \frac{(W_o)^{1-R}-1}{1-R}\] We can solve for \(r\) using numerical methods to find the required average return. This comparison of the risk/reward trade-off illustrates the "equity premium puzzle" mentioned in the exercise.