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Chapter 7: Problem 9

Return to Example \(7.5,\) in which we computed the value of the real option provided by a flexible-fuel car. Continue to assume that the payoff from a fossil-fuel-burning car is \(A_{1}(x)=1-x\). Now assume that the payoff from the biofuel car is higher, \(A_{2}(x)=2 x\). As before, \(x\) is a random variable uniformly distributed between 0 and 1 , capturing the relative availability of biofuels versus fossil fuels on the market over the future lifespan of the car. a. Assume the buyer is risk neutral with von Neumann-Morgenstern utility function \(U(x)=x\). Compute the option value of a flexible-fuel car that allows the buyer to reproduce the payoff from either single-fuel car. b. Repeat the option value calculation for a risk-averse buyer with utility function \(U(x)=\sqrt{x}\) c. Compare your answers with Example \(7.5 .\) Discuss how the increase in the value of the biofuel car affects the option value provided by the flexible- fuel car.

Short Answer

Expert verified
Question: Briefly discuss the effects of the increased value of the biofuel car on the option value of the flexible-fuel car for both risk-neutral and risk-averse buyers. Answer: The increased value of the biofuel car results in higher option values for both risk-neutral and risk-averse buyers. This is because the flexible-fuel car allows buyers to reproduce the payoffs of both single-fuel cars, making it more valuable as the value of the biofuel car increases.

Step by step solution

01

Calculate the expected payoffs for the two individual fuel types

For the fossil fuel car, the payoff is A1(x)=1-x. For the biofuel car, the payoff is A2(x)=2x. Since x is uniformly distributed between 0 and 1, we can calculate the expected payoff for each type of car by integrating each function over the range [0,1] and dividing by the interval length. Expected Payoff Fossil = \(E[A1] = \int_0^1{(1-x)dx} = \left[x-\frac{x^2}{2}\right]_0^1 = 1-\frac{1}{2} = \frac{1}{2}\) Expected Payoff Biofuel = \(E[A2] = \int_₀^1(2x)dx = \left[x^2\right]_0^1 = 1^2 - 0^2 = 1\)
02

Calculate the option value of the flexible-fuel car for a risk-neutral buyer

Since the buyer is risk-neutral, their utility function is U(x)=x. So, we can calculate the option value of the flexible-fuel car as the sum of the expected payoffs of both the fossil fuel car and the biofuel car. Option value (Risk-neutral) = Expected Payoff Fossil + Expected Payoff Biofuel = \(\frac{1}{2} + 1 = \frac{3}{2}\) #Calculating the option values for risk-averse buyer#
03

Calculate the expected utility for the individual fuel types

For the risk-averse buyer, their utility function is U(x)=sqrt(x). We will calculate the expected utility for each type of car by integrating the utility function of each payoff over the range [0,1] and dividing by the interval length. Expected Utility Fossil = \(E[U(A1)] = \int_0^1\sqrt{1-x}dx = \left[-\frac{2}{3}(1-x)^{\frac{3}{2}}\right]_0^1 = \frac{2}{3}\) Expected Utility Biofuel = \(E[U(A2)] = \int_0^1\sqrt{2x}dx = \frac{4}{3}\left[\frac{x^{\frac{3}{2}}}{\frac{3}{2}}\right]_0^1 = \frac{4}{3}\)
04

Calculate the option value of the flexible-fuel car for a risk-averse buyer

Since the buyer is risk-averse, we want to calculate the option value as the sum of the expected utilities for both the fossil fuel car and the biofuel car. Option value (Risk-averse) = Expected Utility Fossil + Expected Utility Biofuel = \(\frac{2}{3} + \frac{4}{3} = 2\) #Comparing with Example 7.5#
05

Discussion of the results

In Example 7.5, the option values for both the risk-neutral buyer and the risk-averse buyer were 1. Here, the option value for the risk-neutral buyer is \(\frac{3}{2}\), which is higher than in the previous example. The option value for the risk-averse buyer is 2, also higher than before. This increase in option values can be explained by the higher payoff of the biofuel car, A2(x) = 2x. Since the biofuel car is more valuable in this example, the flexible-fuel car that allows the buyer to reproduce either single-fuel car's payoffs also becomes more valuable. This demonstrates that the increased value of the biofuel car boosts the option value provided by the flexible-fuel car for both types of buyers.

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Most popular questions from this chapter

For the CRRA utility function (Equation 7.42), 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 before being 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 \(1.055 \mathrm{W}_{0}\) and \(0.955 \mathrm{W}_{0}\) (1) What value of \(R\) is consistent with this behavior? (2) 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 riskreward trade-off illustrates what is called the equity premium puzzle in that risky investments seem actually to 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 ): \(42-71\).

In Equation 7.30 we showed that the amount an individual is willing to pay to avoid a fair gamble \((h)\) is given by \(p=0.5 E\left(h^{2}\right) r(W),\) 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 \(E\left(v^{2}\right) ?\) b. Now consider varying the gamble in part (a) by multiplying each prize by a positive constant \(k\). Let \(h=k v\). What is the value of \(E\left(h^{2}\right) ?\) c. Suppose this person has a logarithmic utility function \(U(W)=\ln W\). What is a general expression for \(r(W) ?\) d. Compute the risk premium ( \(p\) ) for \(k=0.5,1\), and 2 and for \(W=10\) and \(100 .\) What do you conclude by comparing the six values?

The CARA and CRRA utility functions are both members of a more general class of utility functions called harmonic absolute risk aversion (HARA) functions. The general form for this function is \(U(W)=\theta(\mu+W / \gamma)^{1-\gamma}\), where the various parameters obey the following restrictions: \(\bullet$$\gamma \leq 1\) \(\bullet$$\mu+W / \gamma > 0\) \(\bullet$$\theta[(1-\gamma) / \gamma] > 0\) The reasons for the first two restrictions are obvious; the third is required so that \(U^{\prime} > 0\) a. Calculate \(r(W)\) for this function. Show that the reciprocal of this expression is linear in \(W\). This is the origin of the term harmonic in the function's name. b. Show that when \(\mu=0\) and \(\theta=[(1-\gamma) / \gamma]^{\gamma-1},\) this function reduces to the CRRA function given in Chapter 7 (see footnote 17 ). c. Use your result from part (a) to show that if \(\gamma \rightarrow \infty\), then \(r(W)\) is a constant for this function. d. Let the constant found in part (c) be represented by \(A\). Show that the implied form for the utility function in this case is the CARA function given in Equation 7.35 e. Finally, show that a quadratic utility function can be generated from the HARA function simply by setting \(\gamma=-1\) f. Despite the seeming generality of the HARA function, it still exhibits several limitations for the study of behavior in uncertain situations. Describe some of these shortcomings.

Ms. Fogg is planning an around-the-world trip on which she plans to spend \(\$ 10,000\). The utility from the trip is a function of how much she actually spends on it ( \(Y\) ), given by \\[ U(Y)=\ln Y \\] a. If there is a 25 percent probability that Ms. Fogg will lose \(\$ 1,000\) of her cash on the trip, what is the trip's expected utility? b. Suppose that Ms. Fogg can buy insurance against losing the \(\$ 1,000\) (say, by purchasing traveler's checks) at an "actuarially fair" premium of \(\$ 250 .\) Show that her expected utility is higher if she purchases this insurance than if she faces the chance of losing the \(\$ 1,000\) without insurance. c. What is the maximum amount that Ms. Fogg would be willing to pay to insure her \(\$ 1,000 ?\)

An individual purchases a dozen eggs and must take them home. Although making trips home is costless, there is a 50 percent chance that all the eggs carried on any one trip will be broken during the trip. The individual considers two strategies: (1) take all 12 eggs in one trip; or (2) take two trips with 6 eggs in each trip. a. List the possible outcomes of each strategy and the probabilities of these outcomes. Show that, on average, 6 eggs will remain unbroken after the trip home under either strategy. b. Develop a graph to show the utility obtainable under each strategy. Which strategy will be preferable? c. Could utility be improved further by taking more than two trips? How would this possibility be affected if additional trips were costly?
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