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Chapter 7: Problem 10

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 \(O_{1}(x)=1-x\) Now assume that the payoff from the biofuel car is higher, \(\mathrm{O}_{2}(x)=2 x\). 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 NeumannMorgenstern 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
Answer: The increase in the value of the biofuel car raises the option value provided by the flexible-fuel car for both risk-neutral and risk-averse buyers. For the risk-neutral buyer, the option value increases to 1, as the higher value of the biofuel car makes it the dominant choice. For the risk-averse buyer, the option value rises to 2/3, making the flexible-fuel car more beneficial in providing the flexibility to choose between the two car types.

Step by step solution

01

a. Risk Neutral Buyer

We are given a risk-neutral buyer with the utility function \(U(x)=x\). We need to find the option value of a flexible-fuel car that allows the buyer to replicate the payoff from either single-fuel car. First, let's calculate the expected payoffs of each type of car for a risk-neutral buyer. Because x follows a uniform distribution between 0 and 1, we can calculate the expected payoffs by integrating the payoff functions over this interval and then dividing by the range of the interval (1 - 0 = 1): Expected payoff of the fossil-fuel car (O1): \(\int_0^1(1-x)dx\) Expected payoff of the biofuel car (O2): \(\int_0^1(2x)dx\) Now let's compute these integrals: For O1: \(\int_0^1(1-x)dx = [x-\frac{1}{2}x^2]_0^1 = 1 - \frac{1}{2} = \frac{1}{2}\) For O2: \(\int_0^1(2x)dx = [x^2]_0^1 = 1 - 0 = 1\) Since the buyer is risk-neutral, he will choose the car with the higher payoff. In this case, it's the biofuel car with an expected payoff of \(1\). Therefore, the option value of the flexible-fuel car for the risk-neutral buyer is \(1\).
02

b. Risk-Averse Buyer

Now, let's consider a risk-averse buyer with the utility function \(U(x) = \sqrt{x}\). We need to find the option value of the flexible-fuel car for this buyer. First, we'll calculate the utility of each type of car: Utility of the fossil-fuel car (U1): \(\int_0^1 \sqrt{1-x} dx\) Utility of the biofuel car (U2): \(\int_0^1 \sqrt{2x} dx\) Now let's compute these integrals: For U1: \(\int_0^1 \sqrt{1-x} dx = 2[1-\frac{2}{3}(1-x)^{\frac{3}{2}}]_0^1 = 2[(1-\frac{2}{3}) - (1 - 0)] = \frac{2}{3}\) For U2: \(\int_0^1 \sqrt{2x} dx = \frac{4}{3}[x^{\frac{3}{2}}]_0^1 = \frac{4}{3}(1^{\frac{3}{2}} - 0^{\frac{3}{2}}) = \frac{4}{3}\) Since the buyer is risk-averse, he will choose the car that provides the higher utility. In this case, it's the biofuel car with a utility of \(\frac{4}{3}\). The option value of the flexible-fuel car for the risk-averse buyer is the difference between this utility and the utility of the less desirable choice (the fossil-fuel car), which is \(\frac{4}{3} - \frac{2}{3} = \frac{2}{3}\).
03

c. Comparing with Example 7.5

In Example 7.5, the payoffs from the fossil-fuel car (O1) and the biofuel car (O2) were the same, and the option value provided by the flexible-fuel car was the same for both risk-neutral and risk-averse buyers. However, in this exercise, the payoff from the biofuel car is higher (O2 = 2x), which changes the option values. For the risk-neutral buyer (with utility function U(x) = x), the higher value of the biofuel car makes it a strictly dominant choice, as its expected payoff is higher. This results in the option value of the flexible-fuel car being the same as the value of the biofuel car (1). For the risk-averse buyer (with utility function U(x) = √x), the higher payoff of the biofuel car results in a higher utility of the biofuel car relative to the fossil-fuel car, leading to an option value of \(\frac{2}{3}\) for the flexible-fuel car, which is higher than the value found in Example 7.5. In conclusion, the increase in the value of the biofuel car increases the option value provided by the flexible-fuel car, as it makes the choice between the two types of cars less ambiguous and provides a greater benefit to buyers who have the flexibility to choose between the two.

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Most popular questions from this chapter

Two pioneers of the field of behavioral economics, Daniel Kahneman (winner of the Nobel Prize in economics and author of bestselling book Thinking Fast and Slow) and Amos Tversky (deceased before the prize was awarded), conducted an experiment in which they presented different groups of subjects with one of the following two scenarios: \(\bullet\) Scenario 1: In addition to \(\$ 1,000\) up front, the subject must choose between two gambles. Gamble \(A\) offers an even chance of winning \(\$ 1,000\) or nothing. Gamble \(B\) provides \(\$ 500\) with certainty. \(\bullet\) Scenario 2: In addition to \(\$ 2,000\) given up front, the subject must choose between two gambles. Gamble \(C\) offers an even chance of losing \(\$ 1,000\) or nothing. Gamble \(D\) results in the loss of \(\$ 500\) with certainty. a. Suppose Standard Stan makes choices under uncertainty according to expected utility theory. If Stan is risk neutral, what choice would he make in each scenario? b. What choice would Stan make if he is risk averse? c. Kahneman and Tversky found 16 percent of subjects chose \(A\) in the first scenario and 68 percent chose \(C\) in the second scenario. Based on your preceding answers, explain why these findings are hard to reconcile with expected utility theory. d. Kahneman and Tversky proposed an alternative to expected utility theory, called prospect theory, to explain the experimental results. The theory is that people's current income level functions as an "anchor point" for them. They are risk averse over gains beyond this point but sensitive to small losses below this point. This sensitivity to small losses is the opposite of risk aversion: A risk-averse person suffers disproportionately more from a large than a small loss. (1) Prospect Pete makes choices under uncertainty according to prospect theory. What choices would he make in Kahneman and Tversky's experiment? Explain. (2) Draw a schematic diagram of a utility curve over money for Prospect Pete in the first scenario. Draw a utility curve for him in the second scenario. Can the same curve suffice for both scenarios, or must it shift? How do Pete's utility curves differ from the ones we are used to drawing for people like Standard Stan?

In deciding to park in an illegal place, any individual knows that the probability of getting a ticket is \(p\) and that the fine for receiving the ticket is \(f\). Suppose that all individuals are risk averse (i.e., \(U^{\prime \prime}(W)<0\), where \(W\) is the individual's wealth). Will a proportional increase in the probability of being caught or a proportional increase in the fine be a more effective deterrent to illegal parking? Hint: Use the Taylor series approx- \\[ \text { imation } U(W-f)=U(W)-f U^{\prime}(W)+\left(f^{2} / 2\right) U^{\prime \prime}(W) \\]

Maria has \(\$ 1\) she can invest in two assets, \(A\) and \(B\). A dollar invested in \(A\) has a \(50-50\) chance of returning 16 or nothing and in \(B\) has a \(50-50\) chance of returning 9 or nothing. Maria's utility over wealth is given by the function \(U(W)=\sqrt[0]{W}\) a. Suppose the assets' returns are independent. (1) Despite the fact that \(A\) has a much higher expected return than \(B\), show that Maria would prefer to invest half of her money in \(B\) rather than investing everything in \(A\) (2) Let \(a\) be the fraction of the dollar she invests in \(A\) What value would Maria choose if she could pick any \(a\) between 0 and \(1 ?\) Hint: Write down her expected utility as a function of \(a\) and then either graph this function and look for the peak or compute this function over the grid of values \(a=0,0.1,0.2,\) etc. b. Now suppose the assets' returns are perfectly negatively correlated: When \(A\) has a positive return, \(B\) returns nothing and vice versa. (1) Show that Maria is better off investing half her money in each asset now than when the assets' returns were independent. (2) If she can choose how much to invest in each, show that she would choose to invest a greater fraction in \(B\) than when assets' returns were independent.

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 \(0.955 W_{0}\) (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 risk-reward 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.

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.
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