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Chapter 7: Problem 15

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?

Short Answer

Expert verified
Answer: If Standard Stan is risk-averse, he would choose Gamble B in Scenario 1 and Gamble D in Scenario 2, as both of these options have less uncertainty.

Step by step solution

01

Determine the expected utility of each gamble in both scenarios for Standard Stan

To determine the expected utility of each gamble, we need to calculate the expected value (EV) of each gamble. Here's how to find the EV for each gamble: - Gamble A: EV_A = (0.5 * \(1,000) + (0.5 * \)0) = $500 - Gamble B: EV_B = $500 - Gamble C: EV_C = (0.5 * -\(1,000) + (0.5 * \)0) = -$500 - Gamble D: EV_D = -$500
02

Analyze risk-neutral choices in both scenarios

If Standard Stan is risk-neutral, he will make choices based on the expected value (EV) of each gamble. Given that the expected values of A and B, as well as C and D, are equal, Standard Stan would be indifferent between Gambles A and B in Scenario 1 and between Gambles C and D in Scenario 2.
03

Analyze risk-averse choices in both scenarios

If Standard Stan is risk-averse, he will prefer the less risky option. In Scenario 1, Gamble B is less risky since there is no uncertainty. So Stan will choose B. Similarly, in Scenario 2, Gamble D is less risky, as it involves a certain outcome, Stan will choose D.
04

Compare the experimental results with expected utility theory

The experimental results showed that 16% of subjects chose A in Scenario 1 and 68% chose C in Scenario 2. These results are hard to reconcile with expected utility theory because they demonstrate a clear preference for Gamble A (greater risk, less certain gains) in Scenario 1 and Gamble C (greater risk, less certain loss) in Scenario 2. These results do not correspond to the choices predicted by expected utility theory for risk-neutral or risk-averse individuals.
05

Determine Prospect Pete's choices according to prospect theory

According to prospect theory, people are risk-averse over gains but risk-seeking over losses. In Scenario 1, Prospect Pete would choose Gamble B, as he is risk-averse when it comes to potential gains. In Scenario 2, he would choose Gamble C, as he is risk-seeking when it comes to losses.
06

Draw utility curves for Prospect Pete in both scenarios

According to prospect theory, the utility curve for gains (first scenario) exhibits a concave shape indicating risk aversion. In the second scenario, the utility curve for losses exhibits a convex shape indicating risk-seeking behavior. To accurately depict Pete's choices, one would need to draw two different utility curves for both scenarios—concave for Scenario 1 and convex for Scenario 2. This is a significant difference from the utility curves used for Standard Stan, which are either consistently concave for risk-averse individuals or straight lines for risk-neutral individuals.

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Most popular questions from this chapter

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 six eggs in each trip. a. List the possible outcomes of each strategy and the probabilities of these outcomes. Show that, on average, six 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?

Investment in risky assets can be examined in the state-preference framework by assuming that \(W_{0}\) dollars invested in an asset with a certain return \(r\) will yield \(W_{0}(1+r)\) in both states of the world, whereas investment in a risky asset will yield \(W_{0}\left(l+r_{g}\right)\) in good times and \(W_{0}\left(l+r_{b}\right)\) in bad times (where \(\left.r_{g}>r>r_{b}\right)\). a. Graph the outcomes from the two investments. b. Show how a "mixed portfolio" containing both risk-free and risky assets could be illustrated in your graph. How would you show the fraction of wealth invested in the risky asset? c. Show how individuals' attitudes toward risk will determine the mix of risk- free and risky assets they will hold. In what case would a person hold no risky assets? d. If an individual's utility takes the constant relative risk aversion form (Equation 7.42 ), explain why this person will not change the fraction of risky assets held as his or her wealth increases. \(^{26}\)

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.

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) \\]

A farmer believes there is a \(50-50\) chance that the next growing season will be abnormally rainy. His expected utility function has the form \\[ E[U(Y)]=\frac{1}{2} \ln Y_{N R}+\frac{1}{2} \ln Y_{R} \\] where \(Y_{N R}\) and \(Y_{R}\) represent the farmer's income in the states of "normal rain" and "rainy," respectively. a. Suppose the farmer must choose between two crops that promise the following income prospects: \\[ \begin{array}{lcc} \text { Crop } & Y_{N R} & Y_{R} \\ \hline \text { Wheat } & \$ 28,000 & \$ 10,000 \\ \text { Corn } & \$ 19,000 & \$ 15,000 \end{array} \\] Which of the crops will he plant? b. Suppose the farmer can plant half his field with each crop. Would he choose to do so? Explain your result. c. What mix of wheat and corn would provide maximum expected utility to this farmer? d. Would wheat crop insurance-which is available to farmers who grow only wheat and which costs \(\$ 4,000\) and pays off \(\$ 8,000\) in the event of a rainy growing season-cause this farmer to change what he plants?
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