Econ 119-Final Exam solutions. Friday, December 20, 2013. Problem 1: (50 points) True/False, Multiple Choice, and Short Answer. Answer all of the following short-answer, multiple-choice, and true/false questions. Your answers must fit into the space provided. Make sure to provide clear and thorough explanations of your answers. A correct T/F or letter choice without a correct explanation will not receive credit. (a) (15 points) Short Answer: Four behavioral-economic phenomena are listed below. Answer the following two questions for each one: 1) Explain what the phenomenon is. 2) Can it sometimes cause people to pay too much for certain goods? Explain why or why not. If the answer is yes, give an example. A: Naive present-biased preferences. B: Sophisticated present-biased preferences. C: Projection bias. D: Expectations-based reference points. Present-biased preferences is the phenomenon of discounting all future costs and benefits relative to those in the present. Naivete refers to the belief that one will not engage in such discounting in the future (i.e. that one will not have present-biased preferences in the future) while sophistication refers to the (correct) belief that one will have the same present-biased preferences in the future that one has currently. Present-biased preferences can cause people to be willing to pay too much for any good that has immediate benefits and delayed costs, such as cigarettes, because they do not take the long-term costs into consideration adequately, and therefore experience each unit of the good to be worth more to them than it actually is in the long run. If there is no way for a sophisticate to commit themselves to optimal behavior in the future (by buying a commitment contract, for example) then this kind of over-consumption will happen for both the naif and the sophisticate. Additionally, a naif may pay too much for a commitment device that pays them an incentive every time they do something that has short term costs and long term benefits, because they think that they will respond to the incentive more than they actually will. The most common example of this is a gym contract that eliminates the per-visit fee for gym attendance. In advance the naif thinks that they will not have a self-control problem in the future, so they think that they will go to the gym much more than they actually will at zero payment. As a result, they are willing to pay much more for the membership contract than they should have. Projection bias arises when preferences are state dependent, and is the phenomenon of believing that one's future preferences, under a different state than one is currently in, will be the same as one's current preferences (in the case of total projection bias), or that they will be closer to one's current preferences than they actually will (in the case of partial projection bias). Projection bias can cause people to be willing to pay too much for goods that they have a strong desire for in their current state, but that they will actually consume in a different state, in which they have less desire for the good. For example, if someone buys food when they are hungry, but will eat the food when they are not as hungry, then projection bias may cause them to be willing to pay an amount equal to how much they value the food when they are hungry, even though they will not actually get that much value out of it when they are less hungry. 1 Expectations-based reference point is the phenomenon of having reference-dependent preferences with the reference point determined by one's expectations of future consumption (or future wealth, or future health condition, etc). If a person with expectations-based reference-dependent preferences plans to buy a good at a certain price, if they have rational expectations it must mean that they will, in fact buy the good at that price, and that doing so will give them greater utility than if they did not buy at that price. Loss aversion typically results in the individual feeling a loss if they either buy at a price above their consumption value, or choose not to buy at a price below their consumption value, so they will typically not be willing to pay too much for a good just because of expectations-based reference points. (If there is uncertainty about what price they will have to pay, it is possible that they will wind up paying more than their consumption for the good if the price is on the higher end of the range of possible prices. However, we did not learn about that.) 2 (b) (10 points) True or False: If a policy is designed to make people better off according to their own true preferences, without restricting their choices, it must be a Pareto improvement. True: If a policy is correctly designed in such a way that it actually does make behavioral-types better off according to their own true preferences, and if it does not restrict the choices of rational types, then it will necessarily be a Pareto improvement because rational types will continue to choose what is best for them, and behavioral types will be made better off. An answer of false could be considered correct if we allow for the possibility that a policy which is designed to make people better off according to their own preferences might actually fail to do so, or might make some people better off and some people worse off. An example would be a policy that uses default options to \"nudge\" people to do one thing instead of another. Behavioral types will stick with the default, which may make some of them better off, but may make others worse off. Unless the policy maker knows for sure what will make people better off, a libertarian paternalistic policy may unintentionally make some behavioral types worse off. 3 (c) (10 points) True or False: In the stop-and-frisk policy in New York City, police officers believe that the probability that a black person is carrying an illegal gun is greater than the probability for a white person, even though the opposite is true. Which of the following phenomena can explain this situation? [If you believe that either A or B cannot explain the situation, you must explain why not.] A: Base-rate neglect. B: Availability bias. C: Both. D: Neither. Base-rate neglect cannot explain the stop-and-frisk situation because for a given signal, base-rate neglect will either inflate or deflate the probability estimate for any hypothesis, given that signal. If a hypothesis has a disproportionately low base rate (smaller than 1/N where N is the total number of hypotheses), base-rate neglect will inflate the estimate of the probability of that hypothesis, given any signal, and if a hypothesis has a disproportionately high base rate, base-rate neglect will deflate the estimate of the probability of that hypothesis, given any signal. Thus, base-rate neglect cases estimates of the probability of a person of any race carrying an illegal gun to go up, and thus it cannot cause the order of the probabilities to be reversed, as in the stop-and-frisk situation. Availability bias can explain the stop-and-frisk situation because if the hypothesis that comes to mind most easily in response to a signal is not the one that is most likely, the order of probabilities may be reversed in the minds of police officers. In other words, if in reality whites are disproportionately over-represented among people who carry illegal guns, but the hypothesis \"black\" comes to mind most easily when the signal \"gun\" appears (perhaps because depictions and reports in the media of people carrying illegal guns are disproportionately of African Americans), then availability bias would make the numerator of Bayes' rule bigger when computing P (G|B) and smaller when computing P (G|W ), thus causing police officers to think that the first is bigger than the second when in fact the second is bigger than the first. 4 (d) (15 points) [Harder] The assumption of rationality is the \"glue\" that holds the positive and normative dimensions of standard microeconomic models together. Behavioral economics improves the positive power of the models, but causes problems with the normative dimension. Select one of the four options below and explain how it is an example of, or an explanation of, the statement above. There is more than one correct option, but not all of the options are correct. A: Willingness To Pay (WTP) is a predictor of how people will behave, regardless of whether they are rational, but it may or may not be an indicator of people's true preferences, depending on whether they are rational. Willingness to pay is an observable measure of what people choose to do under certain conditions (their state, the time period, the prices they face, etc). If we observe predictable patterns of choices under specific circumstances then we can typically write down a model that explains those observations, and that model should also be able to predict what people will do in similar situations in the future. If a model that relaxes the assumption of rationality (a behavioral model) does a better job of explaining, and thus predicting, people's behavior, then it has improved the positive (predictive) power of the model. But if people's preferences are irrational in some way, then their willingness to pay may not reflect their true preferences, as we saw in problem 1(a), and in lots of other examples throughout the class. In that case, we may have no observable measure of what their true preferences are, and it may be impossible to place a normative value on the choices they make, which means that it may be impossible to judge whether one outcome or policy is better or worse than another. B: If people are not able to predict their own preferences and/or choices, it becomes impossible to model their behavior correctly, even if we know their true preferences. If people cannot predict their own preferences or choices, we typically can still model (i.e. predict) their behavior, provided we can observe their behavior, and since failure to predict preferences or choices is a form of irrationality, it is highly likely that a behavioral model that relaxes the assumption of rational expectations will do a better job of explaining and predicting behavior than the standard model. C: If preferences change depending on what state a person is in, or what time period they are in, it may be possible to predict their behavior, but we may not know which set of preferences to use for normative purposes. If preferences change depending on what state a person is in, it may be an example of irrationality, such as when gain/loss reference points change on the basis of meaningless signals or based on how a question is framed, and also, if people mispredict how their preferences will change from one state to another, it may lead to irrationality, such as when people buy more food than they actually want because of projection bias with respect to hunger. The same applies if intertemporal preferences (discounting) changes from one time period to another, or if people fail to predict how their discounting will change from one period to another. But in all of these cases, we can still predict people's behaviors if we can write down a model that accurately captures their psychological biases and mispredictions, by relaxing assumptions of rationality. But when preferences change in ways that reflect irrationality we often face situations in which people express more than one level of willingness to pay for the same good, and it may be impossible to know which level captures their true value for the good, which in turn means we do not have a valid measure for valuing economic outcomes or policies. 5 D: Irrationality often causes people to fail to maximize their own true preferences, so there is often no way to know what they value most, even if we know what they will choose. Under the assumption of rationality, we always know what they value most and what they will choose. This statement is virtually self-explanatory. WTP is only a valid measure of how much people value goods if they are consistently maximizing their true preferences, so if they aren't maximizing their true preferences, they may express WTP for a good that is above or below their true valuation of the good. In that case, WTP is no longer a valid measure of true value, and thus we have no way to make normative valuations of economic outcomes or policies. And this is true, even in cases where we can, by relaxing assumptions of rationality, write down a model that correctly predicts behavior. 6 Problem 2. (50 points) Suppose that 10% of mutual fund managers are of high ability (H) and 90% are of low ability (L). Suppose that in any given quarter, high-ability managers have a 40% chance of having a good return (G) and a 60% chance of having a bad return (B), and that a low-ability manager has a 20% chance of having a good return and an 80% chance of having a bad return. (a) (10 points) Let's begin by considering rational probability inference. i. What is the formula used to compute the true probability of a hypothesis, h, given a signal, i? Write out the formula in both its short form and its long form. ii. Draw a diagram that illustrates how this formula works, and explain it in words. Make sure to use the terms \"hypothesis\" and \"signal\" correctly. (There is more than one diagram that can be used.) The formula is called \"Bayes' rule\" and its two forms are as follows: P (h|i) = P (h|i) = P (h, i) P (i) P (h) P (i|h) P (h) P (i|h) + P (not h) P (i|not h)| The diagram below illustrates how the formula works. Area 1 is the numerator of Bayes' rule and Area 1+2 is the denominator. Basically 1+2 defines all the cases in which any fund manager would have a good return, and 1 is the subset of those cases that are high-ability managers. 7 (b) (10 points) Now let's apply rational inference to the mutual fund situation. i. What is the true probability of a fund manager being high ability if they have one quarter of good returns? ii. What is the true probability of a fund manager being low ability if they have one quarter of bad returns? P (H|G) = P (L|B) = .1.4 .1.4+.9.2 = 0.1818 .9.8 .9.8+.1.6 = 0.9231 8 (c) (15 points) Now let's consider what happens under base-rate neglect. i. Write down the formula used to compute the perceived probability of a hypothesis, h, given a signal, i, under base-rate neglect, and explain how this formula captures the concept of base-rate neglect. ii. What are the probabilities from part (b) under base-rate neglect? iii. Compare the rational probabilities to the probabilities under base-rate neglect. Why does base-rate neglect cause your answers to change in the direction(s) that they change? The formula under base-rate neglect is as follows: Pbr (h|i) = P (i|h) P (i|h) + P (i|not h)| The formula implicitly sets all base rates equal, and explicitly eliminates them from the Bayes' rule formula. What this means is that individuals will their inferences on how likely the signal is given the hypothesis, ignoring the fact that some hypotheses are much more likely than others. This will inflate the inferred probability of an unlikely hypothesis and deflate the inferred probability of a likely hypothesis. The probabilities under base-rate neglect are as follows: .4 = 0.6667 Pbr (H|G) = .4+.2 .8 Pbr (L|B) = .8+.6 = 0.5714 The first probability has gone up under base-rate neglect, and the second has gone down. The reason is that in the first case, the base rate for H is below .5 so that 9 base-rate neglect makes the individual think that high-ability is more likely given any signal, which the base rate for L is above .5 so that base-rate neglect makes the individual think that low-ability is less likely given any signal. 10 (d) (15 points) Suppose a fund manager has one quarter of good returns. Base-rate neglect will cause an individual to overestimate the probability that the manager is high ability (H) and to underestimate the probability that the manager is low ability (L). Prove this mathematically and explain in words why it is true. .9.2 Under Bayes' rule we get P (L|G) = .9.2+.1.4 = 0.8181, which, not coincidentally, .2 is 1 P (H|G). Under base rate neglect it is Pbr (L|G) = .2+.4 = 0.3333, which, again not coincidentally, is 1 Pbr (H|G). We've already seen that base-rate neglect makes the probability of a good return coming from a high-ability manager go up, and now we know that it makes the probability of a good return coming from a low-ability manager go down. As explained above, this is because H is less than 50% likely while L is more than 50% likely, so base-rate neglect makes people think that H is more likely given any signal and L is less likely given any signal. 11 Problem 3. (80 points) Boris is a motorcyclist who must cross the Bay Bridge from Oakland to San Francisco every day. Each day Boris decides whether to wear a helmet or not. If he wears a helmet he gets zero utility from crossing the bridge, but if he crashes the helmet will protect him perfectly. If he does not wear a helmet, he gets an immediate benefit, b, which is different each day. However, if he crashes without a helmet he will be disabled for the rest of his life. The expected cost of future disability each time he crosses the bridge is c. Boris has , present-biased preferences with < 1, and = 1. He discounts all costs and benefits more than one day in the future by . Assume that c is high enough so that some days it is rational to wear a helmet, but low enough so that some days it is rational to not wear a helmet. Finally, as usual, assume that Boris has utility for money, y, represented by u(y) = y. (a) (5 points) Compute the marginal internality caused by Boris's present bias. (b) (5 points) What does the marginal internality represent? (c) (5 points) Why does the marginal internality not depend on whether or not Boris is a sophisticate? 12 One day the government decides to paternalistically help Boris overcome the internality caused by his present-biased preferences. The first policy they use is to offer Boris an incentive payment, p, every time he wears a helmet. If he is wearing a helmet they send him the incentive payment electronically. (b) (10 points) Suppose the payment is sent immediately, so that Boris receives it in the present time period. What is the optimal payment? Why does this payment overcome the internality? 13 (c) (10 points) Suppose that the payment is sent one week in the future instead of immediately. What is the optimal payment now? Why is it different from the immediate case above? 14 Now assume that Boris also has reference-dependent preferences with respect to money (but not with respect to helmet-wearing utility or future health utility) so that his money utility is now represented by u(y) = v(y r), where v(x) = x for x 0 and v(x) = 2x for x 0, and where r is his reference point for money, which is determined by his status-quo wealth (i.e. his current wealth, not his expectations of future wealth). Assume his reference point adapts to his new wealth each day. (d) (10 points) What is the optimal incentive payment now, assuming that it is paid immediately, as in part (b)? 15 (e) (10 points) Suppose the government decides to charge Boris a fee, f , each time he doesn't wear a helmet, instead of giving him a payment every time he does. What is the optimal fee, assuming that the fee is charged immediately? (Continue to assume Boris has reference dependence over money, as above.) 16 (f) (10 points) Which of the following policies satisfies the criteria of Libertarian Paternalism? Explain your answer thoroughly. i: The incentive payment from (d). ii: The fee from (e). 17 Finally, let's do some benefit-cost analysis. We'll go back to Boris's original preferences, without loss aversion, but now we'll assume that there are many motorcyclists who drive across the bridge, some of whom have preferences identical to Boris's (aka behavioral types), and some of whom have the same benefits and costs, but have = 1 (aka rational types). Suppose that in the average month (30 days) there are ten days when b = 4 and 20 days when b = 2. Let c = 3. (g) (15 points) [Harder] If = 12 , what is the proportion of behavioral types above which the optimal incentive from (b) is a Hicks-Kaldor improvement? (Note: Remember that the incentive payment itself is just a transfer from the government to motorcycle riders, so it is neither a cost nor a benefit to society as a whole.) [A diagram may help you, though it is not required. But be warned, if you use a diagram, it will not look exactly the same as the diagrams you have seen before. If you slow down and think carefully, one step at a time, you can do this problem.] 18