Economics 1030: Problem Set 2

Transcription

1 Economics 1030: Problem Set 2 Due in class on Thursday, February 26 th Note: - No late problem sets will be accepted. - You may collaborate on problem sets but you must acknowledge the identities of your collaborators. Problem sets may be discussed but they must be written up independently. - Be sure to show your work. - Please put your TF s name and your section time on the top of your problem set. Problem 1: Representative Heuristic A manifestation of the representativeness heuristic is that people believe in the Law of Small Numbers. Part (a): Define the Law of Small Numbers, and explain why it is an error. Describe three pieces of evidence from the readings for the Law of Small Numbers. The Law of Small Numbers is the (fallacious) idea that small samples will resemble the population from which they are drawn. This is true for large samples the Law of Large Numbers is a mathematical fact but it need not be true for small samples. In fact, for a small number of independent draws, it is fairly likely to get a sample that looks very much unlike the population from which they are drawn. There are many pieces of evidence from the readings for the Law of Small Numbers. For example, most psychologists believe it is more likely than it is to replicate successfully the results of a small experiment. The gamblers fallacy the (false) expectation that bad luck will be followed by good luck, and vice-versa is another example. Also, the fact that people believe the coin toss sequence HTHTTH is more likely than HHHTTT (even though they are actually equally likely). Part (b): Suppose that basketball players are, during any given game, in one of three states: Hot (they make 75% of their shots), Normal (they make 50% of their shots), or Cold (they make only 25% of their shots). Suppose Paul Pierce is Hot. What is the probability that he will make 3 baskets in a row? What if he is Normal? Cold? If you have no idea what state he ll be in before the game (that is, each state is equally 1

2 likely), what would you believe about the likelihood that he is Hot after he makes his first 3 baskets in a row? If Paul Pierce is Hot, the probability that he ll make 3 baskets in a row is (0.75) 3 = If Normal, the probability that he ll make 3 baskets in a row is (0.50) 3 = If Cold, the probability that he ll make 3 baskets in a row is (0.25) 3 = Since Pr(Hot) = Pr(Normal) = Pr(Cold) = 1/3, we can calculate by Bayes Rule: Pr (Hot 3 in a row) = = Pr (3 Hot) Pr (Hot) Pr (3 Hot) Pr (Hot) + Pr (3 Normal) Pr (Normal) + Pr (3 Cold) Pr (Cold) (0.422) (1/3) 1/3 [ ] = So after Paul makes his first 3 baskets in a row, you would believe that he is 75% likely to be Hot. Part (c): One of the fans at this game believes in the Law of Small Numbers. She has the wrong model of how likely Paul Pierce is to make a basket. Here s how her model works. The fan imagines that there is a deck of 4 cards. When Paul is Hot, 3 of these cards say hit on them, and only 1 says miss. Every time Paul takes a shot, one of these cards is drawn randomly without replacement from the deck, and the outcome is whatever the card says. Therefore, when Paul is Hot, he always makes 3 out of every 4 shots he takes. (When the deck is used up, the 4 cards are replaced, the deck is shuffled, and the process begins again but that isn t important for this problem.) Similarly, when Paul is Normal or Cold, the outcome of every shot is determined by the draw of a card without replacement from a deck of 4 cards. When Paul is Normal, the deck has 2 hit cards and 2 miss cards. When Paul is Cold, the deck has 1 hit card and 3 miss cards. Explain how her model corresponds to the Law of Small Numbers. If Paul Pierce is Hot, what does the fan believe is the probability that Paul will make his first basket? After Paul makes his first basket, what does the fan believe about the probability that Paul will make his next basket? Explain why it is lower than the fan s belief about the probability that Paul will make his first basket. If the fan has no idea what state Paul will be in before the game (that is, each state is equally likely), what would she believe about the likelihood that he is Hot after he makes his first 3 baskets in a row? Explain intuitively why the fan s beliefs differ from the normatively correct probability that you calculated in Part (b). In the fans model, short sequences of outcomes will resemble the population from which they are drawn. That is exactly the idea behind the Law of Small Numbers. If Paul is Hot, the probability of the first basket is 3/4. There are then three cards left in the deck, 2 hits and 1 miss. The probability of another hit is therefore 2/3. This probability is smaller because the fan expects 1 miss out of every 4 shots. The fact that Paul made a shot means that a miss is now more likely than before. (This is the Gamblers Fallacy.) 2

3 After Paul makes 2 baskets, there are 1 hit and 1 miss left in the deck. So the chance of the third basket in a row is 1/2. Hence, the fan believes the probability that Paul will make 3 baskets in a row is (3/4) (2/3) (1/2)=0.25. If Paul is Normal or Cold, it is impossible for him to make 3 baskets in a row because there arent enough hit cards in the deck. So the probabilities are 0. As a result, after Paul makes 3 baskets in a row, the fan with conclude with certainty that Paul is Hot. To see this with Bayes Rule, P (Hot 3 in a row) = (0.25) (1/3) (1/3) [ ] = 1 Intuitively, because of the Law of Small Numbers, the fan believes a string of 3 baskets in a row is just not possible unless Paul is Hot. She therefore over-infers the likelihood that he is Hot (even though she believes the string of 3 baskets in a row is less likely than it really is when Paul is Hot). Part (d): Suppose that, in reality, there is no such thing as being Hot or Cold. Paul Pierce is, in fact, always Normal. Over many games, with what frequencies will Paul score 0, 1, 2, and 3 baskets in his first 3 attempts? Suppose the fan attends many games and observes these frequencies. Explain why the fan would not believe you if you tried to convince her that there is no such thing as being Hot or Cold. Part (e): In this example, the fan s misunderstanding of probability leads her to believe (falsely) in hot hands and cold hands. Something similar may be going on in the mutual fund industry. Even if mutual fund returns are almost entirely due to luck, there will be some mutual funds that have done exceptionally well and others that have done exceptionally poorly in recent years due entirely to chance. Explain how the Law of Small Numbers would lead some investors to conclude that mutual fund managers differ widely in skill. Consider the possible sequences of hits and misses for Paul Pierce s first three attempts. They are HHH, HHM, HMH, MHH, HMM, MHM, MMH, and MMM. 1/8 of the time Paul Pierce makes no baskets, 3/8 of the time Paul Pierce makes 1 basket, 3/8 of the time Paul Pierce makes 2 baskets, and 1/8 of the time Paul Pierce makes 3 baskets. 12.5% of the time, the fan will observe Paul make 3 baskets in a row. She won t believe thats possible unless he s Hot during those games! Also 12.5% of the time, the fan will observe Paul make 0 baskets in a row. She won t believe thats possible unless he s Cold during those games. The fan will therefore insist that Paul is sometimes Hot and sometimes Cold. In the mutual fund industry, investors will expect chance to offset short strings of good performance with strings of bad performance. Therefore, investors will find it hard to believe that funds could do exceptionally well or exceptionally poorly over the course of several years due solely to chance. They will find it much more plausible that funds that do exceptionally well are run by talented managers and that funds that do exceptionally poorly are run by untalented managers. Investors will conclude that managers differ more in skill than they actually do. 3

4 Problem 2: Prospect Theory and Investments Mr. P will retire in 2 years. All of his savings are invested in a mutual fund that he didn t choose. Every year, the value of his portfolio increases by $G with probability p and decreases by $B with probability 1 p. Assume that G > B and that pg (1 p) B > 0. Mr. P is an anxious guy and he isn t sure whether he should check his portfolio at the end of the first year and at the end of the second year or only at the end of the second year when he plans to move to Florida. Mr. P is a Prospect Theory, expected-value maximizer. His value function is defined over gains and losses relative to a reference point 0: x x 0 v (x) = αx x < 0 and his probability weighting function is π (p) = p. where α > 1 Throughout this problem, assume that there is no discounting. That is, from today s point of view, a unit of utility at the end of year 1 has the same value as a unit of utility at the end of year 2. Part (a): For this part, assume the following: 1. Mr. P experiences utility every time he checks his portfolio. He experiences his gains or losses whenever he checks and thus, if he checks his portfolio at the end of year 1, the utility that he will experience at the end of year 2 will depend on the gains or losses experienced between years 1 and Mr. P cannot change his portfolio at the end of year 1 even if he decides to check its performance. Given the assumptions above, will Mr. P prefer to check the performance of his portfolio at the end of both year 1 and year 2 or only at the end of year 2? Explain the intuition. Note: If you can t show the answer for the general case, show what happens if p = 1 4, G = 6, 000, B = 1, 000, and α = 3. You will still receive full credit if you use these parameter values. Answer (a): Given that Mr. P changes his reference point when he checks his portfolio, if he chooses to check it at the end of both year 1 and year 2, his expected value is: E (V both years ) = E (V 0 to 1 ) + E (V 1 to 2 ) = 2 [pg (1 p) αb] = 2 [ ] = 1500 If he decides to check only at the end of year 2, his expected value is: E (V 0 to 2 ) = p 2 2G + 2p (1 p) [G B] (1 p) 2 (α2b) [ 1 = ] = ( ) = 750 > 1, 500 4

5 Then, he will decide to check only after year 2 if and only if: 2 [pg (1 p) αb] < p 2 2G + 2p (1 p) [G B] (1 p) 2 (α2b) pg (1 p) αb < p 2 G + p (1 p) [G B] (1 p) 2 αb pg (1 p) < (1 p) [αb + pg pb αb + pαb] 0 < (α 1) which, given that α > 1, holds always. The intuition is that checking at the end of the second year allows the portfolio to make up for any losses that might have taken place during the first year. The probability of ending up with a loss on any given year is (1 p) whereas the probability of ending up with a loss after 2 years is only (1 p) 2. Given loss aversion, diminishing the probability of losses increases the overall expected value. For the rest of the problem, assume the following: 1. Mr. P utility is determined only by his total gains and losses over the entire 2 year period, whether he checks the balance on his portfolio every year or only at the end of year If he chooses to check his portfolio at the end of year 1, Mr. P can choose to keep his money in the fund for another year or cash out. Part (b): Mr. P decides to check his portfolio at the end of year 1. He finds that his investments have gone up by G. Does he stay in the mutual fund for another year or does he cash out? Show algebraically and explain the intuition. Note: If you can t show the answer for the general case, show what happens if p = 1 4, G = 6, 000, B = 1, 000, and α = 3. You will still receive full credit if you use these parameter values. is: Answer (b): Given that his investments have increased by G, Mr. P s expected utility if he cashes out E (V cash out G) = G = 6000 If he decides to stay in for another year, his expected value is: E (V stay G) = p2g + (1 p) (G B) [ 1 = ] 4 5 [ ( = )] =

6 And thus, he will stay invested in the fund if and only if: G < p2g + (1 p) (G B) 0 < G + p2g + G B pg + pb 0 < pg (1 p) B That is, a first period of G makes Mr. P risk neutral and he decides to stay in as long as the investment has a positive expected payoff. The first year gain assures Mr. P that he will not end up with a loss and thus, by pushing him into the gain region of his utility, he becomes risk neutral. Part (c): Mr. P decides to check his portfolio at the end of year 1. He finds that his investments have gone down by B. Does he stay in the mutual fund for another year or does he cash out? Show algebraically and explain the intuition. Unlike the answers for parts (a) and (b), the result in this case does not hold in general given the restrictions above. To see this, first answer the question assuming p = 1 4, G = 6, 000, B = 1, 000, and α = 3. Then, see what happens if p, G, and B remain the same but α = 2. Show algebraically and explain the intuition. As an optional part of the problem, try to derive a lower bound for p given any α, B, and G such that Mr. P decides to stay invested after a year 1 loss. is: Answer (c): Given that his investments have decreased by B, Mr. P s expected utility if he cashes out E (V cash out B) = αb = 3000 If he decides to stay in for another year, his expected value is: E (V stay B) = p (G B) + (1 p) α ( 2B) [ 5 = ] 4 6 = 3250 And thus given the parameter values, he will try to cut his losses and exit if he loses money on the first year. In general, he will stay invested in the fund if and only if: αb < p (G B) + (1 p) α ( 2B) α < p G 1 2α + 2αp B α < p G B + (2α 1) So that as long as the good outcome is big enough to offset some of the losses, he will stay invested. For α = 3, he doesn t stay in for the second year. For α = 2, he does. An increase in loss aversion pushes the investor to try to minimize his losses and exit the market for any given bet. 6

7 Note that an explanation along the lines of risk seeking behavior doesn t really work here. Although in general the prospect theory value function is convex over losses, this particular utility function is just linear and thus the investor exhibits loss aversion but doesn t exhibit risk-seeking behavior. Part (d): Given what Mr. P would do in case the first year had positive results and given what he would do if the first year had negative results, will he want to check his portfolio at the end of each year or not? For this part, also assume p = 1 4, G = 6, 000, B = 1, 000, and α = 3. Also, without solving algebraically, explain why the answer would be different if throughout the problem we had assumed that α = 2. Answer (c): When Mr. P checks the portfolio at the end of year 1, he will leave the money in if the first period outcome was positive and take it out if it was negative. Thus, his expected utility at the end of the 2 year period would be: E (V ) = 1 4 E (V stay G) E (V cash out B) = = When Mr. P doesn t check his portfolio the first time, he just gets: E (V ) = E (V 0 to 2 ) = 750 So that he will prefer to check the portfolio every year. Again, this is due to the loss aversion. It is easy to see this when one considers that for α = 2, Mr. P actually doesn t pull his money out given a loss. Then, regardless of whether he checks after the first year or not, the expected payoff if the same. Problem 3: Applications of Prospect Theory For each of the following anecdotes, briefly explain: 1. Why is the agents behavior inconsistent with expected utility theory? 2. Why is the agents behavior consistent with prospect theory? 3. Is there any way in which the agents behavior could be reconciled with expected utility theory? Example: Some students who were about to buy season tickets to a campus theater group were randomly selected and given the tickets for free. During the first part of the season, those who paid full price attended significantly more plays than those who got the free tickets. 1. The money paid for season tickets is already a sunk cost. After buying season tickets or getting them for free, a student s decision to attend play or not should only be driven by the expected utility of 7

8 attending the play. For example, assume that a rational expected utility maximizing student can only choose between staying at home reading or going to the play. He should then consider the probability of the play being good, consider the probability of the play being bad, compute his expected utility from attending the play and, if it is greater than the utility he derives from staying at home, he should go. In this particular calculation, there should be no difference between students who randomly got the free tickets and those who didn t. Thus, the attendance of the groups shouldn t be different. 2. We can assume that the student s consider paying for the tickets and attending the theater in the same mental account (see the reading Mental Accounting Matters by Richard Thaler). Then, students that pay for the tickets are pushed into the region of losses of their value functions whereas those that don t pay stay at their reference point. In prospect theory, the value function is convex over losses which implies that those students in the loss region will exhibit risk-seeking behavior whereas those that got the tickets for free will be risk-averse. The difference in attitudes towards risk explains why the students that paid are more willing to attend plays. 3. A possible explanation of this behavior consistent with expected utility theory would be that the behavior of the students is driven by liquidity constraints. Assume that the students can either go to the theater, stay at home, or go to a basketball game and everyone prefers the basketball game to the theater, and the theater to staying at home. However, those students who paid for the theater tickets might not be able to afford the tickets to a basketball game and thus go to the theater rather than staying at home. Students who got the ticket for free on the other hand, choose to go to the basketball game. Part (a): People purchase insurance against damage to their telephone wires at 45 cents a month. On any given month, the probability that they will have to pay the $60 that a repair would cost is 0.4%. (i) The expected cost of repairs is 24 cents. Risk averse people will buy insurance, and it is not a violation of EU to prefer a choice with lower expected return if it is less variable. However, buying insurance against such a small potential loss would require an unrealistic level of risk aversion (recall the argument that Expected Utility maximizers are locally risk neutral and should take small gambles). The extreme risk aversion implied by the decision to buy this insurance would imply that people also turn down other types of gambles that we know they accept. (ii) Prospect theory says that people overestimate the probability of improbable events. A high value of p(:004) could explain why people buy the insurance. Also, if the damage occurs, people may feel the loss intensely. (iii) People are really credit constrained and don t have $60 saved up at any moment. 8

9 Part (b): The price elasticity of a particular good is the percentage by which demand for that good changes as a response to a 1% change in the price of that good. Empirically, economists have found that for some goods the elasticity is asymmetric. In particular, the change in demand due to a 1% increase in a good s price is larger in magnitude than the change due to a 1% decrease in price. (i) The price elasticity of a good is the change in quanity demanded, in percentage terms, divided by the percentage change in its priace. An expected utility maximizer doesn t has no reason to respond asymetrically to increases and decreases in price. In particular, when deciding how much to consume, an expected utility maximizer usually equates his marginal utility of consumption to the relative price of the good. Thus, as long as he has a well-behaved utility function, the elasticity of demand should not depend on the direction in which the price changes. (ii) Loss-averse consumers dislike price increases more than they like the windfall gain from price cuts and will cut back purchases more when prices rise compared with the extra amount they buy when prices falls. Putler (1992) found that asymmetry in price elasticities in consumer purchases of eggs and Hardie, Johnson, and Fader (1993) find similar results with a more sophisticated methodology. (iii) One could think that the goods chosen to measure this are not any goods. For example, assume that when the price of a good A increases the price of all of A s close substitutes increase even more. Also, assume that when the price of A falls the price of A s substitutes stay the same. Then it would be perfectly consistent with expected utility theory for consumers not to reduce their consumption of A too much when its price increases but to increase it by a lot when its price falls. Part (c): Unionized workers at Harvard have their wages set 1 year in advance. When they receive bad news that inform them that their wages will be cut next year, they don t reduce their spending. However, when they receive good news that inform them that their wages will increase the following year, they do increase their spending. (i) This is inconsistent with expected utility theory because in expected utility theory, if next years wage is surprisingly good, then the union workers should spend more now, and if next years wage is surprisingly bad, then the union workers should cut back on their spending now. (ii) This is consistent with prospect theory for two reasons: (1) because they are loss averse, cutting current consumption means they will consume below their reference point this year, which feels awful. (2) Owing to reflection effects, they are willing to gamble that next years wages might not be so low; thus they would rather take a gamble in which they either consume far below their reference point or consume right at it than accept consumption that is modestly below the reference point. These two forces make the union workers reluctant to cut their current consumption after receiving bad news about future income prospects. (iii) The union workers have bought additional cash benefits that kicks in whenever their wage decreases, hence their real wages do not decrease, and thus their spending does not decrease. It is also possible that the 9

10 workers expect severe deflation next year, and their real spending power of next years wages have actually increased. Part (d): Data from a brokerage firm about all the purchases and sales of a large sample of individual investors shows the disposition effect. Investors hold losing a median of 124 days and hold winning stocks only 104 days. That is, people hold on to stocks that have lost value for too long and are too eager to sell stocks that have risen in value. (i) As long as stock returns are unpredictable an expected utility maximizer should not be more likely to sell a losing stock than she is to sell a winning. (ii) If the reference point of an investor is the purchase price of a particular share, once the stock appreciates the investor will evaluate the future returns of the stock in a more concave region of her value function and thus exhibit more risk aversion. This could push her to decline a gamble she had accepted before. On the other hand, if the stock price declines, she will evaluate the future payoffs on a more convex and thus risk-seeking region of her value function. Then, she will continue to hold the stock. (iii) If investors believed that winning stocks are likely to decline in price and losing stocks are likely to increase, then their actions would be justified. However, this belief is irrational as there is no short-term mean reversion in stock returns. Also, they could sell winners to rebalance their portfolio because the transaction costs of trading at lower prices are relatively higher. However, Odean (1997) shows that these motives do not make investor behavior consistent with expected utility. 10