We never talked directly about the next two questions, but THINK about them they are related to everything we ve talked about during the past week:


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1 ECO 220 Intermediate Microeconomics Professor Mike Rizzo Third COLLECTED Problem Set SOLUTIONS This is an assignment that WILL be collected and graded. Please feel free to talk about the assignment with your friends or with your group and I strongly encourage you to submit your assignment as a group. Assigned: Due: Monday, April 25 th Monday, May 2 nd. Suppose the messiness of an apartment is measured on a scale of 0 to 00, with 0 being the cleanest and 00 being the messiest. Suppose that 0% of the apartments fall between 020, 20% between 20 and 40, 40% between 40 and 0, 20% between 0 and 80 and 0% between 80 and 00. Suppose that all parents tried to teach their parents never to let anyone in to see their apartments if they were over 80 on the messiness scale. If such a rule of thumb were widely observed, what would your best estimate of the messiness index of someone who said, You can t come in now, my place is a pit? In a world in which everyone makes use of all available information, would you expect this rule of thumb to be stable? What do you conclude from the fact that people really do sometimes refuse admission on the grounds that their apartments are too messy? If the general threshold for denying admission is 80, and if those who deny admission are uniformly distributed between 80 and 00, then our best estimate of the messiness of someone who denies admission will be 90. The threshold will not be stable. Someone whose messiness index is between 80 and 90 has good reason to let people in rather than be assumed to have an index of 90. A threshold of 90 should be unstable for similar reasons, as indeed will any disclosure threshold less than 00. In practice, the fact that some people do refuse to let others see their messy apartments seems to indicate that actually seeing the mess firsthand will be more damaging than merely having people conclude in the abstract that the apartment is messy. We never talked directly about the next two questions, but THINK about them they are related to everything we ve talked about during the past week: 2. Why should you EXPECT that all WWF wrestling matches are fixed but expect that fewer prize fights are rigged and far fewer baseball games are rigged?
2 Cheating is a primordial economic act trying to get more for less. To think up an answer, we need to think of the incentives for each group of people to cheat. So, why are the incentives larger for WWF wrestling than for boxing and for boxing than for baseball? (Let s ignore bookies and gambling for now and think of the leagues and commissions themselves) I think that the answer lies in what the stakes are for any particular game being played. For a randomly selected major league baseball game, having the outcome predetermined is unlikely to result in a dramatic change in the standings for either team it is only toward the end of the season that it is clear what needs to be done in order for a particular team to gain an advantage. In order for rigging games in baseball to have any impact, a large number of teams would have to be involved. I tend to think that during the season, the benefits of one team winning a game are near equal to the costs of another team losing. Only when a team is way out of it or way ahead may an asymmetry arise. For example, when I am 40 games out if first place, another loss won t do much in fact, another loss may even improve my draft standing. If a team is 20 games ahead in the standings, a loss will not hurt them as much as a win would help a team on the border of making the playoffs. Furthermore, the better team would have an incentive to let the poorer team win, for these teams might meet again in the playoffs. In boxing, the incentive scheme is slightly different. In many fights, the payoffs to each fighter are guaranteed, so the outcome of the fight will not affect the current earnings of a fighter. If the earnings from a fight are a function of who wins, you might expect fighters to make deals with each other as an insurance policy. You might think however, that there are situations where benefits to winning a fight are greater for one fighter than for another. In these cases, you might expect the fight to be thrown. When would this be the case if the payouts are fixed? Well, future payouts depend on the outcomes of previous fights and fan interest is a function of what has happened in previous fights. If two fighters are of equal ability, it might be in the financial interest of each fighter to win any one fight, but it would be in the financial interests of both fighters to ensure that they would fight several times, with each fighter winning a match once. Even in situations where rematches are unlikely, the payoffs to winning may be asymmetrical. If George Foreman makes a comeback, he will only earn money if he wins a fight, because the reason for a comeback is to be able to win future fights and perhaps a championship. If he is fighting an up and coming boxer, the gains to Foreman from winning likely exceed the gains of the upandcomer from winning, after all, he will have many years in the future to make a name for himself and the mere fact that he fought Foreman would give him name recognition for future fights. What about WWF wrestling? I tend to believe that in every single match, there is an asymmetry in payoffs to each side, and the WWF has an interest in ensuring outcomes occur where the payoffs are largest. For example, if Joe Smith is facing Bobby the Brain Beefcake or whatever a name might be very few fans care about 2
3 Joe Smith and the WWF is an organization the requires fans to be dedicated to particular athletes in fact, to a small number of superstars. Joe Smith will never win a fight against Bobby because it would diminish Bobby s star status and the entire WWF is better off when its stars are hugely popular. The WWF can ensure that Bobby wins all of his journeyman fights and then split the extra proceeds with all of the wrestlers. This is not as likely an occurrence in boxing or baseball. 3. What grounds are there for assuming that a randomly chosen social worker is less likely to cheat you in cards than a randomly chosen person is? Because social workers receive very low salaries relative to the amount of education they have, it is a reasonably safe inference that most of them chose that line of work for reasons other than money. And if the primary motive for cheating at cards is monetary gain, it follows that a social worker is not very likely to cheat. In the case of the used car salesperson (or example), by contrast, there is no similar presumption of nonmonetary motivation. Moreover, there is at least some indication that the capacity to dissemble is linked to success in selling. 4. A new motorcycle sells for $9,000, while a used motorcycle sells for $,000. If there is no depreciation and riskneutral consumers know that 20% of all new motorcycles are defective, how much do consumers value a nondefective motorcycle? If there is no depreciation, then a good motorcycle will have the same value new as it will be used. Therefore, the used motorcycles that are selling must be defective (we know from the lemons principle why this is likely). Therefore, the $9,000 selling price of a new motorcycle must be the expected value of a new motorcycle (if customers are risk neutral recall that they are indifferent when facing a fair gamble. If these customers were risk averse, this $9,000 must be LESS than the expected value of the motorcycle). The expected value of a new motorcycle equals: E N = P D ($,000) + (P D ) X = $9,000 where P D equals the proportion of defective motorcycles and the nondefective cycles are priced at $X. Solving yields X = $, What is the expected value of a random toss of a die (fair and six sided)? 3
4 The expected value of one toss of a sixsided die is: EV = () + ( 2) + () 3 + ( 4) + () 5 + ( ) = ( ) = ( 2) = Suppose your current wealth is $00 and your utility function is U(w) = w 2. You have a lottery ticket which pays $0 with probability 0.25 and $0 with probability What is the minimum amount for which you would be willing to sell this ticket? Compute your expected utility with the ticket: EU(with ticket) = ¼(0) 2 + ¾(00) 2 this utility level. = 0,525 utils or whatever you want to call We need to find the wealth level that provides us with the same utility as we have when we have the ticket: Since U = w 2, we know that w = U; so w = 0,525 = $ Thus, you would be willing to sell the ticket for $ Joe has an investment opportunity that pays $33 with probability ½ and loses $33 with probability ½. a. If his current wealth is $ and his utility function is U(w) = w ½, will he make this investment? He is risk averse, so you should be able to eyeball this and say no. But, to determine whether he makes the investment, compare the expected utility of the game with the utility from the sure thing. The utility of the sure thing is = The expected utility of the gamble is: EU(investment) = ½ 44 + ½ 8 = 0.5. Since his expected utility from the gamble (0.5) is less than his utility with no gamble, he will NOT make the investment. b. Will he make it if he has two equal partners? (Be sure to calculate the relevant expected utilities to at least two decimal places) With two equal partners he wins $33 / 3 = $ if all goes well and loses $30/3 = $0 if not. 4
5 His expected utility is now: EU(investment) = ½ (+) + ½ (0) = = This is greater than the utility he gets from no investment (0.54), so he WILL take the gamble. 8. Given a choice between A (a sure win of $00) and B (an 80% chance to win $50 and a 20% chance to win 0), you pick A. But when you are given a choice between C (a 50% chance to win $00 and a 50% chance to win 0) and D (a 40% chance to win 50 and a 0% chance to win 0), you pick D. Are your choices consistent with expected utility maximization? Let w be your initial wealth, and let U be your utility function: Picking A over B U A = U(w + 00) > EU B = 0.8U(w+50) + 0.2U(w) Picking D over C EU D = 0.4U(w+50) + 0.U(w) > EU C = 0.5U(w+00) + 0.5U(w) Rearranging terms of the last inequality you get: 0.5U(w+00) < 0.4U(w+50) + 0.U(w) and dividing both sides by 0.5 gives: U(w+00) < 0.8U(w+50) + 0.2U(w) Which is the reverse order of the inequality implied by the choice of A over B, hence the inconsistency. 9. Suppose Jones gets a disease with probability ½ and that his wealth is $9 million. If he gets the disease, it costs $5 million to cure. a. If his utility function is U(w) = w ½, will he purchase a government insurance policy that $2. million? b. Now, suppose that there is a diagnostic laboratory that offers him a test. The test costs $D if it says that you will get the disease later in life and it costs $N if you will not get the disease. Further, assume that it costs a hospital $,000 to administer the test and that the 5
6 testing industry is perfectly competitive. What are the equilibrium values of D and M if the test gets it right 00% of the time? Interpret the meaning of this answer. To help you answer the question, you need to consider the following information:. What is the budget constraint faced by a competitive firm (i.e. if it charges two different prices for the test, what must the expected value of the prices be in a longrun perfectly competitive equilibrium)? 2. If Jones discovers from the test that he will get the disease, he will want to purchase the insurance. 3. If Jones discovers from the test that he will not get the disease, he will not want to purchase the insurance. 4. You don t need to use the utility function directly to compute the answer, but realizing what form the utility function takes should tell you what Jones wealth should be like in all states of the world. First, since we know that Jones is risk averse, he prefers a situation where his wealth is constant in all states of the world to one where his wealth is uncertain. In other words, his expected utility from a certain amount of wealth exceeds his expected utility from two different amounts of wealth whose expected value is the same. Second, since the testing industry is perfectly competitive, the price of the test should be equal to the minimum long run average total cost of the test in this case $,000. Therefore, if the hospital is going to charge different prices for the test, it wants the expected price of the test to equal $,000. Since the probability of getting a disease is ½ and the probability of diagnosing it correctly is 00%, then you can write this expected price as ½ (D) + ½ (N) =,000. (think about how this calculation would change if the test was inaccurate some percentage of the time). In other words, we know that D+N = $2,000. Third, write down Jones wealth if he gets the disease (again, it is important to remember that the test predicts with 00% accuracy). He will have $9million  $2.million  $D (he busy insurance). Next, write down his wealth if he does not have the disease he will have $9million  $N. Now you have a simple algebra problem: two equations in two unknowns: 9,000,000 2,00,000 D = 9,000,000 N () D+N = 2,000 (2)
7 Solving for N in (2) and plugging into () yields:,400,000 D = 9,000,000 (2,000 D) (3) and a little algebra tells you that: D = $,299,000 and therefore N = $,30,000. If I do not have the disease, I will pay $.30 million for the test. If I do have the disease, the hospital WILL PAY ME $.299 million. You might think that this example is silly, but equity fees on Wall Street follow rules like this instead of cash you often get paid in stock of the company that you do work for. 0. There are two groups, each with a utility function given by U(w) = w ½. Suppose initial wealth levels for all individuals equals 00. Each member of group faces a loss of 3 with probability 0.5. Each member of group two faces the same loss with probability 0.. a. What is the most a member of each group would be willing to pay to insure against the loss? For group, the reservation price of insurance is found by equating the utility you get with insurance when the premium equals the reservation price to the expected utility without insurance: (00R ) = ½ 00 + ½ 4 = 9 Squaring both sides gives 00R = 8 and the reservation price for group is $9. For group 2 do the same thing to get: (00R 2 ) = = 9.8 which results in a reservation price of $3.9 b. In part (a), if it is impossible for outsiders to discover which individuals belong to which group, will it be practical for members of group 2 to insure against this loss in a competitive insurance market? (For simplicity, you may assume that insurance companies charge only enough premiums to cover their expected benefits payments). Explain. 7
8 If members of the two groups are indistinguishable, an insurance company will have to charge the same premium to each. If its policy holders consisted of equal numbers of people from each group, this premium would have to cover the expected loss, which is [½ (3) + 0.(3) ] / 2 = 0.8. Since this exceeds the reservation price of members of group 2, nobody from that group would buy insurance. And with only group members remaining in the insured pool, the premium would have to rise to 8 in order to cover the expected loss for members of that group. c. Now suppose that the insurance companies in part (b) have an imperfect test for identifying which persons belong to which group. If the test says that a person belongs to a particular group, the probability that he really does belong to that group is x <.0. How large must x be in order to alter your answer to part (b)? If a company has the test described, and the test says a person is a member of group 2, then the expected benefit payout from insuring that person is: E(Loss) = x(0.)(3) +(x)(0.5)(3) where the term on the left tells that expected loss from a person if the test is right while the term on the right tells us the expected loss from the person if the test is wrong (he s in the bad group) Thus E(Loss) = 84.4x Setting E(Loss) equal to the reservation price for group 2 tells is that 84.4x = 3.9 x = The test would have to be accurate 97.5% of the time in order for a member of group 2 to find insurance an acceptable buy. 8
.4 120 +.1 80 +.5 100 = 48 + 8 + 50 = 106.
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