Economics 31 - Homework 5 Fall 26 Dickert-Conlin / Conlin Answer Key 1. Suppose Cush Bring-it-Home Cash has a utility function of U = M 2, where M is her income. Suppose Cush s income is $8 and she is offered a bet from a fair six-sided die: If she rolls a 1, she wins $4 and if she rolls anything else (a 2, 3, 4, 5, or 6), she loses $2. a. Graph Cush s utility function, U = M 2 and label it U. This represents Cush s utility if her income is certain. b. What is Cush s expected INCOME [E(M)] if she takes the gamble? Label this on the x-axis. E(M) = 1/6 * (8+4) + 5/6 * (8-2) = 1/6 * (12) + 5/6 (6) = 2+5 =7 c. If Cush accepts the bet, what is her expected utility E(U) from the gamble? Draw the E(U) chord on the graph and label the E(U) that you calculate on the y-axis. EU gamble = 1/6 * (8+4) 2 + 5/6 * (8-2) 2 = 1/6 (12) 2 + 5/6 * (6) 2 = 1/6 * 144 + 5/6 * 36 = 24 + 3 = 54 U, E(U) 15 14 13 12 11 1 9 8 7 6 5 4 3 2 1 Uno gamble=64 EUgamble=54 1 2 3 4 5 6 7 8 9 1 11 12 13 Mloses E(U) 7=E(M) gamble 8=Mno gamble U = M 2 Mwins M, E(M) d. If Cush turns down the bet, what is her utility with this certain income? Label this on the y-axis. EU no gamble = (8) 2 = 64 e. Does Cush take the bet? Why or why not? No, she doesn t take the bet because EU gamble < EU no gamble, 54 < 64 f. Is this a fair gamble? Explain. No because her expected income with the gamble ($7) is less than her expected income without the gamble ($8). g. Does Cush take the bet if the bet changes so that if she rolls a 1 or a 2, she wins $4 and if she rolls anything else (a 3, 4, 5, or 6), she loses $2? Yes, because EU gamble new = 2/6 * (12) 2 + 4/6 * (6) 2 = 1/3 * 144 + 2/3 * 36 = 48 + 24 = 72 > EU no gamble = 64. Think about what this would do to your diagram... h. Is Cush risk averse, risk seeking or risk neutral? How do you know? BE EXPLICIT! Cush is risk seeking because she has increasing marginal utility. M U MU (1-)/(1-)=1 1 1 (4-1)/(2-1)=3 2 4 i. True or false and explain: Risk seeking individuals accept all bets. False, Cush is a risk seeker and turned down the first bet because the expected winnings were too small. The probability of winning was too low and/or the payoff from winning was too small. 1
2. Suppose Dan is a National Football League (NFL) player with a utility function of U= M, where M represents monthly income in dollars. a. Is Dan risk averse, risk seeking or risk U, E(U) neutral? How do you know? BE 14 U = sqrt(m) EXPLICIT! Dan is risk averse because he has 12 diminishing marginal utility. The chance E(U) football player= of losing of $1 outweighs the chance 1 of winning $1. You should be able to show this in a variety of ways. U commentator 8 For example: Dan s first $1 increases his utility by 1 (MU=1) and the 2 nd E(U) $1 he receives only increases his utility by 6.41. Suppose Dan has two job offers. The first is 4 an offer to be an NFL quarterback. With this job there is a.5 probability of earning 2 $16,9/month (if he starts every game), and a.5 probability of earning $4,9 (if he gets hurt and sits on the bench all year). What is Dan s expected INCOME, E(M), if he takes this new job? 2 4 6 M hurt 8 1 M commentator 12 14 E(M) football player 16 M, E(M) M starts EV(M)=1/2*(169)+1/2*(49) =845+245=19 c. Although his expected income is 19, his actual income if he takes this new job will be either 169 or 49. (This may seem simple, I m just being thorough). d. What is Dan s expected utility, E(U), if he takes this quarterbacking job? EU quarterback = ½* 169 + ½* 49 = ½ * 13 + ½ * 7 = 1 f. As an alternative, Dan has been offered a job as a TV commentator. This job offers him an income of $1,/month and he can earn that income next year with certainty. What is Dan s utility, U, if he takes this job? U commentator = 1 = 1 g. Which job should Dan take? Explain. Dan is indifferent between taking the job or not taking the job since the expected utility from taking the job, 1, is the same as the expected utility from not taking the job, 1. h. Graph Dan s utility ( U= M ) with certain income and expected utility (the chord). Also label his expected income and certain income on the x-axis and his expected utility and certain utility on the y-axis. i. Let t be the probability that he becomes a starter on the team and (1-t) be the probability that he sits on the bench. What is the minimum t can be so that Dan should take the quarterbacking job? Hint: Set up an equation with t and (1-t) in it. He will take the quarterbacking job if: EU quarterbacking job > EU commentator job and we know from above that EU commentator job is 1 EU quarterbacking job = t* 169 + (1-t) * 49 > EU commentator job =1 t* 13 + (1-t) * 7 = 1 13t + 7-7t >1 6t > 3 t > 3/6 =.5 In words, if the probability of becoming a starter is greater than 5 percent, he will leave his old job and take his new job. Actually, we knew this from parts d and e. 2
3. Suppose that everybody in Bedrock has a utility function given by U= M. Everyone normally has an income of $1,, but if they get into a car accident their net income, after accident expenses, falls to $4. Half of the drivers in Bedrock are good drivers and have a probability of.25 of having an accident. The other half are bad drivers and have a probability of.75 of having an accident. a. Suppose that Bedrock residents have access to insurance and can pay $p for insurance that would cover their losses in the event of an accident. That is, they can have income of $1,-p whether or not they are in an accident. What is the most good drivers would be willing to pay for this insurance? EU uninsured =. 75* 1 +. 25* 4 = 8 Willing to pay so that their certain income gives them the same amount of satisfaction as their uncertain income: EU P uninsured = 8 = 1 64=1-P P = $36=willing to pay for insurance b. How much would the bad drivers be willing to pay? EU uninsured =. 25* 1 +. 75* 4 = 4 Willing to pay so that their certain income gives them the same amount of satisfaction as their uncertain income: EU P uninsured = 4 = 1 1,6=1-P P = $84 = willing to pay for insurance c. Rubble s Auto Insurance Company is unable to distinguish good drivers from bad drivers and, therefore, must charge the same premium to everybody. What is the expected value of claims/payouts? (Hint: consider the probability of each type of driver and the probability that each type of driver is in an accident. Also consider how much Rubble s has to pay if there is an accident). If there is an accident, the insurance company the claim will be for $1-$4=$96. 5 percent of the drivers are good drivers and there is a.25 chance that they are in an accident. 5 percent of the drivers are bad and there is a.5 chance that they are in an accident. The expected claim is therefore:.5*(.25)*96+(.5)*(.75)*96=$4,8 d. If Rubble s Auto Insurance Company sets its premium (p) equal to the expected value of claims you found above, will all Bloom County drivers be willing to pay this premium? The good drivers won t buy auto insurance from Rubble s because they are only willing to pay $3,6. e. What is likely to happen in this insurance market and why? Is your answer an explanation for why the government might make car insurance mandatory? Only bad drivers will purchase insurance, which will cause the premium the insurance company charges to rise to $7,2 (where did this number come from?). ADVERSE SELECTION! f. Can you suggest ways the drivers might change their behavior once they have car insurance? They might drive more recklessly or they might not lock their car all the time. This is called moral hazard. 3
4. Sperry Speculator has an investment opportunity that pays 33 with probability ½ and loses 3 with probability ½. a. If his current wealth is M=111, and his utility function is U= M, will he make this investment? EU no investment = 111 = 1.536 EU investment = ½ * 111+ 33 + ½ * 111 3 = 6+4.5 = 1.5 EU no investment > EU investment Therefore, he will not make the investment. b. Would Wild Wanda, who is a risk seeker, be more or less likely to make this investment than Sperry. No math please, just use intuition. Wild Wanda is more likely to make this investment than Sperry because she loves the possibility that she might win the 33. She loves winning more than she hates losing.. Her marginal utility is increasing with income. c. Would Plain Jane, who is risk neutral, be more or less likely to make this investment than Sperry. Again, no math please, just intuition (although it is easy to know by looking at the numbers what Jane would do). Plain Jane is more likely to make this investment than Sperry because she doesn t hate losing as much as Sperry. Her marginal utility is constant with income. d. Would Sperry make the investment in part a if he has two equal partners (suppose Wanda and Jane invest with him)? In this case, the gains or losses are split equally among the three partners. EU no investment = 111 = 1.536 EU investment = ½ * 111+ 11 + ½ * 111 1 = 1.55 EU no investment < EU investment Therefore, he will make the investment. e. What is the advantage of pooling risk, as in part d? It shares risk among people and increases the expected utility from making the investment. 5. Ross has $1, in income and a U= M is his utility function. His sister Monica offers him a bet where if the fair coin comes up tails, he loses his entire $1,. What is the minimum amount that she would have to offer him in the event of heads to make the bet a good one for him? Set up an equation. On one side, you want the level of satisfaction that Ross receives if he doesn t bet. On the other side, you want the expression for his expected utility that incorporates his utility if tails comes up and his utility when heads comes up and he has $1,+X in income. Solve for X. EU no gamble = 1 = 1 M if he wins = 1+X. M if he loses = EU gamblet = ½ * + ½ * 1 + X = ½ 1 + X Ross takes the bet if: EU no gamble EU gamble 1 ½ 1 + X multiply both sides by 2: 2 1 + X Square both sides: 4 1+X Subtract 1, from both sides: 3 X That is, if Monica pays Ross at least 3, when the coin comes up heads, he will take the gamble. 6. Problem # 1 of your text book (The question about messy rooms).. Best estimate is expected value of all rooms that are not shown: 9 This is likely to be unstable because all persons w/ rooms less messy than 9 (like 81-89) will not want outsiders assuming they are this messy. We can conclude that these persons are really messy or that it is costly in some other way to show their room. 4
7. Consider the market for houses. Suppose two different types of houses exist. Type S is a structural sound house and Type F is a structurally flawed house. There are sixty Type S houses and forty Type F houses. All owners of the structurally sound houses have reservation value $175, and all owners of the structurally flawed houses have reservation value $75,. (Therefore, the owners of the structurally sound houses are not willing to sell their house for a price less than $175, and the owners of the structurally flawed houses are not willing to sell their house for a price less than $75,.) All potential buyers are willing to pay $2, for a structural sound house and $1, for a structural flawed house. Assume there are thousands of risk-neutral potential buyers and buyers cannot differentiate between a structurally sound and a structurally flawed house. a. Graphically, depict the supply and demand curves for housing. How many houses are sold in equilibrium and at what price are the houses sold for? $175, S $1, $75, D Q 4 1 At a price above $175,, risk-neutral buyers know that both types of houses will be put on the market so they are willing to pay 1,(4/1)+2,(6/1)=16,. Therefore, the quantity demanded at a price above $175, is zero. At a price between $75, and $175,, buyers are willing to pay $1, because they know only structurally flawed houses are being put on the market. Therefore, the quantity demanded at a price above $1, is zero, at a price of $1, is (,infinity) and at a price less than $1, is infinity. The demand curve is depicted above. Only structurally flawed houses are sold and the price of these houses is $1,. b. If the buyers were risk-averse instead of risk-neutral, how would this affect the supply of and demand for houses? Depict on the graph. It would not affect the supply and demand depicted above. The reason is that in the equilibrium above, the buyers know they are buying a flawed house and, therefore, do not incur any risk. c. Suppose the government is considering passing a law that requires the sellers to disclose whether their house is structurally sound or structurally flawed. (There is actually this type of disclosure law.) Surplus is the difference between what people are willing to pay and what they have to pay (the actual price). How much would total surplus increase if this law were passed? Explain. With the law, the structurally sound houses would be sold and total surplus would be increased by 6(2,- 175,)=1,5,. 5