Test 1 solution sketches

1. Test 1 solution sketches Average: 42.11 points (75.2%) 5 tests with score 56

2. #1: Dianne • Dianne’s utility (U) is the fourth root of her income (I), i.e. U = . She has a 16% chance of receiving an income of $6,250,000 and an 84% chance of receiving an income of $810,000. • (a) (3 points) What is the expected income of this gamble? • 0.16($6,250,000) + 0.84($810,000) = $1,680,400 • (b) (3 points) What is the expected utility of this gamble? • 0.16(6,250,000)0.25 + 0.84(810,000)0.25 = 33.2 • (c) (4 points) What is the certainty equivalent of this gamble? • CE = EU4 = 33.24 = $1,214,933

3. #2: The Hollymetal Bowl • The Hollymetal Bowl invites top acts for concerts, but people that live for miles around also enjoy the music. Assume Q is the number of hours of concerts each year. Total cost for these concerts comes out to TC = Q2 +5Q. Demand, which reflects marginal private benefit of consumption, is represented by the equation P = 500 – 2Q. Since people that live in neighboring houses also enjoy the concerts, there is an external benefit of EB = 50Q. • (a) (5 points) What is the actual output by the Hollymetal Bowl if external benefits are not taken into account. Assume a competitive market. • Set MB = MC (excluding external benefit) • MB comes from the demand curve • MC = dTC/dQ = 2Q + 5 • 500 – 2Q = 2Q + 5 • Q = 123.75

4. #2: The Hollymetal Bowl • (b) (5 points) What is the socially efficient output of concerts? • Set SMB = MC • Note that MEB = dEB/dQ = 50 • SMB = (500 – 2Q) + 50 • SMB = 550 – 2Q • 550 – 2Q = 2Q + 5 • Q = 136.25

5. #3: Lester and Jacqueline • Lester and Jacqueline are wondering what to do with educating their child, Hillary. The family’s utility function is U(x,y) = xy3, where x represents $1000s spent on educating Hillary, and y represents $1000s spent on everything else. The household is able to spend $50,000. • (a) (5 points) How much money should be spent on education if there is no public education available? • max xy3 such that x+y=50 • Substitute to get max (50 – y)y3, or max 50y3 – y4 • FOC: 150y2 – 4y3 • Set equal to zero to get y = 0 or y = 37.5 • Note that y = 0 gives a minimum for utility • So choose y = 37.5 • Choose x = 12.5

6. #3: Lester and Jacqueline • (b) (6 points) Suppose that Lester and Jacqueline have an additional option here: Publicly-provided education. Lester and Jacqueline have two choices: • Choose public education, in which $5,000 of taxpayer money is spent to educate Hillary. Note that Lester and Jacqueline pay nothing for education here, but cannot spend any of their money to educate Hillary. Also note that $50,000 of household money is available to be spent whether or not public schooling is chosen. • Spend as much household money as they want to educate Hillary. If they choose this option, they are not allowed to use any taxpayer money for educating Hillary. • What should Lester and Jacqueline do to maximize family utility? Justify your answer. • No public education: Choose answer from (a) to get a utility of 659,180 • With public education: Choose x= 5 and y = 50 to get a utility of 625,000 • Since the higher utility is with no public education, choose this, and spend a value of x = 12.5, y = 37.5

7. #4: Pismo Valley Heights • Two hundred people live in the land of Pismo Valley Heights. Eighty of these people have individual demand curves for a good given by P = 200 – 4Q, where P is price and Q is number of units. One hundred twenty of these people have individual demand curves for the same good given by P = 100 – Q. The marginal cost of this good is 30. • (a) (4 points) How many units of the good are purchased in total (i.e. the total quantity purchased by all 200 people) if this is a private good? Assume a competitive market. • 80 people consume Q such that 200 – 4Q = 30  Q = 42.5 • 120 people consume Q such that 100 – Q = 30  Q = 70 • Total consumption • 80*42.5 + 120*70 = 11,800

8. #4: Pismo Valley Heights • (b) (6 points) How many units of the good are provided in an efficient outcome if this is a public good? • Vertical summation • For Q between 0-50 • WTPtotal = 80(200 – 4Q) + 120(100 – Q) = 28,000 – 440Q • Set WTPtotal = MC • 28,000 – 440Q = 30  Q = 63.568 • DO NOT CHOOSE THIS (Q is not between 0-50) • For Q between 50-100 (note that 80 people have WTP = 0 here) • WTPtotal = 120(100 – Q) = 12,000 – 120Q • Set WTPtotal = MC • 12,000 – 120Q = 30  Q = 99.75 (This is the answer!)

9. #5: 2 routes between Goldville & Silverville • 5. There are two routes to travel between Goldville and Silverville. One route is Wide Back Road, in which anybody traveling on this route has a travel time of 45 minutes. The other route is Narrow Mountain Highway, in which the travel time depends on the number of travelers on the highway. The travel time on the highway is 21 + T, with T representing the total number of drivers on the highway. • (a) (3 points) If there is no regulation on travel between Goldville and Silverville, what is equilibrium if there are 100 travelers total? Make sure to describe what happens on both routes. • Set 21 + T = 45 • T = 24 • # of drivers on Wide Back Rd. is 100 – 24, or 76

10. #5: 2 routes between Goldville & Silverville • (b) (4 points) Suppose that someone on the Goldville town council proposes to improve Narrow Mountain Highway, such that the travel time on the highway is 21 + 0.4T. How does this change equilibrium travel times relative to part (a)? Use math and/or an explanation of 40 words or less to justify your answer. • Set 21 + 0.4T = 45  T = 60  # of drivers on Wide Back Rd. = 100 – 60 = 40 • Note that equilibrium travel times are the same on each route after the expansion (21 + 0.4*60 = 45)

11. #5: 2 routes between Goldville & Silverville • (c) (4 points) Once again use the information before part (a) to solve this part: What is efficient if you are trying to minimize the total number of minutes of travel time of all travelers? Make sure to describe what happens on both routes. • min 45(100 – T) + (21 + T)T • min 4500 – 24T + T2 • Take the FOC and set it equal to 0 • -24 + 2T = 0  T = 12 • # of drivers on Wide Back Rd. = 100 – 12 = 88

12. #5: 2 routes between Goldville & Silverville • (d) (4 points) Once again use the information before part (a) to solve this part: Suppose that you know that everyone traveling between Goldville and Silverville have values of time of $120 per hour. How much of a toll would be charged on one or both routes in order to minimize the total number of minutes of travel time of all travelers in equilibrium? • Value of time is $120 per hour, or $2 per minute • Total cost to travel on either route should be the same • This includes both tolls and travel time cost • Set toll + 2*33 = 2*45  toll = $24 • Note that toll is only on the highway • Another way to think of this is that you need to charge a toll equivalent to 12 minutes of travel time on the highway • Since value of time is $2/min., 12 min. of time = $24 in tolls

13. Level of difficulty for each question • Easy • 1a, 1b, 1c, 2a, 2b, 4a, 5a, 5b • 31 points • Medium • 3a, 3b • 11 points • Hard • 4b, 5c • 10 points • Hard to very hard • 5d • 4 points