Test 2 solution sketches

1. Test 2 solution sketches Average: 41.35 points (81.1%) 5 tests with score 51

2. #1: 150 people @ Lookout Falls • One hundred fiftypeople live in Lookout Falls. Sixty of these people have individual demand curves for a given public good by P = 30 – Q, where P is price and Q is number of units. Fifty-five of these people have individual demand curves for the same good given by P = 50 – 2Q. The final thirty-five people have individual demand curves for this good given by P = 100 – 5Q. The marginal cost of this good is 500. How many units of the good are provided in an efficient outcome?

3. #1: 150 people @ Lookout Falls • For Q between 0-20: • WTP is 60(30 – Q) + 55 (50 – 2Q) + 35(100 – 5Q) = 8050 – 345Q • Set 500 = 8050 – 345Q  Q = 21.884 (NO!) • For Q between 20-25: • WTP is 60(30 – Q) + 55(50 – 2Q) = 4550 – 170Q • Set 500 = 4550 – 170Q  Q = 23.824 (YES!) • For Q between 25-30: • WTP is 60(30 – Q) = 1800 – 60Q • Set 500 = 1800 – 60Q  Q = 21.667 (NO!)

4. #2: Bobby has a utility function… • Bobby has a utility function of U(I,L) = IL – 60L, with I representing income earned today, and L representing number of hours of leisure today. Bobby can work as many hours as he wants today for $10 per hour. Assume that Bobby has 16 hours today for work, leisure, or a combination of the two. • (a) Without any government assistance programs, how many hours will Bobby work today? • Let W = # of hours worked • max (10W)L – 60L s.t. W + L = 16 (or W = 16 – L) • max 10(16 – L)L – 60L • max 100L – 10L2 • Set first order condition equal to zero & solve • 100 – 2L = 0  L = 5  W = 11

5. #2: Bobby has a utility function… • (b) If the government offers a $20 cash transfer to anyone who receives $50 or less in income from working today, how many hours will Bobby work today? You need to justify your answer completely to get all points. (Note: The $20 cash transfer, if taken, counts as income earned for this problem.) • max (10W + 20)L – 60L s.t. W + L = 16 • max 120L – 10L2 • Set first order condition equal to zero & solve • 120 – 20L = 0  L = 6  W = 10 • We can’t have W = 10 (the closest is 5 so use this) • U(70, 11) = 110 VS. U(110, 5) = 250 • Since the utility of working 11 hours today is higher than working 5 hours today with the transfer, Bobby will work 11 hours today

6. #3: Margaret is in the hospital • Margaret is in the hospital from an extreme skiing event that she competed in. The total cost of being in the hospital is 3,000Q + 200Q2, with Q denoting the number of days in the hospital. Margaret’s total benefit of being in the hospital is 12,000Q – 350Q2. How long will Margaret want to stay in the hospital if she must pay 25% of the costs of staying in the hospital? (Note: Fractional days are possible here.) • MC = 3000 + 400Q  Margaret’s MC = 750 + 100Q • MB = 12,000 – 700Q • Set Margaret’s MC equal to MB to get Q = 14.0625

7. #4: Joe and Paul are both grain farmers • Joe and Paul are both grain farmers. They decide that it would be profit maximizing for both farms to build a single grain silo to be used for both farms. They decide to use Lindahlprices to determine the amount of money they will spend on the silo. Specifically, they will negotiate cost sharing until they agree on a cost-sharing distribution that adds up to one for a given amount of spending. If Q denotes the amount of spending on the silo, then Joe’s share is SM = 0.6 – Q/150,000 and Paul’s share is SH = 0.55 – Q/250,000. • (a) How much will be spent on the silo in equilibrium? • SH+ SM = 1  (0.55 – Q/250,000) + (0.6 – Q/150,000) = 1 • Solve for Q to get Q = 14,062.5

8. #4: Joe and Paul are both grain farmers • (b) What combination of Lindahl prices will lead to equilibrium? In other words, what will Joe and Paul’s shares be in equilibrium? • SM = 0.6 – 14,062.5/150,000 = 0.50625 • SH = 0.55 – 14,062.5/250,000 = 0.49375

9. #5: Assume two periods of consumption • Assume that there are two periods of consumption: Working years (time period 1) and retirement (time period 2).  Because of the Social Security system, Valentino receives an endowment of 500 units during his working years and 150 units for his retirement.  Assume that Valentino’s consumption in periods 1 and 2 are c1and c2, respectively, and his utility is c1c21.5 = c1c23/2. Also assume that he can save or borrow between periods 1 and 2 at a rate of 150%? (Notes: 150% of interest is earned for any money saved between the two time periods. In other words, 150% of interest is accrued for any money borrowed between the two time periods. Also, the endowment is how much Valentino has after contributions and benefits from Social Security.)

10. #5: Assume two periods of consumption • (a) How much does he consume in each time period to maximize utility? • max c1c23/2s.t. (1 + 1.5)c1 + c2 = 500(1+1.5) + 150 • Solve constraint for c1 to get c1 = 560 – 0.4c2 • Plug c1 into the max problem • max (560 – 0.4c2)c23/2 • Max 560c23/2 – 0.4c25/2 • Take the FOC and set it equal to 0: 840c21/2–c23/2 = 0 • Solve for c2 to get 0 or 840 • Ignore 0, since this is a minimum • If c2is 840, then c1 is 560 – 0.4(840), or 224

11. #5: Assume two periods of consumption • (b) How much does he save or borrow privately during time period 1? Make sure to clearly state if your number represents units saved or borrowed. • If you have 500 available in period 1 and you consume 224, then you save 500 – 224, or 276

12. #6: The gov’t is planning a new project • The government is planning how much to spend for a new project. Assume that spending on the project is y dollars. The total benefit of the project to society is 1,000,000 ln(y). What is the efficient level of spending on the project? • MC = 1 • MB = 1,000,000/y • Set MC = MB to get y = 1,000,000

13. Level of difficulty for each question • Easy • 31 points: #2a, 3, 4a, 4b, 5b, 6 • Easy/medium • 6 points: #5a • Medium • 7 points: #1 • Hard • 7 points: #2b