Labor Supply

MGT203 – Winter 2019
Problem Set #2
Due Wednesday 3/13/2019 at the end of lecture in class (i.e., each group should turn in one physical copy with all group member names + student ID numbers at the top of the first page).
Instructions
• You may work in groups of up to 6 people. You need only turn in 1 copy per group.
• Please show your work. I will give partial credit for properly setting up the problem even if the numerical answer is ultimately wrong. This is why I need a physical copy rather than a submission through iLearn.
• You may use any resource at your disposal within reason (i.e., calculator, website, Stata, etc.).
Questions
1. Labor Supply
a. Below is an approximation for two wage profiles based on 2019 Federal and CA tax rates. Compute the missing wage information and make a graph of each “budget line” (1 graph for each wage profile).
Gross pay of $50 per hour + $5000 in non-labor income:
Hours
Gross Income
Net Income
Average Tax Rate
Effective Wage Rate
–
5,000
4,400
12%
n/a
500
30,000
25,232
16%
1,000
55,000
43,659
21%
1,500
80,000
60,784
24%
2,000
105,000
77,347
26%
Gross pay of $500 per hour + $5000 in non-labor income:
Hours
Gross Income
Net Income
Average Tax Rate
Effective Wage Rate
–
5,000
4,400
12%
n/a
500
255,000
166,060
35%
1,000
505,000
298,560
41%
1,500
755,000
422,495
44%
2,000
1,005,000
546,245
46%
b. Find the optimal leisure/consumption bundle for each wage profile assuming a utility function of: ?=?×(?−50,000).
Hint: Find the solution in three steps. First, derive the optimal leisure as a function of w (?=?(?)). Second, compute the optimal L for each wage you compute above. Third, find the {L,w} pair that actually lies on the budget line.
c. Is leisure a “normal” or “inferior” good for this worker? Why?
Hint: Compare the two solutions in part b. Wage goes up. What happens to leisure quantity?
2. Principal Agent Problem (Adapted from Problem #20.23 from Perloff)
In the National Basketball Association (NBA), the owners share revenue but not their costs. Suppose that one team, the L.A. Clippers, sells only general admission seats to a home game with the visiting Philadelphia 76ers (Sixers). The inverse demand for the Clippers-Sixers tickets is ?=100−0.004?. The Clippers’ cost function of selling ? tickets and running the franchise is ?=10?.
a. Find the Clippers’ profit-maximizing number of tickets sold and the price if the Clippers must give 50% of their revenue to the Sixers. At the maximum, what are the Clippers’ profit and the Sixers’ share of the revenues?
b. Instead, suppose that the Sixers set the Clippers’ ticket price based on the same revenue-sharing rule. What price will the Sixers set, how many tickets are sold, and what revenue payment will the Sixers receive? Explain why your answers to parts a and b differ.
c. Now suppose that the Clippers must share their profit rather than their revenue. The Clippers keep 45% of their profit and share 55% with the Sixers. The Clippers set the price. Find the Clippers’ profit-maximizing price and determine how many tickets the team sells and its share of the profit. Explain why your answers to parts a and c differ.
d. Now suppose that the Clippers must pay the Sixers a commission of $5 per ticket. The Clippers set the price. Find the Clippers’ profit-maximizing price and determine how many tickets the team sells and its share of the profit. Explain why your answers to parts a and d differ.
e. If you owned the Clippers, which scheme would you pick?
3. Education Signaling (Problem #19.25-26 from Perloff)
Hint: Review Perloff Section 19.6 and Pindyck Rubinfeld 17.2.
Suppose there are high- and low-ability workers. The percentage of the population that is high-ability is ? and the percentage that is low-ability is 1−?. If the employer can distinguish the two types, it pays equal to the output of the two types: ?ℎ and ??, respectively. If the employer cannot distinguish the two types it pays ?̅, which is equal to the expected (i.e., average) output of all workers it employs.
a. Suppose education is a continuous variable, where ?ℎ is the years of schooling of a high-ability worker and ?? is the years of schooling of a lower-ability worker. (Only integer values of e are allowed: 1, 2, 3, etc.) The cost per period of education for these types of workers is ?ℎ and ?? respectively, where ??>?ℎ. Under what conditions is a separating equilibrium possible? How much education will each type of worker get?
Hint: You will need to explain the constraint on both high- and low-ability workers at equilibrium. These two constraints are the “conditions” the question asks for. Use these constraints to solve for the levels of education.
b. Under what conditions is a pooling equilibrium possible?
Hint: You will need to explain the constraint on high-ability workers and on cost generally. These two constraints are the “conditions” the question asks for. Explain (a) what levels of cost always give you a pooling equilibrium and (b) what combinations of cost and percentage of high-ability give you a pooling equilibrium.
4. Public Goods (Problem #18.12 from Pindyck Rubinfeld)
The Georges Bank, a highly productive fishing area off New England, can be divided into two zones in terms of fish population. Zone 1 has the higher population per square mile but is subject to severe diminishing returns to fishing effort. The aggregate daily fish catch (in tons) in Zone 1 is:
F1 = 200(X1) – 2(X1)2
where X1 is the number of boats fishing there. Zone 2 has fewer fish per mile but is larger, and diminishing returns are less of a problem. Its aggregate daily fish catch is:
F2 = 100(X2) – (X2)2
where X2 is the number of boats fishing in Zone 2. The marginal aggregate fish catch MFC in each zone can be represented as:
MFC1 = 200 – 4(X1)
MFC2 = 100 – 2(X2)
There are 100 boats now licensed by the U.S. government to fish in these two zones. The aggregate fish catch is split evenly by all boats fishing in that zone. The fish are sold at $100 per ton. Total cost (capital and operating) per boat is constant at $1000 per day. Answer the following questions about this situation:
a. If the boats are allowed to fish where they want, with no government restriction, how many will fish in each zone? What will be the gross value of the catch?
Hint: Start with some guess such that 100=?1+?2 (e.g., 50/50). Which zone is more profitable at that guess? Boats will shift into that zone until the profits across zones even out. Specify the equation that shows equal per-boat profits in each zone.
b. If the U.S. government can restrict the distribution of the boats, how many should be allocated to each zone assuming the government only cares about the net total profits from all zones (i.e., its tax base)?
Hint: The government maximizes the profit from both zones (revenues from both zones minus costs from both zones) subject to the quantity of boats remaining at 100.
c. If the U.S. government can restrict the number and distribution of the boats, how many should be allocated to each zone assuming the government only cares about the net total profits (i.e., its tax base)? What will be the gross value of the catch?
Hint: The government finds the optimal allocation by maximizing profits zone by zone.