Economics 261
Economics 411
Problem Set #1
Tuesday, Fe
uary 4, 2020
You may submit your completed problem sets during class or you may put them in Juan
Margitic’s mail box by 5:00 pm.
1. A baseball team receives total profit of (Q) + aQ. represents profit from ticket sales and
aQ represents profit from sales of apparel. The demand curve for tickets is P = 100 – (Q/1,000)
and the team has marginal cost equal to zero.
(i) Find the equili
ium quantity and profit from ticket sales when a = 20 and when a = 40.
(ii) Explain why the profit from ticket sales changes the way it does when a rises.
2. A football team has two kinds of fans. Adults demand QA(p) = 1,000(100 – P) tickets when
the price is P. Senior citizens demand QS(p) = 1,000(60 – P). Marginal cost is equal to zero, and
the stadium has capacity K. Find the profit-maximizing ticket prices when K < 80,000 and resale
of tickets is impossible.
3. A soccer team faces demand uncertainty. In the high-demand state, the quantity demanded is
QH(p) = 2,000(50 – P); in the low-demand state, it is QL(p) = 2,000(30 – P). The probability that
demand is high is h [0, ½]. Marginal cost is MC = 0, and the stadium has capacity K = 40,000.
(i) Find the profit-maximizing ticket price, assuming that the team sells tickets before demand
uncertainty is resolved.
(ii) Suppose that the team sells tickets after demand uncertainty is resolved instead. Find the
profit-maximizing prices.
(iii) In which of the two scenarios is the expected value of social surplus higher?
4. A team’s demand curve for tickets is ? = (200 −
?
10,000
) (
?
100
), where P is the ticket price, Q
is the annual attendance, and W is the team’s winning percentage. Marginal cost equals zero and
fixed costs are 8,000W2.
(i) Find the optimal ticket price and quantity when the team’s winning percentage is W.
(ii) Find the optimal winning percentage.
5. Two teams play each other 100 times. Team Alpha receives gate revenue equal to Aw – 5w2
if it wins w games; Team Beta team receives Bw – 5w2 if it wins w games. Assume that A and B
are known constants and 500 < B < A < 500+B. (The inequalities simply guarantee that an
interior solution exists.) The teams have no other sources of revenue.
(i) Find the equili
ium number of wins for each team and the marginal cost of a win. Explain
iefly why Beta’s number of wins and marginal cost of a win change the way they do when B
ises slightly.
(ii) Now suppose that there is revenue sharing. Each team keeps 60% of its revenue and
eceives 40% of the other team’s revenue. Solve for the equili
ium number of wins for each
team and the marginal cost of a win, assuming that A = 1,000 and B = 800.
(iii) Finally, suppose that there is a salary cap but no revenue sharing. Assume that the cap is
inding for both teams. Explain
iefly how the equili
ium here differs from the equili
ium in
(i).
6. Two players on a basketball team share playing time. Mark’s expected number of points
scored is SM(x) = 30x – 5x
2 if he plays the fraction x [0,1] of the game. John’s expected
number is SJ(x) = 25x – 5x
2 points if he plays the fraction x of the game.
(i) Suppose that Mark plays the entire game. How many points does he expect to score? How
many points would John expect to score if he played the fraction x = of the game? Find the
average scoring rate, SJ()/, as approaches zero.
(ii) What fraction of the time should Mark play in order to maximize the total points scored by
Mark and John? Find the expected number of points scored by each player per unit time.