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Question I
a) A firm with a constant marginal cost production technology sells
1,000 units of its product each week at $10 per unit. This price –
output combination maximizes the firm’s profits. For small
changes in the product’s price, a 1% change in price results in a
2% change in quantity sold. The firm has no fixed costs. What are
its weekly profits? (10 points)
b) A consumer’s utility function for eggs and wine is U(x,y) = x.5 y.5,
where x equals the dozens of eggs he consumes and y equals the
bottles of wine he consumes. Eggs cost $2 per dozen; wine costs
$8 per bottle. Determine the optimal quantities of eggs and wine
the consumer should buy when he has $16 he can spend on these
two goods. [Note. The consumer’s marginal utility of eggs
depends on the quantities of both goods that he has: MUx = .5 y.5 /
x.5. Similarly, MUy = .5 x.5 / y.5.] (10 points)
c) Boogle College has a very large stadium, which is rarely filled for
football games. Noting this, the college has decided to offer
students discounted tickets, but with a limit of two discounted
tickets per student. Moreover, the student who bought the
discounted tickets and a guest must use them. This is enforced by
checking student IDs. Students can buy more than two tickets, of
course, but they must pay full price for any tickets beyond the first
two that they buy. In general terms, what do you think is going on
here? (5 points)
Question II
Boogle College’s football stadium holds 60,000 people. Seats sold to
hometown Boogle supporters (Boogle Boosters) can be sold for PB =
$20 – B/2000, where B is the number of seats sold to Boogle
supporters. Seats sold to supporters of the visiting team’s athletes can
be sold for PV = $24 – V/3000, where V is number of seats sold to
visiting team supporters.
a) Determine the maximum revenue Boogle College can get if it fills
the stadium, that is, if it sells all 60,000 seats to a game. (15 points)
b) Is it a good idea for Boogle College to sell all the available seats?
How do you know?
(10 points)
Question III
In a competitive market with many buyers and many sellers, the
amount of product demanded decreases with q, the price buyers pay,
while the amount of product supplied increases with p, the price
sellers receive. The demand and supply relations are given by
D(q) = 2000(10 – q), q < 10
S(p) = 1000(p – 4), p > 4.
Suppose a sales tax of t percent is imposed: if the posted price of the
good is p (which is what sellers receive net of the tax), buyers must
pay q = (1 + t) p.
a) Set up a spreadsheet and show the instructions you would give
Solver to maximize tax revenue. Define the variables in this
problem in Column A below and show how they are calculated in
Column B. (15 points)
Column A
Solver Instructions:
Target Variable: ____________
By changing: _________________
Subject to the constraint(s):
Column B
Max ____ Min ____
b) The burden of the sales tax will not fall entirely on buyers but will
be shared by buyers and sellers. Explain why a sales tax “paid” by
buyers hurts sellers as well as buyers. (10 points)
Question IV (10 points each part)
At its profit-maximizing price of $7 per cake Acme is now selling
250 cakes per week to local residents. Acme’s manager, Ada Water,
is concerned that few cakes are being sold to residents who live in
Zipcode 00100, the poor side of town, and is thinking of mailing
coupons to these residents. Ada’s economic consultant, Marge N.
L’Coste, estimates weekly cake demand of Zipcode 00100 residents
as D(Pz) = 80 – 10Pz where Pz is the price they pay.
In addition, she estimates weekly cake demand of the residential
market as a whole, including Zipcode 00100 residents, as D(P) = 600
– 50 P where P is the price paid by all residents. Ada then figures
that the weekly cake demand of non-Zipcode 00100 residents is D(Po)
= 520 – 40Po where Po is the price these other residents pay.
a) Compute Acme’s constant marginal cost of baking and selling
cake to local residents from its shop on Main Street. Explain the
logic of your calculation. [The correct answer is $2/cake.]
Question IV (continued)
b) Show that it makes sense for Acme to offer discount coupons to
Zipcode 00100 residents.
c) In addition to offering discount coupons to Zipcode 00100
residents, what else should Acme do? Explain or demonstrate the
wisdom of your suggestion.