CAS EC 201.
Intermediate Microeconomic Analysis.
Problem set 8
(due on Thursday April 23, 2020)
1. First, we examine a monopolistic firm. The firm faces a market described by the
(inverse) demand function p(y) = A−B · y, where p is the (maximum) price at which
y units of output can be sold on the market. The firm’s cost function is given by
C(y) = y2/2.
1.1 Find the profit-maximizing price-quantity combination for the monopolist and
the co
esponding profit for this firm.
1.2 Formulate your first-order condition from part 1.1 in terms of the elasticity of
demand. That is, compute the demand elasticity, and use the elasticity formula
to find the profit-maximizing quantity of output.
1.3 The demand is inelastic for prices below some cutoff p. What is p? Equivalently,
the demand is inelastic for quantities above A/B − p/B. Show that the profit is
decreasing whenever y > A/B − p/B. You have just shown that a monopolist
will never choose to operate where the demand curve is inelastic. What is the
intuition?
1.4 Let p∗ and y∗ be respectively the price and the quantity in 1.1, so that p∗ =
A − B · y∗. Now suppose the demand in the market shifts, so that the new
(inverse) demand curve is p = A − By, with A > A and B > B, but such that
it is still the case that p∗ = A − B · y∗. Hence, if the firm does not change its
quantity, its price will also not change. When is it the case that the quantity of
output y∗ is still optimal, given this new demand curve?
2. Now suppose you must design a tax on the monopoly.
2.1 First, suppose you consider a sales tax of 10% on this market. Suppose that the
firm has to pay the tax, as is typically the case with sales taxes. Hence, if the
firm produces quantity y of output, the price paid by consumers is p = A−B · y,
ut the price received by the firm is .9p = .9A − .9B · y, where the .9 appears
ecause the firm gets to keep only ninety percent of the purchase price, paying
the remaining ten percent to the government in taxes. Find the profit maximizing
quantity of output for the firm, the price paid by consumers, the price received
y the firm, and the firm’s profit. Explain how these answers compare to those of
part 1.1. In particular, do consumers pay more as a result of the tax? Does the
price received by the firm fall? Does the firm’s profit fall?
2.2 Instead of a sales tax, the government considers a profits tax. Hence, if the
firm chooses quantity of output y and charges price p = A − B · y, the gov-
ernment collects t (py − y2/2) = t ((A−B · y) y − y2/2) in tax revenue, leaving
1
(1− t) ((A−B · y) y − y2/2) as after-tax profit for the firm. Once again, find the
firm’s profit-maximizing quantity of output and the resulting price. What effect
does the profits tax have on the quantity of output and price? What effect does
it have on the firm’s profit? In light of your answers, given that a fixed amount of
evenue is to be raised, which tax would consumers prefer—a sales tax or profits
tax? Why?
2.3 Now forget about taxes. Suppose the cost function is given by y2/2+F , where F
is a fixed cost. For example, F may be the cost of acquiring the licenses needed to
produce. What effect does F have on the firm’s optimal quantity of output, price,
and profits? Many communication firms are monopolies because the government
gives them exclusive rights to use certain bands of airwave lengths. Sometimes
the government simply gives the firms this exclusive right, while other times it
sells the right to the highest bidder. The latter method has been criticized on
the grounds that it will drive up the firms’ costs and hence the prices charged to
consumers. What do think think of this argument?
3. A monopolist can produce at constant average and marginal costs of c. The firm faces
a market demand curve given by QD = p−ε, where ε > 1.
3.1 Calculate the profit-maximizing price-quantity combination for the monopolist.
Also calculate the monopolist’s profits.
3.2 Identify, in Figure 1, the “deadweight loss” from monopolization.
3.3 (optional) Compute the consumer surplus under monopoly and under perfect
competition. Take their ratio and check whether it is increasing or decreasing in
ε. Why?
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demand
MR
C ′(y)
quantity
$
Figure 1: Exercise 3.2
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