Unlock document.
This document is partially blurred.
Unlock all pages and 1 million more documents.
Get Access
The problems in this chapter deal primarily with marginal revenue-marginal cost calculations in
different contexts. For such problems, students’ primary difficulty is to remember that the
marginal revenue concept requires differentiation with respect to quantity. Often students choose
to differentiate total revenue with respect to price and then get very confused on how to set this
equal to marginal cost. Of course, it is possible to phrase the monopolist’s problem as one of
choosing a profit-maximizing price, but then the inverse demand function must be used to derive
a marginal cost expression.
The analytical and behavioral problems in this chapter introduce students to some state-
of-the-art research on monopoly reflected in recent academic articles.
Comments on Problems
14.1 This problem is a simple marginal revenue-marginal cost and consumer surplus
computation.
14.2 This problem is an example of the MR = MC calculation with three different types of cost
curves.
14.3 This problem is an example of the MR = MC calculation with three different demand and
marginal revenue curves. The problem also illustrates the “inverse elasticity” rule.
14.4 This problem examines graphically the various possible ways in which shift in demand
may affect the market equilibrium in a monopoly.
14.5 This problem introduces advertising expenditures as a choice variable for a monopoly.
The problem also asks the student to view market price as the decision variable for the
monopoly.
14.6 Note: This problem has been subtly revised from the previous edition; the numbers for
production and transportation cost are now different, helping students see where each
distinctly shows up in the calculations. This is a price-discrimination example in which
markets are separated by transport costs, showing how the price differential is
constrained by the extent of those costs. Part (d) asks students to consider a simple two-
part tariff.
14.7 This problem shows how the welfare cost of monopoly may be larger than in the
traditional case if the monopoly has higher costs.
CHAPTER 14:
Monopoly
14.8 This problem examines some issues in the design of subsidies for a monopoly.
14.9 This problem involves quality choice. The result shows that, in this case, the monopoly
and competitive choices are the same (though output levels differ).
Analytical Problems
14.10 Taxation of a monopoly good. This problem focuses mainly on ad valorem taxes on a
monopoly good. The final part of the problem compares ad valorem and specific taxes.
14.11 Flexible functional forms. This problem has students run through the standard
monopoly analysis but for a class of flexible functional forms introduced in a recent
influential paper by Fabinger and Weyl (2015). While slightly complicated, the
functional forms allow for U-shaped average cost curves and realistic demand shapes.
14.12 Welfare possibilities with different market segmentations. This problem illustrates
extreme possibilities for price discrimination to create or destroy welfare identified in the
important recent paper by Bergemann, Brooks, and Morris (2015). To make their results
accessible, takes the simplest case of two consumer types, but the analysis of this case is
done in full generality.
Behavioral Problem
14.13 Shrouded prices. This problem introduces students to the problem of shrouded prices, a
topic that has received wide attention in behavioral economics. See for example, D.
Laibson and X. Gabaix, “Shrouded Attributes, Consumer Myopia, and Information
Suppression in Competitive Markets,” Quarterly Journal of Economics (May 2006):
505–540. More on whether competition uncovers shrouding to come in the next chapter.
Solutions
Chapter 14: Monopoly
161
monopoly.
14.2 Given market demand is
70 ,QP=−
marginal revenue is
70 2 .MR Q=−
b.
2
0.25 5 300C Q Q= − +
implies
0.5 5.MC Q=−
Setting
MC MR=
gives
Chapter 14: Monopoly
162
14.3 a. Given
10AC MC==
and
60 ,QP=−
implying
60 2 .MR Q=−
For profit
Note: Here the inverse elasticity rule is clearly illustrated:
Problem part
,QP
QP
e =
PQ
1
Q,P
P MC
= P
e
−−
Chapter 14: Monopoly
163
d.
The supply curve for a monopoly is a single point, namely, that quantity–price
combination that corresponds to the quantity for which
.MC MR=
Any attempt to
14.4 a.
b. There is no supply curve for monopoly; have to examine
MR MC=
intersection
Chapter 14: Monopoly
164
14.5 Given
( )
( )
2
20 1 0.1 0.01 .Q P A A= − + −
Let
2
1 0.1 0.01 .K A A= + −
Then
0.1 0.02dK dA A=−
and
b. Substituting
15P=
into the profit function,
Chapter 14: Monopoly
165
14.6 a. In the first market,
11
55QP=−
2
1 1 1 1 1
(55 ) 55R Q Q Q Q = − = −
b. If the producer ignores the problem of arbitrage among consumers, the price
differential between the two markets found to be optimal in the previous part
c. Now
12 .P P P==
So
2
140 3 625.PP
= − −
Taking the first-order condition,
d. If the firm adopts a linear tariff of the form
( ) ,
i
ii
T Q = + mQ
it can maximize
profit by setting
5,m=
Chapter 14: Monopoly
166
14.7 a. Under perfect competition,
10.MC =
Under monopoly,
12.MC =
b. Calculations are aided by the graph below. Loss of consumer
c.
14.8 a. The government wishes the monopoly to expand output toward
.P MC=
A lump-
b. A subsidy per unit of output will effectively shift the MC curve downward. The
Chapter 14: Monopoly
167
c. A subsidy (
t
) must be chosen so that the monopoly chooses the socially
optimal quantity, given
t
. Since social optimality requires
P MC=
and profit
maximization requires
14.9 Since consumers only value
,XQ
firms can be treated as selling that commodity
(i.e., batteries of a specific useful life). Firms seek to minimize the cost of
producing
XQ
for any level of that output. Setting up the Lagrangian,
Chapter 14: Monopoly
168
Analytical Problems
14.10 Taxation of a monopoly good
The inverse elasticity rule is
a. With linear demand, e falls (becomes more elastic) as price rises. Hence,
b. With constant elasticity demand, the inequality in part (a) becomes an equality so
c. If the monopoly operates on a negatively sloped portion of its marginal cost curve
we have (in the constant elasticity case)
d. The key part of this question is the requirement of equal tax revenues. That is
,
a a s
tP Q Q
=
where the subscripts refer to the monopoly’s choices under the two
Chapter 14: Monopoly
169
14.11 Flexible functional forms
a. Writing the monopoly profit function as
( ) [ ( ) ( )] ,Q P Q AC Q Q
=−
substituting
the given functional forms yields, after rearranging,
b. Constant average and marginal cost corresponds to
10.c=
Substituting into the
solution from part (a) gives
c. Monopoly profit with this yet more flexible specification is
Chapter 14: Monopoly
170
d. Here is a graph showing possible shapes for the average cost curve.
14.12 Welfare possibilities with different market segmentations
a. The perfectly competitive outcome with marginal-cost pricing (
0
c
p=
) yields the
b. With just two consumer types, the monopolist can achieve perfect price
discrimination by segmenting each type into one of two markets, charging
v
on
Chapter 14: Monopoly
171
c. The single-price monopolist can choose from one of two pricing strategies, either
i. The monopoly price is
,v
quantity is
,q
and profit is
.qv
There is no
ii. The profit from serving just high-value consumers in segment
B
is
,bqv
and from serving all consumers in that segment is
( ) .bq q v+
At
*,b
these
Chapter 14: Monopoly
172
iii.
To get to a point somewhere in the middle of the base of the triangle—in
other words, to move only part way from part (c.i) to part (c.ii)—one
d. The inequality assumed in this part means that the profit-maximizing single price
for the monopolist now equals
.v
Chapter 14: Monopoly
173
ii. The profit from serving just high-value consumers in segment
A
is
,qv
and from serving all consumers in that segment is
( ) .q aq v+
At
*,a
these
iii.
14.13 Shrouded prices
a. Monopoly profit is
( ) (10 )( 6).Q P AC P P
= − = − −
Solving the first-order
Chapter 14: Monopoly
174
b. Monopoly profit is
( ) (10 )( 6).Q P s AC P P s
= + − = − + −
Solving the first-order
c. Gross consumer surplus can be computed as the area of the trapezoid under the
demand curve up to the quantity sold:
d. Welfare is
e. The solution for the monopoly price is exactly as in part (b). The difference here
is that the subsidy expenditure
sQ
comes from the government, whereas the
Chapter 14: Monopoly
175
