Answer Key

EconS425 - Homework #3 (Due on March 7 th , 2013)
1. Exercise 2 from Chapter 7 in Shy (page 165)
Solution:
a) Given a low reservation price (a low B), we’ll attempt to find a monopoly equilibrium
where not all the market is served. Figure 7.2 illustrates the location of the two
indifferent consumers, one on each side of the firm. The indifferent consumer (on
each side) is determined by the reservation utility, that is, B a p 0 , or,
Solution-Figure 7.2: Single-firm location model
a B p . Hence, the monopoly chooses p that solves
max =p 2a 2 p( B p)
p
The solution is given by
2
p M B , a B , and =p 2a
B
2 2 2
1
Now, it is easy to verify that 0B
1 implies that 0 a
, hence, not all the
2
market is served.
b) When substituting B 1 into the solution for ‘a’ found in the previous subquestion,
we get that the indifferent consumers lie “outside” the city. This implies that for
B 1, the monopoly can increase the price and still having the entire street
purchase the product. Thus, the monopoly will pick the highest possible price
subject to having the consumers living at the edges of town purchase the product.
1
Formally, set p to satisfy B p 0. Hence,
2
M 1 1 M 1
p B , a , and B
2 2 2
2. A unit circle representing the dimension of “mouth feel” in breakfast cereals has 200
consumers spread uniformly along it. All consumers buy either one package of cereal a
week or none at all (in which case they eat bread) and incur disutility c = 200 from each
unit distance by which a brand deviates from their favorite mouth feel. They derive
utility v = 20 from eating bread.
1
Instructor: Ana Espinola

a. What is the equation for the demand curve facing cereal makers (Distinguish
the monopolistic and competitive segments, but ignore the super-competitive
segment.)
Solution:
The equation for the monopolistic segment of the demand curve is
2 L
qm
( v p ) 40 2 p
c
and for the competitive segment
L c 200
qc
( p p)
p p
c n n
b. What is the symmetric equilibrium under free entry if the cereal makers’ costs
are C(q) = 128 +4q
Solution:
Checking first whether the zero-profit equilibrium lies on the monopolistic
segment of the demand curve, we note that if C(q) = 218 + 4q, monopoly profits
are
( p m) q F 2( p 4)(20 p) 128
m
m
Setting the derivative of these profits with respect to p equal to zero, we find
that p $12 . Hence, q 16 and $0 , so that this is indeed the zero-profit
m
m
equilibrium. Dividing the number of consumers, 200, by the number of units
each firm sells, 16, we find that n ≤ 12.
c. What if fixed costs fall to 100
Solution:
With lower fixed costs, monopolistic cereal makers earn non-zero profits, so that
we have to look at the competitive segment of the demand curve for
equilibrium. Competitive profits are
200
c
( p m) qc
F ( p 4)( p p) 100
n
Setting the derivative of these profits with respect to p equal to zero and using
that, in the symmetric equilibrium, p p and profits are zero, we find that
p $14 q 10 $0 and n=20 at the equilibrium.
c
c
3. On an atoll in the Pacific island group Saloponesia there is just one street, Ring Street,
along which all people live—one islander per unit distance—and along which all shops
are located. The island group used to be a French colony, which may explain the
islanders’ custom of walking to a nearby bakery every day to buy a fresh chocolate
croissant. Their attachment to this custom is not unbounded, however. Depending upon
the opportunity cost of their time spent walking, they may find croissants too expensive.
In that case they stay home and eat corn flakes instead. Each of the bakers runs exactly
one bakery and, when not baking, plays Bertrand.
2
Instructor: Ana Espinola

a. Set up a model (adopting simplifying assumptions where appropriate) to analyze
this croissant market, and describe the equilibrium if it is given that there are
some islanders who eat corn flakes.
Solution:
One way of setting up the model that omits the u and v-terms in the Salop model
(which cancel out anyway) is as follows. Let customers’ opportunity cost of
walking be c per unit distance (c/2 going and c/2 coming back). If they live at a
distance d from a bakery charging p for a croissant, their perceived cost of
buying from that bakery is p + cd. If this cost exceeds that of buying from the
nearest bakery in the other direction from where they live, they buy from that
bakery instead. If the cost of buying from either bakery exceeds their reservation
price p u , they eat corn flakes.
Assume that the bakeries are identical. In a symmetric equilibrium, they will then
locate at equal distances from each other on Ring Street. As a result, if there are
some islanders who eat corn flakes, the market areas of the bakeries are not
touching, and each bakery is therefore a local monopoly.
If a bakery is a local monopoly, the length xm
of its market in either direction is
defined by
u
p cx p
Beyond this distance no customer buys from it. Given that there is one islander
per unit distance along Ring Street, its daily sales are
u
2( p p)
qm
2xm
(1)
c
and its profits, assuming that each bakery has the same cost function C (q) = F +
mq, are
u
2( p p)
m
( p m)
F
(2)
c
Setting the derivative of these profits with respect to p equal to zero, we find
that it sets
u
p m
p (3)
2
Substituting equation (3) into (1) and (2) yields
u
p m
qm
(4)
c
And
u 2
( p m)
m
F
(5)
2c
Assuming there is free entry into the croissant market, the monopolist bakers
cannot be making positive profits in equilibrium. (Strictly speaking, we follow
Salop’s assumption that bakeries relocate without cost when entry occurs, each
time redistributing themselves evenly around the circle). From setting profits in
(5) to zero and rearranging, we find that
m
3
Instructor: Ana Espinola

u
p c 2Fc
Substituting this expression back into (3) and (4) yields expressions for the
equilibrium price and sales.
4. Exercise 1 and 3 from Chapter 8 in Shy (pages 213-214)
Exercise 1 from Chapter 8
Solution:
a) I4 20 20 20 10 70
b)
I
HH
2 2
4 10 3 20 1600
c) i. After the merger of firms 1 and 2,
I
HH
2 2 2 2
20 10 10 3 20 1800
ii. I HH
1800 1600 200
iii. The merger may be challenged by the FTC or the Justice Department since the
postmerger I
HH
exceeds 1, 000 and I HH
200 100
Exercise3 from Chapter 8
Solution:
a) We look for a Nash equilibrium in prices. The X-seller takes pY
as given and solves
max X pXQ pX ( pX 2 pY
) , yielding pX
pY
pX
2
The Y-seller takes pX
as given and solves
pX
max Y pY 2Q 2 pY ( pX 2 pY
) , yielding pY
pY
4
Hence,
2
N N N 2 N N N
pX , pY , pS , Q , X
Y
3 6 3 3 9
b) We define a system as one unit of X bundled with two units of Y. Then, the merged
firm chooses a system price p that solves
S
2
max S pS ( pS
) , yielding pS QS and
S
pS
2 4
c) To make a welfare judgment on the gains from this merger it is sufficient to compare
prices and profit levels. Before the merger, the price of one system is
0 2
1
pS pX 2pY pS
which is the price after the merger. Also, before the
3 2
2 2
2
1
merger, aggregate profit is
X
Y
S
which is the profit after the
9 4
merger takes place. Since the system price falls and industry profit increases, the
merger is welfare improving.
4
Instructor: Ana Espinola