Answer Key
Answer Key
Answer Key
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EconS425 - Homework #3 (Due on March 7 th , 2013)<br />
1. Exercise 2 from Chapter 7 in Shy (page 165)<br />
Solution:<br />
a) Given a low reservation price (a low B), we’ll attempt to find a monopoly equilibrium<br />
where not all the market is served. Figure 7.2 illustrates the location of the two<br />
indifferent consumers, one on each side of the firm. The indifferent consumer (on<br />
each side) is determined by the reservation utility, that is, B a p 0 , or,<br />
Solution-Figure 7.2: Single-firm location model<br />
a B p . Hence, the monopoly chooses p that solves<br />
max =p 2a 2 p( B p)<br />
p<br />
The solution is given by<br />
2<br />
p M B , a B , and =p 2a<br />
<br />
B<br />
2 2 2<br />
1<br />
Now, it is easy to verify that 0B<br />
1 implies that 0 a<br />
, hence, not all the<br />
2<br />
market is served.<br />
b) When substituting B 1 into the solution for ‘a’ found in the previous subquestion,<br />
we get that the indifferent consumers lie “outside” the city. This implies that for<br />
B 1, the monopoly can increase the price and still having the entire street<br />
purchase the product. Thus, the monopoly will pick the highest possible price<br />
subject to having the consumers living at the edges of town purchase the product.<br />
1<br />
Formally, set p to satisfy B p 0. Hence,<br />
2<br />
M 1 1 M 1<br />
p B , a , and B <br />
2 2 2<br />
2. A unit circle representing the dimension of “mouth feel” in breakfast cereals has 200<br />
consumers spread uniformly along it. All consumers buy either one package of cereal a<br />
week or none at all (in which case they eat bread) and incur disutility c = 200 from each<br />
unit distance by which a brand deviates from their favorite mouth feel. They derive<br />
utility v = 20 from eating bread.<br />
1<br />
Instructor: Ana Espinola
a. What is the equation for the demand curve facing cereal makers (Distinguish<br />
the monopolistic and competitive segments, but ignore the super-competitive<br />
segment.)<br />
Solution:<br />
The equation for the monopolistic segment of the demand curve is<br />
2 L<br />
qm<br />
( v p ) 40 2 p<br />
c<br />
and for the competitive segment<br />
L c 200<br />
qc<br />
( p p)<br />
p p<br />
c n n<br />
b. What is the symmetric equilibrium under free entry if the cereal makers’ costs<br />
are C(q) = 128 +4q<br />
Solution:<br />
Checking first whether the zero-profit equilibrium lies on the monopolistic<br />
segment of the demand curve, we note that if C(q) = 218 + 4q, monopoly profits<br />
are<br />
( p m) q F 2( p 4)(20 p) 128<br />
m<br />
m<br />
Setting the derivative of these profits with respect to p equal to zero, we find<br />
that p $12 . Hence, q 16 and $0 , so that this is indeed the zero-profit<br />
m<br />
m<br />
equilibrium. Dividing the number of consumers, 200, by the number of units<br />
each firm sells, 16, we find that n ≤ 12.<br />
c. What if fixed costs fall to 100<br />
Solution:<br />
With lower fixed costs, monopolistic cereal makers earn non-zero profits, so that<br />
we have to look at the competitive segment of the demand curve for<br />
equilibrium. Competitive profits are<br />
200<br />
<br />
c<br />
( p m) qc<br />
F ( p 4)( p p) 100<br />
n<br />
Setting the derivative of these profits with respect to p equal to zero and using<br />
that, in the symmetric equilibrium, p p and profits are zero, we find that<br />
p $14 q 10 $0 and n=20 at the equilibrium.<br />
c<br />
c<br />
3. On an atoll in the Pacific island group Saloponesia there is just one street, Ring Street,<br />
along which all people live—one islander per unit distance—and along which all shops<br />
are located. The island group used to be a French colony, which may explain the<br />
islanders’ custom of walking to a nearby bakery every day to buy a fresh chocolate<br />
croissant. Their attachment to this custom is not unbounded, however. Depending upon<br />
the opportunity cost of their time spent walking, they may find croissants too expensive.<br />
In that case they stay home and eat corn flakes instead. Each of the bakers runs exactly<br />
one bakery and, when not baking, plays Bertrand.<br />
2<br />
Instructor: Ana Espinola
a. Set up a model (adopting simplifying assumptions where appropriate) to analyze<br />
this croissant market, and describe the equilibrium if it is given that there are<br />
some islanders who eat corn flakes.<br />
Solution:<br />
One way of setting up the model that omits the u and v-terms in the Salop model<br />
(which cancel out anyway) is as follows. Let customers’ opportunity cost of<br />
walking be c per unit distance (c/2 going and c/2 coming back). If they live at a<br />
distance d from a bakery charging p for a croissant, their perceived cost of<br />
buying from that bakery is p + cd. If this cost exceeds that of buying from the<br />
nearest bakery in the other direction from where they live, they buy from that<br />
bakery instead. If the cost of buying from either bakery exceeds their reservation<br />
price p u , they eat corn flakes.<br />
Assume that the bakeries are identical. In a symmetric equilibrium, they will then<br />
locate at equal distances from each other on Ring Street. As a result, if there are<br />
some islanders who eat corn flakes, the market areas of the bakeries are not<br />
touching, and each bakery is therefore a local monopoly.<br />
If a bakery is a local monopoly, the length xm<br />
of its market in either direction is<br />
defined by<br />
u<br />
p cx p<br />
Beyond this distance no customer buys from it. Given that there is one islander<br />
per unit distance along Ring Street, its daily sales are<br />
u<br />
2( p p)<br />
qm<br />
2xm<br />
(1)<br />
c<br />
and its profits, assuming that each bakery has the same cost function C (q) = F +<br />
mq, are<br />
u<br />
2( p p)<br />
<br />
m<br />
( p m)<br />
F<br />
(2)<br />
c<br />
Setting the derivative of these profits with respect to p equal to zero, we find<br />
that it sets<br />
u<br />
p m<br />
p (3)<br />
2<br />
Substituting equation (3) into (1) and (2) yields<br />
u<br />
p m<br />
qm<br />
(4)<br />
c<br />
And<br />
u 2<br />
( p m)<br />
<br />
m<br />
F<br />
(5)<br />
2c<br />
Assuming there is free entry into the croissant market, the monopolist bakers<br />
cannot be making positive profits in equilibrium. (Strictly speaking, we follow<br />
Salop’s assumption that bakeries relocate without cost when entry occurs, each<br />
time redistributing themselves evenly around the circle). From setting profits in<br />
(5) to zero and rearranging, we find that<br />
m<br />
3<br />
Instructor: Ana Espinola
u<br />
p c 2Fc<br />
Substituting this expression back into (3) and (4) yields expressions for the<br />
equilibrium price and sales.<br />
4. Exercise 1 and 3 from Chapter 8 in Shy (pages 213-214)<br />
Exercise 1 from Chapter 8<br />
Solution:<br />
a) I4 20 20 20 10 70<br />
b)<br />
I <br />
HH<br />
2 2<br />
4 10 3 20 1600<br />
c) i. After the merger of firms 1 and 2,<br />
I <br />
HH<br />
2 2 2 2<br />
20 10 10 3 20 1800<br />
ii. I HH<br />
1800 1600 200<br />
iii. The merger may be challenged by the FTC or the Justice Department since the<br />
postmerger I<br />
HH<br />
exceeds 1, 000 and I HH<br />
200 100<br />
Exercise3 from Chapter 8<br />
Solution:<br />
a) We look for a Nash equilibrium in prices. The X-seller takes pY<br />
as given and solves<br />
<br />
max X pXQ pX ( pX 2 pY<br />
) , yielding pX<br />
pY<br />
pX<br />
2<br />
The Y-seller takes pX<br />
as given and solves<br />
pX<br />
max Y pY 2Q 2 pY ( pX 2 pY<br />
) , yielding pY<br />
<br />
pY<br />
4<br />
Hence,<br />
2<br />
N N N 2 N N N<br />
pX , pY , pS , Q , X <br />
<br />
Y<br />
<br />
3 6 3 3 9<br />
b) We define a system as one unit of X bundled with two units of Y. Then, the merged<br />
firm chooses a system price p that solves<br />
S<br />
2<br />
<br />
max S pS ( pS<br />
) , yielding pS QS and <br />
S<br />
<br />
pS<br />
2 4<br />
c) To make a welfare judgment on the gains from this merger it is sufficient to compare<br />
prices and profit levels. Before the merger, the price of one system is<br />
0 2<br />
1<br />
pS pX 2pY pS<br />
which is the price after the merger. Also, before the<br />
3 2<br />
2 2<br />
2<br />
1<br />
merger, aggregate profit is <br />
X<br />
Y <br />
S<br />
which is the profit after the<br />
9 4<br />
merger takes place. Since the system price falls and industry profit increases, the<br />
merger is welfare improving.<br />
4<br />
Instructor: Ana Espinola