Answer Key

EconS425 - Homework #2 (Due on February 12 th , 2013)
1. Exercise 4 from Chapter 5 in Zhy (page 93)
Solution:
a) The monopoly’s profit function is:
( q , q ) (100 q / 2) q (100 q ) q ( q q )
1 2 1 1 2 2 1 2
b) The two first order conditions are given by:
100 q1 2( q1 q2) 0
q
1
100 2q2 2( q1 q2) 0
q
1
M M
M
M
Solving for q1
and q2
yields that q1 25 and q2 12.5 .
c) Substitute the profit maximizing quantity sold in market 1 and 2 into the market
M
M
M
M
demand function yields: p1 100 q1
/ 2 87.5 and p 2
100 q 2
87.5 .
Hence the profit level of the discriminating monopoly is:
M M
2
( q1 , q2
) 87.5 25 87.5 12.5 (25 12.5) 1875
d) Now that each plant sells only in one market, each plant’s cost function only
depends on their own production and is given by:
TC ( q ) q and
2
1 1 1
The profit functions are:
TC ( q ) q .
( q ) (100 q / 2) q q
2
2 2 2
2
1 1 1 1 1
( q ) (100 q ) q q
2
2 2 2 2 2
The two first order conditions are given by:
1
q
1
2
q
2
100 q 2q
0
1 1
100 2q
2q
0
2 2
M M
M
M
Solving for q1
and q2
yields that q1 100 / 3 and q2 25 .
2
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Instructor: Ana Espinola

e) Substitute the profit maximizing quantity sold in market 1 and 2 into the market
M
M
M
M
demand function yields: p1 100 q1
/ 2 250 / 3 and p 2
100 q 2
75 .
Hence the sum of profits of the two plants is:
2
M
M 250 100 100
2
1( q1 )
2( q2
) 75 25 25 2917
3 3 3
f) This decomposition increase the monopoly’s profit since the technology exhibits DRS.
2. Exercise 1, 3 and 5 from Chapter 6 in Zhy (pages 128-129)
Solution of exercise 1 from Chapter 6:
a) By Lemma 4.1, both firms produce a finite amount of output only if and
. Hence, since , if a competitive equilibrium exists then it must be that
. However, if then both firms produce zero output, but at these
prices demand exceeds zero (since
). Hence, if a competitive
equilibrium exists, then . Therefore, and .
b) Each firm i takes the output of its opponent, q j , as given and solves
yielding first order conditions given by
Solving the two equations for the two variables, q 1 and q 2 yields
Hence,
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Instructor: Ana Espinola

c) From the first order conditions of the previous subquestion we can immediately
solve for firm 2’s best-best response function. Hence,
.
In a sequential-moves equilibrium, firm 1 (the leader) takes firm 2’s best-response
function as given and chooses q 1 to
yielding the first order condition
.
Therefore,
Hence,
d) When firm 2 is the leader, by symmetry (replacing c 1 by c 2 and vice versa in the previous
subquestion) we can conclude that
Hence,
Comparing the two sequential-moves equilibria we see that (i) aggregate output
decreases when the less efficient firm (firm 2) moves before firm 1; (ii) price is higher
when firm 2 moves first; (iii) the market share of the less efficient firm increases when it
moves before firm 1; (iv) the production of the more-efficient firm increases when it
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Instructor: Ana Espinola

moves first; and (v) the market share of the more-efficient firm increases when it moves
first.
e) In a Bertrand equilibrium, the more efficient firm (firm 1) completely undercuts the
price set by firm 2. Assuming that money is continuous, firm 1 sets its price to equal the
unit cost of firm 2, thereby ensuring that firm 2 will not find it profitable to produce any
amount. Hence, we can approximate the Bertrand equilibrium by
and
. Thus, and . Notice that when the unit costs are
not equal (the present case), the Bertrand equilibrium price exceeds the competitive
equilibrium price. Also, instructors should point out to the students that this is not really
a Nash-Bertrand equilibrium since the profit of firm 1 increases as decreases.
Solution of exercise 3 from Chapter 6:
In the third period, firm 3 takes q
1
and q2
as given and chooses q
3
to maximize its profit.
The solution to this problem yields firm 3’s best-response function R3 ( q1, q2)
which is
given in (6.8). Hence, using (6.8) we have it that
a c 1 120 c
q1
q2
R3 ( q1, q2) ( q1 q2)
2b
2 2 2
In the second period, firm 2 takes q1
and R3 ( q1, q2)
as given and chooses q2
that solves
max pq cq [120 q q R ( q , q )] q cq
q2
2 2 2 1 2 3 1 2 2 2
2
d
2
The first-order condition is given by 0 60 q1/ 2 q2
c / 2 . Therefore, the best-
dq
response function of firm 2 is given by
120 q1
c
R2( q1)
2
In the first period, firm 1 takes the best-response function of firms 2 and 3 as given and
chooses q1
that solves
3c
q1
max 1 pq1 cq1 [120 q1 R2 ( q1 ) R3 ( q1, R2 ( q1 ))] q1 cq1 (30 ) q1 cq1
q1
4 4
d1
s
The first-order condition is given by 0 30 q1
/ 2 c/ 4. Therefore, solving for q1
dq
s
and then substituting into R 2
( q 1
), and then into 3
( s s
R q 1
, R 2
( q 1
)) yields:
s 120 c s 120 c s 120 c
q1 , q2 , and q3
2 4 8
1
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Instructor: Ana Espinola

s s s s 7(120 c) s 120 7c
Hence Q q1 q2 + q3
, and p
8 8
In this problem, we assume production is costless, c=0, hence,
q 60, q 30, q 15, Q q q + q 105, and p 15
s s s s s s s s
1 2 3 1 2 3
Solution of exercise 5 from Chapter 6:
a) Let
A
pk
denote the (tariff inclusive) price consumers in country A pay for a unit of
import from country k, k B,
C . Therefore, under a uniform specific tariff of $10 per
A
A
unit of import p 70 and p 50 . Clearly consumers buy from country C and the
B
C
government earns revenue of $10 per unit. A FTA with country B reduces
A A A
p to p 60 50 p . Hence, A’s consumers still purchase from country C
B B C
implying that this FTA is ineffective and therefore does not alter A’s welfare.
b) Now, a FTA with country B reduces p A to p A 60 60.1 p
A . Hence, A’s consumers
B B C
switch to buying from country B and no tariff revenues are collected. Note that in
A
A
this case the consumer price falls from p 60.1 to p 60 (a reduction of 1 cent
C
per unit of import). Hence, the FTA hardly changes consumer surplus. However, in
this case the government faces a large reduction in tariff revenues due to the
elimination of a $10 tariff per unit of import. Altogether, with the exception of highly
elastic demand, country A loses from the FTA.
B
3. Two firms have technologies for producing identical paper clips. Assume that all paper
clips are sold in boxes containing 100 paper clips. Firm A can produce each box at unit
cost of c A = $6 whereas firm B (less efficient) at a unit cost of c B = $8.
a. Suppose that the aggregate market demand for boxes of paper clips is p = 12 –
Q/2, where p is the price per box and Q is the number of boxes sold. Solve for
the Nash-Bertrand equilibrium prices p b A and p b B, and the equilibrium profits ∏ A
b
and ∏ B b .
Solution:
The first case to be checked is where the efficient firm A undercuts B by setting
p c 8 , where is a small number. Firm B set p c
8. In this
A
B
case, all consumers buy brand A only, hence, solving 8 12 q A
/ 2 yields qA
8
and qB
0. The profits are then
A
(8 6) 8 16
and 0 .
The second case to be checked is where A sets a monopoly price. Solving
MR 12 q c 6 yields q 6. Hence p 12 6 / 2 $9 $8 p .
A
A
A
Therefore, in this case, firm A can not charge its monopoly price.
A
B
B
B
B
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Instructor: Ana Espinola

Altogether, a Nash-Bertrand equilibrium prices are p 8 and p 8. The
equilibrium profits are therefore
A
(8 6) 8 16
and 0 .
A
B
B
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Instructor: Ana Espinola