Answer Key
Answer Key
Answer Key
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EconS425 - Homework #2 (Due on February 12 th , 2013)<br />
1. Exercise 4 from Chapter 5 in Zhy (page 93)<br />
Solution:<br />
a) The monopoly’s profit function is:<br />
( q , q ) (100 q / 2) q (100 q ) q ( q q )<br />
1 2 1 1 2 2 1 2<br />
b) The two first order conditions are given by:<br />
<br />
100 q1 2( q1 q2) 0<br />
q<br />
1<br />
<br />
100 2q2 2( q1 q2) 0<br />
q<br />
1<br />
M M<br />
M<br />
M<br />
Solving for q1<br />
and q2<br />
yields that q1 25 and q2 12.5 .<br />
c) Substitute the profit maximizing quantity sold in market 1 and 2 into the market<br />
M<br />
M<br />
M<br />
M<br />
demand function yields: p1 100 q1<br />
/ 2 87.5 and p 2<br />
100 q 2<br />
87.5 .<br />
Hence the profit level of the discriminating monopoly is:<br />
<br />
M M<br />
2<br />
( q1 , q2<br />
) 87.5 25 87.5 12.5 (25 12.5) 1875<br />
d) Now that each plant sells only in one market, each plant’s cost function only<br />
depends on their own production and is given by:<br />
TC ( q ) q and<br />
2<br />
1 1 1<br />
The profit functions are:<br />
TC ( q ) q .<br />
( q ) (100 q / 2) q q<br />
2<br />
2 2 2<br />
2<br />
1 1 1 1 1<br />
( q ) (100 q ) q q<br />
2<br />
2 2 2 2 2<br />
The two first order conditions are given by:<br />
1<br />
q<br />
1<br />
<br />
2<br />
q<br />
2<br />
100 q 2q<br />
0<br />
1 1<br />
100 2q<br />
2q<br />
0<br />
2 2<br />
M M<br />
M<br />
M<br />
Solving for q1<br />
and q2<br />
yields that q1 100 / 3 and q2 25 .<br />
2<br />
1<br />
Instructor: Ana Espinola
e) Substitute the profit maximizing quantity sold in market 1 and 2 into the market<br />
M<br />
M<br />
M<br />
M<br />
demand function yields: p1 100 q1<br />
/ 2 250 / 3 and p 2<br />
100 q 2<br />
75 .<br />
Hence the sum of profits of the two plants is:<br />
2<br />
M<br />
M 250 100 100<br />
<br />
2<br />
1( q1 )<br />
2( q2<br />
) 75 25 25 2917<br />
<br />
<br />
3 3 3 <br />
f) This decomposition increase the monopoly’s profit since the technology exhibits DRS.<br />
2. Exercise 1, 3 and 5 from Chapter 6 in Zhy (pages 128-129)<br />
Solution of exercise 1 from Chapter 6:<br />
a) By Lemma 4.1, both firms produce a finite amount of output only if and<br />
. Hence, since , if a competitive equilibrium exists then it must be that<br />
. However, if then both firms produce zero output, but at these<br />
prices demand exceeds zero (since<br />
). Hence, if a competitive<br />
equilibrium exists, then . Therefore, and .<br />
b) Each firm i takes the output of its opponent, q j , as given and solves<br />
yielding first order conditions given by<br />
Solving the two equations for the two variables, q 1 and q 2 yields<br />
Hence,<br />
2<br />
Instructor: Ana Espinola
c) From the first order conditions of the previous subquestion we can immediately<br />
solve for firm 2’s best-best response function. Hence,<br />
.<br />
In a sequential-moves equilibrium, firm 1 (the leader) takes firm 2’s best-response<br />
function as given and chooses q 1 to<br />
yielding the first order condition<br />
.<br />
Therefore,<br />
Hence,<br />
d) When firm 2 is the leader, by symmetry (replacing c 1 by c 2 and vice versa in the previous<br />
subquestion) we can conclude that<br />
Hence,<br />
Comparing the two sequential-moves equilibria we see that (i) aggregate output<br />
decreases when the less efficient firm (firm 2) moves before firm 1; (ii) price is higher<br />
when firm 2 moves first; (iii) the market share of the less efficient firm increases when it<br />
moves before firm 1; (iv) the production of the more-efficient firm increases when it<br />
3<br />
Instructor: Ana Espinola
moves first; and (v) the market share of the more-efficient firm increases when it moves<br />
first.<br />
e) In a Bertrand equilibrium, the more efficient firm (firm 1) completely undercuts the<br />
price set by firm 2. Assuming that money is continuous, firm 1 sets its price to equal the<br />
unit cost of firm 2, thereby ensuring that firm 2 will not find it profitable to produce any<br />
amount. Hence, we can approximate the Bertrand equilibrium by<br />
and<br />
. Thus, and . Notice that when the unit costs are<br />
not equal (the present case), the Bertrand equilibrium price exceeds the competitive<br />
equilibrium price. Also, instructors should point out to the students that this is not really<br />
a Nash-Bertrand equilibrium since the profit of firm 1 increases as decreases.<br />
Solution of exercise 3 from Chapter 6:<br />
In the third period, firm 3 takes q<br />
1<br />
and q2<br />
as given and chooses q<br />
3<br />
to maximize its profit.<br />
The solution to this problem yields firm 3’s best-response function R3 ( q1, q2)<br />
which is<br />
given in (6.8). Hence, using (6.8) we have it that<br />
a c 1 120 c<br />
q1<br />
q2<br />
R3 ( q1, q2) ( q1 q2)<br />
<br />
2b<br />
2 2 2<br />
In the second period, firm 2 takes q1<br />
and R3 ( q1, q2)<br />
as given and chooses q2<br />
that solves<br />
max pq cq [120 q q R ( q , q )] q cq<br />
q2<br />
2 2 2 1 2 3 1 2 2 2<br />
2<br />
d<br />
2<br />
The first-order condition is given by 0 60 q1/ 2 q2<br />
c / 2 . Therefore, the best-<br />
dq<br />
response function of firm 2 is given by<br />
120 q1<br />
c<br />
R2( q1)<br />
<br />
2<br />
In the first period, firm 1 takes the best-response function of firms 2 and 3 as given and<br />
chooses q1<br />
that solves<br />
3c<br />
q1<br />
max 1 pq1 cq1 [120 q1 R2 ( q1 ) R3 ( q1, R2 ( q1 ))] q1 cq1 (30 ) q1 cq1<br />
q1<br />
4 4<br />
d1<br />
s<br />
The first-order condition is given by 0 30 q1<br />
/ 2 c/ 4. Therefore, solving for q1<br />
dq<br />
s<br />
and then substituting into R 2<br />
( q 1<br />
), and then into 3<br />
( s s<br />
R q 1<br />
, R 2<br />
( q 1<br />
)) yields:<br />
s 120 c s 120 c s 120 c<br />
q1 , q2 , and q3<br />
<br />
2 4 8<br />
1<br />
4<br />
Instructor: Ana Espinola
s s s s 7(120 c) s 120 7c<br />
Hence Q q1 q2 + q3<br />
, and p <br />
8 8<br />
In this problem, we assume production is costless, c=0, hence,<br />
q 60, q 30, q 15, Q q q + q 105, and p 15<br />
s s s s s s s s<br />
1 2 3 1 2 3<br />
Solution of exercise 5 from Chapter 6:<br />
a) Let<br />
A<br />
pk<br />
denote the (tariff inclusive) price consumers in country A pay for a unit of<br />
import from country k, k B,<br />
C . Therefore, under a uniform specific tariff of $10 per<br />
A<br />
A<br />
unit of import p 70 and p 50 . Clearly consumers buy from country C and the<br />
B<br />
C<br />
government earns revenue of $10 per unit. A FTA with country B reduces<br />
A A A<br />
p to p 60 50 p . Hence, A’s consumers still purchase from country C<br />
B B C<br />
implying that this FTA is ineffective and therefore does not alter A’s welfare.<br />
b) Now, a FTA with country B reduces p A to p A 60 60.1 p<br />
A . Hence, A’s consumers<br />
B B C<br />
switch to buying from country B and no tariff revenues are collected. Note that in<br />
A<br />
A<br />
this case the consumer price falls from p 60.1 to p 60 (a reduction of 1 cent<br />
C<br />
per unit of import). Hence, the FTA hardly changes consumer surplus. However, in<br />
this case the government faces a large reduction in tariff revenues due to the<br />
elimination of a $10 tariff per unit of import. Altogether, with the exception of highly<br />
elastic demand, country A loses from the FTA.<br />
B<br />
3. Two firms have technologies for producing identical paper clips. Assume that all paper<br />
clips are sold in boxes containing 100 paper clips. Firm A can produce each box at unit<br />
cost of c A = $6 whereas firm B (less efficient) at a unit cost of c B = $8.<br />
a. Suppose that the aggregate market demand for boxes of paper clips is p = 12 –<br />
Q/2, where p is the price per box and Q is the number of boxes sold. Solve for<br />
the Nash-Bertrand equilibrium prices p b A and p b B, and the equilibrium profits ∏ A<br />
b<br />
and ∏ B b .<br />
Solution:<br />
The first case to be checked is where the efficient firm A undercuts B by setting<br />
p c 8 , where is a small number. Firm B set p c<br />
8. In this<br />
A<br />
B<br />
case, all consumers buy brand A only, hence, solving 8 12 q A<br />
/ 2 yields qA<br />
8<br />
and qB<br />
0. The profits are then <br />
A<br />
(8 6) 8 16<br />
and 0 .<br />
The second case to be checked is where A sets a monopoly price. Solving<br />
MR 12 q c 6 yields q 6. Hence p 12 6 / 2 $9 $8 p .<br />
A<br />
A<br />
A<br />
Therefore, in this case, firm A can not charge its monopoly price.<br />
A<br />
B<br />
B<br />
B<br />
B<br />
5<br />
Instructor: Ana Espinola
Altogether, a Nash-Bertrand equilibrium prices are p 8 and p 8. The<br />
equilibrium profits are therefore <br />
A<br />
(8 6) 8 16<br />
and 0 .<br />
A<br />
B<br />
B<br />
6<br />
Instructor: Ana Espinola